Publications/Phd-Urban.ps
author Christian Urban <urbanc@in.tum.de>
Wed, 13 Sep 2017 16:36:57 +0100
changeset 477 65d3f6e3e935
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updated
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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   815
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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   819
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   822
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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   826
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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   839
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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   843
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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   847
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Christian Urban <urbanc@in.tum.de>
parents:
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   848
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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   850
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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   852
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Christian Urban <urbanc@in.tum.de>
parents:
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   853
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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   855
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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   857
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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   866
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Christian Urban <urbanc@in.tum.de>
parents:
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   867
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Christian Urban <urbanc@in.tum.de>
parents:
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   868
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Christian Urban <urbanc@in.tum.de>
parents:
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   869
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Christian Urban <urbanc@in.tum.de>
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   870
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Christian Urban <urbanc@in.tum.de>
parents:
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   871
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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   876
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   877
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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   884
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Christian Urban <urbanc@in.tum.de>
parents:
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   885
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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   888
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   890
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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   892
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Christian Urban <urbanc@in.tum.de>
parents:
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   893
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   894
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   895
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Christian Urban <urbanc@in.tum.de>
parents:
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   896
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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   898
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   899
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   900
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   901
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   902
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   903
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Christian Urban <urbanc@in.tum.de>
parents:
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   904
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   905
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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   908
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Christian Urban <urbanc@in.tum.de>
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   909
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   910
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Christian Urban <urbanc@in.tum.de>
parents:
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   911
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   912
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   913
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   914
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Christian Urban <urbanc@in.tum.de>
parents:
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   915
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   916
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   917
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
   918
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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   921
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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FFF000001FFFE000001FFF8000001FFF8000001FFFE000001FFFF800000007FC00000001
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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00FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00000000
Christian Urban <urbanc@in.tum.de>
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diff changeset
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00FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00000000
Christian Urban <urbanc@in.tum.de>
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diff changeset
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00FF0000000000FF0000000000FF0000000000FF0000000000FF0000000000FF00000000
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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FF00000000FF00000000FF00000000FF00000000FF00000000FF00000000FF00000000FF
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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FFF800000000FFF8FEF800000000FBF8FEFC00000001FBF8FEFC00000001FBF8FE7E0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1930
001F83F8FE0FC000001F83F8FE0FE000003F83F8FE07E000003F03F8FE07E000003F03F8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1931
FE07F000007F03F8FE03F000007E03F8FE03F80000FE03F8FE03F80000FE03F8FE01F800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1932
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1933
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1934
0FE003F8FE003F800FE003F8FE001F800FC003F8FE001FC01FC003F8FE000FC01F8003F8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1935
FE000FC01F8003F8FE000FE03F8003F8FE0007E03F0003F8FE0007E03F0003F8FE0007F0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1936
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1937
FE0001F8FC0003F8FE0000FDF80003F8FE0000FDF80003F8FE0000FFF80003F8FE00007F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1938
F00003F8FE00007FF00003F8FE00003FE00003F8FE00003FE00003F8FE00003FE00003F8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1939
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1940
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1941
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1942
FEFE3FC00000FEFE1FC00000FEFE1FE00000FEFE0FE00000FEFE0FF00000FEFE07F00000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1943
FEFE07F80000FEFE03F80000FEFE03FC0000FEFE01FC0000FEFE01FE0000FEFE00FE0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1944
FEFE00FF0000FEFE007F0000FEFE007F8000FEFE003F8000FEFE003FC000FEFE001FC000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1945
FEFE001FE000FEFE000FE000FEFE000FF000FEFE0007F000FEFE0007F800FEFE0003F800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1946
FEFE0003FC00FEFE0001FC00FEFE0001FE00FEFE0000FE00FEFE0000FF00FEFE00007F00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1947
FEFE00007F80FEFE00003F80FEFE00003FC0FEFE00001FC0FEFE00001FE0FEFE00000FE0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1948
FEFE00000FF0FEFE000007F0FEFE000007F8FEFE000003F8FEFE000003FCFEFE000001FC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1949
FEFE000001FCFEFE000000FEFEFE000000FFFEFE0000007FFEFE0000007FFEFE0000003F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1950
FEFE0000003FFEFE0000001FFEFE0000001FFE2F3F78BE40>I<000001FF80000000000F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1951
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1952
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1953
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1954
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1955
3F8000000001FC7F8000000001FE7F8000000001FE7F8000000001FE7F0000000000FEFF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1956
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1957
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1958
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1959
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1960
07FC1FE000000007F81FF00000000FF80FF00000000FF00FF80000001FF007FC0000003F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1961
E007FC0000003FE003FE0000007FC001FF000000FF8000FF800001FF0000FFC00003FF00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1962
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1963
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1964
FFF000000007FFFF0000001FFFFFC000007FFFFFF80000FFFFFFFC0001FFFFFFFC0003FF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1965
803FFC0007FC0007F8000FF80001F8001FE0000078001FC0000038003FC0000018003F80
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1966
000000003F80000000007F00000000007F00000000007F00000000007F00000000007F00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1967
000000007F00000000007F80000000007F80000000003FC0000000003FE0000000003FF0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1968
000000001FF8000000000FFE000000000FFFC000000007FFFC00000003FFFFC0000001FF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1969
FFF8000000FFFFFE0000003FFFFF8000000FFFFFC0000003FFFFE00000003FFFF0000000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1970
03FFF8000000007FFC000000000FFC0000000007FE0000000001FE0000000001FF000000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1971
0000FF00000000007F00000000007F80000000003F80000000003F80000000003F800000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1972
00003F80000000003F80000000003F80000000003F80000000007F00600000007F007000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1973
00007F0078000000FE007C000001FE007F000003FC00FFC00007FC00FFF0001FF800FFFF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1974
007FF0007FFFFFFFE0001FFFFFFFC00007FFFFFF000001FFFFFE0000003FFFF800000001
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1975
FFC0000029437CC033>83 D<000FF80000FFFF0003FFFF800FFFFFE01FFFFFF01FFFFFF0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1976
1FF00FF81F8003FC1E0001FC180001FE100000FE000000FF0000007F0000007F0000007F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1977
0000007F0000007F0000007F0000007F00007FFF000FFFFF007FFFFF01FFFFFF07FFFFFF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1978
0FFFF07F3FFC007F3FC0007F7F00007FFE00007FFC00007FFC00007FFC00007FFC0000FF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1979
FE0000FFFE0001FF7F8007FF7FE01FFF3FFFFFFF3FFFFFFF1FFFFF7F0FFFFC7F07FFF07F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1980
01FF0000202B7CA92C>97 D<FE00000000FE00000000FE00000000FE00000000FE000000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1981
00FE00000000FE00000000FE00000000FE00000000FE00000000FE00000000FE00000000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1982
FE00000000FE00000000FE00000000FE00000000FE00000000FE00000000FE00000000FE
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1983
00000000FE00000000FE00000000FE00FF0000FE07FFE000FE1FFFF000FE7FFFF800FFFF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1984
FFFC00FFFFFFFE00FFF80FFF00FFE001FF00FF8000FF80FF00007FC0FF00003FC0FE0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1985
1FC0FE00001FE0FE00000FE0FE00000FE0FE00000FE0FE000007F0FE000007F0FE000007
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1986
F0FE000007F0FE000007F0FE000007F0FE000007F0FE000007F0FE000007F0FE000007F0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1987
FE00000FE0FE00000FE0FE00000FE0FE00001FE0FE00003FC0FF00003FC0FF80007F80FF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1988
8000FF80FFE003FF00FFF81FFE00FFFFFFFC00FEFFFFF800FE7FFFF000FE3FFFE000FE0F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1989
FF80000001FE0000244079BE2F>I<00000007F000000007F000000007F000000007F000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1990
000007F000000007F000000007F000000007F000000007F000000007F000000007F00000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1991
0007F000000007F000000007F000000007F000000007F000000007F000000007F0000000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1992
07F000000007F000000007F000000007F00007F807F0003FFF07F0007FFFC7F001FFFFF7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1993
F003FFFFFFF007FFFFFFF007FF80FFF00FFC003FF01FF0001FF01FE0000FF03FC00007F0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1994
3FC00007F07F800007F07F000007F07F000007F07F000007F0FE000007F0FE000007F0FE
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1995
000007F0FE000007F0FE000007F0FE000007F0FE000007F0FE000007F0FE000007F0FE00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1996
0007F07F000007F07F000007F07F000007F07F800007F03F80000FF03FC0000FF01FE000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1997
1FF01FF0003FF00FF8007FF00FFF01FFF007FFFFFFF003FFFFF7F001FFFFE7F000FFFF87
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1998
F0003FFE07F0000FF0000024407DBE2F>100 D<00001FF00000FFFC0001FFFC0007FFFC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  1999
000FFFFC000FFFFC001FE03C003F8004003F0000007F0000007E000000FE000000FE0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2000
00FE000000FE000000FE000000FE000000FE000000FE000000FE000000FE000000FE0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2001
00FE000000FE0000FFFFFF00FFFFFF00FFFFFF00FFFFFF00FFFFFF00FFFFFF0000FE0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2002
00FE000000FE000000FE000000FE000000FE000000FE000000FE000000FE000000FE0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2003
00FE000000FE000000FE000000FE000000FE000000FE000000FE000000FE000000FE0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2004
00FE000000FE000000FE000000FE000000FE000000FE000000FE000000FE000000FE0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2005
00FE000000FE000000FE000000FE000000FE000000FE00001E407FBF1C>102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2006
D<0003FC003F00000FFF03FF00003FFFDFFF00007FFFFFFF8000FFFFFFFF8001FFFFFFFF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2007
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2008
000FC0003F00000FC0003F00000FC0003F00000FC0003F00000FC0003F00000FC0003F00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2009
000FC0003F000007E0007E000007E0007E000007F000FE000003F801FC000001FE07F800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2010
0003FFFFF8000003FFFFF0000007FFFFE0000007FFFFC0000007CFFF0000000FC3FC0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2011
000F80000000000FC0000000000FC00000000007C00000000007E00000000007FFFFF000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2012
0003FFFFFF000003FFFFFFC00007FFFFFFE0000FFFFFFFF8001FFFFFFFF8003FE0001FFC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2013
007F800003FE007F000001FE007F000000FF00FE0000007F00FE0000007F00FE0000007F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2014
00FE0000007F00FE0000007F00FF000000FF007F800001FE007FC00003FE003FF0000FFC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2015
001FFE007FF8000FFFFFFFF00007FFFFFFE00003FFFFFFC00000FFFFFF0000001FFFF800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2016
000003FFC00000293D7EA82D>I<FFFFFFFFFFFFFFFF0000000000000000000000000000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2017
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2018
7F7F7F7F7F083F7ABE16>105 D<FE00000000FE00000000FE00000000FE00000000FE00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2019
000000FE00000000FE00000000FE00000000FE00000000FE00000000FE00000000FE0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2020
0000FE00000000FE00000000FE00000000FE00000000FE00000000FE00000000FE000000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2021
00FE00000000FE00000000FE00000000FE00000000FE0000FF80FE0001FF00FE0003FE00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2022
FE0007FC00FE000FF800FE001FF000FE003FE000FE007FC000FE00FF8000FE01FF0000FE
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2023
03FE0000FE07FC0000FE0FF80000FE0FF00000FE1FE00000FE3FC00000FE7FE00000FEFF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2024
E00000FFFFF00000FFFFF80000FFFBF80000FFF1FC0000FFE0FE0000FFC0FF0000FF807F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2025
0000FF003F8000FE003FC000FC001FC000FC000FE000FC000FF000FC0007F000FC0003F8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2026
00FC0003FC00FC0001FC00FC0000FE00FC00007F00FC00007F80FC00003F80FC00001FC0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2027
FC00001FE0233F79BE2C>107 D<FEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFE
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2028
FEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFEFE
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2029
FEFEFEFEFE073F79BE16>I<FC01FF0000FF8000FC07FFC003FFE000FC1FFFF00FFFF800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2030
FC7FFFF83FFFFC00FCFFFFFC7FFFFE00FDFFFFFCFFFFFE00FFF80FFFFC07FF00FFE003FF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2031
F001FF00FFC001FFE000FF00FF8000FFC0007F80FF8000FFC0007F80FF00007F80003F80
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2032
FF00007F80003F80FF00007F80003F80FE00007F00003F80FE00007F00003F80FE00007F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2033
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2034
FE00007F00003F80FE00007F00003F80FE00007F00003F80FE00007F00003F80FE00007F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2035
00003F80FE00007F00003F80FE00007F00003F80FE00007F00003F80FE00007F00003F80
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2036
FE00007F00003F80FE00007F00003F80FE00007F00003F80FE00007F00003F80FE00007F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2037
00003F80FE00007F00003F80FE00007F00003F80FE00007F00003F80FE00007F00003F80
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2038
FE00007F00003F80FE00007F00003F80392979A848>I<FC01FF00FC07FFC0FC1FFFF0FC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2039
7FFFF8FCFFFFFCFDFFFFFCFFF80FFEFFE003FEFFC001FEFF8000FFFF8000FFFF00007FFF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2040
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2041
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2042
00007FFE00007FFE00007FFE00007FFE00007FFE00007FFE00007FFE00007FFE00007FFE
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2043
00007FFE00007F202979A82F>I<0001FE0000000FFFC000003FFFF000007FFFF80000FF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2044
FFFC0003FFFFFF0003FF03FF0007F8007F800FF0003FC01FE0001FE01FC0000FE03F8000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2045
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2046
FCFE000001FCFE000001FCFE000001FCFE000001FCFE000001FCFE000001FCFE000001FC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2047
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2048
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2049
FFF000000FFFC0000001FE0000262B7DA92D>I<0000FF0000FE07FFE000FE1FFFF000FE
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2050
7FFFF800FFFFFFFC00FFFFFFFE00FFF80FFF00FFE003FF00FF8000FF80FF00007FC0FF00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2051
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2052
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2053
F0FE00000FF0FE00000FE0FE00000FE0FE00001FE0FE00001FE0FE00003FC0FF00007FC0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2054
FF8000FF80FF8001FF80FFE003FF00FFF81FFE00FFFFFFFC00FEFFFFF800FE7FFFF000FE
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2055
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2056
000000FE00000000FE00000000FE00000000FE00000000FE00000000FE00000000FE0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2057
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2058
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2059
00FF8000FF8000FF0000FF0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2060
00FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2061
00FE0000FE0000FE0000FE0000162979A81F>114 D<001FF80000FFFF8003FFFFE007FF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2062
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2063
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2064
FFC000FFFFE0001FFFF00001FFF000001FF8000007F8000003FC000003FC000001FC0000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2065
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2066
FFC01FFFFF8003FFFE00003FF0001E2B7EA923>I<00FC000000FC000000FC000000FC00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2067
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2068
C0FFFFFFC0FFFFFFC0FFFFFFC0FFFFFFC0FFFFFFC000FC000000FC000000FC000000FC00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2069
0000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2070
0000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC000000FC00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2071
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2072
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2073
FE00007FFE00007FFE00007FFE00007FFE00007FFE00007FFE00007FFE00007FFE00007F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2074
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2075
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2076
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2077
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2078
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2079
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2080
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2081
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2082
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2083
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2084
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2085
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2086
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2087
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2088
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2089
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2090
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2091
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2092
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2093
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2094
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2095
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2096
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2097
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Christian Urban <urbanc@in.tum.de>
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  2098
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Christian Urban <urbanc@in.tum.de>
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  2099
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Christian Urban <urbanc@in.tum.de>
parents:
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  2104
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2105
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  2106
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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C01E03801F07000F0E0007FC0001F000152B7EA919>I<01E000000007F8000E000E3C00
Christian Urban <urbanc@in.tum.de>
parents:
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1F001C3E003F00383E003E00303E003E00703E003E00607E007E00E07C007C00C07C007C
Christian Urban <urbanc@in.tum.de>
parents:
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00C0FC007C0080F800FC0000F800F80001F800F80001F000F80001F001F80003F001F000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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C007E0C003C00FC0C003E01FC1C003E03BE18001F071E380007FE0FF00001F803C00231F
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4807
7E9D29>I<003F007C0000FFC1FF0001C1E383800380F703C00700F60FC00E00FE0FC01C
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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00FC0FC01800FC0FC03800FC07003000F800003000F800002001F800000001F000000001
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4809
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4810
03000007C003003807C007007C0FC00600FC0FC00E00FC1FC00C00FC1BC01C00F03BE038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4811
007071E0F0003FE0FFC0000F803F0000221F7E9D28>120 D<01E0000007F8000E0E3C00
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1F1C3E003F383E003E303E003E703E003E607E007EE07C007CC07C007CC0FC007C80F800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4813
FC00F800F801F800F801F000F801F001F803F001F003E001F003E001F003E003F007E003
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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E007C003E007C003E007C007E007C007C003C00FC003E01FC003E03FC001F07F80007FEF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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80001F8F8000001F8000001F0000001F003E003E003E003E007E007C007E00F8007C01F0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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007003E0003007C0003C0F80000FFE000003F00000202C7E9D23>I
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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32 28 42 42 42 42 65 23 42 23 23 42 42 28 37 42 37 42
Christian Urban <urbanc@in.tum.de>
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37 6[51 60 1[78 60 1[51 46 55 1[46 1[60 74 51 60 32 28
Christian Urban <urbanc@in.tum.de>
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60 60 46 51 60 55 55 60 76 5[23 42 2[42 42 42 42 42 42
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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83.022 /Times-Roman rf /Ge 133[53 60 1[86 1[66 40 47
Christian Urban <urbanc@in.tum.de>
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53 66 66 60 66 100 33 66 1[33 66 60 40 53 66 53 66 60
Christian Urban <urbanc@in.tum.de>
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9[120 86 1[80 66 2[73 93 86 1[80 2[47 1[93 73 80 86 86
Christian Urban <urbanc@in.tum.de>
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80 86 6[40 2[60 60 60 60 60 60 60 2[30 40 30 4[40 39[{
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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rf /Gf 134[103 103 2[115 69 80 92 115 115 103 115 172
Christian Urban <urbanc@in.tum.de>
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57 2[57 115 103 69 92 115 92 115 103 13[115 2[126 1[149
Christian Urban <urbanc@in.tum.de>
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1[138 2[80 2[126 138 149 149 138 149 19[69 45[{
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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rf /Gg 32[45 42[30 10[69 30 16[91 45 1[40 40 10[30 13[40
Christian Urban <urbanc@in.tum.de>
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45 45 66 45 45 25 35 30 45 45 45 45 71 25 45 25 25 45
Christian Urban <urbanc@in.tum.de>
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45 30 40 45 40 45 40 30 2[30 1[30 56 66 1[86 66 66 56
Christian Urban <urbanc@in.tum.de>
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51 61 66 51 66 66 81 56 66 35 30 66 66 51 56 66 61 61
Christian Urban <urbanc@in.tum.de>
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diff changeset
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66 1[40 3[25 25 45 45 45 45 45 45 45 45 45 45 1[23 30
Christian Urban <urbanc@in.tum.de>
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23 2[30 30 30 71 4[30 2[30 12[25 30 12[51 51 2[{
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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%%Page: 1 1
Christian Urban <urbanc@in.tum.de>
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TeXDict begin 1 0 bop Black Black 277 851 a @beginspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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-153 @llx -3 @lly 153 @urx 372 @ury 612 @rwi @clip @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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%%BeginDocument: pics/cam.arms.ps
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4852
%!PS-Adobe-2.0 EPSF-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4853
%%Title: Arms of University of Cambridge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4854
%%Creator: Philip Hazel, July 1986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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%%CreationDate: 12:00:00 31-07-86
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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%%BoundingBox: -153 -3 153 372
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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%%EndComments
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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/cuarmsdict 50 dict def cuarmsdict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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/mtrx matrix def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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/lionfix{filled{gsave lioncolour fill grestore}if stroke}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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/toefix{gsave lioncolour fill grestore stroke} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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/ea{/savematrix mtrx currentmatrix def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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savematrix setmatrix}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4869
/ec{/savematrix mtrx currentmatrix def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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translate 5 -1 roll rotate scale 0 0 1 5 -2 roll arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4871
savematrix setmatrix}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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/db /rlineto load def /dt /lineto load def /mt /moveto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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/lion1{0.77 0.77 scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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-200 -600 translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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464 524 45 241 70 arc 464 554 20 36 -133 arcn
Christian Urban <urbanc@in.tum.de>
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diff changeset
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377 396 161 63 97 arc 362 494 62 93 227 arc
Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4888
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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270 -41 185 23 15 265 517 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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260 540 dt -4 20 db -6 10 db -16 -2 db -14 -8 db
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4891
90 -90 90 40 20 200 560 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4892
-14 8 db -16 2 db -6 -10 db -4 -20 db
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4893
90 -5 221 23 15 135 517 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4894
98 55 125 80 55 200 465 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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420 340 285 164 183 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4896
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4898
/lion2{126 299 27 70 126 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4899
98 333 15 271 190 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4900
287 438 230 207 170 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4901
307 0 145 50 25 33 522 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4902
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4903
240 485 240 162 177 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4904
20 496 18 177 270 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4905
96 -10 190 26 12 7 450 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4907
111 -25 180 36 12 15 388 ea 40 350 dt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4909
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4910
127 -15 180 28 7 66 260 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4911
130 213 45 138 120 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4912
120 296 36 -75 -40 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4913
}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4914
Christian Urban <urbanc@in.tum.de>
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diff changeset
  4915
/lion3{69 255 81 11 -37 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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-123 362 300 -33 -47 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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40 57 160 30 11 49 98 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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40 175 350 40 15 57 90 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4920
72 150 345 50 18 115 123 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4921
90 150 330 45 14 155 177 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4922
80 150 340 50 10 186 230 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4923
110 105 240 32 22 218 284 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4924
}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4925
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4926
/lion4{332 247 36 115 210 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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307 160 31 40 -55 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4928
160 390 305 -58 -70 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4929
22 30 160 20 7 243 90 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4930
27 170 330 30 12 250 77 ea 288 90 dt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4931
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4932
60 125 300 45 15 312 100 ea 340 125 dt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4933
72 155 360 30 12 356 140 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4934
380 190 25 210 400 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4935
90 230 50 18 13 383 208 ec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4936
363 240 22 300 356 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4937
}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4938
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4939
/lion5{476 296 108 212 242 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4940
412 171 32 50 -30 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4941
55 90 310 40 20 420 84 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4942
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4943
498 102 54 200 235 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4944
93 200 360 40 11 470 105 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4945
95 165 350 30 8 474 173 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4946
479 205 5 210 360 arc 490 270 dt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4947
470 260 13 350 195 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4948
348 247 108 10 60 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4949
}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4951
/lion6{90 0 360 30 15 72 445 ea toefix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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243 324 108 180 208 arc stroke}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4953
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4954
/lion7{10 0 360 30 20 53 148 ea toefix}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4955
/lion8{-5 0 360 33 13 243 115 ea toefix}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4957
/lion9{50 0 360 27 13 393 103 ea toefix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4958
317 280 mt 335 292 18 235 380 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4959
385 238 dt stroke}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4960
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4961
/lion10{160 360 mt 4 -20 db 16 -20 db 16 20 db 4 20 db 10 -14 db
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4962
10 -6 db 10 6 db 10 24 db 14 0 db 16 10 db stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4963
gsave 200 490 translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4964
1 1 2{pop 16 0 mt 24 40 db -26 -20 db lioneyecolour fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4965
-1 1 scale}for grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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90 90 270 60 25 200 465 ea 200 450 dt closepath 1 setlinewidth stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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500 710 203 240 271 arc stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4970
/lion11{280 280 dt 262 203 18 133 220 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4971
228 156 36 42 -50 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4972
52 38 315 44 23 160 20 ea 195 40 dt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4974
67 135 322 44 17 220 54 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4975
85 100 305 27 20 263 90 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4976
75 130 340 22 10 293 118 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4977
314 137 8 -150 40 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4979
380 197 72 202 148 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4980
250 317 99 323 350 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4981
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4982
422 334 72 208 248 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4983
47 360 240 24 11 394 242 ec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4984
60 60 310 30 20 364 178 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4986
65 110 340 30 15 408 196 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4987
72 150 350 15 8 435 239 ea
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4988
450 260 11 220 40 arc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4989
505 375 108 237 200 arcn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4990
}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4992
/lion12{30 0 360 30 20 135 56 ea toefix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4993
35 0 360 22 15 352 212 ea toefix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4994
280 280 mt 347 300 dt stroke
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4995
}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4996
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4997
/lion13{0 45 -40 31 20 178 242 ec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4998
40 90 310 31 20 137 168 ea 170 174 dt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  4999
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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(close)i(to)f(adequate)i(for)e(ho)n(w)f(much)h(the)o(y)g(ha)n(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5328
(supported)277 3573 y(me.)277 3785 y(I)26 b(w)o(ant)h(to)g(e)o(xpress)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5329
(my)e(gratitude)k(to)c(the)h(German)g(Academic)h(Exchange)g(Service)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5330
(\(D)l(AAD\))d(for)277 3898 y(generously)38 b(supporting)f(me)d(o)o(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5331
(er)g(three)h(years)g(with)g(tw)o(o)f(scholarships.)64
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5332
b(I)34 b(ackno)n(wledge)j(use)277 4011 y(of)29 b(P)o(aul)g(T)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5333
b(aylor')i(s)30 b(diagram)g(package)h(and)f(thank)g(the)f(authors)i(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5334
e(dvips)h(for)g(making)g(their)g(code)277 4124 y(publicly)d(a)n(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5335
(ailable,)g(without)f(which)f(the)g(modi\002cation)i(that)e(allo)n(wed)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5336
g(me)f(to)h(dra)o(w)f(the)h(proofs)h(in)277 4237 y(Appendix)f(A)e(w)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5337
(ould)h(ha)n(v)o(e)g(been)g(impossible.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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TeXDict begin 8 7 bop Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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TeXDict begin 9 8 bop Black Black 277 982 a Gf(Contents)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1368 y Gb(1)92 b(Intr)n(oduction)2551 b(1)414 1481 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5344
Gg(1.1)96 b(Outline)24 b(of)f(the)h(Thesis)73 b(.)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5345
b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5346
(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 134 w(6)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5347
Black 414 1594 a(1.2)96 b(Contrib)n(utions)69 b(.)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5348
b(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5349
(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5350
Black 134 w(8)p Black 277 1798 a Gb(2)92 b(Sequent)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5351
b(Calculi)2431 b(9)414 1911 y Gg(2.1)96 b(Gentzen')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5352
b(s)25 b(Sequent)f(Calculi)78 b(.)46 b(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5353
(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5354
Black 88 w(10)p Black 414 2023 a(2.2)96 b(Cut-Elimination)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5355
b(in)f(Propositional)i(Classical)f(Logic)83 b(.)45 b(.)g(.)g(.)g(.)h(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5356
f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 88 w(21)p Black 414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5357
2136 a(2.3)96 b(Proof)23 b(of)h(Strong)g(Normalisation)72
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5358
b(.)45 b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5359
g(.)g(.)g(.)g(.)h(.)f(.)p Black 88 w(31)p Black 414 2249
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5360
a(2.4)96 b(First-Order)24 b(Classical)h(Logic)47 b(.)f(.)f(.)g(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5361
(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5362
h(.)f(.)p Black 88 w(41)p Black 414 2362 a(2.5)96 b(V)-10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5363
b(ariations)25 b(of)e(the)h(Sequent)h(Calculus)58 b(.)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5364
b(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5365
(.)h(.)f(.)p Black 88 w(44)p Black 414 2475 a(2.6)96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5366
b(Localised)25 b(V)-10 b(ersion)48 b(.)d(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5367
g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(.)g(.)h(.)f(.)p Black 88 w(48)p Black 414 2588 a(2.7)96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5369
b(Notes)79 b(.)46 b(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5370
(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5371
g(.)g(.)h(.)f(.)p Black 88 w(57)p Black 277 2792 a Gb(3)92
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5372
b(Natural)23 b(Deduction)2273 b(65)414 2905 y Gg(3.1)96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5373
b(Intuitionistic)27 b(Natural)d(Deduction)87 b(.)45 b(.)g(.)h(.)f(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5374
(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5375
Black 88 w(66)p Black 414 3018 a(3.2)96 b(The)23 b(Correspondence)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5376
(between)e(Cut-Elimination)g(and)f(Normalisation)i(I)j(.)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5377
b(.)h(.)f(.)p Black 88 w(73)p Black 414 3131 a(3.3)96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5378
b(Classical)25 b(Natural)f(Deduction)87 b(.)45 b(.)g(.)g(.)g(.)h(.)f(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5379
g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5380
Black 88 w(81)p Black 414 3243 a(3.4)96 b(The)23 b(Correspondence)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5381
(between)e(Cut-Elimination)g(and)f(Normalisation)i(II)67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5382
b(.)46 b(.)f(.)p Black 88 w(93)p Black 414 3356 a(3.5)96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5383
b(Notes)79 b(.)46 b(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5384
(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5385
g(.)g(.)h(.)f(.)p Black 88 w(95)p Black 277 3560 a Gb(4)92
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5386
b(A)n(pplications)23 b(of)g(Cut-Elimination)1708 b(101)414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5387
3673 y Gg(4.1)96 b(Implementation)59 b(.)46 b(.)f(.)g(.)g(.)g(.)g(.)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5388
(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5389
g(.)g(.)g(.)g(.)h(.)f(.)p Black 43 w(102)p Black 414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5390
3786 a(4.2)96 b(Non-Deterministic)26 b(Computation:)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5391
b(Case)24 b(Study)79 b(.)45 b(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5392
(.)g(.)h(.)f(.)p Black 43 w(113)p Black 414 3899 a(4.3)96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5393
b(A)22 b(Simple,)h(Non-Deterministic)j(Programming)f(Language)89
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5394
b(.)45 b(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5395
w(118)p Black 414 4012 a(4.4)96 b(Notes)79 b(.)46 b(.)f(.)g(.)g(.)g(.)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5396
(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5397
f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5398
43 w(122)p Black 277 4216 a Gb(5)92 b(Conclusion)2518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5399
b(125)414 4329 y Gg(5.1)96 b(Further)24 b(W)-7 b(ork)74
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5400
b(.)45 b(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5401
g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5402
Black 43 w(126)p Black 277 4532 a Gb(A)71 b(Experimental)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5403
b(Data)2214 b(131)277 4736 y(B)76 b(Details)24 b(f)n(or)f(some)h(Pr)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5404
(oofs)2057 b(141)414 4849 y Gg(B.1)80 b(Proofs)24 b(of)f(Chapter)i(2)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5405
b(.)45 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5406
f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5407
43 w(141)p Black 414 4962 a(B.2)80 b(Proofs)24 b(of)f(Chapter)i(3)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5408
b(.)45 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5409
f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5410
43 w(156)p Black 414 5075 a(B.3)80 b(Proofs)24 b(of)f(Chapter)i(4)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5411
b(.)45 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5412
f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5413
43 w(167)p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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TeXDict begin 11 10 bop Black Black 277 982 a Gf(List)52
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(of)g(Figur)l(es)414 1277 y Gg(1.1)96 b(Ov)o(ervie)n(w)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5419
b(of)h(the)f(thesis)90 b(.)45 b(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5420
(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5421
Black 134 w(7)p Black 414 1473 a(2.1)96 b(Reduction)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5422
b(sequence)h(impossible)g(in)d(LK)2007 1440 y Gc(tq)2141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5423
1473 y Gg(.)45 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5424
g(.)h(.)f(.)p Black 88 w(18)p Black 414 1586 a(2.2)96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5425
b(V)-10 b(ariant)24 b(of)f(Kleene')-5 b(s)25 b(sequent)h(calculus)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5426
(G3a)90 b(.)45 b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5427
h(.)f(.)p Black 88 w(20)p Black 414 1699 a(2.3)96 b(T)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5428
b(erm)22 b(assignment)k(for)d(classical)j(sequent)g(proofs)87
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5429
b(.)45 b(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5430
Black 88 w(23)p Black 414 1812 a(2.4)96 b(Proof)23 b(substitution)56
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5431
b(.)45 b(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5432
h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5433
Black 88 w(27)p Black 414 1925 a(2.5)96 b(Cut-reductions)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5434
b(for)c(logical)j(cuts)e(and)g(commuting)h(cuts)60 b(.)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5435
b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5436
w(29)p Black 414 2038 a(2.6)96 b(Auxiliary)25 b(proof)f(substitution)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5437
b(.)46 b(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5438
h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 88 w(33)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5439
414 2151 a(2.7)96 b(De\002nition)24 b(of)f(the)h(set)g(operators)i(for)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5440
e(propositional)j(connecti)n(v)o(es)39 b(.)45 b(.)g(.)g(.)g(.)g(.)h(.)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5441
(.)p Black 88 w(35)p Black 414 2263 a(2.8)96 b(De\002nition)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5442
b(of)f(the)h(set)g(operators)i(for)e(quanti\002ers)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5443
b(.)45 b(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5444
Black 88 w(45)p Black 414 2376 a(2.9)96 b(Alternati)n(v)o(e)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5445
b(formulation)h(of)d(some)h(inference)i(rules)61 b(.)46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5446
b(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5447
Black 88 w(46)p Black 414 2489 a(2.10)51 b(T)-6 b(erm)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5448
b(assignment)k(for)d(intuitionistic)28 b(sequent)d(proofs)73
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5449
b(.)45 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5450
Black 88 w(47)p Black 414 2602 a(2.11)51 b(Cut-reductions)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5451
b(for)c(labelled)j(cuts)k(.)45 b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5452
f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5453
88 w(50)p Black 414 2715 a(2.12)51 b(T)-6 b(erm)22 b(assignment)k(for)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5454
(the)h(symmetric)h(lambda)f(calculus)k(.)45 b(.)g(.)g(.)h(.)f(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5455
(.)g(.)g(.)h(.)f(.)p Black 88 w(62)p Black 414 2911 a(3.1)96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5456
b(Natural)24 b(deduction)i(calculus)g(for)e(intuitionistic)j(logic)71
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5457
b(.)45 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5458
Black 88 w(67)p Black 414 3024 a(3.2)96 b(T)-6 b(erm)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5459
b(assignment)k(for)d(intuitionistic)28 b(natural)d(deduction)h(proofs)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5460
58 b(.)45 b(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 88 w(70)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5461
Black 414 3137 a(3.3)96 b(T)m(ranslation)25 b(from)f(NJ-proofs)h(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5462
(intuitionistic)28 b(sequent)d(proofs)75 b(.)45 b(.)g(.)g(.)g(.)g(.)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5463
(.)f(.)p Black 88 w(71)p Black 414 3250 a(3.4)96 b(T)m(ranslation)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5464
b(from)f(intuitionistic)j(sequent)f(proofs)f(to)e(NJ-proofs)75
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5465
b(.)45 b(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 88 w(71)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5466
Black 414 3363 a(3.5)96 b(A)22 b(non-terminating)27 b(reduction)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5467
(sequence)57 b(.)45 b(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5468
(.)g(.)g(.)g(.)h(.)f(.)p Black 88 w(75)p Black 414 3476
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5469
a(3.6)96 b(Natural)24 b(deduction)i(calculus)g(for)e(classical)h(logic)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5470
86 b(.)45 b(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5471
Black 88 w(83)p Black 414 3588 a(3.7)96 b(T)-6 b(erm)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5472
b(assignment)k(for)d(classical)j(natural)f(deduction)i(proofs)72
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5473
b(.)45 b(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5474
w(85)p Black 414 3701 a(3.8)96 b(T)m(ranslation)25 b(from)f(natural)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5475
(deduction)h(proofs)f(to)f(sequent)h(proofs)32 b(.)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5476
b(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 88 w(86)p Black 414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5477
3814 a(3.9)96 b(T)m(ranslation)25 b(from)f(sequent)h(proofs)g(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5478
(natural)h(deduction)h(proofs)32 b(.)45 b(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5479
Black 88 w(87)p Black 414 3927 a(3.10)51 b(Proof)23 b(substitution)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5480
(in)d(natural)h(deduction)h(I)67 b(.)45 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5481
(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 88 w(91)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5482
Black 414 4040 a(3.11)51 b(Proof)23 b(substitution)k(in)d(natural)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5483
(deduction)h(II)37 b(.)45 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5484
(.)g(.)g(.)g(.)h(.)f(.)p Black 88 w(92)p Black 414 4236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5485
a(4.1)96 b(Excerpt)24 b(from)f(the)h(reduction)i(system)f(for)e(an)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5486
(outermost-leftmost)k(strate)o(gy)93 b(.)45 b(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5487
43 w(104)p Black 414 4349 a(4.2)96 b(T)-7 b(w)o(o)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5488
b(normal)i(forms)g(reachable)i(by)d(applying)j(dif)n(ferent)g(logical)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5489
(reductions)40 b(.)46 b(.)f(.)p Black 43 w(105)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5490
414 4462 a(4.3)96 b(Code)23 b(for)h(main)f(datatypes)85
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5491
b(.)45 b(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5492
g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 43 w(108)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5493
Black 414 4575 a(4.4)96 b(A)22 b(normal)i(form)g(of)f(Proof)h(\(4.1\))g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5494
(sho)n(wn)g(on)f(P)o(age)g(110)67 b(.)45 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5495
(.)g(.)g(.)h(.)f(.)p Black 43 w(112)p Black 414 4688
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5496
a(4.5)96 b(Sequent)24 b(proofs)h Ga(\034)33 b Gg(and)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5497
b Ga(")1462 4702 y Gc(i)1527 4688 y Gg(.)45 b(.)g(.)h(.)f(.)g(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5498
(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5499
h(.)f(.)p Black 43 w(119)p Black 414 4801 a(4.6)96 b(T)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5500
b(yping)24 b(rules)g(for)g Ga(\025)1282 4764 y F9(+)1391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5501
4801 y Gg(.)45 b(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5502
g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5503
Black 43 w(121)p Black 414 4997 a(5.1)96 b(A)22 b(normal)i(form)g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5504
(the)h(LK-proof)g(gi)n(v)o(en)g(in)g(Example)f(2.1.3)51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5505
b(.)45 b(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5506
w(128)p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5507
%%Page: 12 12
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5508
TeXDict begin 12 11 bop Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5509
%%Page: 1 13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5510
TeXDict begin 1 12 bop Black Black 277 957 a F8(Chapter)44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5511
b(1)277 1388 y Gf(Intr)l(oduction)p Black Black 1140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5512
1805 a Gd(A)20 b(sequent)f(calculus)g(without)g(cut-elimination)f(is)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5513
(lik)o(e)g(a)g(car)g(without)f(engine.)3031 2004 y(\227J.-Y)-11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5514
b(.)20 b(Girard)1871 2103 y(in)h(Linear)e(Logic:)25 b(Its)c(Syntax)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5515
(and)h(Semantics,)f(1995.)277 2416 y Gg(In)f(this)h(thesis)g(we)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5516
(study)h(the)g(proof)g(theory)h(of)e(classical)i(logic)f(and)g(address)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5517
h(the)f(problem)g(of)f(gi)n(ving)277 2528 y(a)35 b(computational)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5518
b(interpretation)f(to)d(classical)i(proofs.)67 b(This)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5519
b(interpretation)40 b(aims)c(to)g(capture)277 2641 y(features)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5520
b(of)d(computation)k(that)d(go)g(be)o(yond)h(what)f(can)g(be)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5521
(xpressed)i(in)e(intuitionistic)j(logic.)k(The)277 2754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5522
y(point)c(of)e(departure)j(for)e(this)g(thesis)g(will)f(be)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5523
F7(Curry-Howar)m(d)g(corr)m(espondence)p Gg(,)k(which)c(gi)n(v)o(es)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5524
2867 y(a)e(computational)k(interpretation)h(to)23 b(intuitionistic)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5525
b(proofs.)418 2997 y(According)k(to)e(the)h(Curry-Ho)n(w)o(ard)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5526
(correspondence,)37 b(the)30 b(simply-typed)j(lambda)e(calculus)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5527
3110 y(can)24 b(be)g(vie)n(wed)g(as)g(a)f(term)h(assignment)i(for)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5528
(intuitionistic)k(proofs)d(formalised)h(in)d(Gentzen')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5529
b(s)26 b(nat-)277 3222 y(ural)32 b(deduction)h(calculus)g(NJ.)d(What)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5530
(mak)o(es)h(this)f(term)g(assignment)i(e)o(xciting)g(is)e(that)g(it)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5531
(relates)277 3335 y(not)24 b(just)g(tw)o(o)f(calculi,)i(b)n(ut)f(also)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5532
(certain)h(concepts)h(within)e(them,)g(as)f(sho)n(wn)h(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5533
b(.)p Black Black 631 3549 a(Gentzen')h(s)25 b(NJ-calculus)614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5534
b(simply-typed)27 b(lambda)d(calculus)895 3732 y(formula)667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5535
b F6(,)585 b Gg(type)940 3845 y(proof)713 b F6(,)580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5536
b Gg(term)789 3958 y(normalisation)563 b F6(,)492 b Gg(reduction)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5537
4169 y(Curry)20 b(and)g(Fe)o(ys)g([1958])h(hinted)g(at)f(these)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5538
(correspondences,)k(b)n(ut)20 b(it)g(took)h(quite)f(some)g(time)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5539
(before)277 4282 y(the)o(y)k(were)f(systematically)k(studied)f(by)d(Ho)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5540
n(w)o(ard)h([1980].)418 4411 y(One)h(consequence)k(of)c(the)g(Curry-Ho)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5541
n(w)o(ard)h(correspondence)k(is)25 b(that)h(we)e(can)h(talk)h(of)f(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5542
F7(com-)277 4524 y(putational)29 b(interpr)m(etation)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5543
Gg(of)c(NJ-proofs.)37 b(The)26 b(simply-typed)j(lambda)e(calculus)h(is)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5544
d(a)h(prototyp-)277 4637 y(ical)i(functional)j(programming)f(language:)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5545
40 b(the)28 b(terms)g(are)g(programs,)i(and)e(reduction)j(is)c(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5546
(form)277 4750 y(of)c(computation,)i(which)e(con)l(v)o(erts)i(a)d(term)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5547
h(to)g(its)f(simplest)i(form,)f(analogous)j(to)c(symbolic)j(e)n(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5548
(alu-)277 4863 y(ation.)33 b(On)23 b(the)i(other)g(hand,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5549
(normalisation)h(is)e(a)f(method)h(for)g(eliminating)i(certain)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5550
(redundancies)277 4976 y(in)32 b(proofs.)57 b(Applied)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5551
b(iterati)n(v)o(ely)-6 b(,)37 b(it)32 b(transforms)i(a)e(proof)i(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5552
(one)h(in)f(normal)h(form.)55 b(Using)33 b(the)277 5088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5553
y(Curry-Ho)n(w)o(ard)d(correspondence)k(we)28 b(see)h(that)g(the)h(tw)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5554
(o)e(notions)j(coincide,)h(and)e(thus)f(the)g(com-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5555
5201 y(putational)i(interpretation)i(of)28 b(an)h(NJ-proof)g(is)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5556
(corresponding)33 b(simply-typed)e(lambda)f(term)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5557
5314 y(representing)d(a)c(functional)k(program.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5558
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5559
%%Page: 2 14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5560
TeXDict begin 2 13 bop Black -144 51 a Gb(2)3152 b(Intr)n(oduction)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5561
-144 88 3691 4 v Black 452 317 a Gg(Because)37 b(of)e(its)h(connection)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5562
j(with)d(computation,)41 b(the)36 b(Curry-Ho)n(w)o(ard)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5563
(correspondence)k(is)321 430 y(quite)35 b(useful)g(in)e(programming)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5564
(language)h(design:)51 b(often)34 b(computer)h(scientists)h(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5565
(logicians)321 543 y(ha)n(v)o(e)29 b(arri)n(v)o(ed)g(independently)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5566
(at)c(the)h(same)f(idea.)43 b(F)o(or)27 b(e)o(xample,)j(Girard)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5567
([1972])i(and)e(Re)o(ynolds)321 656 y([1974])35 b(independently)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5568
(studied)e(the)e(second-order)k(polymorphic)f(lambda)e(calculus,)j(and)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5569
c(v)o(er)n(-)321 769 y(sions)g(of)e(the)h(typing)h(algorithm)g(of)e(ML)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5570
f(were)h(de)n(v)o(eloped)j(independently)h(by)d(Hindle)o(y)g([1969])321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5571
882 y(and)22 b(Milner)f([1978].)30 b(Another)22 b(e)o(xample)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5572
(\(described)i(by)d(Mitchell)h([1996]\))h(is)e(the)g(formulation)j(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5573
321 995 y(sums)h(in)f(\223classical\224)j(ML,)c(which)i(w)o(as)f(not)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5574
(incorrect,)h(b)n(ut)f(relied)h(on)f(e)o(xceptions)i(to)d(ensure)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5575
(type)321 1108 y(security)-6 b(.)31 b(This)22 b(meant)h(a)f(form)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5576
(runtime)i(type)f(check)h(w)o(as)e(necessary)j(to)e(ensure)h(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5577
(programs)321 1221 y(could)33 b(not)e(change)i(their)f(type.)53
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5578
b(Clearly)32 b(this)g(runtime)g(check)h(slo)n(wed)f(do)n(wn)f(the)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5579
(x)o(ecution)j(of)321 1334 y(programs.)42 b(The)27 b(proper)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5580
(formulation)h(introduced)g(later)l(,)f(which)f(does)g(not)g(require)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5581
(an)o(y)f(runtime)321 1446 y(type)d(check,)f(corresponds)j(e)o(xactly)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5582
(to)e(the)h(inference)i(rules)f(for)e F6(_)g Gg(in)g(NJ.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5583
1587 y(In)28 b(practice,)i(the)e(Curry-Ho)n(w)o(ard)h(correspondence)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5584
(is)27 b(often)i(emplo)o(yed)g(in)e(proof)i(assistants)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5585
1700 y(with)d(which)g(man)o(y)g(mathematical)i(proofs)g(can)e(be)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5586
(fully)h(formalised)h(and)e(check)o(ed.)39 b(F)o(ormalised)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5587
1813 y(proofs)25 b(ho)n(we)n(v)o(er)e(tend)h(to)f(be)h(v)o(ery)f(lar)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5588
(ge,)i(and)e(therefore)j(one)d(needs)i(good)f(bookk)o(eeping)j(de)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5589
(vices)321 1926 y(to)c(k)o(eep)g(track)h(of)e(them.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5590
b(NuPrl)22 b(and)h(Coq)f(are)h(e)o(xamples)h(of)e(contemporary)k(proof)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5591
e(assistants)h(that)321 2038 y(mak)o(e)f(use)g(of)f(the)h(Curry-Ho)n(w)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5592
o(ard)h(correspondence)j(by)c(representing)j(proofs)e(as)e(terms.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5593
2179 y(Whilst)28 b(the)f(Curry-Ho)n(w)o(ard)h(correspondence)k(w)o(as)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5594
26 b(originally)k(established)g(for)d(the)h(simply-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5595
2292 y(typed)g(lambda)f(calculus)h(and)e(Gentzen')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5596
b(s)28 b(NJ-calculus)h(only)-6 b(,)27 b(later)g(it)f(has)h(been)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5597
(often)i(studied)g(in)321 2405 y(the)22 b(setting)i(of)e(more)f(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5598
(xpressi)n(v)o(e)j(typed)f(lambda)g(calculi)g(and)f(more)g(e)o(xpressi)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5599
n(v)o(e)i(logics.)29 b(This)22 b(w)o(as)321 2518 y(moti)n(v)n(ated)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5600
b(by)f(the)g(f)o(act)g(that,)i(although)g(the)d(simply-typed)k(lambda)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5601
(calculus)g(can)f(be)g(seen)g(as)g(a)321 2630 y(core)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5602
b(of)e(v)n(arious)i(mainstream)g(functional)i(programming)f(languages,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5603
g(only)e(a)g(limited)g(number)h(of)321 2743 y(features)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5604
b(of)d(those)h(languages)i(\(e.g.)d(pattern)i(matching)g(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5605
(abstract)h(datatypes\))h(can)e(be)f(easily)321 2856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5606
y(encoded)e(into)f(this)f(core.)34 b(Other)25 b(features)i(cannot.)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5607
b(Examples)26 b(are)f(polymorphism)j(and)d(state.)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5608
b(In)321 2969 y(more)23 b(e)o(xpressi)n(v)o(e)i(typed)f(lambda)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5609
(calculi)i(these)e(features)i(are)e(directly)i(accessible.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5610
b(F)o(or)22 b(instance,)321 3082 y(System)31 b(F)f(pro)o(vides)i(f)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5611
(acilities)h(for)e(polymorphism)j([Girard)d(et)g(al.,)f(1989],)j(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5612
(linear)g(lambda)321 3195 y(calculi)g(of)n(fer)e(mechanisms)i(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5613
(manipulate)i(state)e([W)-7 b(adler,)31 b(1990].)49 b(Using)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5614
b(the)g(Curry-Ho)n(w)o(ard)321 3308 y(correspondence)f(one)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5615
b(\002nds)f(the)h(more)g(e)o(xpressi)n(v)o(e)i(lambda)e(calculi)h(gi)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5616
(v)o(e)f(a)f(computational)k(inter)n(-)321 3421 y(pretation)f(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5617
(proofs)h(in)e(more)h(e)o(xpressi)n(v)o(e)h(NJ-calculi,)g(three)g(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5618
(which)h(are)g(listed)g(belo)n(w)-6 b(.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5619
925 3699 a(NJ-calculi)865 b(typed)25 b(lambda)f(calculi)625
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5620
3883 y(\002rst-order)h(NJ-calculus)147 b F6(,)812 b Ga(\025P)56
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5621
b Gg([de)24 b(Bruijn,)g(1980])514 3996 y(second-order)j(NJ-calculus)147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5622
b F6(,)595 b Gg(System)24 b(F)42 b([Girard,)24 b(1972])781
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5623
4109 y(linear)h(NJ-calculus)147 b F6(,)118 b Gg(linear)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5624
b(lambda)f(calculus)46 b([Benton)25 b(et)e(al.,)g(1993])462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5625
4384 y(In)i(this)g(thesis)h(we)e(shall)i(study)f(the)g(Curry-Ho)n(w)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5626
(ard)h(correspondence)j(in)c(the)g(setting)h(of)f(clas-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5627
4496 y(sical)h(logic.)35 b(This)25 b(logic)h(is)f(more)g(e)o(xpressi)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5628
(v)o(e)i(than)f(intuitionistic)j(logic,)e(in)e(the)g(sense)h(that)g(e)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5629
(v)o(ery)321 4609 y(intuitionistic)32 b(proof)d(is)e(a)g(classical)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5630
(proof)f(b)n(ut)f(not)g F7(vice)g(ver)o(sa)p Gg(.)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5631
b(Grif)n(\002n)27 b([1990])i(w)o(as)e(the)h(\002rst)g(to)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5632
4722 y(note)34 b(a)e(connection)j(between)f(classical)g(logic)g(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5633
(functional)i(programming)f(languages)i(with)321 4835
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5634
y(control)30 b(operators.)44 b(W)-7 b(e)27 b(tak)o(e)i(this)f(observ)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5635
(ation)j(as)d(e)n(vidence)i(that)e(a)f(computational)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5636
b(interpreta-)321 4948 y(tion)26 b(for)f(classical)i(proofs)f(can)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5637
(capture)g(features)h(of)e(computation)i(not)f(directly)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5638
(xpressible)i(in)d(a)321 5061 y(typed)g(lambda)f(calculus)i(which)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5639
(corresponds)j(to)c(intuitionistic)28 b(logic.)462 5201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5640
y(Usually)-6 b(,)32 b(the)e(Curry-Ho)n(w)o(ard)g(correspondence)k(is)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5641
b(formulated)j(for)d(natural)i(deduction)h(cal-)321 5314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5642
y(culi,)j(because)g(in)d(intuitionistic)37 b(logic)c(the)o(y)g(ha)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5643
(e)g(compelling)i(adv)n(antages)g(as)e(deduction)i(sys-)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5644
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5645
%%Page: 3 15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5646
TeXDict begin 3 14 bop Black 3922 51 a Gb(3)p 277 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5647
3691 4 v Black 277 317 a Gg(tems.)27 b(Unfortunately)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5648
b(,)23 b(there)d(are)f(v)n(arious)h(technical)h(reasons)2272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5649
284 y F5(1)2330 317 y Gg(which)e(mak)o(e)g(their)h(classical)h(coun-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5650
277 430 y(terparts)k(less)e(suitable)i(as)d(deduction)k(systems)d(for)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5651
(classical)i(logic.)k(Sequent)24 b(calculi)g(seem)f(much)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5652
543 y(better)i(suited.)30 b(Therefore)25 b(in)e(this)h(thesis)g(we)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5653
(shall)h(mainly)g(focus)h(on)e(sequent)j(calculus)f(formula-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5654
656 y(tions)c(of)f(classical)i(logic.)28 b(The)20 b(best-kno)n(wn)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5655
(sequent)f(calculus)h(for)e(classical)j(logic)d(is)g(the)g(calculus)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5656
769 y(LK)g(in)l(v)o(ented)j(by)e(Gentzen)h([1935].)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5657
b(T)-7 b(o)20 b(illustrate)j(some)e(problems)i(concerning)h(a)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5658
(computational)277 882 y(interpretation)28 b(of)23 b(classical)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5659
(logic,)e(we)f(shall)h(describe)i(this)e(calculus)i(brie\003y)e(belo)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5660
(w)-6 b(.)418 1012 y(In)24 b(LK)d(proofs)k(consist)g(of)f(trees)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5661
(labelled)h(by)f F7(sequents)i Gg(of)d(the)h(form)f F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5662
2751 1000 11 41 v 2762 982 46 5 v 96 w(\001)p Gg(,)f(where)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5663
F4(\000)f Gg(and)g F4(\001)277 1125 y Gg(are)i(collections)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5664
(formulae.)33 b(The)23 b(axioms)j(of)e(LK)e(are)j(of)f(the)h(form)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5665
2573 1046 244 4 v 24 w Ga(B)p 2667 1113 11 41 v 2677
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5666
1095 46 5 v 101 w(B)t Gg(.)31 b(The)23 b(general)j(form)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5667
1238 y(of)e(an)f F7(infer)m(ence)j(rule)e Gg(is)f(either)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5668
Black Black 1479 1462 a Ga(P)p 1479 1482 72 4 v 1479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5669
1560 a(C)1744 1494 y Gg(or)2014 1448 y Ga(P)2072 1462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5670
y F9(1)2203 1448 y Ga(P)2261 1462 y F9(2)p 2014 1482
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5671
287 4 v 2122 1560 a Ga(C)277 1785 y Gg(where)i(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5672
Ga(P)13 b Gg(')-5 b(s)25 b(stand)h(for)f(sequents,)i(called)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5673
F7(pr)m(emises)p Gg(,)g(and)g Ga(C)k Gg(stands)d(for)e(a)f(sequent,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5674
(called)f(the)277 1898 y F7(conclusion)p Gg(,)i(which)d(is)g(deduced)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5675
(from)e(the)g(premises.)34 b(There)26 b(are)f(three)g(sorts)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5676
(inference)i(rules)277 2011 y(in)32 b(LK.)e(Logical)j(inference)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5677
(rules)f(introduce)i(a)c(formula)i(in)f(the)g(conclusion)j(and)e(come)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5678
(in)g(tw)o(o)277 2124 y(\003a)n(v)n(ours\227left)h(rules)f(and)f(right)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5679
g(rules.)51 b(F)o(or)30 b(e)o(xample,)j(the)e(left)g(and)g(right)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5680
(rule)g(introducing)j(an)277 2237 y(implicational)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5681
b(formula)d(ha)n(v)o(e)g(the)g(form:)p Black Black 839
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5682
2448 a F4(\000)896 2462 y F9(1)p 956 2436 11 41 v 966
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5683
2418 46 5 v 1032 2448 a F4(\001)1108 2462 y F9(1)1147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5684
2448 y Ga(;)15 b(B)96 b(C)q(;)15 b F4(\000)1515 2462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5685
y F9(2)p 1575 2436 11 41 v 1586 2418 46 5 v 1651 2448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5686
a F4(\001)1727 2462 y F9(2)p 839 2486 928 4 v 877 2564
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5687
a Ga(B)5 b F6(\033)p Ga(C)q(;)15 b F4(\000)1185 2578
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5688
y F9(1)1225 2564 y Ga(;)g F4(\000)1322 2578 y F9(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5689
1381 2552 11 41 v 1392 2534 46 5 v 1458 2564 a F4(\001)1534
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5690
2578 y F9(1)1573 2564 y Ga(;)g F4(\001)1689 2578 y F9(2)1808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5691
2503 y F6(\033)1879 2517 y Gc(L)2300 2448 y Ga(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5692
b F4(\000)p 2491 2436 11 41 v 2502 2418 46 5 v 96 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5693
Ga(;)g(C)p 2285 2486 485 4 v 2285 2564 a F4(\000)p 2362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5694
2552 11 41 v 2373 2534 46 5 v 96 w(\001)p Ga(;)g(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5695
b F6(\033)p Ga(C)2812 2503 y F6(\033)2882 2517 y Gc(R)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5696
2789 y Gg(Structural)24 b(inference)h(rules)e(manipulate)i(sequents,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5697
(in)d(the)h(sense)g(that)g(contraction)j(identi\002es)e(oc-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5698
2902 y(currences)h(of)d(the)h(same)g(formula)g(and)g(weak)o(ening)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5699
(introduces)g(surplus)f(formulae.)30 b(F)o(or)22 b(e)o(xample,)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5700
3015 y(the)i(contraction-left)k(and)c(weak)o(ening-right)k(rule)c(ha)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5701
(v)o(e)g(the)g(form:)p Black Black 1010 3226 a Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5702
b(;)15 b(B)5 b(;)15 b F4(\000)p 1315 3214 11 41 v 1326
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5703
3196 46 5 v 96 w(\001)p 1010 3264 457 4 v 1067 3342 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5704
Ga(B)5 b(;)15 b F4(\000)p 1258 3330 11 41 v 1269 3312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5705
46 5 v 96 w(\001)1509 3288 y Gg(Contr)1715 3302 y Gc(L)2179
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5706
3244 y F4(\000)p 2256 3232 11 41 v 2267 3213 46 5 v 96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5707
w(\001)p 2122 3264 343 4 v 2122 3342 a(\000)p 2199 3330
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5708
11 41 v 2210 3312 46 5 v 96 w(\001)p Ga(;)g(B)2506 3288
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5709
y Gg(W)-7 b(eak)2710 3302 y Gc(R)277 3567 y Gg(The)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5710
b(third)i(sort)f(of)f(inference)j(rule)e(is)g(the)f(cut-rule)1336
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5711
3750 y F4(\000)1393 3764 y F9(1)p 1453 3738 11 41 v 1463
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5712
3720 46 5 v 1529 3750 a F4(\001)1605 3764 y F9(1)1644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5713
3750 y Ga(;)15 b(B)96 b(B)5 b(;)15 b F4(\000)2020 3764
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5714
y F9(2)p 2079 3738 11 41 v 2089 3720 46 5 v 2155 3750
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5715
a F4(\001)2231 3764 y F9(2)p 1336 3788 935 4 v 1503 3867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5716
a F4(\000)1560 3881 y F9(1)1599 3867 y Ga(;)g F4(\000)1696
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5717
3881 y F9(2)p 1756 3855 11 41 v 1767 3836 46 5 v 1832
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5718
3867 a F4(\001)1908 3881 y F9(1)1948 3867 y Ga(;)g F4(\001)2064
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5719
3881 y F9(2)2312 3818 y Gg(Cut)277 4091 y(which)23 b(enables)i(us)e(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5720
g(join)g(tw)o(o)g(LK-proofs)h(together)-5 b(.)30 b(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5721
b(e)22 b(shall)i(often)g(simply)g(refer)f(to)g(an)g(appli-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5722
4204 y(cation)i(of)e(the)h(cut-rule)i(as)d(a)g(cut.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5723
4334 y(One)f(consequence)k(of)d(our)f(use)h(of)f(sequent)j(calculi)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5724
(in)e(place)h(of)f(natural)i(deduction)i(calculi)e(is)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5725
4447 y(that)i(computation)j(corresponds)g(to)d F7(cut-elimination)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5726
Gg(\227a)j(procedure)g(which)d(transforms)i(a)d(gi)n(v)o(en)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5727
4560 y(sequent)32 b(proof)g(containing)h(instances)g(of)d(cut)h(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5728
(one)h(with)f(no)g(instances)j(of)d(cut)h(\(called)h(a)e(cut-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5729
4673 y(free)e(proof\).)40 b(This)27 b(procedure)i(does)f(not)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5730
(eliminate)i(all)e(cuts)g(from)g(a)g(proof)h(immediately)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5731
b(,)29 b(rather)277 4786 y(it)j(replaces)i(e)n(v)o(ery)f(cut)f(with)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5732
(simpler)h(cuts,)i(and)e(by)f(iteration)i(one)f(e)n(v)o(entually)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5733
(ends)e(up)f(with)g(a)277 4899 y(cut-free)25 b(proof.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5734
5029 y(Zuck)o(er)37 b([1974])g(and)f(Pottinger)h([1977])g(ha)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5735
(sho)n(wn)f(that)g(normalisation)j(and)d(cut-elimi-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5736
5142 y(nation)29 b(are)g(closely)g(related)h(in)d(intuitionistic)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5737
b(logic.)43 b(Thus)28 b(the)g(question)i(of)e(the)g(computational)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5738
Black 277 5227 1290 4 v 383 5283 a F3(1)412 5314 y F2(\223One)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5739
b(may)h(doubt)g(that)f(this)g(is)g(the)h(proper)g(w)o(ay)g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5740
(analysing)h(classical)f(inferences.)-5 b(\224)24 b([Pra)o(witz,)16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5741
b(1971,)i(P)o(age)f(244])p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5742
%%Page: 4 16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5743
TeXDict begin 4 15 bop Black -144 51 a Gb(4)3152 b(Intr)n(oduction)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5744
-144 88 3691 4 v Black 321 317 a Gg(meaning)26 b(of)e(cut-elimination)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5745
(is)c(settled)h(in)f(the)h(intuitionistic)j(case.)j(Surprisingly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5746
b(,)27 b(there)e(is)f(little)321 430 y(attention)36 b(paid)e(in)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5747
(literature)h(to)f(an)f(analysis)i(of)f(what)f(cut-elimination)k(in)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5748
(classical)j(logic)321 543 y(means)23 b(computationally)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5749
b(.)33 b(At)21 b(the)i(time)f(of)g(writing)h(we)f(are)g(a)o(w)o(are)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5750
(only)g(of)f(w)o(ork)h(by)f(Danos)h(et)f(al.)321 656
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5751
y([1997],)34 b(which)c(considers)j(a)d(v)n(ariant)i(of)e(Gentzen')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5752
b(s)33 b(sequent)f(calculus)g(LK)d(as)h(a)g(programming)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5753
769 y(language.)51 b(Ho)n(we)n(v)o(er)l(,)31 b(we)e(shall)i(sho)n(w)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5754
(that)g(there)h(are)f(still)h(man)o(y)f(open)h(questions)h(concerning)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5755
321 882 y(the)24 b(relationship)j(between)e(cut-elimination)i(and)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5756
(computation.)462 1011 y(One)e(of)h(the)f(reasons)j(for)d(the)h(little)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5757
g(attention)i(is)d(that,)h(in)f(contrast)j(to)d(cut-elimination)k(in)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5758
(intu-)321 1124 y(itionistic)28 b(logic,)f(cut-elimination)i(in)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5759
(classical)i(logic)f(had)g(been)g(dismissed)h(in)f(the)f(past)h(as)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5760
(being)321 1237 y(not)j(v)o(ery)g(interesting)i(from)e(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5761
(computational)k(point)d(of)g(vie)n(w)-6 b(.)40 b(F)o(or)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5762
b(e)o(xample,)j(Lafont)f(ar)n(gued)i(in)321 1350 y(the)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5763
b(in\003uential)i(book)e([Girard)h(et)e(al.,)g(1989])i(that)f(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5764
(classical)h(logic)g(a)e(correspondence)28 b(between)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5765
1463 y(cut-elimination)g(and)d(computation)h(is)e(unachie)n(v)n(able,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5766
(o)n(wing)e(to)e(an)h(\223inconsistenc)o(y\224.)35 b(He)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5767
b(noted)321 1576 y(that)h(the)g(classical)i(proof)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5768
Black 397 2243 a @beginspecial 179 @llx 453 @lly 318
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5769
@urx 628 @ury 417 @rwi @clip @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5770
%%BeginDocument: pics/lafont-left.ps
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5771
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5772
%%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5773
%%Title: lafont-left.dvi
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5774
%%Pages: 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5775
%%PageOrder: Ascend
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5776
%%BoundingBox: 0 0 596 842
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5777
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5778
%DVIPSWebPage: (www.radicaleye.com)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5779
%DVIPSCommandLine: dvips -o lafont-left.ps lafont-left.dvi
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5780
%DVIPSParameters: dpi=600, compressed
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5781
%DVIPSSource:  TeX output 2000.03.09:0346
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5782
%%BeginProcSet: texc.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5783
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5784
/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5785
N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5786
mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5787
0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5788
landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5789
mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5790
matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5791
exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5792
statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5793
N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5794
/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5795
/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5796
array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5797
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5798
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5799
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5800
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5801
1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5802
1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5803
0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5804
sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5805
rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5806
gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5807
/chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5808
/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5809
A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5810
get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5811
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5812
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5813
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5814
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5815
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5816
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5817
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5818
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5819
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5820
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5821
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5822
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5823
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5824
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5825
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5826
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5827
(LaserWriter 16/600)]{A length product length le{A length product exch 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5828
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5829
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5830
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5831
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5832
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5833
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5834
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5835
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5836
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5837
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5839
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5840
%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5841
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5842
% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5843
% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5844
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5845
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5846
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5847
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5848
/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5849
/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5850
/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5851
/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5852
/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5853
/L /lineto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5854
/T /translate load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5855
/TMatrix { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5856
/RAngle { 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5857
/Atan { /atan load stopped { pop pop 0 } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5858
/Div { dup 0 eq { pop } { div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5859
/NET { neg exch neg exch T } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5860
/Pyth { dup mul exch dup mul add sqrt } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5861
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5862
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5863
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5864
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5865
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5866
/pathforall load stopped { pop pop pop pop } if z } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5867
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5868
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5869
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5870
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5871
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5872
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5873
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5874
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5875
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5876
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5877
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5878
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5879
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5880
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5881
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5882
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5883
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5884
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5885
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5886
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5887
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5888
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5889
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5890
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5891
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5892
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5893
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5894
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5895
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5896
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5897
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5898
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5899
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5900
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5901
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5902
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5903
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5904
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5905
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5906
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5907
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5908
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5909
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5910
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5911
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5912
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5913
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5914
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5915
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5916
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5917
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5918
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5919
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5920
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5921
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5922
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5923
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5924
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5925
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5926
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5927
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5928
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5929
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5930
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5931
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5932
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5933
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5934
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5935
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5936
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5937
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5938
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5939
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5940
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5941
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5942
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5943
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5944
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5945
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5946
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5947
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5948
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5949
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5950
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5951
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5952
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5953
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5954
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5955
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5956
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5957
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5958
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5959
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5960
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5961
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5962
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5963
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5964
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5965
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5966
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5967
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5968
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5969
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5970
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5971
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5972
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5973
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5974
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5975
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5976
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5977
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5978
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5979
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5980
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5981
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5982
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5983
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5984
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5985
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5986
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5987
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5988
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5989
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5990
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5991
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5992
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5993
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5994
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5995
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5996
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5997
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5998
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  5999
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6000
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6001
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6002
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6003
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6004
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6005
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6006
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6007
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6008
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6009
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6010
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6011
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6012
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6013
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6014
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6015
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6016
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6017
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6018
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6019
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6020
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6021
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6022
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6023
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6024
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6025
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6026
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6027
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6028
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6029
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6030
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6031
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6032
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6033
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6034
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6035
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6036
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6037
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6038
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6039
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6040
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6041
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6042
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6043
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6044
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6045
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6046
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6047
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6048
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6049
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6050
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6051
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6052
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6053
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6054
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6055
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6056
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6057
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6058
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6059
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6060
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6061
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6062
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6063
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6064
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6065
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6066
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6067
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6068
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6069
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6070
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6071
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6072
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6073
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6074
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6075
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6076
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6077
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6078
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6079
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6080
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6081
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6082
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6083
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6084
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6085
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6086
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6087
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6088
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6089
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6090
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6091
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6092
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6093
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6094
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6095
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6096
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6097
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6098
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6099
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6100
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6101
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6102
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6103
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6104
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6105
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6106
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6107
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6108
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6109
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6110
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6111
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6112
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6113
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6114
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6115
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6116
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6117
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6118
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6119
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6120
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6121
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6122
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6123
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6124
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6125
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6126
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6127
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6128
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6129
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6130
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6131
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6132
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6133
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6134
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6135
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6136
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6137
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6138
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6139
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6140
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6141
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6142
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6143
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6144
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6145
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6146
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6147
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6148
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6149
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6150
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6151
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6152
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6153
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6154
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6155
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6156
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6157
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6158
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6159
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6160
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6161
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6162
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6163
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6164
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6165
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6166
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6167
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6168
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6169
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6170
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6171
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6172
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6173
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6174
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6175
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6176
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6178
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6179
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6180
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6181
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6182
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6183
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6184
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6185
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6186
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6187
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6188
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6189
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6190
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6191
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6192
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6193
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6194
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6195
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6196
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6197
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6198
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6199
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6200
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6201
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6202
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6203
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6204
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6205
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6206
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6207
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6208
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6209
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6210
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6211
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6212
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6213
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6214
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6215
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6216
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6217
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6218
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6219
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6220
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6221
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6222
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6223
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6224
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6225
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6226
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6227
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6228
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6229
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6230
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6231
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6232
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6233
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6234
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6235
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6236
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6237
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6238
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6239
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6240
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6241
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6242
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6243
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6244
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6245
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6246
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6247
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6248
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6249
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6250
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6251
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6252
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6253
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6254
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6255
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6256
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6257
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6258
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6259
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6260
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6261
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6262
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6263
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6264
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6265
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6266
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6267
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6268
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6269
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6270
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6271
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6272
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6273
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6274
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6275
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6276
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6277
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6278
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6279
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6280
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6281
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6282
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6283
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6284
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6285
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6286
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6287
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6288
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6289
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6290
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6291
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6292
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6293
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6294
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6295
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6296
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6297
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6298
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6299
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6300
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6301
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6302
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6303
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6304
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6305
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6306
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6307
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6308
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6309
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6310
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6311
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6312
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6313
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6314
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6315
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6316
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6317
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6318
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6319
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6320
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6321
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6322
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6323
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6324
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6325
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6326
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6327
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6328
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6329
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6330
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6331
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6332
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6333
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6334
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6335
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6336
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6337
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6338
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6339
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6340
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6341
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6342
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6343
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6344
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6345
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6346
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6347
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6348
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6349
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6350
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6351
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6352
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6353
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6354
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6355
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6356
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6357
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6358
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6359
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6360
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6361
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6362
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6363
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6364
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6365
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6366
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6367
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6368
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6369
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6370
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6371
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6372
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6373
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6374
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6375
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6376
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6377
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6378
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6379
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6380
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6381
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6382
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6383
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6384
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6385
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6386
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6387
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6388
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6389
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6391
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6392
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6393
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6394
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6395
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6396
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6398
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6399
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6400
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6401
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6402
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6403
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6404
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6405
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6406
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6407
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6408
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6409
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6410
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6411
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6412
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6413
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6414
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6416
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6418
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6419
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6420
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6421
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6422
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6423
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6424
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6425
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6426
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6427
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6428
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6429
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6430
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6431
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6432
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6433
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6434
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6435
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6436
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6437
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6438
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6439
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6440
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6441
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6442
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6443
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6444
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6445
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6446
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6447
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6448
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6449
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6450
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6451
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6452
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6453
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6454
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6455
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6456
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6457
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6458
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6459
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6460
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6462
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6463
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6464
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6465
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6466
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6467
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6468
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6469
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6470
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6471
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6472
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6473
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6474
    Char end stringwidth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6475
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6476
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6477
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6478
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6479
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6480
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6481
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6482
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6483
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6484
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6485
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6486
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6487
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6488
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6489
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6491
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6492
%%BeginProcSet: special.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6493
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6494
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6495
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6496
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6497
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6498
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6499
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6500
/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6501
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6502
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6503
maxlength ge{/md md dup length 20 add dict copy def}if end md begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6504
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6505
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6506
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6507
transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6508
curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6509
pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6510
if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6511
-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6512
get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6513
yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6514
neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6515
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6516
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6517
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6518
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6519
2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6520
-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6521
TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6522
Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6523
}if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6524
save N userdict maxlength dict begin/magscale true def normalscale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6525
currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6526
/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6527
psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6528
psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6529
TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6530
psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6531
roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6532
moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6533
begin/SpecialSave save N gsave normalscale currentpoint TR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6534
@SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6535
CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6536
closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6537
sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6538
}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse
Christian Urban <urbanc@in.tum.de>
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ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6695
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6696
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6697
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6698
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6699
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6700
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6701
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6702
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6703
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6704
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6705
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6706
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6707
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6708
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6709
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6710
(LaserWriter 16/600)]{A length product length le{A length product exch 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6711
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6712
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6713
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6714
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6715
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6716
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6717
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6718
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6719
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6720
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6722
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6723
%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6724
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6725
% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6726
% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6727
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6728
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6729
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6730
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6731
/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6732
/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6733
/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6734
/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6735
/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6736
/L /lineto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6737
/T /translate load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6738
/TMatrix { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6739
/RAngle { 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6740
/Atan { /atan load stopped { pop pop 0 } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6741
/Div { dup 0 eq { pop } { div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6742
/NET { neg exch neg exch T } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6743
/Pyth { dup mul exch dup mul add sqrt } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6744
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6745
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6746
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6747
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6748
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6749
/pathforall load stopped { pop pop pop pop } if z } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6750
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6751
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6752
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6753
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6754
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6755
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6756
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6757
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6758
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6759
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6760
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6761
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6762
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6763
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6764
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6765
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6766
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6767
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6768
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6769
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6770
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6771
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6772
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6773
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6774
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6775
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6776
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6777
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6778
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6779
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6780
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6781
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6782
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6783
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6784
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6785
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6786
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6787
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6788
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6789
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6790
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6791
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6792
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6793
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6794
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6795
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6796
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6797
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6798
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6799
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6800
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6801
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6802
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6803
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6804
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6805
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6806
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6807
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6808
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6809
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6810
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6811
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6812
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6813
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6814
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6815
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6816
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6817
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6818
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6819
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6820
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6821
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6822
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6823
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6824
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6825
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6826
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6827
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6828
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6829
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6830
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6831
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6832
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6833
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6834
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6835
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6836
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6837
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6838
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6839
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6840
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6841
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6842
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6843
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6844
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6845
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6846
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6847
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6848
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6849
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6850
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6851
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6852
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6853
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6854
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6855
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6856
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6857
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6858
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6859
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6860
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6861
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6862
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6863
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6864
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6865
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6866
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6867
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6868
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6869
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6870
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6871
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6872
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6873
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6874
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6875
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6876
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6877
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6878
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6879
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6880
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6881
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6882
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6883
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6884
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6885
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6886
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6887
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6888
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6889
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6890
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6891
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6892
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6893
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6894
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6895
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6896
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6897
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6898
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6899
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6900
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6901
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6902
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6903
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6904
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6905
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6906
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6907
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6908
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6909
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6910
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6911
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6912
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6913
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6914
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6915
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6916
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6917
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6918
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6919
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6920
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6921
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6922
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6923
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6924
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6925
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6926
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6927
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6928
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6929
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6930
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6931
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6932
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6933
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6934
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6935
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6936
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6937
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6938
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6939
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6940
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6941
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6942
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6943
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6944
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6945
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6946
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6947
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6948
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6949
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6950
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6951
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6952
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6953
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6954
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6955
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6956
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6957
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6958
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6959
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6960
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6961
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6962
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6963
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6964
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6965
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6966
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6967
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6968
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6969
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6970
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6971
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6972
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6973
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6974
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6975
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6976
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6977
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6978
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6979
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6980
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6981
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6982
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6983
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6984
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6985
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6986
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6987
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6988
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6989
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6990
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6991
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6992
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6993
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6994
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6995
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6996
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6997
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6998
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  6999
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7000
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7001
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7002
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7003
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7004
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7005
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7006
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7007
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7008
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7009
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7010
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7011
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7012
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7013
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7014
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7015
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7016
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7017
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7018
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7019
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7020
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7021
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7022
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7023
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7024
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7025
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7026
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7027
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7028
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7029
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7030
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7031
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7032
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7033
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7034
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7035
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7036
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7037
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7038
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7039
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7040
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7041
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7042
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7043
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7044
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7045
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7046
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7047
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7048
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7049
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7050
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7051
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7052
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7053
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7054
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7055
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7056
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7057
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7058
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7059
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7061
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7062
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7063
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7064
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7065
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7066
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7067
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7068
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7069
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7070
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7071
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7072
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7073
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7074
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7075
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7076
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7077
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7078
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7079
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7080
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7081
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7082
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7083
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7084
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7085
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7086
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7087
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7088
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7089
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7090
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7091
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7092
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7093
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7094
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7095
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7096
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7097
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7098
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7099
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7100
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7101
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7102
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7103
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7104
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7105
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7106
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7107
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7108
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7109
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7110
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7111
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7112
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7113
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7114
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7115
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7116
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7117
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7118
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7119
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7120
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7121
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7122
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7123
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7124
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7125
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7126
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7127
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7128
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7129
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7130
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7131
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7132
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7133
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7134
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7135
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7136
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7137
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7138
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7139
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7140
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7141
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7142
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7143
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7144
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7145
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7146
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7147
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7148
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7149
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7150
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7151
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7152
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7153
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7154
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7155
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7156
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7157
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7158
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7159
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7160
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7161
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7162
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7163
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7164
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7165
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7166
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7167
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7168
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7169
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7170
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7171
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7172
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7173
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7174
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7175
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7176
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7177
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7178
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7179
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7180
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7181
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7182
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7183
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7184
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7185
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7186
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7187
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7188
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7189
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7190
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7191
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7192
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7193
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7194
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7195
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7196
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7197
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7198
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7199
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7200
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7201
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7202
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7203
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7204
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7205
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7206
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7207
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7208
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7209
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7210
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7211
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7212
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7213
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7214
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7215
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7216
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7217
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7218
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7219
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7220
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7221
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7222
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7223
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7224
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7225
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7226
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7227
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7228
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7229
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7230
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7231
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7232
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7233
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7234
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7235
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7236
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7237
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7238
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7239
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7240
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7241
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7242
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7243
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7244
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7245
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7246
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7247
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7248
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7249
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7250
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7251
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7252
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7253
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7254
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7255
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7256
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7257
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7258
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7259
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7260
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7261
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7262
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7263
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7264
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7265
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7266
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7267
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7268
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7269
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7270
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7271
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7272
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7273
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7274
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7275
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7276
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7277
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7278
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7279
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7280
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7281
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7282
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7283
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7284
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7285
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7286
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7287
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7288
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7289
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7290
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7291
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7292
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7293
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7294
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7295
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7296
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7297
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7298
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7299
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7300
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7301
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7302
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7303
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7304
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7305
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7306
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7308
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7309
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7310
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7311
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7312
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7313
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7314
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7315
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7316
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7317
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7318
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7319
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7320
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7321
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7322
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7323
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7324
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7325
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7326
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7327
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7328
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7329
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7330
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7331
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7332
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7333
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7334
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7335
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7336
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7337
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7338
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7339
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7340
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7341
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7342
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7343
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7344
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7345
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7346
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7347
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7348
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7349
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7350
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7351
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7352
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7353
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7354
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7355
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7356
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7357
    Char end stringwidth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7358
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7359
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7360
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7361
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7362
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7363
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7364
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7365
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7366
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7367
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7368
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7369
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7371
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7372
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7374
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7375
%%BeginProcSet: special.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7376
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7377
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7378
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7379
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7380
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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  7533
 @endspecial 1234 2310 11 41 v 1245 2292 46 5 v 1310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7534
2322 a Ga(B)p 1178 2342 242 4 v 1178 2421 a(C)p 1270
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7535
2409 11 41 v 1280 2391 46 5 v 103 w(B)1461 2366 y Gg(W)g(eak)1665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7536
2380 y Gc(L)p 430 2458 990 4 v 803 2525 11 41 v 813 2507
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7537
46 5 v 879 2537 a Ga(B)5 b(;)15 b(B)1461 2489 y Gg(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7538
783 2575 284 4 v 860 2641 11 41 v 870 2623 46 5 v 936
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7539
2653 a Ga(B)1108 2599 y Gg(Contr)1314 2613 y Gc(R)1888
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7540
2464 y F6(\000)-31 b(\000)g(!)2229 2433 y @beginspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7541
179 @llx 453 @lly 318 @urx 628 @ury 417 @rwi @clip @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7542
%%BeginDocument: pics/lafont-left.ps
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7543
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7544
%%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7545
%%Title: lafont-left.dvi
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7546
%%Pages: 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7547
%%PageOrder: Ascend
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7548
%%BoundingBox: 0 0 596 842
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7549
%%EndComments
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7550
%DVIPSWebPage: (www.radicaleye.com)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7551
%DVIPSCommandLine: dvips -o lafont-left.ps lafont-left.dvi
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7552
%DVIPSParameters: dpi=600, compressed
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7553
%DVIPSSource:  TeX output 2000.03.09:0346
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7554
%%BeginProcSet: texc.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7555
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7556
/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7557
N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7558
mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7559
0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7560
landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7561
mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7562
matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7563
exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7564
statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7565
N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7566
/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7567
/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7568
array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7569
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7570
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7571
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7572
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7573
1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7574
1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7575
0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7576
sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7577
rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7578
gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7579
/chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7580
/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7581
A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7582
get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7583
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7584
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7585
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7586
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7587
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7588
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7589
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7590
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7591
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7592
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7593
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7594
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7595
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7596
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7597
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7598
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7599
(LaserWriter 16/600)]{A length product length le{A length product exch 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7600
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7601
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7602
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7603
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7604
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7605
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7606
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7607
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7608
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7609
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7610
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7611
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7612
%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7613
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7614
% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7615
% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7616
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7617
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7618
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7619
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7620
/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7621
/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7622
/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7623
/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7624
/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7625
/L /lineto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7626
/T /translate load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7627
/TMatrix { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7628
/RAngle { 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7629
/Atan { /atan load stopped { pop pop 0 } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7630
/Div { dup 0 eq { pop } { div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7631
/NET { neg exch neg exch T } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7632
/Pyth { dup mul exch dup mul add sqrt } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7633
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7634
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7635
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7636
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7637
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7638
/pathforall load stopped { pop pop pop pop } if z } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7639
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7640
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7641
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7642
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7643
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7644
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7645
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7646
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7647
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7648
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7649
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7650
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7651
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7652
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7653
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7654
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7655
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7656
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7657
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7658
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7659
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7660
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7661
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7662
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7663
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7664
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7665
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7666
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7667
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7668
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7669
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7670
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7671
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7672
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7673
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7674
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7675
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7676
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7677
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7678
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7679
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7680
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7681
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7682
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7683
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7684
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7685
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7686
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7687
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7688
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7689
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7690
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7691
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7692
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7693
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7694
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7695
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7696
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7697
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7698
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7699
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7700
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7701
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7702
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7703
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7704
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7705
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7706
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7707
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7708
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7709
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7710
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7711
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7712
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7713
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7714
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7715
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7716
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7717
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7718
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7719
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7720
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7721
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7722
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7723
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7724
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7725
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7726
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7727
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7728
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7729
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7730
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7731
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7732
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7733
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7734
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7735
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7736
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7737
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7738
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7739
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7740
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7741
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7742
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7743
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7744
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7745
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7746
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7747
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7748
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7749
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7750
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7751
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7752
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7753
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7754
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7755
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7756
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7757
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7758
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7759
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7760
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7761
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7762
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7763
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7764
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7765
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7766
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7767
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7768
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7769
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7770
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7771
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7772
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7773
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7774
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7775
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7776
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7777
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7778
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7779
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7780
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7781
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7782
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7783
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7784
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7785
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7786
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7787
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7788
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7789
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7790
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7791
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7792
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7793
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7794
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7795
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7796
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7797
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7798
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7799
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7800
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7801
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7802
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7803
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7804
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7805
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7806
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7807
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7808
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7809
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7810
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7811
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7812
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7813
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7814
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7815
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7816
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7817
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7818
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7819
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7820
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7821
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7822
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7823
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7824
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7825
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7826
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7827
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7828
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7829
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7830
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7831
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7832
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7833
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7834
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7835
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7836
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7837
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7839
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7840
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7841
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7842
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7843
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7844
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7845
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7846
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7847
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7848
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7849
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7850
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7851
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7852
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7853
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7854
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7855
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7856
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7857
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7858
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7859
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7860
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7861
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7862
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7863
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7864
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7865
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7866
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7867
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7868
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7869
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7870
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7871
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7872
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7873
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7874
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7875
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7876
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7877
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7878
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7879
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7880
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7881
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7882
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7883
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7884
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7885
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7886
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7887
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7888
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7889
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7890
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7891
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7892
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7893
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7894
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7895
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7896
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7897
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7898
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7899
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7900
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7901
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7902
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7903
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7904
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7905
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7906
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7907
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7908
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7909
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7910
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7911
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7912
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7913
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7914
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7915
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7916
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7917
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7918
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7919
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7920
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7921
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7922
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7923
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7924
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7925
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7926
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7927
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7928
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7929
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7930
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7931
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7932
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7933
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7934
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7935
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7936
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7937
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7938
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7939
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7940
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7941
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7942
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7943
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7944
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7945
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7946
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7947
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7948
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7949
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7950
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7951
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7952
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7953
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7954
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7955
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7956
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7957
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7958
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7959
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7960
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7961
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7962
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7963
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7964
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7965
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7966
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7967
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7968
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7969
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7970
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7971
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7972
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7973
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7974
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7975
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7976
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7977
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7978
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7979
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7980
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7981
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7982
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7983
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7984
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7985
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7986
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7987
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7988
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7989
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7990
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7991
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7992
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7993
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7994
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7995
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7996
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7997
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7998
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  7999
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8000
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8001
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8002
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8003
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8004
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8005
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8006
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8007
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8008
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8009
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8010
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8011
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8012
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8013
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8014
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8015
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8016
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8017
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8018
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8019
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8020
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8021
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8022
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8023
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8024
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8025
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8026
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8027
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8028
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8029
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8030
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8031
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8032
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8033
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8034
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8035
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8036
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8037
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8038
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8039
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8040
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8041
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8042
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8043
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8044
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8045
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8046
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8047
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8048
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8049
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8050
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8051
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8052
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8053
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8054
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8055
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8056
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8057
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8058
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8059
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8060
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8061
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8062
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8063
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8064
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8065
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8066
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8067
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8068
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8069
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8070
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8071
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8072
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8073
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8074
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8075
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8076
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8077
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8078
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8079
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8080
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8081
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8082
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8083
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8084
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8085
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8086
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8087
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8088
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8089
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8090
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8091
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8092
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8093
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8094
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8095
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8096
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8097
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8098
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8099
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8100
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8101
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8102
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8103
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8104
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8105
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8106
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8107
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8108
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8109
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8110
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8111
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8112
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8113
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8114
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8115
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8116
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8117
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8118
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8119
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8120
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8121
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8122
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8123
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8124
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8125
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8126
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8127
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8128
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8129
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8130
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8131
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8132
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8133
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8134
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8135
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8136
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8137
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8138
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8139
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8140
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8141
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8142
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8143
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8144
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8145
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8146
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8147
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8148
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8149
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8150
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8151
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8152
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8153
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8154
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8155
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8156
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8157
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8158
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8159
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8160
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8161
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8162
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8163
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8164
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8165
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8166
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8167
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8168
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8170
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8171
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8172
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8173
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8174
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8175
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8176
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8177
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8178
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8179
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8180
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8181
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8182
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8183
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8184
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8185
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8186
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8188
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8189
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8190
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8191
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8192
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8193
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8194
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8195
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8197
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8198
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8199
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8200
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8201
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8202
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8203
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8204
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8205
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8206
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8207
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8208
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8209
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8210
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8211
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8212
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8214
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8215
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8216
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8217
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8218
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8219
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8220
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8221
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8222
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8223
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8224
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8225
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8226
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8227
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8228
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8229
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8230
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8231
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8232
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8233
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8234
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8235
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8236
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8237
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8238
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8239
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8240
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8241
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8242
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8243
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8244
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8245
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8246
    Char end stringwidth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8247
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8248
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8249
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8250
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8251
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8252
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8253
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8254
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8255
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8256
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8257
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8258
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8259
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8260
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8261
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8262
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8263
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8264
%%BeginProcSet: special.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8265
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8266
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8267
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8268
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8269
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8270
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8271
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8272
/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8273
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8274
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8275
maxlength ge{/md md dup length 20 add dict copy def}if end md begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8276
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8277
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8278
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8279
transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8280
curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8281
pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8282
if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8283
-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8284
get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8285
yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8286
neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8287
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8288
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8289
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8290
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8291
2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8292
-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8293
TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8294
Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8295
}if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8296
save N userdict maxlength dict begin/magscale true def normalscale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8297
currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8298
/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8299
psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8300
psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8301
TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8302
psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8303
roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8304
moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8305
begin/SpecialSave save N gsave normalscale currentpoint TR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8306
@SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8307
CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8308
closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8309
sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8310
}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8311
CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8312
lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8313
/@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8314
repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8315
/@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8316
currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8317
moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8318
/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8319
1 startangle endangle arc savematrix setmatrix}N end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8320
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8321
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8322
%%BeginProcSet: color.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8323
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8324
TeXDict begin/setcmykcolor where{pop}{/setcmykcolor{dup 10 eq{pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8325
setrgbcolor}{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8326
}repeat setrgbcolor pop}ifelse}B}ifelse/TeXcolorcmyk{setcmykcolor}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8327
/TeXcolorrgb{setrgbcolor}def/TeXcolorgrey{setgray}def/TeXcolorgray{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8328
setgray}def/TeXcolorhsb{sethsbcolor}def/currentcmykcolor where{pop}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8329
/currentcmykcolor{currentrgbcolor 10}B}ifelse/DC{exch dup userdict exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8330
known{pop pop}{X}ifelse}B/GreenYellow{0.15 0 0.69 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8331
/Yellow{0 0 1 0 setcmykcolor}DC/Goldenrod{0 0.10 0.84 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8332
/Dandelion{0 0.29 0.84 0 setcmykcolor}DC/Apricot{0 0.32 0.52 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8333
setcmykcolor}DC/Peach{0 0.50 0.70 0 setcmykcolor}DC/Melon{0 0.46 0.50 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8334
setcmykcolor}DC/YellowOrange{0 0.42 1 0 setcmykcolor}DC/Orange{0 0.61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8335
0.87 0 setcmykcolor}DC/BurntOrange{0 0.51 1 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8336
/Bittersweet{0 0.75 1 0.24 setcmykcolor}DC/RedOrange{0 0.77 0.87 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8337
setcmykcolor}DC/Mahogany{0 0.85 0.87 0.35 setcmykcolor}DC/Maroon{0 0.87
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8338
0.68 0.32 setcmykcolor}DC/BrickRed{0 0.89 0.94 0.28 setcmykcolor}DC/Red{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8339
0 1 1 0 setcmykcolor}DC/OrangeRed{0 1 0.50 0 setcmykcolor}DC/RubineRed{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8340
0 1 0.13 0 setcmykcolor}DC/WildStrawberry{0 0.96 0.39 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8341
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8342
setcmykcolor}DC/Magenta{0 1 0 0 setcmykcolor}DC/VioletRed{0 0.81 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8343
setcmykcolor}DC/Rhodamine{0 0.82 0 0 setcmykcolor}DC/Mulberry{0.34 0.90
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8344
0 0.02 setcmykcolor}DC/RedViolet{0.07 0.90 0 0.34 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8345
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8346
setcmykcolor}DC/Thistle{0.12 0.59 0 0 setcmykcolor}DC/Orchid{0.32 0.64 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8347
0 setcmykcolor}DC/DarkOrchid{0.40 0.80 0.20 0 setcmykcolor}DC/Purple{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8348
0.45 0.86 0 0 setcmykcolor}DC/Plum{0.50 1 0 0 setcmykcolor}DC/Violet{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8349
0.79 0.88 0 0 setcmykcolor}DC/RoyalPurple{0.75 0.90 0 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8350
/BlueViolet{0.86 0.91 0 0.04 setcmykcolor}DC/Periwinkle{0.57 0.55 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8351
setcmykcolor}DC/CadetBlue{0.62 0.57 0.23 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8352
/CornflowerBlue{0.65 0.13 0 0 setcmykcolor}DC/MidnightBlue{0.98 0.13 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8353
0.43 setcmykcolor}DC/NavyBlue{0.94 0.54 0 0 setcmykcolor}DC/RoyalBlue{1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8354
0.50 0 0 setcmykcolor}DC/Blue{1 1 0 0 setcmykcolor}DC/Cerulean{0.94 0.11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8355
0 0 setcmykcolor}DC/Cyan{1 0 0 0 setcmykcolor}DC/ProcessBlue{0.96 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8356
setcmykcolor}DC/SkyBlue{0.62 0 0.12 0 setcmykcolor}DC/Turquoise{0.85 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8357
0.20 0 setcmykcolor}DC/TealBlue{0.86 0 0.34 0.02 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8358
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8359
setcmykcolor}DC/Emerald{1 0 0.50 0 setcmykcolor}DC/JungleGreen{0.99 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8360
0.52 0 setcmykcolor}DC/SeaGreen{0.69 0 0.50 0 setcmykcolor}DC/Green{1 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8361
1 0 setcmykcolor}DC/ForestGreen{0.91 0 0.88 0.12 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8362
/PineGreen{0.92 0 0.59 0.25 setcmykcolor}DC/LimeGreen{0.50 0 1 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8363
setcmykcolor}DC/YellowGreen{0.44 0 0.74 0 setcmykcolor}DC/SpringGreen{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
  8417
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8450
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
  8530
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8531
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8532
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8533
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8534
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8535
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8536
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8537
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8538
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8539
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8540
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8541
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8542
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8543
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8544
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8545
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8546
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8547
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8548
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8549
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8550
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8551
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8552
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8553
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8554
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8555
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8556
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8557
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8558
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8559
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8560
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8561
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8562
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8563
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8564
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8565
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8566
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8567
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8568
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8569
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8570
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8571
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8572
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8573
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8574
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8575
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8576
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8577
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8578
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8579
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8580
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8581
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8582
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8583
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8584
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8585
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8586
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8587
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8588
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8589
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8590
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8591
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8592
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8593
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8594
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8595
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8596
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8597
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8598
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8599
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8600
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8601
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8602
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8603
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8604
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8605
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8606
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8607
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8608
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8609
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8610
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8611
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8612
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8613
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8614
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8615
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8616
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8617
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8618
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8619
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8620
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8621
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8622
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8623
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8624
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8625
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8626
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8627
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8628
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8629
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8630
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8631
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8632
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8633
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8634
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8635
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8636
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8637
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8638
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8639
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8640
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8641
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8642
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8643
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8644
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8645
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8646
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8647
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8648
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8649
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8650
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8651
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8652
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8653
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8654
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8655
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8656
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8657
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8658
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8659
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8660
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8661
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8662
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8663
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8664
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8665
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8666
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8667
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8668
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8669
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8670
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8671
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8672
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8673
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8674
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8675
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8676
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8677
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8678
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8679
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8680
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8681
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8682
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8683
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8684
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8685
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8686
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8687
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8688
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8689
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8690
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8691
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8692
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8693
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8694
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8695
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8696
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8697
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8698
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8699
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8700
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8701
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8702
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8703
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8704
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8705
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8706
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8707
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8708
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8709
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8710
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8711
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8712
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8713
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8714
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8715
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8716
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8717
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8718
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8719
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8720
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8721
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8722
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8723
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8724
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8725
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8726
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8727
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8728
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8729
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8730
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8731
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8732
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8733
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8734
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8735
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8736
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8737
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8738
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8739
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8740
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8741
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8742
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8743
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8744
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8745
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8746
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8747
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8748
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8749
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8750
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8751
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8752
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8753
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8754
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8755
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8756
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8757
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8758
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8759
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8760
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8761
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8762
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8763
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8764
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8765
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8766
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8767
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8768
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8769
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8770
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8771
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8772
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8773
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8774
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8775
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8776
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8777
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8778
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8779
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8780
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8781
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8782
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8783
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8784
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8785
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8786
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8787
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8788
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8789
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8790
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8791
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8792
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8793
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8794
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8795
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8796
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8797
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8798
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8799
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8800
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8801
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8802
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8803
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8804
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8805
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8806
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8807
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8808
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8809
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8810
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8811
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8812
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8813
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8814
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8815
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8816
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8817
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8818
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8819
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8820
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8821
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8822
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8823
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8824
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8825
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8826
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8827
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8828
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8829
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8830
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8831
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8832
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8833
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8834
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8835
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8836
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8837
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8838
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8839
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8840
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8841
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8842
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8843
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8844
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8845
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8846
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8847
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8848
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8849
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8850
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8851
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8852
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8853
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8854
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8855
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8856
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8857
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8858
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8859
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8860
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8861
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8862
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8863
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8864
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8865
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8866
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8867
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8868
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8869
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8870
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8871
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8872
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8873
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8874
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8875
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8876
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8877
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8878
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8879
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8880
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8881
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8882
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8883
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8884
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8885
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8886
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8887
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8888
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8889
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8890
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8891
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8892
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8893
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8894
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8895
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8896
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8897
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8898
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8899
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8900
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8901
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8902
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8903
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8904
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8905
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8906
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8907
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8908
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8909
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8910
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8911
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8912
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8913
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8914
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8915
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8916
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8917
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8918
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8919
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8920
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8921
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8922
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8923
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8924
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8925
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8926
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8927
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8928
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8929
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8930
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8931
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8932
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8933
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8934
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8935
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8936
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8937
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8938
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8939
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8940
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8941
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8942
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8943
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8944
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8945
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8946
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8947
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8948
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8949
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8950
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8951
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8952
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8953
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8954
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8955
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8956
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8957
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8958
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8959
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8960
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8961
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8962
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8963
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8964
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8965
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8966
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8967
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8968
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8969
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8970
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8971
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8972
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8973
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8974
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8975
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8976
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8977
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8978
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8979
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8980
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8981
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8982
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8983
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8984
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8985
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8986
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8987
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8988
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8989
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8990
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8991
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8992
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8993
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8994
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8995
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8996
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8997
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8998
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  8999
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9000
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9001
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9002
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9003
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9004
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9005
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9006
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9007
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9008
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9009
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9010
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9011
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9012
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9013
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9014
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9015
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9016
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9017
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9018
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9019
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9020
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9021
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9022
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9023
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9024
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9025
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9026
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9027
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9028
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9029
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9030
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9031
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9032
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9033
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9034
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9035
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9036
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9037
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9038
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9039
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9040
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9041
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9042
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9043
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9044
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9045
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9046
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9047
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9048
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9049
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9050
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9051
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9052
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9053
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9054
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9055
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9056
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9057
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9058
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9059
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9060
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9061
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9062
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9063
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9064
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9065
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9066
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9067
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9068
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9069
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9070
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9071
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9072
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9073
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9074
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9076
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9077
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9078
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9079
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9080
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9081
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9082
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9083
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9084
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9085
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9086
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9087
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9088
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9089
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9090
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9091
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9092
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9093
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9095
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9096
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9097
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9098
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9099
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9100
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9101
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9102
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9104
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9105
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9106
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9107
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9108
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9109
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9110
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9111
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9112
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9113
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9114
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9115
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9116
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9117
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9118
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9119
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9120
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9121
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9122
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9124
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9125
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9126
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9127
    Char end stringwidth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9128
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9129
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9130
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9131
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9132
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9133
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9134
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9135
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9136
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9137
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9138
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9139
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9141
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9142
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9144
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9145
%%BeginProcSet: special.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9146
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9147
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9148
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9149
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9150
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9151
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9152
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9153
/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9154
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9155
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9156
maxlength ge{/md md dup length 20 add dict copy def}if end md begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9157
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9158
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9159
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9160
transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9161
curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9162
pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9163
if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9164
-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9165
get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9166
yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9167
neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9168
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9169
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9170
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9171
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9172
2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9173
-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9174
TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9175
Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9176
}if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9177
save N userdict maxlength dict begin/magscale true def normalscale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9178
currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9179
/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9180
psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9181
psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9182
TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9183
psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9184
roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9185
moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9186
begin/SpecialSave save N gsave normalscale currentpoint TR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9187
@SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9188
CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9189
closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9190
sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9191
}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9192
CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9193
lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9194
/@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9195
repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9196
/@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9197
currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9198
moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9199
/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9200
1 startangle endangle arc savematrix setmatrix}N end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9202
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9203
%%BeginProcSet: color.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9204
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9205
TeXDict begin/setcmykcolor where{pop}{/setcmykcolor{dup 10 eq{pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9206
setrgbcolor}{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9207
}repeat setrgbcolor pop}ifelse}B}ifelse/TeXcolorcmyk{setcmykcolor}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9208
/TeXcolorrgb{setrgbcolor}def/TeXcolorgrey{setgray}def/TeXcolorgray{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9209
setgray}def/TeXcolorhsb{sethsbcolor}def/currentcmykcolor where{pop}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9210
/currentcmykcolor{currentrgbcolor 10}B}ifelse/DC{exch dup userdict exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9211
known{pop pop}{X}ifelse}B/GreenYellow{0.15 0 0.69 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9357
(the)f(inference)p 1673 827 11 41 v 1683 809 46 5 v 1749
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9358
839 a Ga(B)p 1597 859 282 4 v 1617 926 11 41 v 1627 907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9359
46 5 v 1693 938 a(B)5 b(;)15 b(C)1920 883 y Gg(W)-7 b(eak)2124
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9360
897 y Gc(R)277 1143 y Gg(and)24 b(in)g(linear)g(logic)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9361
(structural)i(rules)e(are)g(formulated)i(such)e(that)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9362
1320 1314 11 41 v 1330 1296 46 5 v 1396 1326 a Ga(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9363
1222 1346 325 4 v 1242 1414 11 41 v 1253 1396 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9364
1318 1426 a(B)5 b(;)15 b F4(?)p Ga(C)1588 1370 y Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9365
b(eak)1792 1384 y Gc(R)p 2061 1314 11 41 v 2071 1296
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9366
46 5 v 2137 1326 a Ga(B)p 1992 1346 267 4 v 1992 1426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9367
a F4(!)p Ga(C)p 2109 1414 11 41 v 2120 1396 46 5 v 103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9368
w(B)2300 1370 y Gg(W)g(eak)2504 1384 y Gc(L)p 1222 1463
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9369
1037 4 v 1619 1530 11 41 v 1629 1512 46 5 v 1695 1542
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9370
a Ga(B)5 b(;)15 b(B)2300 1494 y Gg(Cut)277 1748 y(is)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9371
b(not)g(a)f(v)n(alid)i(instance)h(of)d(the)i(cut-rule.)55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9372
b(Consequently)-6 b(,)37 b(Lafont')-5 b(s)33 b(e)o(xample)f(cannot)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9373
(arise)e(in)277 1861 y(intuitionistic)c(logic)c(or)g(in)f(linear)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9374
(logic.)418 1990 y(F)o(or)j(intuitionistic)k(logic)d(we)f(ha)n(v)o(e,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9375
(as)e(mentioned)i(earlier)l(,)h(the)e(Curry-Ho)n(w)o(ard)g(correspon-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9376
277 2103 y(dence)24 b(with)f(the)g(simply-typed)j(lambda)d(calculus.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9377
b(F)o(or)22 b(linear)h(logic)h(the)f(situation)i(is)e(unclear)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9378
b(.)30 b(At)277 2216 y(its)g(inception)i(it)d(w)o(as)h(thought)h(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9379
(ha)n(v)o(e)g(direct)h(connection)h(with)e(concurrenc)o(y)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9380
b(,)34 b(and)c(some)g(con-)277 2329 y(nections)i(ha)n(v)o(e)f(been)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9381
(unco)o(v)o(ered)i(by)e(Abramsk)o(y)g([1993].)49 b(Ho)n(we)n(v)o(er)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9382
(,)31 b(it)f(is)f(still)i(f)o(air)f(to)g(say)g(that)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9383
2442 y(in)h(linear)g(logic)h(the)f(question)i(of)d(what)h(kind)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9384
(computation)i(the)e(process)h(of)f(cut-elimination)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9385
2555 y(corresponds)c(is)c(lar)n(gely)j(unanswered.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9386
2684 y(Recently)-6 b(,)38 b(a)33 b(number)h(of)g(other)g(solutions)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9387
(for)c(the)h(\223inconsistenc)o(y\224)k(in)c(Lafont')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9388
b(s)35 b(e)o(xample)277 2797 y(ha)n(v)o(e)26 b(been)g(proposed.)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9389
b(Instead)27 b(of)e(restricting)j(classical)f(logic)f(so)f(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9390
(Proof)f(\(1.1\))g(is)g(not)g(deri)n(v-)277 2910 y(able,)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9391
b(the)o(y)e(restrict)g(the)g(reduction)h(rules)f(of)f(classical)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9392
(logic.)61 b(F)o(or)33 b(e)o(xample)i(in)f(P)o(arigot')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9393
b(s)35 b Ga(\025\026)p Gg(-)277 3023 y(calculus,)23 b(although)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9394
(formulated)f(as)e(a)f(natural)j(deduction)g(calculus,)h(Lafont')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9395
b(s)22 b(proof)g(is)f(deri)n(v)n(able,)277 3136 y(b)n(ut)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9396
b(reduces)i(to)e(one)g(subproof)i(only\227the)f(one)g(on)e(the)h(right)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9397
h([P)o(arigot,)f(1992,)h(Bierman,)e(1999].)277 3249 y(In)h(this)h(w)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9398
(ay)e(Lafont')-5 b(s)29 b(ar)n(gument)h(does)f(not)g(apply;)i(it)d(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9399
(completely)i(bypassed.)44 b(As)28 b(a)f(pleasing)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9400
3362 y(consequence,)33 b(Bierman)c([1998])h(w)o(as)e(able)h(to)g(sho)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9401
(w)f(that)h Ga(\025\026)e Gg(has)i(a)f(simple)h(computational)j(in-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9402
3475 y(terpretation:)47 b(it)30 b(is)h(a)f(simply-typed)k(lambda)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9403
(calculus)i(which)e(is)g(able)g(to)f(sa)n(v)o(e)i(and)f(restore)h(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9404
277 3588 y(runtime)25 b(en)l(vironment.)418 3717 y(Another)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9405
b(solution)h(is)d(gi)n(v)o(en)h(by)g(Danos)g(et)g(al.)f([1997].)51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9406
b(In)31 b(their)g(sequent)i(calculus,)h(named)277 3830
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9407
y(LK)399 3797 y Gc(tq)462 3830 y Gg(,)d(Lafont')-5 b(s)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9408
b(proof)f(is)e(deri)n(v)n(able,)35 b(b)n(ut)c(e)n(v)o(ery)g(formula)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9409
(is)f(required)i(to)e(be)g(annotated)i(with)e(a)277 3943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9410
y(colour)-5 b(.)33 b(In)24 b(ef)n(fect,)h(their)g(solution)i(of)d(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9411
(\223inconsistenc)o(y\224)29 b(is)24 b(similar)h(to)f(the)h(approach)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9412
(tak)o(en)e(by)277 4056 y(P)o(arigot,)h(in)f(the)h(sense)g(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9413
(Proof)g(\(1.1\))g(reduces)h(to)e(one)h(subproof)i(only)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9414
b(.)35 b(Ho)n(we)n(v)o(er)l(,)25 b(whereas)h(in)277 4169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9415
y Ga(\025\026)f Gg(the)h(reduction)i(system)f(is)f(restricted)i(so)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9416
(that)g(Lafont')-5 b(s)27 b(proof)g(reduces)h(to)d(only)i(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9417
(subproof)277 4282 y(on)i(the)h(right,)h(in)e(LK)977
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9418
4249 y Gc(tq)1066 4282 y Gg(this)h(proof)g(can)g(reduce)g(to)f(either)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9419
(subproof,)h(b)n(ut)d(the)h(colour)g(annotation)277 4394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9420
y(determines)g(which)d(one.)38 b(The)26 b(colour)i(annotation)i(seems)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9421
(to)f(correspond)k(to)c(the)h(restriction)i(im-)277 4507
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9422
y(posed)35 b(by)f(linear)h(logic,)j(since)d(e)n(v)o(ery)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9423
(cut-elimination)k(step)c(in)g(LK)2572 4474 y Gc(tq)2668
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9424
4507 y Gg(can)g(be)g(mapped)h(onto)g(a)277 4620 y(series)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9425
b(of)f(cut-elimination)k(steps)d(in)f(linear)h(logic.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9426
b(Ho)n(we)n(v)o(er)l(,)22 b(the)g(precise)i(relationship)h(needs)e(yet)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9427
277 4733 y(to)29 b(be)f(w)o(ork)o(ed)i(out.)44 b(Danos)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9428
b(et)e(al.)g(gi)n(v)o(e)h(a)f(computational)k(interpretation)h(for)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9429
(LK)3011 4700 y Gc(tq)3073 4733 y Gg(-proofs,)j(b)n(ut)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9430
4846 y(their)24 b(proposal)i(is)e(some)n(what)g(blurred.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9431
4976 y(Although)f(Lafont')-5 b(s)22 b(reasoning)i(has)d(found)h(its)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9432
(w)o(ay)g(into)h(a)e(number)i(of)f(treatises)i(on)e(the)g(proof)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9433
5088 y(theory)26 b(of)e(classical)j(logic,)f(for)e(e)o(xample)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9434
([Girard)h(et)e(al.,)g(1989,)h(Girard,)g(1991,)g(Schellinx,)h(1994,)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9435
5201 y(Bierman,)19 b(1999],)h(there)g(is)e(an)h(ob)o(vious)h(question)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9436
(whether)f(the)f(restrictions)i(mentioned)g(abo)o(v)o(e)e(are)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9437
5314 y F7(r)m(eally)j Gg(necessary)i(to)e(solv)o(e)g(the)g(problem)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9438
(with)f(the)h(\223inconsistenc)o(y\224)k(in)21 b(Lafont')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9439
b(s)23 b(proof.)29 b(There)22 b(is)p Black Black eop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9440
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9441
%%Page: 6 18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9442
TeXDict begin 6 17 bop Black -144 51 a Gb(6)3152 b(Intr)n(oduction)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9443
-144 88 3691 4 v Black 321 317 a Gg(a)27 b(fear)g(that)h(certain)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9444
(computational)i(features)f(of)e(classical)i(logic)f(are)f(lost)h(by)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9445
(these)h(restrictions.)321 430 y(In)c(classical)h(logic)g(we)d(do)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9446
(not)g(ha)n(v)o(e)g(access)h(to)e(ne)n(w)g(functions,)i(a)e(f)o(act)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9447
(established)j(se)n(v)o(eral)e(years)321 543 y(ago)j(\(for)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9448
(xample)g(by)f(Friedman)h([1978]\),)h(b)n(ut)f(we)e(do)i(ha)n(v)o(e)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9449
(access)i(to)e(ne)n(w)g(proofs,)i(compared)321 656 y(to)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9450
(intuitionistic)k(logic.)51 b(Thus)31 b(we)f(may)h(hope)g(to)g(ha)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9451
(e)g(access)i(to)d(a)h(substantially)j(lar)n(ger)f(class)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9452
769 y(of)k(programs)h(capturing)i(computational)g(beha)n(viour)g(not)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9453
(e)o(xpressible)j(in)d(intuitionistic)j(typed)321 882
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9454
y(lambda)25 b(calculi.)30 b(Unfortunately)-6 b(,)27 b(none)d(of)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9455
(proposed)i(solutions)g(for)e(Lafont')-5 b(s)24 b(e)o(xample)h(ha)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9456
(e)321 995 y(yet)f(ful\002lled)h(this)f(hope.)462 1132
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9457
y(In)f(this)h(thesis)g(we)f(shall)h(sho)n(w)f(that)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9458
(restrictions)i(are)d F7(not)h Gg(necessary)h(to)e(obtain)i(a)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9459
(strongly)321 1245 y(normalising)33 b(cut-elimination)g(procedure.)51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9460
b(Also,)31 b(we)e(w)o(ant)h(to)f(ar)n(gue)j(that)e(the)g(restrictions)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9461
(are)321 1358 y F7(not)f Gg(required)i(to)d(solv)o(e)h(the)g(problem)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9462
(arising)g(from)e(Lafont')-5 b(s)33 b(e)o(xample.)53
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9463
b(The)31 b(\223inconsistenc)o(y\224)321 1470 y(only)j(appears)h(if)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9464
(we)f(re)o(gard)i(cut-elimination)j(as)c(an)g(equality)i(preserving)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9465
(operation.)60 b(This)33 b(is)321 1583 y(of)28 b(course)h(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9466
(plausible)j(doctrine)f(coming)f(from)g(the)g(simply-typed)i(lambda)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9467
(calculus,)j(b)n(ut)d(there)321 1696 y(are)38 b(man)o(y)e(calculi,)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9468
b(notably)d(calculi)g(for)e(concurrenc)o(y)-6 b(,)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9469
b(where)38 b(reduction)h(is)e F7(not)g Gg(an)g(equality)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9470
1809 y(preserving)28 b(operation.)35 b(T)-7 b(ak)o(e)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9471
b(for)h(e)o(xample)h(a)e(non-deterministic)29 b(choice)e(operator)l(,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9472
(say)e F4(+)p Gg(,)f(with)321 1922 y(the)g(reduction)1448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9473
2174 y Ga(M)30 b F4(+)20 b Ga(N)61 b F6(\000)-32 b(\000)h(!)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9474
b Ga(M)61 b Gg(or)51 b Ga(N)35 b(:)321 2424 y Gg(Clearly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9475
b(,)25 b(we)e(cannot)i(hope)g(that)f(reduction)j(preserv)o(es)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9476
(equality)-6 b(.)32 b(Therefore,)25 b(we)e(shall)i(accept)g(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9477
2537 y(vie)n(w)34 b(that)g(cut-elimination)j(corresponds)g(to)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9478
(computation)k(which)d(may)f(or)h(may)f(not)h(preserv)o(e)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9479
2650 y(equality)c(between)f(proofs)h(and)e(in)g(this)h(w)o(ay)f(a)n(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9480
(oid)h(the)f(\223inconsistenc)o(y\224)33 b(in)28 b(Lafont')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9481
b(s)29 b(e)o(xample.)321 2763 y(Under)g(this)g(assumption)i(it)d(seems)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9482
g(prudent)j(ho)n(we)n(v)o(er)d(to)g(reconsider)k(the)c(proof)i(theory)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9483
(of)g(clas-)321 2876 y(sical)e(logic,)h(despite)g(the)e(long)h(list)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9484
(of)f(te)o(xtbooks,)j(for)d(instance)i([Sch)8 b(\250)-38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9485
b(utte,)27 b(1960,)g(T)-7 b(ak)o(euti,)27 b(1975,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9486
2989 y(Girard,)e(1987b,)i(T)m(roelstra)f(and)f(Schwichtenber)n(g,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9487
(1996,)e(Buss,)e(1998],)j(already)f(written)g(on)f(this)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9488
3102 y(subject.)321 3455 y Ge(1.1)119 b(Outline)31 b(of)f(the)g(Thesis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9489
321 3693 y Gg(W)-7 b(e)24 b(shall)i(introduce)h(se)n(v)o(eral)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9490
(reduction)i(systems)e(in)e(this)i(thesis;)h(Figure)e(1.1)g(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9491
(es)g(an)g(o)o(v)o(ervie)n(w)321 3806 y(and)19 b(roughly)i(indicates)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9492
(the)f(dependencies)k(between)c(them.)27 b(The)18 b(plan)h(of)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9493
(thesis)i(is)e(as)h(follo)n(ws:)p Black 458 4072 a F6(\017)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9494
Black 46 w Gg(Chapter)31 b(2)g(re)n(vie)n(ws)g(Gentzen')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9495
b(s)33 b(sequent)g(calculus)g(LK)c(and)i(his)h(cut-elimination)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9496
(proce-)549 4185 y(dure.)j(Then)26 b(we)g(introduce)i(a)e(ne)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9497
(sequent)i(calculus)g(for)f(classical)h(logic)f(and)g(sho)n(w)f(ho)n(w)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9498
549 4298 y(its)j(proofs)i(can)f(be)f(annotated)j(with)e(terms.)46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9499
b(Ne)o(xt,)31 b(tw)o(o)e(cut-elimination)k(procedures)f(for)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9500
4411 y(this)g(sequent)h(calculus)g(are)f(formulated;)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9501
b(the)o(y)31 b(include)j(a)d(notion)h(of)g(proof)g(substitution.)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9502
4524 y(The)e(principal)j(result)g(of)d(this)i(thesis)g(is)f(a)f(proof)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9503
(that)g(establishes)i(strong)e(normalisation)549 4637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9504
y(for)22 b(both)h(cut-elimination)i(procedures.)32 b(This)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9505
b(result)j(is)d(then)i(e)o(xtended)h(to)e(\002rst-order)i(clas-)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9506
4750 y(sical)29 b(logic,)i(to)e(other)h(formulations)h(of)e(classical)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9507
(logic)f(and)f(to)g(intuitionistic)k(logic.)45 b(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9508
b(e)549 4863 y(also)33 b(pro)o(v)o(e)h(strong)h(normalisation)h(for)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9509
(a)f(cut-elimination)j(procedure)g(where)e(the)f(proof)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9510
4976 y(substitution)e(is)d(replaced)i(with)e(completely)i(local)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9511
(reduction)i(rules.)43 b(Ha)n(ving)30 b(considered)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9512
5088 y(se)n(v)o(eral)d(strongly)h(normalising)g(cut-elimination)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9513
(procedures,)f(we)c(conclude)j(this)f(chapter)549 5201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9514
y(comparing)35 b(our)f(w)o(ork)f(with)g(w)o(ork)h(by)f(Dragalin)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9515
([1988])h(and)f(w)o(ork)g(by)f(Barbanera)i(and)549 5314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9516
y(Berardi)24 b([1994].)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9517
%%Page: 7 19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9518
TeXDict begin 7 18 bop Black 277 51 a Gb(1.1)23 b(Outline)g(of)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9519
(Thesis)2703 b(7)p 277 88 3691 4 v Black Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9520
Black 922 229 2077 4 v 922 2146 4 1918 v 1023 2088 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9521
@beginspecial 81 @llx 332 @lly 495 @urx 732 @ury 2277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9522
@rwi @clip @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9523
%%BeginDocument: pics/1.0.0.Overview.ps
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9524
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9525
%%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9526
%%Title: 1.0.0.Overview.dvi
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9527
%%Pages: 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9528
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Christian Urban <urbanc@in.tum.de>
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diff changeset
  9529
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Christian Urban <urbanc@in.tum.de>
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diff changeset
  9530
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Christian Urban <urbanc@in.tum.de>
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diff changeset
  9531
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Christian Urban <urbanc@in.tum.de>
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diff changeset
  9532
%DVIPSWebPage: (www.radicaleye.com)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9533
%DVIPSCommandLine: dvips 1.0.0.Overview.dvi -o 1.0.0.Overview.ps
Christian Urban <urbanc@in.tum.de>
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diff changeset
  9534
%DVIPSParameters: dpi=600, compressed
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9535
%DVIPSSource:  TeX output 2000.06.05:0717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9536
%%BeginProcSet: texc.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9537
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9538
/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9539
N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9540
mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9541
0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9542
landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9543
mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9544
matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9545
exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9546
statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9547
N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9548
/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9549
/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9550
array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9551
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9552
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9553
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9554
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9555
1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9556
1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9557
0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9558
sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9559
rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9560
gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9561
/chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9562
/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9563
A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9564
get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9565
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9566
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9567
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9568
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9569
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9570
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9571
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9572
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9573
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9574
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9575
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9576
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9577
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9578
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9579
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9580
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9581
(LaserWriter 16/600)]{A length product length le{A length product exch 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9582
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9583
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9584
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9586
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9587
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9588
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9589
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9590
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9591
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9593
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
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diff changeset
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%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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%!
Christian Urban <urbanc@in.tum.de>
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diff changeset
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% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
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diff changeset
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% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
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diff changeset
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% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9599
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9600
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9601
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
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diff changeset
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/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
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diff changeset
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/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
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diff changeset
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/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
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diff changeset
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/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
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diff changeset
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/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
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diff changeset
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/L /lineto load def
Christian Urban <urbanc@in.tum.de>
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diff changeset
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/T /translate load def
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
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diff changeset
  9617
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9618
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
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diff changeset
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0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
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diff changeset
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def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
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diff changeset
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2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9635
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9636
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9637
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9638
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9639
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9640
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9641
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9642
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9643
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9644
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9645
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9646
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9647
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9648
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9649
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9650
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9651
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9652
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9653
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9654
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9655
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9656
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9657
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9658
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9659
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9660
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9661
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9662
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9663
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9664
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9665
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9666
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9667
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9668
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9669
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9670
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9671
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9672
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9673
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9674
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9675
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9676
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9677
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9678
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9679
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9680
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9681
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9682
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9683
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9684
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9685
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9686
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9687
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9688
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9689
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9690
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9691
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9692
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9693
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9694
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9695
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9696
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9697
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9698
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9699
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9700
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9701
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9702
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9703
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9704
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9705
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9706
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9707
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9708
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9709
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9710
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9711
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9712
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9713
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9714
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9715
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9716
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9717
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9718
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9719
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9720
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9721
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9722
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9723
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9724
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9725
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9726
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9727
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9728
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9729
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9730
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9731
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9732
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9733
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9734
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9735
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9736
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9737
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9738
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9739
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9740
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9741
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9742
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9743
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9744
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9745
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9746
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9747
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9748
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9749
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9750
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9751
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9752
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9753
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9754
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9755
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9756
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9757
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9758
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9759
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9760
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9761
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9762
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9763
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9764
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9765
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9766
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9767
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9768
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9769
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9770
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9771
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9772
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9773
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9774
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9775
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9776
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9777
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9778
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9779
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9780
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9781
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9782
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9783
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9784
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9785
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9786
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9787
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9788
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9789
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9790
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9791
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9792
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9793
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9794
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9795
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9796
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9797
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9798
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9799
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9800
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9801
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9802
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9803
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9804
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9805
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9806
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9807
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9808
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9809
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9810
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9811
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9812
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9813
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9814
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9815
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9816
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9817
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9818
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9819
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9821
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9822
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9823
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9824
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9825
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9826
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9827
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9828
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9829
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9830
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9831
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9832
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9833
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9834
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9835
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9836
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9837
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9838
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9839
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9840
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9841
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9842
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9843
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9844
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9845
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9846
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9847
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9848
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9849
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9850
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9851
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9852
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9853
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9854
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9855
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9856
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9857
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9858
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9859
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9860
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9861
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9862
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9863
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9864
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9865
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9866
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9867
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9868
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9869
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9870
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9871
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9872
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9873
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9874
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9875
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9876
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9877
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9878
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9879
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9880
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9881
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9882
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9883
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9884
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9885
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9886
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9887
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9888
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9889
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9890
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9891
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9892
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9893
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9894
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9895
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9896
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9897
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9898
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9899
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9900
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9901
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9902
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9903
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9904
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9905
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9906
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9907
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9908
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9909
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9910
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9911
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9912
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9913
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9914
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9915
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9916
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9917
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9918
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9919
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9920
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9921
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9922
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9923
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9924
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9925
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9926
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9927
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9928
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9929
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9930
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9931
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9932
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9933
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9934
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9935
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9936
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9937
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9938
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9939
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9940
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9941
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9942
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9943
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9944
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9945
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9946
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9947
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9948
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9949
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9950
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9951
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9952
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9953
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9954
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9955
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9956
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9957
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9958
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9959
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9960
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9961
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9962
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9963
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9964
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9965
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9966
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9967
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9968
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9969
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9970
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9971
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9972
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9973
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9974
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9975
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9976
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9977
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9978
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9979
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9980
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9981
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9982
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9983
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9984
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9985
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9986
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9987
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9988
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9989
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9990
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9991
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9992
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9993
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9994
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9995
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9996
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9997
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9998
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
  9999
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10000
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10001
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10002
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10003
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10004
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10005
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10006
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10007
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10008
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10009
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10010
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10011
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10012
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10013
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10014
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10015
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10016
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10017
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10018
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10019
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10020
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10021
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10022
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10023
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10024
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10025
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10026
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10027
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10028
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10029
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10030
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10031
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10032
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10033
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10034
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10035
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10036
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10037
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10038
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10039
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10040
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10041
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10042
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10043
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10044
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10045
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10046
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10047
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10048
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10049
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10050
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10051
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10052
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10053
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10054
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10055
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10056
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10057
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10058
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10059
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10060
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10061
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10062
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10063
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10064
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10065
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10066
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10067
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10068
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10069
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10070
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10071
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10072
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10073
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10074
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10075
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10076
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10077
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10078
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10079
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10080
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10081
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10082
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10083
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10084
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10085
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10086
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10087
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10088
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10089
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10090
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10091
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10092
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10093
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10094
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10095
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10096
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10097
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10098
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10099
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10100
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10101
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10102
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10103
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10104
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10105
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10106
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10107
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10108
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10109
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10110
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10111
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10112
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10113
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10114
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10115
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10116
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10117
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10118
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10119
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10120
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10121
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10122
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10123
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10124
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10125
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10126
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10127
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10128
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10129
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10130
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10131
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10132
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10133
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10134
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10135
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10136
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10137
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10138
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10139
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10140
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10141
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10142
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10143
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10145
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10146
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10147
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10148
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10149
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10150
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10151
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10152
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10153
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10155
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10156
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10157
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10158
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10159
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10160
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10161
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10162
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10163
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10164
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10165
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10166
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10167
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10168
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10170
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10172
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10173
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10174
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10175
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10177
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10179
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10180
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10181
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10182
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10183
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10184
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10185
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10186
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10187
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10188
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10189
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10190
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10191
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10192
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10193
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10194
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10195
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10196
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10197
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10198
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10199
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10200
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10201
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10202
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10203
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10205
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10206
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10207
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10208
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10209
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10210
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10211
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10212
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10213
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10214
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10215
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10216
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10217
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10218
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10219
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10220
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10221
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10222
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10223
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10224
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10225
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10226
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10227
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10228
    Char end stringwidth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10229
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10230
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10231
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10232
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10233
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10234
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10235
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10236
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10237
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10238
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10239
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10240
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10241
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10242
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10243
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10244
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10245
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10246
%%BeginProcSet: 8r.enc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10247
% @@psencodingfile@{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10248
%   author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10249
%   version = "0.6",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10250
%   date = "1 July 1998",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10251
%   filename = "8r.enc",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10252
%   email = "tex-fonts@@tug.org",
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10253
%   docstring = "Encoding for TrueType or Type 1 fonts
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10254
%                to be used with TeX."
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10255
% @}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10256
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10257
% Idea is to have all the characters normally included in Type 1 fonts
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10258
% available for typesetting. This is effectively the characters in Adobe
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10259
% Standard Encoding + ISO Latin 1 + extra characters from Lucida.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10260
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10261
% Character code assignments were made as follows:
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10262
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10263
% (1) the Windows ANSI characters are almost all in their Windows ANSI
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10264
% positions, because some Windows users cannot easily reencode the
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10265
% fonts, and it makes no difference on other systems. The only Windows
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10266
% ANSI characters not available are those that make no sense for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10267
% typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10268
% (173). quotesingle and grave are moved just because it's such an
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10269
% irritation not having them in TeX positions.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10270
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10271
% (2) Remaining characters are assigned arbitrarily to the lower part
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10272
% of the range, avoiding 0, 10 and 13 in case we meet dumb software.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10273
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10274
% (3) Y&Y Lucida Bright includes some extra text characters; in the
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10275
% hopes that other PostScript fonts, perhaps created for public
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10276
% consumption, will include them, they are included starting at 0x12.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10277
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10278
% (4) Remaining positions left undefined are for use in (hopefully)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10279
% upward-compatible revisions, if someday more characters are generally
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10280
% available.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10281
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10282
% (5) hyphen appears twice for compatibility with both 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10283
% ASCII and Windows.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10284
% 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10285
/TeXBase1Encoding [
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10286
% 0x00 (encoded characters from Adobe Standard not in Windows 3.1)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10287
  /.notdef /dotaccent /fi /fl
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10288
  /fraction /hungarumlaut /Lslash /lslash
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10289
  /ogonek /ring /.notdef
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10290
  /breve /minus /.notdef 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10291
% These are the only two remaining unencoded characters, so may as
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10292
% well include them.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10293
  /Zcaron /zcaron 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10294
% 0x10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10295
 /caron /dotlessi 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10296
% (unusual TeX characters available in, e.g., Lucida Bright)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10297
 /dotlessj /ff /ffi /ffl 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10298
 /.notdef /.notdef /.notdef /.notdef
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10299
 /.notdef /.notdef /.notdef /.notdef
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10300
 % very contentious; it's so painful not having quoteleft and quoteright
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10301
 % at 96 and 145 that we move the things normally found there to here.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10302
 /grave /quotesingle 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10303
% 0x20 (ASCII begins)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10304
 /space /exclam /quotedbl /numbersign
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10305
 /dollar /percent /ampersand /quoteright
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10306
 /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10307
% 0x30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10308
 /zero /one /two /three /four /five /six /seven
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10309
 /eight /nine /colon /semicolon /less /equal /greater /question
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10310
% 0x40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10311
 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10312
% 0x50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10313
 /P /Q /R /S /T /U /V /W
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10314
 /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10315
% 0x60
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10316
 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10317
% 0x70
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10318
 /p /q /r /s /t /u /v /w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10319
 /x /y /z /braceleft /bar /braceright /asciitilde
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10320
 /.notdef % rubout; ASCII ends
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10321
% 0x80
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10322
 /.notdef /.notdef /quotesinglbase /florin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10323
 /quotedblbase /ellipsis /dagger /daggerdbl
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10324
 /circumflex /perthousand /Scaron /guilsinglleft
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10325
 /OE /.notdef /.notdef /.notdef
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10326
% 0x90
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10327
 /.notdef /.notdef /.notdef /quotedblleft
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10328
 /quotedblright /bullet /endash /emdash
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10329
 /tilde /trademark /scaron /guilsinglright
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10330
 /oe /.notdef /.notdef /Ydieresis
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10331
% 0xA0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10332
 /.notdef % nobreakspace
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10333
 /exclamdown /cent /sterling
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10334
 /currency /yen /brokenbar /section
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10335
 /dieresis /copyright /ordfeminine /guillemotleft
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10336
 /logicalnot
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10337
 /hyphen % Y&Y (also at 45); Windows' softhyphen
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10338
 /registered
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10339
 /macron
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10340
% 0xD0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10341
 /degree /plusminus /twosuperior /threesuperior
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10342
 /acute /mu /paragraph /periodcentered
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10343
 /cedilla /onesuperior /ordmasculine /guillemotright
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10344
 /onequarter /onehalf /threequarters /questiondown
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10345
% 0xC0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10346
 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10347
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 10791
(with)e(completely)i(implicit)f(structural)i(rules)d(is)g(de)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10792
(eloped)i(for)749 1255 y(classical)d(logic.)30 b(A)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10793
b(term)i(assignment)i(is)d(gi)n(v)o(en)h(for)g(the)g(proofs)h(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10794
(this)h(calculus.)p Black 658 1401 a F6(\017)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10795
46 w Gg(T)-7 b(w)o(o)23 b(strongly)k(normalising)g(cut-elimination)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10796
(procedures)e(are)e(designed.)35 b(The)o(y)24 b(are)749
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10797
1514 y(less)32 b(restricti)n(v)o(e)j(than)d(pre)n(vious)j(strongly)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10798
(normalising)h(procedures)g(and)e(allo)n(w)-6 b(,)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10799
b(in)749 1627 y(general,)25 b(more)e(normal)h(forms)g(to)g(be)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10800
(reached.)p Black 658 1773 a F6(\017)p Black 46 w Gg(A)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10801
b(strongly)k(normalising)g(cut-elimination)h(procedure)f(with)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10802
(completely)j(local)e(re-)749 1886 y(duction)e(rules)f(is)f(de)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10803
(eloped.)41 b(As)26 b(f)o(ar)i(as)f(I)f(am)h(a)o(w)o(are,)g(it)g(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10804
(the)g(only)h(strongly)i(nor)n(-)749 1999 y(malising)f(cut-elimination)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10805
j(procedure)e(for)e(classical)j(logic)e(whose)f(reduction)j(rules)749
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10806
2112 y(are)23 b(local)i(and)f(which)g(allo)n(ws)f(cuts)i(to)e(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10807
(permuted)h(with)e(other)i(cuts.)p Black 321 2299 a Gb(Chapter)e(3:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10808
Black Black Black Black 658 2445 a F6(\017)p Black 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10809
w Gg(The)e(correspondence)27 b(between)c(the)g(standard)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10810
(normalisation)i(procedure)f(of)d(NJ)f(and)749 2558 y(the)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10811
b(intuitionistic)k(v)n(ariant)d(of)f(one)g(of)g(our)g(cut-elimination)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10812
(procedures)f(is)c(estab-)749 2671 y(lished)24 b(in)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10813
F4(\()p F6(\033)p Ga(;)15 b F6(^)p F4(\))p Gg(-fragment.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10814
Black 658 2817 a F6(\017)p Black 46 w Gg(A)24 b(counter)k(e)o(xample)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10815
(is)f(gi)n(v)o(en,)h(which)f(sho)n(ws)g(that)h(certain)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10816
(cut-elimination)j(reduc-)749 2930 y(tions)f(cannot)h(be)e(mapped,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10817
(contrary)f(to)e(commonly)i(accepted)g(belief,)h(onto)e(reduc-)749
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10818
3043 y(tions)e(of)f(the)h(e)o(xplicit)h(substitution)i(calculus)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10819
Ga(\025)p F1(x)p Gg(,)e(pro)o(vided)i(one)f(uses)g(the)f(standard)749
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10820
3156 y(translations)g(between)f(sequent)h(proofs)f(and)f(natural)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10821
(deduction)h(proofs.)p Black 658 3302 a F6(\017)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10822
46 w Gg(A)c(sequence-conclusi)q(on)30 b(natural)25 b(deduction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10823
(calculus)f(for)e(classical)i(logic)f(is)e(intro-)749
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10824
3415 y(duced.)53 b(It)32 b(is)f(sho)n(wn)h(that)g(its)f(normalisation)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10825
(procedure)f(is)d(strongly)j(normalising)749 3528 y(and)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10826
b(that)g(the)h(normal)f(natural)i(deduction)g(proofs)g(respect)f(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10827
(subformula)j(property)-6 b(.)749 3641 y(F)o(or)18 b(this)i(natural)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10828
(deduction)h(calculus)f(the)f(correspondence)k(between)c(normalisation)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10829
749 3754 y(and)k(cut-elimination)j(is)c(established)k(for)d(all)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10830
(connecti)n(v)o(es.)p Black 321 3941 a Gb(Chapter)g(4:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10831
Black Black Black Black 658 4088 a F6(\017)p Black 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10832
w Gg(An)k(implementation)32 b(for)d(a)f(leftmost-outermost)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10833
b(cut-reduction)f(strate)o(gy)e(is)f(de)n(v)o(el-)749
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10834
4200 y(oped.)f(This)21 b(strate)o(gy)i(does)f(not)f(restrict)i(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10835
(normal)h(forms)f(reachable)i(from)e(a)g(proof.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10836
658 4347 a F6(\017)p Black 46 w Gg(The)e(classical)j(proof)e(studied)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10837
(by)e(Barbanera)h(et)f(al.)f([1997])i(is)f(translated)i(into)e(our)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10838
(se-)749 4459 y(quent)h(calculus.)30 b(The)20 b(beha)n(viour)j(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10839
(this)g(proof)g(under)h(cut-elimination)i(is)d(analysed,)749
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10840
4572 y(and)h(it)f(is)h(sho)n(wn)g(that)h(tw)o(o)e(of)h(its)g(normal)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10841
(forms)h(dif)n(fer)f(in)g(\223essential\224)j(features.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10842
b(The)749 4685 y(normal)24 b(forms)g(are)f(gi)n(v)o(en)h(in)g(Appendix)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10843
h(A.)p Black 658 4831 a F6(\017)p Black 46 w Gg(It)h(is)g(sho)n(wn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10844
(that)g(normalisation)i(in)e(the)f(simply-typed)k(lambda)d(calculus)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10845
(e)o(xtended)749 4944 y(with)24 b(an)h(erratic)h(choice)g(operator)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10846
(can)e(be)g(simulated)h(by)f(cut-elimination)j(in)d(classi-)749
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10847
5057 y(cal)e(logic.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10848
%%Page: 9 21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10849
TeXDict begin 9 20 bop Black Black 277 990 a F8(Chapter)44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10850
b(2)277 1455 y Gf(Sequent)50 b(Calculi)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10851
1140 1944 a Gd(In)18 b(order)f(to)i(be)g(able)f(to)h(enunciate)e(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10852
(pro)o(v)o(e)e(the)j F0(Hauptsatz)f Gd(in)g(a)h(con)m(v)o(enient)1140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10853
2044 y(form,)d(I)g(had)f(to)i(pro)o(vide)d(a)i(logical)g(calculus)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10854
(especially)f(suited)h(to)g(the)h(purpose.)1140 2143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10855
y(F)o(or)j(this)g(the)h(natural)e(deduction)f(calculus)i(pro)o(v)o(ed)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10856
(unsuitable.)3041 2276 y(\227G.)j(Gentzen)1904 2376 y(in)g(In)m(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10857
(estigations)c(into)j(Logical)g(Deductions,)e(1935.)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10858
3006 y Gg(T)-7 b(w)o(o)32 b(of)g(Gentzen')-5 b(s)35 b(most)e(striking)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10859
(and)e(highly)i(original)g(in)l(v)o(entions)g(are)f F7(sequent)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10860
(calculi)h Gg(and)277 3119 y F7(cut-elimination)28 b(pr)l(ocedur)m(es)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10861
Gg(.)33 b(In)24 b(this)g(chapter)i(we)d(shall)i(sho)n(w)f(that)h(only)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10862
(a)e(small)h(restriction)j(on)277 3232 y(the)h(standard)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10863
(cut-elimination)i(procedure)f(for)d(classical)i(logic)f(is)f(suf)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10864
(\002cient)h(to)f(obtain)h(strongly)277 3345 y(normalising)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10865
b(proof)e(transformations.)62 b(Prior)34 b(to)f(our)g(w)o(ork,)j(some)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10866
(strongly)j(normalising)g(cut-)277 3458 y(elimination)e(procedures,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10867
(notably)d(in)f([Dragalin,)g(1988,)g(Danos)g(et)f(al.,)g(1997],)j(ha)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10868
(v)o(e)e(been)g(de-)277 3571 y(v)o(eloped,)c(b)n(ut)e(all)g(of)g(them)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10869
(impose)i(some)f(quite)g(strong)i(restrictions.)38 b(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10870
b(e)25 b(shall)i(sho)n(w)e(that)h(these)277 3683 y(restrictions)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10871
b(are)d(unnecessary)j(to)c(ensure)i(strong)g(normalisation.)56
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10872
b(The)31 b(basic)h(idea)g(of)g(our)g(cut-)277 3796 y(elimination)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10873
(procedure)h(is)d(to)f(pro)o(vide)j(some)e(means)g(to)g(pre)n(v)o(ent)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10874
(interference)i(between)e(proof)277 3909 y(transformations)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10875
b(that)e(simply)f(permute)g(cut-rules)i(upw)o(ards)f(in)f(a)f(proof.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10876
418 4072 y(W)-7 b(e)27 b(shall)h(\002rst)f(describe)j(Gentzen')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10877
b(s)29 b(sequent)g(calculi)g(LK)d(and)i(LJ,)e(and)i(the)f(main)h(ideas)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10878
g(be-)277 4185 y(hind)35 b(his)f F7(Hauptsatz)i Gg(\(cut-elimination)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10879
(theorem\).)61 b(Ne)o(xt,)36 b(we)d(shall)i(de)n(vise)g(term)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10880
(annotations)277 4298 y(for)i(proofs)h(of)f(a)f(sequent)i(calculus)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10881
(of)e(classical)h(logic,)j(whose)c(inference)i(rules)e(are)g(inspired)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10882
277 4411 y(by)f(a)f(v)n(ariant)j(of)d(Kleene')-5 b(s)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10883
b(sequent)f(calculus)h(G3.)62 b(The)35 b(main)f(part)i(of)f(this)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10884
(chapter)i(is)d(occu-)277 4524 y(pied)23 b(by)g(an,)f(unfortunately)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10885
b(,)27 b(rather)d(lengthy)g(proof)f(of)g(strong)h(normalisation)h(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10886
(which)g(we)e(adapt)277 4637 y(the)27 b(technique)j(of)c(symmetric)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10887
(reducibility)i(candidates)g(de)n(v)o(eloped)f(by)e(Barbanera)h(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10888
(Berardi)277 4750 y([1994].)43 b(Subsequently)-6 b(,)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10889
b(we)c(e)o(xtend)i(this)f(result)h(to)f(the)g(\002rst-order)h(case)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10890
(and)f(to)f(other)i(formula-)277 4863 y(tions)i(of)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10891
(propositional)j(fragment.)48 b(The)29 b(cut-elimination)k(procedure)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10892
(for)d(which)h(we)f(pro)o(v)o(e)277 4976 y(strong)d(normalisation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10893
(depends)f(on)d(a)g(global,)j(or)d(non-local,)j(notion)f(of)f(proof)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10894
(substitution.)36 b(Us-)277 5088 y(ing)d(results)i(concerning)h(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10895
(xplicit)e(substitution)j(calculi)d(we)e(can)i(replace)g(the)f(global)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10896
(operation)277 5201 y(with)23 b(completely)j(local)e(proof)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10897
(transformations)j(and)d(obtain)g(again)g(a)f(strongly)i(normalising)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10898
(cut-)277 5314 y(elimination)g(procedure.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10899
eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10900
%%Page: 10 22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10901
TeXDict begin 10 21 bop Black -144 51 a Gb(10)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10902
b(Calculi)p -144 88 3691 4 v Black 321 317 a Ge(2.1)119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10903
b(Gentzen')l(s)30 b(Sequent)i(Calculi)321 541 y Gg(In)h(his)f(seminal)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10904
(paper)f([1935])h(Gentzen)g(introduced)h(the)e(sequent)h(calculi)g(LJ)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10905
(and)h(LK)e(for)h(in-)321 654 y(tuitionistic)h(and)d(classical)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10906
(logic,)g(respecti)n(v)o(ely)-6 b(.)50 b(He)29 b(w)o(as)h(able)g(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10907
(sho)n(w)f(that)h(for)g(each)h(sequent)321 767 y(proof)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10908
b(a)f(normal)h(form)f(can)g(be)h(found)g(in)f(which)g(all)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10909
(applications)i(of)d(the)h(cut-rule)h(are)e(eliminated.)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10910
880 y(Gentzen)j(not)e(only)i(pro)o(v)o(ed)f(that)g(all)f(occurrences)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10911
(of)c(this)h(rule)f(can)h(be)f(eliminated,)j(b)n(ut)d(also)h(ga)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10912
(e)321 993 y(a)29 b(simple)h(procedure)i(for)e(doing)h(so.)46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10913
b(Before)30 b(we)e(outline)k(this)e(procedure,)j(let)c(us)h(\002rst)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10914
(e)o(xamine)321 1106 y(LK)22 b(in)i(more)f(detail.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10915
1235 y(A)f F7(sequent)i Gg(in)e(LK)f(is)h(a)g(pair)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10916
F4(\(\000)p Ga(;)15 b F4(\001\))22 b Gg(of)g(\002nite)g(multisets)i(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10917
e(formulae,)i(commonly)g(written)f(as)321 1348 y F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10918
398 1336 11 41 v 409 1318 46 5 v 96 w(\001)p Gg(,)g(where)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10919
(formulae)h(are)f(gi)n(v)o(en)g(by)g(the)f(grammar)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10920
Black Black 1218 1577 a Ga(B)30 b F4(::=)25 b Ga(A)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10921
F6(j)f(:)p Ga(B)30 b F6(j)25 b Ga(B)5 b F6(^)o Ga(B)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10922
b F6(j)25 b Ga(B)5 b F6(_)p Ga(B)29 b F6(j)d Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10923
F6(\033)o Ga(B)30 b Gg(.)321 1806 y(W)-7 b(e)35 b(adopt)h(the)f(con)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10924
(v)o(ention)k(that)c Ga(A)g Gg(stands)h(for)g(atomic)f(formulae,)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10925
(also)d(called)h(propositional)321 1919 y(symbols,)c(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10926
Ga(B)5 b(;)15 b(C)q(;)g(:)g(:)g(:)32 b Gg(for)e(arbitrary)j(formulae.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10927
51 b(The)30 b(turnstile)i(di)n(vides)g(a)e(sequent)j(into)e(tw)o(o)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10928
2032 y(zones,)24 b(called)g(the)g F7(antecedent)i Gg(and)d(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10929
F7(succedent)p Gg(.)31 b(The)23 b(intuiti)n(v)o(e)h(meaning)g(of)f(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10930
(sequent)h(of)f(the)321 2145 y(form)1454 2367 y Ga(B)1523
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10931
2381 y F9(1)1562 2367 y Ga(;)15 b(:)g(:)g(:)i(;)e(B)1833
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10932
2381 y Gc(n)p 1900 2355 11 41 v 1910 2337 46 5 v 1976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10933
2367 a Ga(C)2041 2381 y F9(1)2081 2367 y Ga(;)g(:)g(:)g(:)h(;)f(C)2347
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10934
2381 y Gc(m)321 2588 y Gg(is)20 b(that)h(the)f(conjunction)k(of)c(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10935
(formulae,)i Ga(B)1801 2602 y Gc(i)1829 2588 y Gg(,)d(in)h(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10936
(antecedent)i(implies)e(the)f(disjunction)j(of)d(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10937
2701 y(formulae,)27 b Ga(C)762 2715 y Gc(j)799 2701 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10938
Gg(,)d(in)i(the)f(succedent.)37 b(There)25 b(is)h(no)f(e)o(xplicit)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10939
(constant)g(for)f(f)o(alsum)g(in)f(LK;)f(ho)n(we)n(v)o(er)l(,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10940
2814 y(a)f(sequent)j(with)d(an)h(empty)g(succedent,)i(say)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10941
F4(\000)p 1853 2802 11 41 v 1863 2784 46 5 v 96 w Gg(,)e(can)i(be)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10942
(interpreted)i(as)e F4(\000)e Gg(implies)j(f)o(alsum.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10943
2943 y(The)k(proofs)i(in)e(Gentzen')-5 b(s)31 b(sequent)g(calculi)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10944
(are)e(tree-structures)34 b(where)29 b(the)g(lea)n(v)o(es)i(are)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10945
F7(ax-)321 3056 y(ioms)37 b Gg(of)f(the)g(form)p 999
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10946
2977 244 4 v 37 w Ga(B)p 1092 3044 11 41 v 1103 3026
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10947
46 5 v 101 w(B)k Gg(and)c(the)h(nodes)h(are)e F7(infer)m(ence)j(rules)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10948
Gg(.)68 b(There)36 b(are)h(three)g(kinds)h(of)321 3169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10949
y(inference)28 b(rules:)33 b(logical)27 b(rules,)f(structural)h(rules)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10950
(and)g(the)f(cut-rule.)35 b(The)25 b(logical)h(and)g(structural)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10951
3282 y(inference)g(rules)f(f)o(all)f(into)g(tw)o(o)f(groups\227left)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10952
(rules)f(and)f(right)g(rules\227depending)k(on)c(which)g(half)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10953
3395 y(of)g(the)g(sequent)h(the)o(y)f(act)g(on.)k(Some)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10954
b(of)h(the)g(logical)h(rules)f(are)g(listed)h(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10955
b(.)p Black Black 1016 3676 a Ga(B)1085 3690 y Gc(i)1113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10956
3676 y Ga(;)15 b F4(\000)p 1231 3664 11 41 v 1241 3646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10957
46 5 v 97 w(\001)p 926 3714 547 4 v 926 3792 a Ga(B)995
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10958
3806 y F9(1)1034 3792 y F6(^)p Ga(B)1164 3806 y F9(2)1203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10959
3792 y Ga(;)g F4(\000)p 1321 3780 11 41 v 1331 3762 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10960
5 v 97 w(\001)1514 3727 y F6(^)1575 3741 y Gc(L)1623
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10961
3751 y FZ(i)2190 3676 y F4(\000)p 2267 3664 11 41 v 2277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10962
3646 46 5 v 96 w(\001)p Ga(;)g(B)95 b F4(\000)p 2700
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10963
3664 11 41 v 2711 3646 46 5 v 96 w(\001)p Ga(;)15 b(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10964
2190 3714 775 4 v 2339 3792 a F4(\000)p 2416 3780 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10965
41 v 2427 3762 46 5 v 97 w(\001)p Ga(;)g(B)5 b F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10966
Ga(C)3006 3732 y F6(^)3066 3746 y Gc(R)828 3995 y Ga(B)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10967
b F4(\000)p 1019 3983 11 41 v 1029 3965 46 5 v 96 w(\001)90
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10968
b Ga(C)q(;)15 b F4(\000)p 1445 3983 11 41 v 1456 3965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10969
46 5 v 97 w(\001)p 828 4032 770 4 v 977 4111 a Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10970
b F6(_)p Ga(C)q(;)15 b F4(\000)p 1295 4099 11 41 v 1306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10971
4081 46 5 v 97 w(\001)1639 4051 y F6(_)1699 4065 y Gc(L)2381
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10972
3995 y F4(\000)p 2458 3983 11 41 v 2468 3965 46 5 v 96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10973
w(\001)p Ga(;)g(B)2719 4009 y Gc(i)p 2291 4032 547 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10974
v 2291 4111 a F4(\000)p 2368 4099 11 41 v 2378 4081 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10975
5 v 96 w(\001)p Ga(;)g(B)2629 4125 y F9(1)2668 4111 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10976
F6(_)p Ga(B)2798 4125 y F9(2)2879 4046 y F6(_)2939 4060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10977
y Gc(R)2992 4070 y FZ(i)744 4313 y F4(\000)801 4327 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10978
F9(1)p 860 4301 11 41 v 871 4283 46 5 v 936 4313 a F4(\001)1012
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10979
4327 y F9(1)1051 4313 y Ga(;)g(B)96 b(C)q(;)15 b F4(\000)1419
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10980
4327 y F9(2)p 1480 4301 11 41 v 1490 4283 46 5 v 1556
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10981
4313 a F4(\001)1632 4327 y F9(2)p 744 4351 928 4 v 782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10982
4430 a Ga(B)5 b F6(\033)o Ga(C)q(;)15 b F4(\000)1089
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10983
4444 y F9(1)1129 4430 y Ga(;)g F4(\000)1226 4444 y F9(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10984
1286 4418 11 41 v 1296 4400 46 5 v 1362 4430 a F4(\001)1438
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10985
4444 y F9(1)1477 4430 y Ga(;)g F4(\001)1593 4444 y F9(2)1712
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10986
4368 y F6(\033)1783 4383 y Gc(L)2344 4313 y Ga(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10987
b F4(\000)p 2535 4301 11 41 v 2546 4283 46 5 v 96 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10988
Ga(;)g(C)p 2329 4351 485 4 v 2329 4430 a F4(\000)p 2406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10989
4418 11 41 v 2417 4400 46 5 v 96 w(\001)p Ga(;)g(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10990
b F6(\033)p Ga(C)2856 4368 y F6(\033)2927 4383 y Gc(R)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10991
4733 y Gg(W)-7 b(e)24 b(say)i(the)f F6(^)806 4747 y Gc(L)854
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10992
4757 y FZ(i)884 4733 y Gg(-rule)h(introduces)i(the)e(formula)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10993
Ga(B)2000 4747 y F9(1)2039 4733 y F6(^)p Ga(B)2169 4747
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10994
y F9(2)2208 4733 y Gg(\227the)f F7(main)g(formula)i Gg(of)e(this)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10995
(inference)321 4846 y(rule\227and)k(refer)e(to)g F4(\000)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10996
Gg(and)h F4(\001)e Gg(as)i F7(conte)n(xts)i Gg(\(multisets)g(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10997
(formulae\).)40 b(In)27 b(both)g(the)g(premise)h(and)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10998
4959 y(conclusion)i(of)d(this)h(rule)g(the)f(comma)g(is)g(interpreted)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 10999
(as)d(the)g(multiset)i(union.)40 b(The)27 b(terminology)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11000
5072 y(gi)n(v)o(en)d(for)g F6(^)732 5086 y Gc(L)780 5096
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11001
y FZ(i)833 5072 y Gg(also)g(applies)h(to)f(the)g(other)g(rules.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11002
5201 y(There)e(is)f(a)g(slight)i(peculiarity)h(in)e(Gentzen')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11003
b(s)23 b F6(\033)2016 5215 y Gc(L)2068 5201 y Gg(-rule,)f(in)f(which,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11004
(if)g(we)e(read)i(it)f(from)h(bottom)321 5314 y(to)k(top,)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11005
(conte)o(xts)i(are)e(split)h(up)f(into)h(tw)o(o)f(multisets.)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11006
b(In)26 b(contrast,)i(in)e(the)h(inference)h(rules)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11007
F6(^)3489 5328 y Gc(R)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11008
%%Page: 11 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11009
TeXDict begin 11 22 bop Black 277 51 a Gb(2.1)23 b(Gentzen')m(s)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11010
(Sequent)e(Calculi)2442 b(11)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11011
277 317 a Gg(and)33 b F6(_)500 331 y Gc(L)584 317 y Gg(both)g(premises)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11012
g(share)h(the)e(same)g(conte)o(xt.)56 b(The)32 b(rules)h(sharing)h(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11013
e(conte)o(xts)i(are)f(said)277 430 y(to)28 b(be)h F7(additive)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11014
Gg(,)i(while)d(the)h(rules)g(splitting)h(the)f(conte)o(xts)h(are)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11015
(said)h(to)f(be)g F7(multiplicative)p Gg(.)46 b(Ev)o(ery)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11016
543 y(inference)26 b(rule)e(with)f(tw)o(o)h(premises)h(can)e(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11017
(de\002ned)g(either)h(w)o(ay)-6 b(.)418 673 y(In)26 b(our)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11018
(formulation)i(of)e(Gentzen')-5 b(s)28 b(LK)c(there)i(are)g(four)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11019
(structural)h(rules)f(again)f(di)n(vided)i(into)277 786
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11020
y(left)c(and)g(right)g(rules.)967 753 y F5(1)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11021
Black 1067 956 a F4(\000)p 1144 944 11 41 v 1155 926
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11022
46 5 v 96 w(\001)p 1010 976 343 4 v 1010 1055 a Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11023
b(;)15 b F4(\000)p 1201 1043 11 41 v 1212 1025 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11024
96 w(\001)1395 1000 y Gg(W)-7 b(eak)1599 1014 y Gc(L)2179
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11025
956 y F4(\000)p 2256 944 11 41 v 2267 926 46 5 v 96 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11026
2122 976 343 4 v 2122 1055 a(\000)p 2199 1043 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11027
2210 1025 46 5 v 96 w(\001)p Ga(;)15 b(B)2506 1000 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11028
Gg(W)-7 b(eak)2710 1014 y Gc(R)952 1269 y Ga(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11029
b(B)5 b(;)15 b F4(\000)p 1257 1257 11 41 v 1268 1239
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11030
46 5 v 96 w(\001)p 952 1307 457 4 v 1009 1385 a Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11031
b(;)15 b F4(\000)p 1200 1373 11 41 v 1211 1355 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11032
96 w(\001)1450 1331 y Gg(Contr)1656 1345 y Gc(L)2064
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11033
1269 y F4(\000)p 2141 1257 11 41 v 2151 1239 46 5 v 96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11034
w(\001)p Ga(;)g(B)5 b(;)15 b(B)p 2064 1307 457 4 v 2121
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11035
1385 a F4(\000)p 2198 1373 11 41 v 2208 1355 46 5 v 96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11036
w(\001)p Ga(;)g(B)2562 1331 y Gg(Contr)2768 1345 y Gc(R)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11037
1570 y Gg(The)23 b(only)i(inference)g(rule)g(that)f(is)f(neither)i(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11038
(left)h(nor)g(a)f(right)i(rule)f(is)f(the)h(cut-rule)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11039
Black Black 1336 1741 a F4(\000)1393 1755 y F9(1)p 1453
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11040
1729 11 41 v 1463 1711 46 5 v 1529 1741 a F4(\001)1605
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11041
1755 y F9(1)1644 1741 y Ga(;)15 b(B)96 b(B)5 b(;)15 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11042
F4(\000)2020 1755 y F9(2)p 2079 1729 11 41 v 2089 1711
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11043
46 5 v 2155 1741 a F4(\001)2231 1755 y F9(2)p 1336 1779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11044
935 4 v 1503 1857 a F4(\000)1560 1871 y F9(1)1599 1857
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11045
y Ga(;)g F4(\000)1696 1871 y F9(2)p 1756 1845 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11046
1767 1827 46 5 v 1832 1857 a F4(\001)1908 1871 y F9(1)1948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11047
1857 y Ga(;)g F4(\001)2064 1871 y F9(2)2312 1809 y Gg(Cut)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11048
2042 y(in)24 b(which)f(the)h(formula)h Ga(B)i Gg(is)c(called)i(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11049
F7(cut-formula)p Gg(.)418 2172 y(The)36 b(intuitionistic)k(sequent)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11050
(calculus)g(LJ)d(can)h(be)g(obtained)j(from)d(LK)e(by)i(restricting)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11051
(the)277 2285 y(succedent)33 b(of)c(sequents)j(to)e(at)g(most)g(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11052
(single)i(formula,)h(by)e(dropping)i(the)f(Contr)2992
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11053
2299 y Gc(R)3050 2285 y Gg(-rule)g(and)f(by)277 2398
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11054
y(replacing)c(the)e F6(\033)843 2412 y Gc(L)895 2398
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11055
y Gg(-rule)g(with)p Black Black 1415 2568 a F4(\000)1472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11056
2582 y F9(1)p 1532 2556 11 41 v 1542 2538 46 5 v 1608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11057
2568 a Ga(B)95 b(C)q(;)15 b F4(\000)1935 2582 y F9(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11058
1995 2556 11 41 v 2006 2538 46 5 v 2072 2568 a Ga(D)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11059
1415 2606 735 4 v 1453 2685 a(B)5 b F6(\033)o Ga(C)q(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11060
b F4(\000)1760 2699 y F9(1)1800 2685 y Ga(;)g F4(\000)1897
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11061
2699 y F9(2)p 1957 2673 11 41 v 1968 2654 46 5 v 2033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11062
2685 a Ga(D)2191 2623 y F6(\033)2262 2637 y Gc(L)2339
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11063
2623 y Ga(:)418 2869 y Gg(One)37 b(of)g(Gentzen')-5 b(s)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11064
b(moti)n(v)o(es)f(when)f(designing)j(the)d(sequent)i(calculi)g(LK)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11065
(and)h(LJ)g(w)o(as)f(to)277 2982 y(pro)o(vide)e(deducti)n(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11066
(systems)e(for)g(classical)h(and)f(intuitionistic)k(logic,)e(respecti)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11067
(v)o(ely)-6 b(,)37 b(for)c(which)277 3095 y F7(consistency)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11068
b Gg(could)e(be)e(established)k(by)d(simple)g(methods.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11069
b(Let)20 b(us)h(brie\003y)g(illustrate)i(his)d(ar)n(gument)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11070
3208 y(for)g(consistenc)o(y)-6 b(.)31 b(First)20 b(notice)h(that)f(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11071
(sequent)i(calculi)f(consistenc)o(y)i(coincides)g(with)c(the)h(absence)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11072
277 3321 y(of)35 b(a)g(proof)h(for)f(the)h(empty)f(sequent)i(\(i.e.,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11073
(both)e(the)f(antecedent)j(and)e(succedent)i(are)d(empty\).)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11074
3434 y(Assume)26 b(we)f(w)o(ould)h(be)g(gi)n(v)o(en)g(a)g(proof)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11075
(this)g(sequent,)i(then)f(we)e(can)h(construct)i(a)e(proof)g(of)g(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11076
277 3547 y(sequent)p 606 3535 11 41 v 617 3517 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11077
131 w Ga(B)5 b F6(^)o(:)p Ga(B)35 b Gg(simply)e(by)g(applying)h(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11078
b(eak)1937 3561 y Gc(R)1996 3547 y Gg(.)54 b(In)32 b(f)o(act,)i(we)e(w)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11079
o(ould)h(be)f(able)h(to)f(construct)277 3660 y(for)f(an)o(y)g(formula)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11080
Ga(C)k Gg(a)30 b(proof)i(of)f(the)g(sequent)p 1879 3648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11081
11 41 v 1889 3630 46 5 v 144 w Ga(C)22 b Gg(.)49 b(Ob)o(viously)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11082
b(,)34 b(LK)29 b(and)j(LJ)e(w)o(ould)h(then)g(be)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11083
3773 y(inconsistent.)69 b(F)o(ortunately)-6 b(,)41 b(there)c(is)e(no)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11084
(proof)h(of)f(the)g(empty)g(sequent)i(in)e(either)h(LK)d(or)i(LJ.)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11085
3886 y(Although)28 b(this)e(is)g(not)h(ob)o(vious,)h(it)d(is)h(easy)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11086
(to)f(v)o(erify)h(that)f(there)h(cannot)h(be)e(an)o(y)g(such)h(proof)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11087
(that)277 3998 y(enjo)o(ys)e(the)f(subformula)i(property)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11088
b(.)p Black 277 4186 a Gb(De\002nition)23 b(2.1.1)g Gg(\(Subformula)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11089
(and)e(Subformula)h(Property\))p Gb(:)p Black Black 414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11090
4393 a F6(\017)p Black 45 w Gg(The)j(relation)i Ga(B)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11091
F7(is)d(a)g(subformula)i(of)41 b Ga(C)34 b Gg(is)28 b(de\002ned)h(as)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11092
(the)i(transiti)n(v)o(e,)h(re\003e)o(xi)n(v)o(e)f(closure)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11093
4506 y(of)24 b(the)g(clauses:)p Black 614 4689 a F6(\017)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11094
Black 45 w Ga(B)5 b Gg(,)22 b Ga(C)30 b Gg(are)23 b(subformulae)j(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11095
Ga(B)5 b F6(^)o Ga(C)i Gg(,)22 b Ga(B)5 b F6(_)o Ga(C)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11096
b Gg(and)24 b Ga(B)5 b F6(\033)p Ga(C)i Gg(,)21 b(and)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11097
Black 614 4831 a F6(\017)p Black 45 w Ga(B)28 b Gg(is)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11098
b(a)g(subformula)j(of)d F6(:)p Ga(B)5 b Gg(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11099
414 5014 a F6(\017)p Black 45 w Gg(A)27 b(proof)h Ga(\031)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11100
Gg(of)d(the)g(sequent)j F4(\000)p 1523 5002 11 41 v 1533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11101
4984 46 5 v 96 w(\001)c Gg(enjo)o(ys)j(the)e F7(subformula)j(pr)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11102
(operty)p Gg(,)g(if)d(and)h(only)g(if)f(all)504 5127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11103
y(formulae)e(occurring)h(in)e Ga(\031)i Gg(are)d(subformulae)k(of)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11104
(the)h(formulae)h(in)e F4(\000)g Gg(and)h F4(\001)p Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11105
Black 277 5227 1290 4 v 383 5283 a F3(1)412 5314 y F2(Gentzen)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11106
b(de\002ned)g(sequents)h(using)g(sequences)g(of)f(formulae)g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11107
(therefore)g(also)g(had)g(an)g(e)o(xplicit)g(Exchange-rule.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11108
Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11109
%%Page: 12 24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11110
TeXDict begin 12 23 bop Black -144 51 a Gb(12)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11111
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(The)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11112
b(crux)h(of)f(Gentzen')-5 b(s)29 b(ar)n(gument)g(for)e(consistenc)o(y)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11113
(is)c(that)g(in)g(LJ)g(and)g(LK)f(all)h(cut-free)i(proofs)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11114
430 y(enjo)o(y)e(the)f(subformula)j(property:)35 b(the)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11115
b(premises)g(of)e(the)h(logical)i(and)e(structural)i(inference)h(rules)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11116
321 543 y(contain)37 b(only)f(subformulae)i(of)d(their)h(respecti)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11117
(e)h(conclusion.)67 b(If)34 b(there)i(were)f(a)g(proof)h(of)f(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11118
656 y(empty)26 b(sequent,)h(then)f(according)i(to)d(his)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11119
F7(Hauptsatz)h Gg(there)f(must)f(be)h(one)f(without)i(cuts.)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11120
b(But)25 b(this)321 769 y(cut-free)34 b(proof)e(w)o(ould)g(lack)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11121
(subformula)i(property)-6 b(.)55 b(Consequently)-6 b(,)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11122
b(there)d(cannot)g(be)e(an)o(y)321 882 y(proof)25 b(of)e(the)h(empty)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11123
(sequent)i(in)d(either)i(LK)d(or)h(LJ.)462 1017 y(Since)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11124
b(Gentzen')-5 b(s)31 b(w)o(ork)e(man)o(y)f F7(Haupts)1794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11125
1018 y(\250)1787 1017 y(atze)j Gg(ha)n(v)o(e)f(appeared)h(for)e(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11126
(arious)h(sequent)g(calculus)321 1130 y(formulations.)66
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11127
b(The)o(y)34 b(all)h(use)g(his)g(technique)j(of)d(proof)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11128
(transformations,)41 b(referred)36 b(to)f(as)g F7(cut-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11129
1243 y(r)m(eductions)p Gg(,)24 b(which)e(con)l(v)o(ert)h(a)e(proof)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11130
(containing)i(cuts)e(not)g(immediately)h(to)e(a)f(cut-free)j(proof,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11131
(b)n(ut)321 1356 y(replace)e(the)e(cuts)h(by)f(simpler)h(cuts,)h(or)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11132
(permute)h(the)f(cuts)h(to)n(w)o(ards)g(the)f(lea)n(v)o(es)i(where)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11133
(the)o(y)g(e)n(v)o(entu-)321 1469 y(ally)j(v)n(anish.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11134
b(Iterating)23 b(this)f(process)h(of)e(applying)i(proof)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11135
(transformations)i(will)20 b(lead)i(to)f(a)g(cut-free)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11136
1582 y(proof.)66 b(W)-7 b(e)35 b(shall)h(sk)o(etch)h(the)f(main)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11137
(lines)g(of)g(Gentzen')-5 b(s)37 b F7(Hauptsatz)h Gg(by)d(gi)n(ving)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11138
(some)f(of)f(his)321 1695 y(cut-reductions.)48 b(Ho)n(we)n(v)o(er)l(,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11139
30 b(in)e(what)h(follo)n(ws)g(we)e(shall)j(study)g(se)n(v)o(eral)f(dif)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11140
n(ferent)i(cut-reduction)321 1808 y(systems,)h(and)e(for)g(this)g(it)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11141
(is)g(con)l(v)o(enient)k(to)c(introduce)j(some)e(notation)h(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11142
(terminology)j(com-)321 1921 y(monly)j(used)g(in)f(connection)j(with)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11143
(term)h(re)n(writing)g(\(see)g(for)f(e)o(xample)h([Baader)g(and)g(Nipk)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11144
o(o)n(w)-6 b(,)321 2033 y(1998]\).)p Black 321 2221 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11145
Gb(De\002nition)23 b(2.1.2:)p Black Black 458 2422 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11146
F6(\017)p Black 46 w Gg(A)f F7(r)m(eduction)k(system)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11147
Gg(is)f(the)h(pair)g F4(\()p FY(A)p Ga(;)15 b F6(\000)-30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11148
b(\000)e(!)p F4(\))p Gg(,)23 b(in)g(which)p Black 658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11149
2633 a F6(\017)p Black 46 w FY(A)f Gg(is)i(a)f(set)h(of)f(objects)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11150
(\(in)f(this)g(thesis)h(usually)g(a)e(set)h(of)g(terms)f(or)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11151
(proofs\),)h(and)p Black 658 2791 a F6(\017)p Black 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11152
w(\000)-32 b(\000)h(!)23 b Gg(is)g(a)g(binary)i(relation)g(o)o(v)o(er)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11153
FY(A)p Gg(;)f(it)g(is)h(often)g(referred)i(to)d(as)g(a)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11154
F7(r)m(eduction)p Gg(.)549 3002 y(Instead)33 b(of)e(writing)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11155
F4(\()p Ga(a;)15 b(b)p F4(\))41 b F6(2\000)-32 b(\000)h(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11156
Gg(,)32 b(we)f(write)g Ga(a)41 b F6(\000)-32 b(\000)h(!)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11157
b Ga(b)31 b Gg(and)h(say)f Ga(a)g Gg(reduces)j(to)d Ga(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11158
Gg(.)52 b(The)549 3115 y(transiti)n(v)o(e)24 b(and)e(the)h(re\003e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11159
(xi)n(v)o(e)f(transiti)n(v)o(e)i(closure)g(of)e F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11160
b(\000)f(!)22 b Gg(is)g(written)h(as)f F6(\000)-32 b(\000)h(!)3081
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11161
3082 y F9(+)3161 3115 y Gg(and)23 b F6(\000)-31 b(\000)f(!)3484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11162
3082 y FX(\003)3524 3115 y Gg(,)549 3228 y(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11163
b(.)30 b(W)-7 b(e)19 b(shall)h(often)h(annotate)h(the)d(`)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11164
F6(\000)-31 b(\000)g(!)p Gg(')19 b(to)h(qualify)h(the)f(reduction)i(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11165
e(question.)549 3390 y(Whene)n(v)o(er)29 b(we)f(de\002ne)h(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11166
(reduction,)k(say)d Ga(a)34 b F6(\000)-31 b(\000)g(!)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11167
b Ga(b)p Gg(,)29 b(we)e(automatically)32 b(assume)d(that)g(the)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11168
3503 y(reduction)e(is)e(closed)i(under)f(conte)o(xt)h(formation.)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11169
b(This)25 b(is)f(a)h(standard)i(con)l(v)o(ention)i(in)c(term)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11170
3616 y(re)n(writing.)p Black 458 3827 a F6(\017)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11171
46 w Gg(A)d(term)i(is)g(said)h(to)f(be)g(a)f F7(normal)i(form)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11172
Gg(if)g(and)g(only)h(if)f(it)f(does)i(not)g(reduce.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11173
b(In)24 b(this)h(thesis)g(a)549 3940 y(normal)f(form)f(often)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11174
(corresponds)i(to)c(a)g(cut-free)j(proof.)p Black 458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11175
4151 a F6(\017)p Black 46 w Gg(Let)h Ga(a)h Gg(be)g(an)g(element)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11176
FY(A)p Gg(,)f(then)h Ga(a)e Gg(is)i(said)f(to)h(be)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11177
F7(str)l(ongly)i(normalising)h Gg(if)d(and)h(only)g(if)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11178
4264 y(all)g(reduction)j(sequences)g(starting)g(from)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11179
Ga(a)g Gg(are)g(\002nite.)47 b(In)29 b(this)h(case)g(we)f(write)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11180
Ga(a)c F6(2)g Ga(S)5 b(N)3508 4278 y Gc(r)549 4377 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11181
Gg(and)21 b(use)j FW(M)t(A)t(X)t(R)t(E)t(D)1184 4391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11182
y Gc(r)1222 4377 y F4(\()p Ga(a)p F4(\))d Gg(to)g(denote)i(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11183
(longest)i(reduction)h(sequence)f(starting)g(from)f Ga(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11184
Gg(\(the)549 4490 y(annotation)26 b Ga(r)f Gg(indicates)h(to)e(which)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11185
(reduction)i(these)f(notions)g(refer)f(to\).)29 b(Correspondingly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11186
b(,)549 4603 y(a)25 b(reduction)j(system)e(is)g(said)g(to)f(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11187
(strongly)i(normalising)g(if)d(and)h(only)h(if)e(all)h(elements)h(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11188
549 4716 y FY(A)22 b Gg(are)i(strongly)i(normalising.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11189
4976 y(Cut-reductions)h(come)22 b(in)h(tw)o(o)f(\003a)n(v)n(ours)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11190
(depending)g(on)e(whether)h(the)o(y)f(apply)h(to)e F7(lo)o(gical)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11191
(cuts)321 5088 y Gg(or)j F7(commuting)i(cuts)p Gg(.)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11192
b(A)27 b(cut)i(is)e(said)i(to)f(be)g(a)g(logical)h(cut)g(when)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11193
(cut-formula)j(is)d(introduced)321 5201 y(on)d(both)h(sides)f(by)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11194
(axioms)h(or)e(logical)j(inference)g(rules;)f(otherwise)h(the)e(cut)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11195
(is)f(said)i(to)e(be)h(a)g(com-)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11196
%%Page: 13 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11197
TeXDict begin 13 24 bop Black 277 51 a Gb(2.1)23 b(Gentzen')m(s)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11198
(Sequent)e(Calculi)2442 b(13)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11199
277 317 a Gg(muting)24 b(cut.)29 b(An)23 b(instance)j(of)d(a)g(logical)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11200
j(cut)d(is)h(as)f(follo)n(ws.)p Black Black 929 523 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11201
F4(\000)986 537 y F9(1)p 1046 511 11 41 v 1056 493 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11202
5 v 1122 523 a F4(\001)1198 537 y F9(1)1237 523 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11203
b(B)96 b F4(\000)1499 537 y F9(1)p 1558 511 11 41 v 1569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11204
493 46 5 v 1635 523 a F4(\001)1711 537 y F9(1)1750 523
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11205
y Ga(;)15 b(C)p 929 560 933 4 v 1119 639 a F4(\000)1176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11206
653 y F9(1)p 1235 627 11 41 v 1246 609 46 5 v 1311 639
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11207
a F4(\001)1387 653 y F9(1)1426 639 y Ga(;)g(B)5 b F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11208
Ga(C)1903 579 y F6(^)1964 593 y Gc(R)2176 523 y Ga(B)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11209
b F4(\000)2347 537 y F9(2)p 2406 511 11 41 v 2417 493
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11210
46 5 v 2482 523 a F4(\001)2558 537 y F9(2)p 2112 560
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11211
549 4 v 2112 639 a Ga(B)5 b F6(^)o Ga(C)q(;)15 b F4(\000)2409
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11212
653 y F9(2)p 2470 627 11 41 v 2480 609 46 5 v 2546 639
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11213
a F4(\001)2622 653 y F9(2)2703 574 y F6(^)2763 588 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11214
Gc(L)2811 597 y FV(1)p 1119 677 1543 4 v 1590 756 a F4(\000)1647
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11215
770 y F9(1)1686 756 y Ga(;)g F4(\000)1783 770 y F9(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11216
1843 744 11 41 v 1853 725 46 5 v 1919 756 a F4(\001)1995
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11217
770 y F9(1)2034 756 y Ga(;)g F4(\001)2150 770 y F9(2)2703
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11218
707 y Gg(Cut)277 975 y(Here)21 b(the)h(cut-formula,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11219
Ga(B)5 b F6(^)o Ga(C)i Gg(,)21 b(is)g(the)h(main)f(formula)i(in)e(both)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11220
h(the)g F6(^)2519 989 y Gc(R)2597 975 y Gg(and)g F6(^)2810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11221
989 y Gc(L)2858 998 y FV(1)2897 975 y Gg(-rule.)28 b(According)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11222
1088 y(to)c(Gentzen')-5 b(s)25 b(procedure)h(this)e(cut)g(reduces)h(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11223
p Black Black 1336 1269 a F4(\000)1393 1283 y F9(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11224
1453 1257 11 41 v 1463 1239 46 5 v 1529 1269 a F4(\001)1605
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11225
1283 y F9(1)1644 1269 y Ga(;)15 b(B)96 b(B)5 b(;)15 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11226
F4(\000)2020 1283 y F9(2)p 2079 1257 11 41 v 2089 1239
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11227
46 5 v 2155 1269 a F4(\001)2231 1283 y F9(2)p 1336 1306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11228
935 4 v 1503 1385 a F4(\000)1560 1399 y F9(1)1599 1385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11229
y Ga(;)g F4(\000)1696 1399 y F9(2)p 1756 1373 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11230
1767 1355 46 5 v 1832 1385 a F4(\001)1908 1399 y F9(1)1948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11231
1385 y Ga(;)g F4(\001)2064 1399 y F9(2)2312 1337 y Gg(Cut)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11232
1580 y(where)26 b(the)f(de)o(gree)h(of)f(the)h(cut-formula)h(has)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11233
(decreased)i(\(the)d(de)o(gree)i(of)e(a)f(formula)i(is)f(de\002ned)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11234
(as)277 1693 y(usual\).)k(Another)25 b(instance)g(of)f(a)f(logical)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11235
(cut)f(is)f(as)h(follo)n(ws.)p Black Black 1236 1898
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11236
a F4(\000)p 1313 1886 11 41 v 1323 1868 46 5 v 96 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11237
Ga(;)15 b(B)p 1170 1936 475 4 v 1170 2015 a F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11238
1247 2003 11 41 v 1257 1984 46 5 v 96 w(\001)p Ga(;)g(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11239
b F6(_)o Ga(C)1686 1950 y F6(_)1747 1964 y Gc(R)1800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11240
1973 y FV(1)p 1929 1936 508 4 v 1929 2015 a Ga(B)g F6(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11241
Ga(C)p 2155 2003 11 41 v 2166 1984 46 5 v 103 w(B)g F6(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11242
Ga(C)p 1170 2052 1268 4 v 1566 2131 a F4(\000)p 1643
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11243
2119 11 41 v 1653 2101 46 5 v 96 w(\001)p Ga(;)15 b(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11244
b F6(_)o Ga(C)2478 2083 y Gg(Cut)277 2351 y(Here)24 b(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11245
(cut-formula)i(is)e(introduced)j(by)d(a)f(logical)j(rule)e(and)h(by)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11246
(an)g(axiom.)30 b(Clearly)-6 b(,)25 b(this)f(appli-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11247
2463 y(cation)h(of)e(the)h(cut)g(can)g(disappear)l(,)i(to)e(yield)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11248
(the)g(follo)n(wing)h(proof.)p Black Black 1621 2669
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11249
a F4(\000)p 1698 2657 11 41 v 1709 2639 46 5 v 96 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11250
Ga(;)15 b(B)p 1555 2707 475 4 v 1555 2785 a F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11251
1632 2773 11 41 v 1643 2755 46 5 v 96 w(\001)p Ga(;)g(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11252
b F6(_)p Ga(C)2072 2720 y F6(_)2132 2734 y Gc(R)2185
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11253
2743 y FV(1)418 3005 y Gg(In)31 b(the)h(cases)g(where)g(a)e(cut)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11254
(cannot)h(be)e(transformed)j(by)d(a)g(cut-reduction)k(for)c(logical)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11255
(cuts,)277 3118 y(then)28 b(the)g(cut)g(is)g(a)f(commuting)i(cut,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11256
(for)g(which)g(proof)h(transformations)i(are)d(de\002ned)g(permuting)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11257
277 3231 y(the)c(cut)g(to)n(w)o(ards)g(the)g(lea)n(v)o(es.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11258
b(A)22 b(typical)j(instance)h(of)d(a)h(commuting)g(cut)g(is)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11259
(proof)1011 3431 y Ga(B)5 b(;)15 b F4(\000)1182 3445
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11260
y F9(1)p 1242 3419 11 41 v 1252 3400 46 5 v 1318 3431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11261
a F4(\001)1394 3445 y F9(1)1433 3431 y Ga(;)g(C)q(;)g(D)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11262
996 3468 678 4 v 996 3547 a F4(\000)1053 3561 y F9(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11263
1113 3535 11 41 v 1123 3517 46 5 v 1189 3547 a F4(\001)1265
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11264
3561 y F9(1)1304 3547 y Ga(;)g(B)5 b F6(\033)p Ga(C)q(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11265
b(D)1715 3486 y F6(\033)1785 3500 y Gc(R)1995 3431 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11266
Ga(D)s(;)g(E)5 b(;)15 b F4(\000)2282 3445 y F9(2)p 2342
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11267
3419 11 41 v 2353 3400 46 5 v 2418 3431 a F4(\001)2494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11268
3445 y F9(2)p 1934 3468 661 4 v 1934 3547 a Ga(D)s(;)g(E)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11269
b F6(^)p Ga(F)s(;)15 b F4(\000)2343 3561 y F9(2)p 2403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11270
3535 11 41 v 2413 3517 46 5 v 2479 3547 a F4(\001)2555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11271
3561 y F9(2)2636 3482 y F6(^)2696 3496 y Gc(L)2744 3505
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11272
y FV(1)p 996 3584 1599 4 v 1250 3663 a Ga(E)5 b F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11273
Ga(F)s(;)15 b F4(\000)1541 3677 y F9(1)1581 3663 y Ga(;)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11274
F4(\000)1678 3677 y F9(2)p 1737 3651 11 41 v 1748 3633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11275
46 5 v 1814 3663 a F4(\001)1890 3677 y F9(1)1929 3663
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11276
y Ga(;)g F4(\001)2045 3677 y F9(2)2084 3663 y Ga(;)g(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11277
b F6(\033)p Ga(C)2636 3615 y Gg(Cut)p Black 3328 3663
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11278
a(\(2.1\))p Black 277 3846 a(whose)25 b(cut-formula,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11279
Ga(D)s Gg(,)c(is)g(not)h(introduced)j(by)d(the)g(inference)j(rules)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11280
(directly)i(abo)o(v)o(e)e(the)h(cut.)32 b(In)277 3959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11281
y(this)e(case,)h(the)f(cut)f(is)g(permuted)i(upw)o(ards)f(in)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11282
(proof)i(yielding)g(either)f(one)g(of)f(the)h(follo)n(wing)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11283
4072 y(tw)o(o)23 b(proofs.)p Black Black 280 4394 a Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11284
b(;)15 b F4(\000)451 4408 y F9(1)p 510 4382 11 41 v 521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11285
4364 46 5 v 586 4394 a F4(\001)662 4408 y F9(1)702 4394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11286
y Ga(;)g(C)q(;)g(D)1063 4278 y(D)s(;)g(E)5 b(;)15 b F4(\000)1350
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11287
4292 y F9(2)p 1411 4266 11 41 v 1421 4247 46 5 v 1487
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11288
4278 a F4(\001)1563 4292 y F9(2)p 1002 4315 661 4 v 1002
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11289
4394 a Ga(D)s(;)g(E)5 b F6(^)q Ga(F)s(;)15 b F4(\000)1412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11290
4408 y F9(2)p 1471 4382 11 41 v 1482 4364 46 5 v 1547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11291
4394 a F4(\001)1623 4408 y F9(2)1689 4329 y F6(^)1750
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11292
4343 y Gc(L)1798 4352 y FV(1)p 280 4431 1383 4 v 441
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11293
4510 a Ga(B)5 b(;)15 b(E)5 b F6(^)p Ga(F)s(;)15 b F4(\000)846
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11294
4524 y F9(1)886 4510 y Ga(;)g F4(\000)983 4524 y F9(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11295
1043 4498 11 41 v 1053 4480 46 5 v 1119 4510 a F4(\001)1195
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11296
4524 y F9(1)1234 4510 y Ga(;)g F4(\001)1350 4524 y F9(2)1389
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11297
4510 y Ga(;)g(C)1689 4462 y Gg(Cut)p 426 4548 1091 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11298
v 426 4627 a Ga(E)5 b F6(^)p Ga(F)s(;)15 b F4(\000)717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11299
4641 y F9(1)757 4627 y Ga(;)g F4(\000)854 4641 y F9(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11300
914 4615 11 41 v 924 4596 46 5 v 990 4627 a F4(\001)1066
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11301
4641 y F9(1)1105 4627 y Ga(;)g F4(\001)1221 4641 y F9(2)1260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11302
4627 y Ga(;)g(B)5 b F6(\033)p Ga(C)1558 4565 y F6(\033)1629
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11303
4579 y Gc(R)1911 4278 y Ga(B)g(;)15 b F4(\000)2082 4292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11304
y F9(1)p 2141 4266 11 41 v 2151 4247 46 5 v 2217 4278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11305
a F4(\001)2293 4292 y F9(1)2332 4278 y Ga(;)g(C)q(;)g(D)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11306
1895 4315 678 4 v 1895 4394 a F4(\000)1952 4408 y F9(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11307
2012 4382 11 41 v 2022 4364 46 5 v 2088 4394 a F4(\001)2164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11308
4408 y F9(1)2203 4394 y Ga(;)g(B)5 b F6(\033)p Ga(C)q(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11309
b(D)2599 4333 y F6(\033)2669 4347 y Gc(R)2803 4394 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11310
Ga(D)s(;)g(E)5 b(;)15 b F4(\000)3090 4408 y F9(2)p 3150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11311
4382 11 41 v 3161 4364 46 5 v 3226 4394 a F4(\001)3302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11312
4408 y F9(2)p 1895 4431 1447 4 v 2134 4510 a Ga(E)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11313
b F4(\000)2303 4524 y F9(1)2343 4510 y Ga(;)g F4(\000)2440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11314
4524 y F9(2)p 2500 4498 11 41 v 2510 4480 46 5 v 2576
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11315
4510 a F4(\001)2652 4524 y F9(1)2691 4510 y Ga(;)g F4(\001)2807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11316
4524 y F9(2)2847 4510 y Ga(;)g(B)5 b F6(\033)p Ga(C)3368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11317
4462 y Gg(Cut)p 2073 4548 1091 4 v 2073 4627 a Ga(E)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11318
F6(^)p Ga(F)s(;)15 b F4(\000)2364 4641 y F9(1)2404 4627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11319
y Ga(;)g F4(\000)2501 4641 y F9(2)p 2561 4615 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11320
2571 4596 46 5 v 2637 4627 a F4(\001)2713 4641 y F9(1)2752
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11321
4627 y Ga(;)g F4(\001)2868 4641 y F9(2)2908 4627 y Ga(;)g(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11322
b F6(\033)o Ga(C)3205 4561 y F6(^)3266 4575 y Gc(L)3314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11323
4584 y FV(1)277 4846 y Gg(Usually)-6 b(,)23 b(it)e(is)g(left)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11324
(unspeci\002ed)j(which)d(alternati)n(v)o(e)j(is)d(tak)o(en,)i(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11325
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11326
(non-determinism.)32 b(This)22 b(is)g(a)g(point)h(we)e(shall)i(study)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11327
(more)g(closely)g(in)f(the)h(follo)n(wing)277 5072 y(sections)j(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11328
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11329
(cut-elimination)j(theorem)e(lie)f(in)g(the)g(f)o(act)g(that)h(one)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11330
(has)g(to)g(sho)n(w)g(that)277 5314 y(the)k(process)i(of)e(applying)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11331
(cut-reductions)i(terminates;)i(this)33 b(is)f(complicated)i(in)e(LK)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11332
(and)i(other)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11333
%%Page: 14 26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11334
TeXDict begin 14 25 bop Black -144 51 a Gb(14)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11335
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(sequent)k(calculi)f(by)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11336
e(the)h(presence)i(of)d(contractions.)33 b(Consider)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11337
b(the)f(follo)n(wing)h(commuting)f(cut.)1170 521 y F4(\000)1227
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11338
535 y F9(1)p 1287 509 11 41 v 1297 491 46 5 v 1363 521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11339
a F4(\001)1439 535 y F9(1)1478 521 y Ga(;)15 b(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11340
b(B)p 1170 559 536 4 v 1227 637 a F4(\000)1284 651 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11341
F9(1)p 1344 625 11 41 v 1354 607 46 5 v 1420 637 a F4(\001)1496
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11342
651 y F9(1)1535 637 y Ga(;)g(B)1747 583 y Gg(Contr)1953
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11343
597 y Gc(R)2103 637 y Ga(B)5 b(;)15 b F4(\000)2274 651
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11344
y F9(2)p 2333 625 11 41 v 2344 607 46 5 v 2409 637 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11345
F4(\001)2485 651 y F9(2)p 1227 675 1298 4 v 1576 754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11346
a F4(\000)1633 768 y F9(1)1672 754 y Ga(;)g F4(\000)1769
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11347
768 y F9(2)p 1829 742 11 41 v 1839 724 46 5 v 1905 754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11348
a F4(\001)1981 768 y F9(1)2020 754 y Ga(;)g F4(\001)2136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11349
768 y F9(2)2566 705 y Gg(Cut)p Black 3372 754 a(\(2.2\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11350
Black 321 942 a(This)24 b(cut)f(reduces)j(to)981 1145
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11351
y F4(\000)1038 1159 y F9(1)p 1097 1133 11 41 v 1107 1115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11352
46 5 v 1173 1145 a F4(\001)1249 1159 y F9(1)1288 1145
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11353
y Ga(;)15 b(B)5 b(;)15 b(B)96 b(B)5 b(;)15 b F4(\000)1778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11354
1159 y F9(2)p 1837 1133 11 41 v 1848 1115 46 5 v 1914
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11355
1145 a F4(\001)1990 1159 y F9(2)p 981 1183 1049 4 v 1148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11356
1262 a F4(\000)1205 1276 y F9(1)1244 1262 y Ga(;)g F4(\000)1341
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11357
1276 y F9(2)p 1401 1250 11 41 v 1411 1231 46 5 v 1477
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11358
1262 a F4(\001)1553 1276 y F9(1)1592 1262 y Ga(;)g F4(\001)1708
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11359
1276 y F9(2)1748 1262 y Ga(;)g(B)2070 1213 y Gg(Cut)2293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11360
1262 y Ga(B)5 b(;)15 b F4(\000)2464 1276 y F9(2)p 2523
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11361
1250 11 41 v 2533 1231 46 5 v 2599 1262 a F4(\001)2675
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11362
1276 y F9(2)p 1148 1299 1567 4 v 1485 1378 a F4(\000)1542
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11363
1392 y F9(1)1581 1378 y Ga(;)g F4(\000)1678 1392 y F9(2)1718
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11364
1378 y Ga(;)g F4(\000)1815 1392 y F9(2)p 1875 1366 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11365
41 v 1885 1348 46 5 v 1951 1378 a F4(\001)2027 1392 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11366
F9(1)2066 1378 y Ga(;)g F4(\001)2182 1392 y F9(2)2221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11367
1378 y Ga(;)g F4(\001)2337 1392 y F9(2)2756 1330 y Gg(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11368
1485 1416 893 4 v 1489 1452 884 4 v 1631 1531 a F4(\000)1688
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11369
1545 y F9(1)1727 1531 y Ga(;)g F4(\000)1824 1545 y F9(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11370
1884 1519 11 41 v 1895 1501 46 5 v 1960 1531 a F4(\001)2036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11371
1545 y F9(1)2075 1531 y Ga(;)g F4(\001)2191 1545 y F9(2)2414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11372
1482 y Gg(Contr)p Black 3372 1531 a(\(2.3\))p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11373
1719 a(where)22 b(the)g(right)g(subproof)i(has)e(been)h(duplicated)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11374
(and)e(where)g(the)g(double)h(lines)f(stand)h(for)f(se)n(v)o(eral)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11375
1831 y(contractions.)32 b(The)22 b(reader)h(will)f(see)g(that)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11376
(proof-size)h(of)f(the)f(reduct)i(is)d(v)n(astly)j(bigger)f(than)g(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11377
321 1944 y(proof)k(from)e(which)g(we)f(started.)35 b(In)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11378
b(f)o(act,)f(it)g(is)g(not)h(hard)g(to)f(check)h(that)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11379
(cut-elimination)k(pro-)321 2057 y(cess)g(can)f(result)g(into)h(super)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11380
(-e)o(xponential)k(gro)n(wth)28 b(of)f(the)h(proof-size;)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11381
b(see)28 b(for)f(e)o(xample)i([Girard)321 2170 y(et)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11382
b(al.,)e(1989,)j(P)o(age)e(111].)462 2301 y(When)d(designing)j(a)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11383
(cut-elimination)k(procedure)f(one)f(therefore)g(has)f(to)g(tak)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11384
(care)h(that)f(for)g(e)n(v-)321 2414 y(ery)26 b(sequent)i(proof)f(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11385
(process)i(of)d(applying)k(cut-reductions)h(actually)e(terminates;)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11386
(that)d(means)321 2526 y(it)e(leads)h(to)f(a)f(cut-free)j(proof.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11387
b(In)24 b(this)h(thesis)g(we)e(shall)i(introduce)h(se)n(v)o(eral)f(no)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11388
(v)o(el)g(cut-elimination)321 2639 y(procedures)g(for)d(classical)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11389
(logic.)30 b(The)21 b(design)i(of)f(these)h(procedures)i(is)d(moti)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11390
n(ated)h(by)f(the)g(follo)n(w-)321 2752 y(ing)i(three)h(criteria:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11391
Black 458 2986 a F6(\017)p Black 46 w Gg(\002rst,)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11392
b(the)e(cut-elimination)j(procedure)f(should)f F7(not)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11393
Gg(restrict)h(the)e(collection)k(of)c(normal)549 3099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11394
y(forms)31 b(reachable)j(from)d(a)g(gi)n(v)o(en)h(proof)h(in)e(such)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11395
(a)f(w)o(ay)g(that)h(\223essential\224)i(normal)e(forms)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11396
3212 y(are)23 b(no)h(longer)h(reachable,)p Black 458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11397
3404 a F6(\017)p Black 46 w Gg(second,)f(the)g(cut-elimination)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11398
(procedure)g(should)e(be)e F7(str)l(ongly)j(normalising)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11399
Gg(,)g(i.e.,)c(all)i(pos-)549 3516 y(sible)g(reduction)i(strate)o(gies)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11400
g(should)f(terminate,)g(and)p Black 458 3708 a F6(\017)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11401
Black 46 w Gg(third,)f(the)g(cut-elimination)j(procedure)f(should)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11402
(allo)n(w)f(cuts)g(to)f(pass)i(o)o(v)o(er)e(other)i(cuts.)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11403
3942 y(At)j(the)g(time)g(of)h(writing,)h(we)d(are)i(una)o(w)o(are)g(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11404
f(an)o(y)h(other)g(cut-elimination)j(procedure)f(for)d(a)g(se-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11405
4055 y(quent)37 b(calculus)h(formulation)h(of)c(classical)j(logic)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11406
(that)g(satis\002es)g(all)e(three)i(criteria.)68 b(So)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11407
b(let)h(us)321 4168 y(justify)25 b(these)g(criteria.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11408
4298 y(As)31 b(we)f(ha)n(v)o(e)i(seen)g(abo)o(v)o(e,)h(some)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11409
(cut-reductions)36 b(can)31 b(be)g(applied)i(in)e(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11410
(non-deterministic)321 4411 y(f)o(ashion)36 b(so)d(that)h(applying)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11411
(dif)n(ferent)g(cut-reductions)h(may)d(result)g(in)g(dif)n(ferent)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11412
(normal)g(forms.)321 4524 y(W)l(ith)g(respect)h(to)e(our)h(\002rst)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11413
(criterion,)39 b(most)c(e)o(xisting)h(cut-elimination)i(procedures,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11414
(including)321 4637 y(Gentzen')-5 b(s)29 b(original,)g(are)e(thus)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11415
(quite)h(unsatisf)o(actory)j(since)d(the)o(y)f(terminate)h(only)f(if)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11416
(a)f(particular)321 4750 y(strate)o(gy)20 b(for)f(applying)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11417
(cut-reductions)j(is)18 b(emplo)o(yed\227the)o(y)j(are)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11418
F7(weakly)h(normalising)p Gg(.)30 b(Common)321 4863 y(e)o(xamples)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11419
(being)g(an)e(innermost)i(reduction)h(strate)o(gy)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11420
b(,)36 b(or)d(the)f(elimination)j(of)e(the)f(cut)h(with)f(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11421
4976 y(highest)k(rank.)60 b(An)34 b(unpleasant)i(consequence)i(of)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11422
(these)h(strate)o(gies)h(is)d(that)i(the)o(y)f(surly)h(restrict)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11423
5088 y(the)30 b(number)g(of)f(normal)h(forms)f(reachable)j(from)d(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11424
(gi)n(v)o(en)g(proof.)47 b(As)28 b(we)h(shall)h(demonstrate)i(in)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11425
5201 y(Chapter)f(4,)g(the)g(normal)g(forms)f(reachable)i(from)e(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11426
(proof)h(play)g(ho)n(we)n(v)o(er)f(an)g(important)i(r)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11427
b(\210)-38 b(ole)31 b(in)321 5314 y(in)l(v)o(estigating)h(a)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11428
b(computational)k(interpretation)h(for)c(classical)i(proofs.)44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11429
b(Therefore)29 b(our)f(\002rst)g(tw)o(o)321 5427 y(criteria.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11430
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11431
%%Page: 15 27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11432
TeXDict begin 15 26 bop Black 277 51 a Gb(2.1)23 b(Gentzen')m(s)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11433
(Sequent)e(Calculi)2442 b(15)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11434
418 317 a Gg(As)35 b(a)h(\002rst)g(attempt)h(for)f(a)f(strongly)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11435
(normalising)f(cut-elimination)i(procedure)e(one)f(might)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11436
430 y(tak)o(e)e(simply)g(an)g(unrestricted)i(v)o(ersion)f(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11437
(Gentzen')-5 b(s)36 b(cut-elimination)j(procedure;)j(that)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11438
b(is)f(by)277 543 y(remo)o(ving)21 b(the)f(strate)o(gy)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11439
b(.)29 b(Unfortunately)-6 b(,)23 b(this)d(w)o(ould,)h(as)e(stated)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11440
(earlier)l(,)h(allo)n(w)d(in\002nite)i(reduction)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11441
656 y(sequences,)32 b(one)c(of)h(which)f(is)g(illustrated)j(in)d(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11442
(follo)n(wing)i(e)o(xample)f(gi)n(v)o(en)g(by)f(Gallier)h([1993])277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11443
769 y(and)24 b(Danos)g(et)f(al.)h([1997].)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11444
957 a Gb(Example)f(2.1.3:)p Black 35 w Gg(Consider)i(the)e(proof)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11445
Black Black 1074 1071 213 4 v 1074 1144 a FU(A)p 1154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11446
1132 10 38 v 1164 1116 42 4 v 88 w(A)p 1369 1071 213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11447
4 v 83 w(A)p 1450 1132 10 38 v 1460 1116 42 4 v 89 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11448
1074 1164 509 4 v 1113 1238 a(A)p FT(_)q FU(A)p 1311
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11449
1226 10 38 v 1321 1209 42 4 v 88 w(A;)14 b(A)1623 1181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11450
y FT(_)1679 1193 y FS(L)p 1113 1274 430 4 v 1163 1347
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11451
a FU(A)p FT(_)p FU(A)p 1361 1335 10 38 v 1371 1318 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11452
4 v 89 w(A)1584 1296 y FR(Contr)1797 1308 y FS(R)p 1934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11453
1071 213 4 v 1934 1144 a FU(A)p 2014 1132 10 38 v 2024
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11454
1116 42 4 v 88 w(A)p 2229 1071 213 4 v 83 w(A)p 2310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11455
1132 10 38 v 2319 1116 42 4 v 88 w(A)p 1934 1164 509
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11456
4 v 1973 1238 a(A;)g(A)p 2153 1226 10 38 v 2162 1209
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11457
42 4 v 88 w(A)p FT(^)q FU(A)2483 1181 y FT(^)2538 1193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11458
y FS(R)p 1973 1274 430 4 v 2022 1347 a FU(A)p 2103 1335
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11459
10 38 v 2113 1318 42 4 v 89 w(A)p FT(^)p FU(A)2444 1296
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11460
y FR(Contr)2657 1308 y FS(L)p 1163 1367 1190 4 v 1534
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11461
1440 a FU(A)p FT(_)p FU(A)p 1732 1428 10 38 v 1742 1412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11462
42 4 v 89 w(A)p FT(^)p FU(A)2394 1395 y FR(Cut)277 1607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11463
y Gg(The)22 b(problem)h(lies)g(with)e(the)i(lo)n(wer)f(cut,)g(which)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11464
(needs)g(to)f(be)g(permuted)i(upw)o(ards.)29 b(There)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11465
b(are)h(tw)o(o)277 1720 y(possible)32 b(reductions:)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11466
b(either)30 b(the)g(cut)f(can)h(be)f(permuted)i(upw)o(ards)f(in)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11467
(left)f(proof)i(branch)f(or)277 1833 y(in)d(the)h(right)g(proof)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11468
(branch.)42 b(In)27 b(both)h(cases)h(a)e(subproof)i(needs)g(to)e(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11469
(duplicated.)43 b(If)27 b(one)h(is)f(not)277 1945 y(careful,)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11470
b(applying)f(these)e(reductions)j(in)c(alternation)k(can)d(lead)g(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11471
(arbitrary)i(big)e(normal)g(forms)277 2058 y(and)e(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11472
(non-termination.)47 b(F)o(or)28 b(e)o(xample,)i(consider)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11473
(reduction)i(sequence)g(starting)f(with)e(the)277 2171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11474
y(proof)d(abo)o(v)o(e)f(and)g(continuing)i(as)e(follo)n(ws)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11475
Black Black 730 2395 213 4 v 730 2468 a FU(A)p 811 2456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11476
10 38 v 821 2440 42 4 v 89 w(A)p 1026 2395 213 4 v 83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11477
w(A)p 1107 2456 10 38 v 1116 2440 42 4 v 88 w(A)p 730
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11478
2488 509 4 v 770 2562 a(A)p FT(_)p FU(A)p 968 2550 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11479
38 v 978 2533 42 4 v 89 w(A;)14 b(A)1280 2505 y FT(_)1335
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11480
2517 y FS(L)p 1468 2286 213 4 v 1468 2359 a FU(A)p 1549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11481
2347 10 38 v 1558 2330 42 4 v 88 w(A)p 1763 2286 213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11482
4 v 83 w(A)p 1844 2347 10 38 v 1854 2330 42 4 v 89 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11483
1468 2379 509 4 v 1507 2452 a(A;)g(A)p 1687 2440 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11484
v 1697 2423 42 4 v 89 w(A)p FT(^)p FU(A)2017 2396 y FT(^)2073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11485
2408 y FS(R)p 1507 2488 430 4 v 1557 2562 a FU(A)p 1638
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11486
2550 10 38 v 1647 2533 42 4 v 88 w(A)p FT(^)q FU(A)1978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11487
2510 y Gd(Contr)2168 2522 y FS(L)p 770 2598 1118 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11488
1055 2671 a FU(A)p FT(_)q FU(A)p 1253 2659 10 38 v 1263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11489
2642 42 4 v 88 w(A;)g(A)p FT(^)q FU(A)1929 2625 y Gd(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11490
2300 2395 213 4 v 2300 2468 a FU(A)p 2381 2456 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11491
v 2390 2440 42 4 v 88 w(A)p 2595 2395 213 4 v 83 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11492
2676 2456 10 38 v 2686 2440 42 4 v 89 w(A)p 2300 2488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11493
509 4 v 2339 2562 a(A;)g(A)p 2519 2550 10 38 v 2529 2533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11494
42 4 v 89 w(A)p FT(^)p FU(A)2849 2505 y FT(^)2905 2517
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11495
y FS(R)p 2339 2598 430 4 v 2389 2671 a FU(A)p 2470 2659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11496
10 38 v 2479 2642 42 4 v 88 w(A)p FT(^)q FU(A)2810 2620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11497
y Gd(Contr)3000 2632 y FS(L)p 1055 2707 1665 4 v 1555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11498
2780 a FU(A)p FT(_)p FU(A)p 1753 2768 10 38 v 1763 2752
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11499
42 4 v 89 w(A)p FT(^)p FU(A;)g(A)p FT(^)q FU(A)2761 2735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11500
y Gd(Cut)p 1555 2816 665 4 v 1663 2890 a FU(A)p FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11501
FU(A)p 1861 2878 10 38 v 1871 2861 42 4 v 88 w(A)p FT(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11502
FU(A)2261 2838 y Gd(Contr)2451 2850 y FS(R)277 3056 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11503
Gg(where)34 b(the)f(cut)g(is)g(permuted)i(to)e(the)g(left,)j(creating)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11504
(tw)o(o)e(copies)h(of)f(the)h(right)g(subproof.)60 b(No)n(w)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11505
3169 y(permute)25 b(the)f(upper)g(cut)g(to)g(the)f(right,)i(which)e(gi)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11506
n(v)o(es)h(the)g(follo)n(wing)h(proof.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11507
395 3393 213 4 v 395 3466 a FU(A)p 476 3454 10 38 v 485
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11508
3438 42 4 v 88 w(A)p 690 3393 213 4 v 83 w(A)p 771 3454
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11509
10 38 v 781 3438 42 4 v 89 w(A)p 395 3486 509 4 v 434
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11510
3559 a(A)p FT(_)q FU(A)p 633 3547 10 38 v 642 3531 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11511
4 v 88 w(A;)14 b(A)944 3503 y FT(_)1000 3515 y FS(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11512
1132 3283 213 4 v 1132 3357 a FU(A)p 1213 3345 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11513
v 1223 3328 42 4 v 89 w(A)p 1428 3283 213 4 v 83 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11514
1509 3345 10 38 v 1518 3328 42 4 v 88 w(A)p 1132 3377
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11515
509 4 v 1172 3450 a(A)p FT(_)p FU(A)p 1370 3438 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11516
v 1380 3421 42 4 v 89 w(A;)g(A)1682 3394 y FT(_)1737
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11517
3406 y FS(L)p 1870 3283 213 4 v 1870 3357 a FU(A)p 1951
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11518
3345 10 38 v 1960 3328 42 4 v 88 w(A)p 2166 3283 213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11519
4 v 84 w(A)p 2246 3345 10 38 v 2256 3328 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11520
1870 3377 509 4 v 1909 3450 a(A;)g(A)p 2089 3438 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11521
v 2099 3421 42 4 v 89 w(A)p FT(^)p FU(A)2420 3394 y FT(^)2475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11522
3406 y FS(R)p 1172 3486 1167 4 v 1432 3559 a FU(A)p FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11523
FU(A;)g(A)p 1730 3547 10 38 v 1739 3531 42 4 v 88 w(A;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11524
FT(^)q FU(A)2380 3514 y Gd(Cut)p 434 3596 1645 4 v 825
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11525
3669 a FU(A)p FT(_)p FU(A;)g(A)p FT(_)q FU(A)p 1240 3657
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11526
10 38 v 1250 3640 42 4 v 88 w(A;)g(A;)g(A)p FT(^)q FU(A)2120
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11527
3623 y Gd(Cut)p 825 3705 863 4 v 933 3778 a FU(A)p FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11528
FU(A)p 1132 3766 10 38 v 1141 3750 42 4 v 88 w(A;)g(A;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11529
FT(^)q FU(A)1729 3727 y Gd(Contr)1919 3739 y FS(L)p 933
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11530
3814 647 4 v 983 3888 a FU(A)p FT(_)p FU(A)p 1181 3876
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11531
10 38 v 1191 3859 42 4 v 89 w(A;)g(A)p FT(^)p FU(A)1621
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11532
3837 y FR(Contr)1834 3849 y FS(R)p 2612 3612 213 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11533
2612 3685 a FU(A)p 2693 3673 10 38 v 2703 3656 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11534
89 w(A)p 2908 3612 213 4 v 83 w(A)p 2989 3673 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11535
2998 3656 42 4 v 88 w(A)p 2612 3705 509 4 v 2652 3778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11536
a(A;)g(A)p 2832 3766 10 38 v 2841 3750 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11537
FT(^)q FU(A)3162 3722 y FT(^)3217 3734 y FS(R)p 2652
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11538
3814 430 4 v 2701 3888 a FU(A)p 2782 3876 10 38 v 2792
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11539
3859 42 4 v 89 w(A)p FT(^)p FU(A)3123 3837 y FR(Contr)3336
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11540
3849 y FS(L)p 983 3924 2049 4 v 1675 3997 a FU(A)p FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11541
FU(A)p 1873 3985 10 38 v 1883 3968 42 4 v 88 w(A)p FT(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11542
FU(A;)g(A)p FT(^)q FU(A)3073 3952 y FR(Cut)p 1675 4033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11543
665 4 v 1783 4106 a FU(A)p FT(_)q FU(A)p 1982 4094 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11544
38 v 1991 4078 42 4 v 88 w(A)p FT(^)q FU(A)2381 4055
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11545
y Gd(Contr)2571 4067 y FS(R)277 4273 y Gg(This)22 b(proof)i(contains)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11546
(an)d(instance)j(of)d(the)h(reduction)i(applied)f(in)f(the)g(\002rst)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11547
(step)h(\(bold)g(f)o(ace\).)30 b(Ev)o(en)277 4386 y(w)o(orse,)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11548
b(it)f(is)h(bigger)h(than)f(the)g(proof)h(with)e(which)h(we)f(started,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11549
i(and)f(so)f(in)h(ef)n(fect)g(we)f(can)h(construct)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11550
4499 y(reduction)j(sequences)g(with)e(possibly)i(in\002nitely)f(big)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11551
(normal)g(forms.)418 4753 y(It)g(seems)h(dif)n(\002cult)g(to)f(a)n(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11552
(oid)i(the)f(in\002nite)g(reduction)h(sequence)h(gi)n(v)o(en)e(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11553
(the)h(e)o(xample)g(abo)o(v)o(e)277 4866 y(using)k(an)e(unrestricted)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11554
(Gentzen-lik)o(e)f(formulation)g(of)d(the)h(cut-reductions.)44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11555
b(A)26 b(number)i(of)g(peo-)277 4979 y(ple,)38 b(for)d(e)o(xample)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11556
(Dragalin)h([1988],)g(Herbelin)g([1994],)g(Cichon)g(et)e(al.)h([1996],)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11557
h(Danos)f(et)g(al.)277 5092 y([1997])26 b(and)g(Bittar)f([1999],)h(ha)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11558
(v)o(e)f(managed)i(to)d(de)n(v)o(elop)i(strongly)h(normalising)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11559
(cut-elimination)277 5204 y(procedures,)e(b)n(ut)e(the)o(y)g(all)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11560
(impose)g(f)o(airly)h(strong)g(restrictions)i(on)c(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11561
(cut-reductions.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11562
%%Page: 16 28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11563
TeXDict begin 16 27 bop Black -144 51 a Gb(16)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11564
b(Calculi)p -144 88 3691 4 v Black 462 317 a Gg(Here)j(is)f(one)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11565
(common)g(restriction:)34 b(consider)27 b(the)e(follo)n(wing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11566
(reduction)h(rule,)e(which)g(allo)n(ws)321 430 y(a)e(cut-rule)j(\(Suf)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11567
(\002x)d(2\))g(to)h(pass)g(o)o(v)o(er)g(another)h(cut-rule)g(\(Suf)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11568
(\002x)e(1\).)p Black Black 322 561 a Ga(:)15 b(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11569
464 549 11 41 v 474 531 46 5 v 128 w(:)g(:)g(:)77 b(:)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11570
b(:)g(:)p 878 549 11 41 v 889 531 46 5 v 127 w(:)g(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11571
322 581 754 4 v 530 650 a(:)g(:)g(:)p 671 638 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11572
681 620 46 5 v 127 w(:)g(:)g(:)1102 605 y Gg(Cut)1233
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11573
619 y F9(1)1293 650 y Ga(:)g(:)g(:)p 1434 638 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11574
1445 620 46 5 v 127 w(:)g(:)g(:)p 530 670 1102 4 v 911
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11575
739 a(:)g(:)g(:)p 1053 727 11 41 v 1063 709 46 5 v 128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11576
w(:)g(:)g(:)1658 694 y Gg(Cut)1789 708 y F9(2)1857 692
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11577
y F6(\000)-31 b(\000)f(!)2039 561 y Ga(:)15 b(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11578
2180 549 11 41 v 2191 531 46 5 v 128 w(:)g(:)g(:)77 b(:)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11579
b(:)g(:)p 2595 549 11 41 v 2605 531 46 5 v 127 w(:)g(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11580
2039 581 754 4 v 2246 650 a(:)g(:)g(:)p 2388 638 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11581
v 2398 620 46 5 v 128 w(:)g(:)g(:)2819 605 y Gg(Cut)2950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11582
619 y F9(2)3009 650 y Ga(:)g(:)g(:)p 3151 638 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11583
3161 620 46 5 v 128 w(:)g(:)g(:)p 2246 670 1102 4 v 2628
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11584
739 a(:)g(:)g(:)p 2769 727 11 41 v 2780 709 46 5 v 128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11585
w(:)g(:)g(:)3375 694 y Gg(Cut)3506 708 y F9(1)321 893
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11586
y Gg(Clearly)-6 b(,)25 b(this)f(reduction)i(w)o(ould)f(immediately)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11587
(break)g(strong)g(normalisation)i(because)f(the)e(reduct)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11588
1006 y(is)j(again)g(an)g(instance)i(of)e(this)g(rule,)h(and)f(we)f(can)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11589
h(loop)h(by)e(constantly)k(applying)f(this)f(rule.)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11590
b(Thus)321 1119 y(a)29 b(common)h(restriction)j(is)c(not)h(to)g(allo)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11591
(w)f(a)h(cut-rule)h(to)f(pass)g(o)o(v)o(er)g(another)h(cut-rule)h(in)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11592
(an)o(y)h(cir)n(-)321 1232 y(cumstances.)45 b(Ho)n(we)n(v)o(er)l(,)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11593
b(this)f(has)h(se)n(v)o(eral)g(serious)h(dra)o(wbacks.)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11594
b(In)28 b(the)h(intuitionistic)j(case,)d(for)321 1345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11595
y(e)o(xample,)f(such)g(a)e(restriction)k(limits)d(the)g(correspondence)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11596
k(between)d(cut-elimination)i(and)e(beta-)321 1458 y(reduction.)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11597
b(In)28 b(particular)l(,)33 b(strong)d(normalisation)i(of)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11598
(beta-reduction)33 b(cannot)d(be)f(inferred)i(from)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11599
1571 y(strong)j(normalisation)h(of)d(cut-elimination,)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11600
b(as)32 b(noted)h(by)f(Herbelin)i([1994])f(and)g(by)f(Dyckhof)n(f)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11601
1684 y(and)24 b(Pinto)g([1998].)30 b(Therefore)25 b(our)f(third)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11602
(criterion.)462 1813 y(In)g(the)h(rest)f(of)g(this)g(chapter)i(we)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11603
(shall)i(de)n(v)o(elop)h(strongly)g(normalising)g(cut-elimination)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11604
(pro-)321 1926 y(cedures)36 b(which)f(contain)h(the)e(standard)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11605
(Gentzen-lik)o(e)h(cut-elimination)h(steps)d(for)f(logical)i(cuts)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11606
2039 y(and)26 b(allo)n(w)g(commuting)h(cuts)f(to)g(pass)g(o)o(v)o(er)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11607
(other)g(cuts.)36 b(As)25 b(a)g(pleasing)i(result,)g(we)e(can)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11608
(simulate)321 2152 y(beta-reduction)i(and)d(infer)f(strong)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11609
(normalisation)h(for)d(the)g(simply-typed)j(lambda)e(calculus)h(from)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11610
321 2265 y(the)g(strong)h(normalisation)i(result)e(of)f(one)g(of)g(our)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11611
g(cut-elimination)j(procedures.)39 b(W)-7 b(e)24 b(shall)j(pro)o(v)o(e)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11612
321 2378 y(this)d(result)h(in)f(Chapter)g(3.)462 2507
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11613
y(Danos)i(et)f(al.)g(allo)n(w)h(cut-rules)h(to)f(pass)g(o)o(v)o(er)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11614
(other)h(cut-rules)i(in)d(their)h(strongly)i(normalising)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11615
2620 y(cut-elimination)k(procedure)e(introduced)h(for)d(the)g(sequent)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11616
(calculus)h(LK)2732 2587 y Gc(tq)2822 2620 y Gg([Danos)e(et)g(al.,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11617
(1997,)321 2733 y(Joinet)e(et)d(al.,)h(1998].)29 b(So)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11618
b(this)g(cut-elimination)k(procedure)e(satis\002es)f(our)g(second)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11619
(and)g(third)g(crite-)321 2846 y(rion,)h(b)n(ut)g(as)g(we)e(shall)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11620
(see)g(it)f(violates)j(the)d(\002rst.)31 b(In)24 b(LK)2147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11621
2813 y Gc(tq)2234 2846 y Gg(e)n(v)o(ery)h(formula)g(\(and)g(its)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11622
(subformulae\))321 2959 y(are)j(required)j(to)c(be)h(coloured)j(with)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11623
(either)j(`)p Ga(\()p Gg(')d(or)h(`)p Ga(*)p Gg('.)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11624
b(Here)27 b(is)h(an)g(instance)i(of)e(a)g(cut-rule)i(in)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11625
3072 y(LK)443 3039 y Gc(tq)506 3072 y Gg(.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11626
1291 3156 a FS(\()1293 3230 y FU(A)p 1399 3210 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11627
v 1408 3193 42 4 v 1495 3109 a FS(\()-11 b FQ(\000)g(\000)g(\000)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11628
(\000)1494 3156 y FS(\()1491 3230 y FU(B)1582 3222 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11629
FT(^)1663 3156 y FS(*)1660 3230 y FU(C)1978 3109 y FS(\()g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11630
FQ(\000)g(\000)g(\000)g(\000)1978 3156 y FS(\()1975 3230
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11631
y FU(B)2065 3222 y FT(^)2147 3156 y FS(*)2144 3230 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11632
FU(C)p 2251 3210 10 38 v 2261 3193 42 4 v 2348 3156 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11633
FS(*)2344 3230 y FU(D)p 1291 3250 1125 4 v 1717 3297
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11634
a FS(\()1719 3371 y FU(A)p 1825 3351 10 38 v 1835 3334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11635
42 4 v 1921 3297 a FS(*)1918 3371 y FU(D)2457 3277 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11636
Gd(Cut)321 3513 y Gg(Recall)32 b(the)g(problematic)i(in\002nite)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11637
(reduction)j(sequence)e(in)f(Example)g(2.1.3.)52 b(This)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11638
b(sequence)i(is)321 3612 y(a)n(v)n(oided)e(in)e(LK)856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11639
3579 y Gc(tq)948 3612 y Gg(by)f(de)n(vising)j(a)d(speci\002c)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11640
(protocol)h(for)e(cut-elimination,)35 b(which)30 b(uses)g(the)g(ad-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11641
3712 y(ditional)k(information)g(pro)o(vided)f(by)f(the)g(colours.)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11642
b(If)32 b(in)f(a)g(commuting)i(cut)f(the)g(colour)h(`)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11643
Ga(\()p Gg(')e(is)321 3811 y(attached)25 b(to)e(the)h(cut-formula,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11644
(then)f(the)f(commuting)h(cut)g(is)f(permuted)h(to)f(the)g(left,)h(and)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11645
f(similarly)321 3911 y(for)i(the)g(`)p Ga(*)p Gg('-colour)h(\(hence)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11646
(the)f(use)g(of)f(an)h(arro)n(w)f(to)g(denote)i(a)e(colour!\).)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11647
b(Suppose)26 b(we)d(ha)n(v)o(e)i(the)321 4011 y(follo)n(wing)g(colour)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11648
(annotation)i(for)d(the)f(commuting)i(cut)f(gi)n(v)o(en)g(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11649
(\(2.1\).)1245 4111 y FS(*)1242 4185 y FU(B)p 1351 4165
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11650
10 38 v 1360 4148 42 4 v 1446 4111 a FS(*)1444 4185 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11651
FU(C)1510 4177 y(;)1550 4111 y FS(\()1546 4185 y FU(D)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11652
1178 4213 504 4 v 1197 4374 10 38 v 1206 4358 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11653
1297 4273 a FS(\()-11 b FQ(\000)g(\000)g(\000)g(\000)1292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11654
4321 y FS(*)1289 4394 y FU(B)1379 4386 y FT(\033)1470
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11655
4321 y FS(*)1467 4394 y FU(C)1533 4386 y(;)p 0.75 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11656
0.75 TeXcolorgray 1570 4419 112 154 v 0.75 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11657
Black 1599 4321 a FS(\()1595 4394 y FU(D)p 0.75 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11658
Black 1723 4229 a FT(\033)1788 4241 y FS(R)2125 4111
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11659
y(\()2121 4185 y FU(D)2192 4177 y(;)2232 4111 y FS(*)2229
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11660
4185 y FU(E)p 2337 4165 10 38 v 2347 4148 42 4 v 2017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11661
4213 494 4 v 0.75 TeXcolorgray 0.75 TeXcolorgray 2017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11662
4419 112 154 v 0.75 TeXcolorgray Black 2046 4321 a FS(\()2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11663
4394 y FU(D)p 0.75 TeXcolorgray Black 2129 4386 a(;)2168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11664
4273 y FQ(\000)g(\000)g(\000)g(\000)g FS(*)2168 4321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11665
y(*)2166 4394 y FU(E)2255 4386 y FT(^)2339 4321 y FS(*)2334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11666
4394 y FU(F)p 2441 4374 10 38 v 2451 4358 42 4 v 2552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11667
4225 a FT(^)2607 4238 y FS(L)2653 4246 y FP(1)p 1178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11668
4439 1333 4 v 1541 4500 a FQ(\000)g(\000)g(\000)g(\000)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11669
FS(*)1541 4547 y(*)1538 4621 y FU(E)1628 4613 y FT(^)1712
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11670
4547 y FS(*)1707 4621 y FU(F)p 1814 4601 10 38 v 1823
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11671
4584 42 4 v 1914 4500 a FQ(\000)g(\000)g(\000)g(\000)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11672
FS(*)1909 4547 y(*)1906 4621 y FU(B)1996 4613 y FT(\033)2087
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11673
4547 y FS(*)2085 4621 y FU(C)2552 4467 y Gd(Cut)324 4767
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11674
y Gg(Then)23 b(this)g(cut,)f(because)j(the)d(cut-formula)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11675
Ga(D)g Gg(is)d(annotated)j(with)e(`)p Ga(\()p Gg(',)f(reduces)i(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11676
(the)h(follo)n(wing)321 4880 y(proof)i(instance,)g(only)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11677
b(.)1368 5111 y FS(*)1366 5185 y FU(B)p 1474 5165 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11678
38 v 1484 5148 42 4 v 1570 5111 a FS(*)1567 5185 y FU(C)1633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11679
5177 y(;)1674 5111 y FS(\()1670 5185 y FU(D)1958 4901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11680
y FS(\()1954 4975 y FU(D)2025 4967 y(;)2064 4901 y FS(*)2062
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11681
4975 y FU(E)p 2170 4955 10 38 v 2179 4938 42 4 v 1870
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11682
5003 453 4 v 1874 5111 a FS(\()1870 5185 y FU(D)1941
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11683
5177 y(;)1981 5064 y FQ(\000)-11 b(\000)g(\000)g(\000)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11684
FS(*)1981 5111 y(*)1978 5185 y FU(E)2067 5177 y FT(^)2152
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11685
5111 y FS(*)2146 5185 y FU(F)p 2253 5165 10 38 v 2263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11686
5148 42 4 v 2364 5015 a FT(^)2420 5028 y FS(L)2466 5036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11687
y FP(1)p 1366 5213 958 4 v 1578 5321 a FS(*)1575 5394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11688
y FU(B)1642 5386 y(;)1682 5274 y FQ(\000)g(\000)g(\000)g(\000)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11689
FS(*)1682 5321 y(*)1679 5394 y FU(E)1769 5386 y FT(^)1853
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11690
5321 y FS(*)1848 5394 y FU(F)p 1955 5374 10 38 v 1964
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11691
5358 42 4 v 2050 5321 a FS(*)2048 5394 y FU(C)2364 5240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11692
y Gd(Cut)p 1538 5423 612 4 v 1541 5483 a FQ(\000)g(\000)g(\000)g(\000)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11693
FS(*)1541 5531 y(*)1538 5604 y FU(E)1628 5596 y FT(^)1712
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11694
5531 y FS(*)1707 5604 y FU(F)p 1814 5584 10 38 v 1823
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11695
5568 42 4 v 1914 5483 a FQ(\000)g(\000)g(\000)g(\000)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11696
FS(*)1909 5531 y(*)1906 5604 y FU(B)1996 5596 y FT(\033)2087
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11697
5531 y FS(*)2085 5604 y FU(C)2192 5439 y FT(\033)2256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11698
5451 y FS(R)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11699
%%Page: 17 29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11700
TeXDict begin 17 28 bop Black 277 51 a Gb(2.1)23 b(Gentzen')m(s)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11701
(Sequent)e(Calculi)2442 b(17)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11702
277 451 a Gg(By)29 b(enforcing)j(that)e(commuting)h(cuts)f(can)g(be)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11703
(permuted)h(into)f(one)g(direction)i(only)-6 b(,)32 b(the)e(in\002nite)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11704
277 564 y(reduction)22 b(sequence)f(gi)n(v)o(en)f(in)f(Example)h(2.1.3)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11705
f(cannot)i(be)e(constructed.)31 b(Ho)n(we)n(v)o(er)l(,)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11706
b(there)h(are)g(tw)o(o)277 677 y(anno)o(ying)31 b(restrictions)h(in)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11707
(their)g(cut-elimination)j(procedure)g(for)c(LK)2597
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11708
644 y Gc(tq)2660 677 y Gg(,)h(both)g(of)g(which)g(violate)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11709
790 y(our)24 b(\002rst)f(criterion.)p Black 414 1040
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11710
a F6(\017)p Black 45 w Gg(First,)k(there)g(is)f(a)g(problem)h(with)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11711
(the)h(compositionality)j(of)c(the)h(colour)g(annotation,)j(in)c(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11712
504 1153 y(sense)38 b(that)e(some)g(cuts)h(require)h(the)e(same)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11713
(colour)h(annotation)i(for)e(their)f(cut-formulae:)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11714
1266 y(the)30 b(choice)i(of)d(a)h(colouring)i(can)e(permeate)h(through)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11715
h(a)e(proof.)48 b(In)30 b(particular)l(,)k(the)c(colour)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11716
1378 y(annotation)h(has)d(to)f(respect,)j(using)e(terminology)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11717
(introduced)h(for)c(LK)2832 1346 y Gc(tq)2895 1378 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11718
Gg(,)g F7(identity)j(classes)504 1491 y Gg([Schellinx,)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11719
b(1994,)i(P)o(age)c(107].)61 b(F)o(or)33 b(e)o(xample,)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11720
b(in)c(the)g(follo)n(wing)i(LK-proof)f(with)f(tw)o(o)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11721
1604 y(commuting)25 b(cuts)1208 1940 y(.)1208 1973 y(.)1208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11722
2006 y(.)p 1180 2074 11 41 v 1190 2056 46 5 v 1256 2086
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11723
a Ga(B)p 1709 1802 244 4 v 1709 1881 a(B)p 1802 1869
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11724
11 41 v 1813 1850 46 5 v 101 w(B)1819 1906 y Gg(.)1819
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11725
1940 y(.)1819 1973 y(.)1819 2006 y(.)1602 2086 y Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11726
b(;)15 b F4(\000)p 1793 2074 11 41 v 1803 2056 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11727
96 w(\001)p Ga(;)g(B)p 1109 2123 950 4 v 1412 2202 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11728
F4(\000)p 1489 2190 11 41 v 1500 2172 46 5 v 97 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11729
Ga(;)g(B)2100 2154 y Gg(Cut)2603 2056 y(.)2603 2089 y(.)2603
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11730
2122 y(.)2504 2202 y Ga(B)p 2598 2190 11 41 v 2609 2172
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11731
46 5 v 1412 2240 1313 4 v 1954 2318 a F4(\000)p 2031
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11732
2306 11 41 v 2042 2288 46 5 v 96 w(\001)2766 2270 y Gg(Cut)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11733
2581 y(all)23 b(occurrences)j(of)c Ga(B)27 b Gg(must)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11734
b(in)h(LK)1679 2548 y Gc(tq)1763 2581 y Gg(ha)n(v)o(e)h(the)e(same)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11735
(colour)-5 b(.)30 b(Consequently)-6 b(,)26 b(the)d(normal)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11736
2694 y(forms)d(that)h(arise)f(in)g(LK)e(by)h(permuting)j(both)e(cuts)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11737
(to)n(w)o(ards)f(the)g(axiom,)h(where)f(the)o(y)g(mer)n(ge)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11738
2807 y(into)25 b(a)e(single)i(cut,)e(cannot)i(be)f(obtained)i(in)d(LK)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11739
2053 2774 y Gc(tq)2116 2807 y Gg(.)p Black 414 3011 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11740
F6(\017)p Black 45 w Gg(Second,)29 b(the)f(colour)h(annotation)h(is)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11741
(in)l(v)n(ariant)i(under)e(cut-reductions.)45 b(Thus)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11742
b(whene)n(v)o(er)i(a)504 3124 y(cut)h(is)f(duplicated)j(in)d(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11743
(reduction)j(sequence,)h(the)c(colour)i(annotation)h(pre)n(v)o(ents)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11744
(both)f(in-)504 3237 y(stances)c(from)e(reducing)i(dif)n(ferently)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11745
b(.)33 b(Figure)25 b(2.1)e(gi)n(v)o(es)i(an)f(e)o(xample)h(of)e(such)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11746
(a)f(reduction)504 3350 y(sequence)i(that)e(e)o(xists)h(in)e(LK,)f(b)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11747
(ut)i(not)g(in)f(LK)1999 3317 y Gc(tq)2062 3350 y Gg(.)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11748
3600 y(Making)34 b(the)g(cut-elimination)j(procedure)f(dependent)g(on)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11749
(the)h(colour)g(annotations)j(is)c(a)g(strong)277 3713
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11750
y(restriction:)44 b(certainly)31 b(some)e(normal)h(forms)g(can)f(no)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11751
(longer)g(be)f(reached.)48 b(In)29 b(f)o(act,)i(the)f(colours)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11752
3826 y(constrain)35 b(the)e(cut-elimination)k(procedure)e(to)e(be)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11753
(con\003uent,)j(which)e(is)e(an)h(essential)i(property)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11754
3939 y(in)30 b(the)h(strong)h(normalisation)i(proof)d(of)f(Danos)h(et)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11755
(al.:)42 b(it)30 b(enabled)j(them)d(to)g(e)o(xploit)i(the)f(strong)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11756
4052 y(normalisation)g(result)d(for)g(proof-nets)i(in)d(linear)i(logic)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11757
f([Girard,)g(1987a].)42 b(The)27 b(colours)i(are)f(used)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11758
4164 y(ingeniously)38 b(to)c(map)g(e)n(v)o(ery)h(LK)1383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11759
4131 y Gc(tq)1446 4164 y Gg(-proof)h(to)e(a)g(corresponding)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11760
b(proof-net)e(in)d(linear)h(logic)h(and)277 4277 y(e)n(v)o(ery)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11761
b(cut-elimination)j(step)e(to)e(a)g(series)i(of)e(reductions)k(on)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11762
(proof-nets.)418 4411 y(W)-7 b(e)24 b(shall)h(sho)n(w)f(that)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11763
(colour)g(annotations)j(of)c(LK)2183 4378 y Gc(tq)2269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11764
4411 y Gg(are,)h(in)f(f)o(act,)h(unnecessary)j(to)c(ensure)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11765
4524 y(strong)h(normalisation)h(of)d(cut-elimination.)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11766
b(As)22 b(stated)i(earlier)l(,)h(a)e(pleasing)i(consequence)i(is)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11767
(that,)277 4637 y(in)28 b(general,)j(more)d(normal)g(forms)h(can)f(be)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11768
(reached)i(from)e(a)f(gi)n(v)o(en)i(proof)g(containing)i(cuts.)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11769
b(One)277 4750 y(not-so-pleasing)c(consequence)f(of)d(the)g(generality)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11770
i(of)d(our)h(cut-elimination)j(procedure)f(is)e(that)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11771
4863 y(strong)f(normalisation)h(is)d(more)g(dif)n(\002cult)h(to)f(pro)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11772
(v)o(e;)37 b(it)32 b(cannot,)j(for)e(e)o(xample,)h(be)f(pro)o(v)o(ed)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11773
(by)f(a)277 4976 y(translation)i(into)e(proof-nets.)55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11774
b(In)31 b(the)g(end,)i(we)e(found)h(it)f(e)o(xtremely)i(useful)f(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11775
(de)n(v)o(elop)i(a)e(term)277 5088 y(calculus)d(for)d(sequent)i(deri)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11776
(v)n(ations.)37 b(This)25 b(then)h(allo)n(wed)h(us)e(to)g(adapt)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11777
(directly)g(a)e(po)n(werful)h(proof)277 5201 y(technique)g(from)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11778
(term)f(re)n(writing)i(literature)h([Barbanera)f(and)f(Berardi,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11779
(1994].)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11780
%%Page: 18 30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11781
TeXDict begin 18 29 bop Black -144 51 a Gb(18)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11782
b(Calculi)p -144 88 3691 4 v Black Black -144 508 V -144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11783
3960 4 3452 v 67 731 491 4 v 67 810 a Ga(A)p F6(_)p Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11784
284 798 11 41 v 295 780 46 5 v 96 w(A)p F6(_)p Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11785
743 731 491 4 v 186 w(A)p F6(_)p Ga(A)p 960 798 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11786
v 970 780 46 5 v 96 w(A)p F6(_)p Ga(A)p 67 830 1166 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11787
v 87 915 a F4(\()p Ga(A)p F6(_)p Ga(A)p F4(\))p F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11788
F4(\()p Ga(A)p F6(_)p Ga(A)p F4(\))p 703 903 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11789
713 884 46 5 v 97 w Ga(A)p F6(_)p Ga(A;)15 b(A)p F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11790
Ga(A)1274 848 y F6(_)1335 862 y Gc(L)p 87 957 1127 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11791
v 205 1042 a F4(\()p Ga(A)p F6(_)q Ga(A)p F4(\))p F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11792
F4(\()p Ga(A)p F6(_)p Ga(A)p F4(\))p 822 1030 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11793
832 1012 46 5 v 97 w Ga(A)p F6(_)o Ga(A)1255 981 y Gg(Contr)1461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11794
995 y Gc(R)p 1611 650 233 4 v 1611 729 a Ga(A)p 1699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11795
717 11 41 v 1709 698 46 5 v 96 w(A)p 1934 650 233 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11796
91 w(A)p 2023 717 11 41 v 2033 698 46 5 v 97 w(A)p 1611
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11797
748 557 4 v 1654 827 a(A)p F6(_)o Ga(A)p 1871 815 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11798
41 v 1881 797 46 5 v 97 w(A;)g(A)2208 767 y F6(_)2269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11799
781 y Gc(L)p 1654 865 471 4 v 1708 943 a Ga(A)p F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11800
Ga(A)p 1925 931 11 41 v 1936 913 46 5 v 96 w(A)2165 889
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11801
y Gg(Contr)2371 903 y Gc(R)p 2521 650 233 4 v 2521 729
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11802
a Ga(A)p 2609 717 11 41 v 2620 698 46 5 v 96 w(A)p 2845
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11803
650 233 4 v 92 w(A)p 2933 717 11 41 v 2943 698 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11804
96 w(A)p 2521 748 557 4 v 2564 827 a(A;)g(A)p 2761 815
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11805
11 41 v 2771 797 46 5 v 97 w(A)p F6(^)p Ga(A)3119 767
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11806
y F6(^)3179 781 y Gc(R)p 2564 865 471 4 v 2618 943 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11807
Ga(A)p 2707 931 11 41 v 2717 913 46 5 v 97 w(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11808
Ga(A)3076 889 y Gg(Contr)3282 903 y Gc(L)p 1708 963 1272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11809
4 v 2099 1042 a Ga(A)p F6(_)p Ga(A)p 2316 1030 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11810
v 2326 1012 46 5 v 96 w(A)p F6(^)p Ga(A)3021 994 y Gg(Cut)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11811
FO(?)p 205 1085 2384 4 v 953 1170 a F4(\()p Ga(A)p F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11812
Ga(A)p F4(\))p F6(_)p F4(\()p Ga(A)p F6(_)p Ga(A)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11813
1569 1158 11 41 v 1579 1139 46 5 v 97 w Ga(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11814
Ga(A)2631 1115 y Gg(Cut)1671 1405 y FN(#)p -33 1595 233
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11815
4 v -33 1674 a Ga(A)p 55 1662 11 41 v 66 1643 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11816
97 w(A)p 291 1595 233 4 v 91 w(A)p 379 1662 11 41 v 390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11817
1643 46 5 v 96 w(A)p -33 1694 557 4 v 10 1772 a(A)p F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11818
Ga(A)p 227 1760 11 41 v 238 1742 46 5 v 97 w(A;)g(A)565
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11819
1712 y F6(_)625 1726 y Gc(L)p 10 1810 471 4 v 65 1889
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11820
a Ga(A)p F6(_)o Ga(A)p 282 1877 11 41 v 292 1858 46 5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11821
v 97 w(A)522 1834 y Gg(Contr)728 1848 y Gc(R)p 830 1595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11822
233 4 v 830 1674 a Ga(A)p 919 1662 11 41 v 929 1643 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11823
5 v 97 w(A)p 1154 1595 233 4 v 91 w(A)p 1242 1662 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11824
41 v 1253 1643 46 5 v 96 w(A)p 830 1694 557 4 v 873 1772
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11825
a(A;)g(A)p 1070 1760 11 41 v 1081 1742 46 5 v 97 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11826
F6(^)p Ga(A)1428 1712 y F6(^)1489 1726 y Gc(R)p 873 1810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11827
471 4 v 928 1889 a Ga(A)p 1016 1877 11 41 v 1026 1858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11828
46 5 v 96 w(A)p F6(^)p Ga(A)1385 1834 y Gg(Contr)1591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11829
1848 y Gc(L)p 65 1909 1225 4 v 432 1987 a Ga(A)p F6(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11830
Ga(A)p 649 1975 11 41 v 659 1957 46 5 v 97 w(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11831
Ga(A)1331 1939 y Gg(Cut)g FO(?)p 1735 1595 233 4 v 1735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11832
1674 a Ga(A)p 1823 1662 11 41 v 1834 1643 46 5 v 97 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11833
2059 1595 233 4 v 91 w(A)p 2147 1662 11 41 v 2157 1643
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11834
46 5 v 96 w(A)p 1735 1694 557 4 v 1778 1772 a(A)p F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11835
Ga(A)p 1995 1760 11 41 v 2006 1742 46 5 v 96 w(A;)g(A)2333
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11836
1712 y F6(_)2393 1726 y Gc(L)p 1778 1810 471 4 v 1832
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11837
1889 a Ga(A)p F6(_)p Ga(A)p 2050 1877 11 41 v 2060 1858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11838
46 5 v 97 w(A)2290 1834 y Gg(Contr)2496 1848 y Gc(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11839
2598 1595 233 4 v 2598 1674 a Ga(A)p 2687 1662 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11840
v 2697 1643 46 5 v 97 w(A)p 2922 1595 233 4 v 91 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11841
3010 1662 11 41 v 3021 1643 46 5 v 96 w(A)p 2598 1694
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11842
557 4 v 2641 1772 a(A;)g(A)p 2838 1760 11 41 v 2849 1742
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11843
46 5 v 97 w(A)p F6(^)p Ga(A)3196 1712 y F6(^)3257 1726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11844
y Gc(R)p 2641 1810 471 4 v 2696 1889 a Ga(A)p 2784 1877
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11845
11 41 v 2794 1858 46 5 v 96 w(A)p F6(^)p Ga(A)3153 1834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11846
y Gg(Contr)3359 1848 y Gc(L)p 1832 1909 1225 4 v 2200
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11847
1987 a Ga(A)p F6(_)o Ga(A)p 2417 1975 11 41 v 2427 1957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11848
46 5 v 97 w(A)p F6(^)p Ga(A)3099 1939 y Gg(Cut)g FO(?)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11849
432 2007 2259 4 v 997 2092 a F4(\()p Ga(A)p F6(_)q Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11850
F4(\))p F6(_)p F4(\()p Ga(A)p F6(_)p Ga(A)p F4(\))p 1614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11851
2080 11 41 v 1624 2062 46 5 v 97 w Ga(A)p F6(_)p Ga(A;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11852
F6(_)p Ga(A)2731 2025 y F6(_)2792 2039 y Gc(L)p 997 2135
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11853
1127 4 v 1116 2219 a F4(\()p Ga(A)p F6(_)p Ga(A)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11854
F6(_)p F4(\()p Ga(A)p F6(_)q Ga(A)p F4(\))p 1732 2207
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11855
11 41 v 1743 2189 46 5 v 96 w Ga(A)p F6(_)p Ga(A)2166
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11856
2159 y Gg(Contr)2372 2173 y Gc(R)1671 2454 y FN(#)1686
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11857
2574 y FM(.)1686 2607 y(.)1686 2640 y(.)1671 2828 y FN(#)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11858
2 3018 233 4 v 2 3097 a Ga(A)p 90 3085 11 41 v 101 3067
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11859
46 5 v 97 w(A)p 326 3018 233 4 v 91 w(A)p 414 3085 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11860
41 v 425 3067 46 5 v 96 w(A)p 2 3117 557 4 v 45 3196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11861
a(A;)g(A)p 242 3184 11 41 v 253 3165 46 5 v 97 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11862
F6(^)p Ga(A)600 3135 y F6(^)660 3149 y Gc(R)p 45 3233
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11863
471 4 v 99 3312 a Ga(A)p 188 3300 11 41 v 198 3282 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11864
5 v 97 w(A)p F6(^)p Ga(A)557 3257 y Gg(Contr)763 3271
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11865
y Gc(L)p 860 3018 233 4 v 860 3097 a Ga(A)p 948 3085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11866
11 41 v 959 3067 46 5 v 96 w(A)p 1183 3018 233 4 v 91
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11867
w(A)p 1272 3085 11 41 v 1282 3067 46 5 v 97 w(A)p 860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11868
3117 557 4 v 903 3196 a(A;)g(A)p 1100 3184 11 41 v 1110
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11869
3165 46 5 v 97 w(A)p F6(^)p Ga(A)1457 3135 y F6(^)1518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11870
3149 y Gc(R)p 903 3233 471 4 v 957 3312 a Ga(A)p 1045
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11871
3300 11 41 v 1056 3282 46 5 v 97 w(A)p F6(^)o Ga(A)1414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11872
3257 y Gg(Contr)1620 3271 y Gc(L)p 99 3332 1220 4 v 345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11873
3411 a Ga(A)p F6(_)p Ga(A)p 562 3399 11 41 v 573 3380
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11874
46 5 v 96 w(A)p F6(^)p Ga(A;)g(A)p F6(^)q Ga(A)1360 3350
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11875
y F6(_)1421 3364 y Gc(L)p 345 3448 728 4 v 464 3527 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11876
Ga(A)p F6(_)p Ga(A)p 681 3515 11 41 v 692 3497 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11877
96 w(A)p F6(^)p Ga(A)1114 3472 y Gg(Contr)1320 3486 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11878
Gc(R)p 1717 3018 233 4 v 1717 3097 a Ga(A)p 1806 3085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11879
11 41 v 1816 3067 46 5 v 97 w(A)p 2041 3018 233 4 v 91
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11880
w(A)p 2129 3085 11 41 v 2140 3067 46 5 v 96 w(A)p 1717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11881
3117 557 4 v 1760 3196 a(A)p F6(_)p Ga(A)p 1978 3184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11882
11 41 v 1988 3165 46 5 v 97 w(A;)g(A)2315 3135 y F6(_)2376
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11883
3149 y Gc(L)p 1760 3233 471 4 v 1815 3312 a Ga(A)p F6(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11884
Ga(A)p 2032 3300 11 41 v 2042 3282 46 5 v 97 w(A)2272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11885
3257 y Gg(Contr)2478 3271 y Gc(R)p 2580 3018 233 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11886
2580 3097 a Ga(A)p 2669 3085 11 41 v 2679 3067 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11887
97 w(A)p 2904 3018 233 4 v 91 w(A)p 2992 3085 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11888
3003 3067 46 5 v 96 w(A)p 2580 3117 557 4 v 2623 3196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11889
a(A)p F6(_)p Ga(A)p 2841 3184 11 41 v 2851 3165 46 5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11890
v 97 w(A;)g(A)3178 3135 y F6(_)3239 3149 y Gc(L)p 2623
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11891
3233 471 4 v 2678 3312 a Ga(A)p F6(_)p Ga(A)p 2895 3300
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11892
11 41 v 2905 3282 46 5 v 96 w(A)3135 3257 y Gg(Contr)3341
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11893
3271 y Gc(R)p 1815 3332 1225 4 v 2063 3411 a Ga(A)p F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11894
Ga(A;)g(A)p F6(_)p Ga(A)p 2518 3399 11 41 v 2528 3380
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11895
46 5 v 97 w(A)p F6(^)p Ga(A)3081 3350 y F6(^)3141 3364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11896
y Gc(R)p 2063 3448 728 4 v 2182 3527 a Ga(A)p F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11897
Ga(A)p 2399 3515 11 41 v 2409 3497 46 5 v 96 w(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11898
Ga(A)2832 3472 y Gg(Contr)3038 3486 y Gc(L)p 464 3547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11899
2209 4 v 1005 3632 a F4(\()p Ga(A)p F6(_)p Ga(A)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11900
F6(_)p F4(\()p Ga(A)p F6(_)p Ga(A)p F4(\))p 1621 3620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11901
11 41 v 1631 3601 46 5 v 97 w Ga(A)p F6(^)p Ga(A;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11902
F6(^)p Ga(A)2714 3565 y F6(_)2774 3579 y Gc(L)p 1005
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11903
3674 1127 4 v 1123 3759 a F4(\()p Ga(A)p F6(_)p Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11904
F4(\))p F6(_)q F4(\()p Ga(A)p F6(_)p Ga(A)p F4(\))p 1740
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11905
3747 11 41 v 1750 3729 46 5 v 97 w Ga(A)p F6(^)o Ga(A)2173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11906
3698 y Gg(Contr)2379 3712 y Gc(R)p 3543 3960 4 3452 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11907
-144 3963 3691 4 v 321 4116 a Gg(Figure)23 b(2.1:)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11908
b(The)21 b(displayed)k(LK-proofs)e(are)g(tak)o(en)g(from)f(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11909
(reduction)j(sequence)g(that)d(starts)i(with)321 4229
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11910
y(the)34 b(\002rst)f(proof)h(and)g(ends)g(with)f(the)g(third)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11911
(proof\227a)h(normal)f(form.)58 b(The)32 b(second)j(proof)f(is)g(an)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11912
4342 y(intermediate)e(step.)46 b(The)28 b(cuts)i(in)f(the)g(top)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11913
(proof)g(are)f(eliminated)i(such)f(that)g(\002rst)e(the)i(right)g(sub-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11914
321 4455 y(proof)g(is)f(duplicated)j(creating)e(tw)o(o)f(instances)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11915
(of)e(the)g(cut)g(mark)o(ed)h(with)e(a)g(star)i(\(second)g(proof\).)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11916
4568 y(Subsequently)-6 b(,)25 b(each)d(cop)o(y)h(of)e(this)h(cut)g(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11917
(reduced)i(applying)h(dif)n(ferent)f(reduction)h(rules.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11918
b(This)21 b(re-)321 4681 y(duction)i(sequence)g(is)e(impossible)h(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11919
(a)f(cut-reduction)25 b(system)c(with)g(colour)n(-annotation,)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11920
b(because)321 4794 y(the)e(colours)g(pre)n(v)o(ent)h(the)e(tw)o(o)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11921
(copies)h(of)f(the)g(cut)h(from)f(reducing)i(dif)n(ferently)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11922
b(.)33 b(In)24 b(ef)n(fect,)h(starting)321 4907 y(from)i(the)g(\002rst)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11923
f(proof)i(the)f(gi)n(v)o(en)g(normal)g(form)g(is)f(not)h(reachable)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11924
(in)e(LK)2719 4874 y Gc(tq)2807 4907 y Gg(\(in)g(Section)h(4.4.1)e(we)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11925
321 5020 y(shall)f(sho)n(w)e(that)h(this)g(is)g(an)f(\223essential\224)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11926
k(normal)d(form\).)p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11927
%%Page: 19 31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11928
TeXDict begin 19 30 bop Black 277 51 a Gb(2.1)23 b(Gentzen')m(s)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11929
(Sequent)e(Calculi)2442 b(19)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11930
418 317 a Gg(T)-7 b(o)28 b(be)g(able)h(to)g(present)h(our)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11931
(cut-elimination)j(procedures)g(in)c(a)h(con)l(v)o(enient)i(form,)f(we)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11932
d(shall)277 430 y(introduce)34 b(a)d(non-standard)j(formulation)g(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11933
(the)h(sequent)h(calculus.)53 b(This)31 b(sequent)i(calculus)h(is)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11934
543 y(inspired)d(by)f(a)e(v)n(ariant)j(of)e(Kleene')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11935
b(s)30 b(sequent)h(calculus)g(G3.)45 b(Kleene)30 b([1952a,)i(P)o(age)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11936
(481])i(men-)277 656 y(tioned)f(this)f(v)n(ariant)h(as)f(G3a,)g(b)n(ut)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11937
g(did)g(not)g(formalise)h(it.)41 b(Later)l(,)28 b(a)g(complete)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11938
(formalisation)h(of)e(a)277 769 y(calculus)e(similar)f(to)f(G3a,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11939
(named)i(G3c,)e(w)o(as)h(gi)n(v)o(en)g(by)g(T)m(roelstra)h(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11940
(Schwichtenber)n(g)i([1996].)277 882 y(The)g(distinguishing)k(feature)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11941
(of)e(these)h(sequent)h(calculi)g(is)e(that)g(the)o(y)h(ha)n(v)o(e)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11942
(no)h(e)o(xplicit)g(rules)g(for)277 995 y(weak)o(ening)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11943
b(and)e(contraction.)60 b(In)33 b(our)g(calculus)i(contraction)h(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11944
(weak)o(ening)g(are)f(implicit)h(in)277 1108 y(the)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11945
b(form)g(of)f(the)h(axioms)h(and)f(logical)h(rules,)g(which)f(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11946
(some)n(what)g(analogous)i(to)e(the)g(approach)277 1221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11947
y(tak)o(en)k(by)f(Kleene)h(in)f(G3a.)50 b(On)31 b(one)g(hand,)i(this)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11948
(reduces)h(the)e(number)h(of)f(inference)i(rules)f(and)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11949
1334 y(thus)25 b(simpli\002es)g(the)g(calculus;)i(ho)n(we)n(v)o(er)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11950
(on)h(the)f(other)i(hand,)f(the)f(formalisation)k(of)c(the)h(conte)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11951
(xts)277 1446 y(and)f(inference)i(rules)e(is)g(quite)g(intricate.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11952
1577 y(In)19 b(the)g(sequel)h(we)e(shall)h(re)o(gard)g(conte)o(xts)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11953
(as)d(sets)h(of)g(\(label,formula\))i(pairs,)g(as)d(in)h(type)g(theory)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11954
-6 b(,)277 1690 y(and)29 b F7(not)f Gg(as)g(multisets,)j(as)d(in)g(LK)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11955
(or)i(LJ.)f(This)h(is)g(to)g(a)n(v)n(oid)i(well-kno)n(wn)f(problems)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11956
(reported)g(by)277 1803 y(Lei)n(v)n(ant)e([1979],)j(which)d(w)o(ould)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11957
(break)g(the)f(correspondence)k(between)d(the)g(sequent)g(proofs)h(and)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11958
277 1916 y(our)d(term)g(annotations)k(\(the)c(term)g(annotations)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11959
(will)d(be)g(gi)n(v)o(en)g(in)g(Section)h(2.2\).)39 b(Since)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11960
b(we)e(ha)n(v)o(e)277 2029 y(conte)o(xts)k(on)e(both)h(sides)g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11961
(the)h(turnstile,)i(it)d(is)g(con)l(v)o(enient)j(to)d(separate)i(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11962
(labels)g(into)g F7(names)277 2142 y Gg(and)g F7(co-names)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11963
Gg(;)j Ga(a)p Gg(,)c Ga(b)p Gg(,)g Ga(c)p Gg(,.)14 b(.)g(.)g(stand)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11964
b(for)h(co-names)i(and)e(.)14 b(.)g(.)g(,)21 b Ga(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11965
Gg(,)h Ga(y)s Gg(,)h Ga(z)31 b Gg(for)d(names.)43 b(In)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11966
b(this)g(w)o(ay)-6 b(,)29 b(the)277 2255 y(conte)o(xts)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11967
b(on)e(the)h(left-hand)h(side)f(of)f(the)h(turnstile,)j(called)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11968
(left-conte)o(xts,)k(will)34 b(be)g(b)n(uilt)h(up)f(by)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11969
2367 y(\(name,formula\))i(pairs)f(and)f(on)g(the)g(right-hand)i(side)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11970
(by)f(\(co-name,formula\))j(pairs;)j(we)32 b(shall)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11971
2480 y(refer)24 b(to)g(those)g(conte)o(xts)i(as)d(right-conte)o(xts.)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11972
b(Suppose)25 b(we)d(ha)n(v)o(e)j(the)e(conte)o(xts)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11973
Black Black 1262 2709 a F4(\000)47 b(=)g F6(f)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11974
Ga(x;)15 b(B)5 b F4(\))p Ga(;)15 b F4(\()p Ga(y)s(;)g(C)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11975
b F4(\))p Ga(;)15 b F4(\()p Ga(z)t(;)g(D)s F4(\))p F6(g)51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11976
b Gg(and)1243 2865 y F4(\001)c(=)g F6(f)p F4(\()p Ga(a;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11977
b(E)5 b F4(\))p Ga(;)15 b F4(\()p Ga(b;)g(F)e F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11978
F6(g)277 3097 y Gg(then)20 b(we)e(can)h(form)f(the)h(sequent)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11979
F4(\000)p 1416 3085 11 41 v 1427 3066 46 5 v 96 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11980
Gg(.)26 b(W)l(ith)19 b(some)g(minor)g(ab)n(use)i(of)d(language)j(we)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11981
(say)h F6(f)p Ga(x;)c(y)s(;)g(z)t F6(g)277 3210 y Gg(is)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11982
b(the)f F7(domain)i Gg(of)f F4(\000)p Gg(,)e(written)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11983
Ga(dom)p F4(\(\000\))p Gg(;)h F6(f)p Ga(a;)15 b(b)p F6(g)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11984
b Gg(respecti)n(v)o(ely)i(is)e(the)f(domain)i(of)f F4(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11985
Gg(.)k(In)22 b(the)h(rest)g(of)277 3323 y(the)k(thesis)h(we)e(shall)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11986
(emplo)o(y)g(some)e(shorthand)k(notation)f(for)e(conte)o(xts;)j(rather)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11987
e(than)g(writing)f(for)277 3435 y(e)o(xample)d F6(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11988
F4(\()p Ga(x;)15 b(B)5 b F4(\))p Ga(;)15 b F4(\()p Ga(y)s(;)g(C)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11989
b F4(\))p Ga(;)15 b F4(\()p Ga(z)t(;)g(D)s F4(\))p F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11990
Gg(,)27 b(we)c(shall)h(simply)g(write)g Ga(x)17 b F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11991
Ga(B)5 b(;)15 b(y)20 b F4(:)e Ga(C)q(;)d(z)22 b F4(:)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11992
Ga(D)s Gg(.)418 3566 y(Figure)34 b(2.2)f(gi)n(v)o(es)h(the)f(inference)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11993
i(rules)f(of)g(our)f(sequent)i(calculus.)60 b(Note)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11994
b(that)g(there)h(are)g(a)277 3679 y(number)22 b(of)e(subtleties)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11995
(concerning)g(conte)o(xts)e(implicit)g(in)f(our)g(inference)i(rules.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11996
b(First,)21 b(we)f(assume)277 3792 y(the)29 b(con)l(v)o(ention)j(that)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11997
(a)e(conte)o(xt)i(is)f(ill-formed,)j(if)c(it)h(contains)i(more)d(than)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11998
(one)f(occurrence)j(of)d(a)277 3905 y(name)g(or)g(co-name.)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 11999
b(F)o(or)28 b(e)o(xample,)i(the)f(left-conte)o(xt)j Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12000
b F4(:)h Ga(B)5 b(;)15 b(x)27 b F4(:)g Ga(C)34 b Gg(is)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12001
b(not)g(allo)n(wed.)45 b(Hereafter)l(,)277 4018 y(this)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12002
b(will)e(be)h(referred)i(to)e(as)g(the)g F7(conte)n(xt)i(con)l(vention)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12003
p Gg(,)h(and)e(it)e(will)h(be)g(assumed)h(that)g(all)f(inference)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12004
4131 y(rules)i(ha)n(ving)g(as)e(conte)o(xts)j(sets)e(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12005
(\(label,formula\))k(pairs)d(respect)h(this)g(con)l(v)o(ention.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12006
4261 y(Second,)d(we)d(ha)n(v)o(e)h(the)h(follo)n(wing)g(con)l(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12007
(entions)j(for)c(the)g(commas)g(in)g(Figure)h(2.2:)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12008
b(a)20 b(comma)f(in)277 4374 y(a)24 b(conclusion)j(stands)f(for)e(set)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12009
(union)g(and)g(a)f(comma)g(in)g(a)f(premise)j(stands)f(for)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12010
F7(disjoint)h Gg(set)e(union.)277 4487 y(Consider)h(for)f(e)o(xample)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12011
(the)g F6(\033)1291 4501 y Gc(R)1348 4487 y Gg(-rule.)1468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12012
4689 y Ga(x)17 b F4(:)g Ga(B)5 b(;)15 b F4(\000)p 1770
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12013
4677 11 41 v 1781 4659 46 5 v 96 w(\001)p Ga(;)g(a)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12014
F4(:)f Ga(C)p 1468 4727 674 4 v 1513 4807 a F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12015
1590 4795 11 41 v 1600 4776 46 5 v 96 w(\001)p Ga(;)e(b)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12016
F4(:)h Ga(B)5 b F6(\033)o Ga(C)2183 4744 y F6(\033)2254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12017
4759 y Gc(R)277 5032 y Gg(This)31 b(rule)g(introduces)j(the)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12018
(\(co-name,formula\))j(pair)e Ga(b)e F4(:)h Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12019
F6(\033)p Ga(C)36 b Gg(in)31 b(the)g(conclusion.)54 b(Ho)n(we)n(v)o(er)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12020
l(,)277 5145 y(there)34 b(might)e(be)h(an)f F7(implicit)i(contr)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12021
(action)p Gg(.)58 b(By)32 b(this)h(we)f(mean)g Ga(b)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12022
Gg(might)h(already)h(occur)g(in)e(the)p Black Black eop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12023
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12024
%%Page: 20 32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12025
TeXDict begin 20 31 bop Black -144 51 a Gb(20)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12026
b(Calculi)p -144 88 3691 4 v Black Black 321 229 3226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12027
4 v 321 2138 4 1909 v 1596 321 676 4 v 1596 400 a Ga(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12028
b F4(:)g Ga(B)5 b(;)15 b F4(\000)p 1898 388 11 41 v 1909
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12029
370 46 5 v 96 w(\001)p Ga(;)g(a)j F4(:)f Ga(B)884 591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12030
y F4(\000)p 961 579 11 41 v 971 560 46 5 v 96 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12031
Ga(;)e(a)i F4(:)h Ga(B)p 852 628 515 4 v 852 707 a(x)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12032
F4(:)g F6(:)p Ga(B)5 b(;)15 b F4(\000)p 1215 695 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12033
v 1225 677 46 5 v 96 w(\001)1408 641 y F6(:)1469 655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12034
y Gc(L)2380 591 y Ga(x)j F4(:)f Ga(B)5 b(;)15 b F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12035
2683 579 11 41 v 2693 560 46 5 v 96 w(\001)p 2352 628
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12036
512 4 v 2352 707 a(\000)p 2429 695 11 41 v 2440 677 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12037
5 v 96 w(\001)p Ga(;)g(a)j F4(:)f F6(:)p Ga(B)2905 641
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12038
y F6(:)2966 655 y Gc(R)857 922 y Ga(x)g F4(:)h Ga(B)1038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12039
936 y Gc(i)1066 922 y Ga(;)d F4(\000)p 1183 910 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12040
v 1194 892 46 5 v 96 w(\001)p 769 960 655 4 v 769 1038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12041
a Ga(y)20 b F4(:)d Ga(B)945 1052 y F9(1)985 1038 y F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12042
Ga(B)1114 1052 y F9(2)1154 1038 y Ga(;)e F4(\000)p 1271
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12043
1026 11 41 v 1282 1008 46 5 v 96 w(\001)1465 973 y F6(^)1525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12044
987 y Gc(L)1573 997 y FZ(i)2117 922 y F4(\000)p 2194
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12045
910 11 41 v 2205 892 46 5 v 96 w(\001)p Ga(;)g(a)j F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12046
Ga(B)96 b F4(\000)p 2736 910 11 41 v 2746 892 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12047
96 w(\001)p Ga(;)15 b(b)i F4(:)h Ga(C)p 2117 960 981
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12048
4 v 2321 1038 a F4(\000)p 2398 1026 11 41 v 2408 1008
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12049
46 5 v 96 w(\001)p Ga(;)d(c)i F4(:)h Ga(B)5 b F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12050
Ga(C)3140 978 y F6(^)3200 992 y Gc(R)615 1254 y Ga(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12051
b F4(:)h Ga(B)5 b(;)15 b F4(\000)p 917 1242 11 41 v 928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12052
1223 46 5 v 96 w(\001)90 b Ga(y)20 b F4(:)e Ga(C)q(;)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12053
F4(\000)p 1452 1242 11 41 v 1462 1223 46 5 v 97 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12054
615 1291 989 4 v 821 1370 a Ga(z)22 b F4(:)17 b Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12055
b F6(_)o Ga(C)q(;)15 b F4(\000)p 1245 1358 11 41 v 1256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12056
1340 46 5 v 97 w(\001)1645 1309 y F6(_)1706 1323 y Gc(L)2358
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12057
1253 y F4(\000)p 2435 1241 11 41 v 2445 1222 46 5 v 96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12058
w(\001)p Ga(;)g(a)i F4(:)h Ga(B)2804 1267 y Gc(i)p 2272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12059
1290 646 4 v 2272 1370 a F4(\000)p 2349 1358 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12060
2359 1340 46 5 v 96 w(\001)p Ga(;)d(b)i F4(:)h Ga(B)2709
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12061
1384 y F9(1)2748 1370 y F6(_)p Ga(B)2878 1384 y F9(2)2959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12062
1303 y F6(_)3019 1317 y Gc(R)3072 1327 y FZ(i)610 1585
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12063
y F4(\000)p 687 1573 11 41 v 697 1555 46 5 v 96 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12064
Ga(;)d(a)i F4(:)h Ga(B)95 b(x)17 b F4(:)h Ga(C)q(;)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12065
F4(\000)p 1447 1573 11 41 v 1457 1555 46 5 v 97 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12066
610 1623 989 4 v 810 1701 a Ga(y)20 b F4(:)e Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12067
F6(\033)o Ga(C)q(;)15 b F4(\000)p 1246 1689 11 41 v 1256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12068
1671 46 5 v 97 w(\001)1640 1640 y F6(\033)1711 1654 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12069
Gc(L)2266 1584 y Ga(x)i F4(:)g Ga(B)5 b(;)15 b F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12070
2568 1572 11 41 v 2579 1554 46 5 v 96 w(\001)p Ga(;)g(a)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12071
F4(:)f Ga(C)p 2266 1622 674 4 v 2311 1701 a F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12072
2388 1689 11 41 v 2398 1671 46 5 v 96 w(\001)p Ga(;)e(b)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12073
F4(:)h Ga(B)5 b F6(\033)o Ga(C)2981 1639 y F6(\033)3052
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12074
1653 y Gc(R)1271 1915 y F4(\000)1328 1929 y F9(1)p 1387
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12075
1903 11 41 v 1398 1885 46 5 v 1463 1915 a F4(\001)1539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12076
1929 y F9(1)1578 1915 y Ga(;)15 b(a)j F4(:)f Ga(B)96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12077
b(x)17 b F4(:)g Ga(B)5 b(;)15 b F4(\000)2173 1929 y F9(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12078
2233 1903 11 41 v 2243 1885 46 5 v 2309 1915 a F4(\001)2385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12079
1929 y F9(2)p 1271 1953 1154 4 v 1547 2032 a F4(\000)1604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12080
2046 y F9(1)1644 2032 y Ga(;)g F4(\000)1741 2046 y F9(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12081
1800 2020 11 41 v 1811 2002 46 5 v 1877 2032 a F4(\001)1953
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12082
2046 y F9(1)1992 2032 y Ga(;)g F4(\001)2108 2046 y F9(2)2466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12083
1983 y Gg(Cut)p 3543 2138 4 1909 v 321 2141 3226 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12084
Black 960 2294 a(Figure)24 b(2.2:)30 b(V)-10 b(ariant)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12085
b(of)f(Kleene')-5 b(s)25 b(sequent)g(calculus)h(G3a.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12086
Black Black 321 2725 a(premise.)k(Thus)24 b(the)f(conclusion)k(of)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12087
(the)h F6(\033)1718 2739 y Gc(R)1776 2725 y Gg(-rule)g(is)f(of)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12088
(form)1607 2988 y F4(\000)p 1684 2976 11 41 v 1694 2958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12089
46 5 v 96 w(\001)c F6(\010)g Ga(b)d F4(:)g Ga(B)5 b F6(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12090
Ga(C)321 3232 y Gg(where)33 b F6(\010)f Gg(denotes)i(set)f(union.)57
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12091
b(Note)32 b(that)h Ga(x)h F4(:)g Ga(B)i Gg(and)d Ga(a)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12092
F4(:)g Ga(C)k Gg(are)33 b(not)g(part)g(of)g(the)f(conclusion)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12093
3345 y(because)26 b(the)o(y)e(are)f(intended)j(to)e(be)f(dischar)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12094
(ged.)32 b(Hence)24 b(the)g(premise)h(must)e(be)h(of)f(the)h(form)1526
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12095
3609 y Ga(x)17 b F4(:)h Ga(B)24 b F6(\012)c F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12096
1899 3597 11 41 v 1910 3578 46 5 v 96 w(\001)g F6(\012)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12097
Ga(a)e F4(:)f Ga(C)321 3830 y Gg(where)24 b F6(\012)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12098
Gg(denotes)i(disjoint)h(set)d(union.)462 3965 y(There)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12099
b(is)f(one)h(point)g(w)o(orth)f(mentioning)j(in)d(the)h(cut-rule,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12100
(because)h(it)d(is)g(the)g(only)h(inference)321 4078
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12101
y(rule)33 b(in)g(our)g(sequent)i(calculus)f(that)f(does)h(not)f(share)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12102
(the)f(conte)o(xts,)j(b)n(ut)d(requires)i(that)e(the)g(tw)o(o)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12103
4191 y(conte)o(xts)j(of)d(the)h(premises)h(are)f(joined)h(on)f(each)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12104
(side)h(of)e(the)h(conclusion.)62 b(Thus)34 b(we)f(tak)o(e)h(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12105
4304 y(cut-rule)26 b(to)d(be)h(of)f(the)h(form)1175 4506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12106
y F4(\000)1232 4520 y F9(1)p 1291 4494 11 41 v 1302 4476
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12107
46 5 v 1367 4506 a F4(\001)1443 4520 y F9(1)1503 4506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12108
y F6(\012)c Ga(a)d F4(:)g Ga(B)96 b(x)17 b F4(:)g Ga(B)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12109
b F6(\012)20 b F4(\000)2219 4520 y F9(2)p 2278 4494 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12110
41 v 2289 4476 46 5 v 2354 4506 a F4(\001)2430 4520 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12111
F9(2)p 1175 4540 1295 4 v 1451 4619 a F4(\000)1508 4633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12112
y F9(1)1568 4619 y F6(\010)g F4(\000)1716 4633 y F9(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12113
1775 4607 11 41 v 1786 4589 46 5 v 1851 4619 a F4(\001)1927
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12114
4633 y F9(1)1987 4619 y F6(\010)g F4(\001)2154 4633 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12115
F9(2)2511 4570 y Gg(Cut)26 b Ga(:)321 4863 y Gg(In)i(consequence,)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12116
(this)d(rule)f(is)g(only)h(applicable,)i(if)d(it)f(does)i(not)f(break)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12117
(the)f(conte)o(xt)i(con)l(v)o(ention,)321 4976 y(which)i(can)f(al)o(w)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12118
(ays)h(be)f(achie)n(v)o(ed)i(by)e(renaming)i(some)e(labels)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12119
(appropriately)-6 b(.)56 b(Note,)32 b(ho)n(we)n(v)o(er)l(,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12120
5088 y(we)h(do)g(not)h(require)g(that)g(cuts)g(ha)n(v)o(e)g(to)f(be)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12121
(\223fully\224)i(multiplicati)n(v)o(e:)51 b(the)34 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12122
F4(\000)2851 5102 y Gc(i)2879 5088 y Gg(')-5 b(s)33 b(can)g(share)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12123
(some)321 5201 y(formulae)25 b(\(similarly)g(the)f F4(\001)1258
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12124
5215 y Gc(j)1295 5201 y Gg(')-5 b(s\).)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12125
eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12126
%%Page: 21 33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12127
TeXDict begin 21 32 bop Black 277 51 a Gb(2.2)23 b(Cut-Elimination)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12128
(in)e(Pr)n(opositional)k(Classical)f(Logic)1582 b(21)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12129
277 88 3691 4 v Black 277 317 a Ge(2.2)119 b(Cut-Elimination)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12130
b(in)g(Pr)n(opositional)f(Classical)f(Logic)277 541 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12131
Gg(In)23 b(this)g(section)i(we)d(shall)i(introduce)h(a)d(strongly)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12132
(normalising)h(cut-elimination)g(procedure.)31 b(Usu-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12133
654 y(ally)-6 b(,)29 b(sequent)h(proofs)f(are)f(written)h(pictorially)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12134
(in)c(a)h(tree-lik)o(e)i(f)o(ashion)g(using)f(inference)h(rules)e(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12135
277 767 y(the)k(form)636 729 y Gc(P)681 738 y FV(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12136
636 746 80 4 v 642 798 a Gc(C)756 767 y Gg(or)872 729
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12137
y Gc(P)917 738 y FV(1)971 729 y Gc(P)1016 738 y FV(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12138
872 746 179 4 v 934 798 a Gc(C)1060 767 y Gg(.)51 b(This)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12139
b(is)f(ho)n(we)n(v)o(er)g(a)g(rather)i(space)f(consuming)i(notation.)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12140
b(In)31 b(order)h(to)277 880 y(present)e(our)f(strong)h(normalisation)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12141
(proof)f(in)e(a)g(manageable)j(form,)e(we)f(shall)h(annotate)h(our)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12142
(se-)277 993 y(quent)34 b(proofs)g(with)e(terms)h(and)g(e)o(xpress)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12143
(the)f(cut-elimination)j(procedure)f(as)e(a)f(term)g(re)n(writing)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12144
1105 y(system.)46 b(In)29 b(particular)l(,)34 b(we)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12145
b(are)h(able)h(to)f(e)o(xpress)i(a)d(proof)j(transformation)h(as)d(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12146
(special)i(sort)e(of)277 1218 y(proof)c(substitution.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12147
1348 y(F)o(or)f(sequent)i(calculi,)g(e)n(v)o(en)f(in)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12148
(intuitionistic)j(case,)d(it)g(is)f(not)h(entirely)h(clear)f(what)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12149
(appro-)277 1461 y(priate)31 b(term)e(annotations)k(are.)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12150
b(In)29 b(f)o(act,)j(there)e(are)g(a)f(number)i(of)e(choices,)k(and)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12151
(there)h(is)e(no)h(real)277 1574 y(consensus)j(on)c(the)h(best.)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12152
b(Our)29 b(term)g(calculus)j(encodes)f(the)f(entire)h(structure)h(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12153
(a)g(sequent)j(cal-)277 1687 y(culus)c(proof)h(and)e(thus)h(allo)n(ws)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12154
(us)f(to)g(de\002ne)h(a)e(complete)j(cut-elimination)i(procedure)f(as)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12155
(a)g(term)277 1799 y(re)n(writing)38 b(system.)918 1766
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12156
y F5(2)1022 1799 y Gg(Other)f(proposals)i(use)d(lambda)h(terms)g(as)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12157
(annotation)j([Pottinger,)f(1977,)277 1912 y(Abramsk)o(y)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12158
b(,)32 b(1993,)h(Barendre)o(gt)g(and)f(Ghilezan,)h(2000],)h(which)e(ho)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12159
n(we)n(v)o(er)g(do)g(not)g(fully)h(encode)277 2025 y(the)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12160
b(structure)i(of)d(the)h(sequent)h(proofs)g(and)f(so)g(w)o(ould)g(seem)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12161
g(less)g(useful)h(for)e(our)h(purposes.)p Black 277 2213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12162
a Gb(De\002nition)f(2.2.1)g Gg(\(Ra)o(w)g(T)-6 b(erms\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12163
Gb(:)p Black 33 w Gg(The)23 b(set)h(of)g(ra)o(w)e(terms,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12164
FY(R)p Gg(,)e(is)h(de\002ned)h(by)g(the)g(grammar)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12165
Black 603 2386 a Ga(M)5 b(;)15 b(N)110 b F4(::=)99 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12166
FL(Ax)p F4(\()p Ga(x;)15 b(a)p F4(\))914 b Gg(axiom)967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12167
2499 y F6(j)147 b FL(Cut)p F4(\()1312 2487 y FX(h)1340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12168
2499 y Ga(a)r F4(:)r Ga(B)1491 2487 y FX(i)1518 2499
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12169
y Ga(M)11 b(;)1657 2487 y F9(\()1684 2499 y Ga(x)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12170
Ga(B)1839 2487 y F9(\))1867 2499 y Ga(N)f F4(\))381 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12171
Gg(cut)967 2612 y F6(j)147 b FL(Not)1282 2626 y Gc(R)1340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12172
2612 y F4(\()1375 2600 y F9(\()1403 2612 y Ga(x)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12173
Ga(B)1558 2600 y F9(\))1585 2612 y Ga(M)10 b(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12174
F4(\))560 b Gg(not-right)967 2725 y F6(j)147 b FL(Not)1282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12175
2739 y Gc(L)1334 2725 y F4(\()1369 2713 y FX(h)1397 2725
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12176
y Ga(a)r F4(:)r Ga(B)1548 2713 y FX(i)1575 2725 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12177
b(;)15 b(x)p F4(\))566 b Gg(not-left)967 2838 y F6(j)147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12178
b FL(And)1294 2852 y Gc(R)1352 2838 y F4(\()1387 2826
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12179
y FX(h)1414 2838 y Ga(a)r F4(:)r Ga(B)1565 2826 y FX(i)1593
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12180
2838 y Ga(M)10 b(;)1731 2826 y FX(h)1759 2838 y Ga(b)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12181
F4(:)r Ga(C)1899 2826 y FX(i)1926 2838 y Ga(N)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12182
b(c)p F4(\))243 b Gg(and-right)967 2953 y F6(j)147 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12183
FL(And)1294 2916 y Gc(i)1294 2976 y(L)1346 2953 y F4(\()1381
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12184
2941 y F9(\()1409 2953 y Ga(x)r F4(:)r Ga(B)1564 2941
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12185
y F9(\))1591 2953 y Ga(M)10 b(;)15 b(y)s F4(\))554 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12186
Gg(and-left)2646 2967 y Gc(i)2893 2953 y Ga(i)25 b F4(=)g(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12187
Ga(;)15 b F4(2)967 3067 y F6(j)147 b FL(Or)1239 3030
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12188
y Gc(i)1239 3090 y(R)1296 3067 y F4(\()1331 3055 y FX(h)1359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12189
3067 y Ga(a)r F4(:)r Ga(B)1510 3055 y FX(i)1538 3067
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12190
y Ga(M)10 b(;)15 b(b)p F4(\))616 b Gg(or)n(-right)2639
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12191
3081 y Gc(i)2893 3067 y Ga(i)25 b F4(=)g(1)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12192
b F4(2)967 3180 y F6(j)147 b FL(Or)1239 3194 y Gc(L)1291
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12193
3180 y F4(\()1326 3168 y F9(\()1354 3180 y Ga(x)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12194
Ga(B)1509 3168 y F9(\))1536 3180 y Ga(M)10 b(;)1674 3168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12195
y F9(\()1702 3180 y Ga(y)5 b F4(:)r Ga(C)1851 3168 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12196
F9(\))1878 3180 y Ga(N)10 b(;)15 b(z)t F4(\))284 b Gg(or)n(-left)967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12197
3293 y F6(j)147 b FL(Imp)1284 3315 y Gc(R)1342 3293 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12198
F4(\()1377 3281 y F9(\()1404 3293 y Ga(x)r F4(:)r Ga(B)1559
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12199
3281 y F9(\))q FX(h)1614 3293 y Ga(a)r F4(:)r Ga(C)1763
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12200
3281 y FX(i)1790 3293 y Ga(M)10 b(;)15 b(b)p F4(\))364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12201
b Gg(implication-right)967 3406 y F6(j)147 b FL(Imp)1284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12202
3428 y Gc(L)1336 3406 y F4(\()1371 3394 y FX(h)1399 3406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12203
y Ga(a)r F4(:)r Ga(B)1550 3394 y FX(i)1577 3406 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12204
b(;)1715 3394 y F9(\()1743 3406 y Ga(x)r F4(:)r Ga(C)1896
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12205
3394 y F9(\))1923 3406 y Ga(N)g(;)15 b(y)s F4(\))237
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12206
b Gg(implication-left)277 3578 y(where)30 b Ga(B)j Gg(and)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12207
Ga(C)35 b Gg(are)30 b(types)g(\(formulae\);)35 b Ga(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12208
Gg(,)30 b Ga(y)s Gg(,)f Ga(z)k Gg(are)d(tak)o(en)h(from)e(a)g(set)h(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12209
f(names)h(and)g Ga(a)p Gg(,)g Ga(b)p Gg(,)g Ga(c)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12210
3690 y Gg(from)24 b(a)f(set)g(of)h(co-names.)418 3944
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12211
y(In)34 b(a)f(term)g(we)g(use)g(round)i(brack)o(ets)h(to)d(signify)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12212
(that)f(a)f(name)h(becomes)g(bound)h(and)f(angle)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12213
4057 y(brack)o(ets)f(that)f(a)e(co-name)j(becomes)f(bound.)53
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12214
b(In)31 b(what)g(follo)n(ws)g(we)f(shall)i(often)g(omit)f(the)h(type)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12215
277 4170 y(annotations)g(on)c(bindings,)k(and)c(assume)h(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12216
Ga(F)13 b(N)d F4(\()p Ga(M)g F4(\))28 b Gg(and)h Ga(F)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12217
b(C)7 b F4(\()p Ga(M)j F4(\))28 b Gg(stand,)i(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12218
b(,)32 b(for)277 4283 y(the)27 b(set)f(of)g(free)h(names)g(and)g(free)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12219
(co-names)i(of)e(the)h(term)f Ga(M)10 b Gg(.)36 b(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12220
b(e)26 b(do)g(not)h(gi)n(v)o(e)f(the)h(formal)g(de\002-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12221
4396 y(nitions)d(of)e Ga(F)13 b(N)d F4(\()p 834 4396
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12222
28 4 v 852 4396 V 870 4396 V 65 w(\))22 b Gg(and)h Ga(F)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12223
b(C)7 b F4(\()p 1287 4396 V 1304 4396 V 1322 4396 V 64
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12224
w(\))p Gg(,)22 b(rather)h(gi)n(v)o(e)g(the)f(reader)i(tw)o(o)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12225
(illustrati)n(v)o(e)i(e)o(xamples.)30 b(Consider)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12226
4509 y(the)24 b(follo)n(wing)h(terms.)p Black Black 1205
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12227
4688 a FL(Ax)p F4(\()p Ga(x;)15 b(a)p F4(\))284 b FL(Imp)1947
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12228
4710 y Gc(L)1999 4688 y F4(\()2034 4676 y FX(h)2062 4688
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12229
y Ga(b)2101 4676 y FX(i)2128 4688 y Ga(M)10 b(;)2266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12230
4676 y F9(\()2294 4688 y Ga(y)2342 4676 y F9(\))2369
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12231
4688 y Ga(N)g(;)15 b(z)t F4(\))277 4868 y Gg(Here,)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12232
b Ga(x)f Gg(and)h Ga(z)j Gg(are)c(free)h(names,)h(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12233
Ga(a)f Gg(is)g(a)g(free)h(co-name.)39 b(On)26 b(the)g(other)i(hand,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12234
Ga(y)g Gg(and)f Ga(b)f Gg(are)h(a)277 4980 y(bound)d(name)e(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12235
(bound)i(co-name,)f(respecti)n(v)o(ely)-6 b(.)31 b(F)o(or)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12236
b(con)l(v)o(enience)26 b Ga(F)13 b(N)d F4(\()p 2790 4980
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12237
V 2808 4980 V 2825 4980 V 65 w(\))21 b Gg(and)i Ga(F)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12238
b(C)7 b F4(\()p 3242 4980 V 3259 4980 V 3277 4980 V 64
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12239
w(\))22 b Gg(will)277 5093 y(not)f(be)g(restricted)j(to)c(just)i(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12240
(single)i(term,)f(b)n(ut)h(also)f(be)g(used)h(for)f(sets)g(of)g(terms.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12241
28 b(Notice)21 b(that)h(names)277 5206 y(and)h(co-names)g(are)f(not)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12242
(the)h(same)f(notions)h(as)f(a)g(v)n(ariable)h(in)f(the)g(lambda)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12243
(calculus:)31 b(whilst)22 b(a)g(term)277 5319 y(can)i(be)f(substituted)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12244
j(for)e(a)e(v)n(ariable,)j(a)e(name)g(or)g(a)g(co-name)h(can)g(only)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12245
(be)f(\223renamed\224.)31 b(Re)n(writing)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12246
5384 1290 4 v 383 5439 a F3(2)412 5471 y F2(Pfenning)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12247
b([1995])g(de)n(v)o(eloped)h(similar)e(terms)g(for)h(an)g(encoding)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12248
(of)e(a)h(weakly)g(normalising)g(cut-elimination)g(pro-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12249
5562 y(cedure)f(into)f(LF)-6 b(.)p Black Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12250
eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12251
%%Page: 22 34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12252
TeXDict begin 22 33 bop Black -144 51 a Gb(22)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12253
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(a)e(name)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12254
Ga(x)e Gg(to)i Ga(y)h Gg(in)e Ga(M)30 b Gg(is)20 b(written)h(as)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12255
Ga(M)10 b F4([)p Ga(x)g F6(7!)g Ga(y)s F4(])p Gg(,)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12256
b(and)h(re)n(writing)h(a)d(co-name)j Ga(a)e Gg(to)g Ga(b)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12257
Gg(in)g Ga(M)30 b Gg(is)20 b(written)321 430 y(as)k Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12258
b F4([)p Ga(a)g F6(7!)g Ga(b)p F4(])p Gg(.)28 b(The)c(routine)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12259
(formalisation)h(of)e(these)g(re)n(writing)h(operations)i(is)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12260
(omitted.)462 560 y(T)-7 b(o)26 b(annotate)i(terms)f(to)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12261
(sequent)j(proofs)e(of)g(the)f(calculus)j(gi)n(v)o(en)d(in)h(Figure)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12262
(2.2,)f(we)g(shall)321 673 y(replace)h(e)n(v)o(ery)e(sequent)h(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12263
(the)h(form)g F4(\000)p 1643 661 11 41 v 1653 642 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12264
5 v 96 w(\001)e Gg(with)i(a)f F7(typing)i(judg)o(ement)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12265
Gg(of)d(the)h(form)f F4(\000)3237 661 y Gc(.)3292 673
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12266
y Ga(M)3416 661 y Gc(.)3471 673 y F4(\001)321 786 y Gg(in)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12267
b(which)g Ga(M)41 b Gg(is)31 b(a)h(term,)h F4(\000)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12268
Gg(a)g(left-conte)o(xt)k(and)d F4(\001)e Gg(a)i(right-conte)o(xt.)56
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12269
b(Henceforth)34 b(we)d(shall)i(be)321 898 y(interested)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12270
b(in)e F7(well-typed)h Gg(terms,)f(only;)g(this)g(means)g(those)h(for)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12271
(which)h(there)h(are)e(tw)o(o)g(conte)o(xts,)j F4(\000)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12272
1011 y Gg(and)g F4(\001)p Gg(,)f(such)i(that)f F4(\000)1036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12273
999 y Gc(.)1091 1011 y Ga(M)1214 999 y Gc(.)1269 1011
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12274
y F4(\001)f Gg(holds)i(gi)n(v)o(en)f(the)g(inference)i(rules)f(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12275
(Figure)h(2.3.)35 b(W)-7 b(e)25 b(de\002ne)h(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12276
1124 y(follo)n(wing)f(set)f(of)f(well-typed)j(terms.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12277
Black Black 550 1325 a FY(T)706 1273 y F5(def)713 1325
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12278
y F4(=)891 1224 y FK(n)977 1325 y Ga(M)1122 1220 y FK(\014)1122
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12279
1275 y(\014)1122 1329 y(\014)1200 1325 y Ga(M)35 b F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12280
b FY(R)d Gg(and)i(well-typed)i(by)d(the)h(rules)h(sho)n(wn)e(in)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12281
(Figure)g(2.3)3257 1224 y FK(o)462 1542 y Gg(The)g(de\002nition)h(we)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12282
(gi)n(v)o(e)g(ne)o(xt)h(corresponds)j(to)d(the)g(traditional)i(notion)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12283
(of)d(the)h(main)g(formula)321 1655 y(of)g(an)f(inference)j(rule.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12284
Black 321 1843 a Gb(De\002nition)d(2.2.2:)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12285
458 2035 a F6(\017)p Black 46 w Gg(A)f(term,)h Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12286
b Gg(,)23 b F7(intr)l(oduces)j Gg(the)e(name)f Ga(z)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12287
Gg(or)d(co-name)h Ga(c)p Gg(,)d(if)i(and)g(only)g(if)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12288
Ga(M)33 b Gg(is)24 b(of)f(the)h(form:)p Black Black 807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12289
2205 a(for)g Ga(z)t Gg(:)100 b FL(Ax)o F4(\()p Ga(z)t(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12290
b(c)p F4(\))1107 2318 y FL(Not)1250 2332 y Gc(L)1302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12291
2318 y F4(\()1337 2306 y FX(h)1365 2318 y Ga(a)1413 2306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12292
y FX(i)1440 2318 y Ga(S)5 b(;)15 b(z)t F4(\))1107 2432
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12293
y FL(And)1262 2395 y Gc(i)1262 2455 y(L)1314 2432 y F4(\()1349
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12294
2420 y F9(\()1376 2432 y Ga(x)1428 2420 y F9(\))1456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12295
2432 y Ga(S)5 b(;)15 b(z)t F4(\))1107 2545 y FL(Or)1206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12296
2559 y Gc(L)1258 2545 y F4(\()1293 2533 y F9(\()1321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12297
2545 y Ga(x)1373 2533 y F9(\))1401 2545 y Ga(S)5 b(;)1502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12298
2533 y F9(\()1529 2545 y Ga(y)1577 2533 y F9(\))1605
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12299
2545 y Ga(T)12 b(;)j(z)t F4(\))1107 2658 y FL(Imp)1251
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12300
2680 y Gc(L)1304 2658 y F4(\()1339 2646 y FX(h)1366 2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12301
y Ga(a)1414 2646 y FX(i)1442 2658 y Ga(S)5 b(;)1543 2646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12302
y F9(\()1571 2658 y Ga(x)1623 2646 y F9(\))1650 2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12303
y Ga(T)13 b(;)i(z)t F4(\))2197 2205 y Gg(for)24 b Ga(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12304
Gg(:)100 b FL(Ax)p F4(\()p Ga(z)t(;)15 b(c)p F4(\))2490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12305
2318 y FL(Not)2633 2332 y Gc(R)2691 2318 y F4(\()2726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12306
2306 y F9(\()2753 2318 y Ga(x)2805 2306 y F9(\))2833
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12307
2318 y Ga(S)5 b(;)15 b(c)p F4(\))2490 2430 y FL(And)2645
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12308
2444 y Gc(R)2703 2430 y F4(\()2738 2418 y FX(h)2765 2430
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12309
y Ga(a)2813 2418 y FX(i)2841 2430 y Ga(S)5 b(;)2942 2418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12310
y FX(h)2970 2430 y Ga(b)3009 2418 y FX(i)3036 2430 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12311
Ga(T)13 b(;)i(c)p F4(\))2490 2545 y FL(Or)2590 2508 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12312
Gc(i)2590 2568 y(R)2647 2545 y F4(\()2682 2533 y FX(h)2710
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12313
2545 y Ga(a)2758 2533 y FX(i)2786 2545 y Ga(S)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12314
b(c)p F4(\))2490 2658 y FL(Imp)2635 2680 y Gc(R)2692
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12315
2658 y F4(\()2727 2646 y F9(\()2755 2658 y Ga(x)2807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12316
2646 y F9(\))q FX(h)2862 2658 y Ga(a)2910 2646 y FX(i)2938
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12317
2658 y Ga(S)5 b(;)15 b(c)p F4(\))p Black 458 2825 a F6(\017)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12318
Black 46 w Gg(A)25 b(term,)i Ga(M)10 b Gg(,)27 b F7(fr)m(eshly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12319
Gg(introduces)i(a)c(name,)i(if)e(and)h(only)h(if)e Ga(M)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12320
b Gg(introduces)30 b(this)e(name,)f(b)n(ut)549 2938 y(none)e(of)g(its)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12321
(proper)i(subterms.)34 b(In)24 b(other)i(w)o(ords,)f(the)g(name)f(must)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12322
h(not)g(be)f(free)h(in)g(a)f(proper)549 3050 y(subterm)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12323
Ga(M)10 b Gg(,)22 b(just)i(on)g(the)g(top-le)n(v)o(el)h(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12324
Ga(M)10 b Gg(.)29 b(Similarly)24 b(for)f(co-names.)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12325
3304 y(Let)29 b(us)h(illustrate)i(the)e(de\002nition)i(just)e(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12326
(en)h(and)f(our)g(implicit)h(con)l(v)o(entions)i(for)d(conte)o(xts)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12327
(with)321 3417 y(the)24 b(follo)n(wing)h(e)o(xample:)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12328
b(consider)c(the)e F6(\033)1724 3431 y Gc(R)1782 3417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12329
y Gg(-rule)h(again,)e(this)i(time)e(annotated)j(with)d(terms.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12330
Black Black 1431 3574 a Ga(x)17 b F4(:)g Ga(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12331
b F4(\000)1738 3562 y Gc(.)1793 3574 y Ga(M)1917 3562
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12332
y Gc(.)1972 3574 y F4(\001)p Ga(;)g(a)i F4(:)g Ga(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12333
1195 3611 1308 4 v 1195 3696 a F4(\000)1277 3684 y Gc(.)1332
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12334
3696 y FL(Imp)1476 3718 y Gc(R)1534 3696 y F4(\()1569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12335
3684 y F9(\()1597 3696 y Ga(x)1649 3684 y F9(\))p FX(h)1704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12336
3696 y Ga(a)1752 3684 y FX(i)1779 3696 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12337
b(b)p F4(\))2017 3684 y Gc(.)2072 3696 y F4(\001)p Ga(;)g(b)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12338
F4(:)f Ga(B)5 b F6(\033)o Ga(C)2544 3629 y F6(\033)2615
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12339
3643 y Gc(R)321 3868 y Gg(No)n(w)22 b(this)i(rule)g(introduces)i(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12340
(\(co-name,type\))i(pair)l(,)e Ga(b)18 b F4(:)f Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12341
b F6(\033)o Ga(C)i Gg(,)22 b(in)h(the)h(conclusion,)i(and)e(conse-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12342
3981 y(quently)-6 b(,)24 b Ga(b)d Gg(is)h(a)f(free)h(co-name)h(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12343
FL(Imp)1567 4003 y Gc(R)1625 3981 y F4(\()1660 3969 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12344
F9(\()1688 3981 y Ga(x)1740 3969 y F9(\))p FX(h)1795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12345
3981 y Ga(a)1843 3969 y FX(i)1870 3981 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12346
b(b)p F4(\))p Gg(.)28 b(In)22 b(general,)i(ho)n(we)n(v)o(er)l(,)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12347
Ga(b)f Gg(could)i(already)h(be)321 4094 y(free)g(in)g(the)g(subterm)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12348
Ga(M)10 b Gg(,)23 b(and)h(thus)g(the)g(conclusion)i(of)e(this)g(rule)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12349
(is)f(of)h(the)g(form)1219 4277 y F4(\000)1301 4265 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12350
Gc(.)1356 4277 y FL(Imp)1501 4299 y Gc(R)1558 4277 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12351
F4(\()1593 4265 y F9(\()1621 4277 y Ga(x)1673 4265 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12352
F9(\))q FX(h)1728 4277 y Ga(a)1776 4265 y FX(i)1804 4277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12353
y Ga(M)10 b(;)15 b(b)p F4(\))2042 4265 y Gc(.)2097 4277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12354
y F4(\001)20 b F6(\010)f Ga(b)f F4(:)f Ga(B)5 b F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12355
Ga(C)32 b(:)321 4448 y Gg(In)25 b(case)g(the)g(term)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12356
FL(Imp)1071 4470 y Gc(R)1128 4448 y F4(\()1163 4436 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12357
F9(\()1191 4448 y Ga(x)1243 4436 y F9(\))q FX(h)1298
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12358
4448 y Ga(a)1346 4436 y FX(i)1373 4448 y Ga(M)11 b(;)k(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12359
F4(\))24 b Gg(freshly)i(introduces)i Ga(b)19 b F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12360
Ga(B)5 b F6(\033)o Ga(C)i Gg(,)23 b(then)j(the)f(conclusion)i(is)e(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12361
321 4561 y(the)f(form)1245 4674 y F4(\000)1327 4662 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12362
Gc(.)1382 4674 y FL(Imp)1526 4696 y Gc(R)1584 4674 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12363
F4(\()1619 4662 y F9(\()1646 4674 y Ga(x)1698 4662 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12364
F9(\))q FX(h)1753 4674 y Ga(a)1801 4662 y FX(i)1829 4674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12365
y Ga(M)10 b(;)15 b(b)p F4(\))2067 4662 y Gc(.)2122 4674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12366
y F4(\001)20 b F6(\012)g Ga(b)d F4(:)g Ga(B)5 b F6(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12367
Ga(C)321 4833 y Gg(where)29 b Ga(b)d F4(:)g Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12368
F6(\033)o Ga(C)34 b Gg(is)28 b(not)g(in)g F4(\001)p Gg(.)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12369
b(The)27 b(binders)j Ga(x)e Gg(and)g Ga(a)g Gg(cannot)h(be)g(part)f(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12370
g(the)h(conclusion,)j(and)321 4946 y(therefore)26 b(the)e(premise)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12371
(must)g(be)g(of)f(the)h(form)1420 5136 y Ga(x)17 b F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12372
Ga(B)25 b F6(\012)20 b F4(\000)1798 5124 y Gc(.)1853
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12373
5136 y Ga(M)1976 5124 y Gc(.)2031 5136 y F4(\001)g F6(\012)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12374
Ga(a)d F4(:)h Ga(C)32 b(:)462 5307 y Gg(W)-7 b(e)28 b(add)g(tw)o(o)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12375
(ne)n(w)f(syntactic)j(cate)o(gories)h(of)d(terms,)h(which)f(are)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12376
F7(not)h Gg(proof)g(annotations,)j(b)n(ut)321 5420 y(play)h(an)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12377
(important)i(r)8 b(\210)-38 b(ole)33 b(in)f(the)g(de\002nition)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12378
(substitution)k(and)c(in)g(our)h(strong)g(normalisation)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12379
5533 y(proof.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12380
%%Page: 23 35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12381
TeXDict begin 23 34 bop Black 277 51 a Gb(2.2)23 b(Cut-Elimination)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12382
(in)e(Pr)n(opositional)k(Classical)f(Logic)1582 b(23)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12383
277 88 3691 4 v Black Black 277 229 V 277 2078 4 1849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12384
v 1584 321 1054 4 v 1584 406 a Ga(x)17 b F4(:)h Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12385
b(;)15 b F4(\000)1892 394 y Gc(.)1947 406 y FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12386
Ga(x;)g(a)p F4(\))2286 394 y Gc(.)2341 406 y F4(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12387
Ga(;)g(a)i F4(:)g Ga(B)818 597 y F4(\000)900 585 y Gc(.)954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12388
597 y Ga(M)1078 585 y Gc(.)1133 597 y F4(\001)p Ga(;)e(a)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12389
F4(:)h Ga(B)p 555 634 1138 4 v 555 719 a(x)f F4(:)g F6(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12390
Ga(B)5 b(;)15 b F4(\000)923 707 y Gc(.)978 719 y FL(Not)1121
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12391
733 y Gc(L)1173 719 y F4(\()1208 707 y FX(h)1236 719
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12392
y Ga(a)1284 707 y FX(i)1311 719 y Ga(M)10 b(;)15 b(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12393
F4(\))1562 707 y Gc(.)1617 719 y F4(\001)1734 647 y F6(:)1795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12394
661 y Gc(L)2639 597 y Ga(x)j F4(:)f Ga(B)5 b(;)15 b F4(\000)2947
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12395
585 y Gc(.)3002 597 y Ga(M)3125 585 y Gc(.)3180 597 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12396
F4(\001)p 2378 634 1140 4 v 2378 719 a(\000)2460 707
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12397
y Gc(.)2515 719 y FL(Not)2657 733 y Gc(R)2715 719 y F4(\()2750
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12398
707 y F9(\()2778 719 y Ga(x)2830 707 y F9(\))2857 719
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12399
y Ga(M)c(;)k(a)p F4(\))3105 707 y Gc(.)3159 719 y F4(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12400
Ga(;)g(a)j F4(:)f F6(:)p Ga(B)3559 647 y F6(:)3620 661
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12401
y Gc(R)791 910 y Ga(x)g F4(:)g Ga(B)971 924 y Gc(i)999
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12402
910 y Ga(;)e F4(\000)1122 898 y Gc(.)1177 910 y Ga(M)1300
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12403
898 y Gc(.)1355 910 y F4(\001)p 466 947 1289 4 v 466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12404
1045 a Ga(y)20 b F4(:)e Ga(B)643 1059 y F9(1)682 1045
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12405
y F6(^)p Ga(B)812 1059 y F9(2)851 1045 y Ga(;)d F4(\000)974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12406
1033 y Gc(.)1028 1045 y FL(And)1183 1008 y Gc(i)1183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12407
1068 y(L)1235 1045 y F4(\()1270 1033 y F9(\()1298 1045
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12408
y Ga(x)1350 1033 y F9(\))1377 1045 y Ga(M)10 b(;)15 b(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12409
F4(\))1624 1033 y Gc(.)1679 1045 y F4(\001)1796 960 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12410
F6(^)1857 974 y Gc(L)1905 984 y FZ(i)2212 922 y F4(\000)2294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12411
910 y Gc(.)2349 922 y Ga(M)2472 910 y Gc(.)2527 922 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12412
F4(\001)p Ga(;)g(a)i F4(:)h Ga(B)277 b F4(\000)3179 910
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12413
y Gc(.)3234 922 y Ga(N)3342 910 y Gc(.)3397 922 y F4(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12414
Ga(;)15 b(b)j F4(:)f Ga(C)p 2212 960 1472 4 v 2238 1045
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12415
a F4(\000)2320 1033 y Gc(.)2375 1045 y FL(And)2530 1059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12416
y Gc(R)2587 1045 y F4(\()2622 1033 y FX(h)2650 1045 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12417
Ga(a)2698 1033 y FX(i)2726 1045 y Ga(M)10 b(;)2864 1033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12418
y FX(h)2892 1045 y Ga(b)2931 1033 y FX(i)2958 1045 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12419
Ga(N)g(;)15 b(c)p F4(\))3181 1033 y Gc(.)3236 1045 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12420
F4(\001)p Ga(;)g(c)j F4(:)f Ga(B)5 b F6(^)p Ga(C)3725
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12421
978 y F6(^)3786 992 y Gc(R)384 1248 y Ga(x)17 b F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12422
Ga(B)5 b(;)15 b F4(\000)692 1236 y Gc(.)747 1248 y Ga(M)870
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12423
1236 y Gc(.)925 1248 y F4(\001)272 b Ga(y)20 b F4(:)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12424
Ga(C)q(;)d F4(\000)1570 1236 y Gc(.)1625 1248 y Ga(N)1733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12425
1236 y Gc(.)1788 1248 y F4(\001)p 384 1285 1480 4 v 434
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12426
1370 a Ga(z)21 b F4(:)d Ga(B)5 b F6(_)o Ga(C)q(;)15 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12427
F4(\000)863 1358 y Gc(.)918 1370 y FL(Or)1017 1384 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12428
Gc(L)1069 1370 y F4(\()1104 1358 y F9(\()1132 1370 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12429
Ga(x)1184 1358 y F9(\))1211 1370 y Ga(M)c(;)1350 1358
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12430
y F9(\()1377 1370 y Ga(y)1425 1358 y F9(\))1453 1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12431
y Ga(N)f(;)15 b(z)t F4(\))1683 1358 y Gc(.)1738 1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12432
y F4(\001)1905 1303 y F6(_)1966 1317 y Gc(L)2617 1235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12433
y F4(\000)2699 1223 y Gc(.)2754 1235 y Ga(M)2877 1223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12434
y Gc(.)2932 1235 y F4(\001)p Ga(;)g(a)i F4(:)h Ga(B)3225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12435
1249 y Gc(i)p 2326 1273 1218 4 v 2326 1370 a F4(\000)2408
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12436
1358 y Gc(.)2463 1370 y FL(Or)2562 1333 y Gc(i)2562 1393
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12437
y(R)2620 1370 y F4(\()2655 1358 y FX(h)2683 1370 y Ga(a)2731
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12438
1358 y FX(i)2758 1370 y Ga(M)10 b(;)15 b(b)p F4(\))2996
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12439
1358 y Gc(.)3051 1370 y F4(\001)p Ga(;)g(b)j F4(:)f Ga(B)3335
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12440
1384 y F9(1)3374 1370 y F6(_)p Ga(B)3504 1384 y F9(2)3585
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12441
1286 y F6(_)3646 1300 y Gc(R)3699 1310 y FZ(i)379 1560
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12442
y F4(\000)461 1548 y Gc(.)516 1560 y Ga(M)639 1548 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12443
Gc(.)694 1560 y F4(\001)p Ga(;)e(a)j F4(:)f Ga(B)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12444
b(x)17 b F4(:)h Ga(C)q(;)d F4(\000)1565 1548 y Gc(.)1620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12445
1560 y Ga(N)1728 1548 y Gc(.)1783 1560 y F4(\001)p 379
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12446
1598 1480 4 v 400 1683 a Ga(y)20 b F4(:)d Ga(B)5 b F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12447
Ga(C)q(;)15 b F4(\000)840 1671 y Gc(.)895 1683 y FL(Imp)1040
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12448
1705 y Gc(L)1092 1683 y F4(\()1127 1671 y FX(h)1155 1683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12449
y Ga(a)1203 1671 y FX(i)1230 1683 y Ga(M)10 b(;)1368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12450
1671 y F9(\()1396 1683 y Ga(x)1448 1671 y F9(\))1475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12451
1683 y Ga(N)g(;)15 b(y)s F4(\))1707 1671 y Gc(.)1762
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12452
1683 y F4(\001)1900 1616 y F6(\033)1971 1630 y Gc(L)2525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12453
1560 y Ga(x)i F4(:)g Ga(B)5 b(;)15 b F4(\000)2832 1548
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12454
y Gc(.)2887 1560 y Ga(M)3010 1548 y Gc(.)3065 1560 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12455
F4(\001)p Ga(;)g(a)j F4(:)f Ga(C)p 2289 1598 1308 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12456
2289 1683 a F4(\000)2371 1671 y Gc(.)2426 1683 y FL(Imp)2570
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12457
1705 y Gc(R)2628 1683 y F4(\()2663 1671 y F9(\()2691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12458
1683 y Ga(x)2743 1671 y F9(\))p FX(h)2797 1683 y Ga(a)2845
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12459
1671 y FX(i)2873 1683 y Ga(M)10 b(;)15 b(b)p F4(\))3111
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12460
1671 y Gc(.)3166 1683 y F4(\001)p Ga(;)g(b)i F4(:)h Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12461
b F6(\033)o Ga(C)3638 1616 y F6(\033)3709 1630 y Gc(R)1268
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12462
1873 y F4(\000)1325 1887 y F9(1)1390 1861 y Gc(.)1445
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12463
1873 y Ga(M)1568 1861 y Gc(.)1623 1873 y F4(\001)1699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12464
1887 y F9(1)1738 1873 y Ga(;)15 b(a)j F4(:)f Ga(B)146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12465
b(x)17 b F4(:)h Ga(B)5 b(;)15 b F4(\000)2384 1887 y F9(2)2448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12466
1861 y Gc(.)2503 1873 y Ga(N)2611 1861 y Gc(.)2666 1873
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12467
y F4(\001)2742 1887 y F9(2)p 1268 1911 1514 4 v 1373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12468
1996 a F4(\000)1430 2010 y F9(1)1469 1996 y Ga(;)g F4(\000)1566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12469
2010 y F9(2)1631 1984 y Gc(.)1686 1996 y FL(Cut)p F4(\()1859
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12470
1984 y FX(h)1887 1996 y Ga(a)1935 1984 y FX(i)1962 1996
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12471
y Ga(M)10 b(;)2100 1984 y F9(\()2128 1996 y Ga(x)2180
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12472
1984 y F9(\))2207 1996 y Ga(N)g F4(\))2351 1984 y Gc(.)2406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12473
1996 y F4(\001)2482 2010 y F9(1)2521 1996 y Ga(;)15 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12474
F4(\001)2637 2010 y F9(2)2823 1941 y Gg(Cut)p 3965 2078
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12475
4 1849 v 277 2081 3691 4 v 277 2235 a(Figure)22 b(2.3:)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12476
b(T)-6 b(erm)20 b(assignment)k(for)e(sequent)h(proofs)g(of)f(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12477
(propositional)26 b(fragment)d(of)e(classical)277 2348
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12478
y(logic.)p Black Black 277 2738 a Gb(De\002nition)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12479
b(2.2.3:)p Black 35 w Gg(Let)g Ga(M)38 b Gg(and)30 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12480
Ga(N)38 b Gg(be)29 b(well-typed)i(terms,)g(then)2451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12481
2726 y F9(\()2479 2738 y Ga(x)r F4(:)r Ga(B)2634 2726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12482
y F9(\))2661 2738 y Ga(M)39 b Gg(and)2947 2726 y FX(h)2974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12483
2738 y Ga(a)r F4(:)r Ga(C)3123 2726 y FX(i)3151 2738
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12484
y Ga(N)f Gg(are)29 b(re-)277 2851 y(ferred)f(to)e(as)g(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12485
F7(named)g(term)h Gg(and)f F7(co-named)i(term)p Gg(,)f(respecti)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12486
(ely)-6 b(.)39 b(More)27 b(formally)-6 b(,)28 b(we)d(ha)n(v)o(e)i(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12487
277 2964 y(follo)n(wing)e(tw)o(o)e(f)o(amilies)i(of)e(sets)i(inde)o(x)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12488
(ed)f(by)g(types:)p Black Black 391 3171 a FY(T)448 3190
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12489
y F9(\()p Gc(B)s F9(\))659 3120 y F5(def)666 3171 y F4(=)844
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12490
3070 y FK(n)929 3159 y F9(\()957 3171 y Ga(x)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12491
Ga(B)1112 3159 y F9(\))1139 3171 y Ga(M)1284 3067 y FK(\014)1284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12492
3121 y(\014)1284 3176 y(\014)1362 3171 y Ga(M)35 b F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12493
b FY(T)51 b Gg(with)23 b(the)h(typing)h(judgement)53
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12494
b Ga(x)17 b F4(:)h Ga(B)5 b(;)15 b F4(\000)2994 3159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12495
y Gc(.)3049 3171 y Ga(M)3172 3159 y Gc(.)3227 3171 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12496
F4(\001)3328 3070 y FK(o)392 3382 y FY(T)449 3400 y FX(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12497
Gc(C)5 b FX(i)659 3330 y F5(def)666 3382 y F4(=)844 3281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12498
y FK(n)929 3370 y FX(h)957 3382 y Ga(a)r F4(:)r Ga(C)1106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12499
3370 y FX(i)1133 3382 y Ga(N)1284 3277 y FK(\014)1284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12500
3332 y(\014)1284 3386 y(\014)1362 3382 y Ga(N)50 b F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12501
b FY(T)51 b Gg(with)23 b(the)h(typing)h(judgement)53
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12502
b F4(\000)2768 3370 y Gc(.)2823 3382 y Ga(N)2932 3370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12503
y Gc(.)2986 3382 y F4(\001)p Ga(;)15 b(a)j F4(:)f Ga(C)3322
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12504
3281 y FK(o)277 3585 y Gg(W)-7 b(e)22 b(e)o(xtend)j(the)e(notion)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12505
(free)h(names)f(and)h(free)g(co-names)g(to)f(the)h(named)g(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12506
(co-named)i(terms)277 3698 y(in)f(the)f(ob)o(vious)j(w)o(ay)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12507
b(.)418 3952 y(No)n(w)30 b(let)h(us)f(focus)i(on)f(term)g(re)n(writing)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12508
h(rules.)51 b(One)31 b(reason)h(for)f(introducing)j(terms)d(is)g(that)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12509
277 4065 y(the)o(y)c(greatly)h(simplify)g(the)f(formalisation)j(of)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12510
(the)h(cut-reduction)j(rules,)e(most)f(notably)i(the)d(rules)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12511
4177 y(for)j(commuting)i(cuts.)46 b(In)29 b(the)h(sequent)h(calculus)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12512
(gi)n(v)o(en)f(in)f(Figure)g(2.2)g(there)h(are)g(300)f(dif)n(ferent)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12513
4290 y(cases)k(of)e(commuting)i(cuts!)55 b(T)-7 b(o)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12514
b(consider)k(each)f(case)f(separately)i(is)e(clearly)h(rather)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12515
(laborious,)277 4403 y(and)d(unfortunately)j(simpli\002cations)f(using)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12516
e(sequent)h(proofs)g(seem)e(dif)n(\002cult)h(to)f(obtain.)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12517
b(F)o(or)28 b(e)o(x-)277 4516 y(ample)22 b(Ungar)g([1992,)h(P)o(age)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12518
(192])h(remarks)h(on)e(a)g(cut-elimination)26 b(procedure)e(in)e(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12519
(much)g(simpler)277 4629 y(setting:)p Black Black 504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12520
4815 a(Unfortunately)-6 b(,)33 b(there)c(are)g(a)f(lar)n(ge)i(number)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12521
(of)f(cases)i(to)e(consider)j(and)e(I)f(ha)n(v)o(e)h(not)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12522
4928 y(been)j(able)f(to)f(\002nd)g(a)g(notation)i(that)f(enables)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12523
(more)e(than)i(a)d(couple)j(of)e(these)i(to)e(be)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12524
5041 y(amalgamated.)277 5228 y(Therefore)37 b(traditional)g(treatments)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12525
g(of)e(cut-elimination)j(either)e(omit)f(these)h(permutation)h(rules)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12526
277 5341 y(entirely)e(or)d(present)i(only)g(a)e(handful)j(of)d(cases;)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12527
38 b(for)33 b(e)o(xample)h([Gentzen,)f(1935,)h(Girard)f(et)f(al.,)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12528
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12529
%%Page: 24 36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12530
TeXDict begin 24 35 bop Black -144 51 a Gb(24)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12531
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(1989,)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12532
b(Gallier,)f(1993].)57 b(This)33 b(is)f(unfortunate)k(as)d(a)f(careful)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12533
j(study)f(of)e(these)i(rules)g(sheds)f(some)321 430 y(light)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12534
b(on)e(the)h(problems)h(of)f(non-termination)j(of)d(cut-elimination.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12535
462 591 y(A)j(substitution)k(operation,)g(which)d(we)f(shall)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12536
(introduce)j(for)c(terms,)i(allo)n(ws)f(us)f(to)h(formalise)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12537
704 y(the)e(proof)g(transformations)j(necessary)f(for)d(commuting)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12538
(cuts)f(in)f(only)h(twenty)f(clauses.)36 b(Clearly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12539
b(,)321 817 y(this)28 b(is)f(an)g(adv)n(antage)j(of)d(our)h(use)f(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12540
(terms.)40 b(T)-7 b(o)26 b(gi)n(v)o(e)h(the)h(de\002nition)h(of)e(this)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12541
h(substitution)i(oper)n(-)321 930 y(ation)c(in)f(a)g(compact)h(form,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12542
(we)f(assume)i(the)f(follo)n(wing)h(con)l(v)o(ention,)j(which)c(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12543
(standard)i(in)e(term)321 1042 y(re)n(writing.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12544
321 1230 a Gb(Con)l(v)o(ention)19 b(2.2.4)f Gg(\(Barendre)o(gt-style)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12545
(Naming)c(Con)l(v)o(ention\))p Gb(:)p Black 37 w Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12546
b(e)17 b(assume)i(that)g(terms)f(are)g(equal)321 1343
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12547
y(up)26 b(to)f(alpha-con)l(v)o(ersion;)31 b(that)26 b(means)g(re)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12548
(writing)g(bound)h(names)e(or)h(co-names)g(does)g(not)g(change)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12549
1456 y(terms.)i(This)20 b(con)l(v)o(ersion)k(generates)e(alpha-equi)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12550
(v)n(alence)k(classes)c(of)e(terms.)28 b(When)20 b(writing)h(terms)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12551
1569 y(we)28 b(observ)o(e)j(a)d(Barendre)o(gt-style)k(naming)e(con)l(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12552
o(ention,)j(which)c(says)h(that)f(free)g(and)h(bound)g(\(co-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12553
1682 y(\)names)35 b(are)g(al)o(w)o(ays)g(distinct)h(\(cf.)e(Con)l(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12554
(ention)i(2.1.13)f(gi)n(v)o(en)g(by)f(Barendre)o(gt)i([1981]\).)63
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12555
b(Thus)321 1795 y(we)24 b(can)g(vie)n(w)g(our)g(terms)h(as)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12556
(representati)n(v)o(es)k(of)c(the)g(alpha-equi)n(v)n(alence)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12557
b(classes)d(and)e(w)o(ork)h(with)321 1908 y(these)g(terms)f(in)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12558
(na)m(\250)-27 b(\021v)o(e)24 b(w)o(ay)-6 b(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12559
321 2145 a Gb(Remark)39 b(2.2.5:)p Black 34 w Gg(An)f(alternati)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12560
(e)j(to)e(this)h(con)l(v)o(ention)i(a)n(v)n(oiding)f(the)f(problems)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12561
(arising)h(from)321 2258 y(alpha-con)l(v)o(ersions)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12562
b(w)o(ould)29 b(be)g(to)f(use)h(de)f(Bruijn')-5 b(s)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12563
b(nameless)g(terms.)44 b(This)28 b(an)g(ele)o(gant)i(method)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12564
2371 y(for)c(an)g(implementation,)j(ho)n(we)n(v)o(er)l(,)d(at)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12565
(e)o(xpense)j(of)d(making)i(our)f(terms)g(ille)o(gible)h(for)f(human)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12566
321 2484 y(readers.)67 b(Another)37 b(alternati)n(v)o(e)h(is)d(to)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12567
(de\002ne)g(directly)i(the)e(alpha-equi)n(v)n(alence)k(classes)e(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12568
(our)321 2597 y(terms)22 b(using)h(the)e(meta-language)k(pro)o(vided)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12569
(by)d(Gabbay)h(and)g(Pitts)g([1999].)29 b(Both)22 b(methods)h(seem)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12570
2710 y(less)32 b(useful)g(for)f(our)g(purposes,)k(since)c(the)o(y)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12571
(either)h(complicate)h(the)e(de\002nition)i(of)e(terms)g(or)f(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12572
2822 y(de\002nition)23 b(of)e(substitution.)31 b(Therefore)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12573
b(we)d(shall)i(use)f(the)h(Barendre)o(gt-style)i(naming)e(con)l(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12574
(ention)321 2935 y(throughout)27 b(the)d(thesis.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12575
3220 y(Before)h(we)f(formalise)i(the)e(substitution)k(operation,)f(let)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12576
d(us)h(gi)n(v)o(e)f(some)g(intuition)j(behind)f(this)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12577
3333 y(operation.)31 b(Consider)24 b(the)f(follo)n(wing)i(sequent)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12578
(proof)g(where)f(we)f(ha)n(v)o(e)h(omitted)h(the)f(term)g(annota-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12579
3446 y(tions)i(and)f(labels)h(for)e(bre)n(vity)-6 b(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12580
Black Black 307 3916 a FU(\031)354 3928 y FJ(1)391 3746
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12581
y FI(8)391 3821 y(<)391 3970 y(:)p 472 3865 703 4 v 472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12582
3938 a FU(B)t FT(_)q FU(C)p 678 3926 10 38 v 688 3910
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12583
42 4 v 94 w(D)r FT(\033)p FU(E)5 b(;)14 b(B)t FT(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12584
FU(C)1216 3884 y FH(?)p 1339 3755 670 4 v 1339 3829 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12585
FU(B)t FT(_)q FU(C)q(;)g(D)p 1649 3817 10 38 v 1658 3800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12586
42 4 v 90 w(E)5 b(;)14 b(B)t FT(_)q FU(C)2050 3775 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12587
FH(?)p 1323 3865 703 4 v 1323 3938 a FU(B)t FT(_)p FU(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12588
1529 3926 10 38 v 1539 3910 42 4 v 95 w(D)r FT(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12589
FU(E)5 b(;)14 b(B)t FT(_)q FU(C)2053 3881 y FT(\033)2117
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12590
3893 y FS(R)p 472 3974 1553 4 v 711 4053 a FG(\()p FU(B)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12591
FT(_)q FU(C)6 b FG(\))p FT(_)q FG(\()p FU(B)t FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12592
FU(C)g FG(\))p 1290 4041 10 38 v 1299 4024 42 4 v 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12593
w FU(D)r FT(\033)p FU(E)f(;)14 b(B)t FT(_)p FU(C)2053
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12594
3991 y FT(_)2108 4004 y FS(L)p 2245 3761 427 4 v 2245
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12595
3834 a FU(B)t(;)g(F)p 2433 3822 10 38 v 2442 3806 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12596
4 v 100 w(B)t(;)g(C)p 2755 3761 420 4 v 90 w(C)q(;)g(F)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12597
2935 3822 10 38 v 2945 3806 42 4 v 100 w(B)t(;)g(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12598
2245 3870 929 4 v 2439 3944 a(B)t FT(_)p FU(C)q(;)g(F)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12599
2742 3932 10 38 v 2752 3915 42 4 v 101 w(B)t(;)g(C)3202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12600
3887 y FT(_)3257 3900 y FS(L)3307 3887 y FT(\017)p 2378
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12601
3980 663 4 v 2378 4053 a FU(B)t FT(_)q FU(C)q(;)g(F)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12602
FT(^)q FU(G)p 2802 4041 10 38 v 2812 4024 42 4 v 88 w(B)t(;)i(C)3069
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12603
3992 y FT(^)3124 4005 y FS(L)3170 4013 y FP(1)3355 3746
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12604
y FI(9)3355 3821 y(=)3355 3970 y(;)3429 3916 y FU(\031)3476
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12605
3928 y FJ(2)p 593 4073 2613 4 v 1261 4152 a FG(\()p FU(B)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12606
FT(_)p FU(C)6 b FG(\))p FT(_)q FG(\()p FU(B)t FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12607
FU(C)g FG(\))p FU(;)14 b(F)e FT(^)q FU(G)p 2062 4140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12608
10 38 v 2071 4123 42 4 v 88 w(D)r FT(\033)p FU(E)5 b(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12609
b(B)t(;)g(C)3234 4101 y Gd(Cut)321 4524 y Gg(The)23 b(cut-formula,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12610
Ga(B)5 b F6(_)o Ga(C)i Gg(,)22 b(is)h(neither)i(introduced)i(in)c(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12611
(inference)i(rule)e F6(_)2794 4538 y Gc(L)2869 4524 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12612
Gg(nor)f(in)h F6(^)3166 4538 y Gc(L)3214 4547 y FV(1)3253
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12613
4524 y Gg(.)k(There-)321 4637 y(fore)d(the)g(cut)f(is,)h(by)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12614
(de\002nition,)i(a)e(commuting)i(cut.)31 b(In)24 b Ga(\031)2194
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12615
4651 y F9(1)2257 4637 y Gg(the)h(cut-formula)h(is)f(introduced)i(in)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12616
(the)321 4750 y(axioms)31 b(mark)o(ed)f(with)f(a)g(star)l(,)j(and)e(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12617
f Ga(\031)1664 4764 y F9(2)1704 4750 y Gg(,)g(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12618
b(,)34 b(in)c(the)f(inference)j(rule)e(mark)o(ed)h(with)e(a)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12619
4863 y(disc.)k(Eliminating)26 b(the)f(cut)g(in)g(the)g(proof)h(abo)o(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12620
o(e)f(means)g(to)g(either)h(permute)f(the)g(deri)n(v)n(ation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12621
Ga(\031)3412 4877 y F9(2)3476 4863 y Gg(to)321 4976 y(the)j(places)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12622
(mark)o(ed)h(with)e(a)f(star)i(and)g(replace)h(the)e(corresponding)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12623
b(axioms)c(with)f Ga(\031)3107 4990 y F9(2)3146 4976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12624
y Gg(,)h(or)f(to)g(per)n(-)321 5088 y(mute)23 b Ga(\031)577
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12625
5102 y F9(1)639 5088 y Gg(and)h(\223cut)g(it)e(against\224)j(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12626
(inference)h(rule)f(mark)o(ed)g(with)f(a)g(disc.)29 b(In)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12627
b(the)h(former)f(case)h(the)321 5201 y(deri)n(v)n(ation)i(being)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12628
(permuted)g(is)e(duplicated.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12629
%%Page: 25 37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12630
TeXDict begin 25 36 bop Black 277 51 a Gb(2.2)23 b(Cut-Elimination)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12631
(in)e(Pr)n(opositional)k(Classical)f(Logic)1582 b(25)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12632
277 88 3691 4 v Black 418 294 a Gg(W)-7 b(e)18 b(realise)i(these)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12633
(operations)j(at)c(the)h(term)f(le)n(v)o(el)h(with)f(tw)o(o)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12634
(symmetric)i(forms)f(of)f(substitution,)277 407 y(which)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12635
b(we)f(shall)h(write)g(as)p Black Black 1012 586 a Ga(P)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12636
b F4([)p Ga(x)k F4(:)r Ga(B)31 b F4(:=)1425 574 y FX(h)1452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12637
586 y Ga(a)17 b F4(:)r Ga(B)1618 574 y FX(i)1646 586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12638
y Ga(Q)p F4(])99 b Gg(or)i Ga(S)5 b F4([)p Ga(b)17 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12639
F4(:)r Ga(B)30 b F4(:=)2407 574 y F9(\()2435 586 y Ga(y)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12640
b F4(:)r Ga(B)2601 574 y F9(\))2628 586 y Ga(T)13 b F4(])25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12641
b Gg(.)277 765 y(F)o(or)g(bre)n(vity)j(we)d(shall)i(usually)g(omit)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12642
(the)g(type)h(annotations)i(in)d(the)g(substitution)k(operation,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12643
(pro-)277 878 y(vided)c(the)o(y)f(are)f(clear)i(from)e(the)h(conte)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12644
(xt.)418 1008 y(Returning)d(to)e(our)g(moti)n(v)n(ating)h(e)o(xample,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12645
(assume)e(that)h Ga(M)28 b Gg(and)19 b Ga(N)28 b Gg(are)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12646
b(the)g(terms)g(correspond-)277 1120 y(ing)24 b(to)f(the)h(subproofs)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12647
Ga(\031)1077 1134 y F9(1)1139 1120 y Gg(and)d Ga(\031)1345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12648
1134 y F9(2)1384 1120 y Gg(,)e(which)i(thus)g(ha)n(v)o(e)h(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12649
(typing)j(judgements:)p Black Black 1011 1294 a Ga(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12650
b F4(:)h(\()p Ga(B)5 b F6(_)o Ga(C)i F4(\))p F6(_)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12651
Ga(B)e F6(_)p Ga(C)i F4(\))1786 1282 y Gc(.)1866 1294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12652
y Ga(M)2015 1282 y Gc(.)2095 1294 y Ga(a)17 b F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12653
Ga(D)s F6(\033)o Ga(E)5 b(;)15 b(b)j F4(:)g Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12654
F6(_)o Ga(C)1092 1421 y(y)20 b F4(:)e Ga(B)5 b F6(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12655
Ga(C)q(;)15 b(z)22 b F4(:)c Ga(F)13 b F6(^)o Ga(G)1800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12656
1409 y Gc(.)1888 1421 y Ga(N)2028 1409 y Gc(.)2108 1421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12657
y Ga(c)18 b F4(:)f Ga(B)5 b(;)15 b(d)j F4(:)f Ga(C)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12658
1591 y Gg(Consequently)-6 b(,)27 b(we)22 b(ha)n(v)o(e)j(the)e(term)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12659
(annotation)p Black Black 528 1758 a Ga(x)18 b F4(:)f(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12660
Ga(B)5 b F6(_)o Ga(C)i F4(\))p F6(_)p F4(\()p Ga(B)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12661
F6(_)o Ga(C)i F4(\))p Ga(;)15 b(z)22 b F4(:)c Ga(F)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12662
b F6(^)o Ga(G)1653 1746 y Gc(.)1733 1758 y FL(Cut)p F4(\()1906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12663
1746 y FX(h)1934 1758 y Ga(b)1973 1746 y FX(i)2000 1758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12664
y Ga(M)e(;)2139 1746 y F9(\()2166 1758 y Ga(y)2214 1746
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12665
y F9(\))2242 1758 y Ga(N)f F4(\))2411 1746 y Gc(.)2491
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12666
1758 y Ga(a)17 b F4(:)g Ga(D)s F6(\033)p Ga(E)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12667
b(c)j F4(:)g Ga(B)5 b(;)15 b(d)i F4(:)g Ga(C)277 1943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12668
y Gg(for)27 b(the)g(conclusion)j(of)c(the)h(e)o(xample)h(proof.)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12669
b(The)26 b(term)g Ga(M)10 b F4([)p Ga(b)32 b F4(:=)2474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12670
1931 y F9(\()2501 1943 y Ga(y)2549 1931 y F9(\))2576
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12671
1943 y Ga(N)10 b F4(])26 b Gg(denotes)j(the)e(follo)n(wing)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12672
2056 y(proof,)d(where)g(we)f(ha)n(v)o(e)h(again)g(omitted)h(all)f(term)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12673
f(annotations)k(and)d(labels.)p Black Black 286 2296
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12674
665 4 v 286 2370 a FU(B)t(;)14 b(F)p 474 2358 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12675
483 2341 42 4 v 100 w(D)r FT(\033)p FU(E)5 b(;)14 b(B)t(;)g(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12676
1080 2296 659 4 v 135 w(C)q(;)g(F)p 1261 2358 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12677
1271 2341 42 4 v 101 w(D)r FT(\033)o FU(E)5 b(;)14 b(B)t(;)g(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12678
286 2406 1453 4 v 622 2479 a(B)t FT(_)p FU(C)q(;)g(F)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12679
925 2467 10 38 v 935 2450 42 4 v 101 w(D)r FT(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12680
FU(E)5 b(;)14 b(B)t(;)g(C)1780 2423 y FT(_)1835 2435
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12681
y FS(L)p 562 2515 902 4 v 562 2588 a FU(B)t FT(_)p FU(C)q(;)g(F)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12682
FT(^)q FU(G)p 986 2576 10 38 v 995 2560 42 4 v 88 w(D)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12683
FT(\033)p FU(E)5 b(;)14 b(B)t(;)g(C)1505 2528 y FT(^)1560
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12684
2540 y FS(L)1606 2548 y FP(1)p 1968 2187 628 4 v 1968
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12685
2260 a FU(B)t(;)g(F)r(;)g(D)p 2254 2248 10 38 v 2264
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12686
2232 42 4 v 91 w(E)5 b(;)14 b(B)t(;)g(C)p 2725 2187 622
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12687
4 v 135 w(C)q(;)g(F)r(;)g(D)p 3005 2248 10 38 v 3014
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12688
2232 42 4 v 91 w(E)5 b(;)14 b(B)t(;)g(C)p 1968 2296 1379
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12689
4 v 2285 2370 a(B)t FT(_)q FU(C)q(;)g(F)r(;)g(D)p 2687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12690
2358 10 38 v 2697 2341 42 4 v 91 w(E)5 b(;)14 b(B)t(;)g(C)3388
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12691
2313 y FT(_)3443 2326 y FS(L)p 2220 2406 874 4 v 2220
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12692
2479 a FU(B)t FT(_)q FU(C)q(;)g(F)e FT(^)q FU(G;)i(D)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12693
2752 2467 10 38 v 2762 2450 42 4 v 90 w(E)5 b(;)14 b(B)t(;)g(C)3136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12694
2418 y FT(^)3191 2431 y FS(L)3237 2439 y FP(1)p 2207
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12695
2515 902 4 v 2207 2588 a FU(B)t FT(_)p FU(C)q(;)g(F)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12696
FT(^)q FU(G)p 2631 2576 10 38 v 2640 2560 42 4 v 88 w(D)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12697
FT(\033)p FU(E)5 b(;)14 b(B)t(;)g(C)3150 2531 y FT(\033)3214
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12698
2544 y FS(R)p 562 2624 2547 4 v 1196 2703 a FG(\()p FU(B)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12699
FT(_)p FU(C)6 b FG(\))p FT(_)q FG(\()p FU(B)t FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12700
FU(C)g FG(\))p FU(;)14 b(F)e FT(^)q FU(G)p 1997 2691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12701
10 38 v 2006 2675 42 4 v 88 w(D)r FT(\033)p FU(E)5 b(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12702
b(B)t(;)g(C)3150 2641 y FT(_)3205 2654 y FS(L)277 2901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12703
y Gg(Similarly)-6 b(,)24 b(the)g(symmetric)g(case)h Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12704
b F4([)p Ga(y)28 b F4(:=)1677 2889 y FX(h)1705 2901 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12705
Ga(b)1744 2889 y FX(i)1771 2901 y Ga(M)10 b F4(])23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12706
Gg(denotes)j(the)d(proof)p Black Black 440 3144 703 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12707
v 440 3218 a FU(B)t FT(_)q FU(C)p 647 3206 10 38 v 656
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12708
3189 42 4 v 94 w(D)r FT(\033)p FU(E)5 b(;)14 b(B)t FT(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12709
FU(C)p 1288 3035 670 4 v 1288 3108 a(B)t FT(_)p FU(C)q(;)g(D)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12710
1597 3096 10 38 v 1607 3080 42 4 v 91 w(E)5 b(;)14 b(B)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12711
FT(_)p FU(C)p 1272 3144 703 4 v 1272 3218 a(B)t FT(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12712
FU(C)p 1478 3206 10 38 v 1487 3189 42 4 v 94 w(D)r FT(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12713
FU(E)5 b(;)14 b(B)t FT(_)p FU(C)2015 3160 y FT(\033)2080
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12714
3173 y FS(R)p 440 3254 1534 4 v 670 3333 a FG(\()p FU(B)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12715
FT(_)q FU(C)6 b FG(\))p FT(_)q FG(\()p FU(B)t FT(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12716
FU(C)g FG(\))p 1248 3321 10 38 v 1258 3304 42 4 v 89
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12717
w FU(D)r FT(\033)o FU(E)f(;)14 b(B)t FT(_)q FU(C)2015
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12718
3271 y FT(_)2071 3283 y FS(L)p 2217 3150 427 4 v 2217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12719
3223 a FU(B)t(;)g(F)p 2405 3211 10 38 v 2414 3195 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12720
4 v 100 w(B)t(;)g(C)p 2773 3150 420 4 v 136 w(C)q(;)g(F)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12721
2954 3211 10 38 v 2963 3195 42 4 v 100 w(B)t(;)g(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12722
2217 3259 976 4 v 2434 3333 a(B)t FT(_)p FU(C)q(;)g(F)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12723
2737 3321 10 38 v 2747 3304 42 4 v 101 w(B)t(;)g(C)3234
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12724
3276 y FT(_)3289 3289 y FS(L)p 670 3373 2307 4 v 1244
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12725
3452 a FG(\()p FU(B)t FT(_)q FU(C)6 b FG(\))p FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12726
FG(\()p FU(B)t FT(_)p FU(C)g FG(\))p FU(;)14 b(F)p 1924
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12727
3440 10 38 v 1934 3424 42 4 v 101 w(D)r FT(\033)o FU(E)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12728
b(;)14 b(B)t(;)g(C)3017 3401 y Gd(Cut)p 1183 3493 1280
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12729
4 v 1183 3572 a FG(\()p FU(B)t FT(_)q FU(C)6 b FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12730
FT(_)q FG(\()p FU(B)t FT(_)q FU(C)g FG(\))p FU(;)14 b(E)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12731
b FT(^)q FU(F)p 1985 3560 10 38 v 1995 3543 42 4 v 100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12732
w(D)r FT(\033)o FU(E)g(;)14 b(B)t(;)g(C)2504 3505 y FT(^)2559
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12733
3518 y FS(L)2605 3526 y FP(1)418 3763 y Gg(As)29 b(we)f(ha)n(v)o(e)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12734
(seen,)h(commuting)g(cuts)f(need)g(to)f(permute,)j(or)d(\223jump\224,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12735
(to)d(the)g(places)i(where)277 3876 y(the)k(cut-formula)j(is)d(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12736
(main)h(formula.)64 b(At)34 b(the)h(le)n(v)o(el)h(of)f(terms)g(this)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12737
(means)h(the)f(cuts)h(need)f(to)277 3989 y(be)e(permuted)i(to)e(e)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12738
(ery)h(subterm)h(that)e(introduces)k(the)c(cut-formula.)61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12739
b(Therefore,)37 b(whene)n(v)o(er)d(a)277 4102 y(substitution)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12740
b(is)c(ne)o(xt)h(to)g(a)f(term)g(in)g(which)h(the)g(cut-formula)h(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12741
(introduced,)42 b(the)37 b(substitution)277 4215 y(becomes)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12742
b(an)f(instance)j(of)d(the)g(cut-term)h(constructor)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12743
b(.)34 b(In)23 b(the)i(follo)n(wing)g(tw)o(o)f(e)o(xamples)h(we)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12744
(shall)277 4328 y(write)h F4([)p Ga(\033)s F4(])f Gg(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12745
F4([)p Ga(\034)10 b F4(])23 b Gg(for)h(the)g(substitutions)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12746
F4([)p Ga(c)f F4(:=)1848 4316 y F9(\()1876 4328 y Ga(x)1928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12747
4316 y F9(\))1955 4328 y Ga(P)13 b F4(])23 b Gg(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12748
F4([)p Ga(x)i F4(:=)2452 4316 y FX(h)2479 4328 y Ga(b)2518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12749
4316 y FX(i)2546 4328 y Ga(Q)p F4(])p Gg(,)c(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12750
b(.)p Black Black 540 4521 a FL(And)694 4535 y Gc(R)752
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12751
4521 y F4(\()787 4509 y FX(h)815 4521 y Ga(a)863 4509
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12752
y FX(i)890 4521 y Ga(M)10 b(;)1028 4509 y FX(h)1056 4521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12753
y Ga(b)1095 4509 y FX(i)1123 4521 y Ga(N)g(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12754
F4(\)[)p Ga(\033)s F4(])101 b(=)e FL(Cut)p F4(\()1869
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12755
4509 y FX(h)1897 4521 y Ga(c)1936 4509 y FX(i)1964 4521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12756
y FL(And)2118 4535 y Gc(R)2176 4521 y F4(\()2211 4509
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12757
y FX(h)2239 4521 y Ga(a)2287 4509 y FX(i)2330 4521 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12758
Ga(M)10 b F4([)p Ga(\033)s F4(])p Ga(;)2573 4509 y FX(h)2601
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12759
4521 y Ga(b)2640 4509 y FX(i)2683 4521 y Ga(N)g F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12760
Ga(\033)s F4(])p Ga(;)15 b(c)p F4(\))r Ga(;)3027 4509
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12761
y F9(\()3054 4521 y Ga(x)3106 4509 y F9(\))3134 4521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12762
y Ga(P)e F4(\))539 4657 y FL(Imp)684 4679 y Gc(L)736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12763
4657 y F4(\()771 4645 y FX(h)798 4657 y Ga(a)846 4645
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12764
y FX(i)874 4657 y Ga(M)d(;)1012 4645 y F9(\()1040 4657
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12765
y Ga(y)1088 4645 y F9(\))1115 4657 y Ga(N)g(;)15 b(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12766
F4(\)[)p Ga(\034)10 b F4(])101 b(=)e FL(Cut)p F4(\()1869
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12767
4645 y FX(h)1897 4657 y Ga(b)1936 4645 y FX(i)1964 4657
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12768
y Ga(Q)o(;)2075 4645 y F9(\()2103 4657 y Ga(x)2155 4645
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12769
y F9(\))2183 4657 y FL(Imp)2327 4679 y Gc(L)2379 4657
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12770
y F4(\()2414 4645 y FX(h)2442 4657 y Ga(a)2490 4645 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12771
FX(i)2533 4657 y Ga(M)10 b F4([)p Ga(\034)g F4(])p Ga(;)2771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12772
4645 y F9(\()2799 4657 y Ga(y)2847 4645 y F9(\))2890
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12773
4657 y Ga(N)g F4([)p Ga(\034)g F4(])p Ga(;)15 b(x)p F4(\))q(\))277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12774
4844 y Gg(In)23 b(the)h(\002rst)f(e)o(xample)h(the)g(formula)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12775
(labelled)g(with)e Ga(c)g Gg(is)h(the)f(main)h(formula)g(and)g(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12776
(the)h(second)h(the)277 4957 y(formula)h(labelled)g(with)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12777
Ga(x)g Gg(is)g(the)h(main)g(formula.)32 b(So)24 b(in)g(both)i(cases)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12778
(the)g(substitutions)j(\223e)o(xpand\224)277 5070 y(to)e(cuts,)g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12779
(in)g(addition)h(the)f(substitutions)k(are)25 b(pushed)j(inside)f(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12780
(subterms.)37 b(This)25 b(is)g(because)277 5183 y(there)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12781
b(might)e(be)h(se)n(v)o(eral)g(occurrences)j(of)c Ga(c)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12782
Gg(and)h Ga(x)f Gg(\(both)h(labels)h(need)f(not)g(ha)n(v)o(e)g(been)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12783
(freshly)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12784
%%Page: 26 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12785
TeXDict begin 26 37 bop Black -144 51 a Gb(26)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12786
b(Calculi)p -144 88 3691 4 v Black 321 294 a Gg(introduced\).)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12787
b(An)20 b(e)o(xception)i(applies)g(to)e(axioms,)i(where)f(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12788
(substitution)k(is)c(de\002ned)h(dif)n(ferently)-6 b(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12789
Black Black 1271 467 a FL(Ax)p F4(\()p Ga(x;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12790
F4(\)[)p Ga(x)26 b F4(:=)1809 455 y FX(h)1836 467 y Ga(b)1875
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12791
455 y FX(i)1903 467 y Ga(P)13 b F4(])99 b(=)h Ga(P)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12792
b F4([)p Ga(b)d F6(7!)g Ga(a)p F4(])1266 604 y FL(Ax)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12793
F4(\()p Ga(x;)15 b(a)p F4(\)[)p Ga(a)27 b F4(:=)1799
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12794
592 y F9(\()1826 604 y Ga(y)1874 592 y F9(\))1902 604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12795
y Ga(Q)p F4(])99 b(=)h Ga(Q)p F4([)p Ga(y)13 b F6(7!)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12796
Ga(x)p F4(])321 780 y Gg(Recall)29 b(that)f Ga(P)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12797
F4([)p Ga(b)18 b F6(7!)g Ga(a)p F4(])28 b Gg(stands)h(for)f(the)g(term)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12798
g Ga(P)40 b Gg(where)28 b(e)n(v)o(ery)g(free)h(occurrence)h(of)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12799
(co-name)321 893 y Ga(b)i Gg(is)g(re)n(written)i(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12800
Ga(a)g Gg(\(similarly)i Ga(Q)p F4([)p Ga(y)26 b F6(7!)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12801
Ga(x)p F4(])p Gg(\).)49 b(W)-7 b(e)30 b(are)g(left)h(with)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12802
(cases)h(where)e(the)h(name)g(or)321 1006 y(co-name)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12803
b(that)f(is)f(being)i(substituted)i(for)c(is)h(not)f(a)g(label)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12804
(the)h(main)g(formula.)29 b(In)22 b(these)i(cases)f(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12805
1119 y(substitutions)k(are)c(pushed)h(inside)g(the)f(subterms)h(or)e(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12806
n(anish)i(in)f(case)g(of)f(the)h(axioms.)29 b(Suppose)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12807
b(the)321 1232 y(substitution)j F4([)p Ga(\033)s F4(])d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12808
Gg(is)f F7(not)h Gg(of)g(the)f(form)h F4([)p Ga(z)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12809
b F4(:=)25 b Ga(:)15 b(:)g(:)q F4(])23 b Gg(and)h F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12810
Ga(a)h F4(:=)h Ga(:)15 b(:)g(:)q F4(])p Gg(,)22 b(then)j(we)d(ha)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12811
(e)j(for)e(e)o(xample:)p Black Black 875 1425 a FL(Or)974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12812
1439 y Gc(L)1027 1425 y F4(\()1062 1413 y F9(\()1089
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12813
1425 y Ga(x)1141 1413 y F9(\))1169 1425 y Ga(M)10 b(;)1307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12814
1413 y F9(\()1335 1425 y Ga(y)1383 1413 y F9(\))1410
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12815
1425 y Ga(N)g(;)15 b(z)t F4(\)[)p Ga(\033)s F4(])101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12816
b(=)f FL(Or)2090 1439 y Gc(L)2142 1425 y F4(\()2177 1413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12817
y F9(\()2205 1425 y Ga(x)2257 1413 y F9(\))2310 1425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12818
y Ga(M)10 b F4([)p Ga(\033)s F4(])p Ga(;)2553 1413 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12819
F9(\()2581 1425 y Ga(y)2629 1413 y F9(\))2682 1425 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12820
Ga(N)g F4([)p Ga(\033)s F4(])p Ga(;)15 b(z)t F4(\))1307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12821
1562 y FL(Ax)p F4(\()p Ga(z)t(;)g(a)p F4(\)[)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12822
F4(])101 b(=)f FL(Ax)o F4(\()p Ga(z)t(;)15 b(a)p F4(\))462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12823
1757 y Gg(The)30 b(complete)h(de\002nition)g(of)f(the)g(substitution)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12824
(operation)f(is)e(gi)n(v)o(en)g(in)f(Figure)i(2.4.)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12825
b(W)-7 b(e)28 b(do)321 1870 y(not)g(need)g(to)f(w)o(orry)h(about)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12826
(inserting)i(contraction)g(rules)e(when)f(a)g(term)g(is)g(duplicated,)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12827
(since)d(our)321 1983 y(conte)o(xts)c(are)e(sets)g(of)g(labelled)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12828
(formulae,)h(and)e(thus)g(contractions)k(are)c(made)f(implicitly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12829
b(.)30 b(Another)321 2096 y(simpli\002cation)23 b(is)c(due)i(to)f(our)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12830
(use)g(of)g(the)g(Barendre)o(gt-style)k(naming)d(con)l(v)o(ention,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12831
(because)f(we)d(do)321 2209 y(not)h(need)g(to)g(w)o(orry)f(about)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12832
(possible)h(capture)f(of)e(free)h(names)g(or)f(co-names.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12833
b(Let)19 b(us)g(no)n(w)g(introduce)321 2322 y(some)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12834
b(useful)h(terminology)h(for)e(substitutions.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12835
321 2509 a Gb(T)-8 b(erminology)35 b(2.2.6:)p Black 34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12836
w Gg(W)-7 b(e)33 b(shall)h(use)g(the)g(e)o(xpression)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12837
F4([)p Ga(\033)s F4(])c Gg(to)h(range)g(o)o(v)o(er)g(substitutions)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12838
(of)33 b(the)321 2622 y(form)26 b F4([)p Ga(x)18 b F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12839
Ga(B)30 b F4(:=)880 2610 y FX(h)908 2622 y Ga(a)17 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12840
F4(:)g Ga(B)1089 2610 y FX(i)1131 2622 y Ga(Q)p F4(])26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12841
b Gg(and)g F4([)p Ga(b)18 b F4(:)f Ga(B)30 b F4(:=)1754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12842
2610 y F9(\()1781 2622 y Ga(y)21 b F4(:)c Ga(B)1963 2610
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12843
y F9(\))2005 2622 y Ga(T)c F4(])p Gg(.)35 b(In)26 b(the)h(\002rst)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12844
(case)i Ga(x)17 b F4(:)h Ga(B)29 b Gg(is)d(the)h F7(domain)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12845
Gg(of)321 2735 y F4([)p Ga(\033)s F4(])p Gg(,)d(written)h(as)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12846
Ga(dom)p F4(\([)p Ga(\033)s F4(]\))p Gg(,)h(and)f(the)g(co-named)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12847
(term)2107 2723 y FX(h)2134 2735 y Ga(a)r F4(:)r Ga(B)2285
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12848
2723 y FX(i)2313 2735 y Ga(Q)d Gg(is)h(the)g F7(co-domain)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12849
Gg(of)e F4([)p Ga(\033)s F4(])p Gg(,)g(written)321 2848
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12850
y(as)g Ga(codom)p F4(\([)p Ga(\033)s F4(]\))p Gg(.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12851
b(Similarly)24 b(in)f(the)h(second)h(case.)462 3085 y(Ne)o(xt,)k(let)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12852
(us)g(focus)i(on)e(the)g(cut-reductions)33 b(for)28 b(logical)i(cuts.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12853
43 b(Consider)30 b(an)e(instance)i(of)f(an)321 3198 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12854
F6(^)382 3212 y Gc(R)440 3198 y Ga(=)p F6(^)546 3212
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12855
y Gc(L)594 3221 y FV(1)632 3198 y Gg(-cut)c(for)e(which)h(a)f(na)m
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12856
(\250)-27 b(\021v)o(e)25 b(de\002nition)g(of)e(reduction)j(might)e(be)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12857
(as)f(follo)n(ws.)p Black Black 627 3409 a FL(Cut)p F4(\()800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12858
3397 y FX(h)827 3409 y Ga(c)866 3397 y FX(i)894 3409
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12859
y FL(And)1049 3423 y Gc(R)1107 3409 y F4(\()1142 3397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12860
y FX(h)1169 3409 y Ga(a)1217 3397 y FX(i)1245 3409 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12861
Ga(M)10 b(;)1383 3397 y FX(h)1411 3409 y Ga(b)1450 3397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12862
y FX(i)1477 3409 y Ga(N)g(;)15 b(c)p F4(\))q Ga(;)1715
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12863
3397 y F9(\()1743 3409 y Ga(y)1791 3397 y F9(\))1818
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12864
3409 y FL(And)1973 3372 y F9(1)1973 3432 y Gc(L)2025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12865
3409 y F4(\()2060 3397 y F9(\()2088 3409 y Ga(x)2140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12866
3397 y F9(\))2167 3409 y Ga(P)e(;)i(y)s F4(\))q(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12867
b F6(\000)-31 b(\000)f(!)23 b FL(Cut)p F4(\()2786 3397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12868
y FX(h)2814 3409 y Ga(a)2862 3397 y FX(i)2889 3409 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12869
Ga(M)10 b(;)3027 3397 y F9(\()3055 3409 y Ga(x)3107 3397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12870
y F9(\))3134 3409 y Ga(P)j F4(\))321 3629 y Gg(Ho)n(we)n(v)o(er)l(,)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12871
b(there)g(is)f(a)f(problem)j(with)e(this)g(reduction)j(rule.)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12872
b(In)19 b(our)g(sequent)i(calculus)h(the)d(structural)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12873
3742 y(rules)28 b(are)f F7(implicit)p Gg(:)36 b(this)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12874
b(means)g(that)h(not)f(only)g(does)h(the)e(calculus)j(ha)n(v)o(e)e(fe)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12875
(wer)g(inference)i(rules,)321 3855 y(b)n(ut)c(more)e(importantly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12876
b(,)27 b(we)c(ha)n(v)o(e)h(a)f(v)o(ery)i(con)l(v)o(enient)h(w)o(ay)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12877
(of)g(de\002ning)h(substitution)i(\(we)c(do)h(not)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12878
3968 y(need)g(e)o(xplicit)g(contractions)i(when)d(a)f(term)g(is)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12879
(duplicated\).)31 b(On)22 b(the)h(other)h(hand,)f(there)h(is)e(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12880
(subtle)321 4081 y(side-ef)n(fect)k(of)e(this)g(design)h(decision;)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12881
(consider)g(the)e(follo)n(wing)h(instance)g(of)f(the)f(rede)o(x)3179
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12882
4048 y F5(3)3241 4081 y Gg(abo)o(v)o(e)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12883
325 4290 a FG(\000)377 4302 y FJ(1)437 4278 y FS(.)490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12884
4290 y FU(M)603 4278 y FS(.)656 4290 y FG(\001)725 4302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12885
y FJ(1)762 4290 y FU(;)p 0.75 TeXcolorgray 0.75 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12886
799 4315 330 107 v 0.75 TeXcolorgray Black 39 w(c)16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12887
b FG(:)g FU(B)t FT(^)q FU(C)p 0.75 TeXcolorgray Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12888
31 w(;)e(a)i FG(:)h FU(B)45 b FG(\000)1425 4302 y FJ(1)1485
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12889
4278 y FS(.)1538 4290 y FU(N)1637 4278 y FS(.)1690 4290
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12890
y FG(\001)1759 4302 y FJ(1)1797 4290 y FU(;)14 b(b)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12891
FG(:)i FU(C)p 325 4335 1666 4 v 464 4414 a FG(\000)516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12892
4426 y FJ(1)577 4402 y FS(.)630 4414 y FF(And)771 4426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12893
y FS(R)825 4414 y FG(\()857 4402 y FQ(h)884 4414 y FU(a)928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12894
4402 y FQ(i)955 4414 y FU(M)9 b(;)1082 4402 y FQ(h)1108
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12895
4414 y FU(b)1144 4402 y FQ(i)1171 4414 y FU(N)g(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12896
b(c)p FG(\))1375 4402 y FS(.)1428 4414 y FG(\001)1497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12897
4426 y FJ(1)1534 4414 y FU(;)g(c)i FG(:)h FU(B)t FT(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12898
FU(C)2018 4352 y FT(^)2073 4364 y FS(R)2484 4287 y FU(x)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12899
FG(:)f FU(B)t(;)e FG(\000)2743 4299 y FJ(2)2804 4275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12900
y FS(.)2857 4287 y FU(P)2944 4275 y FS(.)2997 4287 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12901
FG(\001)3066 4299 y FJ(2)p 2211 4323 1168 4 v 2211 4414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12902
a FU(y)k FG(:)f FU(B)t FT(^)p FU(C)q(;)d FG(\000)2581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12903
4426 y FJ(2)2642 4402 y FS(.)2695 4414 y FF(And)2836
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12904
4377 y FJ(1)2836 4434 y FS(L)2886 4414 y FG(\()2918 4402
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12905
y FJ(\()2944 4414 y FU(x)2991 4402 y FJ(\))3017 4414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12906
y FU(P)e(;)i(y)s FG(\))3218 4402 y FS(.)3271 4414 y FG(\001)3340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12907
4426 y FJ(2)3405 4335 y FT(^)3461 4348 y FS(L)3507 4356
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12908
y FP(1)p 464 4455 2914 4 v 795 4546 a FG(\000)847 4558
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12909
y FJ(1)884 4546 y FU(;)g FG(\000)973 4558 y FJ(2)1033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12910
4534 y FS(.)1086 4546 y FF(Cut)p FG(\()1244 4534 y FQ(h)1271
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12911
4546 y FU(c)1307 4534 y FQ(i)1334 4546 y FF(And)1475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12912
4558 y FS(R)1530 4546 y FG(\()1562 4534 y FQ(h)1589 4546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12913
y FU(a)1633 4534 y FQ(i)1660 4546 y FU(M)8 b(;)1786 4534
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12914
y FQ(h)1813 4546 y FU(b)1849 4534 y FQ(i)1875 4546 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12915
FU(N)h(;)14 b(c)p FG(\))p FU(;)2093 4534 y FJ(\()2119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12916
4546 y FU(y)2163 4534 y FJ(\))2189 4546 y FF(And)2330
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12917
4509 y FJ(1)2330 4566 y FS(L)2380 4546 y FG(\()2412 4534
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12918
y FJ(\()2438 4546 y FU(x)2485 4534 y FJ(\))2511 4546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12919
y FU(P)e(;)i(y)s FG(\)\))2744 4534 y FS(.)2797 4546 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12920
FG(\001)2866 4558 y FJ(1)2904 4546 y FU(;)g FG(\001)3010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12921
4558 y FJ(2)3405 4482 y Gd(Cut)321 4762 y Gg(where)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12922
b Ga(c)f Gg(is)h(a)f(free)h(co-name)g(in)g Ga(M)33 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12923
Gg(\(shaded)25 b(box\).)30 b(Our)23 b(na)m(\250)-27 b(\021v)o(e)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12924
b(reduction)i(rule)e(w)o(ould)g(yield)p Black Black 1003
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12925
4947 a FG(\000)1055 4959 y FJ(1)1115 4935 y FS(.)1168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12926
4947 y FU(M)1280 4935 y FS(.)1333 4947 y FG(\001)1402
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12927
4959 y FJ(1)1440 4947 y FU(;)14 b(c)i FG(:)g FU(B)t FT(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12928
FU(C)q(;)e(a)i FG(:)h FU(B)87 b(x)17 b FG(:)f FU(B)t(;)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12929
FG(\000)2297 4959 y FJ(2)2357 4935 y FS(.)2410 4947 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12930
FU(P)2498 4935 y FS(.)2551 4947 y FG(\001)2620 4959 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12931
FJ(2)p 1003 4983 1655 4 v 1049 5081 a FG(\000)1101 5093
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12932
y FJ(1)1138 5081 y FU(;)g FG(\000)1227 5093 y FJ(2)1287
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12933
5069 y FS(.)1340 5081 y FF(Cut)p FG(\()1498 5069 y FQ(h)1525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12934
5081 y FU(a)1569 5069 y FQ(i)1596 5081 y FU(M)9 b(;)1723
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12935
5069 y FJ(\()1749 5081 y FU(x)1796 5069 y FJ(\))1822
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12936
5081 y FU(P)j FG(\))1942 5069 y FS(.)1995 5081 y FG(\001)2064
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12937
5093 y FJ(1)2101 5081 y FU(;)i FG(\001)2207 5093 y FJ(2)2245
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12938
5081 y FU(;)p 0.75 TeXcolorgray 0.75 TeXcolorgray 2282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12939
5106 330 107 v 0.75 TeXcolorgray Black 39 w(c)i FG(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12940
FU(B)t FT(^)q FU(C)p 0.75 TeXcolorgray Black 2699 5010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12941
a Gd(Cut)23 b FU(:)321 5297 y Gg(Unfortunately)-6 b(,)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12942
b Ga(c)31 b Gg(has)g(no)n(w)f(become)i(free)g(in)f(the)g(conclusion!)55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12943
b(The)30 b(problem)i(here)g(is)f(that)g(the)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12944
5362 1290 4 v 427 5417 a F3(3)456 5449 y FE(Rede)o(x)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12945
b F2(refers)f(to)h(a)g(place)g(in)g(a)g(term)g(where)g(a)g(reduction)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12946
(may)f(occur)l(.)p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12947
%%Page: 27 39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12948
TeXDict begin 27 38 bop Black 277 51 a Gb(2.2)23 b(Cut-Elimination)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12949
(in)e(Pr)n(opositional)k(Classical)f(Logic)1582 b(27)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12950
277 88 3691 4 v Black Black 277 537 V 277 4783 4 4247
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12951
v 818 744 a FL(Ax)p F4(\()p Ga(x;)15 b(c)p F4(\)[)p Ga(c)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12952
b F4(:=)1334 732 y F9(\()1362 744 y Ga(y)1410 732 y F9(\))1437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12953
744 y Ga(P)13 b F4(])1580 692 y F5(def)1587 744 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12954
b Ga(P)13 b F4([)p Ga(y)g F6(7!)d Ga(x)p F4(])814 935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12955
y FL(Ax)o F4(\()p Ga(y)s(;)15 b(a)p F4(\)[)p Ga(y)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12956
b F4(:=)1343 923 y FX(h)1370 935 y Ga(c)1409 923 y FX(i)1437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12957
935 y Ga(P)13 b F4(])1580 884 y F5(def)1587 935 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12958
b Ga(P)13 b F4([)p Ga(c)d F6(7!)g Ga(a)p F4(])550 1127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12959
y FL(Not)693 1141 y Gc(R)750 1127 y F4(\()785 1115 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12960
F9(\()813 1127 y Ga(x)865 1115 y F9(\))892 1127 y Ga(M)h(;)k(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12961
F4(\)[)p Ga(a)26 b F4(:=)1334 1115 y F9(\()1362 1127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12962
y Ga(y)1410 1115 y F9(\))1437 1127 y Ga(P)13 b F4(])1580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12963
1076 y F5(def)1587 1127 y F4(=)54 b FL(Cut)p F4(\()1885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12964
1115 y FX(h)1913 1127 y Ga(a)1961 1115 y FX(i)1989 1127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12965
y FL(Not)2131 1141 y Gc(R)2189 1127 y F4(\()2224 1115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12966
y F9(\()2252 1127 y Ga(x)2304 1115 y F9(\))2331 1127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12967
y Ga(M)10 b F4([)p Ga(a)26 b F4(:=)2649 1115 y F9(\()2676
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12968
1127 y Ga(y)2724 1115 y F9(\))2752 1127 y Ga(P)13 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12969
Ga(;)i(a)p F4(\))q Ga(;)3012 1115 y F9(\()3040 1127 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12970
Ga(y)3088 1115 y F9(\))3115 1127 y Ga(P)e F4(\))560 1319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12971
y FL(Not)703 1333 y Gc(L)755 1319 y F4(\()790 1307 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12972
FX(h)818 1319 y Ga(a)866 1307 y FX(i)893 1319 y Ga(M)d(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12973
b(x)p F4(\)[)p Ga(x)26 b F4(:=)1343 1307 y FX(h)1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12974
1319 y Ga(c)1409 1307 y FX(i)1437 1319 y Ga(P)13 b F4(])1580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12975
1268 y F5(def)1587 1319 y F4(=)54 b FL(Cut)p F4(\()1885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12976
1307 y FX(h)1913 1319 y Ga(c)1952 1307 y FX(i)1980 1319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12977
y Ga(P)13 b(;)2091 1307 y F9(\()2119 1319 y Ga(x)2171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12978
1307 y F9(\))2198 1319 y FL(Not)2341 1333 y Gc(L)2393
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12979
1319 y F4(\()2428 1307 y FX(h)2456 1319 y Ga(a)2504 1307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12980
y FX(i)2531 1319 y Ga(M)d F4([)p Ga(x)26 b F4(:=)2853
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12981
1307 y FX(h)2880 1319 y Ga(c)2919 1307 y FX(i)2947 1319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12982
y Ga(P)13 b F4(])p Ga(;)i(x)p F4(\))q(\))342 1511 y FL(And)497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12983
1525 y Gc(R)554 1511 y F4(\()589 1499 y FX(h)617 1511
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12984
y Ga(a)665 1499 y FX(i)693 1511 y Ga(M)10 b(;)831 1499
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12985
y FX(h)859 1511 y Ga(b)898 1499 y FX(i)925 1511 y Ga(N)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12986
b(c)p F4(\)[)p Ga(c)27 b F4(:=)1334 1499 y F9(\()1362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12987
1511 y Ga(y)1410 1499 y F9(\))1437 1511 y Ga(P)13 b F4(])1580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12988
1460 y F5(def)1587 1511 y F4(=)54 b FL(Cut)p F4(\()1885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12989
1499 y FX(h)1913 1511 y Ga(c)1952 1499 y FX(i)1980 1511
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12990
y FL(And)2134 1525 y Gc(R)2192 1511 y F4(\()2227 1499
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12991
y FX(h)2255 1511 y Ga(a)2303 1499 y FX(i)2330 1511 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12992
Ga(M)10 b F4([)p Ga(c)26 b F4(:=)2640 1499 y F9(\()2667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12993
1511 y Ga(y)2715 1499 y F9(\))2742 1511 y Ga(P)13 b F4(])q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12994
Ga(;)2879 1499 y FX(h)2906 1511 y Ga(b)2945 1499 y FX(i)2973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12995
1511 y Ga(N)d F4([)p Ga(c)26 b F4(:=)3267 1499 y F9(\()3294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12996
1511 y Ga(y)3342 1499 y F9(\))3370 1511 y Ga(P)13 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12997
Ga(;)i(c)p F4(\))q Ga(;)3621 1499 y F9(\()3649 1511 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12998
Ga(y)3697 1499 y F9(\))3724 1511 y Ga(P)e F4(\))552 1703
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 12999
y FL(And)707 1666 y Gc(i)707 1726 y(L)759 1703 y F4(\()794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13000
1691 y F9(\()822 1703 y Ga(x)874 1691 y F9(\))901 1703
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13001
y Ga(M)d(;)15 b(y)s F4(\)[)p Ga(y)29 b F4(:=)1343 1691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13002
y FX(h)1370 1703 y Ga(c)1409 1691 y FX(i)1437 1703 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13003
Ga(P)13 b F4(])1580 1652 y F5(def)1587 1703 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13004
b FL(Cut)p F4(\()1885 1691 y FX(h)1913 1703 y Ga(c)1952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13005
1691 y FX(i)1980 1703 y Ga(P)13 b(;)2091 1691 y F9(\()2119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13006
1703 y Ga(y)2167 1691 y F9(\))2194 1703 y FL(And)2349
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13007
1666 y Gc(i)2349 1726 y(L)2401 1703 y F4(\()2436 1691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13008
y F9(\()2463 1703 y Ga(x)2515 1691 y F9(\))2543 1703
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13009
y Ga(M)d F4([)p Ga(y)28 b F4(:=)2861 1691 y FX(h)2888
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13010
1703 y Ga(c)2927 1691 y FX(i)2955 1703 y Ga(P)13 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13011
Ga(;)i(y)s F4(\))q(\))615 1895 y FL(Or)714 1858 y Gc(i)714
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13012
1918 y(R)772 1895 y F4(\()807 1883 y FX(h)834 1895 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13013
Ga(a)882 1883 y FX(i)910 1895 y Ga(M)10 b(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13014
F4(\)[)p Ga(c)27 b F4(:=)1334 1883 y F9(\()1362 1895
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13015
y Ga(y)1410 1883 y F9(\))1437 1895 y Ga(P)13 b F4(])1580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13016
1844 y F5(def)1587 1895 y F4(=)54 b FL(Cut)p F4(\()1885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13017
1883 y FX(h)1913 1895 y Ga(c)1952 1883 y FX(i)1980 1895
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13018
y FL(Or)2079 1858 y Gc(i)2079 1918 y(R)2137 1895 y F4(\()2172
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13019
1883 y FX(h)2200 1895 y Ga(a)2248 1883 y FX(i)2275 1895
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13020
y Ga(M)10 b F4([)p Ga(c)26 b F4(:=)2584 1883 y F9(\()2612
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13021
1895 y Ga(y)2660 1883 y F9(\))2687 1895 y Ga(P)13 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13022
Ga(;)i(c)p F4(\))q Ga(;)2938 1883 y F9(\()2966 1895 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13023
Ga(y)3014 1883 y F9(\))3041 1895 y Ga(P)e F4(\))385 2087
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13024
y FL(Or)484 2101 y Gc(L)536 2087 y F4(\()571 2075 y F9(\()599
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13025
2087 y Ga(x)651 2075 y F9(\))678 2087 y Ga(M)e(;)817
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13026
2075 y F9(\()844 2087 y Ga(y)892 2075 y F9(\))920 2087
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13027
y Ga(N)f(;)15 b(z)t F4(\)[)p Ga(z)30 b F4(:=)1343 2075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13028
y FX(h)1370 2087 y Ga(c)1409 2075 y FX(i)1437 2087 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13029
Ga(P)13 b F4(])1580 2036 y F5(def)1587 2087 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13030
b FL(Cut)p F4(\()1885 2075 y FX(h)1913 2087 y Ga(c)1952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13031
2075 y FX(i)1980 2087 y Ga(P)13 b(;)2091 2075 y F9(\()2119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13032
2087 y Ga(z)2165 2075 y F9(\))2192 2087 y FL(Or)2292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13033
2101 y Gc(L)2344 2087 y F4(\()2379 2075 y F9(\()2407
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13034
2087 y Ga(x)2459 2075 y F9(\))2486 2087 y Ga(M)d F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13035
Ga(z)30 b F4(:=)2802 2075 y FX(h)2830 2087 y Ga(c)2869
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13036
2075 y FX(i)2896 2087 y Ga(P)13 b F4(])q Ga(;)3033 2075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13037
y F9(\()3060 2087 y Ga(y)3108 2075 y F9(\))3136 2087
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13038
y Ga(N)d F4([)p Ga(z)29 b F4(:=)3437 2075 y FX(h)3464
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13039
2087 y Ga(c)3503 2075 y FX(i)3531 2087 y Ga(P)13 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13040
Ga(;)i(z)t F4(\))q(\))463 2279 y FL(Imp)608 2301 y Gc(R)665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13041
2279 y F4(\()700 2267 y F9(\()728 2279 y Ga(x)780 2267
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13042
y F9(\))q FX(h)835 2279 y Ga(a)883 2267 y FX(i)911 2279
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13043
y Ga(M)10 b(;)15 b(b)p F4(\)[)p Ga(b)26 b F4(:=)1334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13044
2267 y F9(\()1362 2279 y Ga(y)1410 2267 y F9(\))1437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13045
2279 y Ga(P)13 b F4(])1580 2228 y F5(def)1587 2279 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13046
F4(=)54 b FL(Cut)p F4(\()1885 2267 y FX(h)1913 2279 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13047
Ga(b)1952 2267 y FX(i)1980 2279 y FL(Imp)2124 2301 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13048
Gc(R)2182 2279 y F4(\()2217 2267 y F9(\()2244 2279 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13049
Ga(x)2296 2267 y F9(\))q FX(h)2351 2279 y Ga(a)2399 2267
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13050
y FX(i)2427 2279 y Ga(M)10 b F4([)p Ga(b)25 b F4(:=)2736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13051
2267 y F9(\()2763 2279 y Ga(y)2811 2267 y F9(\))2838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13052
2279 y Ga(P)13 b F4(])q Ga(;)i(b)p F4(\))p Ga(;)3089
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13053
2267 y F9(\()3117 2279 y Ga(y)3165 2267 y F9(\))3193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13054
2279 y Ga(P)e F4(\))336 2471 y FL(Imp)481 2493 y Gc(L)533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13055
2471 y F4(\()568 2459 y FX(h)596 2471 y Ga(a)644 2459
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13056
y FX(i)671 2471 y Ga(M)d(;)809 2459 y F9(\()837 2471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13057
y Ga(x)889 2459 y F9(\))917 2471 y Ga(N)f(;)15 b(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13058
F4(\)[)p Ga(y)29 b F4(:=)1343 2459 y FX(h)1370 2471 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13059
Ga(c)1409 2459 y FX(i)1437 2471 y Ga(P)13 b F4(])1580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13060
2420 y F5(def)1587 2471 y F4(=)54 b FL(Cut)p F4(\()1885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13061
2459 y FX(h)1913 2471 y Ga(c)1952 2459 y FX(i)1980 2471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13062
y Ga(P)13 b(;)2091 2459 y F9(\()2119 2471 y Ga(y)2167
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13063
2459 y F9(\))2194 2471 y FL(Imp)2338 2493 y Gc(L)2391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13064
2471 y F4(\()2426 2459 y FX(h)2453 2471 y Ga(a)2501 2459
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13065
y FX(i)2529 2471 y Ga(M)d F4([)p Ga(y)28 b F4(:=)2847
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13066
2459 y FX(h)2874 2471 y Ga(c)2913 2459 y FX(i)2941 2471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13067
y Ga(P)13 b F4(])p Ga(;)3077 2459 y F9(\()3105 2471 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13068
Ga(x)3157 2459 y F9(\))3184 2471 y Ga(N)d F4([)p Ga(y)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13069
b F4(:=)3487 2459 y FX(h)3514 2471 y Ga(c)3553 2459 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13070
FX(i)3581 2471 y Ga(P)13 b F4(])p Ga(;)i(y)s F4(\))q(\))315
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13071
2734 y Gb(Otherwise:)1114 2926 y FL(Ax)o F4(\()p Ga(x;)g(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13072
F4(\)[)p Ga(\033)s F4(])1580 2874 y F5(def)1587 2926
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13073
y F4(=)54 b FL(Ax)p F4(\()p Ga(x;)15 b(a)p F4(\))788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13074
3118 y FL(Cut)o F4(\()960 3106 y FX(h)988 3118 y Ga(a)1036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13075
3106 y FX(i)1064 3118 y Ga(M)10 b(;)1202 3106 y F9(\()1230
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13076
3118 y Ga(x)1282 3106 y F9(\))1309 3118 y Ga(N)g F4(\)[)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13077
Ga(\033)s F4(])1580 3066 y F5(def)1587 3118 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13078
b FL(Cut)p F4(\()1885 3106 y FX(h)1913 3118 y Ga(a)1961
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13079
3106 y FX(i)2014 3118 y Ga(M)10 b F4([)p Ga(\033)s F4(])q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13080
Ga(;)2258 3106 y F9(\()2285 3118 y Ga(x)2337 3106 y F9(\))2390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13081
3118 y Ga(N)g F4([)p Ga(\033)s F4(])q(\))863 3310 y FL(Not)1006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13082
3324 y Gc(R)1063 3310 y F4(\()1098 3298 y F9(\()1126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13083
3310 y Ga(x)1178 3298 y F9(\))1205 3310 y Ga(M)h(;)k(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13084
F4(\)[)p Ga(\033)s F4(])1580 3258 y F5(def)1587 3310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13085
y F4(=)54 b FL(Not)1855 3324 y Gc(R)1913 3310 y F4(\()1948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13086
3298 y F9(\()1976 3310 y Ga(x)2028 3298 y F9(\))2080
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13087
3310 y Ga(M)10 b F4([)p Ga(\033)s F4(])q Ga(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13088
F4(\))868 3502 y FL(Not)1011 3516 y Gc(L)1063 3502 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13089
F4(\()1098 3490 y FX(h)1126 3502 y Ga(a)1174 3490 y FX(i)1202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13090
3502 y Ga(M)10 b(;)15 b(x)p F4(\)[)p Ga(\033)s F4(])1580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13091
3450 y F5(def)1587 3502 y F4(=)54 b FL(Not)1855 3516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13092
y Gc(L)1907 3502 y F4(\()1942 3490 y FX(h)1970 3502 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13093
Ga(a)2018 3490 y FX(i)2071 3502 y Ga(M)10 b F4([)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13094
F4(])q Ga(;)15 b(x)p F4(\))646 3694 y FL(And)801 3708
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13095
y Gc(R)859 3694 y F4(\()894 3682 y FX(h)921 3694 y Ga(a)969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13096
3682 y FX(i)997 3694 y Ga(M)10 b(;)1135 3682 y FX(h)1163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13097
3694 y Ga(b)1202 3682 y FX(i)1229 3694 y Ga(N)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13098
b(c)p F4(\)[)p Ga(\033)s F4(])1580 3642 y F5(def)1587
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13099
3694 y F4(=)54 b FL(And)1867 3708 y Gc(R)1925 3694 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13100
F4(\()1960 3682 y FX(h)1987 3694 y Ga(a)2035 3682 y FX(i)2088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13101
3694 y Ga(M)10 b F4([)p Ga(\033)s F4(])q Ga(;)2332 3682
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13102
y FX(h)2360 3694 y Ga(b)2399 3682 y FX(i)2452 3694 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13103
Ga(N)g F4([)p Ga(\033)s F4(])p Ga(;)15 b(c)p F4(\))857
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13104
3886 y FL(And)1011 3849 y Gc(i)1011 3909 y(L)1063 3886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13105
y F4(\()1098 3874 y F9(\()1126 3886 y Ga(x)1178 3874
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13106
y F9(\))1206 3886 y Ga(M)10 b(;)15 b(y)s F4(\)[)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13107
F4(])1580 3834 y F5(def)1587 3886 y F4(=)54 b FL(And)1867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13108
3849 y Gc(i)1867 3909 y(L)1919 3886 y F4(\()1954 3874
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13109
y F9(\()1982 3886 y Ga(x)2034 3874 y F9(\))2087 3886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13110
y Ga(M)10 b F4([)p Ga(\033)s F4(])p Ga(;)15 b(y)s F4(\))919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13111
4077 y FL(Or)1018 4041 y Gc(i)1018 4101 y(R)1076 4077
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13112
y F4(\()1111 4065 y FX(h)1139 4077 y Ga(a)1187 4065 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13113
FX(i)1215 4077 y Ga(M)10 b(;)15 b(b)p F4(\)[)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13114
F4(])1580 4026 y F5(def)1587 4077 y F4(=)54 b FL(Or)1812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13115
4041 y Gc(i)1812 4101 y(R)1869 4077 y F4(\()1904 4065
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13116
y FX(h)1932 4077 y Ga(a)1980 4065 y FX(i)2033 4077 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13117
Ga(M)10 b F4([)p Ga(\033)s F4(])q Ga(;)15 b(b)p F4(\))688
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13118
4269 y FL(Or)787 4283 y Gc(L)839 4269 y F4(\()874 4257
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13119
y F9(\()902 4269 y Ga(x)954 4257 y F9(\))981 4269 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13120
b(;)1119 4257 y F9(\()1147 4269 y Ga(y)1195 4257 y F9(\))1222
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13121
4269 y Ga(N)g(;)15 b(z)t F4(\)[)p Ga(\033)s F4(])1580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13122
4218 y F5(def)1587 4269 y F4(=)54 b FL(Or)1812 4283 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13123
Gc(L)1864 4269 y F4(\()1899 4257 y F9(\()1927 4269 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13124
Ga(x)1979 4257 y F9(\))2031 4269 y Ga(M)10 b F4([)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13125
F4(])q Ga(;)2275 4257 y F9(\()2303 4269 y Ga(y)2351 4257
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13126
y F9(\))2403 4269 y Ga(N)g F4([)p Ga(\033)s F4(])q Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13127
b(z)t F4(\))767 4461 y FL(Imp)912 4483 y Gc(R)969 4461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13128
y F4(\()1004 4449 y F9(\()1032 4461 y Ga(x)1084 4449
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13129
y F9(\))q FX(h)1139 4461 y Ga(a)1187 4449 y FX(i)1215
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13130
4461 y Ga(M)10 b(;)15 b(b)p F4(\)[)p Ga(\033)s F4(])1580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13131
4410 y F5(def)1587 4461 y F4(=)54 b FL(Imp)1857 4483
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13132
y Gc(R)1915 4461 y F4(\()1950 4449 y F9(\()1977 4461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13133
y Ga(x)2029 4449 y F9(\))q FX(h)2084 4461 y Ga(a)2132
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13134
4449 y FX(i)2185 4461 y Ga(M)10 b F4([)p Ga(\033)s F4(])q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13135
Ga(;)15 b(b)p F4(\))641 4653 y FL(Imp)785 4675 y Gc(L)837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13136
4653 y F4(\()872 4641 y FX(h)900 4653 y Ga(a)948 4641
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13137
y FX(i)975 4653 y Ga(M)c(;)1114 4641 y F9(\()1141 4653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13138
y Ga(x)1193 4641 y F9(\))1221 4653 y Ga(N)f(;)15 b(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13139
F4(\)[)p Ga(\033)s F4(])1580 4602 y F5(def)1587 4653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13140
y F4(=)54 b FL(Imp)1857 4675 y Gc(L)1909 4653 y F4(\()1944
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13141
4641 y FX(h)1972 4653 y Ga(a)2020 4641 y FX(i)2073 4653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13142
y Ga(M)10 b F4([)p Ga(\033)s F4(])p Ga(;)2316 4641 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13143
F9(\()2344 4653 y Ga(x)2396 4641 y F9(\))2449 4653 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13144
Ga(N)g F4([)p Ga(\033)s F4(])q Ga(;)15 b(y)s F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13145
3965 4783 V 277 4786 3691 4 v Black 1340 4940 a Gg(Figure)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13146
b(2.4:)29 b(Proof)24 b(substitution.)p Black Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13147
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13148
%%Page: 28 40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13149
TeXDict begin 28 39 bop Black -144 51 a Gb(28)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13150
b(Calculi)p -144 88 3691 4 v Black 321 294 a Gg(original)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13151
b(proof,)g(despite)f(\002rst)f(appearances,)k(is)28 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13152
F7(not)h Gg(a)g(logical)h(cut,)g(b)n(ut)f(is)g(in)g(f)o(act)g(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13153
(commuting)321 407 y(cut,)c(and)g(should)h(really)g(be)e(reduced)j(to)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13154
Black Black 1078 586 a FL(And)1233 600 y Gc(R)1290 586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13155
y F4(\()1325 574 y FX(h)1353 586 y Ga(a)1401 574 y FX(i)1429
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13156
586 y Ga(M)10 b(;)1567 574 y FX(h)1595 586 y Ga(b)1634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13157
574 y FX(i)1661 586 y Ga(N)g(;)15 b(c)p F4(\)[)p Ga(c)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13158
b F4(:=)2070 574 y F9(\()2098 586 y Ga(y)2146 574 y F9(\))2173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13159
586 y FL(And)2327 549 y F9(1)2327 609 y Gc(L)2380 586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13160
y F4(\()2415 574 y F9(\()2442 586 y Ga(x)2494 574 y F9(\))2522
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13161
586 y Ga(P)13 b(;)i(y)s F4(\)])26 b Gg(.)321 765 y(Consequently)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13162
b(,)27 b(we)c(ensure)i(that)f(logical)i(reduction)g(rules)e(apply)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13163
(only)g(where)f(the)g(cut-formula)i(is)321 878 y F7(fr)m(eshly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13164
Gg(introduced.)37 b(Figure)26 b(2.5)f(gi)n(v)o(es)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13165
(cut-reductions)k(for)25 b(logical)i(cuts,)f(denoted)i(by)3428
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13166
841 y Gc(l)3353 878 y F6(\000)-31 b(\000)g(!)p Gg(,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13167
991 y(and)27 b(commuting)h(cuts,)g(denoted)g(by)1614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13168
954 y Gc(c)1544 991 y F6(\000)-31 b(\000)f(!)p Gg(.)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13169
b(W)-7 b(e)25 b(automatically)30 b(assume)d(that)g(the)g(reductions)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13170
(are)321 1104 y(closed)24 b(under)f(conte)o(xt)g(formation,)h(which)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13171
(is)g(a)f(standard)j(con)l(v)o(ention)h(in)d(term)g(re)n(writing.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13172
b(F)o(or)21 b(the)321 1217 y(cut-reduction)28 b F6(\033)899
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13173
1231 y Gc(R)957 1217 y Ga(=)p F6(\033)1073 1231 y Gc(L)1148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13174
1217 y Gg(\(Rule)c(4\))f(there)i(are)f(a)f(fe)n(w)f(remarks)j(w)o(orth)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13175
f(making.)p Black 321 1404 a Gb(Remark)f(2.2.7:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13176
Black 458 1526 a F6(\017)p Black 46 w Gg(First,)37 b(there)f(are)g(tw)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13177
(o)f(w)o(ays)g(to)g(reduce)i(a)d(cut-rule)k(ha)n(ving)f(an)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13178
(implication)i(as)e(the)h(cut-)549 1639 y(formula.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13179
b(Consider)c(the)f(follo)n(wing)h(proof-fragment)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13180
Black 989 1819 a FU(x)17 b FG(:)f FU(B)1182 1807 y FS(.)1235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13181
1819 y FU(M)1348 1807 y FS(.)1401 1819 y FU(a)g FG(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13182
FU(C)p 752 1839 1051 4 v 775 1906 a FS(.)828 1918 y FF(Imp)960
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13183
1938 y FS(R)1015 1918 y FG(\()1047 1906 y FJ(\()1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13184
1918 y FU(x)1120 1906 y FJ(\))p FQ(h)1173 1918 y FU(a)1217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13185
1906 y FQ(i)1244 1918 y FU(M)9 b(;)14 b(b)p FG(\))1461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13186
1906 y FS(.)1514 1918 y FU(b)i FG(:)h FU(B)t FT(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13187
FU(C)1844 1855 y FT(\033)1909 1867 y FS(R)2128 1791 y(.)2181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13188
1803 y FU(N)2280 1791 y FS(.)2333 1803 y FU(c)f FG(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13189
FU(B)254 b(y)19 b FG(:)d FU(C)2928 1791 y FS(.)2981 1803
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13190
y FU(P)3069 1791 y FS(.)p 2046 1839 1135 4 v 2046 1918
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13191
a FU(z)k FG(:)c FU(B)t FT(\033)p FU(C)2364 1906 y FS(.)2417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13192
1918 y FF(Imp)2549 1938 y FS(L)2599 1918 y FG(\()2631
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13193
1906 y FQ(h)2658 1918 y FU(c)2694 1906 y FQ(i)2721 1918
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13194
y FU(N)8 b(;)2833 1906 y FJ(\()2859 1918 y FU(y)2903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13195
1906 y FJ(\))2929 1918 y FU(P)k(;)i(z)t FG(\))3128 1906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13196
y FS(.)3223 1855 y FT(\033)3287 1867 y FS(L)p 752 1958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13197
2429 4 v 1059 2025 a(.)1112 2037 y FF(Cut)p FG(\()1270
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13198
2025 y FQ(h)1297 2037 y FU(b)1333 2025 y FQ(i)1360 2037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13199
y FF(Imp)1492 2057 y FS(R)1546 2037 y FG(\()1578 2025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13200
y FJ(\()1604 2037 y FU(x)1651 2025 y FJ(\))q FQ(h)1705
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13201
2037 y FU(a)1749 2025 y FQ(i)1775 2037 y FU(M)9 b(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13202
b(b)p FG(\))p FU(;)2007 2025 y FJ(\()2033 2037 y FU(z)2076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13203
2025 y FJ(\))2101 2037 y FF(Imp)2233 2057 y FS(L)2282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13204
2037 y FG(\()2314 2025 y FQ(h)2342 2037 y FU(c)2378 2025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13205
y FQ(i)2404 2037 y FU(N)9 b(;)2517 2025 y FJ(\()2543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13206
2037 y FU(y)2587 2025 y FJ(\))2613 2037 y FU(P)i(;)j(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13207
FG(\)\))2844 2025 y FS(.)3223 1986 y Gd(Cut)549 2224
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13208
y Gg(which)23 b(can)h(be)g(reduced)h(either)g(to)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13209
Black 1154 2358 a FS(.)1207 2370 y FU(N)1305 2358 y FS(.)1358
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13210
2370 y FU(c)17 b FG(:)f FU(B)87 b(x)17 b FG(:)f FU(B)1793
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13211
2358 y FS(.)1846 2370 y FU(M)1959 2358 y FS(.)2012 2370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13212
y FU(a)g FG(:)h FU(C)p 1131 2390 1047 4 v 1228 2456 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13213
FS(.)1280 2468 y FF(Cut)p FG(\()1438 2456 y FQ(h)1466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13214
2468 y FU(c)1502 2456 y FQ(i)1528 2468 y FU(N)9 b(;)1641
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13215
2456 y FJ(\()1667 2468 y FU(x)1714 2456 y FJ(\))1740
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13216
2468 y FU(M)g FG(\))1885 2456 y FS(.)1938 2468 y FU(a)16
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13217
b FG(:)h FU(C)2218 2417 y Gd(Cut)2421 2468 y FU(y)i FG(:)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13218
FU(C)2609 2456 y FS(.)2662 2468 y FU(P)2750 2456 y FS(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13219
1204 2509 1599 4 v 1417 2576 a(.)1470 2588 y FF(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13220
FG(\()1628 2576 y FQ(h)1655 2588 y FU(a)1699 2576 y FQ(i)1726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13221
2588 y FF(Cut)p FG(\()1884 2576 y FQ(h)1911 2588 y FU(c)1947
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13222
2576 y FQ(i)1974 2588 y FU(N)8 b(;)2086 2576 y FJ(\()2112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13223
2588 y FU(x)2159 2576 y FJ(\))2186 2588 y FU(M)g FG(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13224
FU(;)2345 2576 y FJ(\()2370 2588 y FU(y)2414 2576 y FJ(\))2440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13225
2588 y FU(P)k FG(\))2560 2576 y FS(.)2844 2537 y Gd(Cut)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13226
2717 y Gg(or)23 b(to)p Black Black 1246 2907 a FS(.)1299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13227
2919 y FU(N)1398 2907 y FS(.)1451 2919 y FU(c)16 b FG(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13228
FU(B)1669 2804 y(x)g FG(:)f FU(B)1862 2792 y FS(.)1915
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13229
2804 y FU(M)2028 2792 y FS(.)2081 2804 y FU(a)g FG(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13230
FU(C)90 b(y)19 b FG(:)d FU(C)2516 2792 y FS(.)2569 2804
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13231
y FU(P)2657 2792 y FS(.)p 1669 2840 1042 4 v 1741 2919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13232
a FU(x)h FG(:)f FU(B)1934 2907 y FS(.)1987 2919 y FF(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13233
FG(\()2145 2907 y FQ(h)2172 2919 y FU(a)2216 2907 y FQ(i)2243
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13234
2919 y FU(M)9 b(;)2370 2907 y FJ(\()2396 2919 y FU(y)2440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13235
2907 y FJ(\))2465 2919 y FU(P)j FG(\))2585 2907 y FS(.)2752
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13236
2867 y Gd(Cut)p 1223 2959 1416 4 v 1344 3026 a FS(.)1397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13237
3038 y FF(Cut)p FG(\()1555 3026 y FQ(h)1582 3038 y FU(c)1618
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13238
3026 y FQ(i)1645 3038 y FU(N)d(;)1758 3026 y FJ(\()1784
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13239
3038 y FU(x)1831 3026 y FJ(\))1857 3038 y FF(Cut)p FG(\()2015
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13240
3026 y FQ(h)2042 3038 y FU(a)2086 3026 y FQ(i)2113 3038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13241
y FU(M)f(;)2239 3026 y FJ(\()2265 3038 y FU(y)2309 3026
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13242
y FJ(\))2335 3038 y FU(P)k FG(\)\))2487 3026 y FS(.)2680
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13243
2987 y Gd(Cut)549 3226 y Gg(Therefore)26 b(we)e(ha)n(v)o(e)i(included)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13244
(tw)o(o)e(reductions,)j(which)d(entails)i(that)e(our)h(cut-elimination)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13245
549 3338 y(procedure)g(is)d(non-deterministic.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13246
458 3509 a F6(\017)p Black 46 w Gg(Second,)38 b(special)e(care)f(needs)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13247
h(to)e(be)h(tak)o(en)h(so)f(that)g(there)g(is)g(no)g(clash)g(between)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13248
(bound)549 3622 y(and)31 b(free)g(\(co-\)names)s(.)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13249
b(The)31 b(term)g FL(Imp)1876 3644 y Gc(R)1934 3622 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13250
F4(\()1969 3610 y F9(\()1996 3622 y Ga(x)2048 3610 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13251
F9(\))q FX(h)2103 3622 y Ga(a)2151 3610 y FX(i)2179 3622
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13252
y Ga(M)10 b(;)15 b(b)p F4(\))31 b Gg(binds)h Ga(x)e Gg(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13253
Ga(a)g Gg(simultaneously;)549 3735 y(ho)n(we)n(v)o(er)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13254
b(in)g(the)h(reducts)h(the)e(cut-rules)j(bind)e Ga(x)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13255
Gg(and)i Ga(a)p Gg(,)e(separately)-6 b(.)34 b(Therefore)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13256
b(in)f(the)h(\002rst)549 3848 y(reduction)33 b(rule)e(we)f(need)i(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13257
(ensure)h(that)g Ga(a)e Gg(is)g(not)h(a)g(free)g(co-name)h(in)2966
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13258
3836 y FX(h)2993 3848 y Ga(c)3032 3836 y FX(i)3060 3848
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13259
y Ga(N)40 b Gg(and)31 b(in)g(the)549 3961 y(second)k(rule)g(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13260
Ga(x)e Gg(is)h(not)g(a)g(free)g(name)g(in)2089 3949 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13261
F9(\()2117 3961 y Ga(y)2165 3949 y F9(\))2192 3961 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13262
Ga(P)13 b Gg(.)59 b(This)34 b(can)h(al)o(w)o(ays)g(be)f(achie)n(v)o(ed)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13263
h(by)549 4074 y(renaming)22 b Ga(a)f Gg(and)h Ga(x)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13264
Gg(appropriately:)32 b(the)o(y)22 b(are)g(binders)h(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13265
FL(Imp)2573 4096 y Gc(R)2631 4074 y F4(\()2666 4062 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13266
F9(\()2694 4074 y Ga(x)2746 4062 y F9(\))p FX(h)2801
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13267
4074 y Ga(a)2849 4062 y FX(i)2876 4074 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13268
b(b)p F4(\))p Gg(.)28 b(W)-7 b(e)21 b(assume)549 4187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13269
y(that)j(the)f(renaming)j(is)d(done)h(implicitly)i(in)d(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13270
(cut-elimination)j(procedure.)321 4412 y(W)-7 b(e)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13271
b(are)g(no)n(w)f(ready)j(to)e(formulate)h(the)g(cut-elimination)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13272
(procedure,)f(which)e(we)e(shall)i(de\002ne)g(in)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13273
4525 y(terms)24 b(of)f(a)h(reduction)i(system.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13274
321 4712 a Gb(De\002nition)d(2.2.8)g Gg(\(Cut-Elimination)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13275
(Procedure\))p Gb(:)p Black 321 4825 a Gg(The)d(cut-elimination)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13276
b(procedure)e F4(\()p FY(T)t Ga(;)1613 4788 y Gc(cut)1583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13277
4825 y F6(\000)-32 b(\000)h(!)p F4(\))23 b Gg(is)g(a)g(reduction)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13278
(system)d(where:)p Black 458 4988 a F6(\017)p Black 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13279
w FY(T)e Gg(is)i(the)g(set)f(of)h(well-typed)h(terms,)f(and)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13280
Black 458 5159 a F6(\017)p Black 579 5122 a Gc(cut)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13281
5159 y F6(\000)-32 b(\000)h(!)23 b Gg(consists)i(of)f(the)f(reductions)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13282
k(for)d(logical)h(cuts)f(and)g(commuting)h(cuts;)f(that)g(is)1662
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13283
5297 y Gc(cut)1631 5334 y F6(\000)-31 b(\000)f(!)1826
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13284
5283 y F5(def)1834 5334 y F4(=)2011 5297 y Gc(l)1937
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13285
5334 y F6(\000)g(\000)h(!)25 b([)2288 5297 y Gc(c)2218
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13286
5334 y F6(\000)-31 b(\000)f(!)51 b Ga(:)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13287
eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13288
%%Page: 29 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13289
TeXDict begin 29 40 bop Black 277 51 a Gb(2.2)23 b(Cut-Elimination)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13290
(in)e(Pr)n(opositional)k(Classical)f(Logic)1582 b(29)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13291
277 88 3691 4 v Black Black 277 663 V 277 4657 4 3995
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13292
v 475 830 a(Logical)24 b(Cuts)46 b(\()p FO(i)30 b FD(=)f(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13293
FO(;)18 b FD(2)p Gb(\):)546 1036 y Gg(1.)99 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13294
F4(\()886 1024 y FX(h)914 1036 y Ga(a)962 1024 y FX(i)990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13295
1036 y FL(Not)1132 1050 y Gc(R)1190 1036 y F4(\()1225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13296
1024 y F9(\()1253 1036 y Ga(x)1305 1024 y F9(\))1332
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13297
1036 y Ga(M)10 b(;)15 b(a)p F4(\))q Ga(;)1594 1024 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13298
F9(\()1622 1036 y Ga(y)1670 1024 y F9(\))1697 1036 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13299
FL(Not)1840 1050 y Gc(L)1892 1036 y F4(\()1927 1024 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13300
FX(h)1955 1036 y Ga(b)1994 1024 y FX(i)2021 1036 y Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13301
b(;)15 b(y)s F4(\))q(\))2363 999 y Gc(l)2288 1036 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13302
b(\000)g(!)25 b FL(Cut)p F4(\()2657 1024 y FX(h)2685
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13303
1036 y Ga(b)2724 1024 y FX(i)2751 1036 y Ga(N)10 b(;)2874
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13304
1024 y F9(\()2902 1036 y Ga(x)2954 1024 y F9(\))2981
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13305
1036 y Ga(M)g F4(\))996 1219 y Gg(if)23 b FL(Not)1217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13306
1233 y Gc(R)1275 1219 y F4(\()1310 1207 y F9(\()1337
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13307
1219 y Ga(x)1389 1207 y F9(\))1417 1219 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13308
b(a)p F4(\))23 b Gg(and)h FL(Not)1958 1233 y Gc(L)2010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13309
1219 y F4(\()2045 1207 y FX(h)2073 1219 y Ga(b)2112 1207
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13310
y FX(i)2140 1219 y Ga(N)9 b(;)15 b(y)s F4(\))24 b Gg(freshly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13311
(introduce)h Ga(a)d Gg(and)h Ga(y)546 1496 y Gg(2.)99
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13312
b FL(Cut)p F4(\()886 1484 y FX(h)914 1496 y Ga(b)953
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13313
1484 y FX(i)980 1496 y FL(And)1135 1510 y Gc(R)1193 1496
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13314
y F4(\()1228 1484 y FX(h)1256 1496 y Ga(a)1304 1510 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13315
F9(1)1343 1484 y FX(i)1370 1496 y Ga(M)1458 1510 y F9(1)1498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13316
1496 y Ga(;)1538 1484 y FX(h)1566 1496 y Ga(a)1614 1510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13317
y F9(2)1653 1484 y FX(i)1681 1496 y Ga(M)1769 1510 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13318
F9(2)1809 1496 y Ga(;)15 b(b)p F4(\))p Ga(;)1963 1484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13319
y F9(\()1991 1496 y Ga(y)2039 1484 y F9(\))2066 1496
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13320
y FL(And)2221 1459 y Gc(i)2221 1519 y(L)2273 1496 y F4(\()2308
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13321
1484 y F9(\()2336 1496 y Ga(x)2388 1484 y F9(\))2415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13322
1496 y Ga(N)10 b(;)15 b(y)s F4(\))q(\))2757 1459 y Gc(l)2682
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13323
1496 y F6(\000)-31 b(\000)g(!)25 b FL(Cut)p F4(\()3051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13324
1484 y FX(h)3079 1496 y Ga(a)3127 1510 y Gc(i)3155 1484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13325
y FX(i)3182 1496 y Ga(M)3270 1510 y Gc(i)3299 1496 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13326
Ga(;)3339 1484 y F9(\()3367 1496 y Ga(x)3419 1484 y F9(\))3446
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13327
1496 y Ga(N)10 b F4(\))996 1679 y Gg(if)23 b FL(And)1229
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13328
1693 y Gc(R)1286 1679 y F4(\()1321 1667 y FX(h)1349 1679
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13329
y Ga(a)1397 1693 y F9(1)1437 1667 y FX(i)1464 1679 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13330
Ga(M)1552 1693 y F9(1)1592 1679 y Ga(;)1632 1667 y FX(h)1660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13331
1679 y Ga(a)1708 1693 y F9(2)1747 1667 y FX(i)1775 1679
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13332
y Ga(M)1863 1693 y F9(2)1902 1679 y Ga(;)15 b(b)p F4(\))24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13333
b Gg(and)g FL(And)2348 1642 y Gc(i)2348 1702 y(L)2400
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13334
1679 y F4(\()2435 1667 y F9(\()2463 1679 y Ga(x)2515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13335
1667 y F9(\))2543 1679 y Ga(N)10 b(;)15 b(y)s F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13336
b Gg(freshly)i(introduce)h Ga(b)d Gg(and)h Ga(y)546 1956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13337
y Gg(3.)99 b FL(Cut)p F4(\()886 1944 y FX(h)914 1956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13338
y Ga(b)953 1944 y FX(i)980 1956 y FL(Or)1080 1919 y Gc(i)1080
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13339
1979 y(R)1137 1956 y F4(\()1172 1944 y FX(h)1200 1956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13340
y Ga(a)1248 1944 y FX(i)1276 1956 y Ga(M)10 b(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13341
F4(\))q Ga(;)1529 1944 y F9(\()1556 1956 y Ga(y)1604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13342
1944 y F9(\))1632 1956 y FL(Or)1731 1970 y Gc(L)1783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13343
1956 y F4(\()1818 1944 y F9(\()1846 1956 y Ga(x)1898
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13344
1970 y F9(1)1937 1944 y(\))1965 1956 y Ga(N)2038 1970
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13345
y F9(1)2077 1956 y Ga(;)2117 1944 y F9(\()2145 1956 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13346
Ga(x)2197 1970 y F9(2)2236 1944 y(\))2264 1956 y Ga(N)2337
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13347
1970 y F9(2)2376 1956 y Ga(;)g(y)s F4(\))q(\))2635 1919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13348
y Gc(l)2561 1956 y F6(\000)-32 b(\000)h(!)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13349
F4(\()2929 1944 y FX(h)2957 1956 y Ga(a)3005 1944 y FX(i)3032
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13350
1956 y Ga(M)10 b(;)3170 1944 y F9(\()3198 1956 y Ga(x)3250
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13351
1970 y Gc(i)3278 1944 y F9(\))3306 1956 y Ga(N)3379 1970
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13352
y Gc(i)3407 1956 y F4(\))996 2139 y Gg(if)23 b FL(Or)1173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13353
2153 y Gc(L)1226 2139 y F4(\()1261 2127 y F9(\()1288
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13354
2139 y Ga(x)1340 2153 y F9(1)1380 2127 y(\))1407 2139
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13355
y Ga(N)1480 2153 y F9(1)1520 2139 y Ga(;)1560 2127 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13356
F9(\()1588 2139 y Ga(x)1640 2153 y F9(2)1679 2127 y(\))1706
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13357
2139 y Ga(N)1779 2153 y F9(2)1819 2139 y Ga(;)15 b(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13358
F4(\))23 b Gg(and)h FL(Or)2218 2102 y Gc(i)2218 2162
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13359
y(R)2276 2139 y F4(\()2311 2127 y FX(h)2339 2139 y Ga(a)2387
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13360
2127 y FX(i)2414 2139 y Ga(M)11 b(;)k(b)p F4(\))23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13361
Gg(freshly)i(introduce)h Ga(y)g Gg(and)e Ga(b)546 2416
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13362
y Gg(4.)99 b FL(Cut)p F4(\()886 2404 y FX(h)914 2416
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13363
y Ga(b)953 2404 y FX(i)980 2416 y FL(Imp)1125 2438 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13364
Gc(R)1183 2416 y F4(\()1218 2404 y F9(\()1245 2416 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13365
Ga(x)1297 2404 y F9(\))q FX(h)1352 2416 y Ga(a)1400 2404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13366
y FX(i)1428 2416 y Ga(M)10 b(;)15 b(b)p F4(\))q Ga(;)1681
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13367
2404 y F9(\()1709 2416 y Ga(z)1755 2404 y F9(\))1782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13368
2416 y FL(Imp)1927 2438 y Gc(L)1979 2416 y F4(\()2014
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13369
2404 y FX(h)2042 2416 y Ga(c)2081 2404 y FX(i)2108 2416
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13370
y Ga(N)10 b(;)2231 2404 y F9(\()2259 2416 y Ga(y)2307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13371
2404 y F9(\))2334 2416 y Ga(P)j(;)i(z)t F4(\))r(\))788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13372
2515 y Gc(l)713 2552 y F6(\000)-31 b(\000)g(!)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13373
F4(\()1082 2540 y FX(h)1110 2552 y Ga(a)1158 2540 y FX(i)1185
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13374
2552 y FL(Cut)p F4(\()1358 2540 y FX(h)1386 2552 y Ga(c)1425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13375
2540 y FX(i)1452 2552 y Ga(N)10 b(;)1575 2540 y F9(\()1603
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13376
2552 y Ga(x)1655 2540 y F9(\))1683 2552 y Ga(M)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13377
Ga(;)1856 2540 y F9(\()1884 2552 y Ga(y)1932 2540 y F9(\))1959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13378
2552 y Ga(P)j F4(\))49 b Gg(or)788 2650 y Gc(l)713 2687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13379
y F6(\000)-31 b(\000)g(!)25 b FL(Cut)p F4(\()1082 2675
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13380
y FX(h)1110 2687 y Ga(c)1149 2675 y FX(i)1176 2687 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13381
Ga(N)10 b(;)1299 2675 y F9(\()1327 2687 y Ga(x)1379 2675
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13382
y F9(\))1407 2687 y FL(Cut)o F4(\()1579 2675 y FX(h)1607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13383
2687 y Ga(a)1655 2675 y FX(i)1683 2687 y Ga(M)g(;)1821
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13384
2675 y F9(\()1849 2687 y Ga(y)1897 2675 y F9(\))1924
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13385
2687 y Ga(P)j F4(\)\))996 2870 y Gg(if)23 b FL(Imp)1219
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13386
2892 y Gc(L)1271 2870 y F4(\()1306 2858 y FX(h)1334 2870
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13387
y Ga(c)1373 2858 y FX(i)1400 2870 y Ga(N)10 b(;)1523
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13388
2858 y F9(\()1551 2870 y Ga(y)1599 2858 y F9(\))1626
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13389
2870 y Ga(P)j(;)i(z)t F4(\))24 b Gg(and)g FL(Imp)2141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13390
2892 y Gc(R)2198 2870 y F4(\()2233 2858 y F9(\()2261
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13391
2870 y Ga(x)2313 2858 y F9(\))q FX(h)2368 2870 y Ga(a)2416
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13392
2858 y FX(i)2443 2870 y Ga(M)11 b(;)k(b)p F4(\))23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13393
Gg(freshly)i(introduce)h Ga(z)h Gg(and)d Ga(b)546 3147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13394
y Gg(5.)99 b FL(Cut)p F4(\()886 3135 y FX(h)914 3147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13395
y Ga(a)962 3135 y FX(i)990 3147 y Ga(M)10 b(;)1128 3135
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13396
y F9(\()1155 3147 y Ga(x)1207 3135 y F9(\))1235 3147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13397
y FL(Ax)o F4(\()p Ga(x;)15 b(b)p F4(\))q(\))1674 3110
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13398
y Gc(l)1600 3147 y F6(\000)-31 b(\000)f(!)26 b Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13399
b F4([)p Ga(a)g F6(7!)g Ga(b)p F4(])996 3330 y Gg(if)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13400
b Ga(M)33 b Gg(freshly)25 b(introduces)i Ga(a)546 3607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13401
y Gg(6.)99 b FL(Cut)p F4(\()886 3595 y FX(h)914 3607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13402
y Ga(a)962 3595 y FX(i)990 3607 y FL(Ax)o F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13403
b(a)p F4(\))q Ga(;)1339 3595 y F9(\()1367 3607 y Ga(x)1419
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13404
3595 y F9(\))1446 3607 y Ga(M)10 b F4(\))1679 3570 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13405
Gc(l)1605 3607 y F6(\000)-31 b(\000)f(!)26 b Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13406
b F4([)p Ga(x)g F6(7!)g Ga(y)s F4(])996 3790 y Gg(if)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13407
b Ga(M)33 b Gg(freshly)25 b(introduces)i Ga(x)475 4067
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13408
y Gb(Commuting)22 b(Cuts:)546 4273 y Gg(7.)99 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13409
F4(\()886 4261 y FX(h)914 4273 y Ga(a)962 4261 y FX(i)990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13410
4273 y Ga(M)10 b(;)1128 4261 y F9(\()1155 4273 y Ga(x)1207
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13411
4261 y F9(\))1235 4273 y Ga(N)g F4(\))783 4372 y Gc(c)713
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13412
4409 y F6(\000)-31 b(\000)g(!)25 b Ga(M)10 b F4([)p Ga(a)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13413
b F4(:=)1227 4397 y F9(\()1254 4409 y Ga(x)1306 4397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13414
y F9(\))1334 4409 y Ga(N)10 b F4(])113 b Gg(if)24 b Ga(M)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13415
b Gg(does)24 b(not)g(freshly)h(introduce)h Ga(a)p Gg(,)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13416
b(or)783 4507 y Gc(c)713 4544 y F6(\000)-31 b(\000)g(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13417
b Ga(N)10 b F4([)p Ga(x)25 b F4(:=)1215 4532 y FX(h)1243
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13418
4544 y Ga(a)1291 4532 y FX(i)1318 4544 y Ga(M)11 b F4(])113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13419
b Gg(if)24 b Ga(N)32 b Gg(does)25 b(not)f(freshly)h(introduce)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13420
Ga(x)p 3965 4657 V 277 4660 3691 4 v Black 730 4814 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13421
Gg(Figure)f(2.5:)k(Cut-reductions)e(for)c(logical)j(cuts)e(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13422
(commuting)h(cuts.)p Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13423
%%Page: 30 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13424
TeXDict begin 30 41 bop Black -144 51 a Gb(30)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13425
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(Notice)j(that)820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13426
280 y Gc(l)746 317 y F6(\000)-31 b(\000)f(!)23 b Gg(and)1163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13427
280 y Gc(c)1093 317 y F6(\000)-31 b(\000)f(!)23 b Gg(are,)h(by)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13428
(assumption,)h(closed)g(under)g(conte)o(xt)g(formation.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13429
b(The)24 b F7(com-)321 430 y(pleteness)f Gg(of)799 393
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13430
y Gc(cut)769 430 y F6(\000)-32 b(\000)h(!)19 b Gg(is)h(simply)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13431
(f)o(act,)g(ob)o(vious)g(by)f(inspection,)k(that)c(e)n(v)o(ery)h(term)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13432
(be)o(ginning)k(with)321 543 y(a)h(cut)h(matches)h(at)e(least)h(one)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13433
(left-hand)i(side)e(of)f(the)h(reduction)i(rules.)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13434
b(So)23 b(each)i(irreducible)j(term,)321 656 y(also)c(called)h(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13435
(normal)i(form,)e(is)g(cut-free.)462 786 y(W)-7 b(e)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13436
b(should)j(pro)o(v)o(e)e(that)h(the)f(cut-reductions)k(satisfy)e(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13437
(subject)i(reduction)g(property)-6 b(,)26 b(which)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13438
898 y(states)i(that)e(well-typed)i(terms)f(reduce)g(to)f(well-typed)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13439
(terms.)37 b(An)25 b(auxiliary)k(lemma)c(required)k(in)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13440
1011 y(this)24 b(proof)h(follo)n(ws.)p Black 321 1199
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13441
a Gb(Lemma)e(2.2.9)g Gg(\(W)-7 b(eak)o(ening)26 b(and)e(Contraction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13442
(Lemma\))p Gb(:)p Black Black 458 1396 a F6(\017)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13443
46 w Gg(Suppose)j F4(\000)p Gg(,)f F4(\000)1049 1363
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13444
y FX(0)1072 1396 y Gg(,)g F4(\001)f Gg(and)h F4(\001)1460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13445
1363 y FX(0)1510 1396 y Gg(are)g(conte)o(xts.)44 b(If)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13446
b Ga(M)37 b Gg(is)28 b(a)f(term)h(with)g(the)g(typing)i(judgement)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13447
1509 y F4(\000)631 1497 y Gc(.)686 1509 y Ga(M)809 1497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13448
y Gc(.)864 1509 y F4(\001)p Gg(,)22 b(then)j(also)f F4(\000)1391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13449
1476 y FX(0)1414 1509 y Ga(;)15 b F4(\000)1537 1497 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13450
Gc(.)1592 1509 y Ga(M)1715 1497 y Gc(.)1770 1509 y F4(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13451
Ga(;)g F4(\001)1962 1476 y FX(0)2008 1509 y Gg(is)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13452
b(a)f(typing)j(judgement)f(for)g Ga(M)10 b Gg(,)22 b(pro)o(vided)k(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13453
549 1621 y(conte)o(xt)f(con)l(v)o(ention)h(is)e(observ)o(ed.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13454
Black 458 1801 a F6(\017)p Black 46 w Gg(If)e Ga(M)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13455
b Gg(is)22 b(a)g(term)g(with)g(the)h(typing)h(judgement)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13456
Ga(x)17 b F4(:)g Ga(B)5 b(;)15 b(y)20 b F4(:)e Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13457
b(;)15 b F4(\000)2588 1789 y Gc(.)2642 1801 y Ga(M)2766
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13458
1789 y Gc(.)2821 1801 y F4(\001)o Gg(,)22 b(then)h(for)g(the)g(term)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13459
1914 y Ga(M)10 b F4([)p Ga(y)27 b F6(7!)d Ga(x)p F4(])30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13460
b Gg(we)g(ha)n(v)o(e)i(a)e(typing)j(judgement)g Ga(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13461
b F4(:)h Ga(B)5 b(;)15 b F4(\000)2356 1902 y Gc(.)2410
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13462
1914 y Ga(M)10 b F4([)p Ga(y)j F6(7!)d Ga(x)p F4(])2795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13463
1902 y Gc(.)2850 1914 y F4(\001)p Gg(.)50 b(If)31 b Ga(M)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13464
b Gg(is)31 b(a)g(term)549 2027 y(with)26 b(the)h(typing)h(judgement)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13465
F4(\000)1622 2015 y Gc(.)1677 2027 y Ga(M)1800 2015 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13466
Gc(.)1855 2027 y F4(\001)p Ga(;)15 b(a)j F4(:)f Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13467
b(;)15 b(b)i F4(:)h Ga(B)t Gg(,)26 b(then)h(for)g(the)g(term)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13468
Ga(M)10 b F4([)p Ga(b)16 b F6(7!)f Ga(a)p F4(])26 b Gg(we)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13469
2139 y(ha)n(v)o(e)e(a)f(typing)i(judgement)g F4(\000)1547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13470
2127 y Gc(.)1601 2139 y Ga(M)10 b F4([)p Ga(b)g F6(7!)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13471
Ga(a)p F4(])1973 2127 y Gc(.)2028 2139 y F4(\001)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13472
b(a)j F4(:)f Ga(B)5 b Gg(.)p Black 321 2352 a F7(Pr)l(oof)o(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13473
Black 34 w Gg(All)23 b(by)g(tri)n(vial)i(induction)h(on)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13474
(structure)i(of)d Ga(M)10 b Gg(.)p 3480 2352 4 62 v 3484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13475
2294 55 4 v 3484 2352 V 3538 2352 4 62 v Black 321 2540
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13476
a Gb(Pr)n(oposition)25 b(2.2.10)f Gg(\(Subject)h(Reduction\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13477
Gb(:)p Black 321 2653 a Gg(Suppose)i Ga(M)35 b Gg(is)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13478
b(a)g(term)g(with)g(the)h(typing)g(judgement)i F4(\000)2187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13479
2641 y Gc(.)2242 2653 y Ga(M)2365 2641 y Gc(.)2420 2653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13480
y F4(\001)c Gg(and)i Ga(M)2834 2615 y Gc(cut)2803 2653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13481
y F6(\000)-31 b(\000)f(!)29 b Ga(N)10 b Gg(,)24 b(then)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13482
Ga(N)35 b Gg(is)25 b(a)321 2765 y(term)f(with)f(the)h(typing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13483
(judgement)g F4(\000)1573 2753 y Gc(.)1627 2765 y Ga(N)1736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13484
2753 y Gc(.)1791 2765 y F4(\001)o Gg(.)p Black 321 2978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13485
a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)e(inspection)j(of)e(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13486
(reduction)k(rules)d(\(see)g(P)o(age)f(141\).)p 3480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13487
2978 V 3484 2920 55 4 v 3484 2978 V 3538 2978 4 62 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13488
462 3182 a(Let)40 b(us)h(mention)h(that)f(the)g(Barendre)o(gt-style)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13489
(naming)c(con)l(v)o(ention)j(is)d(indispensable)j(for)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13490
3295 y(strong)29 b(normalisation)i(of)1236 3258 y Gc(cut)1205
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13491
3295 y F6(\000)-31 b(\000)f(!)p Gg(.)41 b(If)27 b(we)g(do)h(not)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13492
(observ)o(e)h(this)g(con)l(v)o(ention,)i(we)c(lose)i(strong)g(nor)n(-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13493
321 3408 y(malisation.)61 b(T)-7 b(o)32 b(gi)n(v)o(e)i(an)f(e)o(xample)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13494
i(of)e(a)g(non-terminating)k(reduction)f(sequence)g(we)d(need)h(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13495
3521 y(follo)n(wing)25 b(lemma.)p Black 321 3708 a Gb(Lemma)e(2.2.11:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13496
Black 35 w Gg(F)o(or)f(all)i(terms)g Ga(M)35 b F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13497
b FY(T)e Gg(we)f(ha)n(v)o(e)p Black 417 3892 a(\(i\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13498
Black 47 w Ga(M)10 b F4([)p Ga(x)25 b F4(:=)870 3880
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13499
y FX(h)898 3892 y Ga(a)946 3880 y FX(i)973 3892 y FL(Ax)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13500
F4(\()p Ga(y)s(;)15 b(a)p F4(\))q(])1364 3855 y Gc(cut)1333
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13501
3892 y F6(\000)-31 b(\000)g(!)1504 3859 y FX(\003)1568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13502
3892 y Ga(M)10 b F4([)p Ga(x)g F6(7!)g Ga(y)s F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13503
Black 392 4071 a Gg(\(ii\))p Black 47 w Ga(M)g F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13504
Ga(a)25 b F4(:=)866 4059 y F9(\()894 4071 y Ga(x)946
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13505
4059 y F9(\))973 4071 y FL(Ax)p F4(\()p Ga(x;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13506
F4(\))q(])1359 4034 y Gc(cut)1328 4071 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13507
b(\000)g(!)1499 4038 y FX(\003)1563 4071 y Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13508
F4([)p Ga(a)g F6(7!)g Ga(b)p F4(])p Black 321 4284 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13509
F7(Pr)l(oof)o(.)p Black 34 w Gg(By)23 b(routine)i(induction)h(on)e(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13510
g(structure)h(of)f Ga(M)10 b Gg(.)p 3480 4284 V 3484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13511
4226 55 4 v 3484 4284 V 3538 4284 4 62 v Black 321 4546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13512
a Gb(Example)25 b(2.2.12:)p Black 34 w Gg(W)-7 b(e)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13513
b(start)i(with)e(the)h(term)g FL(Not)1957 4560 y Gc(R)2014
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13514
4546 y F4(\()2049 4534 y F9(\()2077 4546 y Ga(x)2129
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13515
4534 y F9(\))2157 4546 y Ga(M)10 b(;)15 b(a)p F4(\)[)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13516
Ga(a)28 b F4(:=)2603 4534 y F9(\()2630 4546 y Ga(y)2678
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13517
4534 y F9(\))2706 4546 y FL(Ax)o F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13518
b(a)p F4(\))q(])24 b Gg(for)h(which)g(we)321 4659 y(assume)d(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13519
(there)h(are)f(se)n(v)o(eral)h(free)f(occurrences)j(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13520
Ga(a)f Gg(in)h Ga(M)10 b Gg(;)21 b(that)h(means)f FL(Not)2938
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13521
4673 y Gc(R)2996 4659 y F4(\()3031 4647 y F9(\()3058
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13522
4659 y Ga(x)3110 4647 y F9(\))3138 4659 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13523
b(a)p F4(\))21 b Gg(does)321 4772 y(not)27 b(freshly)h(introduce)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13524
Ga(a)p Gg(.)37 b(W)-7 b(e)26 b(apply)h(our)g(de\002nition)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13525
(substitution)k(\(see)d(Figure)g(2.4\))g(so)f(as)h(to)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13526
4885 y(yield)916 5039 y FL(Cut)o F4(\()1088 5027 y FX(h)1116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13527
5039 y Ga(a)1164 5027 y FX(i)1192 5039 y FL(Not)1334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13528
5053 y Gc(R)1392 5039 y F4(\()1427 5027 y F9(\()1455
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13529
5039 y Ga(x)1507 5027 y F9(\))1534 5039 y Ga(M)10 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13530
Ga(a)26 b F4(:=)1852 5027 y F9(\()1880 5039 y Ga(y)1928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13531
5027 y F9(\))1955 5039 y FL(Ax)o F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13532
b(a)p F4(\))q(])q Ga(;)g(a)p F4(\))q Ga(;)2454 5027 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13533
F9(\()2481 5039 y Ga(y)2529 5027 y F9(\))2557 5039 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13534
FL(Ax)o F4(\()p Ga(y)s(;)g(a)p F4(\))q(\))26 b Ga(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13535
Black 420 w Gg(\(2.4\))p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13536
%%Page: 31 43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13537
TeXDict begin 31 42 bop Black 277 51 a Gb(2.3)23 b(Pr)n(oof)i(of)e(Str)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13538
n(ong)h(Normalisation)2290 b(31)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13539
277 317 a Gg(According)32 b(to)e(the)h(lemma)e(just)i(gi)n(v)o(en)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13540
(this)f(term)g(reduces,)j(assuming)f(that)f Ga(a)e Gg(is)h(free)g(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13541
Ga(M)10 b Gg(,)31 b(by)277 430 y(some)24 b(reductions)i(to)1071
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13542
619 y FL(Cut)p F4(\()1244 607 y FX(h)1272 619 y Ga(a)1320
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13543
607 y FX(i)1347 619 y FL(Not)1490 633 y Gc(R)1548 619
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13544
y F4(\()1583 607 y F9(\()1611 619 y Ga(x)1663 607 y F9(\))1690
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13545
619 y Ga(M)10 b F4([)p Ga(a)g F6(7!)g Ga(a)p F4(])q Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13546
b(a)p F4(\))q Ga(;)2210 607 y F9(\()2237 619 y Ga(y)2285
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13547
607 y F9(\))2313 619 y FL(Ax)o F4(\()p Ga(y)s(;)g(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13548
F4(\))q(\))26 b Ga(:)277 823 y Gg(Clearly)-6 b(,)22 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13549
Ga(M)10 b F4([)p Ga(a)g F6(7!)g Ga(a)p F4(])22 b Gg(is)f(equi)n(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13550
(alent)j(to)d Ga(M)10 b Gg(,)20 b(which)i(contains)i(some)d(free)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13551
(occurrences)i(of)d Ga(a)g Gg(though.)277 936 y(Therefore)f(we)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13552
(cannot)h(apply)h(a)e(logical)i(rule,)f(b)n(ut)g(ha)n(v)o(e)g(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13553
(apply)i(a)e(reduction)j(rule)d(for)h(commuting)277 1049
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13554
y(cuts.)29 b(This)24 b(gi)n(v)o(es)g(us)1279 1240 y FL(Not)1421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13555
1254 y Gc(R)1479 1240 y F4(\()1514 1228 y F9(\()1542
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13556
1240 y Ga(x)1594 1228 y F9(\))1621 1240 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13557
b(a)p F4(\)[)p Ga(a)27 b F4(:=)2063 1228 y F9(\()2090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13558
1240 y Ga(y)2138 1228 y F9(\))2166 1240 y FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13559
Ga(y)s(;)15 b(a)p F4(\))q(])277 1444 y Gg(and)27 b(we)f(are)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13560
(\223back\224)i(where)f(we)f(started)i(from.)37 b(A)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13561
b(closer)j(analysis)g(of)f(this)g(erroneous)i(reduction)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13562
1557 y(sequence)37 b(re)n(v)o(eals)e(that)g(in)f(Step)g(\(2.4\))h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13563
(co-name)i Ga(a)d Gg(became)j(bound)f(by)g(the)f(cut)h(when)f(we)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13564
1670 y(pushed)27 b(the)f(substitution)j(inside)e(the)f(term)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13565
FL(Not)1856 1684 y Gc(R)1914 1670 y F4(\()1949 1658 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13566
F9(\()1977 1670 y Ga(x)2029 1658 y F9(\))2056 1670 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13567
Ga(M)10 b(;)15 b(a)p F4(\))p Gg(.)35 b(T)-7 b(o)24 b(a)n(v)n(oid)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13568
(this)f(capture)i(of)d(the)h(free)277 1782 y(occurrence)c(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13569
Ga(a)e Gg(in)945 1770 y F9(\()972 1782 y Ga(y)1020 1770
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13570
y F9(\))1047 1782 y FL(Ax)p F4(\()p Ga(y)s(;)d(a)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13571
Gg(,)k(we)f(ha)n(v)o(e)i(to)g(rename)g(the)f(corresponding)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13572
b(binder)-5 b(.)29 b(Accordingly)-6 b(,)277 1895 y(we)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13573
b(should)i(ha)n(v)o(e)f(written)p Black Black 806 2083
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13574
a FL(Not)949 2097 y Gc(R)1007 2083 y F4(\()1042 2071
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13575
y F9(\()1070 2083 y Ga(x)1122 2071 y F9(\))1149 2083
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13576
y Ga(M)10 b(;)15 b(a)p F4(\)[)p Ga(a)26 b F4(:=)1591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13577
2071 y F9(\()1618 2083 y Ga(y)1666 2071 y F9(\))1693
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13578
2083 y FL(Ax)p F4(\()p Ga(y)s(;)15 b(a)p F4(\))q(])636
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13579
2220 y F6(\021)99 b FL(Not)949 2234 y Gc(R)1007 2220
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13580
y F4(\()1042 2208 y F9(\()1070 2220 y Ga(x)1122 2208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13581
y F9(\))1149 2220 y Ga(M)10 b F4([)p Ga(a)g F6(7!)g Ga(a)1479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13582
2187 y FX(0)1503 2220 y F4(])p Ga(;)15 b(a)1616 2187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13583
y FX(0)1640 2220 y F4(\)[)p Ga(a)1748 2187 y FX(0)1797
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13584
2220 y F4(:=)1918 2208 y F9(\()1946 2220 y Ga(y)1994
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13585
2208 y F9(\))2021 2220 y FL(Ax)p F4(\()p Ga(y)s(;)g(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13586
F4(\))q(])636 2357 y(=)99 b FL(Cut)p F4(\()979 2345 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13587
FX(h)1007 2357 y Ga(a)1055 2324 y FX(0)1078 2345 y(i)1106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13588
2357 y FL(Not)1249 2371 y Gc(R)1306 2357 y F4(\()1341
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13589
2345 y F9(\()1369 2357 y Ga(x)1421 2345 y F9(\))1448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13590
2357 y Ga(M)10 b F4([)p Ga(a)g F6(7!)g Ga(a)1778 2324
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13591
y FX(0)1802 2357 y F4(][)p Ga(a)1900 2324 y FX(0)1949
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13592
2357 y F4(:=)2071 2345 y F9(\()2098 2357 y Ga(y)2146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13593
2345 y F9(\))2173 2357 y FL(Ax)p F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13594
b(a)p F4(\))q(])p Ga(;)g(a)2596 2324 y FX(0)2620 2357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13595
y F4(\))p Ga(;)2695 2345 y F9(\()2723 2357 y Ga(y)2771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13596
2345 y F9(\))2798 2357 y FL(Ax)p F4(\()p Ga(y)s(;)g(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13597
F4(\))q(\))277 2570 y Gg(assuming)25 b Ga(a)691 2537
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13598
y FX(0)737 2570 y Gg(does)g(not)f(occur)g(freely)h(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13599
Ga(M)10 b Gg(.)28 b(No)n(w)23 b(we)f(may)i(apply)g(Lemma)f(2.2.11)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13600
(and)g(recei)n(v)o(e)277 2700 y(performing)i(some)d(reductions)896
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13601
2907 y FL(Cut)p F4(\()1069 2895 y FX(h)1096 2907 y Ga(a)1144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13602
2874 y FX(0)1168 2895 y(i)1195 2907 y FL(Not)1338 2921
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13603
y Gc(R)1396 2907 y F4(\()1431 2895 y F9(\()1458 2907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13604
y Ga(x)1510 2895 y F9(\))1538 2907 y Ga(M)10 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13605
Ga(a)g F6(7!)g Ga(a)1868 2874 y FX(0)1892 2907 y F4(][)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13606
Ga(a)1990 2874 y FX(0)2024 2907 y F6(7!)g Ga(a)p F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13607
Ga(;)15 b(a)2286 2874 y FX(0)2310 2907 y F4(\))p Ga(;)2385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13608
2895 y F9(\()2413 2907 y Ga(y)2461 2895 y F9(\))2488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13609
2907 y FL(Ax)p F4(\()p Ga(y)s(;)g(a)p F4(\))q(\))25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13610
Ga(:)277 3111 y Gg(Because)i FL(Not)747 3125 y Gc(R)805
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13611
3111 y F4(\()840 3099 y F9(\()868 3111 y Ga(x)920 3099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13612
y F9(\))947 3111 y Ga(M)10 b F4([)p Ga(a)g F6(7!)g Ga(a)1277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13613
3078 y FX(0)1301 3111 y F4(][)p Ga(a)1399 3078 y FX(0)1433
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13614
3111 y F6(7!)g Ga(a)p F4(])p Ga(;)15 b(a)1695 3078 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13615
FX(0)1719 3111 y F4(\))25 b Gg(freshly)i(introduces)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13616
Ga(a)2507 3078 y FX(0)2531 3111 y Gg(,)d(we)f(can)i(no)n(w)f(apply)h(a)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13617
f(logi-)277 3224 y(cal)f(reduction)i(and)e(obtain)1149
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13618
3413 y FL(Not)1292 3427 y Gc(R)1349 3413 y F4(\()1384
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13619
3401 y F9(\()1412 3413 y Ga(x)1464 3401 y F9(\))1492
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13620
3413 y Ga(M)10 b F4([)p Ga(a)g F6(7!)g Ga(a)1822 3380
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13621
y FX(0)1845 3413 y F4(][)p Ga(a)1943 3380 y FX(0)1977
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13622
3413 y F6(7!)g Ga(a)p F4(])q Ga(;)15 b(a)2240 3375 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13623
FX(0)2263 3413 y F4(\)[)p Ga(a)2371 3375 y FX(0)2405
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13624
3413 y F6(7!)10 b Ga(a)p F4(])26 b Ga(:)277 3600 y Gg(This)d(term)h(is)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13625
f(equi)n(v)n(alent)j(to)e FL(Not)1367 3614 y Gc(R)1425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13626
3600 y F4(\()1460 3588 y F9(\()1488 3600 y Ga(x)1540
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13627
3588 y F9(\))1567 3600 y Ga(M)10 b(;)15 b(a)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13628
Gg(\227the)25 b(term)e(we)g(e)o(xpected)i(gi)n(v)o(en)f(Lemma)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13629
(2.2.11.)277 3901 y Ge(2.3)119 b(Pr)n(oof)29 b(of)h(Str)n(ong)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13630
(Normalisation)277 4125 y Gg(In)c(this)g(section)i(we)d(shall)i(gi)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13631
o(e)f(a)f(direct)i(proof)g(of)f(strong)h(normalisation)i(for)d(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13632
(reduction)j(sys-)277 4238 y(tem)c F4(\()p FY(T)t Ga(;)602
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13633
4201 y Gc(cut)571 4238 y F6(\000)-31 b(\000)f(!)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13634
Gg(.)35 b(The)25 b(proof)h(adapts)h(the)f(technique)i(of)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13635
(symmetric)g(reducibility)j(candidates)3461 4205 y F5(4)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13636
4351 y Gg(de)n(v)o(eloped)j(by)d(Barbanera)i(and)f(Berardi)g([1994].)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13637
b(Unfortunately)-6 b(,)34 b(we)28 b(cannot)j(apply)g(this)e(tech-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13638
4464 y(nique)c(directly)h(to)e(pro)o(v)o(e)g(strong)h(normalisation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13639
(for)d F4(\()p FY(T)t Ga(;)2190 4426 y Gc(cut)2159 4464
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13640
y F6(\000)-31 b(\000)f(!)p F4(\))p Gg(,)23 b(because)j(in)e(order)h(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13641
e(strengthen)277 4576 y(an)h(induction)i(hypothesis,)g(we)c(need)j(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13642
f(follo)n(wing)h(property)897 4817 y Ga(M)10 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13643
Ga(x)26 b F4(:=)1219 4805 y FX(h)1247 4817 y Ga(a)1295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13644
4805 y FX(i)1322 4817 y Ga(P)13 b F4(][)p Ga(b)26 b F4(:=)1629
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13645
4805 y F9(\()1657 4817 y Ga(y)1705 4805 y F9(\))1732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13646
4817 y Ga(Q)p F4(])f F6(\021)g Ga(M)10 b F4([)p Ga(b)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13647
b F4(:=)2259 4805 y F9(\()2287 4817 y Ga(y)2335 4805
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13648
y F9(\))2362 4817 y Ga(Q)p F4(][)p Ga(x)f F4(:=)2683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13649
4805 y FX(h)2710 4817 y Ga(a)2758 4805 y FX(i)2786 4817
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13650
y Ga(P)13 b F4(])277 5038 y Gg(for)24 b Ga(b)e Gg(not)i(free)g(in)864
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13651
5026 y FX(h)891 5038 y Ga(a)939 5026 y FX(i)967 5038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13652
y Ga(P)35 b Gg(and)24 b Ga(x)f Gg(not)h(free)f(in)1685
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13653
5026 y F9(\()1712 5038 y Ga(y)1760 5026 y F9(\))1788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13654
5038 y Ga(Q)o Gg(.)28 b(Ho)n(we)n(v)o(er)23 b(this)h(does)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13655
F7(not)g Gg(hold)h(for)e(the)h(substitu-)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13656
5100 1290 4 v 383 5156 a F3(4)412 5188 y F2(It)g(seems)i(this)f(type)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13657
(of)g(candidates)i(w)o(as)e(\002rst)f(used)i(by)g(Girard)f([1987a])h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13658
(for)f(pro)o(ving)h(strong)g(normalisation)g(of)277 5279
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13659
y(proof-nets.)p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13660
%%Page: 32 44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13661
TeXDict begin 32 43 bop Black -144 51 a Gb(32)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13662
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(tion)h(operation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13663
(de\002ned)e(in)g(Figure)g(2.4.)28 b(In)22 b(ef)n(fect,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13664
(\223independent\224)k(substitutions,)f(in)c(general,)i(do)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13665
430 y(not)g(commute!)30 b(The)23 b(\(only\))i(problematic)h(case)e(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13666
(where)f Ga(M)33 b Gg(is)24 b(of)f(the)h(form)f FL(Ax)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13667
F4(\()p Ga(x;)15 b(b)p F4(\))p Gg(:)789 643 y FL(Ax)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13668
F4(\()p Ga(x;)g(b)p F4(\)[)p Ga(x)27 b F4(:=)1317 631
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13669
y FX(h)1344 643 y Ga(a)1392 631 y FX(i)1420 643 y Ga(P)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13670
b F4(][)p Ga(b)26 b F4(:=)1727 631 y F9(\()1754 643 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13671
Ga(y)1802 631 y F9(\))1830 643 y Ga(Q)p F4(])83 b(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13672
Ga(P)13 b F4([)p Ga(a)d F6(7!)g Ga(b)p F4(][)p Ga(b)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13673
b F4(:=)2694 631 y F9(\()2721 643 y Ga(y)2769 631 y F9(\))2797
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13674
643 y Ga(Q)p F4(])k Ga(;)15 b Gg(b)n(ut)789 792 y FL(Ax)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13675
F4(\()p Ga(x;)g(b)p F4(\)[)p Ga(b)27 b F4(:=)1304 780
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13676
y F9(\()1332 792 y Ga(y)1380 780 y F9(\))1407 792 y Ga(Q)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13677
F4(][)p Ga(x)e F4(:=)1728 780 y FX(h)1755 792 y Ga(a)1803
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13678
780 y FX(i)1831 792 y Ga(P)13 b F4(])83 b(=)g Ga(Q)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13679
F4([)p Ga(y)13 b F6(7!)d Ga(x)p F4(][)p Ga(x)25 b F4(:=)2721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13680
780 y FX(h)2748 792 y Ga(a)2796 780 y FX(i)2823 792 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13681
Ga(P)13 b F4(])i Ga(:)321 1002 y Gg(Clearly)-6 b(,)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13682
b(there)g(is)e(no)h(reason)h(for)e(the)h(tw)o(o)f(resultant)j(terms)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13683
(to)f(be)h(equal.)28 b(T)-7 b(o)17 b(remedy)i(this)g(situation,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13684
1115 y(we)34 b(shall)h(de\002ne)g(an)f(auxiliary)j(cut-reduction)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13685
(system,)f F4(\()p FY(T)t Ga(;)2380 1078 y Gc(aux)2360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13686
1115 y F6(\000)-31 b(\000)f(!)p F4(\))p Gg(,)37 b(which)d(has)h(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13687
(more)g(subtle)321 1228 y(de\002nition)23 b(of)d(substitution)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13688
(including)f(tw)o(o)e(special)h(clauses)g(to)f(handle)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13689
(problematic)i(e)o(xample)321 1341 y(abo)o(v)o(e.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13690
b(Intuiti)n(v)o(ely)-6 b(,)25 b(we)e(should)i(e)o(xpect)923
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13691
1556 y FL(Ax)p F4(\()p Ga(x;)15 b(b)p F4(\)[)p Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13692
b F4(:=)1452 1544 y FX(h)1479 1556 y Ga(a)1527 1544 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13693
FX(i)1555 1556 y Ga(P)13 b F4(][)p Ga(b)25 b F4(:=)1861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13694
1544 y F9(\()1889 1556 y Ga(y)1937 1544 y F9(\))1964
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13695
1556 y Ga(Q)p F4(])923 1705 y FL(Ax)p F4(\()p Ga(x;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13696
b(b)p F4(\)[)p Ga(b)26 b F4(:=)1439 1693 y F9(\()1466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13697
1705 y Ga(y)1514 1693 y F9(\))1541 1705 y Ga(Q)p F4(][)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13698
Ga(x)g F4(:=)1862 1693 y FX(h)1890 1705 y Ga(a)1938 1693
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13699
y FX(i)1965 1705 y Ga(P)13 b F4(])2103 1475 y FK(\))2227
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13700
1631 y F4(=)50 b FL(Cut)p F4(\()2521 1619 y FX(h)2548
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13701
1631 y Ga(a)2596 1619 y FX(i)2624 1631 y Ga(P)13 b(;)2735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13702
1619 y F9(\()2763 1631 y Ga(y)2811 1619 y F9(\))2838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13703
1631 y Ga(Q)p F4(\))26 b Ga(:)321 1903 y Gg(The)h(ne)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13704
(substitution)k(operation,)f(written)e(as)f Ga(M)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13705
F6(f)-7 b Ga(a)33 b F4(:=)2235 1891 y F9(\()2263 1903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13706
y Ga(x)2315 1891 y F9(\))2342 1903 y Ga(N)p F6(g)27 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13707
Gg(and)h Ga(N)5 b F6(f)-7 b Ga(x)33 b F4(:=)2974 1891
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13708
y F9(\()3001 1903 y Ga(a)3049 1891 y F9(\))3077 1903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13709
y Ga(M)p F6(g)p Gg(,)28 b(is)f(gi)n(v)o(en)321 2016 y(in)j(Figure)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13710
(2.6.)49 b(W)-7 b(e)29 b(apply)j(the)e(same)g(terminology)j(as)d(for)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13711
(substitutions)j(of)c(the)h(form)f F4([)p Ga(\033)s F4(])g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13712
Gg(\(see)321 2129 y(T)-6 b(erminology)25 b(2.2.6\).)k(F)o(or)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13713
b F6(f)p 1270 2129 28 4 v 1288 2129 V 1306 2129 V 65
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13714
w(g)g Gg(we)g(ha)n(v)o(e)h(the)g(follo)n(wing)h(substitution)i(lemma.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13715
Black 321 2317 a Gb(Lemma)c(2.3.1)g Gg(\(Substitution)k(Lemma\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13716
Gb(:)p Black 321 2430 a Gg(F)o(or)f(all)h Ga(M)42 b F6(2)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13717
b FY(T)d Gg(and)g(tw)o(o)f(arbitrary)j(proof)f(substitutions,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13718
F6(f)p Ga(\033)s F6(g)d Gg(and)f F6(f)p Ga(\034)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13719
F6(g)p Gg(,)27 b(such)h(that)f Ga(dom)p F4(\()p F6(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13720
Ga(\033)s F6(g)p F4(\))321 2543 y Gg(is)d(not)g(free)g(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13721
Ga(codom)p F4(\()p F6(f)p Ga(\034)10 b F6(g)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13722
Gg(,)25 b(we)d(ha)n(v)o(e)1392 2721 y Ga(M)10 b F6(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13723
Ga(\033)s F6(gf)p Ga(\034)g F6(g)28 b(\021)d Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13724
b F6(f)p Ga(\034)g F6(gf)p Ga(\033)s F6(f)p Ga(\034)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13725
F6(gg)28 b Ga(:)p Black 321 2934 a F7(Pr)l(oof)o(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13726
Black 34 w Gg(By)23 b(induction)j(on)e(the)f(structure)j(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13727
Ga(M)32 b Gg(\(see)25 b(P)o(age)e(141\).)p 3480 2934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13728
4 62 v 3484 2876 55 4 v 3484 2934 V 3538 2934 4 62 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13729
321 3138 a(Since)34 b(we)f(changed)j(the)e(substitution)j(operation)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13730
(we)d(need)i(to)e(change)j(the)e(reduction)i(rule)e(for)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13731
3251 y(commuting)25 b(cuts.)30 b(The)23 b(modi\002ed)h(rule)g(is)f(as)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13732
(follo)n(ws.)p Black Black 427 3469 a FL(Cut)p F4(\()600
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13733
3457 y FX(h)628 3469 y Ga(a)676 3457 y FX(i)704 3469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13734
y Ga(M)10 b(;)842 3457 y F9(\()869 3469 y Ga(x)921 3457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13735
y F9(\))949 3469 y Ga(N)g F4(\))1225 3432 y Gc(c)1256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13736
3409 y FC(0)1167 3469 y F6(\000)-31 b(\000)f(!)100 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13737
Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)1763 3457 y F9(\()1790
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13738
3469 y Ga(x)1842 3457 y F9(\))1870 3469 y Ga(N)p F6(g)114
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13739
b Gg(if)24 b Ga(M)33 b Gg(does)24 b(not)g(freshly)h(introduce)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13740
Ga(a)p Gg(,)d(or)1225 3581 y Gc(c)1256 3557 y FC(0)1167
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13741
3618 y F6(\000)-31 b(\000)f(!)100 b Ga(N)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13742
b Ga(x)26 b F4(:=)1752 3606 y FX(h)1779 3618 y Ga(a)1827
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13743
3606 y FX(i)1855 3618 y Ga(M)p F6(g)114 b Gg(if)24 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13744
Ga(N)32 b Gg(does)25 b(not)f(freshly)h(introduce)h Ga(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13745
Gg(.)321 3848 y(The)d(auxiliary)j(cut-elimination)i(procedure)e(is)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13746
(then)p Black 321 4056 a Gb(De\002nition)g(2.3.2)g Gg(\(Auxiliary)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13747
(Cut-Elimination)g(Procedure\))p Gb(:)p Black 321 4169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13748
a Gg(The)d(auxiliary)j(cut-elimination)i(procedure,)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13749
F4(\()p FY(T)-11 b Ga(;)1956 4132 y Gc(aux)1936 4169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13750
y F6(\000)-31 b(\000)f(!)p F4(\))p Gg(,)23 b(is)h(a)f(reduction)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13751
(system)e(where:)p Black 458 4352 a F6(\017)p Black 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13752
w FY(T)e Gg(is)i(the)g(set)f(of)h(well-typed)h(terms,)f(and)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13753
Black 458 4531 a F6(\017)p Black 569 4494 a Gc(aux)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13754
4531 y F6(\000)-32 b(\000)h(!)26 b Gg(consists)j(of)d(the)h(rules)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13755
(for)f(the)g(logical)h(cuts)g(and)f(the)g(modi\002ed)g(reduction)j(for)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13756
d(com-)549 4644 y(muting)d(cuts;)g(that)g(is)1651 4741
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13757
y Gc(aux)1631 4778 y F6(\000)-31 b(\000)f(!)1826 4726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13758
y F5(def)1834 4778 y F4(=)2011 4741 y Gc(l)1937 4778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13759
y F6(\000)g(\000)h(!)25 b([)2277 4741 y Gc(c)2308 4717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13760
y FC(0)2218 4778 y F6(\000)-31 b(\000)f(!)51 b Ga(:)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13761
5032 y Gg(W)-7 b(e)27 b(ha)n(v)o(e)h(to)g(v)o(erify)h(that)1186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13762
4995 y Gc(aux)1166 5032 y F6(\000)-31 b(\000)f(!)27 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13763
Gg(satis\002es)i(the)f(subject)h(reduction)h(property)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13764
b(.)44 b(Ho)n(we)n(v)o(er)l(,)28 b(gi)n(v)o(en)g(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13765
5145 y(proof)j(of)f(Proposition)i(2.2.10,)f(this)g(is)e(a)h(routine)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13766
(matter)l(,)h(and)e(so)g(we)f(omit)g(the)i(details.)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13767
b(Let)30 b(us)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13768
%%Page: 33 45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13769
TeXDict begin 33 44 bop Black 277 51 a Gb(2.3)23 b(Pr)n(oof)i(of)e(Str)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13770
n(ong)h(Normalisation)2290 b(33)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13771
Black 277 601 V 277 4767 4 4167 v 879 791 a FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13772
Ga(x;)15 b(c)p F4(\))-5 b F6(f)e Ga(c)28 b F4(:=)1403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13773
779 y F9(\()1431 791 y Ga(y)1479 779 y F9(\))1506 791
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13774
y Ga(P)s F6(g)1646 740 y F5(def)1653 791 y F4(=)40 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13775
FL(Cut)p F4(\()1937 779 y FX(h)1965 791 y Ga(c)2004 779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13776
y FX(i)2031 791 y FL(Ax)p F4(\()p Ga(x;)15 b(c)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13777
Ga(;)2376 779 y F9(\()2404 791 y Ga(y)2452 779 y F9(\))2479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13778
791 y Ga(P)e F4(\))874 961 y FL(Ax)p F4(\()p Ga(y)s(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13779
F4(\))-5 b F6(f)e Ga(y)29 b F4(:=)1412 949 y FX(h)1439
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13780
961 y Ga(c)1478 949 y FX(i)1506 961 y Ga(P)s F6(g)1646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13781
909 y F5(def)1653 961 y F4(=)40 b FL(Cut)p F4(\()1937
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13782
949 y FX(h)1965 961 y Ga(c)2004 949 y FX(i)2031 961 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13783
Ga(P)13 b(;)2142 949 y F9(\()2170 961 y Ga(y)2218 949
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13784
y F9(\))2245 961 y FL(Ax)p F4(\()p Ga(y)s(;)i(a)p F4(\))q(\))611
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13785
1130 y FL(Not)753 1144 y Gc(R)811 1130 y F4(\()846 1118
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13786
y F9(\()874 1130 y Ga(x)926 1118 y F9(\))953 1130 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13787
b(;)15 b(a)p F4(\))-5 b F6(f)e Ga(a)27 b F4(:=)1403 1118
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13788
y F9(\()1431 1130 y Ga(y)1479 1118 y F9(\))1506 1130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13789
y Ga(P)s F6(g)1646 1079 y F5(def)1653 1130 y F4(=)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13790
b FL(Cut)p F4(\()1937 1118 y FX(h)1965 1130 y Ga(a)2013
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13791
1118 y FX(i)2040 1130 y FL(Not)2183 1144 y Gc(R)2240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13792
1130 y F4(\()2275 1118 y F9(\()2303 1130 y Ga(x)2355
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13793
1118 y F9(\))2383 1130 y Ga(M)5 b F6(f)-7 b Ga(a)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13794
F4(:=)2709 1118 y F9(\()2736 1130 y Ga(y)2784 1118 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13795
F9(\))2812 1130 y Ga(P)s F6(g)q Ga(;)15 b(a)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13796
Ga(;)3082 1118 y F9(\()3110 1130 y Ga(y)3158 1118 y F9(\))3186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13797
1130 y Ga(P)e F4(\))621 1299 y FL(Not)763 1313 y Gc(L)815
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13798
1299 y F4(\()850 1287 y FX(h)878 1299 y Ga(a)926 1287
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13799
y FX(i)954 1299 y Ga(M)d(;)15 b(x)p F4(\))-5 b F6(f)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13800
Ga(x)26 b F4(:=)1412 1287 y FX(h)1439 1299 y Ga(c)1478
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13801
1287 y FX(i)1506 1299 y Ga(P)s F6(g)1646 1248 y F5(def)1653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13802
1299 y F4(=)40 b FL(Cut)p F4(\()1937 1287 y FX(h)1965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13803
1299 y Ga(c)2004 1287 y FX(i)2031 1299 y Ga(P)13 b(;)2142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13804
1287 y F9(\()2170 1299 y Ga(x)2222 1287 y F9(\))2250
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13805
1299 y FL(Not)2392 1313 y Gc(L)2444 1299 y F4(\()2479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13806
1287 y FX(h)2507 1299 y Ga(a)2555 1287 y FX(i)2583 1299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13807
y Ga(M)5 b F6(f)-7 b Ga(x)26 b F4(:=)2913 1287 y FX(h)2940
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13808
1299 y Ga(c)2979 1287 y FX(i)3007 1299 y Ga(P)t F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13809
Ga(;)15 b(x)p F4(\))q(\))403 1469 y FL(And)557 1483 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13810
Gc(R)615 1469 y F4(\()650 1457 y FX(h)678 1469 y Ga(a)726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13811
1457 y FX(i)753 1469 y Ga(M)10 b(;)891 1457 y FX(h)919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13812
1469 y Ga(b)958 1457 y FX(i)986 1469 y Ga(N)g(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13813
F4(\))-5 b F6(f)e Ga(c)27 b F4(:=)1403 1457 y F9(\()1431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13814
1469 y Ga(y)1479 1457 y F9(\))1506 1469 y Ga(P)s F6(g)1646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13815
1417 y F5(def)1653 1469 y F4(=)40 b FL(Cut)p F4(\()1937
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13816
1457 y FX(h)1965 1469 y Ga(c)2004 1457 y FX(i)2031 1469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13817
y FL(And)2186 1483 y Gc(R)2244 1469 y F4(\()2279 1457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13818
y FX(h)2306 1469 y Ga(a)2354 1457 y FX(i)2382 1469 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13819
Ga(M)5 b F6(f)-7 b Ga(c)26 b F4(:=)2699 1457 y F9(\()2727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13820
1469 y Ga(y)2775 1457 y F9(\))2802 1469 y Ga(P)t F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13821
Ga(;)2949 1457 y FX(h)2977 1469 y Ga(b)3016 1457 y FX(i)3043
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13822
1469 y Ga(N)5 b F6(f)-7 b Ga(c)27 b F4(:=)3346 1457 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13823
F9(\()3373 1469 y Ga(y)3421 1457 y F9(\))3449 1469 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13824
Ga(P)s F6(g)q Ga(;)15 b(c)p F4(\))q Ga(;)3711 1457 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13825
F9(\()3739 1469 y Ga(y)3787 1457 y F9(\))3814 1469 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13826
Ga(P)e F4(\))613 1638 y FL(And)768 1601 y Gc(i)768 1661
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13827
y(L)820 1638 y F4(\()855 1626 y F9(\()883 1638 y Ga(x)935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13828
1626 y F9(\))962 1638 y Ga(M)d(;)15 b(y)s F4(\))-5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13829
F6(f)e Ga(y)29 b F4(:=)1412 1626 y FX(h)1439 1638 y Ga(c)1478
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13830
1626 y FX(i)1506 1638 y Ga(P)s F6(g)1646 1587 y F5(def)1653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13831
1638 y F4(=)40 b FL(Cut)p F4(\()1937 1626 y FX(h)1965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13832
1638 y Ga(c)2004 1626 y FX(i)2031 1638 y Ga(P)13 b(;)2142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13833
1626 y F9(\()2170 1638 y Ga(y)2218 1626 y F9(\))2245
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13834
1638 y FL(And)2400 1601 y Gc(i)2400 1661 y(L)2452 1638
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13835
y F4(\()2487 1626 y F9(\()2515 1638 y Ga(x)2567 1626
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13836
y F9(\))2594 1638 y Ga(M)5 b F6(f)-7 b Ga(y)29 b F4(:=)2920
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13837
1626 y FX(h)2948 1638 y Ga(c)2987 1626 y FX(i)3015 1638
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13838
y Ga(P)s F6(g)q Ga(;)15 b(y)s F4(\)\))675 1807 y FL(Or)775
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13839
1771 y Gc(i)775 1831 y(R)832 1807 y F4(\()867 1795 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13840
FX(h)895 1807 y Ga(a)943 1795 y FX(i)971 1807 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13841
b(;)15 b(c)p F4(\))-5 b F6(f)e Ga(c)27 b F4(:=)1403 1795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13842
y F9(\()1431 1807 y Ga(y)1479 1795 y F9(\))1506 1807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13843
y Ga(P)s F6(g)1646 1756 y F5(def)1653 1807 y F4(=)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13844
b FL(Cut)p F4(\()1937 1795 y FX(h)1965 1807 y Ga(c)2004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13845
1795 y FX(i)2031 1807 y FL(Or)2131 1771 y Gc(i)2131 1831
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13846
y(R)2188 1807 y F4(\()2223 1795 y FX(h)2251 1807 y Ga(a)2299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13847
1795 y FX(i)2327 1807 y Ga(M)5 b F6(f)-7 b Ga(c)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13848
F4(:=)2644 1795 y F9(\()2672 1807 y Ga(y)2720 1795 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13849
F9(\))2747 1807 y Ga(P)s F6(g)q Ga(;)15 b(c)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13850
Ga(;)3009 1795 y F9(\()3037 1807 y Ga(y)3085 1795 y F9(\))3112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13851
1807 y Ga(P)e F4(\))445 1977 y FL(Or)545 1991 y Gc(L)597
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13852
1977 y F4(\()632 1965 y F9(\()660 1977 y Ga(x)712 1965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13853
y F9(\))739 1977 y Ga(M)d(;)877 1965 y F9(\()905 1977
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13854
y Ga(y)953 1965 y F9(\))980 1977 y Ga(N)g(;)15 b(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13855
F4(\))-5 b F6(f)e Ga(z)31 b F4(:=)1412 1965 y FX(h)1439
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13856
1977 y Ga(c)1478 1965 y FX(i)1506 1977 y Ga(P)s F6(g)1646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13857
1925 y F5(def)1653 1977 y F4(=)40 b FL(Cut)p F4(\()1937
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13858
1965 y FX(h)1965 1977 y Ga(c)2004 1965 y FX(i)2031 1977
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13859
y Ga(P)13 b(;)2142 1965 y F9(\()2170 1977 y Ga(z)2216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13860
1965 y F9(\))2244 1977 y FL(Or)2343 1991 y Gc(L)2395
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13861
1977 y F4(\()2430 1965 y F9(\()2458 1977 y Ga(x)2510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13862
1965 y F9(\))2537 1977 y Ga(M)5 b F6(f)-7 b Ga(z)31 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13863
F4(:=)2862 1965 y FX(h)2889 1977 y Ga(c)2928 1965 y FX(i)2956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13864
1977 y Ga(P)t F6(g)p Ga(;)3103 1965 y F9(\()3131 1977
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13865
y Ga(y)3179 1965 y F9(\))3206 1977 y Ga(N)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13866
b Ga(z)30 b F4(:=)3516 1965 y FX(h)3543 1977 y Ga(c)3582
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13867
1965 y FX(i)3610 1977 y Ga(P)s F6(g)q Ga(;)15 b(z)t F4(\))q(\))524
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13868
2146 y FL(Imp)668 2168 y Gc(R)726 2146 y F4(\()761 2134
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13869
y F9(\()789 2146 y Ga(x)841 2134 y F9(\))p FX(h)896 2146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13870
y Ga(a)944 2134 y FX(i)971 2146 y Ga(M)10 b(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13871
F4(\))-5 b F6(f)e Ga(b)27 b F4(:=)1403 2134 y F9(\()1431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13872
2146 y Ga(y)1479 2134 y F9(\))1506 2146 y Ga(P)s F6(g)1646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13873
2095 y F5(def)1653 2146 y F4(=)40 b FL(Cut)p F4(\()1937
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13874
2134 y FX(h)1965 2146 y Ga(b)2004 2134 y FX(i)2031 2146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13875
y FL(Imp)2175 2168 y Gc(R)2233 2146 y F4(\()2268 2134
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13876
y F9(\()2296 2146 y Ga(x)2348 2134 y F9(\))p FX(h)2403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13877
2146 y Ga(a)2451 2134 y FX(i)2478 2146 y Ga(M)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13878
b Ga(b)26 b F4(:=)2796 2134 y F9(\()2823 2146 y Ga(y)2871
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13879
2134 y F9(\))2898 2146 y Ga(P)t F6(g)p Ga(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13880
F4(\))q Ga(;)3160 2134 y F9(\()3188 2146 y Ga(y)3236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13881
2134 y F9(\))3263 2146 y Ga(P)e F4(\))397 2316 y FL(Imp)541
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13882
2337 y Gc(L)594 2316 y F4(\()629 2304 y FX(h)656 2316
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13883
y Ga(a)704 2304 y FX(i)732 2316 y Ga(M)d(;)870 2304 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13884
F9(\()898 2316 y Ga(x)950 2304 y F9(\))977 2316 y Ga(N)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13885
b(y)s F4(\))-5 b F6(f)e Ga(y)29 b F4(:=)1412 2304 y FX(h)1439
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13886
2316 y Ga(c)1478 2304 y FX(i)1506 2316 y Ga(P)s F6(g)1646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13887
2264 y F5(def)1653 2316 y F4(=)40 b FL(Cut)p F4(\()1937
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13888
2304 y FX(h)1965 2316 y Ga(c)2004 2304 y FX(i)2031 2316
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13889
y Ga(P)13 b(;)2142 2304 y F9(\()2170 2316 y Ga(y)2218
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13890
2304 y F9(\))2245 2316 y FL(Imp)2390 2337 y Gc(L)2442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13891
2316 y F4(\()2477 2304 y FX(h)2505 2316 y Ga(a)2553 2304
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13892
y FX(i)2580 2316 y Ga(M)5 b F6(f)-7 b Ga(y)29 b F4(:=)2906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13893
2304 y FX(h)2934 2316 y Ga(c)2973 2304 y FX(i)3001 2316
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13894
y Ga(P)s F6(g)q Ga(;)3148 2304 y F9(\()3175 2316 y Ga(x)3227
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13895
2304 y F9(\))3255 2316 y Ga(N)5 b F6(f)-7 b Ga(y)29 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13896
F4(:=)3566 2304 y FX(h)3593 2316 y Ga(c)3632 2304 y FX(i)3660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13897
2316 y Ga(P)t F6(g)p Ga(;)15 b(y)s F4(\))q(\))314 2532
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13898
y FL(Cut)o F4(\()486 2520 y FX(h)514 2532 y Ga(a)562
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13899
2520 y FX(i)590 2532 y FL(Ax)o F4(\()p Ga(x;)g(a)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13900
Ga(;)943 2520 y F9(\()971 2532 y Ga(y)1019 2520 y F9(\))1046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13901
2532 y Ga(M)c F4(\))-5 b F6(f)e Ga(x)26 b F4(:=)1412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13902
2520 y FX(h)1439 2532 y Ga(b)1478 2520 y FX(i)1506 2532
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13903
y Ga(P)s F6(g)1646 2481 y F5(def)1653 2532 y F4(=)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13904
b FL(Cut)p F4(\()1937 2520 y FX(h)1965 2532 y Ga(b)2004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13905
2520 y FX(i)2031 2532 y Ga(P)13 b(;)2142 2520 y F9(\()2170
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13906
2532 y Ga(y)2218 2520 y F9(\))2245 2532 y Ga(M)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13907
b Ga(x)26 b F4(:=)2575 2520 y FX(h)2603 2532 y Ga(b)2642
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13908
2520 y FX(i)2669 2532 y Ga(P)t F6(g)p F4(\))323 2702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13909
y FL(Cut)o F4(\()495 2690 y FX(h)523 2702 y Ga(a)571
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13910
2690 y FX(i)599 2702 y Ga(M)10 b(;)737 2690 y F9(\()765
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13911
2702 y Ga(x)817 2690 y F9(\))844 2702 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13912
Ga(x;)15 b(b)p F4(\))q(\))-5 b F6(f)e Ga(b)26 b F4(:=)1403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13913
2690 y F9(\()1431 2702 y Ga(y)1479 2690 y F9(\))1506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13914
2702 y Ga(P)s F6(g)1646 2650 y F5(def)1653 2702 y F4(=)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13915
b FL(Cut)p F4(\()1937 2690 y FX(h)1965 2702 y Ga(a)2013
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13916
2690 y FX(i)2040 2702 y Ga(M)5 b F6(f)-7 b Ga(b)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13917
F4(:=)2357 2690 y F9(\()2385 2702 y Ga(y)2433 2690 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13918
F9(\))2460 2702 y Ga(P)t F6(g)p Ga(;)2607 2690 y F9(\()2635
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13919
2702 y Ga(y)2683 2690 y F9(\))2710 2702 y Ga(P)13 b F4(\))330
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13920
2942 y Gb(Otherwise:)1153 3111 y FL(Ax)p F4(\()p Ga(x;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13921
F4(\))p F6(f)p Ga(\033)s F6(g)1646 3060 y F5(def)1653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13922
3111 y F4(=)40 b FL(Ax)o F4(\()p Ga(x;)15 b(a)p F4(\))827
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13923
3281 y FL(Cut)p F4(\()1000 3269 y FX(h)1028 3281 y Ga(a)1076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13924
3269 y FX(i)1103 3281 y Ga(M)10 b(;)1241 3269 y F9(\()1269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13925
3281 y Ga(x)1321 3269 y F9(\))1348 3281 y Ga(N)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13926
F6(f)p Ga(\033)s F6(g)1646 3229 y F5(def)1653 3281 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13927
F4(=)40 b FL(Cut)p F4(\()1937 3269 y FX(h)1965 3281 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13928
Ga(a)2013 3269 y FX(i)2065 3281 y Ga(M)10 b F6(f)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13929
F6(g)r Ga(;)2350 3269 y F9(\()2377 3281 y Ga(x)2429 3269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13930
y F9(\))2482 3281 y Ga(N)g F6(f)p Ga(\033)s F6(g)q F4(\))902
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13931
3450 y FL(Not)1045 3464 y Gc(R)1103 3450 y F4(\()1138
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13932
3438 y F9(\()1165 3450 y Ga(x)1217 3438 y F9(\))1245
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13933
3450 y Ga(M)g(;)15 b(a)p F4(\))p F6(f)p Ga(\033)s F6(g)1646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13934
3398 y F5(def)1653 3450 y F4(=)40 b FL(Not)1907 3464
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13935
y Gc(R)1964 3450 y F4(\()1999 3438 y F9(\()2027 3450
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13936
y Ga(x)2079 3438 y F9(\))2132 3450 y Ga(M)10 b F6(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13937
Ga(\033)s F6(g)q Ga(;)15 b(a)p F4(\))908 3619 y FL(Not)1051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13938
3633 y Gc(L)1103 3619 y F4(\()1138 3607 y FX(h)1165 3619
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13939
y Ga(a)1213 3607 y FX(i)1241 3619 y Ga(M)10 b(;)15 b(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13940
F4(\))p F6(f)p Ga(\033)s F6(g)1646 3568 y F5(def)1653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13941
3619 y F4(=)40 b FL(Not)1907 3633 y Gc(L)1959 3619 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13942
F4(\()1994 3607 y FX(h)2021 3619 y Ga(a)2069 3607 y FX(i)2122
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13943
3619 y Ga(M)10 b F6(f)p Ga(\033)s F6(g)q Ga(;)15 b(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13944
F4(\))686 3789 y FL(And)840 3803 y Gc(R)898 3789 y F4(\()933
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13945
3777 y FX(h)961 3789 y Ga(a)1009 3777 y FX(i)1036 3789
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13946
y Ga(M)10 b(;)1174 3777 y FX(h)1202 3789 y Ga(b)1241
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13947
3777 y FX(i)1269 3789 y Ga(N)g(;)15 b(c)p F4(\))p F6(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13948
Ga(\033)s F6(g)1646 3737 y F5(def)1653 3789 y F4(=)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13949
b FL(And)1918 3803 y Gc(R)1976 3789 y F4(\()2011 3777
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13950
y FX(h)2039 3789 y Ga(a)2087 3777 y FX(i)2140 3789 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13951
Ga(M)10 b F6(f)p Ga(\033)s F6(g)q Ga(;)2424 3777 y FX(h)2452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13952
3789 y Ga(b)2491 3777 y FX(i)2543 3789 y Ga(N)g F6(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13953
Ga(\033)s F6(g)r Ga(;)15 b(c)p F4(\))896 3958 y FL(And)1051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13954
3921 y Gc(i)1051 3981 y(L)1103 3958 y F4(\()1138 3946
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13955
y F9(\()1166 3958 y Ga(x)1218 3946 y F9(\))1245 3958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13956
y Ga(M)10 b(;)15 b(y)s F4(\))p F6(f)p Ga(\033)s F6(g)1646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13957
3906 y F5(def)1653 3958 y F4(=)40 b FL(And)1918 3921
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13958
y Gc(i)1918 3981 y(L)1971 3958 y F4(\()2006 3946 y F9(\()2033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13959
3958 y Ga(x)2085 3946 y F9(\))2138 3958 y Ga(M)10 b F6(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13960
Ga(\033)s F6(g)q Ga(;)15 b(y)s F4(\))959 4127 y FL(Or)1058
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13961
4091 y Gc(i)1058 4150 y(R)1116 4127 y F4(\()1151 4115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13962
y FX(h)1178 4127 y Ga(a)1226 4115 y FX(i)1254 4127 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13963
Ga(M)10 b(;)15 b(b)p F4(\))p F6(f)p Ga(\033)s F6(g)1646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13964
4076 y F5(def)1653 4127 y F4(=)40 b FL(Or)1863 4091 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13965
Gc(i)1863 4150 y(R)1921 4127 y F4(\()1956 4115 y FX(h)1984
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13966
4127 y Ga(a)2032 4115 y FX(i)2084 4127 y Ga(M)10 b F6(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13967
Ga(\033)s F6(g)r Ga(;)15 b(b)p F4(\))727 4297 y FL(Or)826
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13968
4311 y Gc(L)878 4297 y F4(\()913 4285 y F9(\()941 4297
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13969
y Ga(x)993 4285 y F9(\))1021 4297 y Ga(M)10 b(;)1159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13970
4285 y F9(\()1187 4297 y Ga(y)1235 4285 y F9(\))1262
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13971
4297 y Ga(N)g(;)15 b(z)t F4(\))p F6(f)p Ga(\033)s F6(g)1646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13972
4245 y F5(def)1653 4297 y F4(=)40 b FL(Or)1863 4311 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13973
Gc(L)1915 4297 y F4(\()1950 4285 y F9(\()1978 4297 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13974
Ga(x)2030 4285 y F9(\))2083 4297 y Ga(M)10 b F6(f)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13975
F6(g)q Ga(;)2367 4285 y F9(\()2395 4297 y Ga(y)2443 4285
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13976
y F9(\))2495 4297 y Ga(N)g F6(f)p Ga(\033)s F6(g)q Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13977
b(z)t F4(\))807 4466 y FL(Imp)951 4488 y Gc(R)1009 4466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13978
y F4(\()1044 4454 y F9(\()1072 4466 y Ga(x)1124 4454
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13979
y F9(\))p FX(h)1178 4466 y Ga(a)1226 4454 y FX(i)1254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13980
4466 y Ga(M)10 b(;)15 b(b)p F4(\))p F6(f)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13981
F6(g)1646 4415 y F5(def)1653 4466 y F4(=)40 b FL(Imp)1908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13982
4488 y Gc(R)1966 4466 y F4(\()2001 4454 y F9(\()2029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13983
4466 y Ga(x)2081 4454 y F9(\))p FX(h)2136 4466 y Ga(a)2184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13984
4454 y FX(i)2236 4466 y Ga(M)10 b F6(f)p Ga(\033)s F6(g)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13985
Ga(;)15 b(b)p F4(\))680 4635 y FL(Imp)824 4657 y Gc(L)877
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13986
4635 y F4(\()912 4623 y FX(h)939 4635 y Ga(a)987 4623
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13987
y FX(i)1015 4635 y Ga(M)10 b(;)1153 4623 y F9(\()1181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13988
4635 y Ga(x)1233 4623 y F9(\))1260 4635 y Ga(N)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13989
b(y)s F4(\))p F6(f)p Ga(\033)s F6(g)1646 4584 y F5(def)1653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13990
4635 y F4(=)40 b FL(Imp)1908 4657 y Gc(L)1960 4635 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13991
F4(\()1995 4623 y FX(h)2023 4635 y Ga(a)2071 4623 y FX(i)2124
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13992
4635 y Ga(M)10 b F6(f)p Ga(\033)s F6(g)q Ga(;)2408 4623
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13993
y F9(\()2436 4635 y Ga(x)2488 4623 y F9(\))2541 4635
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13994
y Ga(N)g F6(f)p Ga(\033)s F6(g)q Ga(;)15 b(y)s F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13995
3965 4767 V 277 4770 3691 4 v Black 1157 4923 a Gg(Figure)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13996
b(2.6:)29 b(Auxiliary)c(proof)g(substitution.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13997
Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13998
%%Page: 34 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 13999
TeXDict begin 34 45 bop Black -144 51 a Gb(34)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14000
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(outline)k(ho)n(w)d(we)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14001
(shall)j(proceed)g(in)f(our)g(proof)g(of)g(strong)h(normalisation)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14002
(for)e F4(\()p FY(T)t Ga(;)3017 280 y Gc(aux)2997 317
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14003
y F6(\000)-31 b(\000)f(!)p F4(\))p Gg(.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14004
Black Black Black 597 536 a(\(1\))p Black 47 w(De\002ne)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14005
b(the)g(sets)h(of)g(candidates)i(o)o(v)o(er)e(types)g(using)h(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14006
(\002x)o(ed)g(point)i(construction)749 648 y(\(De\002nition)i(2.3.4\).)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14007
p Black 597 791 a(\(2\))p Black 47 w(Pro)o(v)o(e)f(that)h(candidates)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14008
(are)e(closed)h(under)g(reduction)h(\(Lemma)c(2.3.12\).)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14009
Black 597 933 a(\(3\))p Black 47 w(Sho)n(w)35 b(that)i(a)f(named)h(or)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14010
(co-named)i(term)e(in)h(a)f(candidate)i(implies)g(strong)749
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14011
1046 y(normalisation)26 b(for)e(the)g(corresponding)k(term)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14012
b(\(Lemma)g(2.3.13\).)p Black 597 1189 a(\(4\))p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14013
47 w(Extend)40 b(the)g(notion)i(of)e(safe)g(substitution)k(to)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14014
b(simultaneous)k(substitutions)749 1302 y(\(De\002nition)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14015
b(2.3.15\).)p Black 597 1444 a(\(5\))p Black 47 w(Pro)o(v)o(e)f(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14016
(all)f(terms)h(are)g(strongly)i(normalising)g(\(Theorem)e(2.3.19\).)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14017
1662 y(Finally)-6 b(,)32 b(we)d(shall)i(sho)n(w)e(that)i(e)n(v)o(ery)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14018
1604 1625 y Gc(cut)1573 1662 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14019
Gg(-reduction)33 b(maps)d(onto)g(a)f(series)i(of)2987
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14020
1625 y Gc(aux)2967 1662 y F6(\000)-31 b(\000)f(!)p Gg(-reductions)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14021
1775 y(and)24 b(thus)h(pro)o(v)o(e)f(that)g F4(\()p FY(T)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14022
Ga(;)1199 1738 y Gc(cut)1168 1775 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14023
F4(\))23 b Gg(is)g(strongly)j(normalising,)g(too.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14024
1905 y(First,)40 b(we)35 b(shall)j(de\002ne)f(for)f(e)n(v)o(ery)h(type)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14025
g(tw)o(o)f(candidates,)43 b(written)37 b(as)f FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14026
F6(h)p Ga(B)5 b F6(i)p FB(K)35 b Gg(and)i FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14027
Ga(B)5 b F4(\))p FB(K)p Gg(.)321 2018 y(These)40 b(candidates)j(are)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14028
(subsets)h(of)e(named)i(or)e(co-named)i(terms;)49 b(i.e.,)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14029
b FB(J)p F6(h)p Ga(B)5 b F6(i)p FB(K)55 b F6(\022)f FY(T)3264
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14030
2036 y FX(h)p Gc(B)s FX(i)3415 2018 y Gg(and)321 2130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14031
y FB(J)p F4(\()p Ga(B)5 b F4(\))p FB(K)44 b F6(\022)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14032
FY(T)754 2149 y F9(\()p Gc(B)s F9(\))866 2130 y Gg(.)58
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14033
b(Whereas)35 b(traditional)h(notions)f(of)f(candidates)i(are)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14034
(de\002ned)g(by)g(a)f(simple)h(in-)321 2243 y(duction)h(o)o(v)o(er)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14035
(types,)j(our)d(candidates)i(are)e(inducti)n(v)o(ely)j(de\002ned)d(o)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14036
(v)o(er)g(types,)j(b)n(ut)d(also)h(include)321 2356 y(\002x)o(ed)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14037
b(point)i(operations.)31 b(Before)22 b(we)f(gi)n(v)o(e)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14038
(de\002nition)h(of)f(the)f(candidates,)k(we)c(shall)h(introduce)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14039
2469 y(some)30 b(set)f(operators,)k(which)c(\002x)g(certain)i(closure)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14040
(properties)g(for)f(the)f(candidates.)49 b(Each)29 b(of)g(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14041
2582 y(operators)i(is)e(de\002ned)g(o)o(v)o(er)g(sets)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14042
(\(co-\)named)h(terms)f(ha)n(ving)i(a)d(speci\002c)h(term)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14043
(constructor)j(at)321 2695 y(the)24 b(top-le)n(v)o(el.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14044
Black 321 2883 a Gb(De\002nition)f(2.3.3)g Gg(\(Set)h(Operators\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14045
Gb(:)p Black Black Black 745 3192 a FW(A)t(X)t(I)t(O)t(M)t(S)1055
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14046
3210 y F9(\()p Gc(B)s F9(\))1256 3140 y F5(def)1263 3192
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14047
y F4(=)1426 3091 y FK(n)1537 3180 y F9(\()1564 3192 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14048
Ga(x)r F4(:)r Ga(B)1719 3180 y F9(\))1747 3192 y FL(Ax)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14049
F4(\()p Ga(y)s(;)15 b(b)p F4(\))2120 3087 y FK(\014)2120
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14050
3142 y(\014)2120 3196 y(\014)2224 3180 y F9(\()2251 3192
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14051
y Ga(x)r F4(:)r Ga(B)2406 3180 y F9(\))2433 3192 y FL(Ax)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14052
F4(\()p Ga(y)s(;)g(b)p F4(\))26 b F6(2)f FY(T)2902 3210
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14053
y F9(\()p Gc(B)s F9(\))3064 3091 y FK(o)745 3426 y FW(A)t(X)t(I)t(O)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14054
(M)t(S)1055 3445 y FX(h)p Gc(B)s FX(i)1256 3375 y F5(def)1263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14055
3426 y F4(=)1426 3325 y FK(n)1537 3414 y FX(h)1564 3426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14056
y Ga(a)r F4(:)r Ga(B)1715 3414 y FX(i)1743 3426 y FL(Ax)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14057
F4(\()p Ga(y)s(;)15 b(b)p F4(\))2116 3322 y FK(\014)2116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14058
3376 y(\014)2116 3431 y(\014)2220 3414 y FX(h)2247 3426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14059
y Ga(a)r F4(:)r Ga(B)2398 3414 y FX(i)2426 3426 y FL(Ax)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14060
F4(\()p Ga(y)s(;)g(b)p F4(\))26 b F6(2)f FY(T)2894 3445
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14061
y FX(h)p Gc(B)s FX(i)3057 3325 y FK(o)321 3639 y Gg(Note)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14062
b(that)h Ga(x)e Gg(can)h(be)g(equal)h(to)f Ga(y)s Gg(,)i(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14063
Ga(a)g Gg(to)g Ga(b)p Gg(.)68 b(Figure)37 b(2.7)g(gi)n(v)o(es)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14064
(set)f(operators)i(which)321 3752 y(correspond)27 b(to)c(the)h(other)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14065
(term)e(constructors.)32 b(Additionally)27 b(we)22 b(ha)n(v)o(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14066
Black Black 330 3915 a FW(B)t(I)t(N)t(D)t(I)t(N)t(G)665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14067
3933 y F9(\()p Gc(B)s F9(\))782 3915 y F4(\()p Ga(X)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14068
b F4(\))1019 3863 y F5(def)1026 3915 y F4(=)1189 3814
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14069
y FK(n)1274 3903 y F9(\()1302 3915 y Ga(x)r F4(:)r Ga(B)1457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14070
3903 y F9(\))1484 3915 y Ga(M)1607 3810 y FK(\014)1607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14071
3865 y(\014)1607 3919 y(\014)1663 3915 y Gg(for)24 b(all)1908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14072
3903 y FX(h)1935 3915 y Ga(a)r F4(:)r Ga(B)2086 3903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14073
y FX(i)2114 3915 y Ga(P)38 b F6(2)25 b Ga(X)d(:)k(M)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14074
b F6(f)-7 b Ga(x)26 b F4(:=)2774 3903 y FX(h)2801 3915
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14075
y Ga(a)r F4(:)r Ga(B)2952 3903 y FX(i)2980 3915 y Ga(P)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14076
F6(g)g(2)f Ga(S)5 b(N)3332 3929 y Gc(aux)3479 3814 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14077
FK(o)339 4149 y FW(B)t(I)t(N)t(D)t(I)t(N)t(G)674 4168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14078
y FX(h)p Gc(B)s FX(i)791 4149 y F4(\()p Ga(Y)20 b F4(\))1019
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14079
4098 y F5(def)1026 4149 y F4(=)1189 4048 y FK(n)1274
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14080
4137 y FX(h)1302 4149 y Ga(a)r F4(:)r Ga(B)1453 4137
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14081
y FX(i)1480 4149 y Ga(M)1604 4045 y FK(\014)1604 4099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14082
y(\014)1604 4154 y(\014)1659 4149 y Gg(for)k(all)1904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14083
4137 y F9(\()1931 4149 y Ga(x)r F4(:)r Ga(B)2086 4137
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14084
y F9(\))2114 4149 y Ga(P)38 b F6(2)25 b Ga(Y)35 b(:)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14085
b(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)2761 4137 y F9(\()2788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14086
4149 y Ga(x)r F4(:)r Ga(B)2943 4137 y F9(\))2970 4149
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14087
y Ga(P)t F6(g)g(2)e Ga(S)5 b(N)3322 4163 y Gc(aux)3470
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14088
4048 y FK(o)462 4420 y Gg(The)33 b(set)f(operators)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14089
b FW(A)t(X)t(I)t(O)t(M)t(S)f Gg(and)d(the)g(set)g(operators)i(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14090
(en)e(in)f(Figure)i(2.7)e(correspond)k(to)321 4532 y(the)d(properties)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14091
(we)c(need)j(to)e(pro)o(v)o(e)g(that)h(a)f(logical)i(cut)f(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14092
(strongly)i(normalising;)42 b FW(B)t(I)t(N)t(D)t(I)t(N)t(G)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14093
b Gg(is)321 4645 y(suf)n(\002cient)h(to)f(sho)n(w)f(strong)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14094
(normalisation)i(of)c(a)g(commuting)i(cut.)62 b(F)o(or)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14095
b(the)h(de\002nition)h(of)e(the)321 4758 y(candidates)d(we)c(use)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14096
(\002x)o(ed)g(points)h(of)f(increasing)i(set)e(operators.)44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14097
b(A)27 b(set)h(operator)l(,)j Ga(op)p Gg(,)d(is)f(said)i(to)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14098
4871 y(be:)p Black Black 774 5065 a(increasing,)118 b(if)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14099
b(and)h(only)h(if)114 b Ga(S)30 b F6(\022)25 b Ga(S)2114
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14100
5032 y FX(0)2188 5065 y F6(\))50 b Ga(op)p F4(\()p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14101
b F4(\))26 b F6(\022)f Ga(op)p F4(\()p Ga(S)2858 5032
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14102
y FX(0)2881 5065 y F4(\))p Gg(,)e(and)774 5213 y(decreasing,)103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14103
b(if)23 b(and)h(only)h(if)114 b Ga(S)30 b F6(\022)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14104
b Ga(S)2114 5180 y FX(0)2188 5213 y F6(\))50 b Ga(op)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14105
F4(\()p Ga(S)5 b F4(\))26 b F6(\023)f Ga(op)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14106
Ga(S)2858 5180 y FX(0)2881 5213 y F4(\))p Gg(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14107
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14108
%%Page: 35 47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14109
TeXDict begin 35 46 bop Black 277 51 a Gb(2.3)23 b(Pr)n(oof)i(of)e(Str)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14110
n(ong)h(Normalisation)2290 b(35)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14111
Black 277 492 V 277 4876 4 4384 v Black Black 708 834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14112
a F5(N)t(O)q(T)t(R)t(I)t(G)t(H)t(T)1075 849 y FQ(h:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14113
FS(B)s FQ(i)1231 834 y FG(\()p FU(X)7 b FG(\))1471 787
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14114
y F5(def)1481 834 y FG(=)1655 664 y FI(8)1655 738 y(<)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14115
888 y(:)1771 721 y FQ(h)1798 733 y FU(a)r FG(:)r FT(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14116
FU(B)1991 721 y FQ(i)2019 733 y FF(Not)2149 745 y FS(R)2204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14117
733 y FG(\()2236 721 y FJ(\()2262 733 y FU(x)r FG(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14118
FU(B)2403 721 y FJ(\))2430 733 y FU(M)i(;)14 b(a)p FG(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14119
b FT(j)1960 833 y FF(Not)2090 845 y FS(R)2144 833 y FG(\()2176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14120
821 y FJ(\()2203 833 y FU(x)r FG(:)r FU(B)2344 821 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14121
FJ(\))2371 833 y FU(M)9 b(;)14 b(a)p FG(\))23 b Gd(freshly)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14122
(introduces)i FU(a;)1960 920 y FJ(\()1986 932 y FU(x)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14123
FG(:)r FU(B)2127 920 y FJ(\))2154 932 y FU(M)32 b FT(2)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14124
b FU(X)3326 664 y FI(9)3326 738 y(=)3326 888 y(;)759
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14125
1279 y F5(N)t(O)q(T)t(L)t(E)t(F)t(T)1079 1294 y FJ(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14126
FQ(:)p FS(B)s FJ(\))1231 1279 y FG(\()p FU(X)7 b FG(\))1471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14127
1232 y F5(def)1481 1279 y FG(=)1655 1109 y FI(8)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14128
1184 y(<)1655 1333 y(:)1771 1167 y FJ(\()1797 1179 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14129
FU(x)r FG(:)r FT(:)p FU(B)1993 1167 y FJ(\))2020 1179
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14130
y FF(Not)2151 1191 y FS(L)2200 1179 y FG(\()2232 1167
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14131
y FQ(h)2260 1179 y FU(a)r FG(:)r FU(B)2398 1167 y FQ(i)2426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14132
1179 y FU(M)h(;)14 b(x)p FG(\))24 b FT(j)1960 1279 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14133
FF(Not)2090 1291 y FS(L)2140 1279 y FG(\()2172 1267 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14134
FQ(h)2199 1279 y FU(a)r FG(:)r FU(B)2337 1267 y FQ(i)2365
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14135
1279 y FU(M)8 b(;)14 b(x)p FG(\))24 b Gd(freshly)19 b(introduces)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14136
FU(x;)1960 1366 y FQ(h)1987 1378 y FU(a)r FG(:)r FU(B)2125
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14137
1366 y FQ(i)2152 1378 y FU(M)32 b FT(2)23 b FU(X)3326
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14138
1109 y FI(9)3326 1184 y(=)3326 1333 y(;)548 1775 y F5(A)t(N)t(D)t(R)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14139
(I)t(G)t(H)t(T)925 1790 y FQ(h)p FS(B)s FQ(^)p FS(C)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14140
FQ(i)1132 1775 y FG(\()p FU(X)r(;)14 b(Y)19 b FG(\))1471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14141
1728 y F5(def)1481 1775 y FG(=)1655 1555 y FI(8)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14142
1630 y(>)1655 1655 y(>)1655 1680 y(<)1655 1829 y(>)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14143
1854 y(>)1655 1879 y(:)1771 1613 y FQ(h)1798 1625 y FU(c)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14144
FG(:)r FU(B)t FT(^)q FU(C)2049 1613 y FQ(i)2076 1625
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14145
y FF(And)2217 1637 y FS(R)2272 1625 y FG(\()2304 1613
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14146
y FQ(h)2331 1625 y FU(a)r FG(:)r FU(B)2469 1613 y FQ(i)2497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14147
1625 y FU(M)8 b(;)2623 1613 y FQ(h)2650 1625 y FU(b)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14148
FG(:)r FU(C)2778 1613 y FQ(i)2806 1625 y FU(N)h(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14149
b(c)p FG(\))23 b FT(j)1960 1724 y FF(And)2101 1736 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14150
FS(R)2155 1724 y FG(\()2187 1712 y FQ(h)2214 1724 y FU(a)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14151
FG(:)r FU(B)2352 1712 y FQ(i)2380 1724 y FU(M)9 b(;)2507
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14152
1712 y FQ(h)2534 1724 y FU(b)r FG(:)r FU(C)2662 1712
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14153
y FQ(i)2689 1724 y FU(N)g(;)14 b(c)p FG(\))23 b Gd(freshly)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14154
(introduces)h FU(c;)1960 1812 y FQ(h)1987 1824 y FU(a)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14155
FG(:)r FU(B)2125 1812 y FQ(i)2152 1824 y FU(M)32 b FT(2)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14156
b FU(X)r(;)1960 1912 y FQ(h)1987 1924 y FU(b)r FG(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14157
FU(C)2115 1912 y FQ(i)2142 1924 y FU(N)32 b FT(2)24 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14158
FU(Y)3614 1555 y FI(9)3614 1630 y(>)3614 1655 y(>)3614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14159
1680 y(=)3614 1829 y(>)3614 1854 y(>)3614 1879 y(;)636
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14160
2276 y F5(A)t(N)t(D)t(L)t(E)t(F)t(T)966 2246 y FS(i)966
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14161
2303 y FJ(\()p FS(B)1042 2311 y FP(1)1074 2303 y FQ(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14162
FS(B)1169 2311 y FP(2)1201 2303 y FJ(\))1231 2276 y FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14163
FU(X)7 b FG(\))1471 2229 y F5(def)1481 2276 y FG(=)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14164
2106 y FI(8)1655 2181 y(<)1655 2330 y(:)1771 2164 y FJ(\()1797
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14165
2176 y FU(y)e FG(:)r FU(B)1931 2188 y FJ(1)1969 2176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14166
y FT(^)p FU(B)2087 2188 y FJ(2)2124 2164 y(\))2150 2176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14167
y FF(And)2291 2139 y FS(i)2291 2197 y(L)2341 2176 y FG(\()2373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14168
2164 y FJ(\()2399 2176 y FU(x)r FG(:)r FU(B)2536 2188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14169
y FS(i)2565 2164 y FJ(\))2591 2176 y FU(M)k(;)14 b(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14170
FG(\))23 b FT(j)1960 2281 y FF(And)2101 2244 y FS(i)2101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14171
2302 y(L)2150 2281 y FG(\()2182 2269 y FJ(\()2209 2281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14172
y FU(x)r FG(:)r FU(B)2346 2293 y FS(i)2375 2269 y FJ(\))2401
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14173
2281 y FU(M)8 b(;)14 b(y)s FG(\))23 b Gd(freshly)d(introduces)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14174
FU(y)s(;)1960 2369 y FJ(\()1986 2381 y FU(x)r FG(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14175
FU(B)2123 2393 y FS(i)2152 2369 y FJ(\))2178 2381 y FU(M)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14176
b FT(2)24 b FU(X)3355 2106 y FI(9)3355 2181 y(=)3355
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14177
2330 y(;)642 2734 y F5(O)t(R)t(R)t(I)t(G)t(H)t(T)963
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14178
2704 y FS(i)963 2761 y FQ(h)p FS(B)1040 2769 y FP(1)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14179
2761 y FQ(_)p FS(B)1168 2769 y FP(2)1200 2761 y FQ(i)1231
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14180
2734 y FG(\()p FU(X)7 b FG(\))1471 2687 y F5(def)1481
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14181
2734 y FG(=)1655 2564 y FI(8)1655 2638 y(<)1655 2788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14182
y(:)1771 2621 y FQ(h)1798 2633 y FU(b)r FG(:)r FU(B)1924
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14183
2645 y FJ(1)1961 2633 y FT(_)q FU(B)2080 2645 y FJ(2)2117
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14184
2621 y FQ(i)2144 2633 y FF(Or)2234 2596 y FS(i)2234 2654
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14185
y(R)2289 2633 y FG(\()2321 2621 y FQ(h)2348 2633 y FU(a)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14186
FG(:)r FU(B)2482 2645 y FS(i)2511 2621 y FQ(i)2537 2633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14187
y FU(M)i(;)14 b(b)p FG(\))23 b FT(j)1960 2739 y FF(Or)2050
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14188
2702 y FS(i)2050 2759 y(R)2105 2739 y FG(\()2137 2727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14189
y FQ(h)2164 2739 y FU(a)r FG(:)r FU(B)2298 2751 y FS(i)2326
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14190
2727 y FQ(i)2353 2739 y FU(M)9 b(;)14 b(b)p FG(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14191
b Gd(freshly)c(introduces)i FU(b;)1960 2827 y FQ(h)1987
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14192
2839 y FU(a)r FG(:)r FU(B)2121 2851 y FS(i)2149 2827
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14193
y FQ(i)2176 2839 y FU(M)32 b FT(2)23 b FU(X)3291 2564
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14194
y FI(9)3291 2638 y(=)3291 2788 y(;)654 3235 y F5(O)t(R)t(L)t(E)t(F)t(T)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14195
928 3250 y FJ(\()p FS(B)s FQ(_)o FS(C)t FJ(\))1132 3235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14196
y FG(\()p FU(X)r(;)14 b(Y)19 b FG(\))1471 3188 y F5(def)1481
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14197
3235 y FG(=)1655 3015 y FI(8)1655 3090 y(>)1655 3115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14198
y(>)1655 3140 y(<)1655 3289 y(>)1655 3314 y(>)1655 3339
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14199
y(:)1771 3073 y FJ(\()1797 3085 y FU(z)6 b FG(:)r FU(B)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14200
FT(_)p FU(C)2054 3073 y FJ(\))2081 3085 y FF(Or)2171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14201
3097 y FS(L)2221 3085 y FG(\()2253 3073 y FJ(\()2279
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14202
3085 y FU(x)r FG(:)r FU(B)2420 3073 y FJ(\))2448 3085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14203
y FU(M)i(;)2574 3073 y FJ(\()2600 3085 y FU(y)d FG(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14204
FU(C)2736 3073 y FJ(\))2763 3085 y FU(N)k(;)14 b(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14205
FG(\))22 b FT(j)1960 3185 y FF(Or)2050 3197 y FS(L)2100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14206
3185 y FG(\()2132 3173 y FJ(\()2158 3185 y FU(x)r FG(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14207
FU(B)2299 3173 y FJ(\))2327 3185 y FU(M)8 b(;)2453 3173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14208
y FJ(\()2479 3185 y FU(y)d FG(:)r FU(C)2615 3173 y FJ(\))2642
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14209
3185 y FU(N)k(;)14 b(z)t FG(\))22 b Gd(freshly)e(introduces)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14210
FU(z)t(;)1960 3272 y FJ(\()1986 3284 y FU(x)r FG(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14211
FU(B)2127 3272 y FJ(\))2154 3284 y FU(M)32 b FT(2)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14212
b FU(X)r(;)1960 3372 y FJ(\()1986 3384 y FU(y)5 b FG(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14213
FU(C)2122 3372 y FJ(\))2148 3384 y FU(N)32 b FT(2)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14214
b FU(Y)3579 3015 y FI(9)3579 3090 y(>)3579 3115 y(>)3579
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14215
3140 y(=)3579 3289 y(>)3579 3314 y(>)3579 3339 y(;)566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14216
3880 y F5(I)t(M)t(P)t(R)t(I)t(G)t(H)t(T)917 3895 y FQ(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14217
FS(B)s FQ(\033)q FS(C)t FQ(i)1132 3880 y FG(\()p FU(X)r(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14218
b(Y)19 b FG(\))1471 3833 y F5(def)1481 3880 y FG(=)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14219
3561 y FI(8)1655 3635 y(>)1655 3660 y(>)1655 3685 y(>)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14220
3710 y(>)1655 3735 y(>)1655 3760 y(>)1655 3785 y(<)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14221
3934 y(>)1655 3959 y(>)1655 3984 y(>)1655 4009 y(>)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14222
4034 y(>)1655 4059 y(>)1655 4084 y(:)1771 3618 y FQ(h)1798
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14223
3630 y FU(b)r FG(:)r FU(B)t FT(\033)p FU(C)2058 3618
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14224
y FQ(i)2085 3630 y FF(Imp)2217 3651 y FS(R)2271 3630
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14225
y FG(\()2303 3618 y FJ(\()2330 3630 y FU(x)r FG(:)r FU(B)2471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14226
3618 y FJ(\))q FQ(h)2525 3630 y FU(a)r FG(:)r FU(C)2661
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14227
3618 y FQ(i)2689 3630 y FU(M)9 b(;)14 b(b)p FG(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14228
b FT(j)1960 3730 y FF(Imp)2092 3750 y FS(R)2146 3730
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14229
y FG(\()2178 3718 y FJ(\()2204 3730 y FU(x)r FG(:)r FU(B)2345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14230
3718 y FJ(\))r FQ(h)2400 3730 y FU(a)r FG(:)r FU(C)2536
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14231
3718 y FQ(i)2564 3730 y FU(M)8 b(;)14 b(b)p FG(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14232
b Gd(freshly)c(introduces)i FU(b;)1960 3830 y Gd(for)e(all)2183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14233
3818 y FJ(\()2209 3830 y FU(z)6 b FG(:)r FU(C)2344 3818
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14234
y FJ(\))2371 3830 y FU(P)35 b FT(2)23 b FU(Y)42 b(:)2290
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14235
3917 y FJ(\()2316 3929 y FU(x)r FG(:)r FU(B)2457 3917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14236
y FJ(\))2485 3929 y FU(M)t FT(f)-7 b FU(a)22 b FG(:=)2782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14237
3917 y FJ(\()2808 3929 y FU(z)2851 3917 y FJ(\))2876
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14238
3929 y FU(P)r FT(g)h(2)g FU(X)r(;)1960 4029 y Gd(for)c(all)2183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14239
4017 y FQ(h)2210 4029 y FU(c)r FG(:)r FU(B)2340 4017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14240
y FQ(i)2368 4029 y FU(Q)k FT(2)g FU(X)30 b(:)2290 4117
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14241
y FQ(h)2317 4129 y FU(a)r FG(:)r FU(C)2453 4117 y FQ(i)2481
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14242
4129 y FU(M)t FT(f)-7 b FU(x)23 b FG(:=)2782 4117 y FQ(h)2809
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14243
4129 y FU(c)2845 4117 y FQ(i)2871 4129 y FU(Q)-9 b FT(g)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14244
b(2)i FU(Y)3502 3561 y FI(9)3502 3635 y(>)3502 3660 y(>)3502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14245
3685 y(>)3502 3710 y(>)3502 3735 y(>)3502 3760 y(>)3502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14246
3785 y(=)3502 3934 y(>)3502 3959 y(>)3502 3984 y(>)3502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14247
4009 y(>)3502 4034 y(>)3502 4059 y(>)3502 4084 y(;)616
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14248
4525 y F5(I)t(M)t(P)t(L)t(E)t(F)t(T)920 4540 y FJ(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14249
FS(B)s FQ(\033)p FS(C)t FJ(\))1132 4525 y FG(\()p FU(X)r(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14250
b(Y)19 b FG(\))1471 4478 y F5(def)1481 4525 y FG(=)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14251
4305 y FI(8)1655 4380 y(>)1655 4405 y(>)1655 4430 y(<)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14252
4579 y(>)1655 4604 y(>)1655 4629 y(:)1771 4363 y FJ(\()1797
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14253
4375 y FU(y)5 b FG(:)r FU(B)t FT(\033)p FU(C)2065 4363
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14254
y FJ(\))2091 4375 y FF(Imp)2223 4395 y FS(L)2273 4375
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14255
y FG(\()2305 4363 y FQ(h)2332 4375 y FU(a)r FG(:)r FU(B)2470
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14256
4363 y FQ(i)2498 4375 y FU(M)j(;)2624 4363 y FJ(\()2650
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14257
4375 y FU(x)r FG(:)r FU(C)2789 4363 y FJ(\))2817 4375
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14258
y FU(N)h(;)14 b(y)s FG(\))23 b FT(j)1960 4475 y FF(Imp)2092
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14259
4495 y FS(L)2141 4475 y FG(\()2173 4463 y FQ(h)2200 4475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14260
y FU(a)r FG(:)r FU(B)2338 4463 y FQ(i)2366 4475 y FU(M)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14261
b(;)2493 4463 y FJ(\()2519 4475 y FU(x)r FG(:)r FU(C)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14262
4463 y FJ(\))2685 4475 y FU(N)g(;)14 b(y)s FG(\))23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14263
Gd(freshly)c(introduces)i FU(y)s(;)1960 4562 y FQ(h)1987
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14264
4574 y FU(a)r FG(:)r FU(B)2125 4562 y FQ(i)2152 4574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14265
y FU(M)32 b FT(2)23 b FU(X)r(;)1960 4662 y FJ(\()1986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14266
4674 y FU(x)r FG(:)r FU(C)2125 4662 y FJ(\))2152 4674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14267
y FU(N)32 b FT(2)23 b FU(Y)3626 4305 y FI(9)3626 4380
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14268
y(>)3626 4405 y(>)3626 4430 y(=)3626 4579 y(>)3626 4604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14269
y(>)3626 4629 y(;)p 3965 4876 V 277 4879 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14270
582 5032 a Gg(Figure)h(2.7:)29 b(De\002nition)c(of)e(the)h(set)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14271
(operators)i(for)d(propositional)28 b(connecti)n(v)o(es.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14272
Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14273
%%Page: 36 48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14274
TeXDict begin 36 47 bop Black -144 51 a Gb(36)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14275
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14276
b(e)23 b(are)h(no)n(w)f(ready)h(to)g(de\002ne)g(the)f(set)h(operator)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14277
FW(N)t(E)t(G)d Gg(and)f(the)g(candidates.)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14278
505 a Gb(De\002nition)f(2.3.4)g Gg(\(Reducibility)k(Candidates\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14279
Gb(:)p Black 321 618 a Gg(The)c(mutually)i(recursi)n(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14280
(de\002nition)f(o)o(v)o(er)e(types)i(for)h FW(N)t(E)t(G)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14281
Gg(and)f(the)g(candidates)j(is)c(as)h(follo)n(ws:)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14282
781 y FA(NEG)483 798 y Fz(h)p Fy(B)s Fz(i)615 781 y Gb(:)333
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14283
974 y FW(N)t(E)t(G)493 993 y FX(h)p Gc(A)p FX(i)606 974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14284
y F4(\()p Ga(X)7 b F4(\))947 923 y F5(def)954 974 y F4(=)333
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14285
1155 y FW(N)t(E)t(G)493 1173 y FX(h:)p Gc(C)e FX(i)655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14286
1155 y F4(\()p Ga(X)i F4(\))947 1103 y F5(def)954 1155
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14287
y F4(=)333 1336 y FW(N)t(E)t(G)493 1354 y FX(h)p Gc(C)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14288
FX(^)r Gc(D)r FX(i)715 1336 y F4(\()p Ga(X)i F4(\))947
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14289
1284 y F5(def)954 1336 y F4(=)333 1516 y FW(N)t(E)t(G)493
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14290
1535 y FX(h)p Gc(C)570 1544 y FV(1)607 1535 y FX(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14291
Gc(C)704 1544 y FV(2)738 1535 y FX(i)770 1516 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14292
Ga(X)g F4(\))947 1465 y F5(def)954 1516 y F4(=)333 1697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14293
y FW(N)t(E)t(G)493 1715 y FX(h)p Gc(C)e FX(\033)r Gc(D)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14294
FX(i)723 1697 y F4(\()p Ga(X)i F4(\))947 1645 y F5(def)954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14295
1697 y F4(=)1073 863 y FK(9)1073 945 y(>)1073 972 y(>)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14296
999 y(>)1073 1027 y(>)1073 1054 y(>)1073 1081 y(>)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14297
1109 y(>)1073 1136 y(>)1073 1163 y(>)1073 1190 y(>)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14298
1218 y(=)1073 1381 y(>)1073 1409 y(>)1073 1436 y(>)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14299
1463 y(>)1073 1490 y(>)1073 1518 y(>)1073 1545 y(>)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14300
1572 y(>)1073 1599 y(>)1073 1627 y(>)1073 1654 y(;)1238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14301
1151 y FW(A)t(X)t(I)t(O)t(M)t(S)1548 1169 y FX(h)p Gc(B)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14302
FX(i)1690 1151 y F6([)1329 1331 y FW(B)t(I)t(N)t(D)t(I)t(N)t(G)1664
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14303
1350 y FX(h)p Gc(B)s FX(i)1781 1331 y F4(\()p Ga(X)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14304
F4(\))26 b F6([)2102 863 y FK(8)2102 945 y(>)2102 972
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14305
y(>)2102 999 y(>)2102 1027 y(>)2102 1054 y(>)2102 1081
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14306
y(>)2102 1109 y(>)2102 1136 y(>)2102 1163 y(>)2102 1190
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14307
y(>)2102 1218 y(<)2102 1381 y(>)2102 1409 y(>)2102 1436
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14308
y(>)2102 1463 y(>)2102 1490 y(>)2102 1518 y(>)2102 1545
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14309
y(>)2102 1572 y(>)2102 1599 y(>)2102 1627 y(>)2102 1654
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14310
y(:)2224 974 y F6(\000)2226 1155 y FW(N)t(O)q(T)t(R)t(I)t(G)t(H)t(T)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14311
2626 1173 y FX(h:)p Gc(C)5 b FX(i)2790 1155 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14312
FB(J)p F4(\()p Ga(C)i F4(\))p FB(K)o F4(\))2226 1336
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14313
y FW(A)t(N)t(D)t(R)t(I)t(G)t(H)t(T)2638 1354 y FX(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14314
Gc(C)e FX(^)r Gc(D)r FX(i)2861 1336 y F4(\()p FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14315
F6(h)p Ga(C)i F6(i)p FB(K)o Ga(;)15 b FB(J)p F6(h)p Ga(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14316
F6(i)q FB(K)o F4(\))2224 1448 y FK(S)2300 1543 y Gc(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14317
F9(=1)p Gc(;)p F9(2)2490 1516 y FW(O)t(R)t(R)t(I)t(G)t(H)t(T)2841
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14318
1483 y Gc(i)2841 1548 y FX(h)p Gc(C)2918 1557 y FV(1)2955
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14319
1548 y FX(_)p Gc(C)3052 1557 y FV(2)3087 1548 y FX(i)3119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14320
1516 y F4(\()p FB(J)p F6(h)p Ga(C)3291 1530 y Gc(i)3319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14321
1516 y F6(i)p FB(K)p F4(\))2226 1697 y FW(I)t(M)t(P)t(R)t(I)t(G)t(H)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14322
(T)2608 1715 y FX(h)p Gc(C)5 b FX(\033)t Gc(D)r FX(i)2840
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14323
1697 y F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))p FB(K)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14324
b FB(J)p F6(h)p Ga(D)s F6(i)q FB(K)o F4(\))321 1918 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14325
FA(NEG)483 1935 y Fx(\()p Fy(B)s Fx(\))615 1918 y Gb(:)333
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14326
2111 y FW(N)t(E)t(G)493 2130 y F9(\()p Gc(A)p F9(\))606
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14327
2111 y F4(\()p Ga(X)7 b F4(\))947 2060 y F5(def)954 2111
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14328
y F4(=)333 2292 y FW(N)t(E)t(G)493 2310 y F9(\()p FX(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14329
Gc(C)e F9(\))655 2292 y F4(\()p Ga(X)i F4(\))947 2240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14330
y F5(def)954 2292 y F4(=)333 2473 y FW(N)t(E)t(G)493
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14331
2491 y F9(\()p Gc(C)570 2500 y FV(1)607 2491 y FX(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14332
Gc(C)704 2500 y FV(2)738 2491 y F9(\))770 2473 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14333
Ga(X)g F4(\))947 2421 y F5(def)954 2473 y F4(=)333 2653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14334
y FW(N)t(E)t(G)493 2672 y F9(\()p Gc(C)e FX(_)r Gc(D)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14335
F9(\))715 2653 y F4(\()p Ga(X)i F4(\))947 2602 y F5(def)954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14336
2653 y F4(=)333 2834 y FW(N)t(E)t(G)493 2852 y F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14337
Gc(C)e FX(\033)r Gc(D)r F9(\))723 2834 y F4(\()p Ga(X)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14338
F4(\))947 2782 y F5(def)954 2834 y F4(=)1073 2000 y FK(9)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14339
2082 y(>)1073 2109 y(>)1073 2137 y(>)1073 2164 y(>)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14340
2191 y(>)1073 2218 y(>)1073 2246 y(>)1073 2273 y(>)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14341
2300 y(>)1073 2327 y(>)1073 2355 y(=)1073 2518 y(>)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14342
2546 y(>)1073 2573 y(>)1073 2600 y(>)1073 2627 y(>)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14343
2655 y(>)1073 2682 y(>)1073 2709 y(>)1073 2737 y(>)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14344
2764 y(>)1073 2791 y(;)1238 2288 y FW(A)t(X)t(I)t(O)t(M)t(S)1548
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14345
2306 y F9(\()p Gc(B)s F9(\))1690 2288 y F6([)1329 2468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14346
y FW(B)t(I)t(N)t(D)t(I)t(N)t(G)1664 2487 y F9(\()p Gc(B)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14347
F9(\))1781 2468 y F4(\()p Ga(X)g F4(\))26 b F6([)2102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14348
2000 y FK(8)2102 2082 y(>)2102 2109 y(>)2102 2137 y(>)2102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14349
2164 y(>)2102 2191 y(>)2102 2218 y(>)2102 2246 y(>)2102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14350
2273 y(>)2102 2300 y(>)2102 2327 y(>)2102 2355 y(<)2102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14351
2518 y(>)2102 2546 y(>)2102 2573 y(>)2102 2600 y(>)2102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14352
2627 y(>)2102 2655 y(>)2102 2682 y(>)2102 2709 y(>)2102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14353
2737 y(>)2102 2764 y(>)2102 2791 y(:)2224 2111 y F6(\000)2226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14354
2292 y FW(N)t(O)q(T)t(L)t(E)t(F)t(T)2571 2310 y F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14355
FX(:)p Gc(C)5 b F9(\))2737 2292 y F4(\()p FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14356
Ga(C)i F6(i)p FB(K)o F4(\))2224 2404 y FK(S)2300 2500
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14357
y Gc(i)p F9(=1)p Gc(;)p F9(2)2490 2473 y FW(A)t(N)t(D)t(L)t(E)t(F)t(T)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14358
2847 2440 y Gc(i)2847 2504 y F9(\()p Gc(C)2924 2513 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14359
FV(1)2963 2504 y FX(^)p Gc(c)3041 2513 y FV(2)3075 2504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14360
y F9(\))3107 2473 y F4(\()p FB(J)p F4(\()p Ga(C)3279
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14361
2487 y Gc(i)3307 2473 y F4(\))q FB(K)o F4(\))2226 2653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14362
y FW(O)t(R)t(L)t(E)t(F)t(T)2522 2672 y F9(\()p Gc(C)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14363
FX(_)t Gc(D)r F9(\))2747 2653 y F4(\()p FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14364
Ga(C)i F4(\))p FB(K)o Ga(;)15 b FB(J)p F4(\()p Ga(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14365
F4(\))q FB(K)o F4(\))2226 2834 y FW(I)t(M)t(P)t(L)t(E)t(F)t(T)2553
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14366
2852 y F9(\()p Gc(C)5 b FX(\033)h Gc(D)r F9(\))2787 2834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14367
y F4(\()p FB(J)p F6(h)p Ga(C)h F6(i)p FB(K)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14368
b FB(J)p F4(\()p Ga(D)s F4(\))q FB(K)o F4(\))321 3055
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14369
y Gb(Candidates:)574 3228 y FB(J)p F4(\()p Ga(B)5 b F4(\))o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14370
FB(K)891 3176 y F5(def)898 3228 y F4(=)106 b Ga(X)1150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14371
3242 y F9(0)574 3368 y FB(J)p F6(h)p Ga(B)5 b F6(i)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14372
FB(K)891 3316 y F5(def)898 3368 y F4(=)108 b FW(N)t(E)t(G)1237
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14373
3386 y FX(h)p Gc(B)s FX(i)1354 3368 y F4(\()p FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14374
F4(\()p Ga(B)5 b F4(\))p FB(K)o F4(\))321 3586 y Gg(where)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14375
b Ga(X)641 3600 y F9(0)704 3586 y Gg(is)f(the)h(least)g(\002x)o(ed)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14376
(point)i(of)e(the)h(operator)k FW(N)t(E)t(G)2266 3605
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14377
y F9(\()p Gc(B)s F9(\))2382 3586 y F6(\016)40 b FW(N)t(E)t(G)2627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14378
3605 y FX(h)p Gc(B)s FX(i)2744 3586 y Gg(.)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14379
3824 a Gb(Remark)28 b(2.3.5:)p Black 34 w Gg(The)h(least)g(\002x)o(ed)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14380
(point)i(of)e(the)h(operator)k FW(N)t(E)t(G)2434 3842
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14381
y F9(\()p Gc(B)s F9(\))2565 3824 y F6(\016)38 b FW(N)t(E)t(G)2808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14382
3842 y FX(h)p Gc(B)s FX(i)2952 3824 y Gg(is)28 b(de\002ned,)j(since)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14383
3936 y(both)e FW(B)t(I)t(N)t(D)t(I)t(N)t(G)845 3955 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14384
FX(h)p Gc(B)s FX(i)987 3936 y Gg(and)g FW(B)t(I)t(N)t(D)t(I)t(N)t(G)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14385
1481 3955 y F9(\()p Gc(B)s F9(\))1623 3936 y Gg(are)d(decreasing)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14386
(operators.)38 b(Consequently)-6 b(,)31 b FW(N)t(E)t(G)3273
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14387
3955 y FX(h)p Gc(B)s FX(i)3415 3936 y Gg(and)323 4049
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14388
y FW(N)t(E)t(G)483 4068 y F9(\()p Gc(B)s F9(\))622 4049
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14389
y Gg(are)23 b(decreasing.)31 b(But)23 b(then)i FW(N)t(E)t(G)1688
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14390
4068 y F9(\()p Gc(B)s F9(\))1805 4049 y F6(\016)17 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14391
FW(N)t(E)t(G)2027 4068 y FX(h)p Gc(B)s FX(i)2166 4049
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14392
y Gg(must)22 b(be)h(increasing,)j(and)d(the)g(least)h(\002x)o(ed)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14393
4162 y(point)h Ga(X)606 4176 y F9(0)668 4162 y Gg(e)o(xists)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14394
(according)h(to)e(T)-7 b(arski')i(s)24 b(\002x)o(ed)f(point)i(theorem)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14395
([Da)n(v)o(e)o(y)g(and)g(Priestle)o(y,)g(1990].)321 4437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14396
y(T)-7 b(w)o(o)22 b(basic)j(properties)h(of)e(the)g(candidates)i(are)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14397
(as)f(follo)n(ws.)p Black 321 4625 a Gb(Lemma)g(2.3.6:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14398
Black 34 w Gg(F)o(or)g(the)h(candidates)i(we)d(ha)n(v)o(e:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14399
Black Black 722 4776 a(\(i\))100 b FB(J)p F4(\()p Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14400
b F4(\))p FB(K)25 b F4(=)i FW(N)t(E)t(G)1408 4795 y F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14401
Gc(B)s F9(\))1524 4776 y F4(\()p FB(J)p F6(h)p Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14402
b F6(i)q FB(K)o F4(\))355 b Gg(\(ii\))103 b FW(A)t(X)t(I)t(O)t(M)t(S)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14403
2690 4795 y F9(\()p Gc(B)s F9(\))2832 4776 y F6(\022)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14404
b FB(J)p F4(\()p Ga(B)5 b F4(\))p FB(K)907 4925 y(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14405
F6(h)p Ga(B)g F6(i)p FB(K)25 b F4(=)i FW(N)t(E)t(G)1408
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14406
4943 y FX(h)p Gc(B)s FX(i)1524 4925 y F4(\()p FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14407
F4(\()p Ga(B)5 b F4(\))q FB(K)o F4(\))568 b FW(A)t(X)t(I)t(O)t(M)t(S)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14408
2690 4943 y FX(h)p Gc(B)s FX(i)2832 4925 y F6(\022)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14409
b FB(J)p F6(h)p Ga(B)5 b F6(i)p FB(K)p Black 321 5125
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14410
a F7(Pr)l(oof)o(.)p Black 34 w Gg(\(i\))25 b(is)g(immediate)i(from)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14411
(De\002nition)h(2.3.4,)g(and)g(\(ii\))f(holds)i(tri)n(vially)g(since)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14412
FW(N)t(E)t(G)g Gg(is)d(closed)321 5238 y(under)i FW(A)t(X)t(I)t(O)t(M)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14413
(S)r Gg(.)p 3480 5238 4 62 v 3484 5180 55 4 v 3484 5238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14414
V 3538 5238 4 62 v Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14415
%%Page: 37 49
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14416
TeXDict begin 37 48 bop Black 277 51 a Gb(2.3)23 b(Pr)n(oof)i(of)e(Str)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14417
n(ong)h(Normalisation)2290 b(37)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14418
277 294 a Gg(Let)21 b(us)g(analyse)i(some)e(of)g(the)h(moti)n(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14419
(ations)h(behind)g(the)e(completely)j(symmetric)e(de\002nition)h(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14420
(the)277 407 y(candidates.)p Black 277 594 a Gb(Remark)j(2.3.7:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14421
Black 34 w Gg(Gi)n(v)o(en)h(the)g(symmetry)h(stated)g(in)e(Lemma)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14422
(2.3.6\(i\),)h(we)f(ha)n(v)o(e)i(a)e(simple)h(method)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14423
707 y(to)f(check)h(whether)g(a)f(named)h(or)f(co-named)h(term)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14424
(belongs)i(to)e(a)g(candidate.)33 b(F)o(or)23 b(e)o(xample,)i(tak)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14425
(a)277 820 y(co-named)f(term)f(of)f(the)h(form)1275 808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14426
y FX(h)1302 820 y Ga(a)r F4(:)r Ga(B)5 b F6(^)p Ga(C)1586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14427
808 y FX(i)1613 820 y Ga(M)32 b Gg(for)22 b(which)h(we)f(wish)h(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14428
(kno)n(w)h(whether)g(it)g(belongs)h(to)277 933 y FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14429
F6(h)p Ga(B)5 b F6(^)o Ga(C)i F6(i)p FB(K)p Gg(.)38 b(Because)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14430
b(of)f(the)g(equation)i FB(J)p F6(h)p Ga(B)5 b F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14431
Ga(C)i F6(i)p FB(K)31 b F4(=)j FW(N)t(E)t(G)2241 952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14432
y FX(h)p Gc(B)s FX(^)r Gc(C)5 b FX(i)2459 933 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14433
FB(J)p F4(\()p Ga(B)g F6(^)p Ga(C)i F4(\))p FB(K)o F4(\))27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14434
b Gg(it)f(is)h(suf)n(\002cient)h(to)277 1046 y(sho)n(w)f(that)655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14435
1034 y FX(h)682 1046 y Ga(a)r F4(:)r Ga(B)5 b F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14436
Ga(C)966 1034 y FX(i)993 1046 y Ga(M)35 b Gg(belongs)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14437
b(to)f FW(N)t(E)t(G)1683 1064 y FX(h)p Gc(B)s FX(^)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14438
Gc(C)5 b FX(i)1902 1046 y F4(\()p FB(J)p F4(\()p Ga(B)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14439
F6(^)o Ga(C)i F4(\))p FB(K)p F4(\))p Gg(.)37 b(By)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14440
b(de\002nition)i(of)h FW(N)t(E)t(G)3152 1064 y FX(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14441
Gc(B)s FX(^)r Gc(C)5 b FX(i)3396 1046 y Gg(we)277 1159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14442
y(therefore)26 b(ha)n(v)o(e)e(to)g(sho)n(w)f(that)1284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14443
1147 y FX(h)1311 1159 y Ga(a)r F4(:)r Ga(B)5 b F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14444
Ga(C)1595 1147 y FX(i)1622 1159 y Ga(M)33 b Gg(is)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14445
b(in)h(at)f(least)h(one)g(of)g(the)g(follo)n(wing)h(three)f(sets:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14446
Black Black 1234 1373 a(\(i\))102 b FW(A)t(X)t(I)t(O)t(M)t(S)1731
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14447
1391 y FX(h)p Gc(B)s FX(^)s Gc(C)5 b FX(i)1208 1508 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14448
Gg(\(ii\))103 b FW(A)t(N)t(D)t(R)t(I)t(G)t(H)t(T)1833
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14449
1527 y FX(h)p Gc(B)s FX(^)s Gc(C)5 b FX(i)2052 1508 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14450
F4(\()p FB(J)p F6(h)p Ga(B)g F6(i)p FB(K)p Ga(;)15 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14451
FB(J)p F6(h)p Ga(C)7 b F6(i)p FB(K)p F4(\))1183 1644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14452
y Gg(\(iii\))103 b FW(B)t(I)t(N)t(D)t(I)t(N)t(G)1756
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14453
1662 y FX(h)p Gc(B)s FX(^)r Gc(C)5 b FX(i)1975 1644 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14454
F4(\()p FB(J)p F4(\()p Ga(B)g F6(^)p Ga(C)i F4(\))o FB(K)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14455
F4(\))277 1845 y Gg(This)32 b(means)g(that)901 1833 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14456
FX(h)928 1845 y Ga(a)r F4(:)r Ga(B)5 b F6(^)p Ga(C)1212
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14457
1833 y FX(i)1239 1845 y Ga(M)41 b Gg(must)32 b(satisfy)i(certain)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14458
(conditions)i(depending)g(on)d(its)h(top-le)n(v)o(el)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14459
1958 y(term)24 b(constructor)-5 b(.)35 b(F)o(or)23 b(e)o(xample,)j(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14460
(\(i\))g(it)g(is)h(required)h(that)f(the)g(co-named)h(term)2982
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14461
1946 y FX(h)3009 1958 y Ga(a)r F4(:)r Ga(B)5 b F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14462
Ga(C)3293 1946 y FX(i)3320 1958 y Ga(M)34 b Gg(is)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14463
2071 y(of)29 b(the)g(form)725 2059 y FX(h)752 2071 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14464
Ga(a)r F4(:)r Ga(B)5 b F6(^)p Ga(C)1036 2059 y FX(i)1063
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14465
2071 y FL(Ax)o F4(\()p Ga(x;)15 b(b)p F4(\))q Gg(;)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14466
b(in)e(\(ii\))g(of)g(the)g(form)2109 2059 y FX(h)2136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14467
2071 y Ga(a)r F4(:)r Ga(B)5 b F6(^)p Ga(C)2420 2059 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14468
FX(i)2447 2071 y FL(And)2601 2085 y Gc(R)2659 2071 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14469
F4(\()2694 2059 y FX(h)2722 2071 y Ga(b)2761 2059 y FX(i)2788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14470
2071 y Ga(S)g(;)2889 2059 y FX(h)2917 2071 y Ga(c)2956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14471
2059 y FX(i)2984 2071 y Ga(T)13 b(;)i(a)p F4(\))28 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14472
Gg(where)i(it)277 2184 y(is)24 b(presupposed)k(that)998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14473
2172 y FX(h)1026 2184 y Ga(b)1065 2172 y FX(i)1092 2184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14474
y Ga(S)g Gg(and)1331 2172 y FX(h)1358 2184 y Ga(c)1397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14475
2172 y FX(i)1425 2184 y Ga(T)36 b Gg(belong)26 b(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14476
FB(J)p F6(h)p Ga(B)5 b F6(i)p FB(K)23 b Gg(and)h(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14477
FB(J)p F6(h)p Ga(C)7 b F6(i)p FB(K)p Gg(,)23 b(respecti)n(v)o(ely;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14478
(in)d(\(iii\))h(it)f(is)277 2297 y(required)30 b(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14479
Ga(M)38 b Gg(is)28 b(strongly)i(normalising)h(under)e(an)o(y)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14480
(substitution)i(on)e Ga(a)e Gg(with)h(a)g(named)h(term)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14481
2410 y(belonging)d(to)e(the)g(candidate)i FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14482
Ga(B)5 b F6(^)o Ga(C)i F4(\))p FB(K)o Gg(.)277 2664 y(In)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14483
b(the)f(ne)o(xt)h(three)h(lemmas)e(we)g(shall)h(deduce)i(some)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14484
(properties)k(for)2550 2627 y Gc(aux)2530 2664 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14485
b(\000)g(!)p Gg(.)p Black 277 2852 a Gb(Lemma)23 b(2.3.8:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14486
Black 34 w Gg(Suppose)i Ga(M)35 b F6(2)25 b FY(T)e Gg(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14487
Ga(M)1760 2815 y Gc(aux)1740 2852 y F6(\000)-31 b(\000)g(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14488
b Ga(M)2034 2819 y FX(0)2057 2852 y Gg(.)p Black 373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14489
3055 a(\(i\))p Black 46 w(If)f Ga(M)33 b Gg(freshly)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14490
b(introduces)h(the)e(name)g Ga(x)p Gg(,)e(then)i Ga(M)2113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14491
3022 y FX(0)2159 3055 y Gg(freshly)i(introduces)g Ga(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14492
Gg(.)p Black 348 3206 a(\(ii\))p Black 46 w(If)e Ga(M)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14493
b Gg(freshly)25 b(introduces)h(the)e(co-name)h Ga(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14494
Gg(,)d(then)i Ga(M)2225 3173 y FX(0)2272 3206 y Gg(freshly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14495
(introduces)h Ga(a)p Gg(.)p Black 277 3418 a F7(Pr)l(oof)o(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14496
Black 34 w Gg(If)36 b Ga(M)46 b Gg(freshly)39 b(introduces)g(a)e(name)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14497
(or)h(a)f(co-name,)41 b(then)d Ga(M)46 b Gg(cannot)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14498
b(be)f(of)f(the)h(form)277 3531 y FL(Cut)p F4(\()p 452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14499
3531 28 4 v 470 3531 V 488 3531 V 65 w Ga(;)p 557 3531
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14500
V 575 3531 V 593 3531 V 80 w F4(\))22 b Gg(\(see)h(De\002nition)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14501
(2.2.2\).)29 b(The)22 b(lemma)g(follo)n(ws)h(by)g(inspection)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14502
(the)f(reduction)j(rules)277 3644 y(of)396 3607 y Gc(aux)376
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14503
3644 y F6(\000)-32 b(\000)h(!)p Gg(.)p 3436 3644 4 62
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14504
v 3440 3586 55 4 v 3440 3644 V 3494 3644 4 62 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14505
277 3832 a Gb(Lemma)23 b(2.3.9:)p Black 34 w Gg(F)o(or)g(all)g(terms)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14506
Ga(M)35 b F6(2)25 b FY(T)e Gg(we)g(ha)n(v)o(e)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14507
373 4035 a(\(i\))p Black 46 w Ga(M)5 b F6(f)-7 b Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14508
b F4(:=)835 4023 y FX(h)862 4035 y Ga(a)910 4023 y FX(i)938
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14509
4035 y FL(Ax)o F4(\()p Ga(y)s(;)15 b(a)p F4(\))-8 b F6(g)1329
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14510
3998 y Gc(aux)1308 4035 y F6(\000)-31 b(\000)g(!)1479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14511
4002 y FX(\003)1543 4035 y Ga(M)10 b F4([)p Ga(x)g F6(7!)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14512
Ga(y)s F4(])p Black 348 4186 a Gg(\(ii\))p Black 46 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14513
Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)831 4174 y F9(\()858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14514
4186 y Ga(x)910 4174 y F9(\))938 4186 y FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14515
Ga(x;)15 b(b)p F4(\))-8 b F6(g)1324 4148 y Gc(aux)1303
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14516
4186 y F6(\000)-31 b(\000)g(!)1474 4153 y FX(\003)1538
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14517
4186 y Ga(M)10 b F4([)p Ga(a)g F6(7!)g Ga(b)p F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14518
Black 277 4398 a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14519
b(routine)i(induction)h(on)e(the)f(structure)j(of)e Ga(M)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14520
b Gg(\(see)24 b(the)f(proof)i(of)f(Lemma)e(2.2.11\).)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14521
3436 4398 V 3440 4340 55 4 v 3440 4398 V 3494 4398 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14522
62 v Black 277 4586 a Gb(Notation)g(2.3.10:)p Black 35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14523
w Gg(The)f(e)o(xpression)k Ga(M)1626 4549 y Gc(aux)1606
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14524
4586 y F6(\000)-31 b(\000)f(!)1776 4553 y F9(0)p Gc(=)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14525
F9(1)1896 4586 y Ga(M)1994 4553 y FX(0)2039 4586 y Gg(stands)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14526
b(for)f(either)h Ga(M)d F6(\021)10 b Ga(M)2929 4553 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14527
FX(0)2973 4586 y Gg(or)21 b Ga(M)3198 4549 y Gc(aux)3178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14528
4586 y F6(\000)-32 b(\000)h(!)10 b Ga(M)3456 4553 y FX(0)3479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14529
4586 y Gg(.)p Black 277 4807 a Gb(Lemma)23 b(2.3.11:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14530
Black 35 w Gg(F)o(or)f(an)i(arbitrary)h(substitution)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14531
F6(f)p Ga(\033)s F6(g)p Gg(,)p Black Black 504 5010 a(if)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14532
Ga(M)726 4973 y Gc(aux)706 5010 y F6(\000)-31 b(\000)f(!)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14533
b Ga(M)1000 4977 y FX(0)1023 5010 y Gg(,)c(then)j Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14534
b F6(f)p Ga(\033)s F6(g)1537 4973 y Gc(aux)1517 5010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14535
y F6(\000)-31 b(\000)f(!)1687 4977 y F9(0)p Gc(=)p F9(1)1823
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14536
5010 y Ga(M)1921 4977 y FX(0)1944 5010 y F6(f)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14537
F6(g)p Gg(.)p Black 277 5223 a F7(Pr)l(oof)o(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14538
34 w Gg(By)23 b(induction)j(on)d(the)h(structure)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14539
Ga(M)33 b Gg(\(see)24 b(P)o(age)f(142\).)p 3436 5223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14540
V 3440 5165 55 4 v 3440 5223 V 3494 5223 4 62 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14541
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14542
%%Page: 38 50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14543
TeXDict begin 38 49 bop Black -144 51 a Gb(38)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14544
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(In)k(ef)n(fect,)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14545
(substitution)j(operation)f F6(f)p 1674 317 28 4 v 1693
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14546
317 V 1710 317 V 65 w(g)e Gg(beha)n(v)o(es)h(quite)g(dif)n(ferent)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14547
(under)g(reduction)h(in)e(com-)321 430 y(parison)38 b(with)d(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14548
(standard)i(notion)f(of)e(v)n(ariable)j(substitution.)68
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14549
b(Note,)38 b(on)e(the)g(other)h(hand,)i(if)321 543 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14550
Ga(N)450 506 y Gc(aux)430 543 y F6(\000)-32 b(\000)h(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14551
b Ga(N)708 510 y FX(0)731 543 y Gg(,)e(then)h Ga(M)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14552
b F6(f)-7 b Ga(a)26 b F4(:=)1282 531 y F9(\()1310 543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14553
y Ga(x)1362 531 y F9(\))1389 543 y Ga(N)q F6(g)1554 506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14554
y Gc(aux)1533 543 y F6(\000)-31 b(\000)g(!)1704 510 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14555
FX(\003)1768 543 y Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)2095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14556
531 y F9(\()2122 543 y Ga(x)2174 531 y F9(\))2201 543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14557
y Ga(N)2284 510 y FX(0)2298 543 y F6(g)p Gg(,)d(as)h(e)o(xpected.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14558
673 y(The)c(ne)o(xt)f(tw)o(o)h(lemmas)f(establish)j(important)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14559
(properties)i(of)c(the)h(candidates.)30 b(The)19 b(\002rst)g(sho)n(ws)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14560
321 786 y(that)k(the)f(candidates)i(are)e(closed)i(under)f(reductions,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14561
h(and)f(the)f(second)h(sho)n(ws)f(ho)n(w)g(the)g(candidates)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14562
898 y(are)i(link)o(ed)h(to)f(the)f(property)j(of)e(strong)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14563
(normalisation.)p Black 321 1086 a Gb(Lemma)e(2.3.12:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14564
Black Black 417 1262 a Gg(\(i\))p Black 47 w(If)632 1250
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14565
y FX(h)659 1262 y Ga(a)r F4(:)r Ga(B)810 1250 y FX(i)838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14566
1262 y Ga(M)35 b F6(2)25 b FB(J)p F6(h)p Ga(B)5 b F6(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14567
FB(K)22 b Gg(and)i Ga(M)1585 1225 y Gc(aux)1564 1262
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14568
y F6(\000)-31 b(\000)g(!)25 b Ga(M)1858 1229 y FX(0)1881
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14569
1262 y Gg(,)e(then)2106 1250 y FX(h)2133 1262 y Ga(a)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14570
F4(:)r Ga(B)2284 1250 y FX(i)2312 1262 y Ga(M)2410 1229
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14571
y FX(0)2458 1262 y F6(2)i FB(J)p F6(h)p Ga(B)5 b F6(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14572
FB(K)p Gg(.)p Black 392 1404 a(\(ii\))p Black 47 w(If)632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14573
1392 y F9(\()659 1404 y Ga(x)r F4(:)r Ga(B)814 1392 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14574
F9(\))842 1404 y Ga(M)35 b F6(2)25 b FB(J)p F4(\()p Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14575
b F4(\))p FB(K)22 b Gg(and)i Ga(M)1589 1367 y Gc(aux)1568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14576
1404 y F6(\000)-31 b(\000)g(!)25 b Ga(M)1862 1371 y FX(0)1885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14577
1404 y Gg(,)e(then)2110 1392 y F9(\()2137 1404 y Ga(x)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14578
F4(:)r Ga(B)2292 1392 y F9(\))2320 1404 y Ga(M)2418 1371
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14579
y FX(0)2466 1404 y F6(2)i FB(J)p F4(\()p Ga(B)5 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14580
FB(K)p Gg(.)p Black 321 1616 a F7(Pr)l(oof)o(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14581
34 w Gg(By)33 b(simultaneous)38 b(induction)e(on)f(the)f(de)o(gree)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14582
(of)g Ga(B)i Gg(\(the)e(number)g(of)f(logical)i(symbols)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14583
1729 y(in)d Ga(B)5 b Gg(\).)56 b(Gi)n(v)o(en)32 b(Lemma)g(2.3.6\(i\),)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14584
(the)d(proof)h(analyses)h(all)e(possible)i(sets)e(where)3079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14585
1717 y FX(h)3107 1729 y Ga(a)r F4(:)r Ga(B)3258 1717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14586
y FX(i)3285 1729 y Ga(M)42 b Gg(and)321 1830 y F9(\()349
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14587
1842 y Ga(x)r F4(:)r Ga(B)504 1830 y F9(\))531 1842 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14588
Ga(M)33 b Gg(could)25 b(be)e(member)h(in)f(\(see)h(P)o(age)f(142\).)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14589
3480 1842 4 62 v 3484 1784 55 4 v 3484 1842 V 3538 1842
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14590
4 62 v Black 321 2030 a Gb(Lemma)g(2.3.13:)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14591
417 2206 a Gg(\(i\))p Black 47 w(If)632 2194 y FX(h)659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14592
2206 y Ga(a)r F4(:)r Ga(B)810 2194 y FX(i)838 2206 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14593
Ga(M)35 b F6(2)25 b FB(J)p F6(h)p Ga(B)5 b F6(i)p FB(K)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14594
Gg(,)23 b(then)h Ga(M)35 b F6(2)25 b Ga(S)5 b(N)1832
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14595
2220 y Gc(aux)1954 2206 y Gg(.)p Black 392 2348 a(\(ii\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14596
Black 47 w(If)632 2336 y F9(\()659 2348 y Ga(x)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14597
Ga(B)814 2336 y F9(\))842 2348 y Ga(M)35 b F6(2)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14598
FB(J)p F4(\()p Ga(B)5 b F4(\))p FB(K)o Gg(,)23 b(then)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14599
Ga(M)35 b F6(2)25 b Ga(S)5 b(N)1836 2362 y Gc(aux)1958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14600
2348 y Gg(.)p Black 321 2560 a F7(Pr)l(oof)o(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14601
34 w Gg(The)23 b(proof)i(is)e(similar)h(to)g(the)g(one)g(gi)n(v)o(en)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14602
(for)f(Lemma)g(2.3.12)h(\(see)g(P)o(age)f(144\).)p 3480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14603
2560 V 3484 2502 55 4 v 3484 2560 V 3538 2560 4 62 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14604
462 2764 a(W)-7 b(e)31 b(are)h(no)n(w)f(in)g(the)h(position)i(to)d(pro)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14605
o(v)o(e)h(that)h(a)e(cut)h(is)f(strongly)j(normalising)g(pro)o(vided)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14606
(its)321 2877 y(immediate)j(subterms)h(are)e(strongly)i(normalising)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14607
(and)e(in)f(a)f(candidate)k(corresponding)h(to)c(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14608
2990 y(cut-formula.)40 b(This)26 b(is)g(a)g(lengthy)h(lemma:)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14609
b(the)26 b(cases)h(for)g(the)f(logical)i(reductions)h(require)f(rather)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14610
321 3103 y(dif)n(\002cult)d(ar)n(guments.)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14611
3291 a Gb(Lemma)e(2.3.14:)p Black 35 w Gg(If)g Ga(M)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14612
b(;)15 b(N)36 b F6(2)24 b Ga(S)5 b(N)1477 3305 y Gc(aux)1600
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14613
3291 y Gg(,)1645 3279 y FX(h)1673 3291 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14614
Ga(B)1824 3279 y FX(i)1851 3291 y Ga(M)35 b F6(2)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14615
FB(J)p F6(h)p Ga(B)5 b F6(i)p FB(K)22 b Gg(and)2454 3279
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14616
y F9(\()2482 3291 y Ga(x)r F4(:)r Ga(B)2637 3279 y F9(\))2664
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14617
3291 y Ga(N)35 b F6(2)25 b FB(J)p F4(\()p Ga(B)5 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14618
FB(K)p Gg(,)22 b(then)1315 3485 y FL(Cut)p F4(\()1488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14619
3473 y FX(h)1515 3485 y Ga(a)r F4(:)r Ga(B)1666 3473
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14620
y FX(i)1694 3485 y Ga(M)10 b(;)1832 3473 y F9(\()1860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14621
3485 y Ga(x)r F4(:)r Ga(B)2015 3473 y F9(\))2042 3485
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14622
y Ga(N)g F4(\))26 b F6(2)e Ga(S)5 b(N)2405 3499 y Gc(aux)2528
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14623
3485 y Ga(:)p Black 321 3697 a F7(Pr)l(oof)o(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14624
34 w Gg(The)26 b(proof)i(of)f(this)h(lemma)e(is)h(inspired)i(by)e(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14625
(technique)i(applied)g(in)e([Pra)o(witz,)g(1971].)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14626
b(It)321 3810 y(sho)n(ws)34 b(by)g(induction)i(that)e(e)n(v)o(ery)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14627
(immediate)h(reduct)g(of)e(the)h(term)f FL(Cut)p F4(\()2780
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14628
3798 y FX(h)2808 3810 y Ga(a)r F4(:)r Ga(B)2959 3798
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14629
y FX(i)2986 3810 y Ga(M)11 b(;)3125 3798 y F9(\()3152
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14630
3810 y Ga(x)r F4(:)r Ga(B)3307 3798 y F9(\))3335 3810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14631
y Ga(N)f F4(\))33 b Gg(is)321 3923 y(strongly)f(normalising)f(\(see)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14632
(P)o(ages)f(145\226148\).)48 b(The)28 b(induction)k(proceeds)g(o)o(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14633
(er)d(a)f(le)o(xicograph-)321 4036 y(ically)37 b(ordered)g(induction)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14634
(v)n(alue)e(of)f(the)g(form)g F4(\()p Ga(\016)n(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14635
b(\026;)g(\027)6 b F4(\))p Gg(,)39 b(where)c Ga(\016)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14636
Gg(is)c(the)g(de)o(gree)h(of)g(the)f(cut-)321 4149 y(formula)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14637
b Ga(B)5 b Gg(;)30 b Ga(\026)d Gg(and)i Ga(\027)k Gg(are)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14638
b(the)g(longest)i(reduction)h(sequences)g(\(relati)n(v)o(e)f(to)2862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14639
4112 y Gc(aux)2842 4149 y F6(\000)-32 b(\000)h(!)p Gg(\))28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14640
b(starting)i(from)321 4262 y Ga(M)j Gg(and)24 b Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14641
b Gg(,)23 b(respecti)n(v)o(ely\227by)k(assumption)f(both)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14642
Ga(\026)e Gg(and)i Ga(\027)29 b Gg(are)24 b(\002nite.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14643
3480 4262 V 3484 4204 55 4 v 3484 4262 V 3538 4262 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14644
62 v 462 4466 a(It)g(is)f(left)h(to)f(sho)n(w)h(that)g(all)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14645
(well-typed)h(terms)f(are)g(strongly)h(normalising.)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14646
b(In)23 b(order)i(to)e(do)h(so,)321 4579 y(we)19 b(shall)h(consider)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14647
(a)d(special)i(class)f(of)g(substitutions,)k(which)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14648
b(are)h(called)h F7(safe)p Gg(.)28 b(T)-7 b(w)o(o)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14649
b(substitutions,)321 4692 y(say)k F6(f)p Ga(\033)s F6(g)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14650
Gg(and)h F6(f)p Ga(\034)10 b F6(g)p Gg(,)22 b(are)f(safe,)h(if)f(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14651
(only)h(if)f(the)h(domain)g(of)f F6(f)p Ga(\033)s F6(g)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14652
Gg(is)f(not)g(free)h(in)f(the)g(co-domain)j(of)321 4805
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14653
y F6(f)p Ga(\034)10 b F6(g)22 b Gg(and)h(the)f(domain)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14654
F6(f)p Ga(\034)10 b F6(g)22 b Gg(is)g(not)g(free)h(in)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14655
(co-domain)i(of)e F6(f)p Ga(\033)s F6(g)p Gg(.)28 b(F)o(or)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14656
b(e)o(xample)e F6(f)-7 b Ga(x)25 b F4(:=)3336 4793 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14657
FX(h)3364 4805 y Ga(a)3412 4793 y FX(i)3439 4805 y Ga(P)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14658
F6(g)321 4917 y Gg(and)h F6(f)-7 b Ga(b)38 b F4(:=)726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14659
4905 y F9(\()753 4917 y Ga(y)801 4905 y F9(\))828 4917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14660
y Ga(Q)-9 b F6(g)30 b Gg(are)g(safe,)i(pro)o(vided)g(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14661
Ga(x)e Gg(is)h(not)h(free)f(in)2410 4905 y F9(\()2438
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14662
4917 y Ga(y)2486 4905 y F9(\))2513 4917 y Ga(Q)f Gg(and)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14663
Ga(b)e Gg(is)h(not)h(free)g(in)3350 4905 y FX(h)3377
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14664
4917 y Ga(a)3425 4905 y FX(i)3453 4917 y Ga(P)13 b Gg(.)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14665
5030 y(As)25 b(e)o(xplained)j(earlier)l(,)f(the)e(substitution)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14666
(operation)f F6(f)p 2099 5030 28 4 v 2117 5030 V 2135
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14667
5030 V 65 w(g)d Gg(is)h(de\002ned)g(with)f(the)h(property)h(in)e(mind)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14668
321 5143 y(that)f(safe)h(substitutions)i(can)d(commute.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14669
b(As)23 b(a)g(special)i(case)f(of)g(the)g(substitution)j(lemma)c(for)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14670
F6(f)p 3438 5143 V 3456 5143 V 3474 5143 V 65 w(g)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14671
5256 y Gg(\(Lemma)f(2.3.1\))h(we)f(ha)n(v)o(e)h(for)f(all)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14672
Ga(M)33 b Gg(and)24 b(tw)o(o)f(safe)h(substitutions)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14673
F6(f)p Ga(\033)s F6(g)c Gg(and)g F6(f)p Ga(\034)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14674
F6(g)1463 5411 y Ga(M)g F6(f)p Ga(\033)s F6(gf)p Ga(\034)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14675
F6(g)27 b(\021)e Ga(M)10 b F6(f)p Ga(\034)g F6(gf)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14676
F6(g)28 b Ga(:)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14677
%%Page: 39 51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14678
TeXDict begin 39 50 bop Black 277 51 a Gb(2.3)23 b(Pr)n(oof)i(of)e(Str)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14679
n(ong)h(Normalisation)2290 b(39)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14680
418 317 a Gg(W)-7 b(e)22 b(shall)h(no)n(w)g(e)o(xtend)g(the)g(notion)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14681
(of)f(safety)h(from)e(substitutions)27 b(to)22 b(simultaneous)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14682
(substitu-)277 430 y(tions;)f(that)f(is)f(to)h(sets)g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14683
(substitutions.)p Black 277 618 a Gb(De\002nition)g(2.3.15)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14684
Gg(\(Safe)f(Simultaneous)j(Substitution,)g FO(sss)p Gg(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14685
Gb(:)p Black Black Black 1167 775 a F6(;)100 b Gg(is)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14686
b(an)g F7(sss)p Gg(.)915 888 y F4(^)-50 b Ga(\033)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14687
b F6([)d(f)p Ga(\033)s F6(g)101 b Gg(is)23 b(an)g F7(sss)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14688
Gg(,)h(if)f(and)h(only)h(if)1413 1000 y F6(\017)52 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14689
F4(^)-50 b Ga(\033)26 b Gg(is)e(an)f F7(sss)p Gg(,)1413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14690
1113 y F6(\017)47 b Ga(dom)p F4(\()p F6(f)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14691
F6(g)p F4(\))28 b F6(62)c Ga(dom)p F4(\()5 b(^)-50 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14692
Ga(\033)t F4(\))p Gg(,)1413 1226 y F6(\017)47 b Ga(dom)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14693
F4(\()p F6(f)p Ga(\033)s F6(g)p F4(\))25 b Gg(not)f(free)g(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14694
Ga(codom)p F4(\()5 b(^)-50 b Ga(\033)5 b F4(\))p Gg(,)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14695
b(and)1413 1339 y F6(\017)47 b Ga(dom)p F4(\()5 b(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14696
b Ga(\033)t F4(\))23 b Gg(not)h(free)g(in)f Ga(codom)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14697
F4(\()p F6(f)p Ga(\033)s F6(g)p F4(\))p Gg(.)277 1593
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14698
y(In)j(the)g(presence)i(of)d(our)h(Barendre)o(gt-style)k(naming)d(con)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14699
(v)o(ention)h(and)f(alpha-con)l(v)o(ersion,)k(which)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14700
1706 y(we)26 b(assumed)i(in)f(Con)l(v)o(ention)i(2.2.4,)f(an)o(y)f(set)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14701
g(of)g(substitutions)j(can)e(be)f(transformed)i(into)e(a)g(safe)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14702
1819 y(simultaneous)35 b(substitution.)59 b(W)-7 b(e)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14703
b(shall,)k(ho)n(we)n(v)o(er)l(,)g(omit)d(a)g(formal)h(proof)g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14704
(rather)h(gi)n(v)o(e)e(the)277 1932 y(reader)25 b(the)f(follo)n(wing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14705
(e)o(xample.)p Black 277 2120 a Gb(Example)g(2.3.16:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14706
Black 35 w Gg(Suppose)h(we)e(ha)n(v)o(e)i(a)e(term,)h(say)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14707
Ga(M)10 b Gg(,)24 b(and)i(a)e(safe)i(simultaneous)i(substitution,)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14708
2233 y(say)1047 2392 y F4(^)-50 b Ga(\033)29 b F4(=)1219
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14709
2291 y FK(n)1275 2392 y F6(f)-7 b Ga(x)25 b F4(:=)1511
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14710
2380 y FX(h)1539 2392 y Ga(c)1578 2380 y FX(i)1606 2392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14711
y FL(Ax)o F4(\()p Ga(x;)15 b(b)p F4(\))-8 b F6(g)p Ga(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14712
b F6(f)-7 b Ga(a)27 b F4(:=)2215 2380 y F9(\()2242 2392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14713
y Ga(z)2288 2380 y F9(\))2316 2392 y FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14714
Ga(z)t(;)15 b(c)p F4(\))-7 b F6(g)2651 2291 y FK(o)2712
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14715
2392 y Ga(;)277 2613 y Gg(and)27 b(assume)g(we)e(ha)n(v)o(e)i(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14716
(substitution)j F6(f)p Ga(\033)s F6(g)h(\021)26 b(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14717
b Ga(b)30 b F4(:=)2146 2601 y F9(\()2173 2613 y Ga(y)2221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14718
2601 y F9(\))2248 2613 y FL(Ax)p F4(\()p Ga(x;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14719
F4(\))-8 b F6(g)p Gg(.)36 b(Clearly)-6 b(,)33 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14720
b Ga(\033)25 b F6([)d(f)p Ga(\033)s F6(g)27 b Gg(is)f(not)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14721
2726 y(safe,)34 b(and)f(therefore)h Ga(M)10 b F4(\()5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14722
b(^)-50 b Ga(\033)30 b F6([)c(f)p Ga(\033)s F6(g)p F4(\))32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14723
b Gg(is)g(an)g(ill-de\002ned)i(e)o(xpression.)56 b(Ho)n(we)n(v)o(er)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14724
(,)34 b Ga(x)p Gg(,)f Ga(a)e Gg(and)h Ga(b)f Gg(are)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14725
2839 y(considered)25 b(as)d(binders)i(and)e(thus)h(can)f(be)g(re)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14726
(written.)29 b(In)22 b(ef)n(fect,)h(we)e(can)h(form)g(the)g(follo)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14727
(wing)i(safe)277 2951 y(v)o(ersion)h(of)j F4(^)-49 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14728
Ga(\033)23 b F6([)d(f)p Ga(\033)s F6(g)608 3205 y F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14729
b Ga(\033)655 3225 y FE(safe)805 3205 y F4(=)901 3104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14730
y FK(n)957 3205 y F6(f)-7 b Ga(x)1047 3168 y FX(0)1096
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14731
3205 y F4(:=)1217 3193 y FX(h)1244 3205 y Ga(c)1283 3193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14732
y FX(i)1311 3205 y FL(Ax)p F4(\()p Ga(x;)15 b(b)p F4(\))-9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14733
b F6(g)p Ga(;)10 b F6(f)-7 b Ga(a)1772 3168 y FX(0)1822
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14734
3205 y F4(:=)1944 3193 y F9(\()1971 3205 y Ga(z)2017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14735
3193 y F9(\))2045 3205 y FL(Ax)o F4(\()p Ga(z)t(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14736
b(c)p F4(\))-7 b F6(g)p Ga(;)10 b F6(f)-7 b Ga(b)2492
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14737
3168 y FX(0)2542 3205 y F4(:=)2663 3193 y F9(\()2690
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14738
3205 y Ga(y)2738 3193 y F9(\))2766 3205 y FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14739
Ga(x;)15 b(a)p F4(\))-8 b F6(g)3115 3104 y FK(o)277 3461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14740
y Gg(assuming)25 b(that)g Ga(x)855 3428 y FX(0)878 3461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14741
y Gg(,)d Ga(a)971 3428 y FX(0)1018 3461 y Gg(and)i Ga(b)1211
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14742
3428 y FX(0)1257 3461 y Gg(are)g(fresh.)30 b(Subsequently)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14743
b(,)26 b(we)d(need)i(to)e(re)n(write)h(the)g(corresponding)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14744
3574 y(names)36 b(and)f(co-names)i(in)e Ga(M)10 b Gg(.)63
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14745
b(No)n(w)33 b(the)j(e)o(xpression)h Ga(M)10 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14746
Ga(x)32 b F6(7!)g Ga(x)2538 3541 y FX(0)2561 3574 y F4(][)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14747
Ga(a)g F6(7!)f Ga(a)2861 3541 y FX(0)2885 3574 y F4(][)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14748
Ga(b)h F6(7!)f Ga(b)3167 3541 y FX(0)3190 3574 y F4(])21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14749
b(^)-51 b Ga(\033)3282 3593 y FE(safe)3442 3574 y Gg(is)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14750
3686 y(well-de\002ned.)277 3949 y(In)24 b(the)f(ne)o(xt)h(lemma)f(we)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14751
(shall)i(sho)n(w)e(that)h(a)f(speci\002c)i(substitution)i(b)n(uilt)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14752
(up)g(by)g(axioms)g(is)f(an)h(sss.)p Black 277 4137 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14753
Gb(Lemma)f(2.3.17:)p Black 35 w Gg(Let)k F4(^)-50 b Ga(\033)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14754
b Gg(be)e(of)f(the)h(form)672 4265 y FK(8)672 4346 y(<)672
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14755
4510 y(:)830 4365 y([)753 4560 y Gc(i)p F9(=0)p Gc(;:::)n(;n)1003
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14756
4451 y F6(f)-7 b Ga(x)1093 4465 y Gc(i)1147 4451 y F4(:=)1268
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14757
4439 y FX(h)1296 4451 y Ga(c)1335 4439 y FX(i)1362 4451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14758
y FL(Ax)p F4(\()p Ga(x)1552 4465 y Gc(i)1580 4451 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14759
b(c)p F4(\))-8 b F6(g)1731 4265 y FK(9)1731 4346 y(=)1731
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14760
4510 y(;)1832 4451 y F6([)1913 4265 y FK(8)1913 4346
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14761
y(<)1913 4510 y(:)2085 4365 y([)1994 4560 y Gc(j)t F9(=0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14762
Gc(;:::)n(;m)2272 4451 y F6(f)h Ga(a)2358 4465 y Gc(j)2420
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14763
4451 y F4(:=)2542 4439 y F9(\()2569 4451 y Ga(y)2617
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14764
4439 y F9(\))2644 4451 y FL(Ax)p F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14765
b(a)2918 4465 y Gc(j)2955 4451 y F4(\))-9 b F6(g)3026
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14766
4265 y FK(9)3026 4346 y(=)3026 4510 y(;)277 4781 y Gg(where)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14767
b(the)g Ga(x)716 4795 y Gc(i)744 4781 y Gg(')-5 b(s)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14768
b(and)h Ga(a)1037 4795 y Gc(i)1065 4781 y Gg(')-5 b(s)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14769
b(are)g(distinct)h(names)f(and)g(co-names,)i(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14770
b(.)43 b(Substitution)35 b F4(^)-49 b Ga(\033)30 b Gg(is)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14771
4894 y(an)24 b(sss.)p Black 277 5107 a F7(Pr)l(oof)o(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14772
Black 34 w Gg(By)f(induction)j(on)d(the)h(length)h(of)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14773
F4(^)-49 b Ga(\033)s Gg(.)p 3436 5107 4 62 v 3440 5049
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14774
55 4 v 3440 5107 V 3494 5107 4 62 v Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14775
%%Page: 40 52
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14776
TeXDict begin 40 51 bop Black -144 51 a Gb(40)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14777
b(Calculi)p -144 88 3691 4 v Black 462 317 a Gg(No)n(w)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14778
b(we)g(can)h(sho)n(w)f(that)i(e)n(v)o(ery)f(well-typed)i(term)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14779
(together)j(with)e(a)f(closing)i(substitution)j(is)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14780
430 y(strongly)26 b(normalising.)31 b(This)24 b(is)f(again)h(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14781
(rather)i(lengthy)g(proof.)p Black 321 618 a Gb(Lemma)e(2.3.18:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14782
Black Black 458 847 a F6(\017)p Black 46 w Gg(F)o(or)j(e)n(v)o(ery)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14783
(well-typed)h(term)e Ga(M)10 b Gg(\227not)28 b(necessarily)i(strongly)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14784
(normalising\227with)g(a)d(typ-)549 960 y(ing)c(judgement)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14785
F4(\000)1176 948 y Gc(.)1231 960 y Ga(M)1354 948 y Gc(.)1409
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14786
960 y F4(\001)p Gg(,)c(and)p Black 458 1158 a F6(\017)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14787
Black 46 w Gg(for)36 b(e)n(v)o(ery)h(sss,)45 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14788
b Ga(\033)s Gg(,)39 b(such)e(that)g Ga(dom)p F4(\(\000\))31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14789
b F6([)e Ga(dom)p F4(\(\001\))50 b F6(\022)f Ga(dom)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14790
F4(\()5 b(^)-50 b Ga(\033)t F4(\))p Gg(,)39 b(i.e.,)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14791
F4(^)-50 b Ga(\033)39 b Gg(is)e(a)f(closing)549 1271
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14792
y(substitution,)992 1238 y F5(5)1056 1271 y Gg(and)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14793
Black 658 1470 a F6(\017)p Black 46 w Gg(for)23 b(e)n(v)o(ery)1098
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14794
1458 y F9(\()1126 1470 y Ga(x)r F4(:)r Ga(B)1281 1458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14795
y F9(\))1308 1470 y Ga(P)38 b F6(2)25 b Ga(codom)p F4(\()5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14796
b(^)-50 b Ga(\033)t F4(\))23 b Gg(we)g(require)i(that)2461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14797
1458 y F9(\()2489 1470 y Ga(x)r F4(:)r Ga(B)2644 1458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14798
y F9(\))2671 1470 y Ga(P)38 b F6(2)25 b FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14799
Ga(B)5 b F4(\))p FB(K)22 b Gg(and)p Black 658 1622 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14800
F6(\017)p Black 46 w Gg(for)h(e)n(v)o(ery)1098 1610 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14801
FX(h)1126 1622 y Ga(a)r F4(:)r Ga(C)1275 1610 y FX(i)1302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14802
1622 y Ga(Q)i F6(2)g Ga(codom)p F4(\()5 b(^)-50 b Ga(\033)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14803
F4(\))23 b Gg(we)g(require)i(that)2456 1610 y FX(h)2484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14804
1622 y Ga(a)r F4(:)r Ga(C)2633 1610 y FX(i)2660 1622
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14805
y Ga(Q)g F6(2)g FB(J)p F6(h)p Ga(C)7 b F6(i)p FB(K)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14806
Gg(;)p Black Black 549 1821 a(we)22 b(ha)n(v)o(e)i Ga(M)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14807
b F4(^)-50 b Ga(\033)29 b F6(2)c Ga(S)5 b(N)1267 1835
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14808
y Gc(aux)1389 1821 y Gg(.)p Black 321 2034 a F7(Pr)l(oof)o(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14809
Black 34 w Gg(By)23 b(induction)j(o)o(v)o(er)d(the)h(structure)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14810
Ga(M)32 b Gg(\(see)24 b(P)o(ages)g(148\226150\).)p 3480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14811
2034 4 62 v 3484 1976 55 4 v 3484 2034 V 3538 2034 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14812
62 v 321 2241 a(W)-7 b(e)23 b(are)h(no)n(w)f(able)h(to)f(pro)o(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14813
(that)g F4(\()p FY(T)t Ga(;)1580 2203 y Gc(aux)1560 2241
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14814
y F6(\000)-31 b(\000)g(!)o F4(\))24 b Gg(is)f(strongly)j(normalising.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14815
Black 321 2428 a Gb(Theor)n(em)e(2.3.19:)p Black 35 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14816
Gg(The)f(reduction)j(system)e F4(\()p FY(T)t Ga(;)1945
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14817
2391 y Gc(aux)1925 2428 y F6(\000)-31 b(\000)f(!)p F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14818
b Gg(is)h(strongly)h(normalising.)p Black 321 2641 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14819
F7(Pr)l(oof)o(.)p Black 34 w Gg(By)32 b(Lemma)h(2.3.18)h(we)e(kno)n(w)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14820
(that)h(for)g(an)f(arbitrary)j(well-typed)f(term,)g(say)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14821
Ga(M)10 b Gg(,)35 b(with)321 2754 y(the)29 b(typing)h(judgement)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14822
F4(\000)1213 2742 y Gc(.)1268 2754 y Ga(M)1392 2742 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14823
Gc(.)1446 2754 y F4(\001)e Gg(and)h(an)f(arbitrary)j(safe)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14824
(simultaneous)k(substitution,)h(say)g F4(^)-50 b Ga(\033)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14825
Gg(,)321 2867 y(where)32 b(all)g(free)g(names)h(and)f(co-names)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14826
Ga(M)41 b Gg(are)32 b(amongst)h(the)f(domain)h(of)j F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14827
b Ga(\033)t Gg(,)32 b(the)g(term)g Ga(M)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14828
b Ga(\033)321 2979 y Gg(is)27 b(strongly)j(normalising.)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14829
b(T)-7 b(aking)32 b F4(^)-50 b Ga(\033)30 b Gg(to)d(be)h(the)f(safe)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14830
(simultaneous)i(substitution)h(from)d(Lemma)321 3092
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14831
y(2.3.17,)f(we)d(can)i(infer)l(,)h(using)g(Lemma)e(2.3.9,)h(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14832
Ga(M)15 b F4(^)-50 b Ga(\033)2212 3055 y Gc(aux)2192
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14833
3092 y F6(\000)-31 b(\000)f(!)2362 3059 y FX(\003)2431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14834
3092 y Ga(M)10 b Gg(,)25 b(and)h(we)f(therefore)j(ha)n(v)o(e)e(that)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14835
3205 y Ga(M)33 b Gg(is)24 b(strongly)h(normalising.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14836
b(Thus)24 b(we)f(are)g(done.)p 3480 3205 V 3484 3147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14837
55 4 v 3484 3205 V 3538 3205 4 62 v 462 3412 a(From)e(this)h(result)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14838
(we)f(can)h(deduce)h(strong)g(normalisation)h(for)e F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14839
FY(T)t Ga(;)2664 3375 y Gc(cut)2633 3412 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14840
b(\000)g(!)p F4(\))p Gg(,)21 b(which)g(is)h(relati)n(v)o(ely)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14841
3525 y(straightforw)o(ard)31 b(since)c(e)n(v)o(ery)1374
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14842
3488 y Gc(cut)1343 3525 y F6(\000)-31 b(\000)f(!)p Gg(-reduction)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14843
b(maps)c(onto)i(a)e(series)h(of)2738 3488 y Gc(aux)2718
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14844
3525 y F6(\000)-31 b(\000)f(!)p Gg(-reductions.)41 b(First,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14845
3638 y(we)23 b(pro)o(v)o(e)h(that)g Ga(M)10 b F6(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14846
Ga(\033)s F6(g)24 b Gg(reduces)h(to)f(or)f(is)h(equi)n(v)n(alent)i(to)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14847
Ga(M)10 b F4([)p Ga(\033)s F4(])p Gg(.)p Black 321 3826
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14848
a Gb(Lemma)23 b(2.3.20:)p Black 35 w Gg(F)o(or)f(all)i(terms)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14849
Ga(M)5 b(;)15 b(N)35 b F6(2)25 b FY(T)e Gg(we)g(ha)n(v)o(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14850
Black 417 4041 a(\(i\))p Black 47 w Ga(M)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14851
b Ga(a)26 b F4(:=)875 4029 y F9(\()902 4041 y Ga(x)954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14852
4029 y F9(\))982 4041 y Ga(N)p F6(g)1146 4004 y Gc(aux)1126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14853
4041 y F6(\000)-31 b(\000)f(!)1296 4008 y FX(\003)1361
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14854
4041 y Ga(M)10 b F4([)p Ga(a)26 b F4(:=)1679 4029 y F9(\()1706
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14855
4041 y Ga(x)1758 4029 y F9(\))1786 4041 y Ga(N)10 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14856
Black 392 4240 a Gg(\(ii\))p Black 47 w Ga(M)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14857
b Ga(x)26 b F4(:=)879 4228 y FX(h)906 4240 y Ga(a)954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14858
4228 y FX(i)982 4240 y Ga(N)p F6(g)1146 4203 y Gc(aux)1126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14859
4240 y F6(\000)-31 b(\000)f(!)1296 4207 y FX(\003)1361
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14860
4240 y Ga(M)10 b F4([)p Ga(x)25 b F4(:=)1683 4228 y FX(h)1710
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14861
4240 y Ga(a)1758 4228 y FX(i)1786 4240 y Ga(N)10 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14862
Black 321 4453 a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14863
b(induction)j(on)e(the)f(structure)j(of)e Ga(M)32 b Gg(\(see)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14864
b(P)o(age)e(150\).)p 3480 4453 V 3484 4395 55 4 v 3484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14865
4453 V 3538 4453 4 62 v 321 4660 a(The)g(ne)o(xt)h(lemma)f(sho)n(ws)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14866
(that)g(e)n(v)o(ery)1592 4623 y Gc(cut)1561 4660 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14867
b(\000)g(!)p Gg(-reduction)26 b(maps)e(onto)g(a)f(series)i(of)2938
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14868
4623 y Gc(aux)2918 4660 y F6(\000)-31 b(\000)g(!)o Gg(-reductions.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14869
Black 321 4847 a Gb(Lemma)23 b(2.3.21:)p Black 35 w Gg(F)o(or)f(all)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14870
(terms)g Ga(M)10 b Gg(,)22 b Ga(N)35 b F6(2)25 b FY(T)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14871
Gg(,)19 b(if)k Ga(M)2091 4810 y Gc(cut)2060 4847 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14872
b(\000)g(!)25 b Ga(N)10 b Gg(,)22 b(then)j Ga(M)2707
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14873
4810 y Gc(aux)2687 4847 y F6(\000)-31 b(\000)f(!)2857
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14874
4814 y F9(+)2942 4847 y Ga(N)10 b Gg(.)p Black 321 5060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14875
a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)23 b(induction)j(on)e(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14876
(structure)j(of)1791 5023 y Gc(cut)1761 5060 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14877
b(\000)h(!)23 b Gg(\(see)h(P)o(age)f(151\).)p 3480 5060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14878
V 3484 5002 55 4 v 3484 5060 V 3538 5060 4 62 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14879
321 5227 1290 4 v 427 5283 a F3(5)456 5314 y F2(All)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14880
b(free)h(names)g(and)h(co-names)g(of)f Fw(M)27 b F2(are)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14881
b(amongst)h(the)f(domain)g(of)k Fv(^)-42 b Fw(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14882
F2(.)p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14883
%%Page: 41 53
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14884
TeXDict begin 41 52 bop Black 277 51 a Gb(2.4)23 b(First-Order)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14885
(Classical)g(Logic)2399 b(41)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14886
277 317 a Gg(It)23 b(is)h(rather)g(easy)h(no)n(w)e(to)g(sho)n(w)h(that)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14887
g F4(\()p FY(T)t Ga(;)1663 280 y Gc(cut)1632 317 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14888
b(\000)g(!)p F4(\))23 b Gg(is)g(strongly)j(normalising.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14889
Black 277 574 a Gb(Theor)n(em)e(2.3.22:)p Black 34 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14890
Gg(The)g(reduction)i(system)e F4(\()p FY(T)t Ga(;)1911
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14891
537 y Gc(cut)1881 574 y F6(\000)-32 b(\000)h(!)p F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14892
b Gg(is)g(strongly)j(normalising.)p Black 277 821 a F7(Pr)l(oof)o(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14893
Black 34 w Gg(The)c(reduction)i(system)f F4(\()p FY(T)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14894
Ga(;)1489 784 y Gc(aux)1469 821 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14895
F4(\))22 b Gg(is)g(strongly)j(normalising,)f(and)f(whene)n(v)o(er)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14896
Ga(M)3254 784 y Gc(cut)3224 821 y F6(\000)-32 b(\000)h(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14897
b Ga(N)277 934 y Gg(we)18 b(ha)n(v)o(e)i(that)f Ga(M)886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14898
897 y Gc(aux)865 934 y F6(\000)-31 b(\000)g(!)1036 901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14899
y F9(+)1120 934 y Ga(N)10 b Gg(.)26 b(Consequently)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14900
b(,)23 b(the)c(reduction)i(system)f F4(\()p FY(T)t Ga(;)2713
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14901
897 y Gc(cut)2682 934 y F6(\000)-31 b(\000)f(!)p F4(\))18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14902
b Gg(must)h(be)g(strongly)277 1047 y(normalising,)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14903
b(too.)p 3436 1047 4 62 v 3440 988 55 4 v 3440 1047 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14904
3494 1047 4 62 v 277 1481 a Ge(2.4)119 b(First-Order)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14905
b(Classical)g(Logic)277 1723 y Gg(In)e(this)g(section)h(we)e(e)o(xtend)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14906
i(the)f(cut-elimination)k(procedure)e F4(\()p FY(T)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14907
Ga(;)2498 1686 y Gc(cut)2467 1723 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14908
F4(\))27 b Gg(to)h(the)g(\002rst-order)i(frag-)277 1836
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14909
y(ment)c(of)h(classical)h(logic.)38 b(T)-7 b(o)25 b(do)h(so,)h(we)e(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14910
(xtend)j(the)e(notion)i(of)e(a)g(formula)h(by)f(allo)n(wing)i(atomic)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14911
277 1949 y(formulae)d(to)f(ha)n(v)o(e)g(ar)n(guments)i(ranging)f(o)o(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14912
o(er)f F7(e)n(xpr)m(essions)j Gg(and)d(introduce)i(the)e(quanti\002ers)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14913
i F6(8)c Gg(and)277 2062 y F6(9)p Gg(.)57 b(W)-7 b(e)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14914
b(use)i(`e)o(xpression')i(instead)g(of)d(`term')g(\(the)h(standard)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14915
(terminology\))g(in)d(order)i(to)e(a)n(v)n(oid)277 2175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14916
y(confusion)38 b(with)d(our)h(terminology)i(introduced)g(in)d(pre)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14917
(vious)i(sections.)66 b(Moreo)o(v)o(er)l(,)40 b(we)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14918
b(use)i(a)277 2288 y(sans)28 b(serif)g(font)g(for)f(e)o(xpressions)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14919
(to)d(clearly)i(distinguish)i(them)c(from)g(terms.)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14920
b(The)27 b(grammar)g(for)277 2400 y(e)o(xpressions)g(is)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14921
Black Black 1603 2643 a FL(t)e F4(::=)g FL(x)h F6(j)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14922
FL(f)c(t)15 b Ga(:)g(:)g(:)i FL(t)277 2885 y Gg(where)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14923
b FL(x)f Gg(is)g(tak)o(en)i(from)e(a)h(set)f(of)h F7(variables)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14924
Gg(and)e FL(f)30 b Gg(from)24 b(a)h(set)f(of)h(functional)i(symbols.)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14925
b(An)23 b F7(arity)277 2998 y Gg(\(a)33 b(natural)h(number\))h(is)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14926
(assigned)j(to)e(each)h(functional)i(symbol)e(specifying)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14927
(number)f(of)g(its)277 3110 y(ar)n(guments.)39 b(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14928
b(e)26 b(shall)h(often)g(write)g FL(x)p Ga(;)15 b FL(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14929
Ga(;)g FL(z)26 b Gg(for)g(v)n(ariables)j(and)d FL(t)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14930
Gg(for)g(arbitrary)j(e)o(xpressions.)40 b(The)277 3223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14931
y(formulae)25 b(of)e(\002rst-order)j(classical)f(logic)g(are)f(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14932
(en)g(by)f(the)h(grammar)p Black Black 782 3465 a Ga(B)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14933
b F4(::=)c Ga(A)15 b FL(t)1144 3479 y Fu(1)1198 3465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14934
y Ga(:)g(:)g(:)h FL(t)1352 3479 y Fu(n)1418 3465 y F6(j)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14935
b(:)p Ga(B)j F6(j)d Ga(B)5 b F6(^)o Ga(B)29 b F6(j)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14936
Ga(B)5 b F6(_)o Ga(B)30 b F6(j)25 b Ga(B)5 b F6(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14937
Ga(B)29 b F6(j)d(8)p FL(x)p Ga(:B)j F6(j)d(9)p FL(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14938
Ga(:B)277 3707 y Gg(where)21 b Ga(A)e Gg(ranges)j(o)o(v)o(er)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14939
F7(pr)m(edicate)i(symbols)p Gg(,)g(each)e(of)h(which)f(tak)o(e)h(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14940
(\002x)o(ed)f(number)i(of)f(e)o(xpressions)277 3820 y(as)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14941
b(ar)n(guments,)j(and)d(where)h FL(x)e Gg(ranges)i(o)o(v)o(er)f(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14942
(ariables.)36 b(An)24 b(occurrence)k(of)d(the)h(v)n(ariable)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14943
FL(x)d Gg(is)h(said)277 3933 y(to)31 b(be)f F7(bound)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14944
Gg(,)j(if)e(it)f(is)g(inside)i(a)e(formula)i(of)e(the)h(form)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14945
F6(8)p FL(x)p Ga(:B)k Gg(or)d F6(9)p FL(x)p Ga(:B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14946
Gg(.)48 b(W)-7 b(e)29 b(ha)n(v)o(e)i(the)g(standard)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14947
4046 y(notion)c(of)e(free)g(v)n(ariables,)j(and)d(for)h(the)f(set)g(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14948
g(free)h(v)n(ariables)h(of)e(an)g(e)o(xpression,)j(say)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14949
FL(t)p Gg(,)e(and)i(of)f(a)277 4159 y(formula,)f(say)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14950
Ga(B)5 b Gg(,)22 b(we)h(shall)h(write)g Ga(F)13 b(V)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14951
b F4(\()p FL(t)p F4(\))k Gg(and)g Ga(F)13 b(V)20 b F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14952
Ga(B)5 b F4(\))p Gg(,)23 b(respecti)n(v)o(ely)-6 b(.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14953
4298 y(Just)27 b(as)f(with)g(terms,)h(we)f(identify)i(formulae)f(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14954
(dif)n(fer)g(only)g(in)f(the)h(names)g(of)f(their)h(bound)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14955
4411 y(v)n(ariables.)46 b(Moreo)o(v)o(er)l(,)30 b(we)e(shall)h(observ)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14956
(e)h(a)e(Barendre)o(gt-style)k(naming)e(con)l(v)o(ention)h(for)e(bound)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14957
277 4524 y(and)21 b(free)h(v)n(ariables.)30 b(This)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14958
b(is)h(a)g(standard)i(con)l(v)o(ention)h(allo)n(wing)e(us)e(to)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14959
(assume)h(that)f(distinct)i(quan-)277 4637 y(ti\002er)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14960
b(occurrences)k(are)c(follo)n(wed)h(by)g(distinct)h(v)n(ariables.)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14961
b(F)o(or)28 b(e)o(xample,)j(it)e(e)o(xcludes)i(formulae)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14962
4750 y(such)j(as)f F6(9)p FL(x)p Ga(:)p F4(\()p Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14963
FL(x)p F6(^8)p FL(x)p Ga(:B)5 b FL(x)p F4(\))p Gg(,)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14964
b(which)g(can)g(equally)h(be)e(written)h(as)f F6(9)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14965
FL(y)q Ga(:)p F4(\()p Ga(A)p FL(y)q F6(^)q(8)p FL(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14966
Ga(:B)5 b FL(x)p F4(\))p Gg(.)56 b(It)33 b(also)h(al-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14967
4863 y(lo)n(ws)c(a)f(simple)i(de\002nition)h(of)e(the)g(notion)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14968
(capture)j(a)n(v)n(oiding)h(v)n(ariable)f(substitution,)j(written)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14969
4976 y(as)27 b F4([)p FL(x)32 b F4(:=)g FL(t)p F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14970
Gg(,)c(b)n(ut)f(we)g(omit)g(to)g(gi)n(v)o(e)g(the)h(de\002nition)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14971
(\(see)f(for)f(e)o(xample)h([Gallier,)g(1990,)h(P)o(age)d(4]\).)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14972
5088 y(Later)l(,)h(it)f(will)g(be)h(con)l(v)o(enient)i(to)d(e)o(xtend)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14973
(this)f(notion)g(to)g(conte)o(xts.)39 b(F)o(or)25 b(instance)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14974
F4(\000[)p FL(x)j F4(:=)f FL(t)p F4(])c Gg(will)277 5201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14975
y(be)e(tak)o(en)g(to)g(mean)f(that)i(the)e(substitution)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14976
F4([)p FL(x)f F4(:=)f FL(t)p F4(])e Gg(is)g(applied)j(to)d(e)n(v)o(ery)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14977
h(formula)h(in)e F4(\000)p Gg(.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14978
%%Page: 42 54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14979
TeXDict begin 42 53 bop Black -144 51 a Gb(42)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14980
b(Calculi)p -144 88 3691 4 v Black 462 317 a Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14981
b(e)25 b(are)g(no)n(w)f(ready)j(to)e(de\002ne)g(the)g(sequent)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14982
(calculus)h(for)d(\002rst-order)i(classical)g(logic.)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14983
b(There)321 430 y(are)24 b(four)g(rules)h(to)e(go)o(v)o(ern)i(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14984
(quanti\002ers,)j F7(viz.)p Black Black 916 640 a Ga(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14985
b F4(:)g Ga(B)5 b F4([)p FL(x)25 b F4(:=)h FL(t)p F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14986
Ga(;)15 b F4(\000)p 1490 628 11 41 v 1501 610 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14987
96 w(\001)p 916 683 727 4 v 995 763 a Ga(y)20 b F4(:)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14988
F6(8)p FL(x)p Ga(:B)5 b(;)15 b F4(\000)p 1411 751 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14989
41 v 1421 732 46 5 v 96 w(\001)1683 707 y F6(8)1734 721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14990
y Gc(L)2075 640 y F4(\000)p 2152 628 11 41 v 2162 610
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14991
46 5 v 96 w(\001)p Ga(;)g(a)i F4(:)h Ga(B)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14992
FL(x)25 b F4(:=)g FL(y)q F4(])p 2075 683 733 4 v 2161
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14993
763 a(\000)p 2238 751 11 41 v 2249 732 46 5 v 97 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14994
Ga(;)15 b(b)i F4(:)g F6(8)p FL(x)p Ga(:B)2849 707 y F6(8)2900
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14995
721 y Gc(R)911 912 y Ga(x)g F4(:)g Ga(B)5 b F4([)p FL(x)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14996
b F4(:=)g FL(y)q F4(])p Ga(;)15 b F4(\000)p 1495 900
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14997
11 41 v 1506 882 46 5 v 97 w(\001)p 911 955 737 4 v 995
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14998
1034 a Ga(y)20 b F4(:)d F6(9)p FL(x)p Ga(:B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 14999
b F4(\000)p 1411 1022 11 41 v 1421 1004 46 5 v 96 w(\001)1689
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15000
979 y F6(9)1740 993 y Gc(L)2080 912 y F4(\000)p 2157
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15001
900 11 41 v 2167 882 46 5 v 96 w(\001)p Ga(;)g(a)i F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15002
Ga(B)5 b F4([)p FL(x)25 b F4(:=)g FL(t)p F4(])p 2080
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15003
955 723 4 v 2161 1034 a(\000)p 2238 1022 11 41 v 2249
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15004
1004 46 5 v 97 w(\001)p Ga(;)15 b(b)i F4(:)g F6(9)p FL(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15005
Ga(:B)2844 979 y F6(9)2895 993 y Gc(R)321 1253 y Gg(The)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15006
b F6(8)538 1267 y Gc(R)619 1253 y Gg(and)g F6(9)825 1267
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15007
y Gc(L)901 1253 y Gg(rules)g(are)g(subject)i(to)d(the)h(usual)h(pro)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15008
(viso)h(that)e FL(y)g Gg(should)i(not)e(appear)h(freely)g(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15009
F4(\000)321 1366 y Gg(and)k F4(\001)p Gg(.)44 b(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15010
b(e)28 b(obtain)i(the)f(\002rst-order)h(system)g(by)e(adding)j(these)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15011
(rules)f(to)g(the)g(calculus)i(sho)n(wn)e(in)321 1479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15012
y(Figure)24 b(2.2.)29 b(The)23 b(ne)n(w)g(rules)h(gi)n(v)o(e)g(rise)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15013
(to)f(the)h(term)g(annotations)j(gi)n(v)o(en)d(ne)o(xt.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15014
Black 321 1667 a Gb(De\002nition)18 b(2.4.1)h Gg(\(Ra)o(w)e(T)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15015
b(erms)18 b(for)h(Quanti\002ers\))p Gb(:)p Black 36 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15016
Gg(W)-7 b(e)18 b(e)o(xtend)i(the)f(grammar)g(gi)n(v)o(en)g(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15017
(De\002nition)321 1780 y(2.2.1)24 b(with)f(the)h(follo)n(wing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15018
(clauses:)p Black Black 927 1976 a Ga(M)5 b(;)15 b(N)110
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15019
b F4(::=)100 b FL(Exists)1684 1990 y Gc(R)1742 1976 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15020
F4(\()1777 1964 y FX(h)1805 1976 y Ga(a)r F4(:)r Ga(B)1956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15021
1964 y FX(i)1983 1976 y Ga(M)10 b(;)15 b FL(t)p Ga(;)g(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15022
F4(\))265 b Gg(e)o(xists-right)1291 2106 y F6(j)148 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15023
FL(Exists)1684 2120 y Gc(L)1736 2106 y F4(\()1771 2094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15024
y F9(\()1799 2106 y Ga(x)r F4(:)r Ga(B)1954 2094 y F9(\))1981
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15025
2106 y F4([)p FL(y)q F4(])p Ga(M)11 b(;)k(y)s F4(\))237
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15026
b Gg(e)o(xists-left)1291 2235 y F6(j)148 b FL(F)m(o)m(rall)1674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15027
2249 y Gc(R)1732 2235 y F4(\()1767 2223 y FX(h)1795 2235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15028
y Ga(a)r F4(:)r Ga(B)1946 2223 y FX(i)1973 2235 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15029
FL(y)q F4(])p Ga(M)11 b(;)k(b)p F4(\))254 b Gg(forall-right)1291
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15030
2364 y F6(j)148 b FL(F)m(o)m(rall)1674 2378 y Gc(L)1726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15031
2364 y F4(\()1761 2352 y F9(\()1789 2364 y Ga(x)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15032
Ga(B)1944 2352 y F9(\))1971 2364 y Ga(M)10 b(;)15 b FL(t)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15033
Ga(;)g(y)s F4(\))268 b Gg(forall-left)321 2574 y(where)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15034
b FL(y)g Gg(is)f(a)h(v)n(ariable)h(and)f FL(t)e Gg(is)i(an)f(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15035
(xpression;)k(square)e(brack)o(ets)g(denote)h(a)d(ne)n(w)g(binding)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15036
(opera-)321 2687 y(tion)f(for)g(v)n(ariables.)31 b(Thus)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15037
b F4([)p FL(y)q F4(])p Ga(M)33 b Gg(means)24 b(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15038
FL(y)g Gg(becomes)h(bound)g(in)e Ga(M)10 b Gg(.)321 2941
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15039
y(Notice)27 b(that)g(in)f(the)h(clauses)h(for)f(e)o(xists-right)i(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15040
(forall-left)i(we)c(ha)n(v)o(e)i(to)g(record)g(an)g(e)o(xpression)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15041
3054 y(\(this)f(will)e(become)i(clearer)g(when)f(we)f(de\002ne)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15042
(cut-reductions\).)37 b(W)-7 b(e)24 b(can)h(no)n(w)f(gi)n(v)o(e)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15043
(rules)321 3167 y(for)f(forming)h(typing)g(judgements.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15044
Black Black 625 3377 a Ga(x)17 b F4(:)h Ga(B)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15045
FL(x)24 b F4(:=)i FL(t)p F4(])p Ga(;)15 b F4(\000)1204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15046
3365 y Gc(.)1259 3377 y Ga(M)1382 3365 y Gc(.)1437 3377
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15047
y F4(\001)p 403 3420 1332 4 v 403 3505 a Ga(y)20 b F4(:)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15048
F6(8)p FL(x)p Ga(:B)5 b(;)15 b F4(\000)824 3493 y Gc(.)879
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15049
3505 y FL(F)m(o)m(rall)1090 3519 y Gc(L)1142 3505 y F4(\()1177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15050
3493 y F9(\()1205 3505 y Ga(x)1257 3493 y F9(\))1284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15051
3505 y Ga(M)10 b(;)15 b FL(t)p Ga(;)g(y)s F4(\))1604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15052
3493 y Gc(.)1659 3505 y F4(\001)1776 3444 y F6(8)1827
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15053
3458 y Gc(L)2215 3377 y F4(\000)2297 3365 y Gc(.)2352
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15054
3377 y Ga(M)2475 3365 y Gc(.)2530 3377 y F4(\001)p Ga(;)g(a)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15055
F4(:)h Ga(B)5 b F4([)p FL(x)25 b F4(:=)g FL(y)q F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15056
1994 3420 1337 4 v 1994 3505 a(\000)2076 3493 y Gc(.)2131
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15057
3505 y FL(F)m(o)m(rall)2341 3519 y Gc(R)2399 3505 y F4(\()2434
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15058
3493 y FX(h)2462 3505 y Ga(a)2510 3493 y FX(i)2537 3505
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15059
y F4([)p FL(y)q F4(])p Ga(M)11 b(;)k(b)p F4(\))2869 3493
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15060
y Gc(.)2924 3505 y F4(\001)p Ga(;)g(b)i F4(:)h F6(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15061
FL(x)p Ga(:B)3371 3444 y F6(8)3422 3458 y Gc(R)620 3701
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15062
y Ga(x)f F4(:)g Ga(B)5 b F4([)p FL(x)25 b F4(:=)h FL(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15063
F4(])p Ga(;)15 b F4(\000)1209 3689 y Gc(.)1264 3701 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15064
Ga(M)1388 3689 y Gc(.)1443 3701 y F4(\001)p 388 3744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15065
1363 4 v 388 3829 a Ga(y)20 b F4(:)d F6(9)p FL(x)p Ga(:B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15066
b(;)15 b F4(\000)809 3817 y Gc(.)864 3829 y FL(Exists)1084
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15067
3843 y Gc(L)1137 3829 y F4(\()1172 3817 y F9(\()1199
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15068
3829 y Ga(x)1251 3817 y F9(\))1279 3829 y F4([)p FL(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15069
F4(])p Ga(M)c(;)k(y)s F4(\))1619 3817 y Gc(.)1674 3829
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15070
y F4(\001)1792 3768 y F6(9)1843 3782 y Gc(L)2220 3701
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15071
y F4(\000)2302 3689 y Gc(.)2357 3701 y Ga(M)2480 3689
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15072
y Gc(.)2535 3701 y F4(\001)p Ga(;)g(a)i F4(:)h Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15073
b F4([)p FL(x)25 b F4(:=)g FL(t)p F4(])p 1999 3744 1326
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15074
4 v 1999 3829 a(\000)2081 3817 y Gc(.)2136 3829 y FL(Exists)2356
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15075
3843 y Gc(R)2414 3829 y F4(\()2449 3817 y FX(h)2477 3829
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15076
y Ga(a)2525 3817 y FX(i)2552 3829 y Ga(M)11 b(;)k FL(t)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15077
Ga(;)g(b)p F4(\))2864 3817 y Gc(.)2919 3829 y F4(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15078
Ga(;)g(b)i F4(:)h F6(9)p FL(x)p Ga(:B)3366 3768 y F6(9)3417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15079
3782 y Gc(R)321 4048 y Gg(Again,)31 b(the)g F6(8)788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15080
4062 y Gc(R)874 4048 y Gg(and)f F6(9)1085 4062 y Gc(L)1166
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15081
4048 y Gg(rules)h(are)f(subject)i(to)d(the)h(pro)o(viso)i(that)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15082
FL(y)h Gg(does)f(not)h(occur)g(freely)g(in)f F4(\000)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15083
4161 y Gg(and)c F4(\001)p Gg(.)31 b(Since)25 b(the)h(logical)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15084
(cut-reductions)j(for)c(the)h(quanti\002ers)h(include)f(v)n(ariable)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15085
(substitutions,)321 4274 y(we)33 b(need)i(a)e(corresponding)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15086
b(operation)f(for)d(terms.)59 b(Assuming)35 b(the)f(Barendre)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15087
(gt-style)k(naming)321 4386 y(con)l(v)o(ention)27 b(for)d(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15088
(ariables,)h(this)f(operation)i(is)e(de\002ned)g(as)f(follo)n(ws.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15089
Black 321 4574 a Gb(De\002nition)g(2.4.2)g Gg(\(V)-10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15090
b(ariable)25 b(Substitution)i(for)c(T)-6 b(erms\))p Gb(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15091
Black 34 w Gg(The)23 b(four)h(non-tri)n(vial)j(cases)d(are:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15092
Black Black 680 4794 a FL(Exists)900 4808 y Gc(R)958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15093
4794 y F4(\()993 4782 y FX(h)1021 4794 y Ga(a)1069 4782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15094
y FX(i)1096 4794 y Ga(M)10 b(;)15 b FL(s)p Ga(;)g(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15095
F4(\)[)p FL(x)27 b F4(:=)e FL(t)p F4(])1756 4742 y F5(def)1763
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15096
4794 y F4(=)106 b FL(Exists)2161 4808 y Gc(R)2218 4794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15097
y F4(\()2253 4782 y FX(h)2281 4794 y Ga(a)2329 4782 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15098
FX(i)2382 4794 y Ga(M)10 b F4([)p FL(x)25 b F4(:=)h FL(t)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15099
F4(])p Ga(;)15 b FL(s)p F4([)p FL(x)25 b F4(:=)h FL(t)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15100
F4(])p Ga(;)15 b(b)p F4(\))654 4934 y FL(Exists)875 4948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15101
y Gc(L)927 4934 y F4(\()962 4922 y F9(\()990 4934 y Ga(x)1042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15102
4922 y F9(\))1069 4934 y F4([)p FL(y)q F4(])p Ga(M)c(;)k(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15103
F4(\)[)p FL(x)26 b F4(:=)f FL(t)p F4(])1756 4882 y F5(def)1763
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15104
4934 y F4(=)106 b FL(Exists)2161 4948 y Gc(L)2213 4934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15105
y F4(\()2248 4922 y F9(\()2276 4934 y Ga(x)2328 4922
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15106
y F9(\))2355 4934 y F4([)p FL(y)q F4(])26 b Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15107
F4([)p FL(x)25 b F4(:=)h FL(t)p F4(])p Ga(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15108
F4(\))671 5073 y FL(F)m(o)m(rall)882 5087 y Gc(R)940
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15109
5073 y F4(\()975 5061 y FX(h)1002 5073 y Ga(a)1050 5061
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15110
y FX(i)1078 5073 y F4([)p FL(y)q F4(])p Ga(M)c(;)k(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15111
F4(\)[)p FL(x)26 b F4(:=)f FL(t)p F4(])1756 5022 y F5(def)1763
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15112
5073 y F4(=)106 b FL(F)m(o)m(rall)2151 5087 y Gc(R)2208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15113
5073 y F4(\()2243 5061 y FX(h)2271 5073 y Ga(a)2319 5061
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15114
y FX(i)2347 5073 y F4([)p FL(y)q F4(])25 b Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15115
F4([)p FL(x)26 b F4(:=)f FL(t)p F4(])p Ga(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15116
F4(\))683 5213 y FL(F)m(o)m(rall)893 5227 y Gc(L)945
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15117
5213 y F4(\()980 5201 y F9(\()1008 5213 y Ga(x)1060 5201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15118
y F9(\))1087 5213 y Ga(M)c(;)k FL(s)p Ga(;)g(y)s F4(\)[)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15119
FL(x)26 b F4(:=)f FL(t)p F4(])1756 5162 y F5(def)1763
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15120
5213 y F4(=)106 b FL(F)m(o)m(rall)2151 5227 y Gc(L)2203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15121
5213 y F4(\()2238 5201 y F9(\()2265 5213 y Ga(x)2317
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15122
5201 y F9(\))2370 5213 y Ga(M)10 b F4([)p FL(x)26 b F4(:=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15123
FL(t)p F4(])p Ga(;)15 b FL(s)p F4([)p FL(x)26 b F4(:=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15124
FL(t)p F4(])p Ga(;)15 b(b)p F4(\))p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15125
%%Page: 43 55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15126
TeXDict begin 43 54 bop Black 277 51 a Gb(2.4)23 b(First-Order)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15127
(Classical)g(Logic)2399 b(43)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15128
418 317 a Gg(W)-7 b(e)23 b(should)j(point)f(out)g(that)f(our)h(use)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15129
(of)g(the)h(Barendre)o(gt-style)i(naming)e(con)l(v)o(ention)j(for)c(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15130
(ari-)277 430 y(ables)37 b(corresponds)i(to)d(the)g F7(pur)m
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15131
(e-variable)j(con)l(vention)g Gg(commonly)e(used)g(in)e(treatises)j(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15132
e(cut-)277 543 y(elimination)30 b(based)e(on)f(sequent)i(proofs)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15133
(\(for)g(e)o(xample)g([T)m(roelstra)g(and)g(Schwichtenber)n(g,)i(1996,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15134
277 656 y(P)o(age)c(58]\).)36 b(This)26 b(con)l(v)o(ention)j(is)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15135
(vital)h(in)f(the)g(cut-elimination)j(procedure,)g(since)e(without)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15136
(it)f(not)277 769 y(all)e(\002rst-order)h(proofs)g(will)e(ha)n(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15137
(a)f(cut-free)j(normal)e(form.)418 903 y(Whilst)37 b(the)f(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15138
(xtension)j(of)d(the)g(proof)h(substitution)j(sho)n(wn)d(in)f(Figure)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15139
(2.4)g(and)h(the)f(corre-)277 1016 y(sponding)26 b(adaptation)h(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15140
(the)g(reduction)1704 979 y Gc(c)1634 1016 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15141
b(\000)g(!)22 b Gg(for)i(the)g(\002rst-order)i(case)e(are)g(quite)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15142
(easy)-6 b(,)24 b(and)h(thus)277 1129 y(not)k(e)o(xplained,)k(the)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15143
(reduction)i(rules)f(for)f(logical)i(cuts)e(need)h(some)f(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15144
(xplanation.)48 b(Consider)30 b(the)277 1242 y(de\002nition)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15145
b(belo)n(w)-6 b(.)p Black 277 1429 a Gb(De\002nition)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15146
b(2.4.3)g Gg(\(Cut-Reductions)k(for)d(Quanti\002ers\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15147
Gb(:)p Black Black Black 442 1639 a FL(Cut)p F4(\()615
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15148
1627 y FX(h)642 1639 y Ga(b)681 1627 y FX(i)709 1639
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15149
y FL(Exists)929 1653 y Gc(R)987 1639 y F4(\()1022 1627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15150
y FX(h)1050 1639 y Ga(a)1098 1627 y FX(i)1125 1639 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15151
Ga(M)10 b(;)15 b FL(t)p Ga(;)g(b)p F4(\))q Ga(;)1451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15152
1627 y F9(\()1479 1639 y Ga(y)1527 1627 y F9(\))1555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15153
1639 y FL(Exists)1775 1653 y Gc(L)1827 1639 y F4(\()1862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15154
1627 y F9(\()1890 1639 y Ga(x)1942 1627 y F9(\))1969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15155
1639 y F4([)p FL(y)q F4(])p Ga(N)c(;)k(y)s F4(\))q(\))516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15156
1738 y Gc(l)442 1776 y F6(\000)-32 b(\000)h(!)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15157
F4(\()810 1764 y FX(h)838 1776 y Ga(a)886 1764 y FX(i)913
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15158
1776 y Ga(M)10 b(;)1051 1764 y F9(\()1079 1776 y Ga(x)1131
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15159
1764 y F9(\))1159 1776 y Ga(N)g F4([)p FL(y)26 b F4(:=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15160
FL(t)p F4(]\))724 1936 y Gg(if)e FL(Exists)1023 1950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15161
y Gc(R)1081 1936 y F4(\()1116 1924 y FX(h)1143 1936 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15162
Ga(a)1191 1924 y FX(i)1219 1936 y Ga(M)10 b(;)15 b FL(t)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15163
Ga(;)g(b)p F4(\))24 b Gg(and)g FL(Exists)1902 1950 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15164
Gc(L)1954 1936 y F4(\()1989 1924 y F9(\()2017 1936 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15165
Ga(x)2069 1924 y F9(\))2097 1936 y F4([)p FL(y)q F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15166
Ga(N)10 b(;)15 b(y)s F4(\))24 b Gg(freshly)h(introduce)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15167
Ga(a)d Gg(and)h Ga(x)442 2167 y FL(Cut)p F4(\()615 2155
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15168
y FX(h)642 2167 y Ga(b)681 2155 y FX(i)709 2167 y FL(F)m(o)m(rall)919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15169
2181 y Gc(R)977 2167 y F4(\()1012 2155 y FX(h)1040 2167
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15170
y Ga(a)1088 2155 y FX(i)1115 2167 y F4([)p FL(y)q F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15171
Ga(M)11 b(;)k(b)p F4(\))q Ga(;)1462 2155 y F9(\()1490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15172
2167 y Ga(y)1538 2155 y F9(\))1565 2167 y FL(F)m(o)m(rall)1775
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15173
2181 y Gc(L)1827 2167 y F4(\()1862 2155 y F9(\()1890
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15174
2167 y Ga(x)1942 2155 y F9(\))1970 2167 y Ga(N)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15175
b FL(t)p Ga(;)g(y)s F4(\)\))516 2266 y Gc(l)442 2303
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15176
y F6(\000)-32 b(\000)h(!)25 b FL(Cut)p F4(\()810 2291
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15177
y FX(h)838 2303 y Ga(a)886 2291 y FX(i)913 2303 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15178
b F4([)p FL(y)27 b F4(:=)e FL(t)p F4(])p Ga(;)1324 2291
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15179
y F9(\()1352 2303 y Ga(x)1404 2291 y F9(\))1432 2303
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15180
y Ga(N)10 b F4(\))724 2463 y Gg(if)24 b FL(F)m(o)m(rall)1013
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15181
2477 y Gc(R)1071 2463 y F4(\()1106 2451 y FX(h)1133 2463
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15182
y Ga(a)1181 2451 y FX(i)1209 2463 y F4([)p FL(y)q F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15183
Ga(M)11 b(;)k(b)p F4(\))23 b Gg(and)h FL(F)m(o)m(rall)1903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15184
2477 y Gc(L)1955 2463 y F4(\()1990 2451 y F9(\()2017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15185
2463 y Ga(x)2069 2451 y F9(\))2097 2463 y Ga(N)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15186
b FL(t)p Ga(;)g(y)s F4(\))23 b Gg(freshly)j(introduce)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15187
Ga(a)c Gg(and)i Ga(x)277 2713 y Gg(W)-7 b(e)18 b(must)h(ensure)h(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15188
(reducing)i(a)d(v)n(alid)h(proof)h(yields)g(again)g(a)e(v)n(alid)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15189
(proof)h(\(in)f(the)g(sense)h(of)e(being)277 2826 y(deri)n(v)n(able)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15190
(by)f(the)f(inference)j(rules\).)28 b(This)19 b(holds)h(only)g(if)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15191
(some)h(alpha-con)l(v)o(ersions)k(are)c(made.)27 b(An)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15192
2939 y(e)o(xample)i(where)f(such)h(a)f(con)l(v)o(ersion)j(is)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15193
(necessary)k(w)o(as)d(gi)n(v)o(en)g(by)g(Kleene)h([1952a,)h(P)o(age)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15194
(450],)277 3052 y(and)34 b(T)m(roelstra)h(and)e(Schwichtenber)n(g)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15195
([1996,)g(P)o(age)c(58].)58 b(W)-7 b(e)33 b(assume)h(these)g(con)l(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15196
(ersions)j(are)277 3165 y(done)25 b(implicitly)-6 b(.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15197
3299 y(In)37 b(the)h(sequel)g(we)e(shall)i(tak)o(e)1557
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15198
3262 y Gc(l)1483 3299 y F6(\000)-31 b(\000)g(!)o Gg(,)1786
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15199
3262 y Gc(c)1716 3299 y F6(\000)g(\000)f(!)p Gg(,)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15200
b F4([)p Ga(\033)s F4(])e Gg(and)h FY(T)e Gg(to)h(denote)i(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15201
(notions)i(for)e(\002rst-)277 3412 y(order)28 b(classical)i(logic.)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15202
b(The)27 b(de\002nitions)i(are)e(laborious,)j(b)n(ut)e(straightforw)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15203
(ard)j(gi)n(v)o(en)c(the)h(details)277 3524 y(pro)o(vided)f(for)d(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15204
(propositional)k(calculus.)34 b(The)24 b(cut-elimination)k(procedure)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15205
(for)e(the)g(\002rst-order)277 3637 y(system)f(is)g(de\002ned)g(as)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15206
(follo)n(ws.)p Black 277 3825 a Gb(De\002nition)f(2.4.4)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15207
Gg(\(Cut-Elimination)j(for)e(First-Order)h(Classical)g(Logic\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15208
Gb(:)p Black 277 3938 a Gg(The)c(cut-elimination)26 b(procedure)e(for)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15209
(\002rst-order)i(classical)f(logic,)g F4(\()p FY(T)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15210
Ga(;)2630 3901 y Gc(cut)2599 3938 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15211
F4(\))p Gg(,)22 b(is)f(a)g(reduction)k(sys-)277 4051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15212
y(tem)e(where:)p Black 414 4273 a F6(\017)p Black 45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15213
w FY(T)g Gg(is)h(the)f(set)h(of)g(well-typed)h(\(\002rst-order\))h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15214
(terms,)d(and)p Black 414 4478 a F6(\017)p Black 535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15215
4441 a Gc(cut)504 4478 y F6(\000)-31 b(\000)g(!)22 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15216
Gg(consists)k(of)d(the)h(reductions)j(for)c(logical)j(cuts)e(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15217
(commuting)h(cuts;)f(that)g(is)1618 4680 y Gc(cut)1587
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15218
4717 y F6(\000)-31 b(\000)f(!)1782 4666 y F5(def)1789
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15219
4717 y F4(=)1967 4680 y Gc(l)1892 4717 y F6(\000)h(\000)g(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15220
b([)2244 4680 y Gc(c)2174 4717 y F6(\000)-31 b(\000)f(!)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15221
b Ga(:)277 4976 y Gg(W)-7 b(e)30 b(should)j(no)n(w)e(pro)o(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15222
(that)1316 4938 y Gc(cut)1286 4976 y F6(\000)-32 b(\000)h(!)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15223
b Gg(still)i(respects)h(the)f(subject)h(reduction)h(property)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15224
b(.)54 b(Ho)n(we)n(v)o(er)l(,)277 5088 y(this)35 b(proof)g(is)f(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15225
(straightforw)o(ard)38 b(e)o(xtension)e(of)e(the)g(proof)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15226
(Proposition)j(2.2.10.)60 b(The)34 b(only)277 5201 y(additional)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15227
b(lemma)e(we)e(need)j(in)e(this)h(proof)h(is)e(as)h(follo)n(ws.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15228
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15229
%%Page: 44 56
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15230
TeXDict begin 44 55 bop Black -144 51 a Gb(44)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15231
b(Calculi)p -144 88 3691 4 v Black Black 321 317 a(Lemma)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15232
b(2.4.5:)p Black 34 w Gg(If)69 b F4(\000)1105 305 y Gc(.)1160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15233
317 y Ga(M)1283 305 y Gc(.)1338 317 y F4(\001)e Gg(is)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15234
b(a)f(typing)i(judgement,)i(then)d(for)g(an)f(arbitrary)j(substitution)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15235
321 430 y F4([)p FL(x)26 b F4(:=)f FL(t)p F4(])e Gg(also)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15236
F4(\000[)p FL(x)h F4(:=)g FL(t)p F4(])1139 418 y Gc(.)1194
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15237
430 y Ga(M)10 b F4([)p FL(x)25 b F4(:=)g FL(t)p F4(])1589
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15238
418 y Gc(.)1644 430 y F4(\001[)p FL(x)g F4(:=)g FL(t)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15239
F4(])p Gg(.)p Black 321 643 a F7(Pr)l(oof)o(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15240
34 w Gg(By)e(induction)j(on)e(the)f(structure)j(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15241
Ga(M)10 b Gg(.)p 3480 643 4 62 v 3484 585 55 4 v 3484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15242
643 V 3538 643 4 62 v 321 847 a(W)-7 b(e)20 b(conclude)j(this)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15243
(section)h(with)e(the)h(strong)h(normalisation)h(theorem)f(for)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15244
(\002rst-order)h(system.)p Black 321 1035 a Gb(Theor)n(em)j(2.4.6:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15245
Black 34 w Gg(The)g(reduction)i(system)f F4(\()p FY(T)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15246
Ga(;)1915 998 y Gc(cut)1885 1035 y F6(\000)-32 b(\000)h(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15247
F4(\))24 b Gg(for)h(\002rst)g(order)h(classical)h(logic)f(is)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15248
(strongly)321 1148 y(normalising.)p Black 321 1360 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15249
F7(Pr)l(oof)o(.)p Black 34 w Gg(The)g(proof)j(is)e(similar)g(to)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15250
(strong)g(normalisation)i(proof)f(for)e(the)g(propositional)k(calcu-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15251
321 1473 y(lus,)24 b(and)g(for)g(space)g(reasons)i(all)d(b)n(ut)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15252
(follo)n(wing)h(details)g(are)f(omitted.)p Black 458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15253
1663 a F6(\017)p Black 46 w Gg(An)c(auxiliary)j(substitution,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15254
F6(f)p 1538 1663 28 4 v 1556 1663 V 1573 1663 V 65 w(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15255
Gg(,)20 b(and)i(an)f(auxiliary)i(reduction,)2697 1626
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15256
y Gc(aux)2677 1663 y F6(\000)-32 b(\000)h(!)p Gg(,)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15257
b(need)i(to)f(be)g(de\002ned)549 1776 y(along)28 b(the)g(lines)g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15258
(the)h(de\002nitions)h(gi)n(v)o(en)f(for)f(the)h(propositional)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15259
(calculus.)42 b(This)28 b(a)n(v)n(oids)549 1889 y(the)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15260
b(problem)i(of)f F4([)p 1134 1889 V 1152 1889 V 1169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15261
1889 V 64 w(])f Gg(where)h(safe)g(substitutions)k(do)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15262
b(not)h(commute.)p Black 458 2036 a F6(\017)p Black 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15263
w Gg(The)k(candidates)33 b(need)d(to)f(be)g(e)o(xtended)i(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15264
(quanti\002ed)h(formulae;)j(Figure)29 b(2.8)h(gi)n(v)o(es)f(the)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15265
2149 y(corresponding)e(set)d(operators.)p Black 458 2297
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15266
a F6(\017)p Black 46 w Gg(The)g(induction)k(in)d(Lemma)f(2.3.18)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15267
(includes)j(in)d(the)g(\002rst-order)i(system)e(a)g(closing)i(substi-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15268
549 2410 y(tution)d(for)g(v)n(ariables.)p 3480 2410 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15269
62 v 3484 2352 55 4 v 3484 2410 V 3538 2410 4 62 v 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15270
2565 a(Unless)g(otherwise)h(stated,)f(we)f(henceforth)j(assume)e(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15271
F4(\()p FY(T)t Ga(;)2376 2528 y Gc(cut)2345 2565 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15272
b(\000)g(!)o F4(\))23 b Gg(and)h F4(\()p FY(T)t Ga(;)2880
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15273
2528 y Gc(aux)2860 2565 y F6(\000)-32 b(\000)h(!)p F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15274
b Gg(refer)h(to)f(\002rst-)321 2678 y(order)i(classical)h(logic.)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15275
2984 y Ge(2.5)119 b(V)-11 b(ariations)30 b(of)g(the)g(Sequent)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15276
(Calculus)321 3208 y Gg(Designing)25 b(a)d(logical)j(calculus)f(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15277
(lik)o(e)g(composing)i(a)d(fugue:)30 b(there)24 b(are)f(man)o(y)g(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15278
(ariations.)30 b(A)22 b(nat-)321 3321 y(ural)28 b(question)i(is)e(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15279
(what)h(e)o(xtent)g(the)g(strong)h(normalisation)i(result)e(depends)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15280
(on)f(our)g(particular)321 3434 y(formulation)35 b(of)d(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15281
(inference)j(rules.)55 b(Some)31 b(aspects)j(of)e(our)h(rules)g(are)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15282
(crucial)i(in)e(the)g(strong)321 3547 y(normalisation)d(proof;)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15283
(others)f(are)f(not.)35 b(The)25 b(use)h(of)g(sets)g(of)g(labelled)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15284
(formulae)g(as)f(conte)o(xts,)h(for)321 3660 y(e)o(xample,)d(plays)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15285
(an)f(important)h(r)8 b(\210)-38 b(ole.)29 b(Labels)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15286
b(allo)n(w)g(us)g(to)f(distinguish)k(dif)n(ferent)e(formula)g(occur)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15287
(-)321 3772 y(rences)h(in)e(a)g(typing)i(deri)n(v)n(ation.)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15288
b(This)25 b(is)f(an)g(indispensable)29 b(pro)o(viso)d(for)e(a)g(strong)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15289
i(normalisation)321 3885 y(proof)j(b)n(uilt)f(around)h(the)e(notion)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15290
(of)e(candidates;)33 b(see)27 b([Lei)n(v)n(ant,)h(1979].)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15291
b(On)27 b(the)g(other)h(hand,)h(the)321 3998 y(speci\002c)g(form)e(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15292
(our)h(inference)i(rules)f(is)e(irrele)n(v)n(ant,)j(in)d(the)h(sense)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15293
(that)f(our)g(candidate)i(method)321 4111 y(e)o(xtends)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15294
b(to)f(other)g(formulations)j(of)c(the)h(inference)i(rules.)k(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15295
b(e)21 b(shall)j(gi)n(v)o(e)e(some)h(e)o(xamples)h(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15296
b(.)462 4241 y(First,)33 b(there)e(are)g(alternati)n(v)o(e)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15297
(formulations)h(for)c(and-left)j(and)e(or)n(-right.)52
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15298
b(The)31 b(and-left)h(rule,)321 4354 y(for)24 b(instance,)h(may)f(be)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15299
(of)h(the)f(form)p Black Black 781 4543 a Ga(x)17 b F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15300
Ga(B)961 4557 y Gc(i)989 4543 y Ga(;)e F4(\000)p 1107
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15301
4531 11 41 v 1117 4513 46 5 v 97 w(\001)p 693 4580 653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15302
4 v 693 4659 a Ga(z)22 b F4(:)17 b Ga(B)868 4673 y F9(1)908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15303
4659 y F6(^)o Ga(B)1037 4673 y F9(2)1077 4659 y Ga(;)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15304
F4(\000)p 1194 4647 11 41 v 1204 4629 46 5 v 96 w(\001)1387
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15305
4602 y F6(^)1448 4617 y Gc(L)1496 4627 y FZ(i)1577 4602
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15306
y F4(\()p Ga(i)26 b F4(=)f(1)p Ga(;)15 b F4(2\))2078
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15307
4612 y Gg(or)2301 4543 y Ga(x)i F4(:)h Ga(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15308
b(y)k F4(:)f Ga(C)q(;)d F4(\000)p 2818 4531 11 41 v 2828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15309
4513 46 5 v 97 w(\001)p 2301 4580 669 4 v 2347 4659 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15310
Ga(z)22 b F4(:)17 b Ga(B)5 b F6(^)o Ga(C)q(;)15 b F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15311
2771 4647 11 41 v 2782 4629 46 5 v 97 w(\001)3011 4602
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15312
y F6(^)3072 4569 y FX(0)3072 4629 y Gc(L)3149 4602 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15313
Ga(:)321 4863 y Gg(The)24 b(\002rst)f(rule,)h(which)g(we)f(included)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15314
(in)e(our)g(sequent)i(calculus,)f(is)f(often)h(referred)g(to)f(as)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15315
(the)i(addi-)321 4976 y(ti)n(v)o(e)20 b(v)n(ariant)h(of)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15316
(and-left)i(rule,)f(and)f(the)g(second)i(as)e(the)g(multiplicati)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15317
(e)i(v)n(ariant.)29 b(If)20 b(we)f(are)h(only)321 5088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15318
y(interested)29 b(in)e(pro)o(v)n(ability)-6 b(,)30 b(it)c(is)g(not)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15319
(dif)n(\002cult)g(to)g(sho)n(w)f(that)h(in)g(classical)h(logic)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15320
(tw)o(o)f(v)n(ariants)321 5201 y(are)31 b(interderi)n(v)n(able.)55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15321
b(By)30 b(this)i(we)e(mean)h(that)h(one)f(can)h(be)f(deri)n(v)o(ed)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15322
(from)f(the)g(other)-5 b(.)52 b(Ho)n(we)n(v)o(er)l(,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15323
5314 y(their)23 b(tw)o(o)f(beha)n(viours)k(in)c(the)g(process)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15324
(cut-elimination)k(are)c(rather)i(dif)n(ferent.)30 b(Whereas)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15325
b(a)f(cut-)321 5427 y(instance)27 b F6(^)703 5441 y Gc(R)761
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15326
5427 y Ga(=)p F6(^)867 5441 y Gc(L)915 5451 y FZ(i)969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15327
5427 y Gg(is)e(replaced)h(by)f(a)e(single)j(cut)f(on)f(a)g(smaller)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15328
(formula,)f(a)f(cut-instance)k F6(^)3330 5441 y Gc(R)3388
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15329
5427 y Ga(=)p F6(^)3494 5394 y FX(0)3494 5454 y Gc(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15330
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15331
%%Page: 45 57
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15332
TeXDict begin 45 56 bop Black 277 51 a Gb(2.5)23 b(V)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15333
b(ariations)25 b(of)e(the)g(Sequent)f(Calculus)2124 b(45)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15334
277 88 3691 4 v Black Black 277 229 V 277 1974 4 1746
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15335
v 529 472 a F5(F)t(O)t(R)t(A)t(L)t(L)t(L)t(E)t(F)t(T)986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15336
487 y FJ(\()p FQ(8)p Ft(x)p FS(:B)s FJ(\))1180 472 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15337
FG(\()p FU(X)7 b FG(\))1419 425 y F5(def)1429 472 y FG(=)1604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15338
302 y FI(8)1604 376 y(<)1604 526 y(:)1719 359 y FJ(\()1745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15339
371 y FU(y)e FG(:)r FT(8)p FF(x)p FU(:B)1991 359 y FJ(\))2017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15340
371 y FF(F)n(o)n(rall)2209 383 y FS(L)2259 371 y FG(\()2291
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15341
359 y FJ(\()2317 371 y FU(x)r FG(:)r FU(B)t FG([)p FF(x)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15342
b FG(:=)e FF(t)p FG(])2708 359 y FJ(\))2734 371 y FU(M)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15343
b(;)14 b FF(t)p FU(;)g(y)s FG(\))22 b FT(j)1908 471 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15344
FF(F)n(o)n(rall)2100 483 y FS(L)2150 471 y FG(\()2182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15345
459 y FJ(\()2208 471 y FU(x)r FG(:)r FU(B)t FG([)p FF(x)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15346
FG(:=)e FF(t)p FG(])2599 459 y FJ(\))2625 471 y FU(M)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15347
b(;)14 b FF(t)p FU(;)g(y)s FG(\))23 b Gd(freshly)c(introduces)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15348
FU(y)s(;)1908 559 y FJ(\()1934 571 y FU(x)r FG(:)r FU(B)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15349
FG([)p FF(x)k FG(:=)e FF(t)p FG(])2325 559 y FJ(\))2351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15350
571 y FU(M)31 b FT(2)24 b FU(X)3646 302 y FI(9)3646 376
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15351
y(=)3646 526 y(;)479 906 y F5(F)t(O)t(R)t(A)t(L)t(L)t(R)t(I)t(G)t(H)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15352
(T)983 921 y FQ(h8)p Ft(x)p FS(:B)s FQ(i)1180 906 y FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15353
FU(X)7 b FG(\))1419 859 y F5(def)1429 906 y FG(=)1604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15354
735 y FI(8)1604 810 y(<)1604 960 y(:)1719 793 y FJ(\()1745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15355
805 y FU(b)r FG(:)r FT(8)p FF(x)p FU(:B)1983 793 y FJ(\))2009
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15356
805 y FF(F)n(o)n(rall)2201 817 y FS(R)2256 805 y FG(\()2288
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15357
793 y FQ(h)2315 805 y FU(a)r FG(:)r FU(B)t FG([)p FF(x)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15358
b FG(:=)f FF(y)q FG(])2711 793 y FQ(i)2738 805 y FG([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15359
FF(y)q FG(])p FU(M)10 b(;)k(b)p FG(\))22 b FT(j)1908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15360
905 y FF(F)n(o)n(rall)2100 917 y FS(R)2155 905 y FG(\()2187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15361
893 y FQ(h)2214 905 y FU(a)r FG(:)r FU(B)t FG([)p FF(x)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15362
FG(:=)f FF(y)q FG(])2610 893 y FQ(i)2637 905 y FG([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15363
FF(y)q FG(])p FU(M)9 b(;)14 b(b)p FG(\))23 b Gd(freshly)d(introduces)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15364
FU(b;)1908 1004 y Gd(for)f(all)j FF(t)g Gd(s.t.)2305
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15365
992 y FJ(\()2331 1004 y FU(x)r FG(:)r FU(B)t FG([)p FF(x)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15366
FG(:=)d FF(t)p FG(])2721 992 y FJ(\))2747 1004 y FU(M)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15367
b FT(2)23 b FU(X)3661 735 y FI(9)3661 810 y(=)3661 960
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15368
y(;)562 1340 y F5(E)t(X)t(I)t(S)t(T)t(S)t(L)t(E)t(F)t(T)986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15369
1355 y FJ(\()p FQ(9)p Ft(x)p FS(:B)s FJ(\))1180 1340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15370
y FG(\()p FU(X)7 b FG(\))1419 1293 y F5(def)1429 1340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15371
y FG(=)1604 1169 y FI(8)1604 1244 y(<)1604 1394 y(:)1719
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15372
1227 y FJ(\()1745 1239 y FU(y)e FG(:)r FT(9)p FF(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15373
FU(:B)1990 1227 y FJ(\))2017 1239 y FF(Exists)2219 1251
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15374
y FS(L)2268 1239 y FG(\()2300 1227 y FJ(\()2326 1239
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15375
y FU(x)r FG(:)r FU(B)t FG([)p FF(x)25 b FG(:=)e FF(t)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15376
FG(])2717 1227 y FJ(\))2743 1239 y FG([)p FF(y)q FG(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15377
FU(M)9 b(;)14 b(y)s FG(\))23 b FT(j)1908 1339 y FF(Exists)2109
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15378
1351 y FS(L)2159 1339 y FG(\()2191 1327 y FJ(\()2217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15379
1339 y FU(x)r FG(:)r FU(B)t FG([)p FF(x)i FG(:=)e FF(t)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15380
FG(])2608 1327 y FJ(\))2634 1339 y FG([)p FF(y)q FG(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15381
FU(M)9 b(;)14 b(y)s FG(\))23 b Gd(freshly)c(introduces)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15382
FU(y)s(;)1908 1438 y Gd(for)f(all)g FF(t)h Gd(s.t.)2300
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15383
1426 y FJ(\()2326 1438 y FU(x)r FG(:)r FU(B)t FG([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15384
FF(x)k FG(:=)e FF(t)p FG(])2717 1426 y FJ(\))2743 1438
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15385
y FU(M)31 b FT(2)24 b FU(X)3674 1169 y FI(9)3674 1244
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15386
y(=)3674 1394 y(;)512 1774 y F5(E)t(X)t(I)t(S)t(T)t(S)t(R)t(I)t(G)t(H)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15387
(T)983 1789 y FQ(h9)p Ft(x)p FS(:B)s FQ(i)1180 1774 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15388
FG(\()p FU(X)7 b FG(\))1419 1726 y F5(def)1429 1774 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15389
FG(=)1604 1603 y FI(8)1604 1678 y(<)1604 1828 y(:)1719
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15390
1661 y FJ(\()1745 1673 y FU(b)r FG(:)r FT(9)p FF(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15391
FU(:B)1982 1661 y FJ(\))2009 1673 y FF(Exists)2211 1685
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15392
y FS(R)2265 1673 y FG(\()2297 1661 y FQ(h)2324 1673 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15393
FU(a)r FG(:)r FU(B)t FG([)p FF(x)25 b FG(:=)d FF(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15394
FG(])2720 1661 y FQ(i)2748 1673 y FU(M)8 b(;)14 b FF(t)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15395
FU(;)g(b)p FG(\))23 b FT(j)1908 1773 y FF(Exists)2109
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15396
1785 y FS(R)2164 1773 y FG(\()2196 1761 y FQ(h)2223 1773
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15397
y FU(a)r FG(:)r FU(B)t FG([)p FF(x)h FG(:=)f FF(y)q FG(])2619
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15398
1761 y FQ(i)2646 1773 y FU(M)9 b(;)14 b FF(t)p FU(;)g(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15399
FG(\))23 b Gd(freshly)c(introduces)i FU(b;)1908 1860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15400
y FJ(\()1934 1872 y FU(x)r FG(:)r FU(B)t FG([)p FF(x)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15401
FG(:=)e FF(t)p FG(])2325 1860 y FJ(\))2351 1872 y FU(M)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15402
b FT(2)24 b FU(X)3651 1603 y FI(9)3651 1678 y(=)3651
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15403
1828 y(;)p 3965 1974 V 277 1977 3691 4 v Black 854 2131
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15404
a Gg(Figure)g(2.8:)29 b(De\002nition)c(of)e(the)h(set)g(operators)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15405
(for)e(quanti\002ers.)p Black Black 277 2553 a(is)k(replaced)i(by)f(tw)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15406
o(o)f(cuts,)h(and)g(one)g(can)f(choose)i(the)e(order)i(in)e(which)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15407
(these)h(cuts)g(appear)h(in)e(the)277 2666 y(reduct.)h(Gi)n(v)o(en)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15408
b(the)h(term)g(assignment)i(for)d F6(^)1720 2633 y FX(0)1720
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15409
2693 y Gc(L)1791 2666 y Gg(sho)n(wn)h(in)f(Figure)h(2.9,)g(we)f(thus)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15410
(ha)n(v)o(e)f(the)g(reduction)277 2779 y(rule)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15411
Black 434 2991 a FL(Cut)o F4(\()606 2979 y FX(h)634 2991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15412
y Ga(c)673 2979 y FX(i)701 2991 y FL(And)856 3005 y Gc(R)913
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15413
2991 y F4(\()948 2979 y FX(h)976 2991 y Ga(a)1024 2979
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15414
y FX(i)1052 2991 y Ga(M)10 b(;)1190 2979 y FX(h)1217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15415
2991 y Ga(b)1256 2979 y FX(i)1284 2991 y Ga(N)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15416
b(c)p F4(\))q Ga(;)1522 2979 y F9(\()1550 2991 y Ga(z)1596
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15417
2979 y F9(\))1624 2991 y FL(And)1778 2954 y FX(0)1778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15418
3014 y Gc(L)1830 2991 y F4(\()1865 2979 y F9(\()1893
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15419
2991 y Ga(x)1945 2979 y F9(\)\()2000 2991 y Ga(y)2048
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15420
2979 y F9(\))2075 2991 y Ga(P)e(;)i(z)t F4(\))q(\))508
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15421
3102 y Gc(l)434 3140 y F6(\000)-32 b(\000)h(!)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15422
F4(\()802 3128 y FX(h)830 3140 y Ga(a)878 3128 y FX(i)905
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15423
3140 y Ga(M)10 b(;)1043 3128 y F9(\()1071 3140 y Ga(x)1123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15424
3128 y F9(\))1151 3140 y FL(Cut)o F4(\()1323 3128 y FX(h)1351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15425
3140 y Ga(b)1390 3128 y FX(i)1418 3140 y Ga(N)g(;)1541
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15426
3128 y F9(\()1568 3140 y Ga(y)1616 3128 y F9(\))1644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15427
3140 y Ga(P)j F4(\)\))99 b Gg(or)508 3239 y Gc(l)434
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15428
3276 y F6(\000)-32 b(\000)h(!)25 b FL(Cut)p F4(\()802
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15429
3264 y FX(h)830 3276 y Ga(b)869 3264 y FX(i)896 3276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15430
y Ga(N)10 b(;)1019 3264 y F9(\()1047 3276 y Ga(y)1095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15431
3264 y F9(\))1122 3276 y FL(Cut)p F4(\()1295 3264 y FX(h)1323
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15432
3276 y Ga(a)1371 3264 y FX(i)1398 3276 y Ga(M)g(;)1536
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15433
3264 y F9(\()1564 3276 y Ga(x)1616 3264 y F9(\))1644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15434
3276 y Ga(P)j F4(\)\))716 3460 y Gg(if)23 b FL(And)949
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15435
3474 y Gc(R)1007 3460 y F4(\()1042 3448 y FX(h)1069 3460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15436
y Ga(a)1117 3448 y FX(i)1145 3460 y Ga(M)10 b(;)1283
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15437
3448 y FX(h)1311 3460 y Ga(b)1350 3448 y FX(i)1377 3460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15438
y Ga(N)g(;)15 b(c)p F4(\))24 b Gg(and)g FL(And)1907 3423
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15439
y FX(0)1907 3483 y Gc(L)1959 3460 y F4(\()1994 3448 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15440
F9(\()2022 3460 y Ga(x)2074 3448 y F9(\)\()2128 3460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15441
y Ga(y)2176 3448 y F9(\))2204 3460 y Ga(P)13 b(;)i(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15442
F4(\))23 b Gg(freshly)j(introduce)g Ga(c)d Gg(and)h Ga(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15443
Gg(,)277 3694 y(This)f(reduction)i(is)e(similar)h(to)e(that)i(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15444
(en)f(in)g(Figure)g(2.5)g(for)g F6(\033)2342 3708 y Gc(R)2400
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15445
3694 y Ga(=)p F6(\033)2516 3708 y Gc(L)2568 3694 y Gg(,)f(and)h(also)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15446
(requires)h(similar)277 3807 y(side-conditions)40 b(as)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15447
b(speci\002ed)h(in)f(Remark)g(2.2.7.)63 b(Much)35 b(the)h(same)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15448
(discussion)j(as)d(gi)n(v)o(en)g(for)277 3920 y(and-left)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15449
b(applies)f(to)e(or)n(-right,)j(whose)e(multiplicati)n(v)o(e)i(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15450
(ariant,)e F6(_)2413 3887 y FX(0)2413 3947 y Gc(R)2471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15451
3920 y Gg(,)e(is)i(sho)n(wn)f(in)h(Figure)g(2.9.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15452
4051 y(Another)g(source)g(for)f(v)n(ariation)i(is)e(ho)n(w)f(we)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15453
(treat)h(the)g(conte)o(xts)i(in)d(the)h(rules)h(with)e(tw)o(o)h(premi-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15454
277 4164 y(ses:)47 b(these)34 b(conte)o(xts)g(can)f(be)f(either)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15455
(shared)g(or)e(split.)56 b(In)32 b(ef)n(fect,)k(we)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15456
b(ha)n(v)o(e)i(the)g(follo)n(wing)h(tw)o(o)277 4277 y(possibilities)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15457
b(for)d(and-right.)p Black Black 552 4492 a F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15458
629 4480 11 41 v 639 4462 46 5 v 96 w(\001)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15459
b(a)j F4(:)f Ga(B)95 b F4(\000)p 1170 4480 11 41 v 1181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15460
4462 46 5 v 97 w(\001)p Ga(;)15 b(b)i F4(:)g Ga(C)p 552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15461
4530 981 4 v 755 4608 a F4(\000)p 833 4596 11 41 v 843
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15462
4578 46 5 v 97 w(\001)p Ga(;)e(c)i F4(:)h Ga(B)5 b F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15463
Ga(C)1574 4548 y F6(^)1635 4562 y Gc(R)1929 4492 y F4(\000)1986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15464
4506 y F9(1)p 2045 4480 11 41 v 2056 4462 46 5 v 2121
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15465
4492 a F4(\001)2197 4506 y F9(1)2237 4492 y Ga(;)15 b(a)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15466
F4(:)h Ga(B)95 b F4(\000)2606 4506 y F9(2)p 2666 4480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15467
11 41 v 2676 4462 46 5 v 2742 4492 a F4(\001)2818 4506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15468
y F9(2)2857 4492 y Ga(;)15 b(b)j F4(:)f Ga(C)p 1929 4530
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15469
1139 4 v 2026 4608 a F4(\000)2083 4622 y F9(1)2122 4608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15470
y Ga(;)e F4(\000)2219 4622 y F9(2)p 2279 4596 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15471
2289 4578 46 5 v 2355 4608 a F4(\001)2431 4622 y F9(1)2470
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15472
4608 y Ga(;)g F4(\001)2586 4622 y F9(2)2626 4608 y Ga(;)g(c)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15473
F4(:)f Ga(B)5 b F6(^)o Ga(C)3109 4551 y F6(^)3170 4518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15474
y FX(0)3170 4578 y Gc(R)277 4844 y Gg(Note)26 b(that)g(in)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15475
F6(^)932 4812 y FX(0)932 4871 y Gc(R)990 4844 y Gg(-rule)g(the)g(conte)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15476
o(xt)h(are)f(treated)h(as)f(in)f(the)h(cut-rule;)j(that)d(means)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15477
F4(\000)3162 4858 y F9(1)3223 4844 y F6(\\)21 b F4(\000)3362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15478
4858 y F9(2)3426 4844 y Gg(or)277 4957 y F4(\001)353
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15479
4971 y F9(1)410 4957 y F6(\\)c F4(\001)564 4971 y F9(2)625
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15480
4957 y Gg(may)23 b(not)g(be)g(empty)-6 b(.)29 b(Thus)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15481
b F6(^)1586 4924 y FX(0)1586 4984 y Gc(R)1666 4957 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15482
Gg(is)g(actually)h(a)f(generalisation)k(of)c F6(^)2811
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15483
4971 y Gc(R)2869 4957 y Gg(.)28 b(Similar)22 b(remarks)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15484
5070 y(apply)j(to)e(the)h(other)h(inference)h(rules)e(with)f(tw)o(o)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15485
(premises)g(\(see)h(Figure)f(2.9\).)418 5201 y(Our)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15486
b(strong)i(normalisation)i(proof)d(can)g(be)f(easily)i(modi\002ed)f(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15487
f(accommodate)i(the)f(alternati)n(v)o(e)277 5314 y(formulations)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15488
b(of)c(the)h(inference)i(rules)e(mentioned)i(abo)o(v)o(e.)k(In)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15489
b(f)o(act,)h(it)f(is)h(rob)n(ust)h(enough)g(to)f(pro)o(v)o(e)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15490
5427 y(strong)34 b(normalisation)j(for)c(a)f(sequent)j(calculus)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15491
(containing)h(all)d(v)n(ariants)h(and)g(their)f(respecti)n(v)o(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15492
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15493
%%Page: 46 58
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15494
TeXDict begin 46 57 bop Black -144 51 a Gb(46)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15495
b(Calculi)p -144 88 3691 4 v Black Black -144 229 V -144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15496
1636 4 1407 v 168 370 a Ga(x)17 b F4(:)g Ga(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15497
b(y)20 b F4(:)e Ga(C)q(;)d F4(\()p Ga(z)22 b F4(:)c Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15498
b F6(^)o Ga(C)i F4(\))p Ga(;)15 b F4(\000)1112 358 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15499
Gc(.)1167 370 y Ga(M)1290 358 y Gc(.)1345 370 y F4(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15500
139 412 1312 4 v 139 503 a Ga(z)21 b F4(:)d Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15501
F6(^)o Ga(C)q(;)15 b F4(\000)568 491 y Gc(.)622 503 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15502
FL(And)777 466 y FX(0)777 526 y Gc(L)829 503 y F4(\()864
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15503
491 y F9(\()892 503 y Ga(x)944 491 y F9(\)\()999 503
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15504
y Ga(y)1047 491 y F9(\))1074 503 y Ga(M)10 b(;)15 b(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15505
F4(\))1319 491 y Gc(.)1374 503 y F4(\001)1492 434 y F6(^)1552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15506
401 y FX(0)1552 461 y Gc(L)1841 370 y F4(\000)1923 358
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15507
y Gc(.)1978 370 y Ga(M)2102 358 y Gc(.)2156 370 y F4(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15508
Ga(;)g F4(\()p Ga(c)k F4(:)e Ga(B)5 b F6(^)o Ga(C)i F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15509
Ga(;)15 b(a)j F4(:)f Ga(B)5 b(;)15 b(b)i F4(:)h Ga(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15510
1841 412 1241 4 v 1841 503 a F4(\000)1923 491 y Gc(.)1978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15511
503 y FL(Or)2077 466 y FX(0)2077 526 y Gc(R)2134 503
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15512
y F4(\()2169 491 y FX(h)2197 503 y Ga(a)2245 491 y FX(i)q(h)2300
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15513
503 y Ga(b)2339 491 y FX(i)2367 503 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15514
b(c)p F4(\))2605 491 y Gc(.)2660 503 y F4(\001)p Ga(;)g(c)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15515
F4(:)f Ga(B)5 b F6(^)o Ga(C)3122 434 y F6(_)3183 401
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15516
y FX(0)3183 461 y Gc(R)704 718 y F4(\000)761 732 y F9(1)826
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15517
706 y Gc(.)880 718 y Ga(M)1004 706 y Gc(.)1059 718 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15518
F4(\001)1135 732 y F9(1)1174 718 y Ga(;)15 b(a)j F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15519
Ga(B)459 b F4(\000)1907 732 y F9(2)1972 706 y Gc(.)2026
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15520
718 y Ga(N)2135 706 y Gc(.)2190 718 y F4(\001)2266 732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15521
y F9(2)2305 718 y Ga(;)15 b(b)i F4(:)h Ga(C)p 704 755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15522
1812 4 v 715 846 a F4(\000)772 860 y F9(1)811 846 y Ga(;)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15523
F4(\000)908 860 y F9(2)973 834 y Gc(.)1028 846 y FL(And)1182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15524
809 y FX(0)1182 869 y Gc(R)1240 846 y F4(\()1275 834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15525
y FX(h)1303 846 y Ga(a)1351 834 y FX(i)1378 846 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15526
b(;)1516 834 y FX(h)1544 846 y Ga(b)1583 834 y FX(i)1611
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15527
846 y Ga(N)g(;)15 b(c)p F4(\))1834 834 y Gc(.)1889 846
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15528
y F4(\001)1965 860 y F9(1)2004 846 y Ga(;)g F4(\001)2120
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15529
860 y F9(2)2160 846 y Ga(;)g(c)j F4(:)f Ga(B)5 b F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15530
Ga(C)2557 777 y F6(^)2617 744 y FX(0)2617 804 y Gc(R)794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15531
1060 y Ga(x)17 b F4(:)h Ga(B)5 b(;)15 b F4(\000)1077
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15532
1074 y F9(1)1141 1048 y Gc(.)1196 1060 y Ga(M)1319 1048
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15533
y Gc(.)1374 1060 y F4(\001)1450 1074 y F9(1)1762 1060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15534
y Ga(y)20 b F4(:)d Ga(C)q(;)e F4(\000)2032 1074 y F9(2)2098
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15535
1048 y Gc(.)2153 1060 y Ga(N)2261 1048 y Gc(.)2316 1060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15536
y F4(\001)2392 1074 y F9(2)p 737 1097 1751 4 v 737 1188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15537
a Ga(z)22 b F4(:)17 b Ga(B)5 b F6(_)o Ga(C)q(;)15 b F4(\000)1140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15538
1202 y F9(1)1180 1188 y Ga(;)g F4(\000)1277 1202 y F9(2)1342
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15539
1176 y Gc(.)1397 1188 y FL(Or)1496 1151 y FX(0)1496 1211
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15540
y Gc(L)1548 1188 y F4(\()1583 1176 y F9(\()1611 1188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15541
y Ga(x)1663 1176 y F9(\))1691 1188 y Ga(M)10 b(;)1829
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15542
1176 y F9(\()1857 1188 y Ga(y)1905 1176 y F9(\))1932
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15543
1188 y Ga(N)g(;)15 b(z)t F4(\))2162 1176 y Gc(.)2217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15544
1188 y F4(\001)2293 1202 y F9(1)2332 1188 y Ga(;)g F4(\001)2448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15545
1202 y F9(2)2529 1119 y F6(_)2590 1086 y FX(0)2590 1146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15546
y Gc(L)698 1402 y F4(\000)755 1416 y F9(1)819 1390 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15547
Gc(.)874 1402 y Ga(M)998 1390 y Gc(.)1053 1402 y F4(\001)1129
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15548
1416 y F9(1)1168 1402 y Ga(;)g(a)i F4(:)h Ga(B)459 b(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15549
b F4(:)g Ga(C)q(;)e F4(\000)2118 1416 y F9(2)2184 1390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15550
y Gc(.)2239 1402 y Ga(N)2347 1390 y Gc(.)2402 1402 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15551
F4(\001)2478 1416 y F9(2)p 698 1440 1820 4 v 703 1530
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15552
a Ga(y)20 b F4(:)d Ga(B)5 b F6(\033)o Ga(C)q(;)15 b F4(\000)1117
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15553
1544 y F9(1)1158 1530 y Ga(;)g F4(\000)1255 1544 y F9(2)1319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15554
1518 y Gc(.)1374 1530 y FL(Imp)1519 1493 y FX(0)1519
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15555
1553 y Gc(L)1571 1530 y F4(\()1606 1518 y FX(h)1634 1530
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15556
y Ga(a)1682 1518 y FX(i)1709 1530 y Ga(M)10 b(;)1847
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15557
1518 y F9(\()1875 1530 y Ga(x)1927 1518 y F9(\))1955
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15558
1530 y Ga(N)g(;)15 b(y)s F4(\))2186 1518 y Gc(.)2241
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15559
1530 y F4(\001)2317 1544 y F9(1)2356 1530 y Ga(;)g F4(\001)2472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15560
1544 y F9(2)2558 1461 y F6(\033)2629 1428 y FX(0)2629
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15561
1488 y Gc(L)p 3543 1636 4 1407 v -144 1639 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15562
843 1793 a Gg(Figure)25 b(2.9:)k(Alternati)n(v)o(e)c(formulation)h(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15563
(some)h(inference)i(rules.)p Black Black 321 2219 a(cut-reductions.)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15564
b(Since)21 b(the)h(proof)h(does)f(not)g(pose)h(an)o(y)e(ne)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15565
(challenges,)k(the)d(details)h(are)f(omitted.)321 2332
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15566
y(In)28 b(what)f(follo)n(ws)i(we)d(shall)j(emplo)o(y)g(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15567
(formulation)j(of)e(the)g(inference)i(rules)e(that)h(seems)f(most)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15568
2445 y(suited)d(for)f(the)g(problem)g(at)g(hand.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15569
2579 y(The)h(last)g(v)n(ariation)i(for)d(classical)j(logic)f(we)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15570
(shall)h(present)i(arises)e(from)g(the)g(w)o(onderful)i(sym-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15571
2691 y(metry)i(of)f(classical)i(logic.)44 b(Ev)o(ery)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15572
b(sequent)i(of)e(the)h(form)f F4(\000)p 2317 2679 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15573
41 v 2327 2661 46 5 v 96 w(\001)f Gg(can)i(be)f(replaced)i(by)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15574
3237 2679 11 41 v 3247 2661 46 5 v 125 w F6(:)p F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15575
Ga(;)15 b F4(\001)321 2804 y Gg(where)21 b F6(:)p F4(\000)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15576
Gg(is)h(tak)o(en)h(to)e(mean)h(that)h(e)n(v)o(ery)f(formula)h(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15577
F4(\000)e Gg(is)i(ne)o(gated.)29 b(The)20 b(corresponding)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15578
b(sequent)321 2917 y(calculus)38 b(is)d(commonly)i(referred)g(to)f(as)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15579
(single-sided)k(sequent)e(calculus)h(or)d(Gentzen-Sch)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15580
b(\250)-38 b(utte)321 3030 y(system)33 b([Sch)8 b(\250)-38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15581
b(utte,)33 b(1960].)55 b(The)31 b(cut-rule,)36 b(for)c(e)o(xample,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15582
(is)d(in)f(this)i(calculus)h(of)e(the)g(follo)n(wing)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15583
3143 y(form.)p 1412 3327 11 41 v 1423 3309 46 5 v 1488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15584
3339 a F4(\000)p Ga(;)15 b(a)j F4(:)f Ga(B)p 1878 3327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15585
11 41 v 1888 3309 46 5 v 192 w F4(\001)p Ga(;)e(b)i F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15586
F6(:)p Ga(B)p 1392 3377 911 4 v 1733 3444 11 41 v 1743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15587
3426 46 5 v 1809 3456 a F4(\000)p Ga(;)d F4(\001)2344
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15588
3407 y Gg(Cut)321 3692 y(It)30 b(is)f(easy)i(to)f(adapt)h(De\002nition)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15589
f(2.3.4)g(\(the)h(symmetric)f(reducibility)k(candidates\))f(for)d(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15590
(single-)321 3805 y(sided)21 b(sequent)h(calculus,)g(and)e(the)g(proof)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15591
h(of)e(strong)i(normalisation)i(for)d(cut-elimination)j(proceeds)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15592
3918 y(just)h(as)g(the)g(proof)g(gi)n(v)o(en)g(for)1308
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15593
3881 y Gc(cut)1277 3918 y F6(\000)-31 b(\000)f(!)p Gg(.)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15594
b(W)-7 b(e)23 b(shall)h(omit)g(the)g(details.)462 4052
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15595
y(T)-7 b(o)18 b(conclude)k(this)d(section,)i(we)d(shall)i(introduce)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15596
(term)c(annotations)23 b(for)c(intuitionistic)k(sequent)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15597
4165 y(proofs)j(and)f(pro)o(v)o(e)g(a)e(corresponding)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15598
b(strong)d(normalisation)h(theorem.)32 b(F)o(or)24 b(simplicity)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15599
b(,)26 b(we)e(shall)321 4278 y(treat)29 b(only)h(the)e(propositional)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15600
(connecti)n(v)o(es,)g F6(^)p Gg(,)d F6(_)e Gg(and)h F6(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15601
Gg(,)h(and)f(not)h(ne)o(gation)h(and)f(the)f(units,)j(as)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15602
4390 y(the)o(y)24 b(add)g(no)g(substantial)i(ne)n(w)d(issues)i(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15603
(strong)h(normalisation.)462 4524 y(The)g(sequent)i(calculus)g(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15604
(intuitionistic)j(logic)d(can)f(be)g(obtained)j(by)d(restricting)j(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15605
d(succe-)321 4637 y(dent)j(of)f(our)g(classical)j(sequents)f(to)e(at)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15606
(most)g(a)f(single)j(formula.)40 b(So)26 b(our)h(intuitionistic)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15607
(sequents)321 4750 y(are)24 b(of)g(the)g(form)f F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15608
964 4738 11 41 v 975 4719 46 5 v 96 w Ga(a)17 b F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15609
Ga(B)t Gg(.)29 b(W)-7 b(e)22 b(could)j(drop)g(the)f(labels)g(on)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15610
(right-hand)j(side)d(of)f(the)h(turnstile)321 4863 y(when)f(assigning)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15611
(terms)f(to)f(intuitionistic)k(sequents,)e(because)h(on)d(this)h(side)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15612
(contraction)k(is)c(not)g(al-)321 4976 y(lo)n(wed.)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15613
b(Ho)n(we)n(v)o(er)l(,)25 b(we)f(shall)i(k)o(eep)f(this)h(slight)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15614
(incon)l(v)o(enience)j(of)c(ha)n(ving)h(to)f(label)h(this)f(formula:)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15615
321 5088 y(it)i(sa)n(v)o(es)h(us)f(from)g(rede\002ning)i(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15616
(substitution)k(operation)e(and)f(cut-reductions.)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15617
b(Thus)27 b(our)h(intu-)321 5201 y(itionistic)33 b(typing)e(judgements)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15618
h(are)e(of)g(the)g(form)f F4(\000)2050 5189 y Gc(.)2105
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15619
5201 y Ga(M)2229 5189 y Gc(.)2284 5201 y Ga(a)17 b F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15620
Ga(B)5 b Gg(,)30 b(in)g(which)g Ga(M)39 b Gg(is)29 b(a)h(term)f(tak)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15621
(en)321 5314 y(from)f(the)h(set)f(of)g(well-typed)j(terms,)e(denoted)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15622
(by)e FY(I)g Gg(and)h(inducti)n(v)o(ely)h(de\002ned)f(by)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15623
(inference)321 5427 y(rules)d(gi)n(v)o(en)f(in)f(Figure)h(2.10.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15624
b(The)23 b(intuitionistic)28 b(cut-elimination)f(procedure)f(is)e(as)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15625
(follo)n(ws.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15626
%%Page: 47 59
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15627
TeXDict begin 47 58 bop Black 277 51 a Gb(2.5)23 b(V)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15628
b(ariations)25 b(of)e(the)g(Sequent)f(Calculus)2124 b(47)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15629
277 88 3691 4 v Black Black 277 229 V 277 1881 4 1652
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15630
v 1642 321 938 4 v 1642 406 a Ga(x)18 b F4(:)f Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15631
b(;)15 b F4(\000)1950 394 y Gc(.)2005 406 y FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15632
Ga(x;)g(a)p F4(\))2344 394 y Gc(.)2399 406 y Ga(a)i F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15633
Ga(B)859 620 y(x)g F4(:)g Ga(B)1039 634 y Gc(i)1067 620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15634
y Ga(;)e F4(\000)1190 608 y Gc(.)1245 620 y Ga(M)1368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15635
608 y Gc(.)1423 620 y Ga(a)i F4(:)h Ga(C)p 534 658 1393
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15636
4 v 534 755 a(y)i F4(:)e Ga(B)711 769 y F9(1)750 755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15637
y F6(^)p Ga(B)880 769 y F9(2)919 755 y Ga(;)d F4(\000)1042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15638
743 y Gc(.)1097 755 y FL(And)1251 718 y Gc(i)1251 778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15639
y(L)1303 755 y F4(\()1338 743 y F9(\()1366 755 y Ga(x)1418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15640
743 y F9(\))1446 755 y Ga(M)10 b(;)15 b(y)s F4(\))1693
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15641
743 y Gc(.)1747 755 y Ga(a)j F4(:)f Ga(C)1968 671 y F6(^)2029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15642
685 y Gc(L)2077 695 y FZ(i)2397 651 y F4(\000)2479 639
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15643
y Gc(.)2534 651 y Ga(M)2657 639 y Gc(.)2712 651 y Ga(a)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15644
F4(:)f Ga(B)186 b F4(\000)3157 639 y Gc(.)3212 651 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15645
Ga(N)3320 639 y Gc(.)3375 651 y Ga(b)18 b F4(:)f Ga(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15646
2320 671 1303 4 v 2320 755 a F4(\000)2402 743 y Gc(.)2457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15647
755 y FL(And)2611 769 y Gc(R)2669 755 y F4(\()2704 743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15648
y FX(h)2732 755 y Ga(a)2780 743 y FX(i)2807 755 y Ga(M)11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15649
b(;)2946 743 y FX(h)2973 755 y Ga(b)3012 743 y FX(i)3040
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15650
755 y Ga(N)f(;)15 b(c)p F4(\))3263 743 y Gc(.)3318 755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15651
y Ga(c)j F4(:)f Ga(B)5 b F6(^)o Ga(C)3664 689 y F6(^)3725
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15652
703 y Gc(R)440 978 y Ga(x)17 b F4(:)g Ga(B)5 b(;)15 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15653
F4(\000)747 966 y Gc(.)802 978 y Ga(M)926 966 y Gc(.)980
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15654
978 y Ga(a)j F4(:)f Ga(D)185 b(y)20 b F4(:)d Ga(C)q(;)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15655
F4(\000)1644 966 y Gc(.)1699 978 y Ga(N)1807 966 y Gc(.)1862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15656
978 y Ga(a)i F4(:)h Ga(D)p 440 1016 1608 4 v 499 1100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15657
a(z)k F4(:)17 b Ga(B)5 b F6(_)o Ga(C)q(;)15 b F4(\000)928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15658
1088 y Gc(.)983 1100 y FL(Or)1082 1114 y Gc(L)1134 1100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15659
y F4(\()1169 1088 y F9(\()1197 1100 y Ga(x)1249 1088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15660
y F9(\))1276 1100 y Ga(M)c(;)1415 1088 y F9(\()1442 1100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15661
y Ga(y)1490 1088 y F9(\))1518 1100 y Ga(N)f(;)15 b(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15662
F4(\))1748 1088 y Gc(.)1803 1100 y Ga(a)i F4(:)h Ga(D)2089
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15663
1034 y F6(_)2150 1048 y Gc(L)2698 969 y F4(\000)2780
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15664
957 y Gc(.)2835 969 y Ga(M)2959 957 y Gc(.)3014 969 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15665
Ga(a)f F4(:)g Ga(B)3190 983 y Gc(i)p 2408 1003 1102 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15666
v 2408 1100 a F4(\000)2490 1088 y Gc(.)2545 1100 y FL(Or)2644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15667
1063 y Gc(i)2644 1123 y(R)2702 1100 y F4(\()2737 1088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15668
y FX(h)2764 1100 y Ga(a)2812 1088 y FX(i)2840 1100 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15669
Ga(M)10 b(;)15 b(b)p F4(\))3078 1088 y Gc(.)3133 1100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15670
y Ga(b)i F4(:)h Ga(B)3301 1114 y F9(1)3340 1100 y F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15671
Ga(B)3470 1114 y F9(2)3550 1016 y F6(_)3611 1030 y Gc(R)3664
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15672
1040 y FZ(i)552 1315 y F4(\000)634 1303 y Gc(.)689 1315
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15673
y Ga(M)812 1303 y Gc(.)867 1315 y Ga(a)g F4(:)f Ga(B)186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15674
b(x)17 b F4(:)h Ga(C)q(;)d F4(\000)1531 1303 y Gc(.)1586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15675
1315 y Ga(N)1694 1303 y Gc(.)1749 1315 y Ga(b)i F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15676
Ga(D)p 469 1353 1539 4 v 469 1438 a(y)j F4(:)e Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15677
b F6(\033)o Ga(C)q(;)15 b F4(\000)910 1426 y Gc(.)965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15678
1438 y FL(Imp)1109 1460 y Gc(L)1161 1438 y F4(\()1196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15679
1426 y FX(h)1224 1438 y Ga(a)1272 1426 y FX(i)1300 1438
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15680
y Ga(M)10 b(;)1438 1426 y F9(\()1466 1438 y Ga(x)1518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15681
1426 y F9(\))1545 1438 y Ga(N)g(;)15 b(y)s F4(\))1777
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15682
1426 y Gc(.)1832 1438 y Ga(b)i F4(:)g Ga(D)2050 1371
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15683
y F6(\033)2120 1385 y Gc(L)2606 1315 y Ga(x)g F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15684
Ga(B)5 b(;)15 b F4(\000)2914 1303 y Gc(.)2969 1315 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15685
Ga(M)3092 1303 y Gc(.)3147 1315 y Ga(a)i F4(:)h Ga(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15686
2370 1353 1192 4 v 2370 1438 a F4(\000)2453 1426 y Gc(.)2507
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15687
1438 y FL(Imp)2652 1460 y Gc(R)2710 1438 y F4(\()2745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15688
1426 y F9(\()2772 1438 y Ga(x)2824 1426 y F9(\))q FX(h)2879
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15689
1438 y Ga(a)2927 1426 y FX(i)2955 1438 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15690
b(b)p F4(\))3193 1426 y Gc(.)3248 1438 y Ga(b)i F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15691
Ga(B)5 b F6(\033)p Ga(C)3604 1371 y F6(\033)3674 1385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15692
y Gc(R)1319 1653 y F4(\000)1376 1667 y F9(1)1440 1641
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15693
y Gc(.)1495 1653 y Ga(M)1618 1641 y Gc(.)1673 1653 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15694
Ga(a)17 b F4(:)h Ga(B)146 b(x)17 b F4(:)g Ga(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15695
b F4(\000)2278 1667 y F9(2)2343 1641 y Gc(.)2398 1653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15696
y Ga(N)2506 1641 y Gc(.)2561 1653 y Ga(b)i F4(:)g Ga(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15697
1319 1691 1413 4 v 1423 1775 a F4(\000)1480 1789 y F9(1)1519
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15698
1775 y Ga(;)e F4(\000)1616 1789 y F9(2)1681 1763 y Gc(.)1736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15699
1775 y FL(Cut)p F4(\()1909 1763 y FX(h)1937 1775 y Ga(a)1985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15700
1763 y FX(i)2012 1775 y Ga(M)10 b(;)2150 1763 y F9(\()2178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15701
1775 y Ga(x)2230 1763 y F9(\))2258 1775 y Ga(N)g F4(\))2401
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15702
1763 y Gc(.)2456 1775 y Ga(b)17 b F4(:)h Ga(C)2772 1721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15703
y Gg(Cut)p 3965 1881 4 1652 v 277 1884 3691 4 v 277 2038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15704
a(Figure)30 b(2.10:)40 b(T)-6 b(erm)28 b(assignment)k(for)d(sequent)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15705
(proofs)f(in)f(the)h(propositional)j(fragment)d(of)f(intu-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15706
2151 y(itionistic)d(logic.)p Black Black 277 2571 a Gb(De\002nition)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15707
(2.5.1)g Gg(\(Cut-Elimination)j(for)e(Intuitionistic)j(Logic\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15708
Gb(:)p Black 277 2684 a Gg(The)37 b(cut-elimination)k(procedure)e(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15709
(intuitionistic)j(logic,)g F4(\()p FY(I)p Ga(;)2467 2647
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15710
y Gc(int)2428 2684 y F6(\000)-31 b(\000)g(!)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15711
Gg(,)39 b(is)e(a)g(reduction)j(system)277 2797 y(where:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15712
Black 414 3004 a F6(\017)p Black 45 w FY(I)23 b Gg(is)h(the)f(set)h(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15713
g(well-typed)h(intuitionistic)j(terms,)23 b(and)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15714
414 3195 a F6(\017)p Black 543 3158 a Gc(int)504 3195
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15715
y F6(\000)-31 b(\000)g(!)22 b Gg(consists)k(of)d(the)h(reductions)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15716
(for)c(logical)j(cuts)e(and)g(commuting)h(cuts;)f(that)g(is)1626
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15717
3386 y Gc(int)1587 3423 y F6(\000)-31 b(\000)f(!)1782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15718
3371 y F5(def)1789 3423 y F4(=)1967 3386 y Gc(l)1892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15719
3423 y F6(\000)h(\000)g(!)25 b([)2244 3386 y Gc(c)2174
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15720
3423 y F6(\000)-31 b(\000)f(!)50 b Ga(:)277 3677 y Gg(Gi)n(v)o(en)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15721
b(that)h(intuitionistic)k(logic,)d(as)e(we)g(de\002ned)h(it,)h(is)e(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15722
(restricted)j(v)o(ersion)f(of)e(classical)j(logic,)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15723
3790 y(we)23 b(e)o(xpect)h(that)h F4(\()p FY(I)p Ga(;)977
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15724
3753 y Gc(int)939 3790 y F6(\000)-32 b(\000)h(!)p F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15725
b Gg(is)g(strongly)j(normalising.)31 b(This)24 b(is)f(indeed)i(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15726
(case.)p Black 277 4004 a Gb(Theor)n(em)g(2.5.2:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15727
34 w Gg(The)f(reduction)j(system)e F4(\()p FY(I)p Ga(;)1856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15728
3967 y Gc(int)1817 4004 y F6(\000)-31 b(\000)g(!)p F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15729
b Gg(is)g(strongly)j(normalising.)p Black 277 4216 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15730
F7(Pr)l(oof)o(.)p Black 34 w Gg(Assume)31 b(there)h(is)f(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15731
(well-typed)j(intuitionistic)h(term,)d(say)g Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15732
b Gg(,)32 b(from)g(which)f(an)h(in\002nite)277 4329 y(reduction)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15733
(sequence)f(starts.)38 b(Since)26 b(we)f(kno)n(w)h(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15734
F4(\000)2082 4317 y Gc(.)2137 4329 y Ga(M)2260 4317 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15735
Gc(.)2315 4329 y Ga(a)18 b F4(:)f Ga(B)30 b Gg(is)c(deri)n(v)n(able)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15736
(using)f(the)f(rules)277 4442 y(gi)n(v)o(en)36 b(in)f(Figure)h(2.10,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15737
(we)34 b(can)i(translate)i(this)e(deri)n(v)n(ation)i(into)e(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15738
(classical)j(system)e(sho)n(wn)277 4555 y(in)d(Figure)h(2.3)g(\(only)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15739
(the)g F6(\033)1236 4569 y Gc(L)1288 4555 y Gg(-rule)g(requires)h(some)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15740
f(changes\).)61 b(It)33 b(is)g(not)h(hard)g(to)f(see)h(that)g(the)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15741
4668 y(in\002nite)29 b(reduction)h(sequence)g(starting)f(from)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15742
Ga(M)37 b Gg(w)o(ould)28 b(entail)h(an)e(in\002nite)i(reduction)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15743
(sequence)277 4781 y(in)f F4(\()p FY(T)t Ga(;)540 4744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15744
y Gc(cut)509 4781 y F6(\000)-31 b(\000)g(!)o F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15745
Gg(.)46 b(Since)30 b(we)f(pro)o(v)o(ed)h(that)g F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15746
FY(T)t Ga(;)1755 4744 y Gc(cut)1725 4781 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15747
b(\000)h(!)p F4(\))29 b Gg(is)g(strongly)j(normalising,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15748
F4(\()p FY(I)p Ga(;)3016 4744 y Gc(int)2977 4781 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15749
b(\000)h(!)p F4(\))29 b Gg(must)g(be)277 4894 y(strongly)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15750
(normalising,)g(too.)p 3436 4894 4 62 v 3440 4836 55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15751
4 v 3440 4894 V 3494 4894 4 62 v 277 5201 a(W)-7 b(e)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15752
b(shall)h(use)g(this)g(result)h(in)f(Chapter)g(3)f(in)h(order)g(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15753
(pro)o(v)o(e)g(strong)h(normalisation)i(for)c(the)h(simply-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15754
5314 y(typed)k(lambda)f(calculus.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15755
%%Page: 48 60
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15756
TeXDict begin 48 59 bop Black -144 51 a Gb(48)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15757
b(Calculi)p -144 88 3691 4 v Black 321 317 a Ge(2.6)119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15758
b(Localised)30 b(V)-12 b(ersion)321 541 y Gg(In)29 b(the)g(pre)n(vious)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15759
h(section)g(we)e(claimed)i(that)f(our)g(strong)h(normalisation)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15760
(result)f(e)o(xtends)g(to)e(a)h(v)n(a-)321 654 y(riety)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15761
b(of)e(formulations)j(of)e(the)f(inference)j(rules.)62
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15762
b(In)35 b(this)g(section)h(we)d(shall)j(sho)n(w)e(that)h(strong)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15763
767 y(normalisation)g(can)e(also)f(be)g(obtained)j(with)d(a)f(dif)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15764
(ferent)j(set)e(of)g(cut-reduction)k(rules.)55 b(T)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15765
b(o)31 b(sa)n(v)o(e)321 880 y(space,)24 b(we)d(shall)j(restrict)g(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15766
(presentation)k(in)d(this)g(section)h(to)e(only)h(the)g(propositional)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15767
(fragment,)321 993 y(b)n(ut)33 b(it)f(should)i(be)f(emphasised)h(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15768
(the)g(corresponding)k(result)c(for)g(\002rst-order)h(classical)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15769
(logic)321 1105 y(may)24 b(be)f(obtained)j(by)e(a)f(simple)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15770
(adaptation)i(of)e(the)g(proof)g(we)f(shall)h(gi)n(v)o(e.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15771
1235 y(The)d(cut-elimination)26 b(procedures)e(de\002ned)e(so)g(f)o(ar)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15772
f(include)i(global,)g(or)f(non-local,)i(cut-reduc-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15773
1348 y(tions.)32 b(A)23 b(cut-reduction)28 b(is)c(said)h(to)f(be)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15774
F7(local)p Gg(,)h(if)f(only)h(neighbouring)j(inference)f(rules)e(are)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15775
(re)n(writ-)321 1461 y(ten,)30 b(possibly)g(duplicating)i(a)c(subderi)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15776
(v)n(ation.)46 b(Most)28 b(of)g(the)h(traditional)i(cut-elimination)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15777
(proce-)321 1574 y(dures)c(based)h(on)e(Gentzen')-5 b(s)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15778
b F7(Hauptsatz)g Gg(ha)n(v)o(e)e(such)h(local)g(cut-reductions.)44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15779
b(One)26 b(reason)j(for)e(our)321 1687 y(use)h(of)e(the)i(non-local)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15780
(cut-reductions)i(w)o(as)c(to)g(a)n(v)n(oid)h(the)g(in\002nite)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15781
(reduction)j(gi)n(v)o(en)d(in)g(Example)321 1799 y(2.1.3.)h(Using)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15782
b(results)h(concerning)h(e)o(xplicit)g(substitution)h(calculi,)e(we)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15783
(shall)h(sho)n(w)g(that)g(our)g(substi-)321 1912 y(tution)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15784
b(operation)i(can,)d(in)f(f)o(act,)i(be)e(replaced)j(by)e(local)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15785
(cut-reductions)k(without)d(breaking)h(strong)321 2025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15786
y(normalisation.)36 b(The)25 b(local)g(cut-reductions)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15787
b(will)24 b(be)h(de\002ned)h(such)f(that)h(the)o(y)f(bridge)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15788
(gap)g(be-)321 2138 y(tween)f(our)g(global)h(cut-reductions)j(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15789
(en)c(earlier)h(and)f(the)g(Gentzen-lik)o(e)i(local)e(cut-reductions.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15790
462 2268 y(Explicit)k(substitution)i(calculi)e(ha)n(v)o(e)f(been)h(de)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15791
(v)o(eloped)g(in)f(order)g(to)g(internalise)i(the)e(substitu-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15792
2381 y(tion)g(operation,)i(a)c(meta-le)n(v)o(el)i(operation)i(on)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15793
(lambda)g(terms,)h(arising)g(from)f(beta-reductions.)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15794
b(In)321 2493 y(these)26 b(calculi)f(the)g(reductions)i(are)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15795
(completely)h(primiti)n(v)o(e)f(in)f(the)h(sense)g(that)g(the)o(y)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15796
(are)f(part)h(of)f(the)321 2606 y(calculus.)45 b(Note,)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15797
b(ho)n(we)n(v)o(er)l(,)h(if)d(the)i(\(non-local\))i(substitution)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15798
(operation)f(is)d(replaced)i(by)e(a)g(na)m(\250)-27 b(\021v)o(e)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15799
2719 y(de\002nition)28 b(of)d(a)g(\(local\))j(e)o(xplicit)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15800
(substitution)i(operator)l(,)f(the)e(resulting)i(calculus)g(may)d(no)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15801
(longer)321 2832 y(be)f(strongly)i(normalising,)h(as)d(sho)n(wn)g(by)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15802
(Melli)5 b(\036)-35 b(es)26 b([1995].)35 b(Ne)n(v)o(ertheless,)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15803
b(it)e(is)f(possible)k(to)c(gi)n(v)o(e)321 2945 y(an)g(e)o(xplicit)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15804
(substitution)i(operation)f(that)e(preserv)o(es)i(strong)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15805
(normalisation.)462 3075 y(In)30 b Ga(\025)p F1(x)p Gg(,)h(for)f(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15806
(xample,)i(the)e(beta-reduction)35 b F4(\()p Ga(\025x:M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15807
b F4(\))p Ga(N)2401 3037 y Gc(\014)2338 3075 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15808
b(\000)f(!)37 b Ga(M)10 b F4([)p Ga(x)38 b F4(:=)f Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15809
b F4(])29 b Gg(is)h(replaced)i(by)321 3187 y(the)g(reduction)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15810
F4(\()p Ga(\025x:M)10 b F4(\))p Ga(N)1331 3150 y Gc(b)1261
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15811
3187 y F6(\000)-32 b(\000)h(!)41 b Ga(M)10 b F6(h)p Ga(x)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15812
b F4(:=)f Ga(N)10 b F6(i)32 b Gg(where)g(the)g(reduct)h(contains)h(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15813
(ne)n(w)f(syntactic)321 3300 y(constructor)-5 b(.)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15814
b(The)23 b(follo)n(wing)i(reduction)h(rules)f(apply)g(to)e(this)h(term)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15815
f(constructor)-5 b(.)p Black Black 1236 3474 a Ga(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15816
F6(h)p Ga(x)26 b F4(:=)f Ga(P)13 b F6(i)1793 3437 y Gc(x)1728
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15817
3474 y F6(\000)-31 b(\000)f(!)100 b Ga(P)1240 3587 y(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15818
F6(h)p Ga(x)26 b F4(:=)f Ga(P)13 b F6(i)1793 3550 y Gc(x)1728
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15819
3587 y F6(\000)-31 b(\000)f(!)100 b Ga(y)993 3700 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15820
Ga(\025)q(y)s(:M)10 b F4(\))p F6(h)p Ga(x)26 b F4(:=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15821
Ga(P)13 b F6(i)1793 3662 y Gc(x)1728 3700 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15822
b(\000)f(!)100 b Ga(\025y)s(:)15 b(M)10 b F6(h)p Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15823
b F4(:=)f Ga(P)13 b F6(i)1036 3812 y F4(\()p Ga(M)d(N)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15824
F4(\))p F6(h)p Ga(x)27 b F4(:=)e Ga(P)13 b F6(i)1793
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15825
3775 y Gc(x)1728 3812 y F6(\000)-31 b(\000)f(!)100 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15826
Ga(M)10 b F6(h)p Ga(x)26 b F4(:=)f Ga(P)13 b F6(i)i Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15827
b F6(h)p Ga(x)26 b F4(:=)f Ga(P)13 b F6(i)321 3984 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15828
Gg(The)33 b(reader)i(is)e(referred)i(to)e([Rose,)g(1996])i(for)f(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15829
(detailed)k(study)e(of)f Ga(\025)p F1(x)f Gg(including)k(a)d(proof)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15830
(of)321 4097 y(strong)26 b(normalisation.)33 b(Here,)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15831
b(we)f(shall)i(sho)n(w)f(that)g(a)g(similar)g(feat)h(is)f(possible)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15832
(for)e(our)g(term)g(cal-)321 4210 y(culus,)29 b(where)f(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15833
(substitution)k(operation)f(realises)f(the)e(proof)h(transformations)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15834
(corresponding)321 4323 y(to)24 b(the)g(elimination)h(of)f(commuting)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15835
(cuts.)462 4452 y(W)-7 b(e)27 b(be)o(gin)h(by)f(adding)i(to)e(our)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15836
(term)g(calculus)i(tw)o(o)e(ne)n(w)f(constructors)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15837
b(which)c(we)g(refer)h(to)f(as)321 4565 y F7(labelled)f(cuts)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15838
Gg(.)j(The)o(y)23 b(are)h(written:)p Black Black 998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15839
4744 a FL(Cut)1077 4697 y FC( )1138 4744 y F4(\()1173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15840
4732 y FX(h)1200 4744 y Ga(a)1248 4732 y FX(i)1276 4744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15841
y Ga(M)10 b(;)1414 4732 y F9(\()1442 4744 y Ga(x)1494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15842
4732 y F9(\))1521 4744 y Ga(N)g F4(\))205 b Gg(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15843
FL(Cut)2259 4697 y FC(!)2320 4744 y F4(\()2355 4732 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15844
FX(h)2383 4744 y Ga(a)2431 4732 y FX(i)2458 4744 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15845
b(;)2596 4732 y F9(\()2624 4744 y Ga(x)2676 4732 y F9(\))2703
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15846
4744 y Ga(N)g F4(\))26 b Gg(.)321 4924 y(Consequently)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15847
b(,)27 b(we)c(ha)n(v)o(e)h(tw)o(o)f(ne)n(w)g(rules)h(for)g(forming)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15848
(typing)g(judgements.)322 5089 y F4(\000)379 5103 y F9(1)443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15849
5077 y Gc(.)498 5089 y Ga(M)622 5077 y Gc(.)677 5089
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15850
y F4(\001)753 5103 y F9(1)792 5089 y Ga(;)15 b(a)i F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15851
Ga(B)65 b(x)17 b F4(:)h Ga(B)5 b(;)15 b F4(\000)1357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15852
5103 y F9(2)1421 5077 y Gc(.)1476 5089 y Ga(N)1584 5077
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15853
y Gc(.)1639 5089 y F4(\001)1715 5103 y F9(2)p 322 5126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15854
1433 4 v 385 5211 a F4(\000)442 5225 y F9(1)482 5211
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15855
y Ga(;)g F4(\000)579 5225 y F9(2)643 5199 y Gc(.)698
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15856
5211 y FL(Cut)777 5164 y FC( )838 5211 y F4(\()873 5199
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15857
y FX(h)901 5211 y Ga(a)949 5199 y FX(i)976 5211 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15858
b(;)1114 5199 y F9(\()1142 5211 y Ga(x)1194 5199 y F9(\))1222
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15859
5211 y Ga(N)g F4(\))1365 5199 y Gc(.)1420 5211 y F4(\001)1496
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15860
5225 y F9(1)1535 5211 y Ga(;)15 b F4(\001)1651 5225 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15861
F9(2)1781 5159 y Gg(Cut)1853 5112 y FC( )1954 5089 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15862
F4(\000)2011 5103 y F9(1)2075 5077 y Gc(.)2130 5089 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15863
Ga(M)2254 5077 y Gc(.)2309 5089 y F4(\001)2385 5103 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15864
F9(1)2424 5089 y Ga(;)g(a)i F4(:)h Ga(B)65 b(x)17 b F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15865
Ga(B)5 b(;)15 b F4(\000)2989 5103 y F9(2)3053 5077 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15866
Gc(.)3108 5089 y Ga(N)3216 5077 y Gc(.)3271 5089 y F4(\001)3347
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15867
5103 y F9(2)p 1954 5126 V 2017 5211 a F4(\000)2074 5225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15868
y F9(1)2113 5211 y Ga(;)g F4(\000)2210 5225 y F9(2)2275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15869
5199 y Gc(.)2330 5211 y FL(Cut)2409 5164 y FC(!)2470
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15870
5211 y F4(\()2505 5199 y FX(h)2533 5211 y Ga(a)2581 5199
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15871
y FX(i)2608 5211 y Ga(M)10 b(;)2746 5199 y F9(\()2774
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15872
5211 y Ga(x)2826 5199 y F9(\))2854 5211 y Ga(N)g F4(\))2997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15873
5199 y Gc(.)3052 5211 y F4(\001)3128 5225 y F9(1)3167
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15874
5211 y Ga(;)15 b F4(\001)3283 5225 y F9(2)3412 5159 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15875
Gg(Cut)3484 5112 y FC(!)p Black 3372 5315 a Gg(\(2.5\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15876
Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15877
%%Page: 49 61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15878
TeXDict begin 49 60 bop Black 277 51 a Gb(2.6)23 b(Localised)i(V)-9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15879
b(ersion)2779 b(49)p 277 88 3691 4 v Black 277 317 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15880
Gg(Con)l(v)o(ention)39 b(2.2.4)e(concerning)j(the)d(Barendre)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15881
(gt-style)j(naming)e(con)l(v)o(ention)i(and)d(alpha-equi-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15882
430 y(v)n(alence)26 b(e)o(xtends)f(to)f(the)g(ne)n(w)f(terms)h(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15883
(the)g(ob)o(vious)i(w)o(ay)-6 b(.)30 b(Henceforth,)c(we)d(shall)h(only)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15884
h(be)f(inter)n(-)277 543 y(ested)f(in)e(well-typed)i(terms;)g(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15885
(means)f(terms)h Ga(M)31 b Gg(for)21 b(which)h(there)g(are)g(conte)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15886
(xts)h F4(\000)d Gg(and)i F4(\001)f Gg(such)277 656 y(that)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15887
F4(\000)517 644 y Gc(.)572 656 y Ga(M)695 644 y Gc(.)750
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15888
656 y F4(\001)e Gg(is)h(deri)n(v)n(able)i(gi)n(v)o(en)f(the)f(rules)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15889
(in)f(\(2.5\))h(and)f(Figure)h(2.3.)28 b(The)22 b(corresponding)k(set)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15890
277 769 y(of)e(well-typed)h(terms)f(is)f(de\002ned)i(as)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15891
Black Black 428 970 a FY(T)489 936 y FX($)659 918 y F5(def)666
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15892
969 y F4(=)844 868 y FK(n)929 970 y Ga(M)1075 865 y FK(\014)1075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15893
919 y(\014)1075 974 y(\014)1152 970 y Ga(M)33 b Gg(is)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15894
b(well-typed)h(by)f(the)g(rules)g(sho)n(wn)g(in)f(\(2.5\))h(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15895
(Figure)g(2.3)3290 868 y FK(o)277 1167 y Gg(W)l(ith)i(respect)h(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15896
(the)g(terms)h(gi)n(v)o(en)f(earlier)l(,)i(clearly)g(we)e(ha)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15897
FY(T)k F6(\032)f FY(T)2558 1134 y FX($)2629 1167 y Gg(.)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15898
b(In)25 b(the)g(sequel,)i(we)d(shall)277 1280 y(refer)g(to)g(a)f(term)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15899
(whose)h(outermost)i(term)d(constructor)k(is)c(a)g(labelled)j(cut)e(as)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15900
f(a)g F7(labelled)j(term)p Gg(.)418 1409 y(Ne)o(xt,)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15901
b(we)e(replace)k(the)d(non-local)j(reductions)h(of)2052
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15902
1372 y Gc(aux)2032 1409 y F6(\000)-31 b(\000)g(!)19 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15903
Gg(with)h(local)i(ones.)28 b(The)20 b(only)i(reduction)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15904
1522 y(that)33 b(needs)h(to)f(be)f(replaced)j(is)d(the)h(reduction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15905
(for)e(commuting)h(cuts,)h(since)f(it)e(is)h(the)f(only)i(one)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15906
1635 y(which)24 b(reduces)h(a)e(term)h(by)f(applying)j(a)d(proof)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15907
(substitution.)32 b(The)23 b(ne)n(w)g(reductions)k(are:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15908
Black 414 1814 a F6(\017)p Black 45 w Gg(the)d(reduction)1054
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15909
1777 y Gc(c)1085 1754 y FC(00)1004 1814 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15910
b(\000)g(!)p Gg(,)22 b(which)i(\223labels\224)h(cuts)g(when)e(the)o(y)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15911
(need)h(to)e(be)h(commuted,)g F7(viz.)p Black Black 468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15912
1984 a FL(Cut)p F4(\()641 1972 y FX(h)668 1984 y Ga(a)716
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15913
1972 y FX(i)744 1984 y Ga(M)10 b(;)882 1972 y F9(\()910
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15914
1984 y Ga(x)962 1972 y F9(\))989 1984 y Ga(N)g F4(\))1204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15915
1947 y Gc(c)1235 1923 y FC(00)1155 1984 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15916
b(\000)f(!)47 b FL(Cut)1451 1936 y FC( )1512 1984 y F4(\()1547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15917
1972 y FX(h)1575 1984 y Ga(a)1623 1972 y FX(i)1650 1984
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15918
y Ga(M)10 b(;)1788 1972 y F9(\()1816 1984 y Ga(x)1868
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15919
1972 y F9(\))1896 1984 y Ga(N)g F4(\))1467 2096 y Gg(if)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15920
b Ga(M)33 b Gg(does)24 b(not)g(freshly)h(introduce)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15921
Ga(a)d Gg(and)h(is)g(not)g(labelled,)h(or)1204 2196 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15922
Gc(c)1235 2173 y FC(00)1155 2233 y F6(\000)-31 b(\000)f(!)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15923
b FL(Cut)1451 2186 y FC(!)1512 2233 y F4(\()1547 2221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15924
y FX(h)1575 2233 y Ga(a)1623 2221 y FX(i)1650 2233 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15925
Ga(M)10 b(;)1788 2221 y F9(\()1816 2233 y Ga(x)1868 2221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15926
y F9(\))1896 2233 y Ga(N)g F4(\))1467 2346 y Gg(if)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15927
b Ga(N)33 b Gg(does)24 b(not)g(freshly)h(introduce)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15928
Ga(x)d Gg(and)h(is)f(not)h(labelled)p Black 414 2489
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15929
a F6(\017)p Black 45 w Gg(the)41 b(reduction)1103 2452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15930
y Gc(x)1038 2489 y F6(\000)-32 b(\000)h(!)p Gg(,)43 b(which)e(permutes)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15931
g(labelled)i(cuts)e(to)f(the)g(places)i(where)e(the)h(cut-)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15932
2602 y(formula)25 b(is)e(introduced)k(\(Figure)d(2.11)g(gi)n(v)o(es)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15933
(the)g(corresponding)k(reduction)e(rules\).)277 2781
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15934
y(W)-7 b(e)24 b(shall)h(assume)g(that)g(terms)g(ha)n(v)o(e)g(been)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15935
(suitably)h(alpha-con)l(v)o(erted)k(before)25 b(applying)i(an)3367
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15936
2744 y Gc(x)3302 2781 y F6(\000)-32 b(\000)h(!)p Gg(-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15937
2894 y(reduction,)28 b(so)e(that)g(name)g(or)f(co-name)i(capture)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15938
(will)e(not)h(occur)l(,)i(and)e(we)e(assume)j(these)f(con)l(v)o(er)n(-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15939
277 3007 y(sions)f(are)e(done)i(implicitly)-6 b(.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15940
3136 y(Furthermore,)43 b(we)36 b(shall)j(introduce)h(a)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15941
F7(garba)o(g)o(e)j(r)m(eduction)p Gg(,)j(which)38 b(is)f(some)n(what)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15942
(similar)277 3249 y(to)31 b(a)f(traditional)j(cut-reduction)h(where)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15943
(the)g(cut-formula)i(is)d(weak)o(ened,)k(and)d(remo)o(v)o(e)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15944
(side-)277 3362 y(conditions)d(on)c(the)h(logical)i(cut-reductions)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15945
(with)d(axioms.)29 b(Thus)24 b(we)f(ha)n(v)o(e)h(the)g(follo)n(wing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15946
(reduc-)277 3475 y(tions:)p Black 414 3654 a F6(\017)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15947
Black 45 w Gg(the)696 3617 y Gc(g)r(c)644 3654 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15948
b(\000)h(!)p Gg(-reduction)32 b(eliminates)f(labelled)g(cuts)e(pro)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15949
(vided)i(their)f(corresponding)j(name)c(or)504 3767 y(co-name)c(is)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15950
(not)h(free,)g F7(viz.)p Black Black 1066 3930 a FL(Cut)1145
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15951
3882 y FC( )1206 3930 y F4(\()1241 3918 y FX(h)1269 3930
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15952
y Ga(a)1317 3918 y FX(i)1344 3930 y Ga(M)10 b(;)1482
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15953
3918 y F9(\()1510 3930 y Ga(x)1562 3918 y F9(\))1590
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15954
3930 y Ga(N)g F4(\))1859 3892 y Gc(g)r(c)1808 3930 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15955
F6(\000)-32 b(\000)h(!)99 b Ga(M)201 b Gg(if)49 b Ga(a)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15956
b F6(62)g Ga(F)13 b(C)7 b F4(\()p Ga(M)j F4(\))1066 4042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15957
y FL(Cut)1145 3995 y FC(!)1206 4042 y F4(\()1241 4030
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15958
y FX(h)1269 4042 y Ga(a)1317 4030 y FX(i)1344 4042 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15959
Ga(M)g(;)1482 4030 y F9(\()1510 4042 y Ga(x)1562 4030
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15960
y F9(\))1590 4042 y Ga(N)g F4(\))1859 4005 y Gc(g)r(c)1808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15961
4042 y F6(\000)-32 b(\000)h(!)99 b Ga(N)216 b Gg(if)49
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15962
b Ga(x)25 b F6(62)g Ga(F)13 b(N)d F4(\()p Ga(N)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15963
Black 414 4209 a F6(\017)p Black 45 w Gg(the)706 4172
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15964
y Gc(l)728 4148 y FC(0)643 4209 y F6(\000)-32 b(\000)h(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15965
Gg(-reduction)31 b(is)c(just)1534 4172 y Gc(l)1460 4209
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15966
y F6(\000)-31 b(\000)f(!)27 b Gg(e)o(xcept)i(for)f(the)g(logical)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15967
(cuts)f(with)e(axioms,)j(which)e(are)504 4322 y(as)c(follo)n(ws:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15968
Black Black 1206 4473 a FL(Cut)o F4(\()1378 4461 y FX(h)1406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15969
4473 y Ga(a)1454 4461 y FX(i)1482 4473 y FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15970
Ga(x;)15 b(a)p F4(\))q Ga(;)1835 4461 y F9(\()1863 4473
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15971
y Ga(y)1911 4461 y F9(\))1939 4473 y Ga(M)10 b F4(\))2235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15972
4436 y Gc(l)2257 4412 y FC(0)2172 4473 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15973
b(\000)h(!)99 b Ga(M)10 b F4([)p Ga(y)j F6(7!)d Ga(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15974
F4(])1210 4586 y FL(Cut)p F4(\()1383 4574 y FX(h)1411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15975
4586 y Ga(b)1450 4574 y FX(i)1478 4586 y Ga(M)g(;)1616
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15976
4574 y F9(\()1644 4586 y Ga(x)1696 4574 y F9(\))1723
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15977
4586 y FL(Ax)p F4(\()p Ga(x;)15 b(a)p F4(\))q(\))2235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15978
4548 y Gc(l)2257 4525 y FC(0)2172 4586 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15979
b(\000)h(!)99 b Ga(M)10 b F4([)p Ga(b)g F6(7!)g Ga(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15980
F4(])277 4756 y Gg(At)25 b(\002rst)h(sight,)i(it)d(seems)i(as)f(if)g(a)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15981
1408 4719 y Gc(g)r(c)1356 4756 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15982
Gg(-reduction)28 b(can)f(be)f(simulated)i(by)e(some)2965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15983
4719 y Gc(x)2900 4756 y F6(\000)-31 b(\000)g(!)o Gg(-reductions,)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15984
4869 y(b)n(ut)19 b(in)g(general)i(this)e(is)g(not)g(true,)h(because)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15985
(labelled)h(cuts)f(cannot)g(be)f(permuted)h(with)f(other)g(labelled)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15986
4982 y(cuts.)40 b(In)27 b(the)g(same)g(w)o(ay)-6 b(,)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15987
b(a)1261 4945 y Gc(l)1283 4921 y FC(0)1198 4982 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15988
b(\000)f(!)p Gg(-reduction)30 b(cannot,)f(in)e(general,)j(be)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15989
(simulated)h(by)g(some)3376 4945 y Gc(l)3302 4982 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15990
b(\000)h(!)p Gg(-)277 5095 y(and)507 5058 y Gc(x)442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15991
5095 y F6(\000)f(\000)h(!)p Gg(-reductions.)63 b(Since)1390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15992
5058 y Gc(g)r(c)1338 5095 y F6(\000)-31 b(\000)g(!)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15993
b Gg(and)1769 5058 y Gc(l)1791 5034 y FC(0)1706 5095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15994
y F6(\000)-31 b(\000)f(!)34 b Gg(can)g(be)g(easily)h(accommodated)i(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15995
d(our)g(strong)277 5208 y(normalisation)27 b(proof,)d(we)f(include)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15996
(them)f(in)f(the)h(local)h(cut-elimination)i(procedure.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15997
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15998
%%Page: 50 62
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 15999
TeXDict begin 50 61 bop Black -144 51 a Gb(50)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16000
b(Calculi)p -144 88 3691 4 v Black Black -144 318 V -144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16001
5049 4 4732 v Black Black 394 485 a FF(Cut)461 438 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16002
FC( )522 485 y FG(\()554 473 y FQ(h)581 485 y FU(c)617
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16003
473 y FQ(i)643 485 y FF(Ax)q FG(\()p FU(x;)14 b(c)p FG(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16004
FU(;)959 473 y FJ(\()985 485 y FU(y)1029 473 y FJ(\))1054
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16005
485 y FU(P)e FG(\))1236 450 y FS(x)1173 485 y FT(\000)-25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16006
b(\000)g(!)22 b FF(Cut)p FG(\()1516 473 y FQ(h)1543 485
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16007
y FU(c)1579 473 y FQ(i)1606 485 y FF(Ax)p FG(\()p FU(x;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16008
b(c)p FG(\))q FU(;)1921 473 y FJ(\()1947 485 y FU(y)1991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16009
473 y FJ(\))2017 485 y FU(P)d FG(\))-120 621 y FF(Cut)-53
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16010
574 y FC( )8 621 y FG(\()40 609 y FQ(h)67 621 y FU(c)103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16011
609 y FQ(i)130 621 y FF(Cut)o FG(\()287 609 y FQ(h)315
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16012
621 y FU(a)359 609 y FQ(i)385 621 y FU(M)e(;)512 609
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16013
y FJ(\()538 621 y FU(x)585 609 y FJ(\))611 621 y FF(Ax)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16014
FG(\()p FU(x;)14 b(c)p FG(\))q(\))p FU(;)959 609 y FJ(\()985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16015
621 y FU(y)1029 609 y FJ(\))1054 621 y FU(P)e FG(\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16016
586 y FS(x)1173 621 y FT(\000)-25 b(\000)g(!)22 b FF(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16017
FG(\()1516 609 y FQ(h)1543 621 y FU(a)1587 609 y FQ(i)1614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16018
621 y FF(Cut)1680 574 y FC( )1741 621 y FG(\()1773 609
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16019
y FQ(h)1800 621 y FU(c)1836 609 y FQ(i)1863 621 y FU(M)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16020
b(;)1990 609 y FJ(\()2016 621 y FU(y)2060 609 y FJ(\))2085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16021
621 y FU(P)j FG(\))p FU(;)2219 609 y FJ(\()2245 621 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16022
FU(y)2289 609 y FJ(\))2315 621 y FU(P)g FG(\))145 757
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16023
y FF(Cut)212 710 y FC( )273 757 y FG(\()305 745 y FQ(h)332
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16024
757 y FU(a)376 745 y FQ(i)403 757 y FF(Not)533 769 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16025
FS(R)588 757 y FG(\()620 745 y FJ(\()646 757 y FU(x)693
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16026
745 y FJ(\))719 757 y FU(M)d(;)14 b(a)p FG(\))p FU(;)959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16027
745 y FJ(\()985 757 y FU(y)1029 745 y FJ(\))1054 757
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16028
y FU(P)e FG(\))1236 722 y FS(x)1173 757 y FT(\000)-25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16029
b(\000)g(!)22 b FF(Cut)p FG(\()1516 745 y FQ(h)1543 757
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16030
y FU(a)1587 745 y FQ(i)1614 757 y FF(Not)1744 769 y FS(R)1798
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16031
757 y FG(\()1830 745 y FJ(\()1857 757 y FU(x)1904 745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16032
y FJ(\))1930 757 y FF(Cut)1997 710 y FC( )2058 757 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16033
FG(\()2090 745 y FQ(h)2117 757 y FU(a)2161 745 y FQ(i)2188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16034
757 y FU(M)8 b(;)2314 745 y FJ(\()2340 757 y FU(y)2384
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16035
745 y FJ(\))2410 757 y FU(P)j FG(\))q FU(;)p Gd(\))o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16036
FU(;)2594 745 y FJ(\()2620 757 y FU(y)2664 745 y FJ(\))2690
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16037
757 y FU(P)h FG(\))-50 893 y FF(Cut)17 846 y FC( )78
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16038
893 y FG(\()110 881 y FQ(h)137 893 y FU(c)173 881 y FQ(i)200
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16039
893 y FF(And)341 905 y FS(R)395 893 y FG(\()427 881 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16040
FQ(h)454 893 y FU(a)498 881 y FQ(i)525 893 y FU(M)d(;)652
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16041
881 y FQ(h)679 893 y FU(b)715 881 y FQ(i)741 893 y FU(N)g(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16042
b(c)p FG(\))p FU(;)959 881 y FJ(\()985 893 y FU(y)1029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16043
881 y FJ(\))1054 893 y FU(P)e FG(\))1236 858 y FS(x)1173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16044
893 y FT(\000)-25 b(\000)g(!)22 b FF(Cut)p FG(\()1516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16045
881 y FQ(h)1543 893 y FU(c)1579 881 y FQ(i)1606 893 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16046
FF(And)1747 905 y FS(R)1801 893 y FG(\()1833 881 y FQ(h)1860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16047
893 y FU(a)1904 881 y FQ(i)1931 893 y FF(Cut)1998 846
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16048
y FC( )2059 893 y FG(\()2091 881 y FQ(h)2118 893 y FU(c)2154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16049
881 y FQ(i)2181 893 y FU(M)-5 b(;)2294 881 y FJ(\()2319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16050
893 y FU(y)2363 881 y FJ(\))2389 893 y FU(P)12 b FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16051
FU(;)2523 881 y FQ(h)2550 893 y FU(b)2586 881 y FQ(i)2612
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16052
893 y FF(Cut)2679 846 y FC( )2740 893 y FG(\()2772 881
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16053
y FQ(h)2799 893 y FU(c)2835 881 y FQ(i)2862 893 y FU(N)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16054
b(;)2961 881 y FJ(\()2987 893 y FU(y)3031 881 y FJ(\))3056
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16055
893 y FU(P)12 b FG(\))q FU(;)i(c)p FG(\))p FU(;)3296
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16056
881 y FJ(\()3322 893 y FU(y)3366 881 y FJ(\))3391 893
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16057
y FU(P)e FG(\))203 1029 y FF(Cut)269 982 y FC( )330 1029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16058
y FG(\()362 1017 y FQ(h)389 1029 y FU(c)425 1017 y FQ(i)452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16059
1029 y FF(Or)543 992 y FS(i)543 1050 y(R)597 1029 y FG(\()629
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16060
1017 y FQ(h)656 1029 y FU(a)700 1017 y FQ(i)727 1029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16061
y FU(M)d(;)14 b(c)p FG(\))p FU(;)959 1017 y FJ(\()985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16062
1029 y FU(y)1029 1017 y FJ(\))1054 1029 y FU(P)e FG(\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16063
994 y FS(x)1173 1029 y FT(\000)-25 b(\000)g(!)22 b FF(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16064
FG(\()1516 1017 y FQ(h)1543 1029 y FU(c)1579 1017 y FQ(i)1606
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16065
1029 y FF(Or)1696 992 y FS(i)1696 1050 y(R)1751 1029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16066
y FG(\()1783 1017 y FQ(h)1810 1029 y FU(a)1854 1017 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16067
FQ(i)1881 1029 y FF(Cut)1947 982 y FC( )2008 1029 y FG(\()2040
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16068
1017 y FQ(h)2067 1029 y FU(c)2103 1017 y FQ(i)2130 1029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16069
y FU(M)9 b(;)2257 1017 y FJ(\()2283 1029 y FU(y)2327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16070
1017 y FJ(\))2352 1029 y FU(P)j FG(\))p FU(;)i(c)p FG(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16071
FU(;)2592 1017 y FJ(\()2617 1029 y FU(y)2661 1017 y FJ(\))2687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16072
1029 y FU(P)e FG(\))63 1165 y FF(Cut)129 1118 y FC( )190
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16073
1165 y FG(\()222 1153 y FQ(h)249 1165 y FU(b)285 1153
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16074
y FQ(i)312 1165 y FF(Imp)444 1185 y FS(R)498 1165 y FG(\()530
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16075
1153 y FJ(\()557 1165 y FU(x)604 1153 y FJ(\))p FQ(h)657
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16076
1165 y FU(a)701 1153 y FQ(i)728 1165 y FU(M)c(;)14 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16077
FG(\))p FU(;)959 1153 y FJ(\()985 1165 y FU(y)1029 1153
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16078
y FJ(\))1054 1165 y FU(P)e FG(\))1236 1130 y FS(x)1173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16079
1165 y FT(\000)-25 b(\000)g(!)22 b FF(Cut)p FG(\()1516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16080
1153 y FQ(h)1543 1165 y FU(b)1579 1153 y FQ(i)1605 1165
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16081
y FF(Imp)1737 1185 y FS(R)1792 1165 y FG(\()1824 1153
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16082
y FJ(\()1850 1165 y FU(x)1897 1153 y FJ(\))p FQ(h)1950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16083
1165 y FU(a)1994 1153 y FQ(i)2021 1165 y FF(Cut)2088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16084
1118 y FC( )2149 1165 y FG(\()2181 1153 y FQ(h)2208 1165
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16085
y FU(b)2244 1153 y FQ(i)2270 1165 y FU(M)9 b(;)2397 1153
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16086
y FJ(\()2423 1165 y FU(y)2467 1153 y FJ(\))2492 1165
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16087
y FU(P)j FG(\))p FU(;)i(b)p FG(\))p FU(;)2731 1153 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16088
FJ(\()2757 1165 y FU(y)2801 1153 y FJ(\))2827 1165 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16089
FU(P)e FG(\))390 1348 y FF(Cut)456 1301 y FC(!)517 1348
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16090
y FG(\()549 1336 y FQ(h)576 1348 y FU(c)612 1336 y FQ(i)639
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16091
1348 y FU(P)g(;)741 1336 y FJ(\()767 1348 y FU(y)811
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16092
1336 y FJ(\))837 1348 y FF(Ax)p FG(\()p FU(y)s(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16093
FG(\)\))1236 1313 y FS(x)1173 1348 y FT(\000)-25 b(\000)g(!)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16094
b FF(Cut)p FG(\()1516 1336 y FQ(h)1543 1348 y FU(c)1579
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16095
1336 y FQ(i)1606 1348 y FU(P)11 b(;)1707 1336 y FJ(\()1733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16096
1348 y FU(y)1777 1336 y FJ(\))1803 1348 y FF(Ax)q FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16097
FU(y)s(;)j(a)p FG(\)\))-119 1484 y FF(Cut)-52 1437 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16098
FC(!)9 1484 y FG(\()41 1472 y FQ(h)68 1484 y FU(c)104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16099
1472 y FQ(i)131 1484 y FU(P)d(;)232 1472 y FJ(\()258
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16100
1484 y FU(y)302 1472 y FJ(\))328 1484 y FF(Cut)p FG(\()486
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16101
1472 y FQ(h)513 1484 y FU(a)557 1472 y FQ(i)584 1484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16102
y FF(Ax)p FG(\()p FU(y)s(;)j(a)p FG(\))p FU(;)903 1472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16103
y FJ(\()929 1484 y FU(z)972 1472 y FJ(\))997 1484 y FU(M)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16104
b FG(\)\))1236 1449 y FS(x)1173 1484 y FT(\000)-25 b(\000)g(!)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16105
b FF(Cut)p FG(\()1516 1472 y FQ(h)1543 1484 y FU(c)1579
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16106
1472 y FQ(i)1606 1484 y FU(P)11 b(;)1707 1472 y FJ(\()1733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16107
1484 y FU(z)1776 1472 y FJ(\))1801 1484 y FF(Cut)1868
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16108
1437 y FC(!)1929 1484 y FG(\()1961 1472 y FQ(h)1988 1484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16109
y FU(c)2024 1472 y FQ(i)2051 1484 y FU(P)h(;)2153 1472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16110
y FJ(\()2179 1484 y FU(y)2223 1472 y FJ(\))2248 1484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16111
y FU(M)d FG(\)\))152 1620 y FF(Cut)219 1573 y FC(!)280
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16112
1620 y FG(\()312 1608 y FQ(h)339 1620 y FU(c)375 1608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16113
y FQ(i)402 1620 y FU(P)j(;)504 1608 y FJ(\()530 1620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16114
y FU(x)577 1608 y FJ(\))603 1620 y FF(Not)733 1632 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16115
FS(L)783 1620 y FG(\()815 1608 y FQ(h)842 1620 y FU(a)886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16116
1608 y FQ(i)913 1620 y FU(M)d(;)14 b(x)p FG(\)\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16117
1585 y FS(x)1173 1620 y FT(\000)-25 b(\000)g(!)22 b FF(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16118
FG(\()1516 1608 y FQ(h)1543 1620 y FU(c)1579 1608 y FQ(i)1606
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16119
1620 y FU(P)11 b(;)1707 1608 y FJ(\()1733 1620 y FU(x)1780
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16120
1608 y FJ(\))1807 1620 y FF(Not)1937 1632 y FS(L)1987
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16121
1620 y FG(\()2019 1608 y FQ(h)2046 1620 y FU(a)2090 1608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16122
y FQ(i)2117 1620 y FF(Cut)2183 1573 y FC(!)2244 1620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16123
y FG(\()2276 1608 y FQ(h)2303 1620 y FU(c)2339 1608 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16124
FQ(i)2366 1620 y FU(P)h(;)2468 1608 y FJ(\()2494 1620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16125
y FU(x)2541 1608 y FJ(\))2567 1620 y FU(M)d FG(\))p FU(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16126
b(x)p FG(\))q(\))147 1756 y FF(Cut)214 1709 y FC(!)275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16127
1756 y FG(\()307 1744 y FQ(h)334 1756 y FU(c)370 1744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16128
y FQ(i)397 1756 y FU(P)e(;)499 1744 y FJ(\()525 1756
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16129
y FU(y)569 1744 y FJ(\))594 1756 y FF(And)735 1719 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16130
FS(i)735 1777 y(L)785 1756 y FG(\()817 1744 y FJ(\()843
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16131
1756 y FU(x)890 1744 y FJ(\))917 1756 y FU(M)c(;)14 b(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16132
FG(\)\))1236 1721 y FS(x)1173 1756 y FT(\000)-25 b(\000)g(!)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16133
b FF(Cut)p FG(\()1516 1744 y FQ(h)1543 1756 y FU(c)1579
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16134
1744 y FQ(i)1606 1756 y FU(P)11 b(;)1707 1744 y FJ(\()1733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16135
1756 y FU(y)1777 1744 y FJ(\))1803 1756 y FF(And)1944
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16136
1719 y FS(i)1944 1777 y(L)1994 1756 y FG(\()2026 1744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16137
y FJ(\()2052 1756 y FU(x)2099 1744 y FJ(\))2125 1756
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16138
y FF(Cut)2192 1709 y FC(!)2253 1756 y FG(\()2285 1744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16139
y FQ(h)2312 1756 y FU(c)2348 1744 y FQ(i)2375 1756 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16140
FU(P)h(;)2477 1744 y FJ(\()2503 1756 y FU(y)2547 1744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16141
y FJ(\))2572 1756 y FU(M)d FG(\))p FU(;)14 b(y)s FG(\)\))-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16142
1892 y FF(Cut)59 1845 y FC(!)120 1892 y FG(\()152 1880
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16143
y FQ(h)179 1892 y FU(c)215 1880 y FQ(i)242 1892 y FU(P)e(;)344
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16144
1880 y FJ(\()370 1892 y FU(z)413 1880 y FJ(\))438 1892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16145
y FF(Or)529 1904 y FS(L)578 1892 y FG(\()610 1880 y FJ(\()637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16146
1892 y FU(x)684 1880 y FJ(\))710 1892 y FU(M)d(;)837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16147
1880 y FJ(\()862 1892 y FU(y)906 1880 y FJ(\))932 1892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16148
y FU(N)g(;)14 b(z)t FG(\))o(\))1236 1857 y FS(x)1173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16149
1892 y FT(\000)-25 b(\000)g(!)22 b FF(Cut)p FG(\()1516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16150
1880 y FQ(h)1543 1892 y FU(c)1579 1880 y FQ(i)1606 1892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16151
y FU(P)11 b(;)1707 1880 y FJ(\()1733 1892 y FU(z)1776
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16152
1880 y FJ(\))1801 1892 y FF(Or)1892 1904 y FS(L)1942
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16153
1892 y FG(\()1974 1880 y FJ(\()2000 1892 y FU(x)2047
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16154
1880 y FJ(\))2073 1892 y FF(Cut)2140 1845 y FC(!)2201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16155
1892 y FG(\()2233 1880 y FQ(h)2260 1892 y FU(c)2296 1880
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16156
y FQ(i)2323 1892 y FU(P)h(;)2425 1880 y FJ(\()2451 1892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16157
y FU(z)2494 1880 y FJ(\))2519 1892 y FU(M)d FG(\))p FU(;)2678
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16158
1880 y FJ(\()2704 1892 y FU(y)2748 1880 y FJ(\))2773
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16159
1892 y FF(Cut)2840 1845 y FC(!)2901 1892 y FG(\()2933
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16160
1880 y FQ(h)2960 1892 y FU(c)2996 1880 y FQ(i)3023 1892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16161
y FU(P)j(;)3125 1880 y FJ(\()3151 1892 y FU(z)3194 1880
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16162
y FJ(\))3219 1892 y FU(N)d FG(\))p FU(;)14 b(z)t FG(\))o(\))-54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16163
2028 y FF(Cut)13 1981 y FC(!)74 2028 y FG(\()106 2016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16164
y FQ(h)133 2028 y FU(c)169 2016 y FQ(i)196 2028 y FU(P)e(;)298
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16165
2016 y FJ(\()324 2028 y FU(y)368 2016 y FJ(\))393 2028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16166
y FF(Imp)525 2049 y FS(L)575 2028 y FG(\()607 2016 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16167
FQ(h)634 2028 y FU(a)678 2016 y FQ(i)705 2028 y FU(M)c(;)831
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16168
2016 y FJ(\()857 2028 y FU(x)904 2016 y FJ(\))931 2028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16169
y FU(N)g(;)14 b(y)s FG(\)\))1236 1993 y FS(x)1173 2028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16170
y FT(\000)-25 b(\000)g(!)22 b FF(Cut)p FG(\()1516 2016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16171
y FQ(h)1543 2028 y FU(c)1579 2016 y FQ(i)1606 2028 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16172
FU(P)11 b(;)1707 2016 y FJ(\()1733 2028 y FU(y)1777 2016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16173
y FJ(\))1803 2028 y FF(Imp)1935 2049 y FS(L)1985 2028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16174
y FG(\()2017 2016 y FQ(h)2044 2028 y FU(a)2088 2016 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16175
FQ(i)2114 2028 y FF(Cut)2181 1981 y FC(!)2242 2028 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16176
FG(\()2274 2016 y FQ(h)2301 2028 y FU(c)2337 2016 y FQ(i)2364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16177
2028 y FU(P)h(;)2466 2016 y FJ(\()2492 2028 y FU(y)2536
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16178
2016 y FJ(\))2561 2028 y FU(M)d FG(\))p FU(;)2720 2016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16179
y FJ(\()2746 2028 y FU(x)2793 2016 y FJ(\))2820 2028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16180
y FF(Cut)2886 1981 y FC(!)2947 2028 y FG(\()2979 2016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16181
y FQ(h)3006 2028 y FU(c)3042 2016 y FQ(i)3069 2028 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16182
FU(P)j(;)3171 2016 y FJ(\()3197 2028 y FU(y)3241 2016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16183
y FJ(\))3267 2028 y FU(N)c FG(\))q FU(;)14 b(y)s FG(\))o(\))-111
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16184
2212 y FR(Otherwise:)386 2324 y FF(Cut)453 2277 y FC( )514
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16185
2324 y FG(\()546 2312 y FQ(h)573 2324 y FU(b)609 2312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16186
y FQ(i)636 2324 y FF(Ax)p FG(\()p FU(x;)g(a)p FG(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16187
FU(;)959 2312 y FJ(\()985 2324 y FU(y)1029 2312 y FJ(\))1054
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16188
2324 y FU(P)e FG(\))1236 2288 y FS(x)1173 2324 y FT(\000)-25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16189
b(\000)g(!)22 b FF(Ax)p FG(\()p FU(x;)14 b(a)p FG(\))83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16190
2460 y FF(Cut)150 2413 y FC( )211 2460 y FG(\()243 2448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16191
y FQ(h)270 2460 y FU(b)306 2448 y FQ(i)332 2460 y FF(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16192
FG(\()490 2448 y FQ(h)517 2460 y FU(a)561 2448 y FQ(i)588
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16193
2460 y FU(M)9 b(;)715 2448 y FJ(\()741 2460 y FU(x)788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16194
2448 y FJ(\))814 2460 y FU(N)g FG(\))p FU(;)959 2448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16195
y FJ(\()985 2460 y FU(y)1029 2448 y FJ(\))1054 2460 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16196
FU(P)j FG(\))1236 2424 y FS(x)1173 2460 y FT(\000)-25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16197
b(\000)g(!)22 b FF(Cut)p FG(\()1516 2448 y FQ(h)1543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16198
2460 y FU(a)1587 2448 y FQ(i)1614 2460 y FF(Cut)1680
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16199
2413 y FC( )1741 2460 y FG(\()1773 2448 y FQ(h)1800 2460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16200
y FU(b)1836 2448 y FQ(i)1863 2460 y FU(M)9 b(;)1990 2448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16201
y FJ(\()2015 2460 y FU(y)2059 2448 y FJ(\))2085 2460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16202
y FU(P)j FG(\))p FU(;)2219 2448 y FJ(\()2245 2460 y FU(x)2292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16203
2448 y FJ(\))2318 2460 y FF(Cut)2385 2413 y FC( )2446
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16204
2460 y FG(\()2478 2448 y FQ(h)2505 2460 y FU(b)2541 2448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16205
y FQ(i)2568 2460 y FU(N)d(;)2681 2448 y FJ(\()2706 2460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16206
y FU(y)2750 2448 y FJ(\))2776 2460 y FU(P)j FG(\)\))154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16207
2596 y FF(Cut)220 2549 y FC( )281 2596 y FG(\()313 2584
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16208
y FQ(h)340 2596 y FU(b)376 2584 y FQ(i)403 2596 y FF(Not)533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16209
2608 y FS(R)588 2596 y FG(\()620 2584 y FJ(\()646 2596
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16210
y FU(x)693 2584 y FJ(\))719 2596 y FU(M)d(;)14 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16211
FG(\))p FU(;)959 2584 y FJ(\()985 2596 y FU(y)1029 2584
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16212
y FJ(\))1054 2596 y FU(P)e FG(\))1236 2560 y FS(x)1173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16213
2596 y FT(\000)-25 b(\000)g(!)22 b FF(Not)1488 2608 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16214
FS(R)1542 2596 y FG(\()1574 2584 y FJ(\()1601 2596 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16215
FU(x)1648 2584 y FJ(\))1674 2596 y FF(Cut)1741 2549 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16216
FC( )1802 2596 y FG(\()1834 2584 y FQ(h)1861 2596 y FU(b)1897
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16217
2584 y FQ(i)1924 2596 y FU(M)8 b(;)2050 2584 y FJ(\()2076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16218
2596 y FU(y)2120 2584 y FJ(\))2146 2596 y FU(P)j FG(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16219
FU(;)j(a)p FG(\))156 2732 y FF(Cut)223 2685 y FC( )284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16220
2732 y FG(\()316 2720 y FQ(h)343 2732 y FU(b)379 2720
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16221
y FQ(i)406 2732 y FF(Not)536 2744 y FS(L)586 2732 y FG(\()618
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16222
2720 y FQ(h)645 2732 y FU(a)689 2720 y FQ(i)716 2732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16223
y FU(M)8 b(;)14 b(x)p FG(\))q FU(;)959 2720 y FJ(\()985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16224
2732 y FU(y)1029 2720 y FJ(\))1054 2732 y FU(P)e FG(\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16225
2696 y FS(x)1173 2732 y FT(\000)-25 b(\000)g(!)22 b FF(Not)1488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16226
2744 y FS(L)1538 2732 y FG(\()1570 2720 y FQ(h)1597 2732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16227
y FU(a)1641 2720 y FQ(i)1668 2732 y FF(Cut)1734 2685
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16228
y FC( )1795 2732 y FG(\()1827 2720 y FQ(h)1855 2732 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16229
FU(b)1891 2720 y FQ(i)1917 2732 y FU(M)9 b(;)2044 2720
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16230
y FJ(\()2069 2732 y FU(y)2113 2720 y FJ(\))2139 2732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16231
y FU(P)j FG(\))p FU(;)i(x)p FG(\))-57 2868 y FF(Cut)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16232
2821 y FC( )70 2868 y FG(\()102 2856 y FQ(h)130 2868
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16233
y FU(d)173 2856 y FQ(i)200 2868 y FF(And)341 2880 y FS(R)395
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16234
2868 y FG(\()427 2856 y FQ(h)454 2868 y FU(a)498 2856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16235
y FQ(i)525 2868 y FU(M)9 b(;)652 2856 y FQ(h)679 2868
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16236
y FU(b)715 2856 y FQ(i)741 2868 y FU(N)g(;)14 b(c)p FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16237
FU(;)959 2856 y FJ(\()985 2868 y FU(y)1029 2856 y FJ(\))1054
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16238
2868 y FU(P)e FG(\))1236 2832 y FS(x)1173 2868 y FT(\000)-25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16239
b(\000)g(!)22 b FF(And)1499 2880 y FS(R)1553 2868 y FG(\()1585
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16240
2856 y FQ(h)1612 2868 y FU(a)1656 2856 y FQ(i)1683 2868
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16241
y FF(Cut)1750 2821 y FC( )1811 2868 y FG(\()1843 2856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16242
y FQ(h)1870 2868 y FU(d)1913 2856 y FQ(i)1940 2868 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16243
FU(M)9 b(;)2067 2856 y FJ(\()2093 2868 y FU(y)2137 2856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16244
y FJ(\))2162 2868 y FU(P)j FG(\))p FU(;)2296 2856 y FQ(h)2323
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16245
2868 y FU(b)2359 2856 y FQ(i)2386 2868 y FF(Cut)2453
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16246
2821 y FC( )2513 2868 y FG(\()2545 2856 y FQ(h)2573 2868
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16247
y FU(d)2616 2856 y FQ(i)2643 2868 y FU(N)c(;)2755 2856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16248
y FJ(\()2781 2868 y FU(y)2825 2856 y FJ(\))2851 2868
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16249
y FU(P)k FG(\))p FU(;)i(c)p FG(\))141 3004 y FF(Cut)208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16250
2957 y FC( )269 3004 y FG(\()301 2992 y FQ(h)328 3004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16251
y FU(a)372 2992 y FQ(i)398 3004 y FF(And)540 2967 y FS(i)540
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16252
3024 y(L)589 3004 y FG(\()621 2992 y FJ(\()648 3004 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16253
FU(x)695 2992 y FJ(\))721 3004 y FU(M)9 b(;)14 b(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16254
FG(\))o FU(;)960 2992 y FJ(\()986 3004 y FU(z)1029 2992
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16255
y FJ(\))1054 3004 y FU(P)e FG(\))1236 2968 y FS(x)1173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16256
3004 y FT(\000)-25 b(\000)g(!)22 b FF(And)1499 2967 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16257
FS(i)1499 3024 y(L)1549 3004 y FG(\()1581 2992 y FJ(\()1607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16258
3004 y FU(x)1654 2992 y FJ(\))1680 3004 y FF(Cut)1747
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16259
2957 y FC( )1808 3004 y FG(\()1840 2992 y FQ(h)1867 3004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16260
y FU(a)1911 2992 y FQ(i)1938 3004 y FU(M)8 b(;)2064 2992
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16261
y FJ(\()2090 3004 y FU(z)2133 2992 y FJ(\))2158 3004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16262
y FU(P)k FG(\))q FU(;)i(y)s FG(\))199 3140 y FF(Cut)266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16263
3093 y FC( )327 3140 y FG(\()359 3128 y FQ(h)386 3140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16264
y FU(c)422 3128 y FQ(i)449 3140 y FF(Or)539 3103 y FS(i)539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16265
3160 y(R)594 3140 y FG(\()626 3128 y FQ(h)653 3140 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16266
FU(a)697 3128 y FQ(i)724 3140 y FU(M)8 b(;)14 b(b)p FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16267
FU(;)955 3128 y FJ(\()981 3140 y FU(x)1028 3128 y FJ(\))1054
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16268
3140 y FU(P)e FG(\))1236 3104 y FS(x)1173 3140 y FT(\000)-25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16269
b(\000)g(!)22 b FF(Or)1448 3103 y FS(i)1448 3160 y(R)1503
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16270
3140 y FG(\()1535 3128 y FQ(h)1562 3140 y FU(a)1606 3128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16271
y FQ(i)1633 3140 y FF(Cut)1700 3093 y FC( )1760 3140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16272
y FG(\()1792 3128 y FQ(h)1820 3140 y FU(c)1856 3128 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16273
FQ(i)1882 3140 y FU(M)9 b(;)2009 3128 y FJ(\()2035 3140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16274
y FU(x)2082 3128 y FJ(\))2108 3140 y FU(P)j FG(\))p FU(;)i(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16275
FG(\))-16 3276 y FF(Cut)50 3229 y FC( )111 3276 y FG(\()143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16276
3264 y FQ(h)170 3276 y FU(a)214 3264 y FQ(i)241 3276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16277
y FF(Or)332 3288 y FS(L)382 3276 y FG(\()414 3264 y FJ(\()440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16278
3276 y FU(x)487 3264 y FJ(\))513 3276 y FU(M)9 b(;)640
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16279
3264 y FJ(\()666 3276 y FU(y)710 3264 y FJ(\))735 3276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16280
y FU(N)g(;)14 b(z)t FG(\))o FU(;)959 3264 y FJ(\()985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16281
3276 y FU(v)1028 3264 y FJ(\))1054 3276 y FU(P)e FG(\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16282
3240 y FS(x)1173 3276 y FT(\000)-25 b(\000)g(!)22 b FF(Or)1448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16283
3288 y FS(L)1498 3276 y FG(\()1530 3264 y FJ(\()1556
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16284
3276 y FU(x)1603 3264 y FJ(\))1630 3276 y FF(Cut)1697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16285
3229 y FC( )1757 3276 y FG(\()1789 3264 y FQ(h)1817 3276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16286
y FU(a)1861 3264 y FQ(i)1887 3276 y FU(M)9 b(;)2014 3264
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16287
y FJ(\()2040 3276 y FU(v)2083 3264 y FJ(\))2109 3276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16288
y FU(P)j FG(\))p FU(;)2243 3264 y FJ(\()2269 3276 y FU(y)2313
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16289
3264 y FJ(\))2339 3276 y FF(Cut)2405 3229 y FC( )2466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16290
3276 y FG(\()2498 3264 y FQ(h)2525 3276 y FU(a)2569 3264
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16291
y FQ(i)2596 3276 y FU(N)d(;)2709 3264 y FJ(\()2735 3276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16292
y FU(v)2778 3264 y FJ(\))2804 3276 y FU(P)j FG(\))p FU(;)i(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16293
FG(\))62 3412 y FF(Cut)129 3365 y FC( )190 3412 y FG(\()222
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16294
3400 y FQ(h)249 3412 y FU(c)285 3400 y FQ(i)312 3412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16295
y FF(Imp)444 3432 y FS(R)498 3412 y FG(\()530 3400 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16296
FJ(\()557 3412 y FU(x)604 3400 y FJ(\))p FQ(h)657 3412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16297
y FU(a)701 3400 y FQ(i)728 3412 y FU(M)8 b(;)14 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16298
FG(\))p FU(;)959 3400 y FJ(\()985 3412 y FU(y)1029 3400
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16299
y FJ(\))1054 3412 y FU(P)e FG(\))1236 3376 y FS(x)1173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16300
3412 y FT(\000)-25 b(\000)g(!)22 b FF(Imp)1490 3432 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16301
FS(R)1544 3412 y FG(\()1576 3400 y FJ(\()1602 3412 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16302
FU(x)1649 3400 y FJ(\))q FQ(h)1703 3412 y FU(a)1747 3400
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16303
y FQ(i)1773 3412 y FF(Cut)1840 3365 y FC( )1901 3412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16304
y FG(\()1933 3400 y FQ(h)1960 3412 y FU(c)1996 3400 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16305
FQ(i)2023 3412 y FU(M)9 b(;)2150 3400 y FJ(\()2175 3412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16306
y FU(y)2219 3400 y FJ(\))2245 3412 y FU(P)j FG(\))p FU(;)i(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16307
FG(\))-52 3548 y FF(Cut)15 3501 y FC( )76 3548 y FG(\()108
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16308
3536 y FQ(h)135 3548 y FU(b)171 3536 y FQ(i)197 3548
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16309
y FF(Imp)329 3568 y FS(L)379 3548 y FG(\()411 3536 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16310
FQ(h)438 3548 y FU(a)482 3536 y FQ(i)509 3548 y FU(M)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16311
b(;)636 3536 y FJ(\()661 3548 y FU(x)708 3536 y FJ(\))735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16312
3548 y FU(N)g(;)14 b(y)s FG(\))o FU(;)960 3536 y FJ(\()986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16313
3548 y FU(z)1029 3536 y FJ(\))1054 3548 y FU(P)e FG(\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16314
3512 y FS(x)1173 3548 y FT(\000)-25 b(\000)g(!)22 b FF(Imp)1490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16315
3568 y FS(L)1539 3548 y FG(\()1571 3536 y FQ(h)1598 3548
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16316
y FU(a)1642 3536 y FQ(i)1669 3548 y FF(Cut)1736 3501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16317
y FC( )1797 3548 y FG(\()1829 3536 y FQ(h)1856 3548 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16318
FU(b)1892 3536 y FQ(i)1919 3548 y FU(M)8 b(;)2045 3536
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16319
y FJ(\()2071 3548 y FU(z)2114 3536 y FJ(\))2139 3548
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16320
y FU(P)k FG(\))p FU(;)2273 3536 y FJ(\()2299 3548 y FU(x)2346
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16321
3536 y FJ(\))2373 3548 y FF(Cut)2440 3501 y FC( )2500
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16322
3548 y FG(\()2532 3536 y FQ(h)2560 3548 y FU(b)2596 3536
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16323
y FQ(i)2622 3548 y FU(N)d(;)2735 3536 y FJ(\()2761 3548
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16324
y FU(z)2804 3536 y FJ(\))2829 3548 y FU(P)j FG(\))p FU(;)i(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16325
FG(\))386 3731 y FF(Cut)453 3684 y FC(!)514 3731 y FG(\()546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16326
3719 y FQ(h)573 3731 y FU(b)609 3719 y FQ(i)636 3731
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16327
y FU(P)d(;)737 3719 y FJ(\()763 3731 y FU(y)807 3719
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16328
y FJ(\))833 3731 y FF(Ax)p FG(\()p FU(x;)j(a)p FG(\))q(\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16329
3696 y FS(x)1173 3731 y FT(\000)-25 b(\000)g(!)22 b FF(Ax)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16330
FG(\()p FU(x;)14 b(a)p FG(\))83 3867 y FF(Cut)150 3820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16331
y FC(!)211 3867 y FG(\()243 3855 y FQ(h)270 3867 y FU(b)306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16332
3855 y FQ(i)332 3867 y FU(P)e(;)434 3855 y FJ(\()460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16333
3867 y FU(y)504 3855 y FJ(\))530 3867 y FF(Cut)o FG(\()687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16334
3855 y FQ(h)715 3867 y FU(a)759 3855 y FQ(i)785 3867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16335
y FU(M)d(;)912 3855 y FJ(\()938 3867 y FU(x)985 3855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16336
y FJ(\))1011 3867 y FU(N)g FG(\)\))1236 3832 y FS(x)1173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16337
3867 y FT(\000)-25 b(\000)g(!)22 b FF(Cut)p FG(\()1516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16338
3855 y FQ(h)1543 3867 y FU(a)1587 3855 y FQ(i)1614 3867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16339
y FF(Cut)1680 3820 y FC(!)1741 3867 y FG(\()1773 3855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16340
y FQ(h)1800 3867 y FU(b)1836 3855 y FQ(i)1863 3867 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16341
FU(P)12 b(;)1965 3855 y FJ(\()1991 3867 y FU(y)2035 3855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16342
y FJ(\))2060 3867 y FU(M)d FG(\))p FU(;)2219 3855 y FJ(\()2245
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16343
3867 y FU(x)2292 3855 y FJ(\))2318 3867 y FF(Cut)2385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16344
3820 y FC(!)2446 3867 y FG(\()2478 3855 y FQ(h)2505 3867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16345
y FU(b)2541 3855 y FQ(i)2568 3867 y FU(P)j(;)2670 3855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16346
y FJ(\()2695 3867 y FU(y)2739 3855 y FJ(\))2765 3867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16347
y FU(N)d FG(\)\))154 4003 y FF(Cut)220 3956 y FC(!)281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16348
4003 y FG(\()313 3991 y FQ(h)340 4003 y FU(b)376 3991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16349
y FQ(i)403 4003 y FU(P)j(;)505 3991 y FJ(\()531 4003
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16350
y FU(y)575 3991 y FJ(\))600 4003 y FF(Not)731 4015 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16351
FS(R)785 4003 y FG(\()817 3991 y FJ(\()843 4003 y FU(x)890
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16352
3991 y FJ(\))917 4003 y FU(M)c(;)14 b(a)p FG(\)\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16353
3968 y FS(x)1173 4003 y FT(\000)-25 b(\000)g(!)22 b FF(Not)1488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16354
4015 y FS(R)1542 4003 y FG(\()1574 3991 y FJ(\()1601
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16355
4003 y FU(x)1648 3991 y FJ(\))1674 4003 y FF(Cut)1741
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16356
3956 y FC(!)1802 4003 y FG(\()1834 3991 y FQ(h)1861 4003
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16357
y FU(b)1897 3991 y FQ(i)1924 4003 y FU(P)11 b(;)2025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16358
3991 y FJ(\()2051 4003 y FU(y)2095 3991 y FJ(\))2121
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16359
4003 y FU(M)d FG(\))q FU(;)14 b(a)p FG(\))156 4139 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16360
FF(Cut)223 4092 y FC(!)284 4139 y FG(\()316 4127 y FQ(h)343
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16361
4139 y FU(b)379 4127 y FQ(i)406 4139 y FU(P)e(;)508 4127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16362
y FJ(\()533 4139 y FU(y)577 4127 y FJ(\))603 4139 y FF(Not)733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16363
4151 y FS(L)783 4139 y FG(\()815 4127 y FQ(h)842 4139
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16364
y FU(a)886 4127 y FQ(i)913 4139 y FU(M)d(;)14 b(x)p FG(\)\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16365
4104 y FS(x)1173 4139 y FT(\000)-25 b(\000)g(!)22 b FF(Not)1488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16366
4151 y FS(L)1538 4139 y FG(\()1570 4127 y FQ(h)1597 4139
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16367
y FU(a)1641 4127 y FQ(i)1668 4139 y FF(Cut)1734 4092
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16368
y FC(!)1795 4139 y FG(\()1827 4127 y FQ(h)1855 4139 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16369
FU(b)1891 4127 y FQ(i)1917 4139 y FU(P)12 b(;)2019 4127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16370
y FJ(\()2045 4139 y FU(y)2089 4127 y FJ(\))2114 4139
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16371
y FU(M)d FG(\))p FU(;)14 b(x)p FG(\))-57 4275 y FF(Cut)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16372
4228 y FC(!)70 4275 y FG(\()102 4263 y FQ(h)130 4275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16373
y FU(d)173 4263 y FQ(i)200 4275 y FU(P)e(;)302 4263 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16374
FJ(\()327 4275 y FU(y)371 4263 y FJ(\))397 4275 y FF(And)538
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16375
4287 y FS(R)593 4275 y FG(\()625 4263 y FQ(h)652 4275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16376
y FU(a)696 4263 y FQ(i)723 4275 y FU(M)c(;)849 4263 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16377
FQ(h)876 4275 y FU(b)912 4263 y FQ(i)938 4275 y FU(N)h(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16378
b(c)p FG(\)\))1236 4240 y FS(x)1173 4275 y FT(\000)-25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16379
b(\000)g(!)22 b FF(And)1499 4287 y FS(R)1553 4275 y FG(\()1585
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16380
4263 y FQ(h)1612 4275 y FU(a)1656 4263 y FQ(i)1683 4275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16381
y FF(Cut)1750 4228 y FC(!)1811 4275 y FG(\()1843 4263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16382
y FQ(h)1870 4275 y FU(d)1913 4263 y FQ(i)1940 4275 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16383
FU(P)12 b(;)2042 4263 y FJ(\()2068 4275 y FU(y)2112 4263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16384
y FJ(\))2138 4275 y FU(M)c FG(\))p FU(;)2296 4263 y FQ(h)2323
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16385
4275 y FU(b)2359 4263 y FQ(i)2386 4275 y FF(Cut)2453
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16386
4228 y FC(!)2513 4275 y FG(\()2545 4263 y FQ(h)2573 4275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16387
y FU(d)2616 4263 y FQ(i)2643 4275 y FU(P)j(;)2744 4263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16388
y FJ(\()2770 4275 y FU(y)2814 4263 y FJ(\))2840 4275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16389
y FU(N)e FG(\))p FU(;)14 b(c)p FG(\))141 4411 y FF(Cut)208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16390
4364 y FC(!)269 4411 y FG(\()301 4399 y FQ(h)328 4411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16391
y FU(a)372 4399 y FQ(i)398 4411 y FU(P)e(;)500 4399 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16392
FJ(\()526 4411 y FU(z)569 4399 y FJ(\))594 4411 y FF(And)735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16393
4374 y FS(i)735 4432 y(L)785 4411 y FG(\()817 4399 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16394
FJ(\()843 4411 y FU(x)890 4399 y FJ(\))917 4411 y FU(M)c(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16395
b(y)s FG(\)\))1236 4376 y FS(x)1173 4411 y FT(\000)-25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16396
b(\000)g(!)22 b FF(And)1499 4374 y FS(i)1499 4432 y(L)1549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16397
4411 y FG(\()1581 4399 y FJ(\()1607 4411 y FU(x)1654
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16398
4399 y FJ(\))1680 4411 y FF(Cut)1747 4364 y FC(!)1808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16399
4411 y FG(\()1840 4399 y FQ(h)1867 4411 y FU(a)1911 4399
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16400
y FQ(i)1938 4411 y FU(P)12 b(;)2040 4399 y FJ(\()2066
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16401
4411 y FU(z)2109 4399 y FJ(\))2134 4411 y FU(M)c FG(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16402
FU(;)14 b(y)s FG(\))199 4547 y FF(Cut)266 4500 y FC(!)327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16403
4547 y FG(\()359 4535 y FQ(h)386 4547 y FU(c)422 4535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16404
y FQ(i)449 4547 y FU(P)e(;)551 4535 y FJ(\()576 4547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16405
y FU(x)623 4535 y FJ(\))650 4547 y FF(Or)740 4510 y FS(i)740
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16406
4568 y(R)795 4547 y FG(\()827 4535 y FQ(h)854 4547 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16407
FU(a)898 4535 y FQ(i)925 4547 y FU(M)c(;)14 b(b)p FG(\)\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16408
4512 y FS(x)1173 4547 y FT(\000)-25 b(\000)g(!)22 b FF(Or)1448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16409
4510 y FS(i)1448 4568 y(R)1503 4547 y FG(\()1535 4535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16410
y FQ(h)1562 4547 y FU(a)1606 4535 y FQ(i)1633 4547 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16411
FF(Cut)1700 4500 y FC(!)1760 4547 y FG(\()1792 4535 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16412
FQ(h)1820 4547 y FU(c)1856 4535 y FQ(i)1882 4547 y FU(P)12
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16413
b(;)1984 4535 y FJ(\()2010 4547 y FU(x)2057 4535 y FJ(\))2083
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16414
4547 y FU(M)d FG(\))p FU(;)14 b(b)p FG(\))-16 4683 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16415
FF(Cut)50 4636 y FC(!)111 4683 y FG(\()143 4671 y FQ(h)170
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16416
4683 y FU(a)214 4671 y FQ(i)241 4683 y FU(P)e(;)343 4671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16417
y FJ(\()369 4683 y FU(v)412 4671 y FJ(\))438 4683 y FF(Or)529
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16418
4695 y FS(L)578 4683 y FG(\()610 4671 y FJ(\()637 4683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16419
y FU(x)684 4671 y FJ(\))710 4683 y FU(M)d(;)837 4671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16420
y FJ(\()862 4683 y FU(y)906 4671 y FJ(\))932 4683 y FU(N)g(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16421
b(z)t FG(\))o(\))1236 4648 y FS(x)1173 4683 y FT(\000)-25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16422
b(\000)g(!)22 b FF(Or)1448 4695 y FS(L)1498 4683 y FG(\()1530
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16423
4671 y FJ(\()1556 4683 y FU(x)1603 4671 y FJ(\))1630
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16424
4683 y FF(Cut)1697 4636 y FC(!)1757 4683 y FG(\()1789
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16425
4671 y FQ(h)1817 4683 y FU(a)1861 4671 y FQ(i)1887 4683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16426
y FU(P)12 b(;)1989 4671 y FJ(\()2015 4683 y FU(v)2058
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16427
4671 y FJ(\))2084 4683 y FU(M)d FG(\))p FU(;)2243 4671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16428
y FJ(\()2269 4683 y FU(y)2313 4671 y FJ(\))2339 4683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16429
y FF(Cut)2405 4636 y FC(!)2466 4683 y FG(\()2498 4671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16430
y FQ(h)2525 4683 y FU(a)2569 4671 y FQ(i)2596 4683 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16431
FU(P)j(;)2698 4671 y FJ(\()2724 4683 y FU(v)2767 4671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16432
y FJ(\))2793 4683 y FU(N)d FG(\))p FU(;)14 b(z)t FG(\))62
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16433
4819 y FF(Cut)129 4772 y FC(!)190 4819 y FG(\()222 4807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16434
y FQ(h)249 4819 y FU(c)285 4807 y FQ(i)312 4819 y FU(P)e(;)414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16435
4807 y FJ(\()440 4819 y FU(y)484 4807 y FJ(\))509 4819
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16436
y FF(Imp)641 4839 y FS(R)696 4819 y FG(\()728 4807 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16437
FJ(\()754 4819 y FU(x)801 4807 y FJ(\))p FQ(h)854 4819
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16438
y FU(a)898 4807 y FQ(i)925 4819 y FU(M)c(;)14 b(b)p FG(\)\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16439
4784 y FS(x)1173 4819 y FT(\000)-25 b(\000)g(!)22 b FF(Imp)1490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16440
4839 y FS(R)1544 4819 y FG(\()1576 4807 y FJ(\()1602
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16441
4819 y FU(x)1649 4807 y FJ(\))q FQ(h)1703 4819 y FU(a)1747
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16442
4807 y FQ(i)1773 4819 y FF(Cut)1840 4772 y FC(!)1901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16443
4819 y FG(\()1933 4807 y FQ(h)1960 4819 y FU(c)1996 4807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16444
y FQ(i)2023 4819 y FU(P)12 b(;)2125 4807 y FJ(\()2151
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16445
4819 y FU(y)2195 4807 y FJ(\))2220 4819 y FU(M)d FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16446
FU(;)14 b(b)p FG(\))-52 4955 y FF(Cut)15 4908 y FC(!)76
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16447
4955 y FG(\()108 4943 y FQ(h)135 4955 y FU(b)171 4943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16448
y FQ(i)197 4955 y FU(P)e(;)299 4943 y FJ(\()325 4955
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16449
y FU(z)368 4943 y FJ(\))393 4955 y FF(Imp)525 4975 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16450
FS(L)575 4955 y FG(\()607 4943 y FQ(h)634 4955 y FU(a)678
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16451
4943 y FQ(i)705 4955 y FU(M)c(;)831 4943 y FJ(\()857
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16452
4955 y FU(x)904 4943 y FJ(\))931 4955 y FU(N)g(;)14 b(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16453
FG(\)\))1236 4919 y FS(x)1173 4955 y FT(\000)-25 b(\000)g(!)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16454
b FF(Imp)1490 4975 y FS(L)1539 4955 y FG(\()1571 4943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16455
y FQ(h)1598 4955 y FU(a)1642 4943 y FQ(i)1669 4955 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16456
FF(Cut)1736 4908 y FC(!)1797 4955 y FG(\()1829 4943 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16457
FQ(h)1856 4955 y FU(b)1892 4943 y FQ(i)1919 4955 y FU(P)11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16458
b(;)2020 4943 y FJ(\()2046 4955 y FU(z)2089 4943 y FJ(\))2115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16459
4955 y FU(M)d FG(\))p FU(;)2273 4943 y FJ(\()2299 4955
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16460
y FU(x)2346 4943 y FJ(\))2373 4955 y FF(Cut)2440 4908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16461
y FC(!)2500 4955 y FG(\()2532 4943 y FQ(h)2560 4955 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16462
FU(b)2596 4943 y FQ(i)2622 4955 y FU(P)k(;)2724 4943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16463
y FJ(\()2750 4955 y FU(z)2793 4943 y FJ(\))2818 4955
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16464
y FU(N)d FG(\))p FU(;)14 b(y)s FG(\))p 3543 5049 V -144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16465
5052 3691 4 v Black 1112 5206 a Gg(Figure)24 b(2.11:)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16466
b(Cut-reductions)e(for)d(labelled)i(cuts.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16467
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16468
%%Page: 51 63
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16469
TeXDict begin 51 62 bop Black 277 51 a Gb(2.6)23 b(Localised)i(V)-9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16470
b(ersion)2779 b(51)p 277 88 3691 4 v Black Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16471
317 a(De\002nition)23 b(2.6.1)g Gg(\(Local)h(Cut-Elimination)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16472
(Procedure\))p Gb(:)p Black 277 430 a Gg(The)d(local)i(cut-elimination)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16473
i(procedure,)f F4(\()p FY(T)1724 397 y FX($)1795 430
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16474
y Ga(;)1877 393 y Gc(l)q(oc)1835 430 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16475
F4(\))p Gg(,)22 b(is)i(a)f(reduction)j(system)e(where:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16476
Black 414 615 a F6(\017)p Black 45 w FY(T)565 582 y FX($)659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16477
615 y Gg(is)f(the)h(set)g(of)f(terms)h(well-typed)i(by)d(the)h(rules)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16478
(sho)n(wn)g(in)g(\(2.5\))g(and)g(Figure)g(2.3,)f(and)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16479
Black 414 795 a F6(\017)p Black 546 758 a Gc(l)q(oc)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16480
795 y F6(\000)-31 b(\000)g(!)28 b Gg(consists)k(of)d(the)h(rules)g(for)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16481
f(logical,)k(commuting)d(and)g(labelled)h(cuts)f(as)g(well)f(as)g(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16482
504 908 y(rules)c(for)f(garbage)h(collection;)h(that)e(is)1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16483
1073 y Gc(l)q(oc)1328 1111 y F6(\000)-31 b(\000)g(!)1514
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16484
1059 y F5(def)1521 1111 y F4(=)1677 1073 y Gc(l)1699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16485
1050 y FC(0)1614 1111 y F6(\000)f(\000)h(!)25 b([)1944
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16486
1073 y Gc(c)1975 1050 y FC(00)1895 1111 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16487
b(\000)f(!)25 b([)2242 1073 y Gc(x)2176 1111 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16488
b(\000)g(!)25 b([)2509 1073 y Gc(g)r(c)2458 1111 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16489
b(\000)h(!)25 b Ga(:)277 1365 y Gg(As)j(usual,)j(we)d(assume)i(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16490
(reductions)i(are)e(closed)i(under)f(conte)o(xt)g(formation.)46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16491
b(An)28 b(immediate)277 1477 y(observ)n(ation)35 b(is)c(that)1030
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16492
1440 y Gc(l)q(oc)988 1477 y F6(\000)-31 b(\000)g(!)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16493
b Gg(is)i(complete,)j(in)c(the)h(sense)h(that)f(it)f(eliminates)j(all)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16494
(cuts)i(\(routine)g(by)277 1590 y(inspection)k(of)d(the)g(reduction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16495
(rules\).)60 b(Another)35 b(is)f(that)2237 1553 y Gc(l)q(oc)2195
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16496
1590 y F6(\000)-31 b(\000)g(!)32 b Gg(respects)k(the)e(subject)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16497
(reduction)277 1703 y(property)-6 b(.)p Black 277 1891
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16498
a Gb(Pr)n(oposition)34 b(2.6.2)e Gg(\(Subject)i(Reduction\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16499
Gb(:)p Black 110 w Gg(Suppose)g Ga(M)42 b Gg(belongs)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16500
b(to)d FY(T)2794 1858 y FX($)2897 1891 y Gg(and)h(has)g(the)g(typ-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16501
2004 y(ing)f(judgement)i F4(\000)921 1992 y Gc(.)976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16502
2004 y Ga(M)1099 1992 y Gc(.)1154 2004 y F4(\001)p Gg(.)53
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16503
b(Let)31 b Ga(M)1638 1967 y Gc(l)q(oc)1596 2004 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16504
b(\000)g(!)40 b Ga(N)10 b Gg(,)33 b(then)f Ga(N)51 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16505
F6(2)40 b FY(T)2419 1971 y FX($)2520 2004 y Gg(with)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16506
b(the)g(typing)h(judgement)277 2117 y F4(\000)359 2105
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16507
y Gc(.)414 2117 y Ga(N)522 2105 y Gc(.)577 2117 y F4(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16508
Gg(.)p Black 277 2329 a F7(Pr)l(oof)o(.)p Black 34 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16509
Gg(By)19 b(inspection)k(of)d(the)g(reduction)j(rules;)f(analogous)h(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16510
d(proof)h(of)f(Proposition)j(2.2.10.)p 3436 2329 4 62
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16511
v 3440 2271 55 4 v 3440 2329 V 3494 2329 4 62 v 418 2533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16512
a(T)-7 b(o)18 b(pro)o(v)o(e)i(strong)g(normalisation)i(for)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16513
F4(\()p FY(T)1737 2500 y FX($)1808 2533 y Ga(;)1890 2496
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16514
y Gc(l)q(oc)1848 2533 y F6(\000)-31 b(\000)f(!)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16515
Gg(,)19 b(let)h(us)f(\002rst)f(e)o(xamine)i(if)f(there)h(is)f(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16516
(straight-)277 2646 y(forw)o(ard)25 b(method)g(inferring)h(this)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16517
(property)h(from)d(the)i(strong)g(normalisation)i(result)e(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16518
F4(\()p FY(T)t Ga(;)3294 2609 y Gc(aux)3274 2646 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16519
b(\000)f(!)p F4(\))p Gg(.)277 2759 y(A)21 b(na)m(\250)-27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16520
b(\021v)o(e)22 b(attempt)h(might)f(use)g(the)g(follo)n(wing)i(lemma,)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16521
(often)i(applied)h(successfully)h(in)d(the)g(e)o(xplicit)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16522
2872 y(substitution)30 b(community)e(\(see)f(for)f(e)o(xample)i([Bloo,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16523
e(1997]\).)39 b(It)26 b(establishes)j(strong)f(normalisa-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16524
2985 y(tion)f(for)f(a)g(reduction)i(system)f(with)f(the)g(help)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16525
(another)i(one,)f(for)f(which)h(strong)g(normalisation)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16526
3098 y(is)c(already)j(kno)n(wn)e(to)f(hold.)1195 3065
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16527
y F5(6)p Black 277 3286 a Gb(Lemma)d(2.6.3:)p Black 34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16528
w Gg(Let)g F4(\()p FY(R)p Ga(;)1181 3249 y Gc(R)1123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16529
3286 y F6(\000)-31 b(\000)f(!)p F4(\))20 b Gg(and)h F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16530
FY(S)p Ga(;)1686 3249 y Gc(S)1624 3286 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16531
b(\000)g(!)p F4(\))20 b Gg(be)h(tw)o(o)f(reduction)j(systems)f(where)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16532
3086 3249 y Gc(S)3024 3286 y F6(\000)-31 b(\000)f(!)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16533
b Gg(consists)277 3399 y(of)27 b(tw)o(o)f(reductions,)1015
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16534
3361 y Gc(S)1058 3370 y FV(1)969 3399 y F6(\000)-32 b(\000)h(!)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16535
b Gg(and)1368 3361 y Gc(S)1411 3370 y FV(2)1322 3399
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16536
y F6(\000)-31 b(\000)f(!)p Gg(,)27 b(and)g(suppose)i(that)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16537
F4(\()p FY(R)p Ga(;)2380 3361 y Gc(R)2322 3399 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16538
b(\000)h(!)p F4(\))26 b Gg(and)i F4(\()p FY(S)p Ga(;)2882
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16539
3361 y Gc(S)2925 3370 y FV(1)2836 3399 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16540
b(\000)h(!)p F4(\))26 b Gg(are)h(strongly)277 3511 y(normalising.)44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16541
b(Let)27 b F6(j)p 950 3511 28 4 v 968 3511 V 986 3511
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16542
V 65 w(j)1038 3478 y FX(\017)1105 3511 y Gg(be)g(a)h(translation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16543
(which)f(maps)e(elements)j(of)d FY(S)g Gg(to)h(elements)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16544
FY(R)f Gg(such)277 3624 y(that)d(\(for)g(all)g Ga(a;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16545
b(b)26 b F6(2)e FY(S)p Gg(\):)p Black Black 1254 3823
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16546
a F6(\017)74 b Gg(if)23 b Ga(a)1571 3786 y Gc(S)1614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16547
3795 y FV(1)1524 3823 y F6(\000)-31 b(\000)g(!)25 b Ga(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16548
Gg(,)d(then)j F6(j)p Ga(a)p F6(j)2082 3790 y FX(\017)2203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16549
3786 y Gc(R)2144 3823 y F6(\000)-31 b(\000)g(!)2315 3790
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16550
y FX(\003)2377 3823 y F6(j)p Ga(b)p F6(j)2466 3790 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16551
FX(\017)1254 3971 y F6(\017)74 b Gg(if)23 b Ga(a)1571
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16552
3934 y Gc(S)1614 3943 y FV(2)1524 3971 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16553
b(\000)g(!)25 b Ga(b)p Gg(,)d(then)j F6(j)p Ga(a)p F6(j)2082
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16554
3938 y FX(\017)2203 3934 y Gc(R)2144 3971 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16555
b(\000)g(!)2315 3938 y F9(+)2396 3971 y F6(j)p Ga(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16556
F6(j)2485 3938 y FX(\017)277 4198 y Gg(Then)24 b(we)e(ha)n(v)o(e)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16557
(that)f F4(\()p FY(S)p Ga(;)1153 4161 y Gc(S)1091 4198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16558
y F6(\000)-32 b(\000)h(!)p F4(\))23 b Gg(is)g(strongly)j(normalising.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16559
Black 277 4436 a F7(Pr)l(oof)o(.)p Black 34 w Gg(Assume)e(for)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16560
(sak)o(e)g(of)g(deri)n(ving)h(a)e(contradiction)29 b(that)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16561
F4(\()p FY(S)p Ga(;)2639 4399 y Gc(S)2577 4436 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16562
b(\000)f(!)p F4(\))24 b Gg(is)h(not)g(strongly)i(nor)n(-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16563
4548 y(malising.)57 b(W)l(ithout)34 b(loss)f(of)f(generality)j(we)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16564
(may)g(assume)h(that)g(e)n(v)o(ery)g(reduction)i(sequence)f(in)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16565
4661 y F4(\()p FY(S)p Ga(;)464 4624 y Gc(S)402 4661 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16566
F6(\000)-31 b(\000)g(!)o F4(\))23 b Gg(is)h(of)f(the)h(form)1269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16567
4843 y Ga(a)1388 4806 y Gc(S)1431 4815 y FV(1)1342 4843
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16568
y F6(\000)-31 b(\000)f(!)1512 4805 y FX(\003)1577 4843
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16569
y Ga(b)1688 4806 y Gc(S)1731 4815 y FV(2)1641 4843 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16570
F6(\000)h(\000)g(!)25 b Ga(c)1948 4806 y Gc(S)1991 4815
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16571
y FV(1)1901 4843 y F6(\000)-31 b(\000)g(!)2072 4805 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16572
FX(\003)2136 4843 y Ga(d)2255 4806 y Gc(S)2298 4815 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16573
FV(2)2209 4843 y F6(\000)g(\000)f(!)25 b Ga(:)15 b(:)g(:)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16574
5041 y Gg(Suppose)24 b(the)f(abo)o(v)o(e)g(reduction)j(sequence)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16575
(denotes)f(an)f(in\002nite)h(reduction)h(sequence,)g(then)e(using)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16576
Black 277 5114 1290 4 v 383 5170 a F3(6)412 5201 y F2(W)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16577
b(e)18 b(will)g(not)h(use)g(this)g(lemma)g(in)g(this)g(section,)g(b)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16578
(ut)f(in)h(a)g(later)f(one.)p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16579
%%Page: 52 64
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16580
TeXDict begin 52 63 bop Black -144 51 a Gb(52)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16581
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(the)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16582
b(translation)i F6(j)p 906 317 28 4 v 923 317 V 941 317
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16583
V 64 w(j)993 284 y FX(\017)1064 317 y Gg(we)c(can)i(map)e(it)h(onto)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16584
(an)f(in\002nite)h(reduction)h(sequence)h(in)d F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16585
FY(R)p Ga(;)3267 280 y Gc(R)3209 317 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16586
F4(\))p Gg(,)34 b(as)321 430 y(sho)n(wn)24 b(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16587
b(.)1041 565 y Ga(a)1225 528 y Gc(S)1268 537 y FV(1)1143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16588
565 y F6(\000)-21 b(\000)g(\000)g(!)1384 527 y FX(\003)1482
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16589
565 y Ga(b)1681 528 y Gc(S)1724 537 y FV(2)1599 565 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16590
F6(\000)g(\000)g(\000)g(!)78 b Ga(c)2097 528 y Gc(S)2140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16591
537 y FV(1)2016 565 y F6(\000)-21 b(\000)g(\000)g(!)2257
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16592
527 y FX(\003)2350 565 y Ga(d)2554 528 y Gc(S)2597 537
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16593
y FV(2)2472 565 y F6(\000)g(\000)h(\000)e(!)26 b Ga(:)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16594
b(:)g(:)996 929 y F6(j)p Ga(a)p F6(j)1094 891 y FX(\017)1003
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16595
747 y(\017)1043 810 y F6(#)p 1063 746 4 149 v 1237 892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16596
a Gc(R)1143 929 y F6(\000)-21 b(\000)g(\000)g(!)1384
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16597
891 y FX(\003)1437 929 y F6(j)p Ga(b)p F6(j)1526 891
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16598
y FX(\017)1439 747 y(\017)1480 810 y F6(#)p 1500 746
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16599
V 1663 892 a Gc(R)1570 929 y F6(\000)f(\000)i(\000)f(!)1811
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16600
891 y F9(+)1873 929 y F6(j)p Ga(c)p F6(j)1962 891 y FX(\017)1875
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16601
747 y(\017)1916 810 y F6(#)p 1936 746 V 2109 892 a Gc(R)2016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16602
929 y F6(\000)g(\000)g(\000)g(!)2257 891 y FX(\003)2305
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16603
929 y F6(j)p Ga(d)p F6(j)2402 891 y FX(\017)2312 747
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16604
y(\017)2353 810 y F6(#)p 2373 746 V 2537 892 a Gc(R)2443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16605
929 y F6(\000)g(\000)g(\000)g(!)2684 891 y F9(+)2768
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16606
929 y Ga(:)15 b(:)g(:)321 1165 y Gg(By)30 b(assumption)j(we)d(kno)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16607
(that)h(e)n(v)o(ery)1706 1128 y Gc(S)1749 1137 y FV(1)1660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16608
1165 y F6(\000)-31 b(\000)f(!)1830 1132 y FX(\003)1870
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16609
1165 y Gg(-reduction)33 b(is)d(\002nite.)50 b(Thus)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16610
b(we)g(kno)n(w)h(that)g(there)321 1278 y(must)g(be)f(in\002nitely)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16611
(man)o(y)1278 1241 y Gc(S)1321 1250 y FV(2)1231 1278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16612
y F6(\000)-31 b(\000)g(!)p Gg(-reductions.)52 b(Ho)n(we)n(v)o(er)l(,)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16613
b(this)f(implies)g(that)g(we)f(ha)n(v)o(e)h(in\002nitely)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16614
1391 y(man)o(y)598 1354 y Gc(R)539 1391 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16615
b(\000)g(!)710 1358 y F9(+)769 1391 y Gg(-reductions,)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16616
b(which)c(contradicts)k(our)c(assumption)j(about)e F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16617
FY(R)p Ga(;)2864 1354 y Gc(R)2806 1391 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16618
b(\000)f(!)p F4(\))18 b Gg(being)h(strongly)321 1504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16619
y(normalising.)p 3480 1504 4 62 v 3484 1446 55 4 v 3484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16620
1504 V 3538 1504 4 62 v 462 1708 a(Let)30 b(us)h(return)h(to)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16621
(question)h(of)f(whether)g(or)g(not)g F4(\()p FY(T)2325
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16622
1675 y FX($)2396 1708 y Ga(;)2478 1671 y Gc(l)q(oc)2436
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16623
1708 y F6(\000)-31 b(\000)f(!)p F4(\))30 b Gg(is)h(strongly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16624
(normalising.)321 1821 y(It)j(seems)g(as)g(if)f(the)h(lemma)g(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16625
(en)g(abo)o(v)o(e)g(should)i(be)d(helpful,)39 b(since)d(it)f(is)f(easy)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16626
i(to)e(sho)n(w)h(that)321 1934 y F4(\()p FY(T)417 1901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16627
y FX($)488 1934 y Ga(;)594 1897 y Gc(x)529 1934 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16628
b(\000)f(!)p F4(\))21 b Gg(and)g F4(\()p FY(T)1002 1901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16629
y FX($)1073 1934 y Ga(;)1165 1897 y Gc(g)r(c)1114 1934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16630
y F6(\000)-32 b(\000)h(!)p F4(\))20 b Gg(are)i(strongly)h(normalising)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16631
(\(we)e(do)g(not)g(allo)n(w)-6 b(,)22 b(using)g(terminology)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16632
2047 y(from)27 b(e)o(xplicit)i(substitution)h(calculi,)f(substitution)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16633
(composition\).)42 b(Furthermore,)29 b(we)d(should)j(be)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16634
2159 y(able)21 b(to)g(\002nd)f(a)g(translation)j(which)e(maps)1726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16635
2122 y Gc(x)1660 2159 y F6(\000)-31 b(\000)g(!)p Gg(-reductions)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16636
b(and)2462 2122 y Gc(g)r(c)2410 2159 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16637
Gg(-reductions)24 b(onto)d(identities,)321 2272 y(and)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16638
2235 y Gc(l)571 2212 y FC(0)486 2272 y F6(\000)-32 b(\000)h(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16639
Gg(-)33 b(and)933 2235 y Gc(c)964 2212 y FC(00)883 2272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16640
y F6(\000)-31 b(\000)g(!)p Gg(-reductions)37 b(onto)d(a)g(series)h(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16641
2133 2235 y Gc(aux)2112 2272 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16642
Gg(-reductions.)62 b(Whilst)35 b(this)f(approach)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16643
2385 y(seems)i(plausible,)k(in)35 b(f)o(act,)k(it)c(f)o(ails)h(when)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16644
(using)i(the)e(ob)o(vious)i(candidate)h(for)d(the)h(translation,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16645
2498 y(namely)25 b(the)e(one)h(that)h(replaces)g(labelled)h(cuts)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16646
(with)f(substitutions,)k(as)d(indicated)i(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16647
b(.)p Black Black 1119 2714 a F6(j)p FL(Cut)1223 2666
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16648
y FC( )1284 2714 y F4(\()1319 2702 y FX(h)1347 2714 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16649
Ga(a)1395 2702 y FX(i)1422 2714 y Ga(M)11 b(;)1561 2702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16650
y F9(\()1588 2714 y Ga(x)1640 2702 y F9(\))1668 2714
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16651
y Ga(N)f F4(\))p F6(j)1811 2681 y FX(\017)1891 2662 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16652
F5(def)1898 2714 y F4(=)47 b F6(j)p Ga(M)10 b F6(j)2164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16653
2681 y FX(\017)2200 2714 y F6(f)-7 b Ga(a)25 b F4(:=)2433
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16654
2702 y F9(\()2460 2714 y Ga(x)2512 2702 y F9(\))2540
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16655
2714 y F6(j)p Ga(N)10 b F6(j)2673 2681 y FX(\017)2703
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16656
2714 y F6(g)1119 2886 y(j)p FL(Cut)1223 2839 y FC(!)1284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16657
2886 y F4(\()1319 2874 y FX(h)1347 2886 y Ga(a)1395 2874
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16658
y FX(i)1422 2886 y Ga(M)h(;)1561 2874 y F9(\()1588 2886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16659
y Ga(x)1640 2874 y F9(\))1668 2886 y Ga(N)f F4(\))p F6(j)1811
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16660
2853 y FX(\017)1891 2835 y F5(def)1898 2886 y F4(=)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16661
b F6(j)p Ga(N)10 b F6(j)2149 2853 y FX(\017)2185 2886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16662
y F6(f)-7 b Ga(x)25 b F4(:=)2421 2874 y FX(h)2449 2886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16663
y Ga(a)2497 2874 y FX(i)2524 2886 y F6(j)p Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16664
F6(j)2672 2853 y FX(\017)2703 2886 y F6(g)321 3078 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16665
Gg(Using)29 b(this)g(translation,)j(the)d(proof)g(f)o(ails)g(because)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16666
(in)f F4(\()p FY(T)2216 3045 y FX($)2287 3078 y Ga(;)2369
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16667
3041 y Gc(l)q(oc)2327 3078 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16668
F4(\))28 b Gg(we)g(can)g(perform)i(reductions)321 3191
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16669
y(that)d(cannot)i(be)d(\223mirrored\224)j(by)e(an)o(y)f(reduction)j(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16670
e F4(\()p FY(T)t Ga(;)2179 3154 y Gc(aux)2159 3191 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16671
F6(\000)-31 b(\000)f(!)p F4(\))p Gg(.)37 b(Consider)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16672
b(for)d(e)o(xample)i(the)f(term)321 3304 y Ga(M)36 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16673
F6(\021)25 b FL(Cut)619 3257 y FC( )680 3304 y F4(\()715
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16674
3292 y FX(h)743 3304 y Ga(a)791 3292 y FX(i)819 3304
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16675
y Ga(S)5 b(;)920 3292 y F9(\()947 3304 y Ga(x)999 3292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16676
y F9(\))1027 3304 y Ga(T)13 b F4(\))31 b Gg(with)g Ga(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16677
Gg(not)i(free)g(in)f Ga(S)5 b Gg(.)52 b(So)31 b Ga(M)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16678
b Gg(is)31 b(translated)j(to)e F6(j)p Ga(S)5 b F6(j)2928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16679
3271 y FX(\017)2963 3304 y F6(f)-7 b Ga(a)40 b F4(:=)3225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16680
3292 y F9(\()3252 3304 y Ga(x)3304 3292 y F9(\))3332
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16681
3304 y F6(j)p Ga(T)13 b F6(j)3448 3271 y FX(\017)3478
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16682
3304 y F6(g)p Gg(,)321 3417 y(which)25 b(is)e(just)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16683
F6(j)p Ga(S)5 b F6(j)916 3384 y FX(\017)979 3417 y Gg(because)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16684
b(of)e(the)g(assumption)i(about)g Ga(a)p Gg(.)j(Clearly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16685
b(,)25 b(there)g(is)e(no)h(reason)i(that)e(an)o(y)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16686
3530 y(reduction)k Ga(T)37 b Gg(may)25 b(perform)h(can)g(be)f(mirrored)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16687
i(by)e F6(j)p Ga(S)5 b F6(j)2107 3497 y FX(\017)2147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16688
3530 y Gg(,)24 b(and)i(consequently)-6 b(,)30 b(we)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16689
b(cannot)j(guaran-)321 3643 y(tee)j(that)f(an)g(in\002nite)h(reduction)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16690
h(sequence)h(occurs)e(in)f(\(using)i(again)e(terminology)j(from)d(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16691
(xplicit)321 3756 y(substitution)d(calculi\))e(garbage)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16692
(substitutions.)33 b(This)22 b(is)f(ho)n(w)h(Melli)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16693
b(\036)-35 b(es)23 b([1995])h(constructed)h(a)d(non-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16694
3869 y(terminating)k(reduction)g(sequence)g(for)e Ga(\025\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16695
Gg(.)462 3998 y(Because)k(of)e(this)g(f)o(ailure)i(our)f(strong)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16696
(normalisation)j(proof)d(for)f F4(\()p FY(T)2684 3965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16697
y FX($)2755 3998 y Ga(;)2838 3961 y Gc(l)q(oc)2796 3998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16698
y F6(\000)-32 b(\000)h(!)p F4(\))26 b Gg(is)g(again)g(rather)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16699
4111 y(in)l(v)n(olv)o(ed.)55 b(An)31 b(important)i(f)o(act)e(in)h(this)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16700
g(proof)g(is)f(that)2212 4074 y Gc(x)2147 4111 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16701
b(\000)f(!)31 b Gg(is)g(con\003uent,)k(in)c(contrast)i(to)3395
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16702
4074 y Gc(l)q(oc)3353 4111 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16703
Gg(,)321 4224 y(which)24 b(clearly)h(is)f(not.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16704
321 4412 a Gb(Lemma)f(2.6.4:)p Black 34 w Gg(The)g(reduction)1483
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16705
4375 y Gc(x)1417 4412 y F6(\000)-31 b(\000)g(!)22 b Gg(is)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16706
Black 417 4587 a(\(i\))p Black 47 w(strongly)j(normalising)h(and)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16707
Black 392 4763 a(\(ii\))p Black 47 w(con\003uent.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16708
321 4976 a F7(Pr)l(oof)o(.)p Black 34 w Gg(The)20 b(measure)i(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16709
(en)f(on)g(P)o(age)f(151)h(implies)h Fs([)p Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16710
Fs(])26 b Ga(>)f Fs([)p Ga(N)10 b Fs(])21 b Gg(whene)n(v)o(er)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16711
Ga(M)3041 4938 y Gc(x)2976 4976 y F6(\000)-32 b(\000)h(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16712
b Ga(N)10 b Gg(.)27 b(There-)321 5088 y(fore)561 5051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16713
y Gc(x)495 5088 y F6(\000)-31 b(\000)g(!)28 b Gg(must)g(be)h(strongly)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16714
(normalising.)46 b(Con\003uence)30 b(of)2449 5051 y Gc(x)2383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16715
5088 y F6(\000)-31 b(\000)g(!)27 b Gg(follo)n(ws)j(from)e(local)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16716
(con\003u-)321 5201 y(ence,)37 b(which)d(can)g(easily)h(be)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16717
(established)j(by)d(inspection)i(of)e(the)g(reduction)i(rules,)h(and)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16718
(strong)321 5314 y(normalisation)27 b(of)1008 5277 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16719
Gc(x)942 5314 y F6(\000)-31 b(\000)g(!)p Gg(.)p 3480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16720
5314 V 3484 5256 55 4 v 3484 5314 V 3538 5314 4 62 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16721
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16722
%%Page: 53 65
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16723
TeXDict begin 53 64 bop Black 277 51 a Gb(2.6)23 b(Localised)i(V)-9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16724
b(ersion)2779 b(53)p 277 88 3691 4 v Black 277 317 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16725
Gg(As)33 b(a)h(consequence)j(of)d(this)h(lemma,)h(we)d(may)g(de\002ne)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16726
(the)f(unique)h Ga(x)p Gg(-normal)g(form)f(of)g(a)g(term)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16727
430 y(belonging)26 b(to)e FY(T)818 397 y FX($)889 430
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16728
y Gg(.)p Black 277 618 a Gb(De\002nition)f(2.6.5:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16729
34 w Gg(The)g(unique)i Ga(x)p Gg(-normal)g(form)e(of)h(a)f(term)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16730
Ga(M)36 b F6(2)24 b FY(T)2546 585 y FX($)2640 618 y Gg(is)f(denoted)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16731
(by)d F6(j)p Ga(M)10 b F6(j)3295 632 y Gc(x)3340 618
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16732
y Gg(.)277 855 y(Before)24 b(we)f(continue,)i(tw)o(o)f(quick)g(observ)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16733
(ations)j(about)e Ga(x)p Gg(-normal)g(forms.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16734
277 1043 a Gb(Lemma)e(2.6.6:)p Black 34 w Gg(If)g(and)h(only)h(if)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16735
Ga(M)33 b Gg(is)23 b(an)h(element)g(of)g FY(T)t Gg(,)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16736
b(then)24 b F6(j)p Ga(M)10 b F6(j)2494 1057 y Gc(x)2564
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16737
1043 y F6(\021)25 b Ga(M)10 b Gg(.)p Black 277 1256 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16738
F7(Pr)l(oof)o(.)p Black 34 w Gg(If)23 b Ga(M)35 b F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16739
b FY(T)t Gg(,)18 b(then)24 b Ga(M)33 b Gg(cannot)24 b(contain)h(an)o(y)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16740
f(instance)h(of)e FL(Cut)2433 1208 y FC( )2516 1256 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16741
Gg(or)g FL(Cut)2693 1208 y FC(!)2754 1256 y Gg(,)f(and)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16742
F7(vice)g(ver)o(sa)p Gg(.)p 3436 1256 4 62 v 3440 1197
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16743
55 4 v 3440 1256 V 3494 1256 4 62 v Black 277 1443 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16744
Gb(Lemma)f(2.6.7:)p Black 34 w Gg(F)o(or)g(all)g(unlabelled)k(terms,)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16745
(for)h(e)o(xample)g FL(And)2366 1457 y Gc(R)2424 1443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16746
y F4(\()2459 1431 y FX(h)2486 1443 y Ga(a)2534 1431 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16747
FX(i)2562 1443 y Ga(M)10 b(;)2700 1431 y FX(h)2728 1443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16748
y Ga(b)2767 1431 y FX(i)2794 1443 y Ga(N)g(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16749
F4(\))p Gg(,)24 b(we)f(ha)n(v)o(e)881 1667 y F6(j)p FL(And)1061
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16750
1681 y Gc(R)1119 1667 y F4(\()1154 1655 y FX(h)1182 1667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16751
y Ga(a)1230 1655 y FX(i)1257 1667 y Ga(M)10 b(;)1395
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16752
1655 y FX(h)1423 1667 y Ga(b)1462 1655 y FX(i)1489 1667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16753
y Ga(N)g(;)15 b(c)p F4(\))p F6(j)1711 1681 y Gc(x)1782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16754
1667 y F6(\021)25 b FL(And)2032 1681 y Gc(R)2090 1667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16755
y F4(\()2125 1655 y FX(h)2153 1667 y Ga(a)2201 1655 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16756
FX(i)2228 1667 y F6(j)p Ga(M)10 b F6(j)2376 1681 y Gc(x)2421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16757
1667 y Ga(;)2461 1655 y FX(h)2489 1667 y Ga(b)2528 1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16758
y FX(i)2555 1667 y F6(j)p Ga(N)g F6(j)2688 1681 y Gc(x)2732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16759
1667 y Ga(;)15 b(c)p F4(\))27 b Ga(:)277 1891 y Gg(Similarly)d(for)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16760
(the)g(other)g(unlabelled)j(terms.)p Black 277 2104 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16761
F7(Pr)l(oof)o(.)p Black 34 w Gg(By)c(inspection)j(of)d(the)1358
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16762
2067 y Gc(x)1292 2104 y F6(\000)-31 b(\000)g(!)o Gg(-reductions.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16763
3436 2104 V 3440 2046 55 4 v 3440 2104 V 3494 2104 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16764
62 v 277 2314 a(Ne)o(xt,)23 b(we)g(shall)h(pro)o(v)o(e)g(that)1272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16765
2277 y Gc(x)1207 2314 y F6(\000)-31 b(\000)g(!)22 b Gg(correctly)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16766
(simulates)f(the)f(substitution)j(operation.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16767
277 2502 a Gb(Lemma)c(2.6.8:)p Black 34 w Gg(F)o(or)g(all)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16768
Ga(M)5 b(;)15 b(N)36 b F6(2)25 b FY(T)e Gg(we)f(ha)n(v)o(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16769
Black 373 2733 a(\(i\))p Black 46 w FL(Cut)583 2685 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16770
FC( )644 2733 y F4(\()679 2721 y FX(h)707 2733 y Ga(a)755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16771
2721 y FX(i)782 2733 y Ga(M)11 b(;)921 2721 y F9(\()948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16772
2733 y Ga(x)1000 2721 y F9(\))1028 2733 y Ga(N)f F4(\))1237
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16773
2695 y Gc(x)1171 2733 y F6(\000)-31 b(\000)g(!)1342 2700
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16774
y F9(+)1426 2733 y Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)1752
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16775
2721 y F9(\()1780 2733 y Ga(x)1832 2721 y F9(\))1859
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16776
2733 y Ga(N)p F6(g)p Black 348 2946 a Gg(\(ii\))p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16777
46 w FL(Cut)583 2899 y FC(!)644 2946 y F4(\()679 2934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16778
y FX(h)707 2946 y Ga(a)755 2934 y FX(i)782 2946 y Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16779
b(;)905 2934 y F9(\()933 2946 y Ga(x)985 2934 y F9(\))1013
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16780
2946 y Ga(M)g F4(\))1237 2909 y Gc(x)1171 2946 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16781
b(\000)g(!)1342 2913 y F9(+)1426 2946 y Ga(M)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16782
b Ga(x)26 b F4(:=)1756 2934 y FX(h)1783 2946 y Ga(a)1831
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16783
2934 y FX(i)1859 2946 y Ga(N)p F6(g)p Black 277 3159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16784
a F7(Pr)l(oof)o(.)p Black 34 w Gg(Both)d(cases)i(by)e(induction)k(on)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16785
(the)h(structure)i(of)d Ga(M)10 b Gg(;)23 b(note)i(we)d(assumed)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16786
Ga(M)5 b(;)15 b(N)36 b F6(2)25 b FY(T)t Gg(.)p 3436 3159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16787
V 3440 3101 55 4 v 3440 3159 V 3494 3159 4 62 v 277 3370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16788
a(If)e(we)g(apply)i F6(j)p 741 3370 28 4 v 759 3370 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16789
776 3370 V 65 w(j)829 3384 y Gc(x)896 3370 y Gg(to)e(the)h(lemma)f(abo)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16790
o(v)o(e,)h(we)e(ha)n(v)o(e)j(the)e(follo)n(wing)i(corollary)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16791
b(.)p Black 277 3557 a Gb(Cor)n(ollary)26 b(2.6.9:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16792
Black 34 w Gg(F)o(or)d(all)h Ga(M)5 b(;)15 b(N)35 b F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16793
b FY(T)e Gg(we)g(thus)h(ha)n(v)o(e)p Black 373 3788 a(\(i\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16794
Black 46 w F6(j)p FL(Cut)608 3741 y FC( )669 3788 y F4(\()704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16795
3776 y FX(h)732 3788 y Ga(a)780 3776 y FX(i)808 3788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16796
y Ga(M)10 b(;)946 3776 y F9(\()974 3788 y Ga(x)1026 3776
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16797
y F9(\))1053 3788 y Ga(N)g F4(\))p F6(j)1196 3802 y Gc(x)1266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16798
3788 y F6(\021)25 b Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)1688
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16799
3776 y F9(\()1715 3788 y Ga(x)1767 3776 y F9(\))1795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16800
3788 y Ga(N)p F6(g)p Black 348 4002 a Gg(\(ii\))p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16801
46 w F6(j)p FL(Cut)608 3955 y FC(!)669 4002 y F4(\()704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16802
3990 y FX(h)732 4002 y Ga(a)780 3990 y FX(i)808 4002
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16803
y Ga(M)10 b(;)946 3990 y F9(\()974 4002 y Ga(x)1026 3990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16804
y F9(\))1053 4002 y Ga(N)g F4(\))p F6(j)1196 4016 y Gc(x)1266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16805
4002 y F6(\021)25 b Ga(N)5 b F6(f)-7 b Ga(x)25 b F4(:=)1677
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16806
3990 y FX(h)1704 4002 y Ga(a)1752 3990 y FX(i)1780 4002
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16807
y Ga(M)p F6(g)277 4262 y Gg(This)i(corollary)i(concerns)g(subterms)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16808
(belonging)h(to)d FY(T)t Gg(,)22 b(b)n(ut)28 b(we)e(can)h(no)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16809
(easily)h(generalise)i(it)c(to)277 4375 y(all)e(terms)f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16810
FY(T)775 4342 y FX($)846 4375 y Gg(.)p Black 277 4563
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16811
a Gb(Lemma)f(2.6.10:)p Black 35 w Gg(F)o(or)f(all)i Ga(M)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16812
b(;)15 b(N)35 b F6(2)25 b FY(T)1538 4530 y FX($)1632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16813
4563 y Gg(we)e(ha)n(v)o(e)p Black 373 4793 a(\(i\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16814
Black 46 w F6(j)p FL(Cut)608 4746 y FC( )669 4793 y F4(\()704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16815
4781 y FX(h)732 4793 y Ga(a)780 4781 y FX(i)808 4793
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16816
y Ga(M)10 b(;)946 4781 y F9(\()974 4793 y Ga(x)1026 4781
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16817
y F9(\))1053 4793 y Ga(N)g F4(\))p F6(j)1196 4807 y Gc(x)1291
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16818
4793 y F6(\021)50 b(j)p Ga(M)10 b F6(j)1560 4807 y Gc(x)1600
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16819
4793 y F6(f)-7 b Ga(a)26 b F4(:=)1833 4781 y F9(\()1860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16820
4793 y Ga(x)1912 4781 y F9(\))1955 4793 y F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16821
b F6(j)2088 4807 y Gc(x)2123 4793 y F6(g)p Black 348
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16822
5007 a Gg(\(ii\))p Black 46 w F6(j)p FL(Cut)608 4960
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16823
y FC(!)669 5007 y F4(\()704 4995 y FX(h)732 5007 y Ga(a)780
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16824
4995 y FX(i)808 5007 y Ga(M)g(;)946 4995 y F9(\()974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16825
5007 y Ga(x)1026 4995 y F9(\))1053 5007 y Ga(N)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16826
F6(j)1196 5021 y Gc(x)1291 5007 y F6(\021)50 b(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16827
b F6(j)1545 5021 y Gc(x)1585 5007 y F6(f)-7 b Ga(x)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16828
b F4(:=)1822 4995 y FX(h)1849 5007 y Ga(a)1897 4995 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16829
FX(i)1940 5007 y F6(j)p Ga(M)10 b F6(j)2088 5021 y Gc(x)2123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16830
5007 y F6(g)p Black 277 5220 a F7(Pr)l(oof)o(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16831
34 w Gg(By)23 b(calculation)j(using)f(Lemma)d(2.6.6)i(and)g(Corollary)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16832
(2.6.9)f(\(see)g(P)o(age)f(151\).)p 3436 5220 4 62 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16833
3440 5162 55 4 v 3440 5220 V 3494 5220 4 62 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16834
eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16835
%%Page: 54 66
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16836
TeXDict begin 54 65 bop Black -144 51 a Gb(54)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16837
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(No)n(w)g(we)h(are)h(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16838
f(a)g(position)j(to)e(sho)n(w)f(that)h(the)1894 280 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16839
Gc(l)q(oc)1852 317 y F6(\000)-31 b(\000)f(!)p Gg(-reductions)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16840
b(project)e(onto)2934 280 y Gc(aux)2913 317 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16841
b(\000)g(!)p Gg(-reductions.)p Black 321 505 a Gb(Lemma)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16842
b(2.6.11:)p Black 35 w Gg(F)o(or)f(all)i Ga(M)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16843
b(N)36 b F6(2)24 b FY(T)1582 472 y FX($)1679 505 y Gg(if)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16844
Ga(M)1922 468 y Gc(l)q(oc)1880 505 y F6(\000)-31 b(\000)f(!)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16845
b Ga(N)10 b Gg(,)22 b(then)i F6(j)p Ga(M)10 b F6(j)2531
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16846
519 y Gc(x)2621 468 y(aux)2601 505 y F6(\000)-31 b(\000)f(!)2771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16847
472 y FX(\003)2836 505 y F6(j)p Ga(N)10 b F6(j)2969 519
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16848
y Gc(x)3013 505 y Gg(.)p Black 321 718 a F7(Pr)l(oof)o(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16849
Black 34 w Gg(By)23 b(induction)j(on)e(the)f(de\002nition)j(of)1833
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16850
680 y Gc(l)q(oc)1791 718 y F6(\000)-31 b(\000)f(!)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16851
b Gg(\(see)h(P)o(age)f(151\).)p 3480 718 4 62 v 3484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16852
660 55 4 v 3484 718 V 3538 718 4 62 v 321 922 a(As)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16853
b(e)o(xplained)j(earlier)l(,)g(this)e(lemma)f(is)h(not)g(strong)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16854
(enough)g(to)f(pro)o(v)o(e)g(strong)h(normalisation)h(for)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16855
1035 y F4(\()p FY(T)417 1002 y FX($)488 1035 y Ga(;)571
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16856
998 y Gc(l)q(oc)529 1035 y F6(\000)-31 b(\000)f(!)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16857
Gg(.)47 b(In)30 b(the)g(proof,)i(which)e(we)f(shall)i(gi)n(v)o(e)f(ne)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16858
(xt,)h(we)e(mak)o(e)h(use)g(of)g(the)g(recursi)n(v)o(e)i(path)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16859
1148 y(ordering)26 b(theorem)f(by)e(Dersho)n(witz.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16860
Black 321 1335 a Gb(De\002nition)g(2.6.12)h Gg(\(Recursi)n(v)o(e)h(P)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16861
(ath)e(Ordering,)h(Dersho)n(witz,)h(1982\))p Gb(:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16862
2660 1302 a F5(7)321 1448 y Gg(Let)e Ga(s)i F6(\021)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16863
Ga(f)10 b F4(\()p Ga(s)762 1462 y F9(1)801 1448 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16864
b(:)g(:)g(:)h(;)f(s)1045 1462 y Gc(m)1112 1448 y F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16865
b Gg(and)h Ga(t)h F6(\021)g Ga(g)s F4(\()p Ga(t)1592
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16866
1462 y F9(1)1632 1448 y Ga(;)15 b(:)g(:)g(:)i(;)e(t)1867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16867
1462 y Gc(n)1914 1448 y F4(\))23 b Gg(be)h(terms,)f(then)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16868
Ga(s)h(>)2646 1415 y Gc(r)r(po)2779 1448 y Ga(t)p Gg(,)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16869
Black Black 451 1633 a(if)e(and)h(only)g(if)806 1757
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16870
y(\(i\))119 b Ga(s)1053 1771 y Gc(i)1106 1757 y F6(\025)1177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16871
1724 y Gc(r)r(po)1309 1757 y Ga(t)23 b Gg(for)g(some)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16872
Ga(i)i F4(=)f(1)p Ga(;)15 b(:)g(:)g(:)i(;)e(m)550 b Gg(\(subterm\))451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16873
1881 y(or)780 2005 y(\(ii\))120 b Ga(f)34 b F6(\035)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16874
b Ga(g)i Gg(and)d Ga(s)g(>)1567 1972 y Gc(r)r(po)1700
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16875
2005 y Ga(t)1733 2019 y Gc(j)1792 2005 y Gg(for)g(all)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16876
Ga(j)31 b F4(=)25 b(1)p Ga(;)15 b(:)g(:)g(:)i(;)e(n)238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16877
b Gg(\(decreasing)27 b(heads\))451 2130 y(or)755 2254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16878
y(\(iii\))120 b Ga(f)34 b F4(=)25 b Ga(g)h Gg(and)e F6(f)-24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16879
b(j)p Ga(s)1497 2268 y F9(1)1537 2254 y Ga(;)15 b(:)g(:)g(:)i(;)e(s)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16880
1782 2268 y Gc(m)1849 2254 y F6(j)-24 b(g)26 b Ga(>)1992
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16881
2210 y Gc(r)r(po)1992 2283 y(mul)q(t)2172 2254 y F6(f)-24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16882
b(j)p Ga(t)2251 2268 y F9(1)2291 2254 y Ga(;)15 b(:)g(:)g(:)i(;)e(t)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16883
2526 2268 y Gc(n)2573 2254 y F6(j)-24 b(g)119 b Gg(\(equal)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16884
b(heads\))321 2439 y(where)h F6(\035)e Gg(is)i(a)e(precedence)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16885
b(de\002ned)d(o)o(v)o(er)f(term)g(constructors,)k Ga(>)2496
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16886
2395 y Gc(r)r(po)2496 2468 y(mul)q(t)2675 2439 y Gg(is)d(the)f(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16887
(xtension)j(of)d Ga(>)3439 2406 y Gc(r)r(po)321 2552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16888
y Gg(to)f(\002nite)f(multisets)i(and)f F6(\025)1200 2519
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16889
y Gc(r)r(po)1331 2552 y Gg(means)f Ga(>)1656 2519 y Gc(r)r(po)1787
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16890
2552 y Gg(or)g(equi)n(v)n(alent)j(up)e(to)f(permutation)j(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16891
(subterms.)p Black 321 2789 a Gb(Theor)n(em)h(2.6.13)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16892
Gg(\(Dersho)n(witz,)g(1982\))p Gb(:)p Black 36 w Gg(The)e(recursi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16893
(e)j(path)f(ordering)h Ga(>)2822 2756 y Gc(r)r(po)2954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16894
2789 y Gg(is)e(well-founded,)321 2902 y(if)f(and)g(only)g(if)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16895
(precedence)j F6(\035)22 b Gg(is)i(well-founded.)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16896
3156 y(Unfortunately)-6 b(,)27 b(tw)o(o)c(problems)i(preclude)h(a)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16897
(direct)i(application)h(of)e(Dersho)n(witz')-5 b(s)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16898
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16899
b(the)f(theorem)h(requires)h(a)d(well-founded)k(precedence)g(for)d(our)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16900
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16901
b(our)c(reduction)i(rules)e(include)i(the)d(tw)o(o)g(reductions)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16902
(\(written)d(schemati-)549 3581 y(cally\))p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16903
1327 3771 a FL(Cut)p F4(\()p 1502 3771 28 4 v 1520 3771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16904
V 1537 3771 V 65 w Ga(;)p 1607 3771 V 1625 3771 V 1642
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16905
3771 V 80 w F4(\))1854 3734 y Gc(c)1885 3711 y FC(00)1805
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16906
3771 y F6(\000)-32 b(\000)h(!)100 b FL(Cut)2154 3724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16907
y FC( )2214 3771 y F4(\()p 2251 3771 V 2269 3771 V 2287
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16908
3771 V 65 w Ga(;)p 2357 3771 V 2374 3771 V 2392 3771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16909
V 80 w F4(\))1012 3884 y FL(Cut)1091 3837 y FC( )1151
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16910
3884 y F4(\()p FL(Cut)q F4(\()p 1362 3884 V 1380 3884
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16911
V 1397 3884 V 64 w Ga(;)p 1467 3884 V 1485 3884 V 1502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16912
3884 V 80 w F4(\))p Ga(;)p 1607 3884 V 1625 3884 V 1642
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16913
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16914
b(\000)h(!)100 b FL(Cut)o F4(\()p FL(Cut)2326 3837 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16915
FC( )2387 3884 y F4(\()p 2424 3884 V 2443 3884 V 2460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16916
3884 V 65 w Ga(;)p 2530 3884 V 2547 3884 V 2565 3884
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16917
V 80 w F4(\))p Ga(;)15 b FL(Cut)2746 3837 y FC( )2808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16918
3884 y F4(\()p 2845 3884 V 2863 3884 V 2880 3884 V 65
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16919
w Ga(;)p 2950 3884 V 2968 3884 V 2985 3884 V 80 w F4(\)\))549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16920
4071 y Gg(and)24 b(consequently)j(we)c(ha)n(v)o(e)h(the)g(c)o(ycle)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16921
Black Black 1643 4556 a @beginspecial 115 @llx 600 @lly
Christian Urban <urbanc@in.tum.de>
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diff changeset
 16922
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16923
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Christian Urban <urbanc@in.tum.de>
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 16924
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16925
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16926
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16927
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16928
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Christian Urban <urbanc@in.tum.de>
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 16929
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16930
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16932
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16933
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16934
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16935
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16936
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 16950
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16951
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16952
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16953
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16954
1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16955
1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16956
0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16957
sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16958
rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16959
gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16960
/chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16961
/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16962
A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16963
get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16964
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16965
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16966
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16967
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16968
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16969
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16970
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16971
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16972
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16973
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16974
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16975
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16976
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16977
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16978
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16979
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16980
(LaserWriter 16/600)]{A length product length le{A length product exch 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16981
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16982
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16983
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16984
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16985
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16986
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16987
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16988
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16989
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16990
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16992
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16993
%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16994
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16995
% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16996
% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16997
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16998
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 16999
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17000
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17001
/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17002
/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17003
/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17004
/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17005
/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17006
/L /lineto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17007
/T /translate load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17008
/TMatrix { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17009
/RAngle { 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17010
/Atan { /atan load stopped { pop pop 0 } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17011
/Div { dup 0 eq { pop } { div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17012
/NET { neg exch neg exch T } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17013
/Pyth { dup mul exch dup mul add sqrt } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17014
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17015
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17016
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17017
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17018
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17019
/pathforall load stopped { pop pop pop pop } if z } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17020
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17021
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17022
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17023
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17024
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17025
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17026
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17027
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17028
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17029
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17030
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17031
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17032
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17033
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17034
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17035
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17036
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17037
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17038
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17039
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17040
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17041
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17042
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17043
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17044
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17045
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17046
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17047
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17048
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17049
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17050
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17051
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17052
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17053
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17054
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17055
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17056
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17057
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17058
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17059
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17060
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17061
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17062
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17063
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17064
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17065
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17066
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17067
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17068
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17069
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17070
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17071
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17072
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17073
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17074
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17075
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17076
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17077
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17078
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17079
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17080
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17081
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17082
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17083
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17084
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17085
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17086
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17087
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17088
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17089
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17090
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17091
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17092
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17093
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17094
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17095
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17096
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17097
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17098
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17099
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17100
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17101
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17102
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17103
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17104
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17105
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17106
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17107
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17108
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17109
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17110
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17111
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17112
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17113
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17114
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17115
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17116
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17117
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17118
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17119
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17120
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17121
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17122
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17123
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17124
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17125
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17126
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17127
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17128
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17129
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17130
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17131
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17132
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17133
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17134
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17135
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17136
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17137
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17138
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17139
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17140
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17141
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17142
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17143
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17144
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17145
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17146
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17147
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17148
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17149
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17150
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17151
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17152
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17153
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17154
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17155
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17156
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17157
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17158
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17159
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17160
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17161
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17162
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17163
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17164
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17165
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17166
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17167
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17168
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17169
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17170
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17171
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17172
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17173
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17174
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17175
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17176
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17177
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17178
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17179
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17180
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17181
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17182
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17183
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17184
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17185
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17186
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17187
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17188
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17189
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17190
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17191
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17192
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17193
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17194
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17195
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17196
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17197
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17198
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17199
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17200
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17201
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17202
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17203
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17204
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17205
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17206
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17207
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17208
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17209
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17210
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17211
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17212
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17213
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17214
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17215
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17216
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17217
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17218
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17219
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17220
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17221
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17222
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17223
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17224
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17225
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17226
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17227
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17228
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17229
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17230
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17231
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17232
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17233
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17234
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17235
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17236
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17237
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17238
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17239
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17240
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17241
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17242
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17243
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17244
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17245
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17246
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17247
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17248
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17249
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17250
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17251
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17252
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17253
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17254
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17255
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17256
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17257
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17258
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17259
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17260
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17261
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17262
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17263
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17264
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17265
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17266
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17267
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17268
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17269
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17270
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17271
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17272
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17273
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17274
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17275
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17276
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17277
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17278
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17279
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17280
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17281
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17282
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17283
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17284
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17285
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17286
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17287
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17288
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17289
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17290
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17291
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17292
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17293
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17294
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17295
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17296
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17297
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17298
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17299
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17300
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17301
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17302
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17303
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17304
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17305
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17306
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17307
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17308
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17309
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17310
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17311
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17312
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17313
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17314
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17315
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17316
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17317
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17318
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17319
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17320
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17321
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17322
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17323
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17324
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17325
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17326
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17327
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17328
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17329
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17330
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17331
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17332
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17333
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17334
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17335
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17336
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17337
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17338
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17339
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17340
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17341
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17342
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17343
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17344
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17345
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17346
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17347
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17348
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17349
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17350
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17351
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17352
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17353
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17354
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17355
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17356
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17357
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17358
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17359
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17360
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17361
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17362
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17363
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17364
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17365
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17366
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17367
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17368
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17369
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17370
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17371
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17372
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17373
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17374
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17375
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17376
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17377
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17378
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17379
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17380
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17381
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17382
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17383
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17384
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17385
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17386
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17387
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17388
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17389
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17390
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17391
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17392
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17393
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17394
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17395
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17396
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17397
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17398
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17399
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17400
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17401
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17402
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17403
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17404
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17405
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17406
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17407
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17408
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17409
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17410
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17411
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17412
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17413
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17414
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17415
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17416
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17417
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17418
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17419
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17420
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17421
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17422
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17423
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17424
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17425
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17426
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17427
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17428
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17429
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17430
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17431
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17432
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17433
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17434
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17435
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17436
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17437
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17438
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17439
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17440
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17441
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17442
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17443
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17444
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17445
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17446
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17447
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17448
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17449
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17450
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17451
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17452
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17453
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17454
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17455
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17456
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17457
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17458
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17459
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17460
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17461
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17462
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17463
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17464
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17465
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17466
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17467
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17468
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17469
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17470
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17471
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17472
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17473
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17474
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17475
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17476
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17477
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17478
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17479
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17480
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17481
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17482
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17483
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17484
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17485
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17486
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17487
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17488
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17489
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17490
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17491
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17492
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17493
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17494
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17495
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17496
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17497
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17498
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17499
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17500
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17501
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17502
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17503
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17504
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17505
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17506
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17507
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17508
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17509
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17510
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17511
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17512
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17513
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17514
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17515
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17516
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17517
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17518
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17519
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17520
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17521
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17522
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17523
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17524
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17525
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17526
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17527
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17528
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17529
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17530
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17531
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17532
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17533
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17534
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17535
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17536
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17537
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17538
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17539
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17540
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17541
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17542
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17544
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17545
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17546
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17547
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17548
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17549
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17550
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17551
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17552
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17553
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17554
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17555
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17556
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17557
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17558
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17559
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17560
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17561
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17562
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17563
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17564
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17565
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17566
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17567
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17569
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17570
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17571
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17572
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17573
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17574
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17575
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17576
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17577
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17578
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17579
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17580
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17581
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17582
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17583
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17584
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17585
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17586
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17587
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17588
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17589
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17590
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17591
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17592
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17593
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17594
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17595
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17596
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17597
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17598
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17599
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17600
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17601
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17602
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17603
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17604
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17605
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17606
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17607
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17608
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17609
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17610
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17611
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17612
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17613
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17615
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17616
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17617
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17618
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17619
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17620
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17621
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17622
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17623
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17624
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17625
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17626
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17627
    Char end stringwidth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17628
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17629
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17630
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17631
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17632
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17633
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17634
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17635
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17636
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17637
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17638
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17639
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17640
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17641
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17642
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17643
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17644
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17645
%%BeginProcSet: special.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17646
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17647
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17844
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 17904
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17915
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17917
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17920
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17921
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17922
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17923
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17924
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17925
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17926
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17927
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17928
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17929
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17930
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17931
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17932
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17933
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 17936
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 17938
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17939
y Ga(a)2795 4254 y FX(i)2822 4266 y Ga(S)5 b(;)2923 4254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17940
y F9(\()2951 4266 y Ga(x)3003 4254 y F9(\))3031 4266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17941
y Ga(T)12 b F4(\))p F6(j)3156 4280 y Gc(x)3201 4266 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17942
F4(\))493 4413 y FL(Cut)572 4365 y FC(!)633 4413 y F4(\()668
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17943
4401 y FX(h)696 4413 y Ga(a)744 4401 y FX(i)771 4413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17944
y Ga(S)5 b(;)872 4401 y F9(\()900 4413 y Ga(x)952 4401
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17945
y F9(\))980 4413 y Ga(T)12 b F4(\))p 493 4450 588 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17946
1128 4361 a F5(def)1135 4413 y F4(=)54 b F6(h)p Ga(S)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17947
1295 4428 61 4 v 5 w F6(i)1391 4427 y Gc(n)1439 4413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17948
y Ga(T)p 1439 4428 66 4 v 294 w(n)1887 4361 y F5(def)1894
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17949
4413 y F4(=)i FW(M)t(A)t(X)t(R)t(E)t(D)2361 4427 y Gc(aux)2484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17950
4413 y F4(\()p F6(j)p FL(Cut)2623 4365 y FC(!)2684 4413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17951
y F4(\()2719 4401 y FX(h)2747 4413 y Ga(a)2795 4401 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17952
FX(i)2822 4413 y Ga(S)5 b(;)2923 4401 y F9(\()2951 4413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17953
y Ga(x)3003 4401 y F9(\))3031 4413 y Ga(T)12 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17954
F6(j)3156 4427 y Gc(x)3201 4413 y F4(\))277 4630 y Gg(Otherwise)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17955
Black Black 930 4743 a FL(Ax)p F4(\()p Ga(x;)j(a)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17956
930 4780 314 4 v 1344 4692 a F5(def)1351 4743 y F4(=)106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17957
b Ga(?)679 4890 y FL(Not)822 4904 y Gc(R)880 4890 y F4(\()915
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17958
4878 y F9(\()943 4890 y Ga(x)995 4878 y F9(\))1022 4890
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17959
y Ga(M)10 b(;)15 b(a)p F4(\))p 679 4927 565 4 v 1344
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17960
4838 a F5(def)1351 4890 y F4(=)106 b FB(L)p Ga(M)p 1563
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17961
4905 99 4 v 10 w FB(M)463 5037 y FL(And)617 5051 y Gc(R)675
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17962
5037 y F4(\()710 5025 y FX(h)738 5037 y Ga(a)786 5025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17963
y FX(i)813 5037 y Ga(M)11 b(;)952 5025 y FX(h)979 5037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17964
y Ga(b)1018 5025 y FX(i)1046 5037 y Ga(N)f(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17965
F4(\))p 463 5074 782 4 v 1344 4985 a F5(def)1351 5037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17966
y F4(=)106 b FB(L)p Ga(M)p 1563 5052 99 4 v 10 w(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17967
b(N)p 1701 5052 83 4 v 11 w FB(M)736 5184 y FL(Or)835
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17968
5147 y Gc(i)835 5207 y(R)893 5184 y F4(\()928 5172 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17969
FX(h)956 5184 y Ga(a)1004 5172 y FX(i)1031 5184 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17970
b(;)15 b(b)p F4(\))p 736 5221 509 4 v 1344 5132 a F5(def)1351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17971
5184 y F4(=)106 b FB(L)p Ga(M)p 1563 5199 99 4 v 10 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17972
FB(M)584 5331 y FL(Imp)728 5353 y Gc(R)786 5331 y F4(\()821
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17973
5319 y F9(\()849 5331 y Ga(x)901 5319 y F9(\))p FX(h)956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17974
5331 y Ga(a)1004 5319 y FX(i)1031 5331 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17975
b(b)p F4(\))p 584 5368 661 4 v 1344 5279 a F5(def)1351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17976
5331 y F4(=)106 b FB(L)p Ga(M)p 1563 5346 99 4 v 10 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17977
FB(M)2170 4817 y FL(Not)2313 4831 y Gc(L)2365 4817 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17978
F4(\()2400 4805 y FX(h)2428 4817 y Ga(a)2476 4805 y FX(i)2504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17979
4817 y Ga(M)10 b(;)15 b(x)p F4(\))p 2170 4854 559 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17980
2829 4765 a F5(def)2836 4817 y F4(=)106 b FB(L)p Ga(M)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17981
3048 4832 99 4 v 11 w FB(M)2159 4963 y FL(And)2313 4927
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17982
y Gc(i)2313 4986 y(L)2365 4963 y F4(\()2400 4951 y F9(\()2428
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17983
4963 y Ga(x)2480 4951 y F9(\))2508 4963 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17984
b(y)s F4(\))p 2159 5001 571 4 v 2829 4912 a F5(def)2836
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17985
4963 y F4(=)106 b FB(L)p Ga(M)p 3048 4978 99 4 v 11 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17986
FB(M)1990 5111 y FL(Or)2089 5125 y Gc(L)2141 5111 y F4(\()2176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17987
5099 y F9(\()2204 5111 y Ga(x)2256 5099 y F9(\))2283
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17988
5111 y Ga(M)10 b(;)2421 5099 y F9(\()2449 5111 y Ga(y)2497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17989
5099 y F9(\))2524 5111 y Ga(N)g(;)15 b(z)t F4(\))p 1990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17990
5148 740 4 v 2829 5059 a F5(def)2836 5111 y F4(=)106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17991
b FB(L)p Ga(M)p 3048 5126 99 4 v 11 w(;)15 b(N)p 3187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17992
5126 83 4 v 10 w FB(M)1943 5257 y FL(Imp)2087 5279 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17993
Gc(L)2139 5257 y F4(\()2174 5245 y FX(h)2202 5257 y Ga(a)2250
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17994
5245 y FX(i)2277 5257 y Ga(M)c(;)2416 5245 y F9(\()2443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17995
5257 y Ga(x)2495 5245 y F9(\))2523 5257 y Ga(N)f(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17996
b(y)s F4(\))p 1943 5295 787 4 v 2829 5206 a F5(def)2836
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17997
5257 y F4(=)106 b FB(L)p Ga(M)p 3048 5272 99 4 v 11 w(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17998
b(N)p 3187 5272 83 4 v 10 w FB(M)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 17999
%%Page: 56 68
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18000
TeXDict begin 56 67 bop Black -144 51 a Gb(56)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18001
b(Calculi)p -144 88 3691 4 v Black 321 317 a Gg(Because)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18002
b(of)f(the)h(restriction)i(put)d(on)g FY(T)1617 284 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18003
FX($)1613 340 y Gc(<)p FX(1)1739 317 y Gg(,)h(it)f(is)f(not)i(dif)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18004
(\002cult)g(to)f(pro)o(v)o(e)g(that)h(this)g(translation)i(is)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18005
430 y(well-de\002ned.)p Black 321 618 a Gb(Lemma)23 b(2.6.18:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18006
Black 35 w Gg(The)g(translation)p 1503 633 60 4 v 111
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18007
w F4(:)i FY(T)1699 585 y FX($)1695 640 y Gc(<)p FX(1)1846
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18008
618 y F6(!)h FY(H)c Gg(is)h(well-de\002ned.)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18009
830 a F7(Pr)l(oof)o(.)p Black 34 w Gg(Assume)32 b Ga(M)52
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18010
b F6(2)41 b FY(T)1206 797 y FX($)1202 853 y Gc(<)p FX(1)1328
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18011
830 y Gg(.)55 b(W)-7 b(e)31 b(ha)n(v)o(e)i(by)g(de\002nition)h(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18012
FY(T)2427 797 y FX($)2423 853 y Gc(<)p FX(1)2580 830
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18013
y Gg(that)h(the)g Ga(x)p Gg(-normal)g(form)g(of)321 943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18014
y(e)n(v)o(ery)25 b(subterm,)h Ga(N)10 b Gg(,)24 b(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18015
Ga(M)34 b Gg(is)25 b(an)f(element)i(in)e Ga(S)5 b(N)1968
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18016
957 y Gc(aux)2091 943 y Gg(.)31 b(Consequently)-6 b(,)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18017
b FW(M)t(A)t(X)t(R)t(E)t(D)3028 957 y Gc(aux)3150 943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18018
y F4(\()p F6(j)p Ga(N)10 b F6(j)3318 957 y Gc(x)3362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18019
943 y F4(\))25 b Gg(is)f(a)321 1056 y(natural)h(number)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18020
b(.)p 3480 1056 4 62 v 3484 998 55 4 v 3484 1056 V 3538
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18021
1056 4 62 v 462 1260 a(The)32 b(ne)o(xt)h(lemma)f(is)g(useful)h(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18022
(comparing)j(labels)e(of)f(terms)g(belonging)j(to)e FY(H)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18023
Gg(;)h(it)e(relates)321 1373 y(tw)o(o)27 b(terms)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18024
FY(T)875 1340 y FX($)871 1396 y Gc(<)p FX(1)1024 1373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18025
y Gg(according)j(to)e(the)f(number)i(of)2071 1336 y Gc(aux)2051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18026
1373 y F6(\000)-31 b(\000)f(!)p Gg(-reductions)31 b(their)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18027
Ga(x)p Gg(-normal)h(form)e(can)321 1486 y(perform.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18028
Black 321 1674 a Gb(Lemma)c(2.6.19:)p Black 35 w Gg(F)o(or)f(all)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18029
(terms)g Ga(M)5 b(;)15 b(N)35 b F6(2)25 b FY(T)1807 1641
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18030
y FX($)1803 1696 y Gc(<)p FX(1)1952 1674 y Gg(we)e(ha)n(v)o(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18031
Black 417 1846 a(\(i\))p Black 49 w FW(M)t(A)t(X)t(R)t(E)t(D)891
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18032
1860 y Gc(aux)1013 1846 y F4(\()p F6(j)p Ga(M)10 b F6(j)1196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18033
1860 y Gc(x)1241 1846 y F4(\))25 b F6(\025)i FW(M)t(A)t(X)t(R)t(E)t(D)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18034
1739 1860 y Gc(aux)1862 1846 y F4(\()p F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18035
b F6(j)2030 1860 y Gc(x)2074 1846 y F4(\))p Gg(,)23 b(pro)o(vided)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18036
Ga(M)2665 1809 y Gc(l)q(oc)2623 1846 y F6(\000)-31 b(\000)f(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18037
b Ga(N)10 b Gg(.)p Black 392 1990 a(\(ii\))p Black 49
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18038
w FW(M)t(A)t(X)t(R)t(E)t(D)891 2004 y Gc(aux)1013 1990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18039
y F4(\()p F6(j)p Ga(M)g F6(j)1196 2004 y Gc(x)1241 1990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18040
y F4(\))35 b F6(\025)h FW(M)t(A)t(X)t(R)t(E)t(D)1758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18041
2004 y Gc(aux)1880 1990 y F4(\()p F6(j)p Ga(N)10 b F6(j)2048
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18042
2004 y Gc(x)2093 1990 y F4(\))p Gg(,)29 b(pro)o(vided)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18043
Ga(N)38 b Gg(is)28 b(an)h(immediate)g(subterm)549 2103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18044
y(of)23 b Ga(M)10 b Gg(,)23 b(and)h Ga(M)32 b Gg(is)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18045
b(unlabelled.)p Black 367 2247 a(\(iii\))p Black 49 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18046
FW(M)t(A)t(X)t(R)t(E)t(D)891 2261 y Gc(aux)1013 2247
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18047
y F4(\()p F6(j)p Ga(M)10 b F6(j)1196 2261 y Gc(x)1241
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18048
2247 y F4(\))k Ga(>)i FW(M)t(A)t(X)t(R)t(E)t(D)1718 2261
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18049
y Gc(aux)1840 2247 y F4(\()p F6(j)p Ga(N)10 b F6(j)2008
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18050
2261 y Gc(x)2053 2247 y F4(\))p Gg(,)25 b(pro)o(vided)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18051
Ga(M)2674 2210 y Gc(l)2696 2186 y FC(0)2611 2247 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18052
b(\000)f(!)30 b Ga(N)35 b Gg(or)26 b Ga(M)3197 2210 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18053
Gc(c)3228 2186 y FC(00)3148 2247 y F6(\000)-32 b(\000)h(!)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18054
b Ga(N)35 b Gg(on)549 2360 y(the)23 b(outermost)j(le)n(v)o(el.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18055
Black 321 2572 a F7(Pr)l(oof)o(.)p Black 34 w Gg(\(i\))31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18056
b(follo)n(ws)h(from)f(Lemma)f(2.6.11;)36 b(for)31 b(\(ii\))h(note)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18057
(that)f(all)h(reductions)i(which)d F6(j)p Ga(N)10 b F6(j)3345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18058
2586 y Gc(x)3420 2572 y Gg(can)321 2685 y(perform)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18059
b(can)e(be)h(performed)h(by)e F6(j)p Ga(M)10 b F6(j)1556
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18060
2699 y Gc(x)1601 2685 y Gg(;)23 b(\(iii\))h(is)f(by)g(simple)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18061
(calculation)j(using)d(Lemma)f(2.6.7)g(and)321 2798 y(the)h(f)o(act)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18062
(that)g(the)g(side)g(conditions)j(put)d(on)1779 2761
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18063
y Gc(c)1810 2737 y FC(00)1729 2798 y F6(\000)-31 b(\000)g(!)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18064
b Gg(ensures)k(that)e F6(j)p Ga(M)10 b F6(j)2525 2812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18065
y Gc(x)2615 2761 y(aux)2595 2798 y F6(\000)-32 b(\000)h(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18066
b(j)p Ga(N)10 b F6(j)2923 2812 y Gc(x)2967 2798 y Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18067
3480 2798 V 3484 2740 55 4 v 3484 2798 V 3538 2798 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18068
62 v 321 3002 a(No)n(w)26 b(we)g(are)h(in)g(the)h(position)h(to)e(pro)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18069
(v)o(e)g(the)h(\(main\))f(lemma)g(that)g(relates)i(e)n(v)o(ery)3044
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18070
2965 y Gc(l)q(oc)3002 3002 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18071
Gg(-reduction)321 3115 y(to)21 b(a)g(pair)h(of)f(terms)h(that)f(belong)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18072
i(to)e FY(H)g Gg(and)g(which)h(are)f(ordered)i(decreasingly)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18073
(according)f(to)d Ga(>)3416 3082 y Gc(r)r(po)3524 3115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18074
y Gg(.)p Black 321 3303 a Gb(Lemma)i(2.6.20:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18075
35 w Gg(F)o(or)f(all)i Ga(M)5 b(;)15 b(N)36 b F6(2)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18076
b FY(T)1582 3270 y FX($)1578 3325 y Gc(<)p FX(1)1704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18077
3303 y Gg(,)f(if)g Ga(M)1993 3265 y Gc(l)q(oc)1952 3303
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18078
y F6(\000)-32 b(\000)h(!)25 b Ga(N)10 b Gg(,)22 b(then)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18079
Ga(M)p 2455 3318 99 4 v 35 w(>)2649 3270 y Gc(r)r(po)2782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18080
3303 y Ga(N)p 2782 3318 83 4 v 10 w Gg(.)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18081
3515 a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)e(induction)j(on)e(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18082
(de\002nition)j(of)1833 3478 y Gc(l)q(oc)1791 3515 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18083
F6(\000)-31 b(\000)f(!)23 b Gg(\(see)h(P)o(ages)g(152\226155\).)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18084
3480 3515 4 62 v 3484 3457 55 4 v 3484 3515 V 3538 3515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18085
4 62 v 321 3719 a(Using)j(this)g(lemma,)f(we)f(can)i(sho)n(w)f(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18086
(e)n(v)o(ery)1943 3682 y Gc(l)q(oc)1901 3719 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18087
b(\000)h(!)p Gg(-reduction)29 b(sequence)g(starting)f(with)e(a)g(term)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18088
321 3832 y(of)e FY(T)f Gg(is)g(terminating.)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18089
4020 a Gb(Lemma)32 b(2.6.21:)p Black 35 w Gg(Ev)o(ery)1231
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18090
3983 y Gc(l)q(oc)1189 4020 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18091
Gg(-reduction)35 b(sequence)g(starting)g(with)d(a)g(term)h(belonging)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18092
(to)e FY(T)f Gg(is)321 4133 y(terminating.)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18093
4345 a F7(Pr)l(oof)o(.)p Black 34 w Gg(Suppose)22 b(for)e(the)h(sak)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18094
(e)g(of)f(deri)n(ving)i(a)e(contradiction)k(that)d(from)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18095
Ga(M)29 b Gg(the)21 b(follo)n(wing)h(in\002nite)321 4458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18096
y(reduction)k(sequence)g(starts.)1074 4624 y Ga(M)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18097
b F6(\021)25 b Ga(M)1382 4638 y F9(1)1488 4587 y Gc(l)q(oc)1447
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18098
4624 y F6(\000)-32 b(\000)h(!)25 b Ga(M)1730 4638 y F9(2)1837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18099
4587 y Gc(l)q(oc)1795 4624 y F6(\000)-31 b(\000)f(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18100
b Ga(M)2078 4638 y F9(3)2185 4587 y Gc(l)q(oc)2143 4624
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18101
y F6(\000)-31 b(\000)g(!)25 b Ga(M)2427 4638 y F9(4)2534
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18102
4587 y Gc(l)q(oc)2492 4624 y F6(\000)-32 b(\000)h(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18103
b Ga(:)15 b(:)g(:)321 4806 y Gg(By)25 b(Lemma)g(2.6.6)h(we)f(ha)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18104
(e)h(for)g(all)g(subterms)h Ga(N)35 b Gg(of)25 b Ga(M)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18105
b Gg(that)26 b F6(j)p Ga(N)10 b F6(j)2511 4820 y Gc(x)2585
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18106
4806 y F6(\021)29 b Ga(N)35 b Gg(\()p Ga(M)g Gg(does)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18107
b(not)f(contain)321 4919 y(an)o(y)21 b(labelled)h(cut\),)f(and)g(by)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18108
(Theorem)h(2.3.19)g(we)e(kno)n(w)h(that)h(each)g(of)f(them)g(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18109
(strongly)j(normalis-)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18110
%%Page: 57 69
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18111
TeXDict begin 57 68 bop Black 277 51 a Gb(2.7)23 b(Notes)3248
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18112
b(57)p 277 88 3691 4 v Black 277 353 a Gg(ing)24 b(under)666
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18113
316 y Gc(aux)645 353 y F6(\000)-31 b(\000)g(!)p Gg(.)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18114
b(Consequently)-6 b(,)26 b(e)n(v)o(ery)g FW(M)t(A)t(X)t(R)t(E)t(D)1969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18115
367 y Gc(aux)2091 353 y F4(\()p F6(j)p Ga(N)10 b F6(j)2259
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18116
367 y Gc(x)2304 353 y F4(\))23 b Gg(is)g(\002nite,)g(and)h(thus)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18117
Ga(M)33 b Gg(belongs)25 b(to)277 466 y FY(T)338 433 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18118
FX($)334 488 y Gc(<)p FX(1)460 466 y Gg(.)j(By)21 b(Lemmas)h(2.6.16)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18119
(and)h(2.6.20)g(we)e(ha)n(v)o(e)i(that)g(the)g(in\002nite)g(reduction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18120
(sequence)f(starting)277 579 y(from)g Ga(M)32 b Gg(can)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18121
b(be)g(mapped)g(onto)h(the)f(decreasing)i(chain)1124
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18122
802 y Ga(M)1212 816 y F9(1)p 1124 830 128 4 v 1277 802
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18123
a Ga(>)1348 765 y Gc(r)r(po)1480 802 y Ga(M)1568 816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18124
y F9(2)p 1480 830 V 1633 802 a Ga(>)1704 765 y Gc(r)r(po)1837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18125
802 y Ga(M)1925 816 y F9(3)p 1837 830 V 1989 802 a Ga(>)2060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18126
765 y Gc(r)r(po)2193 802 y Ga(M)2281 816 y F9(4)p 2193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18127
830 V 2346 802 a Ga(>)2417 765 y Gc(r)r(po)2549 802 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18128
Ga(:)15 b(:)g(:)277 1034 y Gg(which)33 b(ho)n(we)n(v)o(er)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18129
(contradicts)i(the)d(well-foundedness)k(of)c Ga(>)2297
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18130
1001 y Gc(r)r(po)2404 1034 y Gg(.)56 b(Thus)33 b(all)2862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18131
997 y Gc(l)q(oc)2820 1034 y F6(\000)-31 b(\000)g(!)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18132
Gg(-reduction)36 b(se-)277 1147 y(quences)26 b(starting)f(with)e(a)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18133
(term)f(that)h(is)f(an)h(element)g(in)g FY(T)f Gg(must)g(terminate.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18134
3436 1147 4 62 v 3440 1089 55 4 v 3440 1147 V 3494 1147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18135
4 62 v 418 1351 a(Ne)o(xt,)h(we)f(e)o(xtend)i(this)g(lemma)f(to)g(all)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18136
(terms)h(of)f FY(T)2053 1318 y FX($)2123 1351 y Gg(.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18137
b(T)-7 b(o)23 b(do)h(so,)g(we)g(shall)h(\002rst)f(sho)n(w)f(that)i(for)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18138
277 1464 y(e)n(v)o(ery)h Ga(M)39 b F6(2)28 b FY(T)777
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18139
1431 y FX($)873 1464 y Gg(there)e(is)f(a)g(term)g Ga(N)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18140
b F6(2)29 b FY(T)t Gg(,)20 b(such)27 b(that)f Ga(N)2232
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18141
1427 y Gc(l)q(oc)2190 1464 y F6(\000)-31 b(\000)g(!)2361
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18142
1431 y FX(\003)2429 1464 y Ga(M)10 b Gg(.)34 b(Because)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18143
b Ga(N)35 b Gg(is)25 b(an)g(element)277 1577 y(in)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18144
b FY(T)t Gg(,)c(we)j(ha)n(v)o(e)i(that)f Ga(N)40 b Gg(is)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18145
b(strongly)j(normalising)g(by)e(the)g(lemma)g(just)g(gi)n(v)o(en,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18146
(and)e(so)g Ga(M)10 b Gg(,)31 b(too,)277 1690 y(must)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18147
b(be)f(strongly)j(normalising.)p Black 277 1877 a Gb(Lemma)d(2.6.22:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18148
Black 35 w Gg(F)o(or)h(e)n(v)o(ery)g(term)g Ga(M)37 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18149
F6(2)26 b FY(T)1722 1844 y FX($)1816 1877 y Gg(with)e(the)h(typing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18150
(judgement)g F4(\000)2881 1865 y Gc(.)2936 1877 y Ga(M)3059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18151
1865 y Gc(.)3114 1877 y F4(\001)p Gg(,)d(there)j(is)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18152
1990 y(a)d(term)h Ga(N)35 b F6(2)25 b FY(T)d Gg(with)i(the)g(typing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18153
(judgement)g F4(\000)1840 1957 y FX(0)1863 1990 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18154
b F4(\000)1986 1978 y Gc(.)2041 1990 y Ga(N)2149 1978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18155
y Gc(.)2204 1990 y F4(\001)p Ga(;)g F4(\001)2396 1957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18156
y FX(0)2442 1990 y Gg(such)24 b(that)g Ga(N)2940 1953
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18157
y Gc(l)q(oc)2898 1990 y F6(\000)-31 b(\000)g(!)3069 1957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18158
y FX(\003)3133 1990 y Ga(M)10 b Gg(.)p Black 277 2203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18159
a F7(Pr)l(oof)o(.)p Black 34 w Gg(W)-7 b(e)23 b(construct)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18160
Ga(N)33 b Gg(by)24 b(inducti)n(v)o(ely)j(replacing)f(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18161
Ga(M)34 b Gg(all)24 b(occurrences)k(of)c FL(Cut)3009
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18162
2155 y FC( )3094 2203 y Gg(and)g FL(Cut)3327 2155 y FC(!)3411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18163
2203 y Gg(by)277 2316 y(some)g(instances)i(of)d FL(Cut)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18164
Gg(\(see)h(P)o(age)f(155\).)p 3436 2316 V 3440 2257 55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18165
4 v 3440 2316 V 3494 2316 4 62 v 277 2520 a(No)n(w)f(the)i(strong)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18166
(normalisation)i(theorem)d(is)g(by)f(a)h(simple)g(contradiction)j(ar)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18167
(gument.)p Black 277 2707 a Gb(Theor)n(em)d(2.6.23:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18168
Black 34 w Gg(The)g(reduction)i(system)e F4(\()p FY(T)1844
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18169
2674 y FX($)1915 2707 y Ga(;)1997 2670 y Gc(l)q(oc)1955
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18170
2707 y F6(\000)-31 b(\000)g(!)p F4(\))23 b Gg(is)g(strongly)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18171
(normalising.)p Black 277 2920 a F7(Pr)l(oof)o(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18172
34 w Gg(Suppose)21 b Ga(M)35 b F6(2)25 b FY(T)1132 2887
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18173
y FX($)1222 2920 y Gg(is)20 b(not)h(strongly)h(normalising.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18174
b(Then)20 b(by)g(the)h(lemma)e(just)i(gi)n(v)o(en)f(there)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18175
3033 y(is)27 b(a)g(term)g Ga(N)42 b F6(2)32 b FY(T)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18176
b Gg(such)j(that)e Ga(N)1428 2996 y Gc(l)q(oc)1387 3033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18177
y F6(\000)-32 b(\000)h(!)1557 3000 y FX(\003)1628 3033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18178
y Ga(M)10 b Gg(.)40 b(Clearly)-6 b(,)28 b(if)f Ga(M)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18179
b Gg(is)27 b(not)h(strongly)h(normalising,)i(then)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18180
3146 y(so)d(is)g Ga(N)10 b Gg(,)29 b(which)f(ho)n(we)n(v)o(er)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18181
(contradicts)i(Lemma)c(2.6.21.)44 b(Consequently)-6 b(,)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18182
b Ga(M)38 b Gg(must)28 b(be)g(strongly)277 3259 y(normalising.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18183
3436 3259 V 3440 3201 55 4 v 3440 3259 V 3494 3259 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18184
62 v 277 3463 a(While)k(Gentzen-lik)o(e)i(local)f(reductions)h(tend)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18185
(to)e(break)i(strong)g(normalisation,)j(in)c(this)g(section)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18186
3576 y(we)23 b(ha)n(v)o(e)h(sho)n(wn)g(that)g(a)e(strongly)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18187
(normalising)g(cut-elimination)h(procedure)f(can)e(be)g(obtained)h(by)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18188
277 3689 y(replacing)h(our)e(global)h(operation)h(of)e(proof)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18189
(substitution)j(with)d(appropriate)i(local)f(reductions.)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18190
3995 y Ge(2.7)119 b(Notes)277 4218 y Gg(Se)n(v)o(eral)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18191
b(cut-elimination)k(procedures)f(ha)n(v)o(e)d(been)h(introduced)i(in)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18192
(this)h(chapter)l(,)j(and)d(we)e(ha)n(v)o(e)277 4331
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18193
y(focused)e(on)e(sho)n(wing,)i(in)d(each)i(case,)g(the)f(property)i(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18194
e(strong)h(normalisation.)51 b(In)29 b(this)i(section)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18195
4444 y(we)25 b(shall)h(tak)o(e)g(a)f(step)h(back)g(and)g(consider)i(a)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18196
(more)g(global)i(picture.)36 b(W)-7 b(e)24 b(shall)j(\002rst)e(moti)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18197
(v)n(ate)h(our)277 4557 y(use)31 b(of)g(tw)o(o)f(slightly)j(dif)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18198
(ferent)g(notions)g(of)d(proof)i(substitution.)54 b(Ne)o(xt,)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18199
b(we)e(shall)i(compare)g(our)277 4670 y(w)o(ork)23 b(to)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18200
(cut-elimination)j(procedure)e(by)e(Dragalin)h(and)g(to)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18201
(symmetric)h(lambda)g(calculus)277 4783 y(by)g(Barbanera)h(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18202
(Berardi.)277 5046 y Fq(2.7.1)99 b(Wh)o(y)24 b(T)-7 b(w)o(o)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18203
b(Notions)f(of)h(Pr)n(oof)g(Substitution?)277 5237 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18204
Gg(The)33 b(proof)h(substitution)i(denoted)f(by)e F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18205
1606 5237 28 4 v 1624 5237 V 1642 5237 V 65 w(])f Gg(w)o(as)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18206
(introduced)j(in)d(order)h(to)f(a)n(v)n(oid)h(in\002nite)g(reduc-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18207
5350 y(tion)i(sequences)j(arising)e(from)e(the)h(too)g(liberal)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18208
(de\002nition)h(of)d(Gentzen')-5 b(s)37 b(reduction)i(rules)d(for)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18209
Black Black eop end
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 18210
%%Page: 58 70
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parents:
diff changeset
 18211
TeXDict begin 58 69 bop Black -144 51 a Gb(58)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18212
b(Calculi)p -144 88 3691 4 v Black 321 353 a Gg(commuting)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18213
b(cuts.)40 b(W)-7 b(e)26 b(modi\002ed)i(this)f(proof)i(substitution)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18214
(in)d(Section)h(2.3)f(so)g(that)h(independent)321 466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18215
y(substitutions)i(can)d(commute.)36 b(The)26 b(modi\002ed)g(v)n(ariant)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18216
h(w)o(as)f(referred)i(to)e(as)f(auxiliary)k(proof)e(sub-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18217
579 y(stitution)33 b(and)f(denoted)g(by)f F6(f)p 1296
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18218
579 28 4 v 1315 579 V 1332 579 V 65 w(g)p Gg(.)51 b(Both)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18219
b(substitution)35 b(operations)f(are)d(applied)h(when)f(instances)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18220
692 y(of)h(commuting)h(cuts)f(are)g(reduced:)47 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18221
1576 692 V 1594 692 V 1612 692 V 65 w(])31 b Gg(is)g(used)i(in)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18222
(cut-elimination)k(procedure)e F4(\()p FY(T)t Ga(;)3372
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18223
654 y Gc(cut)3341 692 y F6(\000)-32 b(\000)h(!)p F4(\))321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18224
804 y Gg(and)25 b F6(f)p 523 804 V 541 804 V 559 804
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18225
V 65 w(g)f Gg(in)g F4(\()p FY(T)t Ga(;)902 767 y Gc(aux)882
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18226
804 y F6(\000)-32 b(\000)h(!)p F4(\))p Gg(.)30 b(In)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18227
b(Section)h(2.6)g(we)e(deri)n(v)o(ed)i(from)f F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18228
FY(T)t Ga(;)2447 767 y Gc(aux)2427 804 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18229
b(\000)g(!)o F4(\))24 b Gg(the)h(local)g(cut-elimination)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18230
917 y(procedure)30 b F4(\()p FY(T)807 884 y FX($)878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18231
917 y Ga(;)960 880 y Gc(l)q(oc)918 917 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18232
b(\000)g(!)p F4(\))p Gg(,)27 b(where)g(in)f(place)i(of)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18233
(substitution)k(operation)e F6(f)p 2842 917 V 2860 917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18234
V 2878 917 V 65 w(g)p Gg(,)e(local)h(reduction)321 1030
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18235
y(rules)23 b(are)e(applied.)30 b(This)21 b(cut-elimination)k(procedure)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18236
f(could)f(also)f(be)g(deri)n(v)o(ed)g(from)f F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18237
FY(T)t Ga(;)3257 993 y Gc(cut)3226 1030 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18238
b(\000)g(!)o F4(\))q Gg(,)21 b(in)321 1143 y(which)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18239
b(case)g(we)f(w)o(ould)h(ha)n(v)o(e)g(a)g(slightly)h(dif)n(ferent)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18240
(local)e(procedure.)40 b(All)26 b(procedures)k(are)d(com-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18241
1256 y(plete)f(in)f(the)g(sense)g(of)g(eliminating)i(all)e(cuts)g(from)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18242
g(a)f(classical)j(proof.)34 b(Thus)25 b(the)g(question)i(arises,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18243
1369 y(why)22 b(did)g(we)g(introduce)j(tw)o(o)d(cut-elimination)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18244
(procedures)f(for)d FY(T)t Gg(,)c(or)k(more)g(speci\002cally)j(why)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18245
(did)321 1482 y(we)i(introduce)i(tw)o(o)e(proof)i(substitutions?)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18246
b(In)23 b(the)g(follo)n(wing)i(we)d(will)h(discuss)i(the)f(trade-of)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18247
(fs)h(that)321 1595 y(led)f(us)g(to)f(gi)n(v)o(e)h(the)g(tw)o(o)f(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18248
(ersions)i(of)f(substitution)j(and)d(cut-elimination.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18249
1724 y(Whilst)e(our)f(sequent)i(calculus)g(gi)n(v)o(en)e(in)g(Section)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18250
(2.2)g(is)f(rather)i(dif)n(ferent)h(from)e(Gentzen')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18251
b(s)22 b(se-)321 1837 y(quent)i(calculus)g(LK)d(\(all)i(structural)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18252
(rules)g(are)e(implicit\),)i(the)e(cut-elimination)k(rules)e(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18253
F4(\()p FY(T)t Ga(;)3372 1800 y Gc(cut)3341 1837 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18254
b(\000)h(!)p F4(\))321 1950 y Gg(are)32 b(v)o(ery)f(reminiscent)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18255
(his)h(rules,)h(e)o(xcept)f(that)g(we)e(allo)n(w)h(cuts)h(to)f(pass)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18256
(o)o(v)o(er)f(other)h(cuts.)52 b(In)321 2063 y(contrast,)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18257
b(the)c(rules)g(of)g F4(\()p FY(T)t Ga(;)1274 2026 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18258
Gc(aux)1254 2063 y F6(\000)-31 b(\000)f(!)p F4(\))31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18259
b Gg(dif)n(fer)i(some)n(what)f(from)g(Gentzen')-5 b(s)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18260
b(rules.)55 b(The)31 b(dif)n(ferences)321 2176 y(arise)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18261
b(from)e(the)h(tw)o(o)f(clauses)p Black Black 692 2390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18262
a FL(Cut)p F4(\()865 2378 y FX(h)892 2390 y Ga(a)940
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18263
2378 y FX(i)968 2390 y FL(Ax)p F4(\()p Ga(x;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18264
F4(\))q Ga(;)1322 2378 y F9(\()1349 2390 y Ga(y)1397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18265
2378 y F9(\))1425 2390 y Ga(M)10 b F4(\))-5 b F6(f)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18266
Ga(x)26 b F4(:=)1790 2378 y FX(h)1818 2390 y Ga(b)1857
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18267
2378 y FX(i)1884 2390 y Ga(P)t F6(g)2015 2338 y F5(def)2022
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18268
2390 y F4(=)k FL(Cut)p F4(\()2296 2378 y FX(h)2324 2390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18269
y Ga(b)2363 2378 y FX(i)2390 2390 y Ga(P)13 b(;)2501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18270
2378 y F9(\()2529 2390 y Ga(y)2577 2378 y F9(\))2604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18271
2390 y Ga(M)5 b F6(f)-7 b Ga(x)26 b F4(:=)2935 2378 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18272
FX(h)2962 2390 y Ga(b)3001 2378 y FX(i)3029 2390 y Ga(P)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18273
F6(g)q F4(\))701 2530 y FL(Cut)p F4(\()874 2518 y FX(h)902
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18274
2530 y Ga(a)950 2518 y FX(i)977 2530 y Ga(M)10 b(;)1115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18275
2518 y F9(\()1143 2530 y Ga(x)1195 2518 y F9(\))1222
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18276
2530 y FL(Ax)p F4(\()p Ga(x;)15 b(b)p F4(\))q(\))-5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18277
F6(f)e Ga(b)26 b F4(:=)1781 2518 y F9(\()1809 2530 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18278
Ga(y)1857 2518 y F9(\))1884 2530 y Ga(P)t F6(g)2015 2478
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18279
y F5(def)2022 2530 y F4(=)k FL(Cut)p F4(\()2296 2518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18280
y FX(h)2324 2530 y Ga(a)2372 2518 y FX(i)2399 2530 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18281
Ga(M)5 b F6(f)-7 b Ga(b)26 b F4(:=)2717 2518 y F9(\()2744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18282
2530 y Ga(y)2792 2518 y F9(\))2819 2530 y Ga(P)t F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18283
Ga(;)2966 2518 y F9(\()2994 2530 y Ga(y)3042 2518 y F9(\))3069
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18284
2530 y Ga(P)13 b F4(\))321 2732 y Gg(which)24 b(allo)n(w)g(us,)f(for)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18285
(e)o(xample,)g(to)f(reduce)i(the)f(lo)n(wer)f(cut-instance)k(in)d(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18286
(proof)843 3083 y F4(\000)925 3071 y Gc(.)980 3083 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18287
Ga(N)1088 3071 y Gc(.)1143 3083 y Ga(b)17 b F4(:)g Ga(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18288
1406 2871 841 4 v 1406 2955 a(x)g F4(:)g Ga(B)1616 2943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18289
y Gc(.)1671 2955 y FL(Ax)p F4(\()p Ga(x;)e(a)p F4(\))2010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18290
2943 y Gc(.)2065 2955 y Ga(a)i F4(:)h Ga(B)95 b(y)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18291
b F4(:)d Ga(B)2543 2943 y Gc(.)2598 2955 y Ga(M)2721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18292
2943 y Gc(.)2776 2955 y F4(\001)p 1406 2998 1447 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18293
1485 3083 a Ga(x)g F4(:)h Ga(B)1696 3071 y Gc(.)1750
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18294
3083 y FL(Cut)p F4(\()1923 3071 y FX(h)1951 3083 y Ga(a)1999
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18295
3071 y FX(i)2027 3083 y FL(Ax)o F4(\()p Ga(x;)d(a)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18296
Ga(;)2380 3071 y F9(\()2408 3083 y Ga(y)2456 3071 y F9(\))2483
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18297
3083 y Ga(M)10 b F4(\))2642 3071 y Gc(.)2697 3083 y F4(\001)2894
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18298
3028 y Gg(Cut)p 843 3125 1931 4 v 962 3210 a F4(\000)1044
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18299
3198 y Gc(.)1099 3210 y FL(Cut)p F4(\()1272 3198 y FX(h)1299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18300
3210 y Ga(b)1338 3198 y FX(i)1366 3210 y Ga(N)g(;)1489
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18301
3198 y F9(\()1517 3210 y Ga(x)1569 3198 y F9(\))1596
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18302
3210 y FL(Cut)p F4(\()1769 3198 y FX(h)1797 3210 y Ga(a)1845
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18303
3198 y FX(i)1872 3210 y FL(Ax)p F4(\()p Ga(x;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18304
F4(\))q Ga(;)2226 3198 y F9(\()2254 3210 y Ga(y)2302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18305
3198 y F9(\))2329 3210 y Ga(M)10 b F4(\)\))2523 3198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18306
y Gc(.)2578 3210 y F4(\001)2814 3156 y Gg(Cut)321 3413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18307
y(in)24 b(a)f(single)i(step)f(to)1283 3595 y F4(\000)1365
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18308
3583 y Gc(.)1420 3595 y Ga(N)1528 3583 y Gc(.)1583 3595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18309
y Ga(b)17 b F4(:)h Ga(B)95 b(y)20 b F4(:)e Ga(B)2052
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18310
3583 y Gc(.)2107 3595 y Ga(M)2231 3583 y Gc(.)2285 3595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18311
y F4(\001)p 1283 3632 1079 4 v 1362 3717 a(\000)1444
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18312
3705 y Gc(.)1499 3717 y FL(Cut)p F4(\()1672 3705 y FX(h)1700
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18313
3717 y Ga(b)1739 3705 y FX(i)1766 3717 y Ga(N)10 b(;)1889
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18314
3705 y F9(\()1917 3717 y Ga(y)1965 3705 y F9(\))1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18315
3717 y Ga(M)g F4(\))2151 3705 y Gc(.)2206 3717 y F4(\001)2403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18316
3663 y Gg(Cut)25 b Ga(:)321 3920 y Gg(This)30 b(kind)g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18317
(cut-reduction,)35 b(which)30 b(\223mer)n(ges\224)h(tw)o(o)e(cuts)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18318
(into)g(a)f(single)i(cut)f(has,)h(as)e(f)o(ar)h(as)f(we)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18319
4033 y(are)c(a)o(w)o(are,)g(not)g(been)g(considered)j(else)n(where.)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18320
b(Therefore)26 b(comparing)h(the)e(cut-elimination)j(pro-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18321
4146 y(cedure)k F4(\()p FY(T)t Ga(;)746 4109 y Gc(aux)726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18322
4146 y F6(\000)-32 b(\000)h(!)p F4(\))30 b Gg(with)g(traditional)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18323
(cut-elimination)h(procedures,)g(for)c(e)o(xample,)j(according)g(to)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18324
4259 y(which)h(normal)f(forms)g(can)h(be)e(reached)j(from)e(a)f(proof,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18325
k(is)d(rather)h(dif)n(\002cult.)58 b(Much)33 b(simpler)h(is)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18326
4372 y(the)26 b(comparison)i(with)d F4(\()p FY(T)t Ga(;)1256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18327
4335 y Gc(cut)1225 4372 y F6(\000)-31 b(\000)g(!)o F4(\))26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18328
b Gg(because)h(its)f(reduction)i(rules)e(are)g(easier)h(to)e(relate)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18329
(to)e(standard)321 4485 y(reduction)h(rules.)462 4614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18330
y(Another)l(,)g(more)f(important)h(reason)g(for)f(presenting)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18331
F4(\()p FY(T)t Ga(;)2348 4577 y Gc(cut)2317 4614 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18332
b(\000)f(!)p F4(\))24 b Gg(is)h(the)f(close)i(correspondence)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18333
4727 y(with)i(normalisation)j(in)d(natural)h(deduction.)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18334
b(As)27 b(we)g(shall)h(see)h(in)e(Chapter)i(3,)f(we)f(ha)n(v)o(e)i(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18335
f(fol-)321 4840 y(lo)n(wing)c(correspondences)29 b(in)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18336
b(the)h F4(\()p F6(\033)p Ga(;)15 b F6(^)p F4(\))p Gg(-fragment)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18337
b(of)d(intuitionistic)28 b(logic)p Black Black 1190 5032
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18338
a Ga(M)1427 4995 y Gc(int)1352 5032 y F6(\000)-21 b(\000)h(\000)f(!)64
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18339
b Ga(N)1179 5395 y(M)1277 5358 y FX(0)1176 5214 y Fu(n)1218
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18340
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18341
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18342
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18343
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18344
5032 y F6(\000)g(\000)h(\000)f(!)76 b Ga(T)2148 5395
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18345
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18346
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18347
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18348
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18349
5207 V Black Black eop end
Christian Urban <urbanc@in.tum.de>
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diff changeset
 18350
%%Page: 59 71
Christian Urban <urbanc@in.tum.de>
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diff changeset
 18351
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 18352
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18353
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18354
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 18355
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 18356
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18357
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18358
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 18359
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 18360
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18361
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18362
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18363
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18364
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18365
3045 542 y Gc(aux)3025 579 y F6(\000)-31 b(\000)g(!)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18366
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18367
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18368
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18370
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18371
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18372
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18373
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18374
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18375
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18377
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18378
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18379
b(\000)h(!)28 b Gg(is)g(the)h(f)o(act)g(that)g(if)g(we)e(are)i
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 18381
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 18382
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18383
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18384
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18385
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 18386
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18387
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18388
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18389
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18390
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 18392
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18393
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 18394
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18395
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18396
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18397
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18398
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18399
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18400
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18401
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18402
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18403
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18404
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18405
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18406
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18407
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18408
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18409
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18410
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18411
3144 y(abo)o(v)o(e)g(the)g(cut-rule,)i(as)e(illustrated)i(in)e(\(2.2\))
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18412
g(on)f(P)o(age)h(14.)38 b(Reducing)28 b(this)f(kind)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18413
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18414
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18415
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18416
(without)e(a)g(decrease)h(of)f(the)f(height)j(of)d(the)h(subproofs)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18417
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18418
b(is)g(v)o(ery)h(anno)o(ying)i(in)d(a)g(strong)i(normalisation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18419
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18420
b(the)g(proof)h(size)g(or)f(on)g(the)g(proof)h(height)g(are)f(not)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18421
(applicable.)35 b(F)o(ollo)n(wing)25 b(Gentzen,)i(Dragalin)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18422
3708 y(o)o(v)o(ercame)f(this)f(dif)n(\002culty)h(by)f(using)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18423
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18424
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18425
(\(2.2\))h(to)g(a)f(proof)i(with)e(tw)o(o)g(ne)n(w)g(cuts,)h(one)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18426
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18427
(the)e(corresponding)k(inde)o(x)c(of)f(the)h(multicut,)h(as)e(sho)n(wn)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18428
h(belo)n(w)-6 b(.)p Black Black 288 4225 a FG(\000)340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18429
4237 y FJ(1)p 396 4213 10 38 v 405 4196 42 4 v 465 4225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18430
a FG(\001)534 4237 y FJ(1)572 4225 y FU(;)14 b(C)674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18431
4195 y FS(n)p FJ(+1)p 288 4261 515 4 v 330 4334 a FG(\000)382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18432
4346 y FJ(1)p 438 4322 10 38 v 447 4306 42 4 v 507 4334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18433
a FG(\001)576 4346 y FJ(1)614 4334 y FU(;)g(C)716 4304
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18434
y FS(n)831 4283 y Gd(Contr)1021 4295 y FS(R)1116 4334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18435
y FU(C)1181 4304 y FS(m)1244 4334 y FU(;)g FG(\000)1333
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18436
4346 y FJ(2)p 1389 4322 10 38 v 1398 4306 42 4 v 1458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18437
4334 a FG(\001)1527 4346 y FJ(2)p 330 4370 1235 4 v 671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18438
4444 a FG(\000)723 4456 y FJ(1)760 4444 y FU(;)g FG(\000)849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18439
4456 y FJ(2)p 905 4432 10 38 v 914 4415 42 4 v 974 4444
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18440
a FG(\001)1043 4456 y FJ(1)1081 4444 y FU(;)g FG(\001)1187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18441
4456 y FJ(2)1592 4398 y Gd(Multicut)1937 4396 y FT(\000)-25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18442
b(\000)g(!)2158 4334 y FG(\000)2210 4346 y FJ(1)p 2266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18443
4322 10 38 v 2275 4306 42 4 v 2335 4334 a FG(\001)2404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18444
4346 y FJ(1)2442 4334 y FU(;)14 b(C)2544 4304 y FS(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18445
FJ(+1)2729 4334 y FU(C)2794 4304 y FS(m)2857 4334 y FU(;)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18446
FG(\000)2946 4346 y FJ(2)p 3001 4322 10 38 v 3011 4306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18447
42 4 v 3071 4334 a FG(\001)3140 4346 y FJ(2)p 2158 4370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18448
1020 4 v 2391 4444 a FG(\000)2443 4456 y FJ(1)2481 4444
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18449
y FU(;)g FG(\000)2570 4456 y FJ(2)p 2625 4432 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18450
2635 4415 42 4 v 2695 4444 a FG(\001)2764 4456 y FJ(1)2801
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18451
4444 y FU(;)g FG(\001)2907 4456 y FJ(2)3205 4398 y Gd(Multicut)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18452
4739 y Gg(Another)31 b(dif)n(\002culty)g(arises)f(from)g(reduction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18453
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18454
4838 y(with)h(other)h(\(multi\)cuts:)46 b(the)o(y)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18455
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18456
b(Unfor)n(-)277 4938 y(tunately)-6 b(,)38 b(Dragalin)c(did)f(not)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18457
(consider)h(an)o(y)e(of)g(these)h(restricted)i(rules,)g(and)e(without)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18458
(them)f(the)277 5038 y(correpondence)d(between)c(normalisation)i(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18459
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18460
(serious)277 5137 y(shortcoming)k(of)d(his)g(cut-elimination)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18461
(procedure,)f(not)e(just)f(because)i(we)e(\223lose\224)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18462
(correspon-)277 5237 y(dence)32 b(with)f(normalisation)j(\(see)e(ne)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18463
(xt)f(chapter\),)k(b)n(ut)c(also)h(the)o(y)g(are)f(required)i(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18464
(interpret)i(La-)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18465
%%Page: 60 72
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18466
TeXDict begin 60 71 bop Black -144 51 a Gb(60)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18467
b(Calculi)p -144 88 3691 4 v Black 321 388 a Gg(font')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18468
b(s)27 b(e)o(xample)f(as)g(an)f(instance)j(of)e(a)f(non-deterministic)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18469
30 b(choice.)36 b(Although)27 b(Lafont')-5 b(s)27 b(e)o(xample)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18470
488 y(\(translated)g(into)d(Dragalin')-5 b(s)25 b(system\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18471
1399 675 10 38 v 1409 658 42 4 v 1469 687 a FU(B)p 1311
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18472
707 295 4 v 1329 779 10 38 v 1339 762 42 4 v 1399 791
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18473
a(B)t(;)14 b(C)1568 761 y FJ(1)1647 729 y Gd(W)-7 b(eak)1834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18474
741 y FS(R)p 2041 675 10 38 v 2051 658 42 4 v 2111 687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18475
a FU(B)p 1971 707 258 4 v 1971 791 a(C)2036 761 y FJ(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18476
2092 779 10 38 v 2102 762 42 4 v 2162 791 a FU(B)2271
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18477
729 y Gd(W)g(eak)2458 741 y FS(L)p 1311 827 919 4 v 1659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18478
888 10 38 v 1668 872 42 4 v 1728 900 a FU(B)t(;)14 b(B)2271
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18479
855 y Gd(Multicut)p 1640 936 260 4 v 1711 998 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18480
1721 981 42 4 v 1780 1010 a FU(B)1941 958 y Gd(Contr)2131
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18481
970 y FS(R)321 1220 y Gg(can)24 b(reduce)g(non-deterministicall)q(y)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18482
(to)23 b(either)i(subproof,)g(the)e(lo)n(wer)g(cut)g(in)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18483
(\223nested\224)h(v)o(ersion)321 1333 y(of)f(Lafont')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18484
b(s)24 b(e)o(xample)p 909 1671 11 41 v 919 1653 46 5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18485
v 985 1683 a Ga(B)p 813 1703 322 4 v 833 1783 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18486
844 1765 46 5 v 909 1795 a(B)5 b(;)15 b(C)1095 1762 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18487
F9(1)1176 1727 y Gg(W)-7 b(eak)1380 1741 y Gc(R)p 1683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18488
1541 11 41 v 1694 1523 46 5 v 1760 1553 a Ga(B)p 1584
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18489
1573 328 4 v 1605 1653 11 41 v 1615 1635 46 5 v 1681
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18490
1665 a(B)5 b(;)15 b(D)1873 1632 y F9(1)1953 1597 y Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18491
b(eak)2157 1611 y Gc(R)p 1529 1703 439 4 v 1529 1795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18492
a Ga(C)1601 1762 y F9(1)p 1660 1783 11 41 v 1671 1765
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18493
46 5 v 1736 1795 a Ga(B)5 b(;)15 b(D)1928 1762 y F9(1)2009
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18494
1727 y Gg(W)-7 b(eak)2213 1741 y Gc(L)p 813 1833 1155
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18495
4 v 1190 1913 11 41 v 1200 1895 46 5 v 1266 1925 a Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18496
b(;)15 b(B)5 b(;)15 b(D)1572 1892 y F9(1)2009 1863 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18497
Gg(Multicut)p 2492 1801 11 41 v 2502 1783 46 5 v 2568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18498
1813 a Ga(B)p 2413 1833 288 4 v 2413 1925 a(D)2491 1892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18499
y F9(1)p 2550 1913 11 41 v 2561 1895 46 5 v 2626 1925
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18500
a Ga(B)2741 1857 y Gg(W)-7 b(eak)2945 1871 y Gc(L)p 1169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18501
1963 1531 4 v 1756 2029 11 41 v 1767 2011 46 5 v 1832
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18502
2041 a Ga(B)5 b(;)15 b(B)5 b(;)15 b(B)2741 1993 y Gg(Multicut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18503
1736 2079 398 4 v 1751 2116 369 4 v 1870 2182 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18504
1880 2164 46 5 v 1946 2194 a Ga(B)2160 2140 y Gg(Contr)2366
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18505
2154 y Gc(R)321 2444 y Gg(cannot.)40 b(Using)27 b(Dragalin')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18506
b(s)28 b(reduction)i(rules,)e(the)f(lo)n(wer)f(multicut)i(can)g(reduce)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18507
g(to)e(its)h(right)h(sub-)321 2557 y(proof,)g(b)n(ut)e(not)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18508
(immediately)i(to)e(its)g(left)g(one.)37 b(F)o(or)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18509
b(this)h(the)g(upper)h(multicut)h(has)e(to)g(reduce)h(\002rst.)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18510
2670 y(Consequently)-6 b(,)28 b(the)c(lo)n(wer)g(multicut)i(is)e(not)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18511
(a)e(genuine)k(non-deterministic)i(choice)c(as)f(it)g(depends)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18512
2783 y(on)30 b(another)i(multicut.)48 b(Clearly)-6 b(,)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18513
b(to)e(remedy)g(this)g(problem)h(we)e(need)i(reduction)h(rules)f(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18514
(per)n(-)321 2896 y(mute)25 b(the)g(lo)n(wer)g(multicut)h(with)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18515
(upper)h(one.)33 b(As)24 b(mentioned)i(earlier)l(,)h(the)e(na)m(\250)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18516
-27 b(\021v)o(e)25 b(formulation)321 3009 y(of)f(these)g(reduction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18517
(rules,)f(e.g.)p Black Black 445 3238 a FU(:)14 b(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18518
574 3226 10 38 v 583 3210 42 4 v 115 w(:)g(:)g(:)83 b(:)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18519
b(:)g(:)p 966 3226 10 38 v 976 3210 42 4 v 116 w(:)g(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18520
445 3258 702 4 v 641 3324 a(:)g(:)g(:)p 770 3312 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18521
v 780 3295 42 4 v 115 w(:)g(:)g(:)1188 3280 y Gd(Multicut)1475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18522
3292 y FJ(2)1523 3324 y FU(:)g(:)g(:)p 1652 3312 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18523
v 1662 3295 42 4 v 116 w(:)g(:)g(:)p 641 3344 1192 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18524
v 1082 3410 a(:)g(:)g(:)p 1211 3398 10 38 v 1221 3381
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18525
42 4 v 116 w(:)g(:)g(:)1874 3366 y Gd(Multicut)2161 3378
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18526
y FJ(1)893 3731 y F6(\000)-31 b(\000)g(!)1254 3654 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18527
FU(:)14 b(:)g(:)p 1383 3642 10 38 v 1393 3625 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18528
116 w(:)g(:)g(:)83 b(:)14 b(:)g(:)p 1776 3642 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18529
1785 3625 42 4 v 115 w(:)g(:)g(:)p 1254 3674 702 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18530
1450 3740 a(:)g(:)g(:)p 1580 3728 10 38 v 1589 3711 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18531
4 v 116 w(:)g(:)g(:)1997 3696 y Gd(Multicut)2284 3708
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18532
y FJ(1)2357 3654 y FU(:)g(:)g(:)p 2486 3642 10 38 v 2495
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18533
3625 42 4 v 115 w(:)g(:)g(:)83 b(:)14 b(:)g(:)p 2878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18534
3642 10 38 v 2888 3625 42 4 v 115 w(:)g(:)g(:)p 2357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18535
3674 702 4 v 2553 3740 a(:)g(:)g(:)p 2682 3728 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18536
v 2691 3711 42 4 v 115 w(:)g(:)g(:)3100 3696 y Gd(Multicut)3387
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18537
3708 y FJ(1)p 1450 3760 1412 4 v 2002 3825 a FU(:)g(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18538
2131 3813 10 38 v 2140 3797 42 4 v 115 w(:)g(:)g(:)2904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18539
3782 y Gd(Multicut)3191 3794 y FJ(2)321 4078 y Gg(breaks)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18540
b(strong)g(normalisation.)462 4215 y(As)k(f)o(ar)i(as)e(I)h(ha)n(v)o(e)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18541
g(been)h(able)f(to)g(ascertain,)j(the)e(cut-elimination)i(procedure)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18542
(gi)n(v)o(en)d(in)g(Sec-)321 4328 y(tion)c(2.6)f(is)h(the)f(only)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18543
(strongly)i(normalising)g(cut-elimination)h(procedure)f(that)e(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18544
(both)h(local)g(and)321 4440 y(allo)n(ws)20 b(cut-instances)k(to)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18545
b(be)h(permuted)h(with)f(other)h(cut-instances.)2505
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18546
4408 y F5(8)2573 4440 y Gg(Of)e(course)i(we)e(can)h(easily)h(re-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18547
4553 y(formulate)j(Dragalin')-5 b(s)24 b(cut-elimination)i(procedure,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18548
(along)d(the)g(lines)g(of)g F4(\()p FY(T)2825 4520 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18549
FX($)2896 4553 y Ga(;)2978 4516 y Gc(l)q(oc)2936 4553
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18550
y F6(\000)-31 b(\000)f(!)p F4(\))p Gg(,)22 b(to)h(include)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18551
4666 y(reduction)31 b(rules)e(that)g(permute)g(multicuts)h(with)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18552
(other)h(multicuts)h(so)e(that)h(strong)g(normalisation)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18553
4779 y(still)f(holds.)39 b(But)27 b(a)f(strong)i(normalisation)i(proof)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18554
e(is)e(then)i(much)f(harder:)37 b(these)28 b(rules)f(do)g(not)g(de-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18555
4892 y(crease)g(the)f(proof)g(size)g(or)g(proof)g(height.)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18556
b(It)25 b(is)g(not)h(ob)o(vious)h(ho)n(w)e(Dragalin')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18557
b(s)27 b(proof)g(method)f(can)321 5005 y(be)e(adapted)h(to)f(pro)o(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18558
(e)g(strong)h(normalisation)h(in)e(this)g(case.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18559
321 5115 1290 4 v 427 5171 a F3(8)456 5202 y F2(Zuck)o(er)j([1974])g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18560
(obtained)g(a)f(strongly)h(normalising)g(cut-elimination)g(procedure)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18561
(for)f(intuitionistic)g(logic)g(with)321 5294 y(local)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18562
b(reduction)h(rules)e(by)h(considering)h FE(pr)m(oper)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18563
F2(reductions,)f(i.e.,)f(non-repeating)j(and)e(termination)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18564
(reductions,)g(b)o(ut)g(he)321 5385 y(did)h(not)h(restrict)e(his)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18565
(reduction)h(rules)f(accordingly)-5 b(.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18566
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18567
%%Page: 61 73
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18568
TeXDict begin 61 72 bop Black 277 51 a Gb(2.7)23 b(Notes)3248
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18569
b(61)p 277 88 3691 4 v Black 277 388 a Fq(2.7.3)99 b(Comparison)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18570
b(with)g(the)h(Symmetric)f(Lambda)h(Calculus)277 589
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18571
y Gg(In)38 b(this)g(section)h(we)d(compare)j(our)f(w)o(ork)f(with)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18572
(the)f(symmetric)i(lambda)f(calculus,)43 b Ga(\025)3209
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18573
552 y Gc(S)t(y)r(m)3396 589 y Gg(for)277 702 y(short,)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18574
b(de)n(v)o(eloped)f(by)e(Barbanera)i(and)e(Berardi)h([1994].)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18575
b(This)29 b(comparison)k(is)d(w)o(orthwhile)h(be-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18576
815 y(cause)41 b(we)d(adapted)k(their)e(proof)g(technique)i(based)f(on)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18577
e(the)h(notion)h(of)e(symmetric)i(reducibil-)277 928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18578
y(ity)c(candidates,)43 b(and)38 b(moreo)o(v)o(er)g(the)o(y)f(also)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18579
(tak)o(e)h(the)f(vie)n(w)g(that)h(cut-elimination)i(in)d(classical)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18580
1041 y(logic)30 b(corresponds)i(via)d(the)g(Curry-Ho)n(w)o(ard)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18581
(correspondence)k(to)28 b(non-deterministic)34 b(computa-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18582
1154 y(tion)24 b([Barbanera)i(et)d(al.,)g(1997].)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18583
1288 y(The)g(formulae)i(of)f Ga(\025)1085 1251 y Gc(S)t(y)r(m)1258
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18584
1288 y Gg(are)g(gi)n(v)o(en)g(by)f(the)h(grammar)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18585
Black 1218 1542 a Ga(B)5 b(;)15 b(C)32 b F4(::=)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18586
Ga(A)50 b F6(j)h Ga(A)1838 1509 y FX(?)1948 1542 y F6(j)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18587
Ga(B)5 b F6(^)p Ga(C)57 b F6(j)50 b Ga(B)5 b F6(_)o Ga(C)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18588
1808 y Gg(where)23 b Ga(A)g Gg(stands)h(for)f(atomic)h(formulae.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18589
b(The)23 b(ne)o(gation)h(in)f Ga(\025)2281 1771 y Gc(S)t(y)r(m)2454
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18590
1808 y Gg(is)f(not)i(a)e(primiti)n(v)o(e)i(connecti)n(v)o(e,)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18591
1921 y(as)g(in)f(our)h(sequent)h(calculus,)h(instead)f(it)e(is)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18592
(de\002ned)g(by)g(the)f(clauses:)p Black Black 1138 2196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18593
a F4(\()p Ga(A)p F4(\))1276 2163 y FX(?)1371 2144 y F5(def)1379
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18594
2196 y F4(=)42 b Ga(A)1560 2163 y FX(?)1855 2196 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18595
Ga(B)5 b F6(^)p Ga(C)i F4(\))2132 2163 y FX(?)2226 2144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18596
y F5(def)2233 2196 y F4(=)42 b Ga(B)2420 2163 y FX(?)2494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18597
2196 y F6(_)15 b Ga(C)2642 2163 y FX(?)1079 2336 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18598
Ga(A)1182 2303 y FX(?)1242 2336 y F4(\))1277 2303 y FX(?)1371
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18599
2284 y F5(def)1379 2336 y F4(=)42 b Ga(A)295 b F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18600
Ga(B)5 b F6(_)p Ga(C)i F4(\))2132 2303 y FX(?)2226 2284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18601
y F5(def)2233 2336 y F4(=)42 b Ga(B)2420 2303 y FX(?)2494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18602
2336 y F6(^)15 b Ga(C)2642 2303 y FX(?)277 2601 y Gg(The)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18603
b(terms)h(of)f Ga(\025)817 2564 y Gc(S)t(y)r(m)991 2601
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18604
y Gg(are)g(gi)n(v)o(en)h(by)g(the)g(grammar)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18605
942 2855 a Ga(M)5 b(;)15 b(N)35 b F4(::=)26 b Ga(x)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18606
b F6(j)h(h)p Ga(M)5 b(;)15 b(N)10 b F6(i)41 b(j)g Ga(\033)1933
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18607
2869 y Gc(i)1961 2855 y F4(\()p Ga(M)10 b F4(\))41 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18608
F6(j)g Ga(\025x:M)51 b F6(j)41 b Ga(M)30 b(?)21 b(N)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18609
3109 y Gg(where)31 b Ga(x)f Gg(is)h(tak)o(en)g(from)g(a)f(set)h(of)g(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18610
n(ariables.)52 b(There)31 b(are)g(tw)o(o)g(kinds)g(of)g(typing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18611
(judgements)h(in)277 3222 y Ga(\025)330 3186 y Gc(S)t(y)r(m)481
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18612
3222 y Gg(,)22 b F7(viz.)1275 3350 y F4(\000)1357 3338
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18613
y Gc(.)1412 3350 y Ga(M)28 b F4(:)17 b Ga(B)186 b Gg(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18614
F4(\000)2221 3338 y Gc(.)2275 3350 y Ga(M)28 b F4(:)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18615
b F6(?)p Black 824 w Gg(\(2.6\))p Black 277 3556 a(where)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18616
b(in)g(the)g(latter)h F6(?)d Gg(is)i(a)f(distinguished)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18617
b(atomic)c(formula.)40 b(In)26 b(the)h(sequel)i(we)d(are)g(only)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18618
(inter)n(-)277 3669 y(ested)f(in)f(well-typed)h(terms,)g(that)f(means)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18619
(those)h(for)f(which)g(there)h(is)e(a)h(conte)o(xt)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18620
F4(\000)e Gg(and)h(a)f(formula)277 3782 y Ga(B)i Gg(such)c(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18621
(either)g F4(\000)1031 3770 y Gc(.)1086 3782 y Ga(M)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18622
F4(:)17 b Ga(B)27 b Gg(or)22 b F4(\000)1518 3770 y Gc(.)1573
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18623
3782 y Ga(M)28 b F4(:)17 b F6(?)22 b Gg(is)g(deri)n(v)n(able)j(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18624
(en)e(the)h(inference)h(rules)e(sho)n(wn)g(in)277 3895
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18625
y(Figure)j(2.12.)34 b(Because)26 b(ne)o(gation)h(is)e(not)h(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18626
(primiti)n(v)o(e)j(connecti)n(v)o(e,)g(the)f(inference)h(rules)g(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18627
Ga(\025)3352 3858 y Gc(S)t(y)r(m)277 4008 y Gg(are)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18628
b(quite)g(hard)g(to)g(read,)i(in)d(particular)j(the)e(inference)i(rule)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18629
e Ga(?)p Gg(.)55 b(An)31 b(e)o(xample)j(may)e(clarify)i(this)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18630
4121 y(inference)26 b(rule.)j(Consider)c(the)f(follo)n(wing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18631
Ga(\025)1742 4084 y Gc(S)t(y)r(m)1893 4121 y Gg(-proof.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18632
740 4310 417 4 v 740 4383 a FU(x)17 b FG(:)f FU(B)934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18633
4371 y FS(.)987 4383 y FU(x)g FG(:)h FU(B)p 1240 4310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18634
406 4 v 87 w(y)i FG(:)d FU(C)1428 4371 y FS(.)1481 4383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18635
y FU(y)j FG(:)d FU(C)p 726 4419 934 4 v 726 4518 a(x)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18636
FG(:)f FU(B)t(;)e(y)19 b FG(:)d FU(C)1121 4506 y FS(.)1174
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18637
4518 y FT(h)p FU(x;)e(y)s FT(i)j FG(:)p 0.75 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18638
0.75 TeXcolorgray 1422 4542 238 107 v 0.75 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18639
Black 41 w FU(B)t FT(^)p FU(C)p 0.75 TeXcolorgray Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18640
1701 4440 a FT(h)p 1736 4440 25 4 v 1752 4440 V 1768
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18641
4440 V 60 w FU(;)p 1833 4440 V 1849 4440 V 1865 4440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18642
V 74 w FT(i)p 2207 4299 519 4 v 2207 4386 a FU(z)j FG(:)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18643
FU(B)2372 4356 y FQ(?)2451 4374 y FS(.)2504 4386 y FU(z)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18644
FG(:)c FU(B)2669 4356 y FQ(?)p 2006 4406 922 4 v 2006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18645
4518 a FU(z)j FG(:)d FU(B)2170 4487 y FQ(?)2250 4506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18646
y FS(.)2303 4518 y FU(\033)2350 4530 y FJ(1)2387 4518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18647
y FG(\()p FU(z)t FG(\))g(:)p 0.75 TeXcolorgray 0.75 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18648
2550 4542 378 121 v 0.75 TeXcolorgray Black 42 w FU(B)2642
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18649
4487 y FQ(?)2712 4518 y FT(_)e FU(C)2846 4487 y FQ(?)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18650
0.75 TeXcolorgray Black 2969 4417 a FU(\033)3016 4429
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18651
y FJ(1)p 726 4562 2202 4 v 1185 4649 a FU(x)i FG(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18652
FU(B)t(;)d(y)19 b FG(:)d FU(C)q(;)e(z)20 b FG(:)d FU(B)1754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18653
4619 y FQ(?)1833 4637 y FS(.)1886 4649 y FT(h)p FU(x;)d(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18654
FT(i)19 b FU(?)f(\033)2204 4661 y FJ(1)2241 4649 y FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18655
FU(z)t FG(\))f(:)f FT(?)2969 4582 y FU(?)p Black 3328
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18656
4649 a Gg(\(2.7\))p Black 277 4855 a(The)22 b(last)g(step)h(in)e(this)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18657
(proof)g(is)f(a)f(v)n(alid)i(instance)h(of)e Ga(?)p Gg(,)f(since)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18658
F4(\()p Ga(B)5 b F6(^)p Ga(C)i F4(\))2551 4822 y FX(?)2630
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18659
4855 y Gg(is)22 b(de\002ned)h(as)f Ga(B)3172 4822 y FX(?)3246
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18660
4855 y F6(_)14 b Ga(C)3393 4822 y FX(?)3452 4855 y Gg(.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18661
4990 y(Barbanera)24 b(and)e(Berardi)h(call)f(a)f(term)h(of)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18662
(form)f Ga(M)j(?)14 b(N)32 b F7(symmetric)22 b(application)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18663
Gg(,)j(because)277 5103 y(if)e Ga(M)29 b(?)19 b(N)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18664
b Gg(is)23 b(well-typed,)i(then)f(so)f(is)g Ga(N)29 b(?)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18665
b(M)10 b Gg(.)28 b(Ho)n(we)n(v)o(er)l(,)23 b(one)g(can)h(also)g(re)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18666
(gard)f Ga(?)g Gg(as)g(an)g(instance)277 5216 y(of)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18667
b(the)h(cut-rule.)51 b(T)-7 b(o)29 b(illustrate)k(this,)f(let)f(us)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18668
(reformulate)j(the)d(proof)i(just)f(gi)n(v)o(en)f(in)h(our)f(sequent)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18669
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18670
%%Page: 62 74
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18671
TeXDict begin 62 73 bop Black -144 51 a Gb(62)2987 b(Sequent)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18672
b(Calculi)p -144 88 3691 4 v Black Black 321 300 3226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18673
4 v 321 1258 4 958 v 1601 392 548 4 v 1601 471 a Ga(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18674
b F4(:)h Ga(B)5 b(;)15 b F4(\000)1909 459 y Gc(.)1963
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18675
471 y Ga(x)j F4(:)f Ga(B)692 699 y F4(\000)774 687 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18676
Gc(.)829 699 y Ga(M)27 b F4(:)18 b Ga(B)95 b F4(\000)1233
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18677
687 y Gc(.)1288 699 y Ga(N)27 b F4(:)18 b Ga(C)p 692
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18678
719 811 4 v 753 804 a F4(\000)835 792 y Gc(.)890 804
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18679
y F6(h)p Ga(M)5 b(;)15 b(N)10 b F6(i)18 b F4(:)f Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18680
b F6(^)o Ga(C)1544 741 y F6(h)p 1581 741 28 4 v 1599
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18681
741 V 1616 741 V 65 w Ga(;)p 1686 741 V 1704 741 V 1721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18682
741 V 80 w F6(i)2400 685 y F4(\000)2482 673 y Gc(.)2537
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18683
685 y Ga(M)27 b F4(:)18 b Ga(B)2764 699 y Gc(i)p 2235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18684
719 723 4 v 2235 804 a F4(\000)2317 792 y Gc(.)2372 804
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18685
y Ga(\033)2424 818 y Gc(i)2452 804 y F4(\()p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18686
b F4(\))18 b(:)f Ga(B)2749 818 y F9(1)2789 804 y F6(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18687
Ga(B)2918 818 y F9(2)2999 731 y Ga(\033)3051 745 y Gc(i)895
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18688
1019 y Ga(x)h F4(:)f Ga(B)5 b(;)15 b F4(\000)1203 1007
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18689
y Gc(.)1258 1019 y Ga(M)27 b F4(:)18 b F6(?)p 895 1056
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18690
591 4 v 912 1152 a F4(\000)994 1140 y Gc(.)1049 1152
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18691
y Ga(\025x:M)28 b F4(:)17 b Ga(B)1411 1119 y FX(?)1528
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18692
1088 y Ga(\025)2138 1039 y F4(\000)2195 1053 y F9(1)2260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18693
1027 y Gc(.)2315 1039 y Ga(M)27 b F4(:)18 b Ga(B)95 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18694
F4(\000)2694 1053 y F9(2)2758 1027 y Gc(.)2813 1039 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18695
Ga(N)27 b F4(:)18 b Ga(B)3030 1006 y FX(?)p 2138 1072
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18696
951 4 v 2258 1152 a F4(\000)2315 1166 y F9(1)2355 1152
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18697
y Ga(;)d F4(\000)2452 1166 y F9(2)2516 1140 y Gc(.)2571
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18698
1152 y Ga(M)31 b(?)20 b(N)27 b F4(:)18 b F6(?)3130 1093
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18699
y Ga(?)p 3543 1258 4 958 v 321 1261 3226 4 v Black 735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18700
1415 a Gg(Figure)24 b(2.12:)30 b(T)-6 b(erm)22 b(assignment)k(for)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18701
(the)h(symmetric)h(lambda)f(calculus.)p Black Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18702
1828 a(calculus.)65 b(Here)35 b(is)g(the)g(sequent)i(proof)f(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18703
(corresponds)k(to)34 b(\(2.7\))i(\(for)f(bre)n(vity)h(we)f(omit)f(all)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18704
321 1941 y(labels\).)p 1192 2201 244 4 v 1192 2280 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18705
Ga(B)p 1285 2268 11 41 v 1296 2250 46 5 v 100 w(B)p 1526
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18706
2201 240 4 v 96 w(C)p 1618 2268 11 41 v 1628 2250 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18707
5 v 103 w(C)p 1192 2300 574 4 v 1235 2379 a(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18708
b(C)p 1440 2367 11 41 v 1451 2349 46 5 v 102 w(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18709
F6(^)p Ga(C)1807 2318 y F6(^)1867 2332 y Gc(R)p 2130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18710
2085 244 4 v 2130 2164 a Ga(B)p 2224 2152 11 41 v 2234
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18711
2134 46 5 v 101 w(B)p 2079 2184 345 4 v 2079 2262 a(B)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18712
b F6(:)p Ga(B)p 2348 2250 11 41 v 2358 2232 46 5 v 2465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18713
2196 a F6(:)2526 2210 y Gc(L)p 2016 2300 472 4 v 2016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18714
2379 a Ga(B)5 b F6(^)o Ga(C)q(;)15 b F6(:)p Ga(B)p 2411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18715
2367 11 41 v 2422 2349 46 5 v 2529 2314 a F6(^)2589 2328
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18716
y Gc(L)2637 2337 y FV(1)p 1235 2416 1253 4 v 1635 2495
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18717
a Ga(B)5 b(;)15 b(C)q(;)g F6(:)p Ga(B)p 2010 2483 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18718
41 v 2021 2465 46 5 v 2529 2447 a Gg(Cut)321 2708 y(As)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18719
b(the)h(reader)h(can)f(see,)h(e)n(v)o(en)e(if)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18720
Ga(\033)1628 2722 y F9(1)1667 2708 y Gg(-rule)g(introduces)j(an)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18721
F6(_)o Gg(,)g(it)f(corresponds)k(to)d(an)f(instance)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18722
2821 y(of)d(the)g F6(^)614 2835 y Gc(L)662 2844 y FV(1)701
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18723
2821 y Gg(-rule,)g(and)g(the)g Ga(?)p Gg(-rule)h(is)e(clearly)i(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18724
(cut.)462 2950 y(There)f(are)f(a)g(number)h(of)f(reduction)j(rules)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18725
(associated)i(with)d Ga(\025)2471 2913 y Gc(S)t(y)r(m)2622
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18726
2950 y Gg(,)f(some)i(of)f(which)h(are)f(gi)n(v)o(en)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18727
3063 y(belo)n(w)-6 b(.)p Black Black 1194 3160 a Ga(\025x:M)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18728
b(?)21 b(N)1704 3123 y Gc(sy)r(m)1691 3160 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18729
b(\000)h(!)99 b Ga(M)10 b F4([)p Ga(x)26 b F4(:=)f Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18730
b F4(])1194 3297 y Ga(M)30 b(?)21 b(\025x:N)1704 3260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18731
y Gc(sy)r(m)1691 3297 y F6(\000)-32 b(\000)h(!)99 b Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18732
b F4([)p Ga(x)26 b F4(:=)f Ga(M)10 b F4(])917 3445 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18733
F6(h)p Ga(M)1040 3459 y F9(1)1080 3445 y Ga(;)15 b(M)1208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18734
3459 y F9(2)1248 3445 y F6(i)21 b Ga(?)f(\033)1421 3459
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18735
y Gc(i)1449 3445 y F4(\()p Ga(N)10 b F4(\))1699 3408
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18736
y Gc(sy)r(m)1686 3445 y F6(\000)-31 b(\000)f(!)83 b Ga(M)2027
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18737
3459 y Gc(i)2076 3445 y Ga(?)21 b(N)917 3581 y(\033)969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18738
3595 y Gc(i)997 3581 y F4(\()p Ga(N)10 b F4(\))21 b Ga(?)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18739
F6(h)p Ga(M)1360 3595 y F9(1)1400 3581 y Ga(;)15 b(M)1528
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18740
3595 y F9(2)1568 3581 y F6(i)1699 3544 y Gc(sy)r(m)1686
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18741
3581 y F6(\000)-31 b(\000)f(!)83 b Ga(N)31 b(?)20 b(M)2196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18742
3595 y Gc(i)2539 3358 y FK(\))2718 3513 y Ga(i)26 b F4(=)f(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18743
Ga(;)15 b F4(2)321 3788 y Gg(As)30 b(usual,)i(the)e(e)o(xpression)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18744
Ga(M)10 b F4([)p Ga(x)38 b F4(:=)f Ga(N)10 b F4(])29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18745
b Gg(stands)j(for)e(the)g(substitution)k(of)c(the)g(v)n(ariable)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18746
Ga(x)p Gg(.)47 b(Ac-)321 3901 y(cording)33 b(to)d(the)h(reduction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18747
(rules)e(of)g Ga(\025)1608 3864 y Gc(sy)r(m)1744 3901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18748
y Gg(,)g(the)g(term)f Ga(\025x:M)36 b(?)26 b(\025y)s(:N)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18749
b Gg(reduces)33 b(to)d(either)i Ga(M)40 b Gg(or)321 4014
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18750
y Ga(N)35 b Gg(assuming)28 b(that)f Ga(x)e Gg(and)h Ga(y)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18751
Gg(are)f(not)f(free)g(v)n(ariables)i(in)e Ga(M)36 b Gg(and)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18752
b Ga(N)10 b Gg(,)26 b(respecti)n(v)o(ely)-6 b(.)39 b(So)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18753
b(the)h(reduc-)321 4127 y(tion)e(system)f(of)g Ga(\025)910
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18754
4090 y Gc(S)t(y)r(m)1082 4127 y Gg(is)g(non-deterministic,)k(analogous)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18755
f(to)c(our)h(cut-elimination)k(procedure)e(for)321 4239
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18756
y(classical)h(logic.)462 4369 y(Because)h(of)f(the)g(close)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18757
(correspondence)31 b(with)25 b(our)h(sequent)i(calculus,)g(one)e(may)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18758
(reasonably)321 4482 y(ask)38 b(whether)g(our)f(strong)i(normalisation)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18759
h(results)f(for)e F4(\()p FY(T)t Ga(;)2345 4445 y Gc(cut)2314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18760
4482 y F6(\000)-31 b(\000)f(!)p F4(\))37 b Gg(and)h F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18761
FY(T)t Ga(;)2877 4445 y Gc(aux)2856 4482 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18762
b(\000)g(!)p F4(\))36 b Gg(follo)n(w)i(from)321 4595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18763
y(Barbanera)d(and)e(Berardi')-5 b(s)34 b(strong)f(normalisation)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18764
(result,)g(for)d(e)o(xample,)i(by)d(a)h(simple)g(transla-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18765
4708 y(tion.)42 b(Ho)n(we)n(v)o(er)l(,)29 b(it)e(does)i(not)f(appear)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18766
(that)f(such)h(a)e(simple)h(translation)j(e)o(xists.)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18767
b(Primarily)28 b(this)h(is)321 4821 y(because)g(our)f(term)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18768
(constructors)31 b(include)e(multiple)f(binders;)j(for)c(e)o(xample,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18769
(the)f(term)g(construc-)321 4933 y(tor)j FL(Imp)596 4955
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18770
y Gc(R)653 4933 y F4(\()688 4921 y F9(\()716 4933 y Ga(x)768
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18771
4921 y F9(\))q FX(h)823 4933 y Ga(a)871 4921 y FX(i)898
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18772
4933 y Ga(M)11 b(;)k(b)p F4(\))29 b Gg(and)h FL(And)1455
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18773
4897 y FX(0)1455 4957 y Gc(L)1507 4933 y F4(\()1542 4921
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18774
y F9(\()1570 4933 y Ga(x)1622 4921 y F9(\)\()1677 4933
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18775
y Ga(y)1725 4921 y F9(\))1752 4933 y Ga(M)10 b(;)15 b(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18776
F4(\))p Gg(.)47 b(Owing)30 b(to)f(the)h(speci\002c)h(form)e(of)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18777
(typing)321 5046 y(judgements)c(sho)n(wn)e(in)g(\(2.6\),)f(where)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18778
(only)g(a)g(single)h(formula)f(on)g(the)g(right-hand)i(side)e(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18779
(permit-)321 5159 y(ted,)d(it)e(is)h(dif)n(\002cult)h(to)e(imagine)i(a)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18780
f(translation)i(that)f(encodes)h(the)e(typing)h(judgements)h(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18781
(these)h(term)321 5272 y(constructors)28 b(into)c Ga(\025)1006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18782
5235 y Gc(S)t(y)r(m)1157 5272 y Gg(.)k(Another)d(obstacle)h(to)e(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18783
(proof)i(by)f(translation)j(are)d(our)g(cut-reductions)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18784
5385 y(that)g(replace)h(a)e(single)h(cut-instance)j(with)c(tw)o(o)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18785
(nested)i(cut-instances.)32 b(In)23 b(f)o(act,)h(we)e(had)i(to)f(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18786
(xtend)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18787
%%Page: 63 75
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18788
TeXDict begin 63 74 bop Black 277 51 a Gb(2.7)23 b(Notes)3248
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18789
b(63)p 277 88 3691 4 v Black 277 388 a Gg(the)27 b(proof)g(technique)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18790
(of)e(Barbanera)i(and)f(Berardi)g(to)f(deal)h(with)f(such)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18791
(cut-reductions.)41 b(It)26 b(seems)277 501 y(highly)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18792
b(unlik)o(ely)h(that)e(this)g(dynamic)h(beha)n(viour)i(of)c(our)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18793
(cut-elimination)j(procedure)g(could)d(be)277 614 y(captured)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18794
b(by)d(a)h(simple)g(translation.)418 744 y(Ev)o(en)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18795
b(when)g(restricting)j(our)e(calculus)h(to)e(just)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18796
F4(\()p F6(^)o Ga(;)15 b F6(_)p F4(\))p Gg(-fragment,)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18797
b(i.e.,)c(to)h(the)f(fragment)277 856 y(where)20 b(term)f(constructors)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18798
k(bind)d(only)h(a)e(single)i(name)e(or)h(co-name)g(and)g(where)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18799
(reduction)i(rules)e(do)277 969 y(not)28 b(create)i(nested)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18800
(cut-instances,)k(the)28 b(strong)h(normalisation)i(proof)e(by)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18801
(translation)j(f)o(ails)e(using)277 1082 y(the)24 b(natural)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18802
(candidate,)h F6(j)p 1107 1082 28 4 v 1125 1082 V 1142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18803
1082 V 65 w(j)1195 1049 y Fu(Sym)1327 1082 y Gg(,)c(as)i(translation.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18804
32 b(The)23 b(clauses)i(of)e F6(j)p 2480 1082 V 2498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18805
1082 V 2516 1082 V 65 w(j)2568 1049 y Fu(Sym)2723 1082
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18806
y Gg(are)h(gi)n(v)o(en)g(belo)n(w)-6 b(.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18807
1211 1310 a F6(j)p FL(Ax)o F4(\()p Ga(x;)15 b(a)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18808
F6(j)1573 1277 y Fu(Sym)1806 1259 y F5(def)1813 1310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18809
y F4(=)107 b Ga(x)20 b(?)g(a)743 1450 y F6(j)p FL(And)923
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18810
1464 y Gc(R)981 1450 y F4(\()1016 1438 y FX(h)1043 1450
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18811
y Ga(a)1091 1438 y FX(i)1119 1450 y Ga(M)10 b(;)1257
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18812
1438 y FX(h)1285 1450 y Ga(b)1324 1438 y FX(i)1351 1450
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18813
y Ga(N)g(;)15 b(c)p F4(\))p F6(j)1573 1417 y Fu(Sym)1806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18814
1399 y F5(def)1813 1450 y F4(=)107 b F6(h)p Ga(\025a:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18815
F6(j)p Ga(M)10 b F6(j)2300 1417 y Fu(Sym)2433 1450 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18816
Ga(;)15 b(\025b:)p F6(j)p Ga(N)10 b F6(j)2723 1417 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18817
Fu(Sym)2856 1450 y F6(i)20 b Ga(?)h(c)954 1590 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18818
FL(And)1133 1553 y Gc(i)1133 1613 y(L)1186 1590 y F4(\()1221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18819
1578 y F9(\()1248 1590 y Ga(x)1300 1578 y F9(\))1328
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18820
1590 y Ga(M)10 b(;)15 b(y)s F4(\))p F6(j)1574 1557 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18821
Fu(Sym)1806 1539 y F5(def)1813 1590 y F4(=)107 b Ga(\033)2043
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18822
1604 y Gc(i)2071 1590 y F4(\()p Ga(\025x:)p F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18823
b F6(j)2384 1557 y Fu(Sym)2517 1590 y F4(\))20 b Ga(?)h(y)1016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18824
1730 y F6(j)p FL(Or)1141 1693 y Gc(i)1141 1753 y(R)1198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18825
1730 y F4(\()1233 1718 y FX(h)1261 1730 y Ga(a)1309 1718
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18826
y FX(i)1337 1730 y Ga(M)10 b(;)15 b(b)p F4(\))p F6(j)1574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18827
1697 y Fu(Sym)1806 1678 y F5(def)1813 1730 y F4(=)107
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18828
b Ga(\033)2043 1744 y Gc(i)2071 1730 y F4(\()p Ga(\025a:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18829
F6(j)p Ga(M)10 b F6(j)2380 1697 y Fu(Sym)2513 1730 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18830
F4(\))21 b Ga(?)f(b)784 1870 y F6(j)p FL(Or)909 1884
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18831
y Gc(L)961 1870 y F4(\()996 1858 y F9(\()1024 1870 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18832
Ga(x)1076 1858 y F9(\))1103 1870 y Ga(M)10 b(;)1241 1858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18833
y F9(\()1269 1870 y Ga(y)1317 1858 y F9(\))1344 1870
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18834
y Ga(N)g(;)15 b(z)t F4(\))p F6(j)1573 1837 y Fu(Sym)1806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18835
1818 y F5(def)1813 1870 y F4(=)107 b F6(h)p Ga(\025x:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18836
F6(j)p Ga(M)10 b F6(j)2304 1837 y Fu(Sym)2437 1870 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18837
Ga(;)15 b(\025y)s(:)p F6(j)p Ga(N)10 b F6(j)2736 1837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18838
y Fu(Sym)2869 1870 y F6(i)20 b Ga(?)h(z)884 2009 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18839
FL(Cut)p F4(\()1082 1997 y FX(h)1110 2009 y Ga(a)1158
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18840
1997 y FX(i)1186 2009 y Ga(M)10 b(;)1324 1997 y F9(\()1352
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18841
2009 y Ga(x)1404 1997 y F9(\))1431 2009 y Ga(N)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18842
F6(j)1574 1976 y Fu(Sym)1806 1958 y F5(def)1813 2009
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18843
y F4(=)107 b Ga(\025a:)p F6(j)p Ga(M)10 b F6(j)2265 1976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18844
y Fu(Sym)2418 2009 y Ga(?)20 b(\025x:)p F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18845
b F6(j)2746 1976 y Fu(Sym)277 2226 y Gg(The)20 b(translation)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18846
(respects)f(the)f(typing)h(judgements)g(of)e FY(T)2130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18847
2193 y F9(\()p FX(^)m Gc(;)p FX(_)o F9(\))2299 2226 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18848
Gg(,)g(as)g(stated)h(in)g(the)f(follo)n(wing)i(propo-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18849
2339 y(sition.)p Black 277 2526 a Gb(Pr)n(oposition)j(2.7.1:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18850
Black 34 w Gg(If)f Ga(M)36 b F6(2)25 b FY(T)1348 2493
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18851
y F9(\()p FX(^)m Gc(;)p FX(_)p F9(\))1540 2526 y Gg(with)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18852
(typing)i(judgement)f F4(\000)2604 2514 y Gc(.)2658 2526
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18853
y Ga(M)2782 2514 y Gc(.)2837 2526 y F4(\001)o Gg(,)e(then)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18854
F6(j)p Ga(M)10 b F6(j)3286 2493 y Fu(Sym)3442 2526 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18855
Gg(is)277 2639 y(well-typed)31 b(in)e Ga(\025)847 2602
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18856
y Gc(S)t(y)r(m)1025 2639 y Gg(with)g(the)h(typing)h(judgement)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18857
F4(\000)p Ga(;)15 b F4(\001)2201 2606 y FX(?)2285 2627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18858
y Gc(.)2340 2639 y F6(j)p Ga(M)10 b F6(j)2488 2606 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18859
Fu(Sym)2638 2639 y F4(:)17 b F6(?)p Gg(,)29 b(where)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18860
F4(\001)3130 2606 y FX(?)3217 2639 y Gg(is)f(tak)o(en)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18861
2752 y(to)24 b(mean)f(that)h(all)g(formulae)h(of)e F4(\001)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18862
Gg(are)h(ne)o(gated.)p Black 277 2989 a F7(Pr)l(oof.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18863
Black 46 w Gg(By)f(induction)j(on)e(the)g(structure)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18864
Ga(M)10 b Gg(.)p 3436 2989 4 62 v 3440 2931 55 4 v 3440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18865
2989 V 3494 2989 4 62 v 277 3194 a(But)18 b F6(j)p 453
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18866
3194 28 4 v 471 3194 V 489 3194 V 65 w(j)541 3161 y Fu(Sym)691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18867
3194 y Gg(does)i F7(not)f Gg(respect)i(the)e(reductions)i(in)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18868
(system)h F4(\()p FY(T)2396 3161 y F9(\()p FX(^)m Gc(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18869
FX(_)o F9(\))2565 3194 y Ga(;)2636 3157 y Gc(cut)2605
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18870
3194 y F6(\000)-31 b(\000)g(!)o F4(\))p Gg(.)27 b(T)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18871
b(o)18 b(gi)n(v)o(e)h(a)f(counter)n(-)277 3307 y(e)o(xample,)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18872
(consider)h(the)e(term)g FL(Cut)o F4(\()1445 3295 y FX(h)1473
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18873
3307 y Ga(a)1521 3295 y FX(i)1549 3307 y Ga(M)10 b(;)1687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18874
3295 y F9(\()1715 3307 y Ga(x)1767 3295 y F9(\))1794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18875
3307 y Ga(N)g F4(\))20 b Gg(and)i(assume)f Ga(M)30 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18876
Gg(introduces)24 b Ga(a)p Gg(,)c(b)n(ut)i(not)f(freshly)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18877
3419 y(\(i.e.,)37 b(there)e(is)f(an)h(implicit)g(contraction\).)65
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18878
b(According)37 b(to)d(our)h(reduction)i(rules)f(this)f(term)f(can)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18879
3532 y(reduce)25 b(as)f(follo)n(ws.)1193 3754 y FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18880
F4(\()1366 3742 y FX(h)1394 3754 y Ga(a)1442 3742 y FX(i)1469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18881
3754 y Ga(M)10 b(;)1607 3742 y F9(\()1635 3754 y Ga(x)1687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18882
3742 y F9(\))1714 3754 y Ga(N)g F4(\))1889 3717 y Gc(cut)1858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18883
3754 y F6(\000)-31 b(\000)f(!)25 b Ga(M)10 b F4([)p Ga(a)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18884
b F4(:=)2371 3742 y FX(h)2399 3754 y Ga(x)2451 3742 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18885
FX(i)2478 3754 y Ga(N)10 b F4(])277 3975 y Gg(Since)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18886
b Ga(M)33 b Gg(introduces)26 b Ga(a)p Gg(,)c(the)h(reduct)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18887
Ga(M)10 b F4([)p Ga(a)25 b F4(:=)1816 3963 y FX(h)1844
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18888
3975 y Ga(x)1896 3963 y FX(i)1923 3975 y Ga(N)10 b F4(])23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18889
b Gg(is)g(of)g(the)g(form)g FL(Cut)p F4(\()2740 3963
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18890
y FX(h)2768 3975 y Ga(a)2816 3963 y FX(i)2858 3975 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18891
Ga(:)15 b(:)g(:)q(;)3004 3963 y F9(\()3032 3975 y Ga(x)3084
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18892
3963 y F9(\))3112 3975 y Ga(N)10 b F4(\))p Gg(,)22 b(mean-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18893
4088 y(ing)34 b(that)g(the)g(outermost)i(cut-instance)h(does)d(not)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18894
(change.)60 b(Gi)n(v)o(en)34 b(the)g(translation)i F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18895
3133 4088 V 3151 4088 V 3169 4088 V 65 w(j)3221 4055
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18896
y Fu(Sym)3353 4088 y Gg(,)f(no)277 4201 y(reduction)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18897
Ga(\025)816 4164 y Gc(S)t(y)r(m)1000 4201 y Gg(can)g(\223mirror\224)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18898
(this)e(beha)n(viour)-5 b(.)64 b(The)34 b(e)o(xistence)i(of)e(a)g(more)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18899
g(complicated)277 4314 y(translation)26 b(of)d(our)g(sequent)h(proofs)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18900
(into)e Ga(\025)1694 4277 y Gc(S)t(y)r(m)1845 4314 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18901
Gg(,)e(which)j(gi)n(v)o(es)f(a)f(strong)i(normalisation)i(proof)e(for)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18902
277 4427 y FY(T)338 4394 y F9(\()p FX(^)m Gc(;)p FX(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18903
F9(\))507 4427 y Gg(,)e(is)i(possible,)h(b)n(ut)f(we)f(conjecture)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18904
(unlik)o(ely)-6 b(.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18905
%%Page: 64 76
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18906
TeXDict begin 64 75 bop Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18907
%%Page: 65 77
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18908
TeXDict begin 65 76 bop Black Black 277 1004 a F8(Chapter)44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18909
b(3)277 1435 y Gf(Natural)52 b(Deduction)f(Calculi)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18910
Black Black 1140 1853 a Gd(I)19 b(intended)e(\002rst)i(to)g(set)g(up)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18911
(a)g(formal)e(system)i(which)f(comes)g(as)i(close)e(as)i(possi-)1140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18912
1952 y(ble)15 b(to)h(actual)f(reasoning.)21 b(The)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18913
b(result)g(w)o(as)h(a)g(\223calculus)f(of)g(natural)f(deduction\224.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18914
3041 2152 y(\227G.)21 b(Gentzen)1904 2251 y(in)g(In)m(v)o(estigations)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18915
(into)j(Logical)g(Deductions,)e(1935.)277 2497 y F7(Natur)o(al)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18916
b(deduction)i(calculi)p Gg(,)e(lik)o(e)g(sequent)h(calculi,)f(were)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18917
(in)l(v)o(ented)i(by)e(Gentzen)i([1935].)30 b(Whilst)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18918
2610 y(sequent)k(calculi)g(of)n(fered)f(a)f(con)l(v)o(enient)j(w)o(ay)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18919
(to)g(state)h(his)f F7(Hauptsatz)p Gg(,)k(natural)e(deduction)h(cal-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18920
277 2723 y(culi)c(of)n(fered)i(a)d(con)l(v)o(enient)k(approach)f(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18921
(reasoning)i(by)e(mimicking)h(the)f(reasoning)j(emplo)o(yed)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18922
2836 y(by)24 b(mathematicians.)33 b(Pra)o(witz)23 b([1965])j(pro)o(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18923
(ed)f(later)g(that)f(a)g(v)o(ersion)h(of)f(the)g(cut-elimination)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18924
(the-)277 2948 y(orem)23 b(also)h(holds)g(for)f(NJ)f(and)i(NK,)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18925
(Gentzen')-5 b(s)25 b(natural)f(deduction)i(calculi)f(for)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18926
(intuitionistic)k(and)277 3061 y(classical)f(logic,)e(respecti)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18927
(ely)-6 b(.)418 3191 y(In)25 b(this)h(chapter)h(we)d(shall)i(study)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18928
(the)g(striking)h(similarities)g(between)f(natural)h(deduction)h(cal-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18929
277 3304 y(culi)21 b(and)g(sequent)h(calculi.)29 b(In)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18930
b(particular)l(,)i(we)d(shall)h(sho)n(w)f(a)g(close)h(correspondence)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18931
(between)d(cut-)277 3417 y(elimination)31 b(and)e(normalisation)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18932
(using)e(slight)g(v)n(ariants)g(of)f(the)f(standard)j(translations)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18933
(between)277 3530 y(natural)22 b(deduction)h(proofs)f(and)f(sequent)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18934
(proofs.)29 b(This)21 b(gi)n(v)o(es,)g(for)g(e)o(xample,)g(in)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18935
(classical)i(case)277 3642 y(the)h(correspondences)k(illustrated)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18936
(with)c(the)h(diagrams)p Black Black 1182 3799 a Ga(S)1411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18937
3762 y Gc(\024)1310 3799 y F6(\000)-21 b(\000)h(\000)f(!)64
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18938
b Ga(T)1171 4163 y(S)1232 4125 y FX(0)1146 3981 y Fu(S)1191
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18939
4037 y F6(#)p 1211 3974 4 142 v 1353 4125 a Gc(cut)1281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18940
4163 y F6(\000)-21 b(\000)g(\000)g(!)1522 4125 y F9(+)1605
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18941
4163 y Ga(T)1671 4125 y FX(0)1675 3981 y Fu(S)1628 4037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18942
y F6(#)p 1648 3974 V 2116 3799 a Ga(M)2335 3762 y Gc(cut)2263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18943
3799 y F6(\000)g(\000)g(\000)g(!)56 b Ga(N)2104 4163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18944
y(M)2202 4125 y FX(0)2088 3981 y Fu(N)2144 4037 y F6(#)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18945
2164 3974 V 2358 4125 a Gc(\024)2258 4163 y F6(\000)-21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18946
b(\000)h(\000)e(!)2494 4125 y FX(\003)2561 4163 y Ga(N)2644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18947
4125 y FX(0)2627 3981 y Fu(N)2580 4037 y F6(#)p 2600
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18948
3974 V 277 4316 a Gg(where)34 b F6(j)p 559 4316 28 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18949
v 577 4316 V 595 4316 V 65 w(j)647 4283 y Fu(S)724 4316
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18950
y Gg(is)g(a)f(translation)k(from)d(natural)h(deduction)i(proofs)e(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18951
(sequent)i(proofs,)i F6(j)p 3193 4316 V 3211 4316 V 3228
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18952
4316 V 65 w(j)3281 4283 y Fu(N)3368 4316 y Gg(is)c(a)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18953
4429 y(translation)c(from)d(sequent)i(proofs)f(to)f(natural)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18954
(deduction)h(proofs,)f(and)2717 4392 y Gc(\024)2652 4429
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18955
y F6(\000)-31 b(\000)g(!)26 b Gg(is)h(a)f(normalisation)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18956
4542 y(procedure)i(for)e(our)f(natural)i(deduction)h(calculus.)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18957
b(In)26 b(contrast)h(to)e(earlier)i(w)o(ork,)e(we)g(obtain)i(such)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18958
4655 y(tight)c(correspondences)k(because)d(of)d(the)h(fe)n(wer)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18959
(constraints)j(in)d(our)g(cut-elimination)k(procedures.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18960
4784 y(W)-7 b(e)20 b(shall)i(be)o(gin)g(with)f(sho)n(wing)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18961
(correspondence)k(between)d(our)g(intuitionistic)j(cut-elimi-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18962
4897 y(nation)k(procedure)h(and)e(the)g(standard)h(normalisation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18963
(procedure)f(of)d(NJ.)f(This)i(part)g(will)f(closely)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18964
5010 y(follo)n(w)34 b(e)o(xisting)h(w)o(ork.)60 b(The)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18965
b(ne)o(xt)h(step)g(will)f(be)h(to)g(e)o(xtend)g(the)g(correspondence)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18966
b(to)33 b(classical)277 5123 y(logic.)51 b(Unfortunately)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18967
b(,)36 b(it)30 b(is)h(v)o(ery)g(impractical)i(to)e(do)f(the)h(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18968
(xtension)i(using)f(Gentzen')-5 b(s)33 b(natural)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18969
5236 y(deduction)j(calculus)f(NK,)c(because)k(this)e(calculus)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18970
(requires)g(double)g(ne)o(gation)f(translations)i(for)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18971
5349 y(encoding)24 b(classical)f(proofs.)29 b(So)20 b(instead)j(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18972
(using)j(NK,)c(we)h(shall)i(introduce)h(a)d(v)n(ariant)j(of)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18973
(nat-)277 5462 y(ural)f(deduction)i(calculus)f(de)n(v)o(eloped)h(by)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18974
(Bori)5 b(\020)-35 b(ci)5 b(\264)-35 b(c)20 b([1985])h(and)e(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18975
(e)h(normalisation)i(rules)e(inspired)277 5575 y(by)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18976
b(the)h(rules)f(P)o(arigot)h([1992])g(de\002ned)g(for)g(his)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18977
Ga(\025\026)p Gg(-calculus.)41 b(No)26 b(claim)i(is)e(made)i(about)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18978
(superi-)277 5688 y(ority)i(of)e(our)h(natural)i(deduction)g(calculus)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18979
(o)o(v)o(er)e(other)g(natural)i(deduction)g(calculi)f(for)f(classical)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18980
277 5801 y(logic.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18981
%%Page: 66 78
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18982
TeXDict begin 66 77 bop Black -144 51 a Gb(66)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18983
b(Deduction)p -144 88 3691 4 v Black 321 365 a Ge(3.1)119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18984
b(Intuitionistic)31 b(Natural)f(Deduction)321 588 y Gg(Lik)o(e)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18985
(sequent)j(proofs,)f(natural)g(deduction)h(proofs)f(are)e(written)h(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18986
f(a)g(tree-lik)o(e)i(f)o(ashion,)h(and)e(tw)o(o)321 701
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18987
y(styles)d(of)f(writing)g(them)f(can)h(be)g(identi\002ed)h(in)f(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18988
(literature.)p Black 458 880 a F6(\017)p Black 46 w Gg(One,)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18989
b(used)h(for)e(e)o(xample)i(in)e([Gentzen,)i(1935])g(and)f(in)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18990
(classic)j(w)o(ork)d(on)h(natural)h(de-)549 993 y(duction)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18991
b([Pra)o(witz,)e(1965],)i(is)e(to)g(write)h(trees)g(of)g(the)g(form)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18992
2019 1165 y F4(\000)2036 1191 y Gg(.)2036 1224 y(.)2036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18993
1257 y(.)2036 1290 y(.)2011 1370 y Ga(B)549 1554 y Gg(where)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18994
b(the)g(formula)h Ga(B)j Gg(is)c(the)h(conclusion)i(and)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18995
F4(\000)g Gg(stands)h(for)f(a)g(\002nite)g(number)h(of)f(as-)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18996
1667 y(sumption)d(pack)o(ets.)53 b(The)31 b(assumption)j(pack)o(ets)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18997
(are)f(multisets)g(containing)j(formulae)d(all)549 1780
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18998
y(of)27 b(which)i(are)f(annotated)i(with)e(a)g(unique)h(\(natural)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 18999
(number\))f(label.)43 b(The)27 b(inference)k(rules)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19000
1892 y(that)26 b(dischar)n(ge,)j(or)d(\223close\224,)j(assumption)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19001
(pack)o(ets)g(are)e(annotated)j(with)d(the)g(labels)h(of)f(the)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19002
2005 y(pack)o(ets)f(the)o(y)f(dischar)n(ge.)p Black 458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19003
2176 a F6(\017)p Black 46 w Gg(The)19 b(other)l(,)j(used)e(for)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19004
(xample)h(in)f([Gentzen,)g(1936],)i(is)e(to)f(write)h(natural)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19005
(deduction)g(proofs)549 2289 y(in)27 b(\223sequent-style\224)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19006
b(where)28 b(in)f(e)n(v)o(ery)i(stage)f(of)g(the)g(proof)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19007
(undischar)n(ged)k(assumption)549 2402 y(pack)o(ets)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19008
b(are)f(recorded)h(e)o(xplicitly)-6 b(.)32 b(Thus)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19009
b(the)h(proofs)h(are)f(of)f(the)h(form)2036 2527 y(.)2036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19010
2560 y(.)2036 2593 y(.)2036 2626 y(.)1934 2706 y F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19011
2011 2694 11 41 v 2022 2676 46 5 v 96 w Ga(B)549 2889
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19012
y Gg(Throughout)d(this)f(chapter)g(we)f(shall)g(use)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19013
(sequent-style)k(for)c(writing)h(natural)h(deduction)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19014
3002 y(proofs.)66 b(W)-7 b(e)35 b(shall)i(tak)o(e)f F4(\000)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19015
Gg(to)g(be)h(a)f(set)h(of)g(labelled)i(formulae)f(analogous)h(to)e(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19016
g(left-)549 3115 y(conte)o(xts)24 b(de\002ned)f(in)f(connection)k(with)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19017
c(sequent)i(calculi.)30 b(Ho)n(we)n(v)o(er)22 b(in)h(this)g(section)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19018
(and)f(in)549 3228 y(the)j(ne)o(xt,)h(the)g(standard)h(terminology)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19019
(for)e(natural)h(deduction)h(will)d(be)g(adopted,)j(and)e(thus)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19020
3341 y(the)19 b(labels)i(are)f(referred)h(to)e(as)g F7(variables)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19021
Gg(\(these)f(tw)o(o)e(sections)j(deal)e(with)f(the)h(propositional)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19022
3454 y(fragment)25 b(of)f(intuitionistic)k(logic,)d(only;)h(so)e(no)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19023
(confusion)j(can)d(occur)i(with)e(the)g(notion)i(of)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19024
3567 y(v)n(ariables)f(introduced)i(in)c(Section)i(2.4)e(for)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19025
(\002rst-order)h(classical)h(logic\).)321 3746 y(Figure)d(3.1)e(gi)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19026
o(es)h(the)g(inference)i(rules)f(of)e(Gentzen')-5 b(s)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19027
b(intuitionistic)j(natural)d(deduction)i(calculus,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19028
3859 y(called)i(NJ.)687 3826 y F5(1)750 3859 y Gg(The)e(inference)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19029
(rules)f(come)f(in)g(tw)o(o)g(\003a)n(v)n(ours,)h(those)g(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19030
F7(intr)l(oducing)k Gg(a)25 b(connecti)n(v)o(e)321 3972
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19031
y(and)32 b(those)g(for)f F7(eliminating)j Gg(a)c(connecti)n(v)o(e.)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19032
b(Writing)32 b(the)f(proofs)i(in)e(sequent-style)j(means)e(that)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19033
4085 y(the)23 b(inference)i(rules)f(of)f(natural)h(deduction)h(are)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19034
(solely)h(concerned)i(with)c(manipulating)k(formulae)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19035
4198 y(in)g(the)h(succedent,)i(unlik)o(e)f(in)e(sequent)i(calculi)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19036
(where)e(inference)j(rules)e(manipulate)h(formulae)g(in)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19037
4311 y(both)e(the)f(antecedent)j(and)d(the)h(succedent.)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19038
b(This)25 b(is)g(why)f(it)h(is)f(often)i(said)g(that)f(natural)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19039
(deduction)321 4424 y(calculi)e(are)f(asymmetric.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19040
4553 y(There)i(are)g(well-kno)n(wn)h(translations)i(going)e(back)f(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19041
(Gentzen)h([1935])g(between)g(NJ-proofs)321 4666 y(and)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19042
(intuitionistic)j(sequent)d(proofs.)30 b(Belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19043
b(,)23 b(we)f(shall)h(de\002ne)h(the)f(translation)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19044
F6(j)p 2946 4666 28 4 v 2964 4666 V 2981 4666 V 65 w(j)3034
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19045
4633 y Fu(s)3065 4666 y Gg(,)c(which)h(maps)321 4779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19046
y(NJ-proofs)28 b(to)e(sequent)i(proofs,)f(and)g(the)f(translation)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19047
F6(j)p 2118 4779 V 2136 4779 V 2153 4779 V 65 w(j)2206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19048
4746 y Fu(n)2246 4779 y Gg(,)d(which)g(maps)g(sequent)i(proofs)f(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19049
(NJ-)321 4892 y(proofs.)55 b(It)31 b(will)h(be)f(con)l(v)o(enient)k(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19050
d(de\002ne)g(both)h(o)o(v)o(er)e(terms,)j(rather)f(than)f(proofs.)55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19051
b(One)31 b(slight)321 5005 y(complication)39 b(arises)e(from)e(the)h(f)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19052
o(act)g(that)g(we)f(ha)n(v)o(e)h(chosen)h(intuitionistic)j(sequents)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19053
(to)d(be)h(of)321 5118 y(the)e(form)f F4(\000)p 751 5106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19054
11 41 v 761 5087 46 5 v 96 w Ga(a)17 b F4(:)h Ga(B)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19055
b Gg(\(see)e(P)o(age)f(46\).)57 b(This)33 b(choice)i(spared)f(us)g(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19056
f(w)o(ork)g(of)g(ha)n(ving)i(to)e(de\002ne)321 5230 y(speci\002c)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19057
b(cut-elimination)i(rules)e(for)f(intuitionistic)k(sequent)d(proofs,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19058
(b)n(ut)e(it)f(causes)j(the)e(translation)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19059
5295 1290 4 v 427 5351 a F3(1)456 5382 y F2(W)-6 b(e)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19060
b(are)h(not)g(concerned)i(here)e(with)g(the)g(quanti\002ers)g(and)h(ne)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19061
o(gation.)p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19062
%%Page: 67 79
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19063
TeXDict begin 67 78 bop Black 277 51 a Gb(3.1)23 b(Intuitionistic)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19064
(Natural)f(Deduction)2216 b(67)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19065
Black 277 300 3226 4 v 277 1564 4 1265 v 1486 392 453
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19066
4 v 1486 471 a Ga(x)18 b F4(:)f Ga(B)5 b(;)15 b F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19067
1789 459 11 41 v 1799 441 46 5 v 96 w Ga(B)609 685 y(x)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19068
F4(:)f Ga(B)5 b(;)15 b F4(\000)p 912 673 11 41 v 922
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19069
655 46 5 v 96 w Ga(C)p 609 723 451 4 v 650 801 a F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19070
727 789 11 41 v 738 771 46 5 v 96 w Ga(B)5 b F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19071
Ga(C)1101 740 y F6(\033)1172 754 y Gc(I)1980 699 y F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19072
2057 687 11 41 v 2067 669 46 5 v 96 w Ga(B)g F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19073
Ga(C)98 b F4(\000)p 2517 687 11 41 v 2527 669 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19074
96 w Ga(B)p 1980 723 687 4 v 2211 801 a F4(\000)p 2288
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19075
789 11 41 v 2298 771 46 5 v 96 w Ga(C)2708 740 y F6(\033)2778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19076
754 y Gc(E)568 1029 y F4(\000)p 645 1017 11 41 v 656
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19077
999 46 5 v 97 w Ga(B)d F4(\000)p 963 1017 11 41 v 973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19078
999 46 5 v 96 w Ga(C)p 568 1049 543 4 v 660 1128 a F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19079
737 1116 11 41 v 748 1098 46 5 v 96 w Ga(B)5 b F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19080
Ga(C)1152 1067 y F6(^)1213 1081 y Gc(I)2101 1016 y F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19081
2179 1004 11 41 v 2189 985 46 5 v 97 w Ga(B)2324 1030
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19082
y F9(1)2363 1016 y F6(^)p Ga(B)2493 1030 y F9(2)p 2101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19083
1049 431 4 v 2192 1128 a F4(\000)p 2269 1116 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19084
2279 1098 46 5 v 96 w Ga(B)2414 1142 y Gc(i)2574 1062
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19085
y F6(^)2634 1076 y Gc(E)2686 1086 y FZ(i)704 1346 y F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19086
781 1334 11 41 v 791 1316 46 5 v 96 w Ga(B)926 1360 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19087
Gc(i)p 614 1380 431 4 v 614 1458 a F4(\000)p 691 1446
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19088
11 41 v 701 1428 46 5 v 96 w Ga(B)836 1472 y F9(1)875
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19089
1458 y F6(_)p Ga(B)1005 1472 y F9(2)1086 1393 y F6(_)1146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19090
1407 y Gc(I)1177 1417 y FZ(i)1607 1342 y F4(\000)p 1684
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19091
1330 11 41 v 1694 1312 46 5 v 96 w Ga(B)g F6(_)o Ga(C)98
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19092
b(x)17 b F4(:)g Ga(B)5 b(;)15 b F4(\000)p 2359 1330 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19093
41 v 2370 1312 46 5 v 96 w Ga(D)94 b(y)20 b F4(:)d Ga(C)q(;)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19094
F4(\000)p 2895 1330 11 41 v 2906 1312 46 5 v 97 w Ga(D)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19095
1607 1380 1443 4 v 2213 1458 a F4(\000)p 2290 1446 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19096
41 v 2300 1428 46 5 v 96 w Ga(D)3091 1398 y F6(_)3151
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19097
1412 y Gc(E)p 3499 1564 4 1265 v 277 1567 3226 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19098
770 1721 a Gg(Figure)24 b(3.1:)29 b(Natural)24 b(deduction)i(calculus)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19099
(for)e(intuitionistic)j(logic.)p Black Black 277 2107
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19100
a F6(j)p 304 2107 28 4 v 322 2107 V 340 2107 V 65 w(j)392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19101
2074 y Fu(s)456 2107 y Gg(to)34 b(be)f(slightly)j(messy:)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19102
b(we)33 b(ha)n(v)o(e)h(to)g(e)o(xplicitly)i(record)f(the)e(co-name)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19103
(of)f(the)g(formula)g(in)277 2219 y(the)c(succedent.)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19104
b(Before)30 b(gi)n(ving)g(the)g(de\002nition)h(of)e F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19105
2025 2219 V 2043 2219 V 2060 2219 V 64 w(j)2112 2186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19106
y Fu(s)2172 2219 y Gg(and)h F6(j)p 2359 2219 V 2377 2219
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19107
V 2395 2219 V 65 w(j)2447 2186 y Fu(n)2488 2219 y Gg(,)f(we)g(shall)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19108
(\002rst)e(illustrate)k(the)277 2332 y(translations)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19109
b(on)d(the)f(le)n(v)o(el)h(of)g(proofs.)p Black 277 2512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19110
a Gb(Axioms:)p Black 47 w Gg(The)f(sequent)i(proof)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19111
1723 2636 560 4 v 1723 2715 a Ga(x)18 b F4(:)f Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19112
b(;)15 b F4(\000)p 2026 2703 11 41 v 2036 2685 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19113
96 w Ga(a)i F4(:)h Ga(B)504 2898 y Gg(is)24 b(translated)i(to)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19114
1752 3004 453 4 v 1752 3083 a Ga(x)17 b F4(:)h Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19115
b(;)15 b F4(\000)p 2054 3071 11 41 v 2065 3053 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19116
96 w Ga(B)29 b(:)504 3266 y Gg(The)h(re)n(v)o(erse)h(translation)h(is)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19117
(analogous,)k(b)n(ut)c(we)f(ha)n(v)o(e)h(to)g(record,)j(as)c(mentioned)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19118
j(abo)o(v)o(e,)504 3379 y(the)24 b(co-name)h Ga(a)p Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19119
Black 277 3550 a Gb(Right)e(Rules:)p Black 46 w Gg(A)f(sequent)k(proof)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19120
e(ending)i(with)d(an)g(instance)j(of)e(the)f F6(^)2595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19121
3564 y Gc(R)2653 3550 y Gg(-rule)1671 3689 y Ga(\031)1723
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19122
3703 y F9(1)1549 3781 y F4(\000)p 1626 3769 11 41 v 1637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19123
3751 46 5 v 96 w Ga(a)17 b F4(:)h Ga(B)2090 3688 y(\031)2142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19124
3702 y F9(2)1974 3781 y F4(\000)p 2051 3769 11 41 v 2062
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19125
3751 46 5 v 96 w Ga(b)g F4(:)f Ga(C)p 1549 3801 749 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19126
v 1695 3880 a F4(\000)p 1772 3868 11 41 v 1782 3850 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19127
5 v 96 w Ga(c)g F4(:)h Ga(B)5 b F6(^)o Ga(C)2339 3819
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19128
y F6(^)2400 3833 y Gc(R)504 4063 y Gg(is)24 b(translated)i(to)1658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19129
4211 y F6(j)p Ga(\031)1735 4225 y F9(1)1775 4211 y F6(j)1800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19130
4179 y Fu(n)1636 4313 y F4(\000)p 1713 4301 11 41 v 1723
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19131
4283 46 5 v 96 w Ga(B)1975 4211 y F6(j)p Ga(\031)2052
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19132
4225 y F9(2)2091 4211 y F6(j)2116 4179 y Fu(n)1953 4313
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19133
y F4(\000)p 2030 4301 11 41 v 2041 4283 46 5 v 97 w Ga(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19134
1636 4333 543 4 v 1728 4412 a F4(\000)p 1805 4400 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19135
41 v 1815 4381 46 5 v 96 w Ga(B)5 b F6(^)o Ga(C)2220
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19136
4351 y F6(^)2280 4365 y Gc(I)2345 4351 y Ga(:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19137
277 4595 a Gb(Left)23 b(Rules:)p Black 46 w Gg(A)g(sequent)i(proof)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19138
(ending)g(with)e(an)h(instance)h(of)f(the)g F6(_)2539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19139
4609 y Gc(L)2591 4595 y Gg(-rule)1563 4733 y Ga(\031)1615
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19140
4747 y F9(1)1331 4825 y Ga(x)17 b F4(:)g Ga(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19141
b F4(\000)p 1633 4813 11 41 v 1644 4795 46 5 v 97 w Ga(c)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19142
F4(:)h Ga(D)2204 4733 y(\031)2256 4747 y F9(2)1977 4825
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19143
y Ga(y)i F4(:)e Ga(C)q(;)d F4(\000)p 2269 4813 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19144
v 2279 4795 46 5 v 97 w Ga(c)i F4(:)h Ga(D)p 1331 4863
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19145
1191 4 v 1588 4941 a(z)j F4(:)d Ga(B)5 b F6(_)o Ga(C)q(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19146
b F4(\000)p 2012 4929 11 41 v 2022 4911 46 5 v 97 w Ga(c)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19147
F4(:)h Ga(D)2563 4881 y F6(_)2624 4895 y Gc(L)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19148
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19149
%%Page: 68 80
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19150
TeXDict begin 68 79 bop Black -144 51 a Gb(68)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19151
b(Deduction)p -144 88 3691 4 v Black 549 365 a Gg(is)g(translated)j(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19152
p 725 573 706 4 v 725 652 a Ga(z)c F4(:)17 b Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19153
F6(_)o Ga(C)q(;)15 b F4(\000)p 1149 640 11 41 v 1160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19154
621 46 5 v 97 w Ga(B)5 b F6(_)o Ga(C)1832 550 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19155
Ga(\031)1909 564 y F9(1)1949 550 y F6(j)1974 517 y Fu(n)1522
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19156
652 y Ga(x)17 b F4(:)g Ga(B)5 b(;)15 b(z)22 b F4(:)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19157
b Ga(B)5 b F6(_)o Ga(C)q(;)15 b F4(\000)p 2171 640 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19158
41 v 2182 621 46 5 v 97 w Ga(D)2721 550 y F6(j)p Ga(\031)2798
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19159
564 y F9(2)2837 550 y F6(j)2862 517 y Fu(n)2416 652 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19160
Ga(y)20 b F4(:)d Ga(C)q(;)e(z)23 b F4(:)17 b Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19161
F6(_)o Ga(C)q(;)15 b F4(\000)p 3054 640 11 41 v 3065
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19162
621 46 5 v 97 w Ga(D)p 725 689 2483 4 v 1678 768 a(z)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19163
b F4(:)d Ga(B)5 b F6(_)o Ga(C)q(;)15 b F4(\000)p 2102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19164
756 11 41 v 2112 738 46 5 v 97 w Ga(D)3249 707 y F6(_)3310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19165
721 y Gc(E)549 989 y Gg(where)23 b Ga(z)f F4(:)17 b Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19166
b F6(_)o Ga(C)30 b Gg(might)24 b(need)g(to)f(be)h(added)h(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19167
F6(j)p Ga(\031)2169 1003 y F9(1)2209 989 y F6(j)2234
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19168
956 y Fu(n)2297 989 y Gg(or)h(to)f F6(j)p Ga(\031)2566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19169
1003 y F9(2)2606 989 y F6(j)2631 956 y Fu(n)2672 989
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19170
y Gg(.)p Black 321 1170 a Gb(The)g(Cut-Rule:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19171
45 w Gg(A)f(sequent)k(proof)e(ending)h(with)f(an)f(instance)j(of)d(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19172
h(cut-rule)1576 1346 y Ga(\031)1628 1360 y F9(1)1435
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19173
1439 y F4(\000)1492 1453 y F9(1)p 1551 1427 11 41 v 1561
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19174
1409 46 5 v 1627 1439 a Ga(a)17 b F4(:)h Ga(B)2148 1345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19175
y(\031)2200 1359 y F9(2)1899 1439 y Ga(x)f F4(:)h Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19176
b(;)15 b F4(\000)2182 1453 y F9(2)p 2241 1427 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19177
2252 1409 46 5 v 2317 1439 a Ga(b)j F4(:)f Ga(C)p 1435
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19178
1476 1053 4 v 1711 1556 a F4(\000)1768 1570 y F9(1)1808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19179
1556 y Ga(;)e F4(\000)1905 1570 y F9(2)p 1964 1544 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19180
41 v 1975 1526 46 5 v 2041 1556 a Ga(b)i F4(:)g Ga(C)2529
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19181
1507 y Gg(Cut)549 1777 y(is)23 b(translated)j(to)1470
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19182
1963 y F6(j)p Ga(\031)1547 1977 y F9(1)1586 1963 y F6(j)1611
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19183
1930 y Fu(n)1374 2065 y F4(\000)1431 2079 y F9(1)p 1490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19184
2053 11 41 v 1501 2035 46 5 v 1567 2065 a Ga(a)17 b F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19185
Ga(B)2042 1962 y F6(j)p Ga(\031)2119 1976 y F9(2)2158
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19186
1962 y F6(j)2183 1929 y Fu(n)1839 2065 y Ga(x)g F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19187
Ga(B)5 b(;)15 b F4(\000)2121 2079 y F9(2)p 2181 2053
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19188
11 41 v 2191 2035 46 5 v 2257 2065 a Ga(b)i F4(:)g Ga(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19189
1374 2102 1053 4 v 1651 2182 a F4(\000)1708 2196 y F9(1)1747
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19190
2182 y Ga(;)e F4(\000)1844 2196 y F9(2)p 1904 2170 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19191
41 v 1914 2152 46 5 v 1980 2182 a Ga(b)i F4(:)h Ga(C)2468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19192
2133 y Gg(Subst)27 b Ga(:)549 2403 y Gg(There)c(are)g(tw)o(o)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19193
(possible)j(w)o(ays)e(of)g(dealing)i(with)e(the)g(substitution:)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19194
b(either)24 b(it)f(is)g(performed)549 2516 y(immediately)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19195
b(,)33 b(or)e(it)f(is)g(\223suspended\224)k(and)d(performed)h(later)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19196
b(.)50 b(W)-7 b(e)29 b(shall)j(emplo)o(y)f(the)g(\002rst)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19197
2629 y(method)24 b(and)g(mention)h(some)e(problems)i(with)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19198
(second.)p Black 321 2811 a Gb(Intr)n(oduction)g(Rules:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19199
Black 46 w Gg(An)f(NJ-proof)i(ending)g(with)f(an)f(instance)j(of)d(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19200
h F6(^)2792 2825 y Gc(I)2832 2811 y Gg(-rule)1764 3009
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19201
y F6(D)1834 3023 y F9(1)1705 3101 y F4(\000)p 1782 3089
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19202
11 41 v 1793 3071 46 5 v 96 w Ga(B)2080 3009 y F6(D)2150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19203
3023 y F9(2)2023 3101 y F4(\000)p 2100 3089 11 41 v 2110
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19204
3071 46 5 v 96 w Ga(C)p 1705 3121 543 4 v 1797 3200 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19205
F4(\000)p 1874 3188 11 41 v 1885 3169 46 5 v 96 w Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19206
b F6(^)o Ga(C)2289 3139 y F6(^)2350 3153 y Gc(I)549 3420
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19207
y Gg(is)23 b(translated)j(to)1660 3612 y F6(jD)1755 3626
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19208
y F9(1)1794 3612 y F6(j)1819 3579 y Fu(s)1819 3634 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19209
Gc(a)1593 3713 y F4(\000)p 1670 3701 11 41 v 1681 3683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19210
46 5 v 96 w Ga(a)18 b F4(:)f Ga(B)2083 3606 y F6(jD)2178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19211
3620 y F9(2)2218 3606 y F6(j)2243 3573 y Fu(s)2243 3634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19212
y Gc(b)2019 3713 y F4(\000)p 2096 3701 11 41 v 2106 3683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19213
46 5 v 96 w Ga(b)g F4(:)g Ga(C)p 1593 3733 749 4 v 1739
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19214
3812 a F4(\000)p 1816 3800 11 41 v 1826 3782 46 5 v 96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19215
w Ga(c)g F4(:)h Ga(B)5 b F6(^)o Ga(C)2383 3751 y F6(^)2444
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19216
3765 y Gc(R)549 4033 y Gg(where)23 b Ga(c)g Gg(is)h(a)f(fresh)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19217
(co-name.)p Black 321 4214 a Gb(Elimination)g(Rules:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19218
Black 46 w Gg(An)e(NJ-proof)j(ending)h(with)d(an)g(instance)j(of)e(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19219
f F6(_)2758 4228 y Gc(E)2818 4214 y Gg(-rule)1370 4412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19220
y F6(D)1440 4426 y F9(1)1246 4505 y F4(\000)p 1323 4493
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19221
11 41 v 1333 4475 46 5 v 96 w Ga(B)5 b F6(_)o Ga(C)1869
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19222
4412 y F6(D)1939 4426 y F9(2)1695 4505 y Ga(x)17 b F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19223
Ga(B)5 b(;)15 b F4(\000)p 1998 4493 11 41 v 2008 4475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19224
46 5 v 96 w Ga(D)2410 4412 y F6(D)2480 4426 y F9(3)2242
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19225
4505 y Ga(y)21 b F4(:)c Ga(C)q(;)e F4(\000)p 2534 4493
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19226
11 41 v 2544 4475 46 5 v 97 w Ga(D)p 1246 4542 1443 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19227
v 1851 4621 a F4(\000)p 1928 4609 11 41 v 1939 4591 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19228
5 v 96 w Ga(D)2729 4561 y F6(_)2790 4575 y Gc(E)549 4842
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19229
y Gg(is)23 b(translated)j(to)1229 5144 y F6(jD)1324 5158
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19230
y F9(1)1364 5144 y F6(j)1389 5111 y Fu(s)1389 5166 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19231
Gc(a)1096 5245 y F4(\000)p 1173 5233 11 41 v 1184 5215
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19232
46 5 v 97 w Ga(a)17 b F4(:)g Ga(B)5 b F6(_)o Ga(C)1834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19233
5027 y F6(jD)1929 5041 y F9(2)1969 5027 y F6(j)1994 4994
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19234
y Fu(s)1994 5050 y Gc(c)1654 5129 y Ga(x)17 b F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19235
Ga(B)5 b(;)15 b F4(\000)p 1956 5117 11 41 v 1967 5099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19236
46 5 v 96 w Ga(c)j F4(:)f Ga(D)2475 5027 y F6(jD)2570
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19237
5041 y F9(3)2610 5027 y F6(j)2635 4994 y Fu(s)2635 5050
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19238
y Gc(c)2300 5129 y Ga(y)j F4(:)e Ga(C)q(;)d F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19239
2591 5117 11 41 v 2602 5099 46 5 v 96 w Ga(c)j F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19240
Ga(D)p 1654 5166 1191 4 v 1911 5245 a(z)k F4(:)d Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19241
b F6(_)o Ga(C)q(;)15 b F4(\000)p 2334 5233 11 41 v 2345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19242
5215 46 5 v 97 w Ga(c)i F4(:)h Ga(D)2886 5185 y F6(_)2946
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19243
5199 y Gc(L)p 1096 5283 1491 4 v 1677 5361 a F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19244
1754 5349 11 41 v 1764 5331 46 5 v 96 w Ga(c)g F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19245
Ga(D)2629 5313 y Gg(Cut)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19246
%%Page: 69 81
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19247
TeXDict begin 69 80 bop Black 277 51 a Gb(3.1)23 b(Intuitionistic)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19248
(Natural)f(Deduction)2216 b(69)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19249
504 365 a Gg(where)25 b Ga(z)j Gg(is)d(a)f(fresh)i(name.)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19250
b(This)24 b(is)g(the)h(standard)i(translation)h(for)d(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19251
F6(_)2895 379 y Gc(E)2955 365 y Gg(-rule;)h(ho)n(we)n(v)o(er)l(,)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19252
478 y(we)g(shall)h(slightly)h(modify)g(this)e(translation)k(by)c(using)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19253
i(our)e(proof)i(substitution)h(instead)f(of)504 590 y(the)c(cut-rule.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19254
418 812 y(As)32 b(mentioned)j(earlier)l(,)i(it)c(will)f(be)h(con)l(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19255
(enient)j(to)d(de\002ne)g(the)g(translations)k F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19256
3032 812 28 4 v 3050 812 V 3067 812 V 65 w(j)3120 779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19257
y Fu(s)3183 812 y Gg(and)d F6(j)p 3374 812 V 3391 812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19258
V 3409 812 V 64 w(j)3461 779 y Fu(n)277 925 y Gg(o)o(v)o(er)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19259
b(terms.)29 b(T)-7 b(o)22 b(do)i(so,)f(NJ-proofs)i(will)e(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19260
(annotated)i(with)d(lambda)i(terms.)p Black 277 1113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19261
a Gb(De\002nition)30 b(3.1.1)h Gg(\(Ra)o(w)f(Lambda)h(T)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19262
b(erms\))p Gb(:)p Black 34 w Gg(The)31 b(set)g(of)g(ra)o(w)f(lambda)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19263
(terms)f(is)g(de\002ned)h(by)f(the)277 1225 y(grammar)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19264
Black Black 482 1417 a Ga(M)5 b(;)15 b(N)5 b(;)15 b(P)113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19265
b F4(::=)100 b Ga(x)1232 b Gg(v)n(ariable)952 1530 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19266
F6(j)148 b Ga(\025x)17 b F4(:)g Ga(B)5 b(:M)933 b Gg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19267
(implication-introducti)q(on)952 1643 y F6(j)148 b Ga(M)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19268
b(N)1098 b Gg(implication-elimination)952 1756 y F6(j)148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19269
b(h)p Ga(M)5 b(;)15 b(N)10 b F6(i)998 b Gg(and-introduction)952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19270
1869 y F6(j)148 b Fp(fst)o F4(\()p Ga(M)10 b F4(\))973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19271
b Gg(and)2539 1836 y F9(1)2579 1869 y Gg(-elimination)952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19272
1982 y F6(j)148 b Fp(snd)o F4(\()p Ga(M)10 b F4(\))973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19273
b Gg(and)2539 1949 y F9(2)2579 1982 y Gg(-elimination)952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19274
2095 y F6(j)148 b Fp(inl)o F4(\()p Ga(M)10 b F4(\))973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19275
b Gg(or)2484 2062 y F9(1)2524 2095 y Gg(-introduction)952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19276
2208 y F6(j)148 b Fp(inr)o F4(\()p Ga(M)10 b F4(\))973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19277
b Gg(or)2484 2175 y F9(2)2524 2208 y Gg(-introduction)952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19278
2321 y F6(j)148 b Fp(case)o F4(\()p Ga(P)s(;)15 b(\025x)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19279
F4(:)f Ga(B)5 b(:M)g(;)15 b(\025y)20 b F4(:)e Ga(C)q(:N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19280
b F4(\))190 b Gg(or)n(-elimination)277 2566 y(Let)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19281
b(us)g(brie\003y)h(describe)i(the)e(notions)h(and)f(con)l(v)o(entions)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19282
(we)c(shall)h(use)g(for)f(lambda)i(terms:)43 b(the)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19283
2679 y(notions)30 b(of)d(bound)i(and)g(free)f(v)n(ariables)i(will)d(be)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19284
h(as)f(usual;)k(for)d(bre)n(vity)h(we)e(shall)i(often)f(omit)g(the)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19285
2792 y(typings)35 b(on)e(binders)h(and)g(simply)f(write)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19286
Ga(\025x:M)10 b Gg(;)37 b(a)c(Barendre)o(gt-style)j(naming)e(con)l(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19287
(ention)i(for)277 2905 y(v)n(ariables)c(will)e(al)o(w)o(ays)h(be)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19288
(observ)o(ed;)36 b(and)30 b(lambda)h(terms)f(will)g(be)g(re)o(garded)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19289
(as)e(equi)n(v)n(alent)i(up)277 3018 y(to)25 b(renaming)h(of)f(bound)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19290
(v)n(ariables.)34 b(A)24 b(pleasing)j(consequence)i(of)c(these)g(con)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19291
(v)o(entions)k(is)c(that)g(the)277 3131 y(notion)e(of)f(v)n(ariable)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19292
(substitution)i(can)d(be)g(de\002ned)g(without)g(w)o(orrying)i(about)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19293
(clashes)i(and)e(capture)277 3243 y(of)j(v)n(ariables.)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19294
b(Since)25 b(it)f(will)h(al)o(w)o(ays)g(be)g(clear)h(from)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19295
(conte)o(xt)h(whether)g(we)e(deal)h(with)g(a)f(proof)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19296
3356 y(substitution)i(or)c(a)g(v)n(ariable)j(substitution,)h(no)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19297
(confusion)j(will)d(occur)i(when)f(we)e(emplo)o(y)j(the)e(same)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19298
3469 y(shorthand)k(notation)g(for)e(both)g(substitutions;)k(that)c(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19299
F4([)p Ga(\033)s F4(])p Gg(.)418 3599 y(T)-7 b(o)28 b(annotate)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19300
(natural)e(deduction)i(proofs)f(with)d(lambda)i(terms,)h(we)d(shall)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19301
(replace)h(e)n(v)o(ery)e(se-)277 3712 y(quent)f(of)e(the)g(form)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19302
F4(\000)p 1021 3700 11 41 v 1032 3681 46 5 v 96 w Ga(B)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19303
Gg(with)c(a)g(typing)i(judgement)g(of)e(the)h(form)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19304
F4(\000)2638 3700 y Gc(.)2693 3712 y Ga(M)h F4(:)18 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19305
Ga(B)30 b Gg(in)c(which)g F4(\000)g Gg(is)g(a)277 3825
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19306
y(conte)o(xt,)i Ga(M)35 b Gg(is)26 b(a)g(lambda)g(term)g(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19307
Ga(B)i Gg(is)d(a)g(formula.)37 b(In)26 b(the)g(sequel)i(we)d(are)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19308
(interested)j(in)d(only)277 3937 y(simply-typed)34 b(lambda)e(terms,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19309
(i.e.,)g(those)f(for)f(which)h(there)g(is)f(a)g(conte)o(xt)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19310
F4(\000)d Gg(and)i(a)e(formula)j Ga(B)277 4050 y Gg(such)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19311
b(that)f F4(\000)708 4038 y Gc(.)763 4050 y Ga(M)j F4(:)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19312
b Ga(B)27 b Gg(is)d(deri)n(v)n(able)h(gi)n(v)o(en)g(the)f(inferences)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19313
(sho)n(wn)e(in)g(Figure)g(3.2.)30 b(The)23 b(follo)n(wing)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19314
4163 y(set)h(contains)i(all)d(simply-typed)k(lambda)d(terms.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19315
Black Black 495 4406 a F4(\003)658 4354 y F5(def)665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19316
4406 y F4(=)843 4305 y FK(n)928 4406 y Ga(M)1074 4301
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19317
y FK(\014)1074 4356 y(\014)1074 4410 y(\014)1151 4406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19318
y F4(\000)1233 4394 y Gc(.)1288 4406 y Ga(M)k F4(:)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19319
b Ga(B)27 b Gg(is)d(deri)n(v)n(able)h(using)g(the)f(rules)g(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19320
(en)g(in)g(Figure)g(3.2)3246 4305 y FK(o)418 4650 y Gg(No)n(w)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19321
b(we)h(return)h(to)f(the)h(translations)i F6(j)p 1660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19322
4650 28 4 v 1678 4650 V 1696 4650 V 65 w(j)1748 4617
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19323
y Fu(s)1799 4650 y Gg(and)d F6(j)p 1976 4650 V 1994 4650
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19324
V 2012 4650 V 65 w(j)2064 4617 y Fu(n)2125 4650 y Gg(whose)g(clauses)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19325
(are)f(gi)n(v)o(en)f(in)g(Figures)i(3.3)277 4763 y(and)g(3.4,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19326
(respecti)n(v)o(ely)-6 b(.)30 b(Clearly)-6 b(,)22 b(we)f(e)o(xpect)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19327
(that)f(both)h(translations)i(respect)f(typing)f(judgements.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19328
Black 277 4951 a Gb(Pr)n(oposition)j(3.1.2:)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19329
97 w Gg(\(i\))p Black 46 w(If)35 b Ga(M)56 b F6(2)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19330
b F4(\003)34 b Gg(with)h(the)f(typing)j(judgement)f F4(\000)2735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19331
4939 y Gc(.)2790 4951 y Ga(M)28 b F4(:)17 b Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19332
Gg(,)36 b(then)f(for)g(all)504 5063 y(co-names)25 b Ga(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19333
Gg(we)g(ha)n(v)o(e)h F6(j)p Ga(M)10 b F6(j)1414 5030
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19334
y Fu(s)1414 5086 y Gc(a)1482 5063 y F6(2)25 b FY(I)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19335
Gg(with)i(the)g(typing)h(judgement)g F4(\000)2691 5051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19336
y Gc(.)2746 5063 y F6(j)p Ga(M)10 b F6(j)2894 5030 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19337
Fu(s)2894 5086 y Gc(a)2962 5051 y(.)3017 5063 y Ga(a)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19338
b F4(:)g Ga(B)5 b Gg(.)p Black 348 5249 a(\(ii\))p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19339
46 w(If)28 b Ga(M)42 b F6(2)33 b FY(I)27 b Gg(with)g(the)h(typing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19340
(judgement)g F4(\000)1960 5237 y Gc(.)2014 5249 y Ga(M)2138
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19341
5237 y Gc(.)2193 5249 y Ga(a)17 b F4(:)g Ga(B)5 b Gg(,)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19342
b(then)i(we)d(ha)n(v)o(e)j F6(j)p Ga(M)10 b F6(j)3084
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19343
5216 y Fu(n)3157 5249 y F6(2)33 b F4(\003)27 b Gg(with)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19344
5361 y(the)d(typing)h(judgement)h F4(\000)1382 5349 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19345
Gc(.)1437 5361 y F6(j)p Ga(M)10 b F6(j)1585 5329 y Fu(n)1643
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19346
5361 y F4(:)18 b Ga(B)t Gg(.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19347
%%Page: 70 82
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19348
TeXDict begin 70 81 bop Black -144 51 a Gb(70)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19349
b(Deduction)p -144 88 3691 4 v Black Black 321 277 3226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19350
4 v 321 1863 4 1587 v 1483 369 548 4 v 1483 447 a Ga(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19351
b F4(:)g Ga(B)5 b(;)15 b F4(\000)1790 435 y Gc(.)1845
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19352
447 y Ga(x)i F4(:)h Ga(B)536 638 y(x)f F4(:)g Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19353
b(;)15 b F4(\000)843 626 y Gc(.)898 638 y Ga(M)27 b F4(:)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19354
b Ga(C)p 511 676 641 4 v 511 755 a F4(\000)593 743 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19355
Gc(.)648 755 y Ga(\025x:M)28 b F4(:)17 b Ga(B)5 b F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19356
Ga(C)1193 693 y F6(\033)1264 707 y Gc(I)2030 653 y F4(\000)2112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19357
641 y Gc(.)2167 653 y Ga(M)27 b F4(:)18 b Ga(B)5 b F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19358
Ga(C)97 b F4(\000)2713 641 y Gc(.)2768 653 y Ga(N)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19359
b F4(:)18 b Ga(B)p 2030 677 955 4 v 2275 755 a F4(\000)2357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19360
743 y Gc(.)2412 755 y Ga(M)25 b(N)i F4(:)18 b Ga(C)3026
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19361
694 y F6(\033)3096 708 y Gc(E)431 983 y F4(\000)514 971
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19362
y Gc(.)568 983 y Ga(M)28 b F4(:)17 b Ga(B)96 b F4(\000)973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19363
971 y Gc(.)1028 983 y Ga(N)27 b F4(:)17 b Ga(C)p 431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19364
1003 811 4 v 492 1088 a F4(\000)574 1076 y Gc(.)629 1088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19365
y F6(h)p Ga(M)5 b(;)15 b(N)10 b F6(i)18 b F4(:)g Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19366
b F6(^)o Ga(C)1283 1021 y F6(^)1344 1035 y Gc(I)1760
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19367
969 y F4(\000)1842 957 y Gc(.)1897 969 y Ga(M)28 b F4(:)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19368
b Ga(B)2124 983 y F9(1)2164 969 y F6(^)o Ga(B)2293 983
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19369
y F9(2)p 1738 1003 617 4 v 1738 1088 a F4(\000)1820 1076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19370
y Gc(.)1875 1088 y Fp(fst)o F4(\()p Ga(M)10 b F4(\))18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19371
b(:)g Ga(B)2316 1102 y F9(1)2396 1017 y F6(^)2457 1031
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19372
y Gc(E)2509 1040 y FV(1)2661 969 y F4(\000)2743 957 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19373
Gc(.)2798 969 y Ga(M)27 b F4(:)18 b Ga(B)3025 983 y F9(1)3064
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19374
969 y F6(^)p Ga(B)3194 983 y F9(2)p 2638 1003 V 2638
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19375
1088 a F4(\000)2721 1076 y Gc(.)2775 1088 y Fp(snd)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19376
F4(\()p Ga(M)10 b F4(\))18 b(:)f Ga(B)3216 1102 y F9(2)3297
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19377
1017 y F6(^)3358 1031 y Gc(E)3410 1040 y FV(2)1066 1302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19378
y F4(\000)1148 1290 y Gc(.)1203 1302 y Ga(M)27 b F4(:)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19379
b Ga(B)1429 1316 y F9(1)p 874 1335 786 4 v 874 1420 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19380
F4(\000)956 1408 y Gc(.)1011 1420 y Fp(inl)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19381
Ga(M)10 b F4(\))18 b(:)g Ga(B)1452 1434 y F9(1)1491 1420
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19382
y F6(_)p Ga(B)1621 1434 y F9(2)1702 1349 y F6(_)1762
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19383
1363 y Gc(I)1793 1372 y FV(1)2228 1302 y F4(\000)2310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19384
1290 y Gc(.)2364 1302 y Ga(M)28 b F4(:)17 b Ga(B)2591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19385
1316 y F9(2)p 2036 1335 V 2036 1420 a F4(\000)2118 1408
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19386
y Gc(.)2173 1420 y Fp(inr)o F4(\()p Ga(M)10 b F4(\))18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19387
b(:)g Ga(B)2614 1434 y F9(1)2653 1420 y F6(_)p Ga(B)2783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19388
1434 y F9(2)2864 1349 y F6(_)2924 1363 y Gc(I)2955 1372
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19389
y FV(2)941 1634 y F4(\000)1023 1622 y Gc(.)1078 1634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19390
y Ga(P)30 b F4(:)17 b Ga(B)5 b F6(_)o Ga(C)98 b(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19391
b F4(:)g Ga(B)5 b(;)15 b F4(\000)1812 1622 y Gc(.)1867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19392
1634 y Ga(M)28 b F4(:)17 b Ga(D)94 b(y)20 b F4(:)d Ga(C)q(;)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19393
F4(\000)2490 1622 y Gc(.)2545 1634 y Ga(N)27 b F4(:)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19394
b Ga(D)p 941 1672 1825 4 v 1298 1757 a F4(\000)1380 1745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19395
y Gc(.)1435 1757 y Fp(case)o F4(\()p Ga(P)s(;)e(\025x:M)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19396
b(;)15 b(\025)q(y)s(:N)10 b F4(\))18 b(:)f Ga(D)2807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19397
1690 y F6(_)2867 1704 y Gc(E)p 3543 1863 4 1587 v 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19398
1866 3226 4 v Black 636 2019 a Gg(Figure)24 b(3.2:)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19399
b(T)-6 b(erm)22 b(assignment)k(for)d(intuitionistic)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19400
b(natural)d(deduction)h(proofs.)p Black Black Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19401
2465 a F7(Pr)l(oof)o(.)p Black 34 w Gg(Both)d(by)h(routine)h(induction)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19402
i(on)c(the)h(structure)i(of)d Ga(M)10 b Gg(.)p 3480 2465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19403
4 62 v 3484 2407 55 4 v 3484 2465 V 3538 2465 4 62 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19404
Black 321 2653 a Gb(Remark)35 b(3.1.3:)p Black 34 w Gg(It)g(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19405
(well-kno)n(wn)h(that)g F6(j)p 1768 2653 28 4 v 1786
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19406
2653 V 1804 2653 V 65 w(j)1856 2620 y Fu(n)1897 2653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19407
y Gg(,)h(the)e(translation)j(from)d(sequent)i(proofs)g(to)e(NJ-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19408
2766 y(proofs,)25 b(is)e(man)o(y-to-one.)31 b(T)-7 b(ak)o(e)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19409
b(for)f(e)o(xample)i(the)f(sequent)h(proof)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19410
1320 2960 233 4 v 1320 3039 a Ga(A)p 1408 3027 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19411
v 1418 3008 46 5 v 96 w(A)p 1252 3058 367 4 v 1252 3137
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19412
a(A)p F6(^)p Ga(B)p 1475 3125 11 41 v 1485 3107 46 5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19413
v 101 w(A)1661 3072 y F6(^)1721 3086 y Gc(L)1769 3095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19414
y FV(1)p 1899 2960 233 4 v 1899 3039 a Ga(A)p 1987 3027
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19415
11 41 v 1998 3008 46 5 v 97 w(A)p 2223 2960 233 4 v 91
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19416
w(A)p 2311 3027 11 41 v 2321 3008 46 5 v 96 w(A)p 1899
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19417
3058 557 4 v 1996 3137 a(A)p 2085 3125 11 41 v 2095 3107
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19418
46 5 v 97 w(A)p F6(^)p Ga(A)2497 3077 y F6(^)2557 3091
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19419
y Gc(R)p 1252 3157 1106 4 v 1557 3236 a Ga(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19420
Ga(B)p 1780 3224 11 41 v 1790 3206 46 5 v 101 w(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19421
Ga(A)2399 3187 y Gg(Cut)p 1451 3256 708 4 v 1472 3329
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19422
11 41 v 1482 3310 46 5 v 1548 3341 a F4(\()p Ga(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19423
Ga(B)5 b F4(\))p F6(\033)o F4(\()p Ga(A)p F6(^)q Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19424
F4(\))2201 3273 y F6(\033)2271 3287 y Gc(R)321 3591 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19425
Gg(and)24 b(the)g(tw)o(o)f(normal)i(forms)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19426
852 3766 233 4 v 852 3845 a Ga(A)p 940 3833 11 41 v 950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19427
3815 46 5 v 96 w(A)p 1175 3766 233 4 v 91 w(A)p 1263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19428
3833 11 41 v 1274 3815 46 5 v 97 w(A)p 852 3865 557 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19429
v 949 3943 a(A)p 1037 3931 11 41 v 1048 3913 46 5 v 96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19430
w(A)p F6(^)p Ga(A)1449 3883 y F6(^)1510 3897 y Gc(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19431
882 3963 496 4 v 882 4042 a Ga(A)p F6(^)p Ga(B)p 1104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19432
4030 11 41 v 1115 4012 46 5 v 100 w(A)p F6(^)p Ga(A)1419
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19433
3977 y F6(^)1480 3991 y Gc(L)1528 4000 y FV(1)p 776 4062
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19434
708 4 v 796 4135 11 41 v 806 4117 46 5 v 872 4147 a F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19435
Ga(A)p F6(^)p Ga(B)5 b F4(\))p F6(\033)p F4(\()p Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19436
F6(^)p Ga(A)p F4(\))1525 4080 y F6(\033)1596 4094 y Gc(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19437
1957 3766 233 4 v 1957 3845 a Ga(A)p 2045 3833 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19438
v 2056 3815 46 5 v 96 w(A)p 1890 3865 367 4 v 1890 3943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19439
a(A)p F6(^)o Ga(B)p 2112 3931 11 41 v 2123 3913 46 5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19440
v 101 w(A)2298 3878 y F6(^)2359 3892 y Gc(L)2407 3901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19441
y FV(1)p 2603 3766 233 4 v 2603 3845 a Ga(A)p 2692 3833
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19442
11 41 v 2702 3815 46 5 v 97 w(A)p 2536 3865 367 4 v 2536
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19443
3943 a(A)p F6(^)p Ga(B)p 2759 3931 11 41 v 2769 3913
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19444
46 5 v 101 w(A)2945 3878 y F6(^)3005 3892 y Gc(L)3053
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19445
3901 y FV(1)p 1890 3963 1014 4 v 2149 4042 a Ga(A)p F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19446
Ga(B)p 2371 4030 11 41 v 2381 4012 46 5 v 101 w(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19447
Ga(A)2945 3982 y F6(^)3005 3996 y Gc(R)p 2042 4062 708
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19448
4 v 2063 4135 11 41 v 2073 4117 46 5 v 2139 4147 a F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19449
Ga(A)p F6(^)p Ga(B)g F4(\))p F6(\033)o F4(\()p Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19450
F6(^)q Ga(A)p F4(\))2792 4080 y F6(\033)2862 4094 y Gc(R)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19451
4397 y Gg(reachable)25 b(from)e(it)f(\(for)h(bre)n(vity)h(we)e(omitted)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19452
i(all)f(labels\).)30 b(All)22 b(three)h(sequent)i(proofs)f(are)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19453
(mapped)321 4510 y(by)h F6(j)p 462 4510 28 4 v 480 4510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19454
V 497 4510 V 65 w(j)550 4477 y Fu(n)613 4510 y Gg(to)g(the)g(same)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19455
(lambda)h(term,)g(namely)1522 4755 y Ga(\025x:)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19456
Fp(fst)p F4(\()p Ga(x)p F4(\))p Ga(;)15 b Fp(snd)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19457
Ga(x)p F4(\))p F6(i)26 b Ga(:)462 5023 y Gg(Ne)o(xt)i(we)g(shall)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19458
(de\002ne)g(the)g(reduction)i(rules)e(for)g(NJ-proofs.)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19459
b(Recall)29 b(that)f(one)h(of)g(the)f(moti-)321 5136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19460
y(v)n(ations)e(of)e(cut-elimination)j(are)d(to)g(transform)i(a)d(proof)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19461
i(containing)i(cuts)e(to)f(one)g(in)g(normal)h(form)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19462
5249 y(satisfying)h(the)d(subformula)i(property)g(\(see)f(De\002nition)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19463
f(2.1.1\).)29 b(Whereas)24 b(in)f(sequent)i(calculi)f(one)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19464
5361 y(can)h(check)g(by)g(simple)g(inspection)i(of)d(a)g(proof)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19465
(whether)h(the)e(subformula)j(property)f(holds,)g(this)f(is)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19466
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19467
%%Page: 71 83
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19468
TeXDict begin 71 82 bop Black 277 51 a Gb(3.1)23 b(Intuitionistic)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19469
(Natural)f(Deduction)2216 b(71)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19470
Black 277 533 V 277 2392 4 1860 v 587 711 a Gg(\(1\))926
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19471
b F6(j)p Ga(x)p F6(j)1720 678 y Fu(s)1720 733 y Gc(c)1855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19472
659 y F5(def)1862 711 y F4(=)106 b FL(Ax)p F4(\()p Ga(x;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19473
b(c)p F4(\))587 851 y Gg(\(2\))750 b F6(j)p Ga(\025x:M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19474
b F6(j)1720 818 y Fu(s)1720 873 y Gc(c)1855 799 y F5(def)1862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19475
851 y F4(=)106 b FL(Imp)2184 873 y Gc(R)2242 851 y F4(\()2277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19476
839 y F9(\()2304 851 y Ga(x)2356 839 y F9(\))q FX(h)2411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19477
851 y Ga(a)2459 839 y FX(i)2502 851 y F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19478
b F6(j)2650 818 y Fu(s)2650 873 y Gc(a)2692 851 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19479
b(c)p F4(\))587 990 y Gg(\(3\))691 b F6(jh)p Ga(M)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19480
b(N)10 b F6(ij)1719 957 y Fu(s)1719 1013 y Gc(c)1855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19481
939 y F5(def)1862 990 y F4(=)106 b FL(And)2194 1004 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19482
Gc(R)2252 990 y F4(\()2287 978 y FX(h)2315 990 y Ga(a)2363
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19483
978 y FX(i)2405 990 y F6(j)p Ga(M)10 b F6(j)2553 957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19484
y Fu(s)2553 1013 y Gc(a)2595 990 y Ga(;)2635 978 y FX(h)2663
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19485
990 y Ga(b)2702 978 y FX(i)2745 990 y F6(j)p Ga(N)g F6(j)2878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19486
957 y Fu(s)2878 1018 y Gc(b)2913 990 y Ga(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19487
F4(\))587 1130 y Gg(\(4\))666 b F6(j)p Fp(inl)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19488
Ga(M)10 b F4(\))p F6(j)1720 1097 y Fu(s)1720 1153 y Gc(c)1855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19489
1079 y F5(def)1862 1130 y F4(=)106 b FL(Or)2139 1093
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19490
y F9(1)2139 1153 y Gc(R)2196 1130 y F4(\()2231 1118 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19491
FX(h)2259 1130 y Ga(a)2307 1118 y FX(i)2350 1130 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19492
Ga(M)10 b F6(j)2498 1097 y Fu(s)2498 1153 y Gc(a)2540
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19493
1130 y Ga(;)15 b(c)p F4(\))587 1270 y Gg(\(5\))666 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19494
F6(j)p Fp(inr)p F4(\()p Ga(M)10 b F4(\))p F6(j)1720 1237
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19495
y Fu(s)1720 1293 y Gc(c)1855 1219 y F5(def)1862 1270
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19496
y F4(=)106 b FL(Or)2139 1233 y F9(2)2139 1293 y Gc(R)2196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19497
1270 y F4(\()2231 1258 y FX(h)2259 1270 y Ga(a)2307 1258
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19498
y FX(i)2350 1270 y F6(j)p Ga(M)10 b F6(j)2498 1237 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19499
Fu(s)2498 1293 y Gc(a)2540 1270 y Ga(;)15 b(c)p F4(\))587
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19500
1528 y Gg(\(6\))772 b F6(j)p Ga(M)35 b(N)10 b F6(j)1720
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19501
1495 y Fu(s)1720 1550 y Gc(c)1855 1476 y F5(def)1862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19502
1528 y F4(=)106 b F6(j)p Ga(M)10 b F6(j)2187 1495 y Fu(s)2187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19503
1550 y Gc(a)2230 1528 y F4([)p Ga(a)25 b F4(:=)2449 1516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19504
y F9(\()2477 1528 y Ga(y)2525 1516 y F9(\))2552 1528
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19505
y FL(Imp)2697 1550 y Gc(L)2749 1528 y F4(\()2784 1516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19506
y FX(h)2812 1528 y Ga(b)2851 1516 y FX(i)2878 1528 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19507
F6(j)p Ga(N)10 b F6(j)3011 1495 y Fu(s)3011 1556 y Gc(b)3046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19508
1528 y Ga(;)3086 1516 y F9(\()3114 1528 y Ga(z)3160 1516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19509
y F9(\))3187 1528 y FL(Ax)p F4(\()p Ga(z)t(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19510
F4(\))r Ga(;)g(y)s F4(\)])587 1668 y Gg(\(7\))666 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19511
Fp(fst)p F4(\()p Ga(M)10 b F4(\))p F6(j)1720 1635 y Fu(s)1720
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19512
1690 y Gc(c)1855 1616 y F5(def)1862 1668 y F4(=)106 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19513
F6(j)p Ga(M)10 b F6(j)2187 1635 y Fu(s)2187 1690 y Gc(a)2230
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19514
1668 y F4([)p Ga(a)25 b F4(:=)2449 1656 y F9(\()2477
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19515
1668 y Ga(y)2525 1656 y F9(\))2552 1668 y FL(And)2707
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19516
1631 y F9(1)2707 1691 y Gc(L)2759 1668 y F4(\()2794 1656
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19517
y F9(\()2822 1668 y Ga(x)2874 1656 y F9(\))2901 1668
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19518
y FL(Ax)p F4(\()p Ga(x;)15 b(c)p F4(\))q Ga(;)g(y)s F4(\))q(])587
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19519
1808 y Gg(\(8\))666 b F6(j)p Fp(snd)p F4(\()p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19520
b F4(\))p F6(j)1720 1775 y Fu(s)1720 1830 y Gc(c)1855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19521
1756 y F5(def)1862 1808 y F4(=)106 b F6(j)p Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19522
F6(j)2187 1775 y Fu(s)2187 1830 y Gc(a)2230 1808 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19523
Ga(a)25 b F4(:=)2449 1796 y F9(\()2477 1808 y Ga(y)2525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19524
1796 y F9(\))2552 1808 y FL(And)2707 1771 y F9(2)2707
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19525
1831 y Gc(L)2759 1808 y F4(\()2794 1796 y F9(\()2822
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19526
1808 y Ga(x)2874 1796 y F9(\))2901 1808 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19527
Ga(x;)15 b(c)p F4(\))q Ga(;)g(y)s F4(\))q(])587 1947
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19528
y Gg(\(9\))142 b F6(j)p Fp(case)p F4(\()p Ga(P)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19529
b(\025x:M)5 b(;)15 b(\025)q(y)s(:N)10 b F4(\))p F6(j)1720
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19530
1914 y Fu(s)1720 1970 y Gc(c)1855 1896 y F5(def)1862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19531
1947 y F4(=)106 b F6(j)p Ga(P)13 b F6(j)2160 1914 y Fu(s)2160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19532
1970 y Gc(a)2203 1947 y F4([)p Ga(a)25 b F4(:=)2422 1935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19533
y F9(\()2450 1947 y Ga(z)2496 1935 y F9(\))2524 1947
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19534
y FL(Or)2623 1961 y Gc(L)2675 1947 y F4(\()2710 1935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19535
y F9(\()2738 1947 y Ga(x)2790 1935 y F9(\))2817 1947
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19536
y F6(j)p Ga(M)10 b F6(j)2965 1914 y Fu(s)2965 1970 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19537
Gc(c)3000 1947 y Ga(;)3040 1935 y F9(\()3068 1947 y Ga(y)3116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19538
1935 y F9(\))3144 1947 y F6(j)p Ga(N)g F6(j)3277 1914
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19539
y Fu(s)3277 1970 y Gc(c)3312 1947 y Ga(;)15 b(z)t F4(\))q(])1289
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19540
2173 y Gg(where)24 b(in:)101 b(\(6\))114 b Ga(y)28 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19541
F6(62)d Ga(F)13 b(N)d F4(\()2297 2161 y FX(h)2325 2173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19542
y Ga(b)2364 2161 y FX(i)2392 2173 y F6(j)p Ga(N)g F6(j)2525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19543
2140 y Fu(s)2525 2201 y Gc(b)2559 2173 y F4(\))1730 2286
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19544
y Gg(\(9\))114 b Ga(z)30 b F6(62)25 b Ga(F)13 b(N)d F4(\()2296
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19545
2274 y F9(\()2324 2286 y Ga(x)2376 2274 y F9(\))2403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19546
2286 y F6(j)p Ga(M)g F6(j)2551 2253 y Fu(s)2551 2309
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19547
y Gc(c)2586 2286 y Ga(;)2626 2274 y F9(\()2654 2286 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19548
Ga(y)2702 2274 y F9(\))2729 2286 y F6(j)p Ga(N)g F6(j)2862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19549
2253 y Fu(s)2862 2309 y Gc(c)2898 2286 y F4(\))p 3965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19550
2392 V 277 2395 3691 4 v Black 601 2549 a Gg(Figure)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19551
b(3.3:)29 b(T)m(ranslation)d(from)d(NJ-proofs)i(to)f(intuitionistic)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19552
(sequent)f(proofs.)p Black Black Black 277 3151 V 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19553
4929 4 1779 v 595 3329 a(\(1\))661 b F6(j)p FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19554
Ga(x;)15 b(a)p F4(\))p F6(j)1724 3296 y Fu(n)1865 3277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19555
y F5(def)1872 3329 y F4(=)107 b Ga(x)595 3468 y Gg(\(2\))335
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19556
b F6(j)p FL(Cut)p F4(\()1233 3456 y FX(h)1261 3468 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19557
Ga(a)1309 3456 y FX(i)1336 3468 y Ga(M)10 b(;)1474 3456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19558
y F9(\()1502 3468 y Ga(x)1554 3456 y F9(\))1582 3468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19559
y Ga(N)g F4(\))p F6(j)1725 3435 y Fu(n)1865 3417 y F5(def)1872
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19560
3468 y F4(=)107 b F6(j)p Ga(N)10 b F6(j)2183 3435 y Fu(n)2224
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19561
3468 y F4([)p Ga(x)25 b F4(:=)h F6(j)p Ga(M)10 b F6(j)2596
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19562
3435 y Fu(n)2637 3468 y F4(])595 3726 y Gg(\(3\))194
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19563
b F6(j)p FL(And)1073 3740 y Gc(R)1131 3726 y F4(\()1166
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19564
3714 y FX(h)1194 3726 y Ga(a)1242 3714 y FX(i)1269 3726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19565
y Ga(M)11 b(;)1408 3714 y FX(h)1435 3726 y Ga(b)1474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19566
3714 y FX(i)1502 3726 y Ga(N)f(;)15 b(c)p F4(\))p F6(j)1724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19567
3693 y Fu(n)1865 3675 y F5(def)1872 3726 y F4(=)107 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19568
F6(hj)p Ga(M)10 b F6(j)2233 3693 y Fu(n)2274 3726 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19569
b F6(j)p Ga(N)10 b F6(j)2447 3693 y Fu(n)2489 3726 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19570
F6(i)595 3866 y Gg(\(4\))467 b F6(j)p FL(Or)1291 3829
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19571
y F9(1)1291 3889 y Gc(R)1349 3866 y F4(\()1384 3854 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19572
FX(h)1412 3866 y Ga(a)1460 3854 y FX(i)1487 3866 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19573
b(;)15 b(b)p F4(\))p F6(j)1724 3833 y Fu(n)1865 3815
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19574
y F5(def)1872 3866 y F4(=)107 b Fp(inl)o F4(\()p F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19575
Ga(M)10 b F6(j)2376 3833 y Fu(n)2418 3866 y F4(\))595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19576
4006 y Gg(\(5\))467 b F6(j)p FL(Or)1291 3969 y F9(2)1291
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19577
4029 y Gc(R)1349 4006 y F4(\()1384 3994 y FX(h)1412 4006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19578
y Ga(a)1460 3994 y FX(i)1487 4006 y Ga(M)10 b(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19579
F4(\))p F6(j)1724 3973 y Fu(n)1865 3954 y F5(def)1872
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19580
4006 y F4(=)107 b Fp(inr)o F4(\()p F6(j)p Ga(M)10 b F6(j)2376
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19581
3973 y Fu(n)2418 4006 y F4(\))595 4146 y Gg(\(6\))314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19582
b F6(j)p FL(Imp)1184 4168 y Gc(R)1242 4146 y F4(\()1277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19583
4134 y F9(\()1304 4146 y Ga(x)1356 4134 y F9(\))q FX(h)1411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19584
4146 y Ga(a)1459 4134 y FX(i)1487 4146 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19585
b(c)p F4(\))p F6(j)1724 4113 y Fu(n)1865 4094 y F5(def)1872
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19586
4146 y F4(=)107 b Ga(\025x:)p F6(j)p Ga(M)10 b F6(j)2328
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19587
4113 y Fu(n)595 4404 y Gg(\(7\))404 b F6(j)p FL(And)1284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19588
4367 y F9(1)1284 4427 y Gc(L)1336 4404 y F4(\()1371 4392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19589
y F9(\()1399 4404 y Ga(x)1451 4392 y F9(\))1478 4404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19590
y Ga(M)10 b(;)15 b(y)s F4(\))p F6(j)1724 4371 y Fu(n)1865
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19591
4352 y F5(def)1872 4404 y F4(=)107 b F6(j)p Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19592
F6(j)2198 4371 y Fu(n)2239 4404 y F4([)p Ga(x)25 b F4(:=)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19593
Fp(fst)o F4(\()p Ga(y)s F4(\)])595 4543 y Gg(\(8\))404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19594
b F6(j)p FL(And)1284 4507 y F9(2)1284 4566 y Gc(L)1336
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19595
4543 y F4(\()1371 4531 y F9(\()1399 4543 y Ga(x)1451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19596
4531 y F9(\))1478 4543 y Ga(M)10 b(;)15 b(y)s F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19597
F6(j)1724 4510 y Fu(n)1865 4492 y F5(def)1872 4543 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19598
F4(=)107 b F6(j)p Ga(M)10 b F6(j)2198 4510 y Fu(n)2239
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19599
4543 y F4([)p Ga(x)25 b F4(:=)h Fp(snd)o F4(\()p Ga(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19600
F4(\)])595 4683 y Gg(\(9\))235 b F6(j)p FL(Or)1059 4697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19601
y Gc(L)1111 4683 y F4(\()1146 4671 y F9(\()1174 4683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19602
y Ga(x)1226 4671 y F9(\))1254 4683 y Ga(M)10 b(;)1392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19603
4671 y F9(\()1420 4683 y Ga(y)1468 4671 y F9(\))1495
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19604
4683 y Ga(N)g(;)15 b(z)t F4(\))p F6(j)1724 4650 y Fu(n)1865
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19605
4632 y F5(def)1872 4683 y F4(=)107 b Fp(case)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19606
Ga(z)t(;)15 b(\025)q(x:)p F6(j)p Ga(M)10 b F6(j)2641
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19607
4650 y Fu(n)2682 4683 y Ga(;)15 b(\025)q(y)s(:)p F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19608
Ga(N)10 b F6(j)2982 4650 y Fu(n)3023 4683 y F4(\))595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19609
4823 y Gg(\(10\))143 b F6(j)p FL(Imp)1058 4845 y Gc(L)1110
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19610
4823 y F4(\()1145 4811 y FX(h)1172 4823 y Ga(a)1220 4811
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19611
y FX(i)1248 4823 y Ga(M)10 b(;)1386 4811 y F9(\()1414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19612
4823 y Ga(x)1466 4811 y F9(\))1493 4823 y Ga(N)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19613
b(y)s F4(\))p F6(j)1724 4790 y Fu(n)1865 4771 y F5(def)1872
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19614
4823 y F4(=)107 b F6(j)p Ga(N)10 b F6(j)2183 4790 y Fu(n)2224
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19615
4823 y F4([)p Ga(x)25 b F4(:=)h(\()p Ga(y)33 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19616
Ga(M)10 b F6(j)2709 4790 y Fu(n)2750 4823 y F4(\)])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19617
3965 4929 V 277 4932 3691 4 v Black 601 5086 a Gg(Figure)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19618
b(3.4:)29 b(T)m(ranslation)d(from)d(intuitionistic)k(sequent)f(proofs)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19619
(to)e(NJ-proofs.)p Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19620
%%Page: 72 84
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19621
TeXDict begin 72 83 bop Black -144 51 a Gb(72)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19622
b(Deduction)p -144 88 3691 4 v Black 321 365 a Gg(more)h(dif)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19623
(\002cult)g(in)f(NJ.)f(Consider)i(the)g(proof)g(fragment)h(where)e(an)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19624
(implication)j(is)d(introduced)j(and)321 478 y(eliminated)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19625
(immediately)f(afterw)o(ards.)1558 696 y F6(D)1628 710
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19626
y F9(1)1388 788 y Ga(x)17 b F4(:)h Ga(B)5 b(;)15 b F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19627
1691 776 11 41 v 1701 758 46 5 v 96 w Ga(C)p 1388 825
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19628
451 4 v 1429 904 a F4(\000)p 1506 892 11 41 v 1516 874
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19629
46 5 v 96 w Ga(B)5 b F6(\033)o Ga(C)1880 843 y F6(\033)1950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19630
857 y Gc(I)2140 812 y F6(D)2210 826 y F9(2)2081 904 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19631
F4(\000)p 2158 892 11 41 v 2169 874 46 5 v 96 w Ga(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19632
1429 928 880 4 v 1756 1006 a F4(\000)p 1833 994 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19633
v 1843 976 46 5 v 96 w Ga(C)2349 945 y F6(\033)2420 959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19634
y Gc(E)p Black 3372 1006 a Gg(\(3.1\))p Black 321 1210
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19635
a(Here)21 b(the)h(formula)g Ga(B)5 b F6(\033)o Ga(C)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19636
b Gg(violates)c(the)f(subformula)h(property)-6 b(.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19637
b(Ho)n(we)n(v)o(er)l(,)21 b(we)g(e)o(xpect)h(that)g F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19638
3399 1198 11 41 v 3409 1180 46 5 v 96 w Ga(C)321 1323
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19639
y Gg(is)g(pro)o(v)n(able)h(without)g(introducing)i(this)e(formula.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19640
b(Indeed,)23 b(we)e(can)h(construct)j(a)c(proof)i(roughly)h(as)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19641
1436 y(follo)n(ws:)37 b(some)27 b(lea)n(v)o(es)h(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19642
F6(D)1286 1450 y F9(1)1352 1436 y Gg(are)h(of)g(the)g(form)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19643
1931 1357 569 4 v 27 w Ga(x)17 b F4(:)g Ga(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19644
b F4(\000)p Ga(;)g F4(\001)p 2349 1424 11 41 v 2360 1406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19645
46 5 v 96 w Ga(B)5 b Gg(\227replace)28 b(all)f(those)h(instances)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19646
1549 y(with)23 b(the)g(natural)h(deduction)h(proof)f(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19647
F4(\001)p Ga(;)15 b F4(\000)p 1802 1537 11 41 v 1812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19648
1519 46 5 v 97 w Ga(B)26 b Gg(that)d(is)g(obtained)i(by)d(adding)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19649
F4(\001)e Gg(to)h(all)f(sequents)321 1662 y(of)32 b F6(D)498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19650
1676 y F9(2)568 1662 y Gg(and)g(by)f(possibly)j(renaming)f(some)e(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19651
(ariables;)38 b(\002nally)32 b(remo)o(v)o(e)f(all)h(assumptions)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19652
Ga(x)e F4(:)g Ga(B)321 1775 y Gg(in)k F6(D)497 1789 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19653
F9(1)536 1775 y Gg(.)64 b(Belo)n(w)-6 b(,)37 b(we)e(shall)h(introduce)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19654
(the)e(reduction)i(rules,)h(commonly)e(referred)g(to)e(as)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19655
F7(beta-)321 1888 y(r)m(eductions)p Gg(,)26 b(realising)g(this)e(kind)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19656
(of)g(proof)h(transformations.)p Black 321 2075 a Gb(De\002nition)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19657
(3.1.4)g Gg(\(Beta-Reductions\))p Gb(:)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19658
Black 1667 2285 a F4(\()p Ga(\025x:M)10 b F4(\))15 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19659
Ga(N)2227 2248 y Gc(\014)2164 2285 y F6(\000)-32 b(\000)h(!)99
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19660
b Ga(M)10 b F4([)p Ga(x)26 b F4(:=)f Ga(N)10 b F4(])1563
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19661
2422 y Fp(fst)o F4(\()p F6(h)p Ga(M)5 b(;)15 b(N)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19662
F6(i)p F4(\))2227 2385 y Gc(\014)2164 2422 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19663
b(\000)h(!)99 b Ga(M)1563 2558 y Fp(snd)o F4(\()p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19664
Ga(M)5 b(;)15 b(N)10 b F6(i)p F4(\))2227 2521 y Gc(\014)2164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19665
2558 y F6(\000)-32 b(\000)h(!)99 b Ga(N)1004 2695 y Fp(case)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19666
F4(\()p Fp(inl)p F4(\()p Ga(P)13 b F4(\))p Ga(;)i(\025)q(x:M)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19667
b(;)15 b(\025)q(y)s(:N)10 b F4(\))2227 2658 y Gc(\014)2164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19668
2695 y F6(\000)-32 b(\000)h(!)99 b Ga(M)10 b F4([)p Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19669
b F4(:=)f Ga(P)13 b F4(])1004 2831 y Fp(case)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19670
Fp(inr)p F4(\()p Ga(P)g F4(\))p Ga(;)i(\025)q(x:M)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19671
b(\025)q(y)s(:N)10 b F4(\))2227 2794 y Gc(\014)2164 2831
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19672
y F6(\000)-32 b(\000)h(!)99 b Ga(N)10 b F4([)p Ga(y)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19673
b F4(:=)c Ga(P)13 b F4(])321 3081 y Gg(The)28 b(\002rst)f(reduction)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19674
(rule)d(deals)h(with)f(proof)h(instances)h(as)e(sho)n(wn)g(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19675
(\(3.1\).)42 b(Since)28 b(all)g(rules)g(are)321 3194
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19676
y(rather)d(standard,)g(we)e(do)h(not)g(e)o(xplain)h(them)e(further)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19677
b(.)462 3328 y(Unfortunately)f(,)40 b(to)34 b(ensure)h(that)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19678
(subformula)i(property)g(holds)f(for)f(normal)h(NJ-proofs,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19679
3441 y(the)30 b(beta-reductions)j(are)d(not)f(suf)n(\002cient.)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19680
b(F)o(or)28 b(this)i(property)h(to)e(hold,)j(further)e(reduction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19681
(rules,)321 3554 y(referred)37 b(to)f(as)f F7(commuting)i(r)m
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19682
(eductions)p Gg(,)j(are)c(required.)66 b(These)36 b(rules)g(are)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19683
(often)i(criticised)h(as)321 3667 y(being)e(inele)o(gant,)i(if)c(not)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19684
(catastrophic)k([Girard)d(et)f(al.,)f(1989,)38 b(P)o(age)c(74].)61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19685
b(Certainly)-6 b(,)38 b(the)o(y)c(are)321 3780 y(more)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19686
b(complicated)i(than)e(beta-reductions)k(and)c(seem)f(to)g(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19687
(necessary)h(just)f(because)i(of)d(certain)321 3893 y(defects)f(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19688
(the)h(syntax)h(based)g(on)f(terms,)g(b)n(ut)g(we)f(shall)h(sho)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19689
(that)g(the)o(y)g(correspond)j(to)c(the)h(\(quite)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19690
4005 y(natural\))j(commuting)f(reductions)h(of)e(sequent)h(calculi.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19691
Black 321 4193 a Gb(De\002nition)e(3.1.5)g Gg(\(Commuting)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19692
(Reductions\))p Gb(:)p Black Black Black 964 4403 a Fp(case)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19693
F4(\()p Ga(P)s(;)15 b(\025)q(x:M)5 b(;)15 b(\025y)s(:N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19694
b F4(\))15 b Ga(Q)2036 4366 y Gc(\015)1971 4403 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19695
b(\000)g(!)99 b Fp(case)o F4(\()p Ga(P)s(;)15 b(\025)q(x:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19696
F4(\()p Ga(M)26 b(Q)p F4(\))p Ga(;)15 b(\025y)s(:)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19697
Ga(N)26 b(Q)p F4(\)\))837 4540 y Fp(fst)o F4(\()p Fp(case)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19698
F4(\()p Ga(P)s(;)15 b(\025x:M)5 b(;)15 b(\025)q(y)s(:N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19699
b F4(\)\))2036 4502 y Gc(\015)1971 4540 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19700
b(\000)g(!)99 b Fp(case)o F4(\()p Ga(P)s(;)15 b(\025)q(x:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19701
Fp(fst)o F4(\()p Ga(M)10 b F4(\))p Ga(;)15 b(\025)q(y)s(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19702
Fp(fst)p F4(\()p Ga(N)10 b F4(\)\))837 4676 y Fp(snd)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19703
F4(\()p Fp(case)p F4(\()p Ga(P)s(;)15 b(\025x:M)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19704
b(\025)q(y)s(:N)10 b F4(\)\))2036 4639 y Gc(\015)1971
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19705
4676 y F6(\000)-31 b(\000)g(!)99 b Fp(case)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19706
Ga(P)s(;)15 b(\025)q(x:)p Fp(snd)o F4(\()p Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19707
F4(\))p Ga(;)15 b(\025)q(y)s(:)p Fp(snd)p F4(\()p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19708
b F4(\)\))331 4822 y Fp(case)o F4(\()p Fp(case)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19709
Ga(P)s(;)15 b(\025)q(x:M)5 b(;)15 b(\025)q(y)s(:N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19710
b F4(\))p Ga(;)15 b(\025)q(u:S;)g(\025v)s(:T)e F4(\))2036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19711
4785 y Gc(\015)1971 4822 y F6(\000)-31 b(\000)g(!)2240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19712
4813 y Fp(case)o F4(\()p Ga(P)s(;)15 b(\025)q(x:)p Fp(case)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19713
F4(\()p Ga(M)5 b(;)15 b(\025)q(u:S;)g(\025)q(v)s(:T)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19714
F4(\))p Ga(;)2566 4925 y(\025y)s(:)p Fp(case)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19715
Ga(N)5 b(;)15 b(\025)q(u:S;)g(\025)q(v)s(:T)e F4(\)\))321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19716
5175 y Gg(W)-7 b(e)26 b(shall,)h(as)f(usual,)h(assume)g(that)1546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19717
5138 y Gc(\014)1482 5175 y F6(\000)-31 b(\000)f(!)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19718
b Gg(and)1899 5138 y Gc(\015)1834 5175 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19719
b(\000)f(!)26 b Gg(are)g(closed)i(under)f(conte)o(xt)g(formation.)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19720
b(Thus)321 5288 y(we)23 b(can)h(de\002ne)g(the)g(normalisation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19721
(procedure)h(of)c(NJ-proofs)i(as)f(follo)n(ws.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19722
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19723
%%Page: 73 85
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19724
TeXDict begin 73 84 bop Black 277 51 a Gb(3.2)23 b(The)g(Corr)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19725
(espondence)h(between)f(Cut-Elimination)g(and)f(Normalisation)k(I)849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19726
b(73)p 277 88 3691 4 v Black Black 277 365 a(De\002nition)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19727
b(3.1.6)g Gg(\(Normalisation)j(Procedure)g(of)d(NJ-Proofs\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19728
Gb(:)p Black 277 478 a Gg(The)g(normalisation)k(procedure)f(for)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19729
(NJ-proofs,)h F4(\(\003)p Ga(;)2099 440 y Gc(\023)2027
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19730
478 y F6(\000)-31 b(\000)f(!)p F4(\))p Gg(,)23 b(is)g(a)g(reduction)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19731
(system)f(where:)p Black 414 676 a F6(\017)p Black 45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19732
w F4(\003)e Gg(is)h(the)f(set)h(of)g(simply-typed)i(lambda)f(terms,)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19733
(and)p Black 414 861 a F6(\017)p Black 577 824 a Gc(\023)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19734
861 y F6(\000)-31 b(\000)g(!)22 b Gg(consists)k(of)d(beta-reductions)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19735
b(and)24 b(commuting)h(reductions;)h(that)e(is)1659 1042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19736
y Gc(\023)1587 1079 y F6(\000)-31 b(\000)f(!)1782 1028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19737
y F5(def)1789 1079 y F4(=)1956 1042 y Gc(\014)1892 1079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19738
y F6(\000)h(\000)g(!)25 b([)2239 1042 y Gc(\015)2174
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19739
1079 y F6(\000)-31 b(\000)f(!)50 b Ga(:)277 1333 y Gg(It)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19740
b(is)h(simple)g(to)f(sho)n(w)h(that)1241 1296 y Gc(\023)1169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19741
1333 y F6(\000)-31 b(\000)f(!)23 b Gg(satis\002es)h(the)g(subject)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19742
(reduction)g(property)-6 b(.)p Black 277 1521 a Gb(Pr)n(oposition)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19743
b(3.1.7)e Gg(\(Subject)i(Reduction,)k(Pra)o(witz,)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19744
b(1965\))p Gb(:)p Black 35 w Gg(Suppose)i Ga(M)47 b Gg(is)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19745
b(a)g(simply-typed)277 1634 y(lambda)28 b(term)f(with)h(the)f(typing)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19746
(judgement)g F4(\000)1841 1622 y Gc(.)1896 1634 y Ga(M)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19747
F4(:)17 b Ga(B)31 b Gg(and)d Ga(M)2515 1597 y Gc(\023)2442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19748
1634 y F6(\000)-31 b(\000)g(!)32 b Ga(N)10 b Gg(,)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19749
b(then)h Ga(N)36 b Gg(is)27 b(a)g(simply-)277 1747 y(typed)e(lambda)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19750
(term)g(with)f(the)h(typing)h(judgement)g F4(\000)2043
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19751
1735 y Gc(.)2098 1747 y Ga(M)j F4(:)17 b Ga(B)5 b Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19752
Black 277 1984 a F7(Pr)l(oof.)p Black 46 w Gg(Analogous)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19753
b(to)d(proof)i(of)e(Proposition)j(2.2.10.)p 3436 1984
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19754
4 62 v 3440 1926 55 4 v 3440 1984 V 3494 1984 4 62 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19755
277 2187 a(Furthermore,)k(the)e(reduction)1363 2150 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19756
Gc(\023)1290 2187 y F6(\000)-31 b(\000)g(!)27 b Gg(is)g(complete)i(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19757
(the)g(sense)h(that)f(the)g(normal)g(NJ-proofs)i(sat-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19758
2300 y(isfy)j(the)f(subformula)j(property)-6 b(.)56 b(Whilst)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19759
b(in)f(the)h(setting)g(of)f(sequent)i(calculi)g(the)f(proof)g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19760
(this)277 2413 y(property)d(is)d(by)g(tri)n(vial)h(inspection)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19761
(the)h(inference)h(rules,)g(the)e(proof)i(for)e(NJ)g(is)g(more)g(dif)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19762
(\002cult.)277 2526 y(F)o(or)k(e)o(xample,)i(a)e(simple)h(inspection)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19763
(of)c(the)h(inference)i(rules)e(is)f(not)h(suf)n(\002cient;)k(for)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19764
(the)g(details)277 2639 y(of)25 b(the)g(proof)h(we)e(refer)h(to)g([Pra)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19765
o(witz,)f(1965].)34 b(One)24 b(interesting)k(feature)e(of)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19766
(natural)i(deduction)277 2752 y(calculus)g(we)d(shall)h(introduce)j(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19767
c(Section)i(3.3)e(for)h(classical)i(logic)f(is)e(that)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19768
(subformula)i(prop-)277 2865 y(erty)j(for)g(normal)g(proofs)h(can)e(be)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19769
h(pro)o(v)o(ed,)h(lik)o(e)f(in)f(sequent)j(calculi,)g(by)d(a)g(simple)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19770
(inspection)i(of)277 2978 y(the)24 b(inference)i(rules.)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19771
3286 y Ge(3.2)119 b(The)48 b(Corr)n(espondence)i(between)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19772
(Cut-Elimination)g(and)g(Nor)l(-)546 3435 y(malisation)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19773
b(I)277 3659 y Gg(In)f(this)g(section,)i(we)d(address)i(the)f(question)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19774
i(whether)f(there)f(is)g(a)f(correspondence)32 b(between)d(cut-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19775
3772 y(elimination)h(and)e(normalisation,)j(or)c(more)h(precisely)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19776
(whether)e(the)g(translations)i(de\002ned)f(in)e(the)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19777
3885 y(preceding)j(section)e(respect)h(reduction.)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19778
b(This)27 b(\223correspondence)32 b(question\224)d(w)o(as)e(tackled)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19779
(pre)n(vi-)277 3997 y(ously)23 b(by)e(Zuck)o(er)h([1974])h(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19780
(Pottinger)h([1977],)g(who)e(wrote)g(the)h(classic)h(w)o(orks)f(on)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19781
(the)h(subject.)277 4110 y(Zuck)o(er)g(mentioned)g(that)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19782
(correspondence)k(question)e(goes)f(back)f(to)g(the)g(follo)n(wing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19783
(remark)f(by)277 4223 y(Kreisel)j([1971,)h(P)o(age)e(113],)h(which)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19784
(both)g(Zuck)o(er)h(and)f(Pottinger)h(dispro)o(v)o(e.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19785
Black Black 504 4445 a(Consider)9 b(.)14 b(.)g(.)g(a)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19786
b(calculus)h(of)d(sequents)k(and)d(a)f(system)i(of)f(natural)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19787
(deduction.)63 b(In-)504 4558 y(spection)39 b(sho)n(ws)d(that)12
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19788
b(.)i(.)g(.)g(the)36 b(normalization)k(procedure)e(for)f(systems)g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19789
(natural)504 4671 y(deduction)31 b(does)d(not)g(correspond)i(to)d(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19790
(particularly)k(natural)e(cut)f(elimination)h(pro-)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19791
4784 y(cedure.)277 5006 y(Ho)n(we)n(v)o(er)l(,)h(there)g(are)f(some)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19792
(subtle)g(points)h(in)e(their)h(w)o(orks,)g(which)g(mak)o(e)f(their)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19793
(answers)g(not)f(so)277 5119 y(clear)n(-cut.)i(Let)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19794
b(us)h(analyse)h(in)e(turn)i(these)f(subtleties.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19795
5249 y(Zuck)o(er)32 b(studied)g(the)f(correspondence)k(question)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19796
(between)f(NJ)e(and)h(a)f(v)n(ariant)i(of)f(LJ.)e(Since)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19797
5361 y(all)38 b(reduction)h(sequences)i(of)c(NJ)f(are)i(terminating,)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19798
b(one)38 b(pro)o(viso)g(for)g(the)g(correspondence)k(is)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19799
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19800
%%Page: 74 86
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19801
TeXDict begin 74 85 bop Black -144 51 a Gb(74)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19802
b(Deduction)p -144 88 3691 4 v Black 321 365 a Gg(that)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19803
b(all)g(reduction)h(sequences)i(in)c(the)g(sequent)j(calculus)f(are)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19804
(terminating,)k(too.)40 b(Indeed,)30 b(a)d(non-)321 478
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19805
y(terminating)c(reduction)f(sequence)g(in)e(the)h(sequent)g(calculus)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19806
(w)o(ould)f(hamper)g(the)f(correspondence)321 590 y(with)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19807
b(NJ.)f(Another)j(pro)o(viso)f(is)f(that)h(the)g(cut-elimination)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19808
(procedure)e(has)f(to)f(include)i(reduction)321 703 y(rules)k(which)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19809
(allo)n(w)f(cuts)i(to)e(pass)i(o)o(v)o(er)e(other)i(cuts.)60
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19810
b(Gentzen)35 b([1935])g(did)f(not)g(consider)i(such)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19811
816 y(rules,)23 b(as)f(the)o(y)g(are)h(usually)g(the)g(source)g(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19812
(loops)h(and)g(therefore)h(for)e(non-terminating)k(reduction)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19813
929 y(sequences;)h(ho)n(we)n(v)o(er)l(,)d(the)o(y)g(are)f(crucial)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19814
(for)f(the)g(correspondence)k(question.)462 1059 y(One)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19815
b(\(rather)h F7(ad)e(hoc)p Gg(\))h(w)o(ay)g(round)g(the)g(problem)h(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19816
e(loops)i(is)e(to)g(consider)j(only)e F7(pr)l(oper)h(r)m(educ-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19817
1172 y(tion)f(sequences)h Gg([Zuck)o(er,)f(1974,)g(P)o(age)e(93],)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19818
(which)f(do)g(not)g(include)h(repetitions.)31 b(This)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19819
b(restriction)321 1284 y(is)i(suf)n(\002cient)h(to)f(ensure)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19820
(termination)g(of)e(Zuck)o(er')-5 b(s)24 b(cut-elimination)j(procedure)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19821
f(in)c(the)i F4(\()p F6(^)p Ga(;)15 b F6(\033)p Ga(;)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19822
F6(8)p F4(\))p Gg(-)321 1397 y(fragment)24 b(of)d(intuitionistic)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19823
b(logic,)d(and)f(so)g(he)g(could)h(answer)g(the)f(correspondence)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19824
b(question)d(pos-)321 1510 y(iti)n(v)o(ely)34 b(for)e(this)h(fragment.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19825
56 b(Using)33 b(the)f F6(_)1739 1524 y Gc(L)1791 1510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19826
y Gg(-rule,)j(ho)n(we)n(v)o(er)l(,)g(he)d(constructed)k(a)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19827
(non-repeating,)321 1623 y(non-terminating)40 b(reduction)d(sequence,)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19828
(and)35 b(thus)h(he)g(f)o(ailed)g(to)f(ensure)i(the)e(\002rst)g(pro)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19829
(viso.)66 b(A)321 1736 y(simpli\002ed)31 b(v)o(ersion)g(of)f(this)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19830
(reduction)h(sequence)g(is)e(sho)n(wn)g(in)g(Figure)g(3.5.)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19831
b(Consequently)-6 b(,)35 b(he)321 1849 y(ga)n(v)o(e)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19832
b(a)g(ne)o(gati)n(v)o(e)g(answer)h(to)e(the)i(correspondence)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19833
(question)e(for)e(the)g(fragment)h(that)g(includes)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19834
F6(_)321 1962 y Gg(and)i F6(9)p Gg(.)52 b(Zuck)o(er)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19835
b(concluded)j(that)d(the)g(lack)g(of)g(termination)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19836
(cut-elimination)36 b(w)o(as)31 b(responsi-)321 2075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19837
y(ble)e(for)f(this)g(f)o(ailure,)j(b)n(ut)d(we)g(shall)h(sho)n(w)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19838
(that)g(e)n(v)o(en)g(if)g(one)h(does)f(ha)n(v)o(e)h(a)f(strongly)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19839
(normalising)321 2188 y(cut-elimination)j(procedure)f(\(which)e(is)f(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19840
o(ery)h(similar)g(to)f(the)h(one)g(used)g(by)f(Zuck)o(er\),)j(the)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19841
(corre-)321 2301 y(spondence)i(between)e(cut-elimination)i(and)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19842
(normalisation)j(f)o(ails)d(for)g(the)g(standard)i(formulation)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19843
2414 y(of)h(NJ:)g(certain)i(commuting)f(reductions)i(of)d(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19844
(sequent)h(calculus)g(cannot)g(be)e(encoded)j(in)d(NJ.)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19845
2526 y(In)c(Section)g(3.3)g(we)f(shall)h(mak)o(e)g(appropriate)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19846
(changes)e(to)f(the)g(standard)h(formulation)h(of)e(natural)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19847
2639 y(deduction)i(and)d(impro)o(v)o(e)h(Zuck)o(er')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19848
b(s)27 b(result)g(by)g(sho)n(wing)g(the)f(correspondence)k(in)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19849
(classical)j(logic)321 2752 y(for)24 b(all)g(connecti)n(v)o(es.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19850
2882 y(In)g(some)h(sense)g(Pottinger)h(solv)o(ed)f(Zuck)o(er')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19851
b(s)25 b(problem)h(with)e(non-terminating)k(reduction)e(se-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19852
2995 y(quences,)j(and)e(thus)h(he)e(could)i(sho)n(w)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19853
(correspondence)32 b(for)26 b(all)h(connecti)n(v)o(es.)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19854
b(Unfortunately)-6 b(,)321 3108 y(his)25 b(method)g(had)g(a)f(subtle)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19855
(defect:)32 b(he)25 b(annotated)i(sequent)f(proofs)g(with)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19856
(lambda-terms)i(and)f(then)321 3220 y(formalised)30 b(cut-elimination)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19857
(as)c(normalisation.)43 b(As)27 b(already)i(noted)g(by)e(himself,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19858
(the)f(notion)h(of)321 3333 y(normality)e(for)f(lambda)g(terms)f(does)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19859
F7(not)g Gg(coincide)i(with)d(the)g(notion)i(of)e(cut-freedom)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19860
([Pottinger,)321 3446 y(1977,)d(P)o(age)e(323].)29 b(Consider)c(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19861
(e)o(xample)g(the)g(follo)n(wing)h(sequent)h(proof.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19862
1235 3619 244 4 v 1235 3698 a Ga(B)p 1329 3686 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19863
v 1339 3668 46 5 v 101 w(B)p 1569 3619 244 4 v 95 w(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19864
1663 3686 11 41 v 1673 3668 46 5 v 101 w(B)p 1235 3718
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19865
578 4 v 1402 3797 a(B)p 1496 3785 11 41 v 1506 3767 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19866
5 v 101 w(B)1854 3748 y Gg(Cut)p 2091 3602 354 4 v 2091
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19867
3680 a Ga(B)5 b(;)15 b(C)p 2297 3668 11 41 v 2308 3650
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19868
46 5 v 103 w(C)p 2076 3718 384 4 v 2076 3797 a(B)p 2170
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19869
3785 11 41 v 2181 3767 46 5 v 101 w(C)7 b F6(\033)o Ga(C)2501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19870
3736 y F6(\033)2572 3750 y Gc(R)p 1402 3820 1058 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19871
1739 3899 a Ga(B)p 1833 3887 11 41 v 1843 3869 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19872
101 w(C)g F6(\033)o Ga(C)2501 3851 y Gg(Cut)321 4103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19873
y(Using)31 b(Pottinger')-5 b(s)33 b(term)e(assignment,)k(this)c(proof)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19874
(is)e(annotated)k(with)c(the)h(lambda)h(term)e Ga(\025x:x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19875
Gg(.)321 4216 y(Clearly)-6 b(,)33 b(this)e(term)g(is)f(normal,)j(b)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19876
(ut)e(the)f(sequent)j(proof)e(contains)i(tw)o(o)d(cuts.)51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19877
b(Because)31 b(the)g(cut-)321 4329 y(elimination)23 b(reductions)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19878
(that)d(start)h(from)f(the)g(proof)h(abo)o(v)o(e)g(are)f(not)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19879
(\223mirrored\224)h(by)e(an)o(y)g(reduction)321 4442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19880
y(of)26 b(the)h(lambda)g(term,)f(the)g(question)j(whether)e(these)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19881
(cut-reductions)j(break)d(the)g(correspondence)321 4555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19882
y(remains)d(open.)29 b(W)-7 b(e)22 b(shall)i(a)n(v)n(oid)g(this)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19883
(problem)g(by)f(using)h(our)f(term)f(assignment,)j(which)f(encodes)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19884
4667 y(precisely)i(the)e(structure)i(of)d(sequent)j(proofs.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19885
4797 y(No)n(w)32 b(we)g(shall)i(be)o(gin)f(by)g(sho)n(wing)h(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19886
(normalisation)i(in)c(NJ)h(can)g(be)g(simulated)h(by)f(our)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19887
4910 y(intuitionistic)39 b(cut-elimination)g(procedure.)65
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19888
b(T)-7 b(o)34 b(do)h(so,)i(we)d(shall)i(sho)n(w)e(that)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19889
(beta-reductions)321 5023 y(map)28 b(onto)h(a)f(series)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19890
(cut-reductions)j(and)d(that)g(commuting)g(reductions)i(map)d(onto)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19891
(identities.)321 5136 y(F)o(or)35 b(the)h(\002rst)f(part)h(we)f(ha)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19892
o(e)h(to)g(pro)o(v)o(e)g(three)h(lemmas.)65 b(The)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19893
b(\002rst)g(sho)n(ws)h(that)g(a)f(form)h(of)f(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19894
5249 y(substitution)e(lemma)d(holds)h(for)e(the)h(proof)h(substitution)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19895
i F4([)p 2257 5249 28 4 v 2275 5249 V 2293 5249 V 65
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19896
w(])c Gg(\(the)h(\223usual\224)i(substitution)h(lemma)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19897
5361 y(does,)24 b(in)g(general,)h(not)f(hold)g(for)g(this)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19898
(substitution;)j(cf.)c(Lemma)g(2.3.1\).)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19899
eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19900
%%Page: 75 87
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19901
TeXDict begin 75 86 bop Black 277 51 a Gb(3.2)23 b(The)g(Corr)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19902
(espondence)h(between)f(Cut-Elimination)g(and)f(Normalisation)k(I)849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19903
b(75)p 277 88 3691 4 v Black Black 277 990 V 277 3907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19904
4 2917 v Black Black 1093 1331 a Ga(\031)1145 1345 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19905
F9(1)1020 1423 y Ga(C)p 1111 1411 11 41 v 1122 1393 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19906
5 v 102 w(F)1426 1331 y(\031)1478 1345 y F9(2)1349 1423
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19907
y Ga(D)p 1447 1411 11 41 v 1458 1393 46 5 v 99 w(F)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19908
1020 1443 575 4 v 1118 1522 a(C)7 b F6(_)p Ga(D)p 1349
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19909
1510 11 41 v 1359 1492 46 5 v 99 w(F)1636 1461 y F6(_)1697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19910
1475 y Gc(L)1912 1215 y Ga(\031)1964 1229 y F9(3)1840
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19911
1307 y Ga(A)p 1928 1295 11 41 v 1939 1277 46 5 v 96 w(F)2241
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19912
1215 y(\031)2293 1229 y F9(4)2166 1307 y Ga(B)p 2260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19913
1295 11 41 v 2270 1277 46 5 v 101 w(F)p 1840 1327 568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19914
4 v 1939 1406 a(A)p F6(_)o Ga(B)p 2161 1394 11 41 v 2172
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19915
1375 46 5 v 101 w(F)2449 1345 y F6(_)2509 1359 y Gc(L)2777
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19916
1313 y Ga(\031)2829 1327 y F9(5)2652 1406 y Ga(F)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19917
b(F)p 2845 1394 11 41 v 2855 1375 46 5 v 110 w(G)p 1939
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19918
1443 1054 4 v 2225 1522 a(A)p F6(_)p Ga(B)5 b(;)15 b(F)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19919
2559 1510 11 41 v 2569 1492 46 5 v 109 w(G)3034 1474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19920
y Gb(Cut)3181 1441 y Fy(?)p 1118 1559 1588 4 v 1602 1638
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19921
a Ga(C)7 b F6(_)o Ga(D)s(;)15 b(A)p F6(_)p Ga(B)p 2075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19922
1626 11 41 v 2085 1608 46 5 v 101 w(G)2748 1590 y Gb(Cut)2093
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19923
1868 y FN(#)898 2304 y Ga(\031)950 2318 y F9(1)825 2397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19924
y Ga(C)p 916 2385 11 41 v 927 2366 46 5 v 102 w(F)1231
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19925
2304 y(\031)1283 2318 y F9(2)1154 2397 y Ga(D)p 1252
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19926
2385 11 41 v 1263 2366 46 5 v 100 w(F)p 825 2417 575
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19927
4 v 924 2495 a(C)7 b F6(_)o Ga(D)p 1154 2483 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19928
1164 2465 46 5 v 99 w(F)1441 2435 y F6(_)1502 2449 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19929
Gc(L)1717 2054 y Ga(\031)1769 2068 y F9(3)1645 2146 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19930
Ga(A)p 1733 2134 11 41 v 1744 2116 46 5 v 96 w(F)2096
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19931
2054 y(\031)2148 2068 y F9(5)1971 2146 y Ga(F)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19932
b(F)p 2164 2134 11 41 v 2174 2116 46 5 v 110 w(G)p 1645
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19933
2184 667 4 v 1804 2263 a(A;)g(F)p 2004 2251 11 41 v 2015
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19934
2232 46 5 v 110 w(G)2353 2214 y Gg(Cut)2650 2054 y Ga(\031)2702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19935
2068 y F9(4)2575 2146 y Ga(B)p 2669 2134 11 41 v 2680
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19936
2116 46 5 v 101 w(F)3032 2054 y(\031)3084 2068 y F9(5)2907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19937
2146 y Ga(F)s(;)g(F)p 3100 2134 11 41 v 3110 2116 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19938
5 v 110 w(G)p 2575 2184 673 4 v 2735 2263 a(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19939
b(F)p 2940 2251 11 41 v 2951 2232 46 5 v 109 w(G)3289
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19940
2214 y Gg(Cut)p 1804 2300 1284 4 v 2155 2379 a Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19941
F6(_)o Ga(B)5 b(;)15 b(F)s(;)g(F)p 2590 2367 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19942
2600 2349 46 5 v 110 w(G)3129 2318 y F6(_)3190 2332 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19943
Gc(L)p 2155 2417 583 4 v 2205 2495 a Ga(A)p F6(_)p Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19944
b(;)15 b(F)p 2539 2483 11 41 v 2550 2465 46 5 v 109 w(G)2779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19945
2440 y Gg(Contr)2985 2454 y Gc(L)p 924 2533 1764 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19946
1495 2612 a Ga(C)7 b F6(_)o Ga(D)s(;)15 b(A)p F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19947
Ga(B)p 1968 2600 11 41 v 1978 2581 46 5 v 101 w(G)2728
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19948
2565 y Gg(Cut)2859 2532 y Fz(\017)2093 2841 y FN(#)488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19949
3278 y Ga(\031)540 3292 y F9(1)415 3370 y Ga(C)p 506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19950
3358 11 41 v 517 3340 46 5 v 102 w(F)821 3278 y(\031)873
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19951
3292 y F9(2)744 3370 y Ga(D)p 842 3358 11 41 v 853 3340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19952
46 5 v 99 w(F)p 415 3390 575 4 v 513 3469 a(C)7 b F6(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19953
Ga(D)p 744 3457 11 41 v 754 3438 46 5 v 99 w(F)1031 3408
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19954
y F6(_)1092 3422 y Gc(L)1309 3161 y Ga(\031)1361 3175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19955
y F9(1)1235 3254 y Ga(C)p 1326 3242 11 41 v 1337 3223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19956
46 5 v 103 w(F)1641 3161 y(\031)1693 3175 y F9(2)1565
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19957
3254 y Ga(D)p 1663 3242 11 41 v 1673 3223 46 5 v 99 w(F)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19958
1235 3274 575 4 v 1334 3352 a(C)g F6(_)o Ga(D)p 1564
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19959
3340 11 41 v 1574 3322 46 5 v 99 w(F)1851 3292 y F6(_)1912
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19960
3306 y Gc(L)2127 3027 y Ga(\031)2179 3041 y F9(3)2055
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19961
3120 y Ga(A)p 2143 3108 11 41 v 2154 3089 46 5 v 96 w(F)2506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19962
3027 y(\031)2558 3041 y F9(5)2381 3120 y Ga(F)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19963
b(F)p 2574 3108 11 41 v 2585 3089 46 5 v 110 w(G)p 2055
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19964
3157 667 4 v 2215 3236 a(A;)g(F)p 2414 3224 11 41 v 2425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19965
3206 46 5 v 110 w(G)2763 3188 y Gg(Cut)3060 3027 y Ga(\031)3112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19966
3041 y F9(4)2985 3120 y Ga(B)p 3079 3108 11 41 v 3090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19967
3089 46 5 v 101 w(F)3442 3027 y(\031)3494 3041 y F9(5)3317
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19968
3120 y Ga(F)s(;)g(F)p 3510 3108 11 41 v 3520 3089 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19969
5 v 110 w(G)p 2985 3157 673 4 v 3145 3236 a(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19970
b(F)p 3350 3224 11 41 v 3361 3206 46 5 v 109 w(G)3699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19971
3188 y Gg(Cut)p 2215 3274 1284 4 v 2565 3352 a Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19972
F6(_)p Ga(B)5 b(;)15 b(F)s(;)g(F)p 3000 3340 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19973
3011 3322 46 5 v 109 w(G)3539 3292 y F6(_)3600 3306 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19974
Gc(L)p 1334 3390 1815 4 v 1875 3469 a Ga(C)7 b F6(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19975
Ga(D)s(;)15 b(A)p F6(_)p Ga(B)5 b(;)15 b(F)p 2459 3457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19976
11 41 v 2469 3438 46 5 v 109 w(G)3189 3420 y Gb(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19977
513 3506 2094 4 v 1125 3585 a Ga(C)7 b F6(_)o Ga(D)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19978
b(C)7 b F6(_)o Ga(D)s(;)15 b(A)p F6(_)p Ga(B)p 1848 3573
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19979
11 41 v 1858 3555 46 5 v 101 w(G)2648 3537 y Gb(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19980
1125 3623 871 4 v 1250 3701 a Ga(C)7 b F6(_)o Ga(D)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19981
b(A)p F6(_)p Ga(B)p 1723 3689 11 41 v 1733 3671 46 5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19982
v 101 w(G)2037 3647 y Gg(Contr)2243 3661 y Gc(L)p 3965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19983
3907 4 2917 v 277 3910 3691 4 v 277 4064 a Gg(Figure)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19984
b(3.5:)62 b(A)38 b(non-terminating)44 b(reduction)e(sequence)h(in)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19985
(Zuck)o(er')-5 b(s)41 b(sequent)h(calculus)g(\(see)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19986
4177 y([Ungar,)22 b(1992]\).)30 b(In)22 b(the)g(\002rst)g(step,)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19987
(cut)g(mark)o(ed)h(with)f(a)g(star)g(is)g(reduced)i(creating)g(tw)o(o)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19988
(copies)277 4289 y(of)i Ga(\031)428 4303 y F9(5)491 4289
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19989
y Gg(as)g(well)g(as)g(introducing)j(a)d(contraction.)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19990
b(In)24 b(the)h(second)h(step,)e(the)h(cut)f(mark)o(ed)h(with)f(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19991
(disk)277 4402 y(is)32 b(permuted)i(with)e(the)g(contraction)k(rule)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19992
(creating)j(tw)o(o)c(copies)j(of)e Ga(\031)2606 4416
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19993
y F9(1)2677 4402 y Gg(and)g Ga(\031)2891 4416 y F9(2)2931
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19994
4402 y Gg(.)53 b(No)n(w)31 b(the)i(cuts)277 4515 y(written)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19995
b(in)e(bold)i(f)o(ace)f(ha)n(v)o(e)h(the)f(same)f(shape,)i(and)g(we)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19996
(can)h(repeat)h(the)f(reduction)i(applied)f(in)f(the)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19997
4628 y(\002rst)d(step)h(and)g(so)g(forth)g F7(ad)g(in\002nitum.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19998
Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 19999
%%Page: 76 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20000
TeXDict begin 76 87 bop Black -144 51 a Gb(76)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20001
b(Deduction)p -144 88 3691 4 v Black Black 321 365 a(Lemma)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20002
b(3.2.1:)p Black 34 w Gg(F)o(or)f(all)i Ga(M)51 b F6(2)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20003
b FY(J)32 b Gg(and)h(tw)o(o)e(substitutions)37 b(ha)n(ving)d(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20004
(form)g F4([)p Ga(a)42 b F4(:=)3195 353 y F9(\()3222
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20005
365 y Ga(y)3270 353 y F9(\))3297 365 y Ga(S)5 b F4(])32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20006
b Gg(and)321 478 y F4([)p Ga(x)26 b F4(:=)545 466 y FX(h)572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20007
478 y Ga(b)611 466 y FX(i)639 478 y Ga(T)13 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20008
Gg(,)22 b(we)h(ha)n(v)o(e)684 694 y Ga(M)10 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20009
Ga(a)26 b F4(:=)1002 682 y F9(\()1029 694 y Ga(y)1077
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20010
682 y F9(\))1105 694 y Ga(S)5 b F4(][)p Ga(x)25 b F4(:=)1415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20011
682 y FX(h)1442 694 y Ga(b)1481 682 y FX(i)1508 694 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20012
Ga(T)13 b F4(])1664 657 y Gc(int)1625 694 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20013
b(\000)h(!)1795 656 y FX(\003)1860 694 y Ga(M)10 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20014
Ga(x)25 b F4(:=)2181 682 y FX(h)2209 694 y Ga(b)2248
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20015
682 y FX(i)2275 694 y Ga(T)13 b F4(][)p Ga(a)26 b F4(:=)2586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20016
682 y F9(\()2614 694 y Ga(y)2662 682 y F9(\))2689 694
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20017
y Ga(S)5 b F4([)p Ga(x)25 b F4(:=)2973 682 y FX(h)3001
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20018
694 y Ga(b)3040 682 y FX(i)3067 694 y Ga(T)13 b F4(]])321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20019
910 y Gg(pro)o(vided)26 b Ga(a)d Gg(is)g(not)h(free)g(in)1216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20020
898 y FX(h)1244 910 y Ga(b)1283 898 y FX(i)1310 910 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20021
Ga(T)13 b Gg(,)22 b(and)i Ga(S)k Gg(freshly)d(introduces)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20022
Ga(y)s Gg(,)22 b(b)n(ut)i(is)g(not)f(an)h(axiom.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20023
321 1122 a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)f(induction)j(on)e(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20024
(structure)j(of)e Ga(M)32 b Gg(\(see)25 b(P)o(age)e(156\).)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20025
3480 1122 4 62 v 3484 1064 55 4 v 3484 1122 V 3538 1122
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20026
4 62 v 321 1330 a(The)e(ne)o(xt)h(lemma)f(concerns)j(the)d(freshness)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20027
(of)d(a)g(co-name,)i(say)f Ga(c)p Gg(,)f(when)g(forming)i(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20028
(translation)321 1443 y F6(j)p 348 1443 28 4 v 366 1443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20029
V 384 1443 V 65 w(j)436 1410 y Fu(s)436 1466 y Gc(c)471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20030
1443 y Gg(;)h(it)g(will)h(be)f(applied)i(to)f(determine)h(whether)g(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20031
(logical)i(cut-reduction)i(can)d(be)g(performed.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20032
321 1631 a Gb(Lemma)f(3.2.2:)p Black 34 w Gg(F)o(or)g(all)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20033
Ga(M)35 b F6(2)25 b F4(\003)e Gg(and)h(for)f(all)h(co-names)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20034
Ga(c)p Gg(,)e(if)g Ga(M)33 b Gg(is)23 b(of)h(the)g(form)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20035
Black Black 1168 1851 a Ga(x)51 b F6(j)g Ga(\025x:S)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20036
F6(j)c(h)p Ga(S;)15 b(T)e F6(i)51 b(j)g Fp(inl)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20037
Ga(S)5 b F4(\))52 b F6(j)e Fp(inr)p F4(\()p Ga(S)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20038
F4(\))321 2071 y Gg(then)25 b F6(j)p Ga(M)10 b F6(j)649
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20039
2038 y Fu(s)649 2093 y Gc(c)707 2071 y Gg(freshly)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20040
b(introduces)h Ga(c)p Gg(.)p Black 321 2283 a F7(Pr)l(oof)o(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20041
Black 34 w Gg(By)21 b(inspection)j(of)e(the)f(inference)j(rules)f(sho)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20042
(wn)e(in)h(Figure)g(3.2)f(and)h(the)g(translations)j(gi)n(v)o(en)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20043
2396 y(in)f(Figure)g(3.3.)p 3480 2396 4 62 v 3484 2338
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20044
55 4 v 3484 2396 V 3538 2396 4 62 v 321 2604 a(The)f(v)n(ariable)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20045
(substitution)h(and)d(proof)g(substitution)j(are)d(related)h(as)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20046
(follo)n(ws.)p Black 321 2792 a Gb(Lemma)f(3.2.3:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20047
34 w Gg(F)o(or)g(all)h Ga(M)5 b(;)15 b(N)35 b F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20048
b F4(\003)e Gg(and)h(for)g(all)f(co-names)i Ga(a)p Gg(,)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20049
Ga(b)p Gg(,)f(we)h(ha)n(v)o(e)1167 3008 y F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20050
b F6(j)1315 2971 y Fu(s)1315 3031 y Gc(a)1357 3008 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20051
F4([)p Ga(x)26 b F4(:=)1581 2996 y FX(h)1608 3008 y Ga(b)1647
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20052
2996 y FX(i)1675 3008 y F6(j)p Ga(N)10 b F6(j)1808 2975
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20053
y Fu(s)1808 3036 y Gc(b)1843 3008 y F4(])1932 2971 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20054
Gc(int)1893 3008 y F6(\000)-31 b(\000)f(!)2063 2971 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20055
FX(\003)2128 3008 y F6(j)p Ga(M)10 b F4([)p Ga(x)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20056
F4(:=)f Ga(N)10 b F4(])p F6(j)2608 2971 y Fu(s)2608 3031
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20057
y Gc(a)2675 3008 y Ga(:)p Black 321 3221 a F7(Pr)l(oof)o(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20058
Black 34 w Gg(By)23 b(induction)j(on)e(the)f(structure)j(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20059
Ga(M)32 b Gg(\(see)25 b(P)o(age)e(157\).)p 3480 3221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20060
V 3484 3163 55 4 v 3484 3221 V 3538 3221 4 62 v 321 3429
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20061
a(No)n(w)c(we)g(are)h(in)g(a)g(position)i(to)e(sho)n(w)g(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20062
(beta-reductions)j(map)c(onto)h(a)e(series)j(of)e(cut-reductions.)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20063
3542 y(T)-7 b(o)29 b(mak)o(e)g(the)h(induction)i(go)d(through,)j(we)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20064
(ha)n(v)o(e)h(to)f(pro)o(v)o(e)h(this)g(in)f(conjunction)k(with)c(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20065
(slightly)321 3655 y(more)24 b(general)h(statement.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20066
Black 321 3842 a Gb(Pr)n(oposition)32 b(3.2.4:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20067
34 w Gg(F)o(or)e(all)h Ga(M)5 b(;)15 b(M)1551 3809 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20068
FX(0)1614 3842 y F6(2)38 b F4(\003)p Gg(,)31 b(for)g(all)g(co-names)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20069
Ga(c)p Gg(,)g(and)g(for)e(an)o(y)h(substitution)k(of)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20070
3955 y(the)h(form)e F4([)p Ga(c)47 b F4(:=)931 3943 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20071
F9(\()958 3955 y Ga(y)1006 3943 y F9(\))1033 3955 y Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20072
b F4(])35 b Gg(where)g Ga(N)44 b Gg(freshly)37 b(introduces)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20073
Ga(y)f Gg(and)e(is)g(not)g(an)g(axiom,)j(we)c(ha)n(v)o(e)i(if)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20074
4068 y Ga(M)508 4031 y Gc(\014)445 4068 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20075
b(\000)h(!)25 b Ga(M)738 4035 y FX(0)784 4068 y Gg(then)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20076
Black 417 4288 a(\(i\))p Black 47 w F6(j)p Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20077
F6(j)697 4255 y Fu(s)697 4311 y Gc(c)796 4251 y(int)757
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20078
4288 y F6(\000)-31 b(\000)f(!)927 4255 y F9(+)1012 4288
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20079
y F6(j)p Ga(M)1135 4255 y FX(0)1158 4288 y F6(j)1183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20080
4255 y Fu(s)1183 4311 y Gc(c)p Black 392 4446 a Gg(\(ii\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20081
Black 47 w F6(j)p Ga(M)10 b F6(j)697 4413 y Fu(s)697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20082
4469 y Gc(c)732 4446 y F4([)p Ga(c)26 b F4(:=)943 4434
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20083
y F9(\()970 4446 y Ga(y)1018 4434 y F9(\))1046 4446 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20084
Ga(N)10 b F4(])1218 4409 y Gc(int)1179 4446 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20085
b(\000)f(!)1349 4413 y F9(+)1434 4446 y F6(j)p Ga(M)1557
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20086
4413 y FX(0)1580 4446 y F6(j)1605 4413 y Fu(s)1605 4469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20087
y Gc(c)1640 4446 y F4([)p Ga(c)26 b F4(:=)1851 4434 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20088
F9(\()1879 4446 y Ga(y)1927 4434 y F9(\))1954 4446 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20089
Ga(N)10 b F4(])p Black 321 4659 a F7(Pr)l(oof)o(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20090
34 w Gg(By)23 b(induction)j(on)e(the)f(de\002nition)j(of)1855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20091
4622 y Gc(\014)1791 4659 y F6(\000)-31 b(\000)f(!)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20092
b Gg(\(see)h(P)o(age)f(157\).)p 3480 4659 V 3484 4601
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20093
55 4 v 3484 4659 V 3538 4659 4 62 v 321 4867 a(It)h(remains)g(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20094
(check)g(whether)h(commuting)g(reductions)h(translate)g(onto)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20095
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 20229
605 y Ga(x)2550 593 y F9(\))2577 605 y Ga(N)10 b F4(\))321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20230
843 y Gg(with)24 b Ga(x)e Gg(is)i(not)g(free)g(in)f Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20231
b Gg(,)22 b(then)1105 1095 y F6(j)p FL(Cut)q F4(\()1304
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20232
1083 y FX(h)1331 1095 y Ga(a)1379 1083 y FX(i)1407 1095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20233
y Ga(M)10 b(;)1545 1083 y F9(\()1573 1095 y Ga(x)1625
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20234
1083 y F9(\))1652 1095 y Ga(N)g F4(\))p F6(j)1795 1057
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20235
y Fu(n)1862 1095 y F6(\021)25 b(j)p FL(Cut)p F4(\()2156
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20236
1083 y FX(h)2183 1095 y Ga(a)2231 1083 y FX(i)2259 1095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20237
y Ga(M)2357 1062 y FX(0)2380 1095 y Ga(;)2420 1083 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20238
F9(\()2448 1095 y Ga(x)2500 1083 y F9(\))2528 1095 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20239
Ga(N)10 b F4(\))p F6(j)2671 1057 y Fu(n)2737 1095 y Ga(:)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20240
1334 y Gg(W)-7 b(e)29 b(achie)n(v)o(e)i(a)e(some)n(what)h(closer)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20241
(correspondence)j(between)d(cut-elimination)i(and)e(normalisa-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20242
1446 y(tion,)h(if)e(we)f(translate)j(the)e(sequent)i(calculus)g(into)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20243
(a)g(simply-typed)j(lambda)e(calculus)h(e)o(xtended)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20244
1559 y(with)39 b(an)f(e)o(xplicit)i(substitution)j(operator)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20245
b(.)76 b(This)38 b(w)o(as)g(proposed,)45 b(for)39 b(e)o(xample,)k(by)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20246
b(Herbelin)321 1672 y([1994],)30 b(and)d(Barendre)o(gt)i(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20247
(Ghilezan)h([2000].)41 b(On)27 b(one)g(hand,)i(the)f(e)o(xplicit)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20248
(substitution)j(op-)321 1785 y(erator)24 b(ensures)g(that)f(more)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20249
(cut-elimination)j(reductions)f(match)e(with)g(normalisation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20250
(reductions;)321 1898 y(on)33 b(the)h(other)l(,)i(we)d(are)g(not)g(a)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20251
(w)o(are)g(of)g(an)o(y)h(w)o(ork)f(that)g(points)i(out)e(that)h(there)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20252
(is)f(a)g(discrepanc)o(y)321 2011 y(pre)n(v)o(enting)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20253
b(a)d(perfect)j(match.)k(W)-7 b(e)21 b(shall)h(demonstrate)i(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20254
(discrepanc)o(y)j(using)d Ga(\025)p Fp(x)f Gg([Rose,)h(1996].)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20255
Black 321 2199 a Gb(Remark)37 b(3.2.9:)p Black 34 w Gg(First)h(let)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20256
(us)h(remind)g(the)g(reader)g(of)g(the)f(e)o(xplicit)i(substitution)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20257
(calculus)f Ga(\025)p F1(x)321 2312 y Gg(brie\003y)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20258
b(described)i(on)e(P)o(age)f(48.)29 b(The)23 b(terms)h(are)g(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20259
(en)g(by)f(the)h(grammar)1256 2570 y Ga(x)50 b F6(j)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20260
Ga(\025x:M)61 b F6(j)51 b Ga(M)25 b(N)60 b F6(j)51 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20261
Ga(M)10 b F6(h)p Ga(x)26 b F4(:=)f Ga(N)10 b F6(i)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20262
b Ga(:)321 2809 y Gg(In)d(contrast)h(to)f(the)g(simply-typed)j(lambda)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20263
(calculus,)i(in)d Ga(\025)p Fp(x)g Gg(beta-reductions)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20264
b(are)c(split)g(into)h(more)321 2922 y(atomic)35 b(steps:)52
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20265
b(one)34 b(step)h(is)f(the)g(reduction)j F4(\()p Ga(\025x:M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20266
b F4(\))p Ga(N)2315 2885 y Gc(b)2245 2922 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20267
b(\000)f(!)45 b Ga(M)10 b F6(h)p Ga(x)45 b F4(:=)g Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20268
b F6(i)34 b Gg(introducing)j(the)321 3035 y(e)o(xplicit)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20269
b(substitution)h(operator)l(,)f(and)e(the)g(others)h(in)l(v)n(olv)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20270
(reductions)g(that)e(permute)h(this)f(operator)321 3147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20271
y(inside)37 b(the)e(term)g Ga(M)10 b Gg(.)62 b(As)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20272
b(a)h(consequence,)41 b(the)36 b(translation)i(from)d(sequent)h(proofs)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20273
h(to)e(natural)321 3260 y(deduction)27 b(proofs)e(can)e(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20274
(modi\002ed,)g(so)f(as)h(to)f(incorporate)k(the)d(ne)n(w)f(term)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20275
(constructor)-5 b(.)p Black Black 1076 3533 a F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20276
FL(Cut)p F4(\()1274 3521 y FX(h)1302 3533 y Ga(a)1350
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20277
3521 y FX(i)1377 3533 y Ga(M)10 b(;)1515 3521 y F9(\()1543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20278
3533 y Ga(x)1595 3521 y F9(\))1622 3533 y Ga(N)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20279
F6(j)1765 3500 y Fu(n)1801 3476 y FC(0)1929 3481 y F5(def)1936
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20280
3533 y F4(=)106 b F6(j)p Ga(N)10 b F6(j)2246 3500 y Fu(n)2282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20281
3476 y FC(0)2309 3533 y F6(h)p Ga(x)26 b F4(:=)f F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20282
Ga(M)10 b F6(j)2691 3500 y Fu(n)2727 3476 y FC(0)2755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20283
3533 y F6(i)1055 3673 y(j)p FL(Imp)1225 3694 y Gc(R)1282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20284
3673 y F4(\()1317 3661 y F9(\()1345 3673 y Ga(x)1397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20285
3661 y F9(\))q FX(h)1452 3673 y Ga(a)1500 3661 y FX(i)1528
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20286
3673 y Ga(M)g(;)15 b(c)p F4(\))p F6(j)1765 3640 y Fu(n)1801
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20287
3616 y FC(0)1929 3621 y F5(def)1936 3673 y F4(=)106 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20288
Ga(\025x:)p F6(j)p Ga(M)10 b F6(j)2391 3640 y Fu(n)2427
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20289
3616 y FC(0)929 3812 y F6(j)p FL(Imp)1098 3834 y Gc(L)1151
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20290
3812 y F4(\()1186 3800 y FX(h)1213 3812 y Ga(a)1261 3800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20291
y FX(i)1289 3812 y Ga(M)g(;)1427 3800 y F9(\()1455 3812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20292
y Ga(x)1507 3800 y F9(\))1534 3812 y Ga(N)g(;)15 b(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20293
F4(\))p F6(j)1765 3779 y Fu(n)1801 3756 y FC(0)1929 3761
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20294
y F5(def)1936 3812 y F4(=)106 b F6(j)p Ga(N)10 b F6(j)2246
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20295
3779 y Fu(n)2282 3756 y FC(0)2309 3812 y F6(h)p Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20296
b F4(:=)f(\()p Ga(y)34 b F6(j)p Ga(M)10 b F6(j)2805 3779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20297
y Fu(n)2841 3756 y FC(0)2868 3812 y F4(\))p F6(i)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20298
4055 y Gg(T)-7 b(o)19 b(sho)n(w)g(the)h(discrepanc)o(y)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20299
b(,)23 b(we)18 b(shall)j(no)n(w)d(consider)k(the)e(logical)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20300
(cut-reduction)i(for)d(cuts)g(with)321 4168 y(a)k(cut-formula)j(ha)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20301
(ving)g(an)d(implication)j(as)d(top-most)i(connecti)n(v)o(e.)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20302
b(On)24 b(P)o(age)g(29)h(we)f(de\002ned)h(this)321 4281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20303
y(cut-reduction)j(as)1009 4516 y FL(Cut)p F4(\()1182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20304
4504 y FX(h)1210 4516 y Ga(b)1249 4504 y FX(i)1276 4516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20305
y FL(Imp)1421 4538 y Gc(R)1478 4516 y F4(\()1513 4504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20306
y F9(\()1541 4516 y Ga(x)1593 4504 y F9(\))q FX(h)1648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20307
4516 y Ga(a)1696 4504 y FX(i)1723 4516 y Ga(M)11 b(;)k(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20308
F4(\))p Ga(;)1976 4504 y F9(\()2004 4516 y Ga(z)2050
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20309
4504 y F9(\))2078 4516 y FL(Imp)2222 4538 y Gc(L)2275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20310
4516 y F4(\()2310 4504 y FX(h)2337 4516 y Ga(c)2376 4504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20311
y FX(i)2404 4516 y Ga(N)10 b(;)2527 4504 y F9(\()2555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20312
4516 y Ga(y)2603 4504 y F9(\))2630 4516 y Ga(P)j(;)i(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20313
F4(\))q(\))1265 4627 y Gc(l)1191 4664 y F6(\000)-31 b(\000)f(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20314
b FL(Cut)p F4(\()1559 4652 y FX(h)1587 4664 y Ga(a)1635
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20315
4652 y FX(i)1663 4664 y FL(Cut)o F4(\()1835 4652 y FX(h)1863
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20316
4664 y Ga(c)1902 4652 y FX(i)1930 4664 y Ga(N)10 b(;)2053
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20317
4652 y F9(\()2081 4664 y Ga(x)2133 4652 y F9(\))2160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20318
4664 y Ga(M)g F4(\))q Ga(;)2334 4652 y F9(\()2362 4664
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20319
y Ga(y)2410 4652 y F9(\))2437 4664 y Ga(P)j F4(\))51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20320
b Gg(or)1265 4740 y Gc(l)1191 4777 y F6(\000)-31 b(\000)f(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20321
b FL(Cut)p F4(\()1559 4765 y FX(h)1587 4777 y Ga(c)1626
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20322
4765 y FX(i)1654 4777 y Ga(N)10 b(;)1777 4765 y F9(\()1805
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20323
4777 y Ga(x)1857 4765 y F9(\))1884 4777 y FL(Cut)p F4(\()2057
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20324
4765 y FX(h)2085 4777 y Ga(a)2133 4765 y FX(i)2160 4777
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20325
y Ga(M)g(;)2298 4765 y F9(\()2326 4777 y Ga(y)2374 4765
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20326
y F9(\))2401 4777 y Ga(P)j F4(\))q(\))p Black 3372 4647
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20327
a Gg(\(3.2\))p Black 321 5023 a(including)25 b(a)d(choice)h(of)f(ho)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20328
(w)f(the)i(cuts)f(are)h(arranged)h(in)e(the)g(reduct.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20329
b(There)22 b(is)g(no)g(reason)h(to)f(prefer)321 5136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20330
y(one)28 b(choice)i(o)o(v)o(er)d(the)h(other)h(\(the)o(y)f(e)n(v)o(en)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20331
(lead)g(to)g(dif)n(ferent)i(normal)e(forms)g(as)g(we)e(shall)j(sho)n(w)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20332
f(in)321 5249 y(Chapter)c(4\).)29 b(Unfortunately)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20333
b(,)26 b(using)e(the)f(translation)j F6(j)p 2092 5249
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20334
28 4 v 2110 5249 V 2128 5249 V 65 w(j)2180 5216 y Fu(n)2216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20335
5192 y FC(0)2265 5249 y Gg(gi)n(v)o(en)e(abo)o(v)o(e)f(only)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20336
(\002rst)g(reduction)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20337
%%Page: 79 91
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20338
TeXDict begin 79 90 bop Black 277 51 a Gb(3.2)23 b(The)g(Corr)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20339
(espondence)h(between)f(Cut-Elimination)g(and)f(Normalisation)k(I)849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20340
b(79)p 277 88 3691 4 v Black 277 365 a Gg(is)29 b(matched)i(by)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20341
(reductions)j(in)e Ga(\025)p Fp(x)o Gg(;)i(the)e(other)g(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20342
F7(not)p Gg(.)47 b(The)28 b(translations)33 b(of)c(the)h(reducts)h(sho)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20343
n(wn)277 478 y(in)24 b(\(3.2\))f(are:)p Black Black 631
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20344
702 a F6(j)p FL(Cut)p F4(\()829 690 y FX(h)857 702 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20345
Ga(a)905 690 y FX(i)932 702 y FL(Cut)p F4(\()1105 690
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20346
y FX(h)1133 702 y Ga(c)1172 690 y FX(i)1200 702 y Ga(N)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20347
b(;)1322 690 y F9(\()1350 702 y Ga(x)1402 690 y F9(\))1430
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20348
702 y Ga(M)h F4(\))p Ga(;)1603 690 y F9(\()1631 702 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20349
Ga(y)1679 690 y F9(\))1706 702 y Ga(P)j F4(\))p F6(j)1837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20350
669 y Fu(n)1873 646 y FC(0)1926 702 y F4(=)25 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20351
Ga(P)13 b F6(j)2143 669 y Fu(n)2179 646 y FC(0)2207 702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20352
y F6(h)p Ga(y)28 b F4(:=)d F6(j)p Ga(M)10 b F6(j)2584
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20353
669 y Fu(n)2620 646 y FC(0)2648 702 y F6(h)p Ga(x)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20354
b F4(:=)h F6(j)p Ga(N)10 b F6(j)3015 669 y Fu(n)3051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20355
646 y FC(0)3078 702 y F6(ii)631 868 y(j)p FL(Cut)p F4(\()829
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20356
856 y FX(h)857 868 y Ga(c)896 856 y FX(i)923 868 y Ga(N)g(;)1046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20357
856 y F9(\()1074 868 y Ga(x)1126 856 y F9(\))1154 868
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20358
y FL(Cut)o F4(\()1326 856 y FX(h)1354 868 y Ga(a)1402
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20359
856 y FX(i)1430 868 y Ga(M)g(;)1568 856 y F9(\()1596
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20360
868 y Ga(y)1644 856 y F9(\))1671 868 y Ga(P)j F4(\)\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20361
F6(j)1837 835 y Fu(n)1873 811 y FC(0)1926 868 y F4(=)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20362
b F6(j)p Ga(P)13 b F6(j)2143 835 y Fu(n)2179 811 y FC(0)2207
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20363
868 y F6(h)p Ga(y)28 b F4(:=)d F6(j)p Ga(M)10 b F6(j)2584
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20364
835 y Fu(n)2620 811 y FC(0)2648 868 y F6(ih)p Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20365
b F4(:=)f F6(j)p Ga(N)10 b F6(j)3050 835 y Fu(n)3086
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20366
811 y FC(0)3113 868 y F6(i)277 1089 y Gg(W)l(ith)24 b(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20367
(\(standard\))i(reduction)g(rules)f(of)e Ga(\025)p Fp(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20368
Gg(,)f(only)j(the)f(\002rst)f(reduct)i(can)f(be)f(reached.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20369
Black Black 908 1318 a F6(j)p FL(Cut)p F4(\()1106 1306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20370
y FX(h)1134 1318 y Ga(b)1173 1306 y FX(i)1201 1318 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20371
FL(Imp)1345 1340 y Gc(R)1403 1318 y F4(\()1438 1306 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20372
F9(\()1466 1318 y Ga(x)1518 1306 y F9(\))p FX(h)1572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20373
1318 y Ga(a)1620 1306 y FX(i)1648 1318 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20374
b(b)p F4(\))q Ga(;)1901 1306 y F9(\()1929 1318 y Ga(z)1975
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20375
1306 y F9(\))2002 1318 y FL(Imp)2147 1340 y Gc(L)2199
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20376
1318 y F4(\()2234 1306 y FX(h)2262 1318 y Ga(c)2301 1306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20377
y FX(i)2328 1318 y Ga(N)10 b(;)2451 1306 y F9(\()2479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20378
1318 y Ga(y)2527 1306 y F9(\))2555 1318 y Ga(P)j(;)i(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20379
F4(\))q(\))p F6(j)2808 1285 y Fu(n)2844 1261 y FC(0)1008
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20380
1459 y F4(=)93 b F6(j)p Ga(P)13 b F6(j)1293 1426 y Fu(n)1329
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20381
1403 y FC(0)1357 1459 y F6(h)p Ga(y)28 b F4(:=)d Ga(z)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20382
b F6(j)p Ga(N)10 b F6(j)1781 1426 y Fu(n)1817 1403 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20383
FC(0)1844 1459 y F6(ih)p Ga(z)30 b F4(:=)c Ga(\025x:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20384
F6(j)p Ga(M)10 b F6(j)2386 1426 y Fu(n)2422 1403 y FC(0)2449
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20385
1459 y F6(i)908 1601 y(\000)-31 b(\000)f(!)94 b(j)p Ga(P)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20386
b F6(j)1293 1568 y Fu(n)1329 1545 y FC(0)1357 1601 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20387
F6(h)p Ga(y)28 b F4(:=)d Ga(z)20 b F6(j)p Ga(N)10 b F6(j)1781
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20388
1568 y Fu(n)1817 1545 y FC(0)1844 1601 y F6(h)p Ga(z)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20389
b F4(:=)25 b Ga(\025x:)p F6(j)p Ga(M)10 b F6(j)2350 1568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20390
y Fu(n)2386 1545 y FC(0)2414 1601 y F6(ii)908 1743 y(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20391
b(\000)f(!)1097 1710 y FX(\003)1172 1743 y F6(j)p Ga(P)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20392
b F6(j)1293 1710 y Fu(n)1329 1686 y FC(0)1357 1743 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20393
F6(h)p Ga(y)28 b F4(:=)d(\()p Ga(\025)q(x:)p F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20394
b F6(j)1900 1710 y Fu(n)1936 1686 y FC(0)1963 1743 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20395
F4(\))15 b F6(j)p Ga(N)10 b F6(j)2146 1710 y Fu(n)2182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20396
1686 y FC(0)2210 1743 y F6(i)908 1884 y(\000)-31 b(\000)f(!)94
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20397
b(j)p Ga(P)13 b F6(j)1293 1851 y Fu(n)1329 1828 y FC(0)1357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20398
1884 y F6(h)p Ga(y)28 b F4(:=)d F6(j)p Ga(M)10 b F6(j)1734
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20399
1851 y Fu(n)1770 1828 y FC(0)1798 1884 y F6(h)p Ga(x)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20400
b F4(:=)h F6(j)p Ga(N)10 b F6(j)2165 1851 y Fu(n)2201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20401
1828 y FC(0)2228 1884 y F6(ii)277 2152 y Gg(The)33 b(discrepanc)o(y)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20402
(appears)f(if)e(we)f(try)h(to)g(reach)h(the)g(second)g(reduct.)59
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20403
b(Here)33 b(we)f(w)o(ould)i(need)g(a)277 2265 y(reduction)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20404
b(of)e(the)f(form)h(\()p Ga(x)h F6(62)g Ga(F)13 b(V)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20405
b F4(\()p Ga(P)13 b F4(\))p Gg(\))993 2509 y Ga(P)g F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20406
Ga(y)29 b F4(:=)c Ga(M)10 b F6(h)p Ga(x)25 b F4(:=)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20407
Ga(N)10 b F6(ii)26 b(\000)-32 b(\000)h(!)25 b Ga(P)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20408
b F6(h)p Ga(y)28 b F4(:=)e Ga(M)10 b F6(ih)p Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20409
b F4(:=)f Ga(N)10 b F6(i)277 2730 y Gg(which,)29 b(as)e(f)o(ar)g(as)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20410
(we)e(kno)n(w)-6 b(,)28 b(has)g(not)g(been)g(considered)i(for)e(an)o(y)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20411
f(e)o(xplicit)i(substitution)i(calculus)277 2842 y(and)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20412
b(indeed)h(w)o(ould)f(be)g(rather)h(strange.)418 2972
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20413
y(What)39 b(is)g(sho)n(wn)g(by)g(this)g(e)o(xample)g(is)g(that)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20414
(ob)o(vious)i(candidate)g(for)e(a)f(simply-typed)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20415
3085 y(lambda)29 b(calculus)g(with)f(an)g(e)o(xplicit)h(substitution)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20416
(operator)f(and)e(a)f(natural)j(translation)h(from)c(se-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20417
3198 y(quent)34 b(proofs)f(to)f(this)h(lambda)g(calculus)h(gi)n(v)o(es)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20418
f(a)f(closer)h(correspondence)k(between)d(cut-elimi-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20419
3311 y(nation)e(and)e(normalisation,)35 b(b)n(ut)30 b(not)g(all)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20420
(cut-reductions)j(are)c(\223mirrored\224.)50 b(In)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20421
b(the)g(ne)o(xt)h(section)277 3424 y(we)26 b(shall)i(de)n(v)o(elop)g(a)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20422
e(natural)i(deduction)i(calculus)f(with)d(a)h(symmetric)g(e)o(xplicit)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20423
(substitution)h(op-)277 3536 y(erator)-5 b(.)58 b(F)o(or)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20424
b(this)h(calculus)i(we)d(shall)i(gi)n(v)o(e)f(translations)j(so)d(that)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20425
g F7(all)g Gg(cut-reductions)k(map)c(onto)277 3649 y(normalisation)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20426
b(reductions.)418 3903 y(In)h(the)h(rest)g(of)f(this)g(section)i(we)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20427
(shall)h(analyse)h(whether)f(we)f(can)g(e)o(xtend)i(Proposition)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20428
(3.2.8)277 4016 y(to)22 b(all)h(connecti)n(v)o(es)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20429
(intuitionistic)k(logic.)j(As)22 b(mentioned)i(earlier)l(,)g(Zuck)o(er)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20430
f(concluded)i(that)e(this)277 4129 y(cannot)h(be)e(done)h(for)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20431
(fragment)g(including)i F6(_)c Gg(or)h F6(9)p Gg(,)f(because)j(\(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20432
(his)h(setting\))h(cut-elimination)277 4242 y(f)o(ails)i(to)e(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20433
(strongly)h(normalising.)35 b(Our)24 b(cut-elimination)k(procedure)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20434
F4(\()p FY(I)p Ga(;)2756 4205 y Gc(int)2717 4242 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20435
b(\000)f(!)p F4(\))24 b F7(is)h Gg(strongly)i(nor)n(-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20436
4355 y(malising,)g(and)f(thus)f(it)g(is)g(interesting)k(to)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20437
(re-address)j(the)d(question)j(of)d(the)g(correspondence.)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20438
b(Un-)277 4468 y(fortunately)-6 b(,)28 b(it)d(turns)h(out)f(that)h(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20439
f F6(_)p Gg(-connecti)n(v)o(e)i(is)e(still)h(problematic)h(with)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20440
(respect)h(to)f(the)h(sim-)277 4581 y(ulation)f(of)f(cut-elimination.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20441
418 4710 y(Clearly)-6 b(,)34 b(to)d(eliminate)i(all)e(cut-instances)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20442
(from)c(sequent)i(proofs)f(we)f(need)h(the)f(commuting)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20443
4823 y(reduction)26 b(of)e(the)f(form)1193 5028 y FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20444
F4(\()1366 5016 y FX(h)1394 5028 y Ga(a)1442 5016 y FX(i)1469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20445
5028 y Ga(M)10 b(;)1607 5016 y F9(\()1635 5028 y Ga(x)1687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20446
5016 y F9(\))1714 5028 y Ga(N)g F4(\))1928 4990 y Gc(c)1858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20447
5027 y F6(\000)-31 b(\000)f(!)2053 5028 y Ga(M)10 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20448
Ga(a)26 b F4(:=)2371 5016 y F9(\()2399 5028 y Ga(x)2451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20449
5016 y F9(\))2478 5028 y Ga(N)10 b F4(])p Black 742 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20450
Gg(\(3.3\))p Black 277 5249 a(which)25 b(may)f(be)g(applied)i(if)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20451
Ga(M)33 b Gg(does)25 b(not)g(freshly)h(introduce)h Ga(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20452
Gg(.)j(In)24 b(the)g F4(\()p F6(^)p Ga(;)15 b F6(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20453
F4(\))p Gg(-fragment)27 b(consid-)277 5361 y(ered)f(abo)o(v)o(e,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20454
(this)g(reduction)h(is)f(mapped)g(by)f F6(j)p 1783 5361
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20455
28 4 v 1801 5361 V 1819 5361 V 65 w(j)1871 5329 y Fu(n)1936
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20456
5361 y Gg(onto)h(an)f(identity)j(\(see)d(Lemma)g(3.2.7\).)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20457
b(But)24 b(in)277 5474 y(case)f Ga(M)33 b Gg(is)22 b(an)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20458
F6(_)826 5488 y Gc(L)879 5474 y Gg(-rule,)g(then)g F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20459
1300 5474 V 1318 5474 V 1336 5474 V 65 w(j)1388 5441
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20460
y Fu(n)1451 5474 y Gg(does)g(not)h(translate)g(\(3.3\))g(onto)f(an)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20461
(identity)-6 b(.)30 b(In)23 b(consequence,)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20462
eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20463
%%Page: 80 92
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20464
TeXDict begin 80 91 bop Black -144 51 a Gb(80)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20465
b(Deduction)p -144 88 3691 4 v Black 321 365 a Gg(the)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20466
b(correspondence)k(f)o(ails.)48 b(The)29 b(issues)i(are)f(best)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20467
(xplained)i(with)e(an)f(e)o(xample.)48 b(Consider)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20468
b(the)321 478 y(follo)n(wing)25 b(sequent)h(proof)e(\(where)g(for)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20469
(bre)n(vity)h(we)e(ha)n(v)o(e)h(omitted)h(all)e(labels\).)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20470
758 635 233 4 v 758 714 a Ga(A)p 846 702 11 41 v 857
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20471
684 46 5 v 97 w(A)p 1082 635 244 4 v 91 w(B)p 1175 702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20472
11 41 v 1186 684 46 5 v 101 w(B)p 758 734 567 4 v 801
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20473
813 a(A;)15 b(B)p 1004 801 11 41 v 1014 782 46 5 v 102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20474
w(A)p F6(^)o Ga(B)1367 752 y F6(^)1427 766 y Gc(R)p 1576
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20475
635 233 4 v 1576 714 a Ga(A)p 1664 702 11 41 v 1675 684
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20476
46 5 v 96 w(A)p 1899 635 244 4 v 91 w(B)p 1993 702 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20477
41 v 2004 684 46 5 v 101 w(B)p 1576 734 567 4 v 1619
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20478
813 a(A;)g(B)p 1821 801 11 41 v 1832 782 46 5 v 101 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20479
F6(^)p Ga(B)2184 752 y F6(^)2245 766 y Gc(R)p 801 850
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20480
1299 4 v 1089 929 a Ga(A)p F6(_)o Ga(B)5 b(;)15 b(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20481
b(;)15 b(A)p 1534 917 11 41 v 1544 899 46 5 v 97 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20482
F6(^)p Ga(B)2141 868 y F6(_)2202 882 y Gc(L)p 2393 653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20483
233 4 v 2393 732 a Ga(A)p 2482 720 11 41 v 2492 701 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20484
5 v 97 w(A)p 2717 653 233 4 v 91 w(A)p 2805 720 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20485
v 2816 701 46 5 v 97 w(A)p 2393 752 557 4 v 2491 830
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20486
a(A)p 2579 818 11 41 v 2590 800 46 5 v 96 w(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20487
Ga(A)2991 770 y F6(^)3052 784 y Gc(R)p 2424 850 496 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20488
v 2424 929 a Ga(A)p F6(^)p Ga(B)p 2646 917 11 41 v 2657
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20489
899 46 5 v 100 w(A)p F6(^)p Ga(A)2961 864 y F6(^)3021
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20490
878 y Gc(L)3069 887 y FV(1)p 1089 967 1831 4 v 1645 1045
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20491
a Ga(A)p F6(_)p Ga(B)5 b(;)15 b(B)5 b(;)15 b(A)p 2090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20492
1033 11 41 v 2100 1015 46 5 v 96 w(A)p F6(^)p Ga(A)2961
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20493
997 y Gg(Cut)p Black 3372 1045 a(\(3.4\))p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20494
1204 a(T)-7 b(o)23 b(eliminate)i(the)f(cut,)f(we)g(need)h(to)g(apply)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20495
(the)g(rule)g(sho)n(wn)g(in)g(\(3.3\).)29 b(This)23 b(gi)n(v)o(es)h(us)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20496
p 355 1502 233 4 v 355 1580 a Ga(A)p 443 1568 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20497
453 1550 46 5 v 96 w(A)p 678 1502 244 4 v 91 w(B)p 772
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20498
1568 11 41 v 782 1550 46 5 v 101 w(B)p 355 1600 567 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20499
v 398 1679 a(A;)15 b(B)p 600 1667 11 41 v 610 1649 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20500
5 v 101 w(A)p F6(^)p Ga(B)963 1618 y F6(^)1024 1633 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20501
Gc(R)p 1172 1403 233 4 v 1172 1482 a Ga(A)p 1261 1470
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20502
11 41 v 1271 1451 46 5 v 97 w(A)p 1496 1403 233 4 v 91
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20503
w(A)p 1584 1470 11 41 v 1595 1451 46 5 v 96 w(A)p 1172
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20504
1502 557 4 v 1270 1580 a(A)p 1358 1568 11 41 v 1369 1550
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20505
46 5 v 96 w(A)p F6(^)p Ga(A)1770 1520 y F6(^)1831 1534
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20506
y Gc(R)p 1203 1600 496 4 v 1203 1679 a Ga(A)p F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20507
Ga(B)p 1425 1667 11 41 v 1436 1649 46 5 v 101 w(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20508
Ga(A)1740 1614 y F6(^)1800 1628 y Gc(L)1848 1637 y FV(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20509
398 1717 1301 4 v 810 1795 a Ga(A;)g(B)p 1013 1783 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20510
41 v 1023 1765 46 5 v 102 w(A)p F6(^)o Ga(A)1740 1747
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20511
y Gg(Cut)p 1979 1502 233 4 v 1979 1580 a Ga(A)p 2068
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20512
1568 11 41 v 2078 1550 46 5 v 97 w(A)p 2303 1502 244
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20513
4 v 91 w(B)p 2397 1568 11 41 v 2407 1550 46 5 v 101 w(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20514
1979 1600 567 4 v 2022 1679 a(A;)g(B)p 2225 1667 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20515
v 2235 1649 46 5 v 102 w(A)p F6(^)p Ga(B)2588 1618 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20516
F6(^)2648 1633 y Gc(R)p 2797 1403 233 4 v 2797 1482 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20517
Ga(A)p 2885 1470 11 41 v 2896 1451 46 5 v 96 w(A)p 3121
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20518
1403 233 4 v 92 w(A)p 3209 1470 11 41 v 3219 1451 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20519
5 v 96 w(A)p 2797 1502 557 4 v 2894 1580 a(A)p 2983 1568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20520
11 41 v 2993 1550 46 5 v 97 w(A)p F6(^)p Ga(A)3395 1520
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20521
y F6(^)3455 1534 y Gc(R)p 2827 1600 496 4 v 2827 1679
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20522
a Ga(A)p F6(^)p Ga(B)p 3050 1667 11 41 v 3060 1649 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20523
5 v 101 w(A)p F6(^)p Ga(A)3364 1614 y F6(^)3425 1628
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20524
y Gc(L)3473 1637 y FV(1)p 2022 1717 1301 4 v 2435 1795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20525
a Ga(A;)g(B)p 2637 1783 11 41 v 2648 1765 46 5 v 101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20526
w(A)p F6(^)p Ga(A)3364 1747 y Gg(Cut)p 810 1833 2101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20527
4 v 1499 1912 a Ga(A)p F6(_)o Ga(B)5 b(;)15 b(B)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20528
b(A)p 1944 1900 11 41 v 1954 1881 46 5 v 97 w(A)p F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20529
Ga(B)2952 1851 y F6(_)3012 1865 y Gc(L)p Black 3372 2025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20530
a Gg(\(3.5\))p Black 321 2154 a(where)33 b(tw)o(o)g(copies)h(of)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20531
(cut)h(ha)n(v)o(e)f(been)h(created.)58 b(Let)33 b(us)g(gi)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20532
(the)g(corresponding)k(natural)321 2267 y(deduction)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20533
b(proofs.)j(F)o(orming)23 b F6(j)p Gg(\(3.4\))r F6(j)1545
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20534
2234 y Fu(n)1608 2267 y Gg(we)g(ha)n(v)o(e)h(the)g(natural)h(deduction)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20535
h(proof)p -104 2538 458 4 v -104 2612 a FU(A)p FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20536
FU(B)p 99 2600 10 38 v 109 2583 42 4 v 92 w(A)p FT(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20537
FU(B)p 409 2445 213 4 v 409 2518 a(A)p 490 2506 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20538
v 499 2490 42 4 v 88 w(A)p 691 2445 223 4 v 70 w(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20539
776 2506 10 38 v 786 2490 42 4 v 92 w(B)p 409 2538 504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20540
4 v 441 2612 a(A;)14 b(B)p 626 2600 10 38 v 636 2583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20541
42 4 v 93 w(A)p FT(^)p FU(B)927 2555 y FT(^)982 2568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20542
y FS(I)p 1075 2445 213 4 v 1075 2518 a FU(A)p 1156 2506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20543
10 38 v 1166 2490 42 4 v 89 w(A)p 1357 2445 223 4 v 69
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20544
w(B)p 1443 2506 10 38 v 1452 2490 42 4 v 92 w(B)p 1075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20545
2538 504 4 v 1108 2612 a(A;)g(B)p 1293 2600 10 38 v 1302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20546
2583 42 4 v 92 w(A)p FT(^)q FU(B)1593 2555 y FT(^)1649
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20547
2568 y FS(I)p -104 2648 1651 4 v 391 2721 a FU(A)p FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20548
FU(B)t(;)g(A;)g(B)p 798 2709 10 38 v 807 2692 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20549
92 w(A)p FT(^)q FU(B)1561 2665 y FT(_)1616 2677 y FS(E)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20550
391 2757 661 4 v 452 2830 a FU(A)p FT(_)q FU(B)t(;)g(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20551
859 2818 10 38 v 868 2802 42 4 v 92 w(A)1093 2770 y FT(^)1149
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20552
2782 y FS(E)1198 2790 y FP(1)p 1714 2538 458 4 v 1714
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20553
2612 a FU(A)p FT(_)q FU(B)p 1918 2600 10 38 v 1927 2583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20554
42 4 v 92 w(A)p FT(_)q FU(B)p 2227 2445 213 4 v 2227
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20555
2518 a(A)p 2308 2506 10 38 v 2317 2490 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20556
2509 2445 223 4 v 70 w(B)p 2594 2506 10 38 v 2604 2490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20557
42 4 v 92 w(B)p 2227 2538 504 4 v 2260 2612 a(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20558
2444 2600 10 38 v 2454 2583 42 4 v 92 w(A)p FT(^)p FU(B)2745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20559
2555 y FT(^)2800 2568 y FS(I)p 2894 2445 213 4 v 2894
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20560
2518 a FU(A)p 2974 2506 10 38 v 2984 2490 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20561
3175 2445 223 4 v 69 w(B)p 3261 2506 10 38 v 3270 2490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20562
42 4 v 92 w(B)p 2894 2538 504 4 v 2926 2612 a(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20563
3111 2600 10 38 v 3120 2583 42 4 v 92 w(A)p FT(^)q FU(B)3411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20564
2555 y FT(^)3467 2568 y FS(I)p 1714 2648 1651 4 v 2209
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20565
2721 a FU(A)p FT(_)q FU(B)t(;)g(A;)g(B)p 2616 2709 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20566
38 v 2625 2692 42 4 v 92 w(A)p FT(^)q FU(B)3379 2665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20567
y FT(_)3434 2677 y FS(E)p 2209 2757 661 4 v 2271 2830
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20568
a FU(A)p FT(_)p FU(B)t(;)g(A;)g(B)p 2677 2818 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20569
2687 2802 42 4 v 93 w(A)2912 2770 y FT(^)2967 2782 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20570
FS(E)3016 2790 y FP(1)p 452 2866 2357 4 v 1303 2940 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20571
FU(A)p FT(_)p FU(B)t(;)g(A;)g(B)p 1709 2928 10 38 v 1719
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20572
2911 42 4 v 93 w(A)p FT(^)p FU(A)2850 2883 y FT(^)2906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20573
2896 y FS(I)321 3153 y Gg(in)35 b(which)h(the)f F6(^)888
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20574
3167 y Gc(E)940 3176 y FV(1)979 3153 y Gg(-rules)h(can)f(be)g(permuted)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20575
i(with)e(the)g F6(_)2280 3167 y Gc(E)2339 3153 y Gg(-rules.)65
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20576
b(This)34 b(gi)n(v)o(es)i(the)f(follo)n(wing)321 3266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20577
y(proof.)p -104 3644 458 4 v -104 3717 a FU(A)p FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20578
FU(B)p 99 3705 10 38 v 109 3689 42 4 v 92 w(A)p FT(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20579
FU(B)p 409 3441 213 4 v 409 3515 a(A)p 490 3503 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20580
v 499 3486 42 4 v 88 w(A)p 691 3441 223 4 v 70 w(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20581
776 3503 10 38 v 786 3486 42 4 v 92 w(B)p 409 3535 504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20582
4 v 441 3608 a(A;)14 b(B)p 626 3596 10 38 v 636 3579
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20583
42 4 v 93 w(A)p FT(^)p FU(B)927 3552 y FT(^)982 3564
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20584
y FS(I)p 441 3644 440 4 v 503 3717 a FU(A;)g(B)p 687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20585
3705 10 38 v 697 3689 42 4 v 92 w(A)894 3657 y FT(^)950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20586
3669 y FS(E)999 3677 y FP(1)p 1090 3441 213 4 v 1090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20587
3515 a FU(A)p 1171 3503 10 38 v 1180 3486 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20588
1372 3441 223 4 v 70 w(B)p 1457 3503 10 38 v 1467 3486
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20589
42 4 v 92 w(B)p 1090 3535 504 4 v 1123 3608 a(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20590
1307 3596 10 38 v 1317 3579 42 4 v 92 w(A)p FT(^)q FU(B)1608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20591
3552 y FT(^)1663 3564 y FS(I)p 1123 3644 440 4 v 1184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20592
3717 a FU(A;)g(B)p 1369 3705 10 38 v 1378 3689 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20593
92 w(A)1576 3657 y FT(^)1631 3669 y FS(E)1680 3677 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20594
FP(1)p -104 3754 1605 4 v 429 3827 a FU(A)p FT(_)q FU(B)t(;)g(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20595
836 3815 10 38 v 845 3798 42 4 v 92 w(A)1514 3770 y FT(_)1570
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20596
3783 y FS(E)p 1744 3644 458 4 v 1744 3717 a FU(A)p FT(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20597
FU(B)p 1947 3705 10 38 v 1957 3689 42 4 v 93 w(A)p FT(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20598
FU(B)p 2257 3441 213 4 v 2257 3515 a(A)p 2337 3503 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20599
38 v 2347 3486 42 4 v 88 w(A)p 2538 3441 223 4 v 69 w(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20600
2624 3503 10 38 v 2633 3486 42 4 v 92 w(B)p 2257 3535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20601
504 4 v 2289 3608 a(A;)g(B)p 2474 3596 10 38 v 2483 3579
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20602
42 4 v 92 w(A)p FT(^)q FU(B)2774 3552 y FT(^)2830 3564
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20603
y FS(I)p 2289 3644 440 4 v 2350 3717 a FU(A;)g(B)p 2535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20604
3705 10 38 v 2545 3689 42 4 v 93 w(A)2742 3657 y FT(^)2797
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20605
3669 y FS(E)2846 3677 y FP(1)p 2938 3441 213 4 v 2938
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20606
3515 a FU(A)p 3019 3503 10 38 v 3028 3486 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20607
3220 3441 223 4 v 70 w(B)p 3305 3503 10 38 v 3315 3486
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20608
42 4 v 92 w(B)p 2938 3535 504 4 v 2970 3608 a(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20609
3155 3596 10 38 v 3165 3579 42 4 v 93 w(A)p FT(^)p FU(B)3456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20610
3552 y FT(^)3511 3564 y FS(I)p 2970 3644 440 4 v 3032
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20611
3717 a FU(A;)g(B)p 3216 3705 10 38 v 3226 3689 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20612
92 w(A)3423 3657 y FT(^)3479 3669 y FS(E)3528 3677 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20613
FP(1)p 1744 3754 1605 4 v 2277 3827 a FU(A)p FT(_)q FU(B)t(;)g(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20614
2683 3815 10 38 v 2693 3798 42 4 v 92 w(A)3362 3770 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20615
FT(_)3417 3783 y FS(E)p 429 3863 2386 4 v 1294 3936 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20616
FU(A)p FT(_)q FU(B)t(;)g(A;)g(B)p 1701 3924 10 38 v 1710
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20617
3908 42 4 v 92 w(A)p FT(^)q FU(A)2857 3880 y FT(^)2912
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20618
3892 y FS(I)321 4156 y Gg(Here)24 b(we)e(are)i(stuck!)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20619
b(There)24 b(is)g(no)f(reduction)j(that)e(w)o(ould)g(gi)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20620
F6(j)p Gg(\(3.5\))r F6(j)2640 4123 y Fu(n)2680 4156 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20621
Gg(\227the)g(proof)h(belo)n(w)-6 b(.)p 85 4646 458 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20622
v 85 4720 a FU(A)p FT(_)q FU(B)p 288 4708 10 38 v 298
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20623
4691 42 4 v 92 w(A)p FT(_)p FU(B)p 626 4334 213 4 v 626
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20624
4408 a(A)p 706 4396 10 38 v 716 4379 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20625
921 4334 223 4 v 83 w(B)p 1007 4396 10 38 v 1016 4379
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20626
42 4 v 92 w(B)p 626 4427 518 4 v 665 4501 a(A;)14 b(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20627
850 4489 10 38 v 859 4472 42 4 v 92 w(A)p FT(^)q FU(B)1185
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20628
4444 y FT(^)1240 4457 y FS(I)p 665 4537 440 4 v 726 4610
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20629
a FU(A;)g(B)p 911 4598 10 38 v 920 4582 42 4 v 92 w(A)1145
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20630
4550 y FT(^)1201 4562 y FS(E)1250 4570 y FP(1)p 1369
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20631
4334 213 4 v 1369 4408 a FU(A)p 1450 4396 10 38 v 1459
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20632
4379 42 4 v 88 w(A)p 1665 4334 223 4 v 84 w(B)p 1750
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20633
4396 10 38 v 1760 4379 42 4 v 92 w(B)p 1369 4427 518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20634
4 v 1408 4501 a(A;)g(B)p 1593 4489 10 38 v 1603 4472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20635
42 4 v 93 w(A)p FT(^)p FU(B)1928 4444 y FT(^)1984 4457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20636
y FS(I)p 1408 4537 440 4 v 1470 4610 a FU(A;)g(B)p 1654
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20637
4598 10 38 v 1664 4582 42 4 v 92 w(A)1889 4550 y FT(^)1944
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20638
4562 y FS(E)1993 4570 y FP(1)p 726 4646 1061 4 v 1039
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20639
4720 a FU(A;)g(B)p 1224 4708 10 38 v 1233 4691 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20640
92 w(A)p FT(^)q FU(A)1828 4663 y FT(^)1883 4676 y FS(I)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20641
2113 4334 213 4 v 2113 4408 a FU(A)p 2193 4396 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20642
v 2203 4379 42 4 v 88 w(A)p 2408 4334 223 4 v 83 w(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20643
2494 4396 10 38 v 2503 4379 42 4 v 92 w(B)p 2113 4427
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20644
518 4 v 2152 4501 a(A;)g(B)p 2337 4489 10 38 v 2346 4472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20645
42 4 v 92 w(A)p FT(^)q FU(B)2672 4444 y FT(^)2727 4457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20646
y FS(I)p 2152 4537 440 4 v 2213 4610 a FU(A;)g(B)p 2398
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20647
4598 10 38 v 2408 4582 42 4 v 93 w(A)2633 4550 y FT(^)2688
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20648
4562 y FS(E)2737 4570 y FP(1)p 2856 4334 213 4 v 2856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20649
4408 a FU(A)p 2937 4396 10 38 v 2947 4379 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20650
3152 4334 223 4 v 84 w(B)p 3237 4396 10 38 v 3247 4379
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20651
42 4 v 92 w(B)p 2856 4427 518 4 v 2896 4501 a(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20652
3080 4489 10 38 v 3090 4472 42 4 v 92 w(A)p FT(^)p FU(B)3415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20653
4444 y FT(^)3471 4457 y FS(I)p 2896 4537 440 4 v 2957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20654
4610 a FU(A;)g(B)p 3142 4598 10 38 v 3151 4582 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20655
92 w(A)3376 4550 y FT(^)3431 4562 y FS(E)3480 4570 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20656
FP(1)p 2213 4646 1061 4 v 2526 4720 a FU(A;)g(B)p 2711
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20657
4708 10 38 v 2721 4691 42 4 v 93 w(A)p FT(^)p FU(A)3315
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20658
4663 y FT(^)3370 4676 y FS(I)p 85 4756 2876 4 v 1195
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20659
4829 a FU(A)p FT(_)p FU(B)t(;)g(B)t(;)g(A)p 1601 4817
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20660
10 38 v 1611 4800 42 4 v 89 w(A)p FT(^)p FU(A)3002 4773
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20661
y FT(_)3057 4785 y FS(E)321 5043 y Gg(What)27 b(is)g(needed)h(is)e(a)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20662
(reduction)i(rule)e(which)g(permutes)h F6(^)2271 5057
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20663
y Gc(I)2336 5043 y Gg(with)f F6(_)2584 5057 y Gc(E)2644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20664
5043 y Gg(.)37 b(On)26 b(the)h(le)n(v)o(el)g(of)g(lambda)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20665
5156 y(terms)d(the)g(corresponding)k(reduction)e(rule)e(is)f(as)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20666
(follo)n(ws.)p Black Black 706 5329 a F6(h)p Fp(case)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20667
F4(\()p Ga(P)s(;)15 b(\025)q(x:M)5 b(;)15 b(\025)q(y)s(:N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20668
b F4(\))p Ga(;)15 b Fp(case)o F4(\()p Ga(P)s(;)g(\025)q(z)t(:S;)g(\025)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20669
q(w)r(:T)e F4(\))p F6(i)1129 5477 y(\000)-32 b(\000)h(!)76
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20670
b Fp(case)n F4(\()p Ga(P)s(;)15 b(\025)q(x:)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20671
Ga(M)5 b(;)15 b(S)5 b F4([)p Ga(z)31 b F4(:=)25 b Ga(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20672
F4(])p F6(i)p Ga(;)15 b(\025)q(y)s(:)p F6(h)p Ga(N)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20673
b(;)15 b(T)e F4([)p Ga(w)29 b F4(:=)c Ga(y)s F4(])p F6(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20674
F4(\))p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20675
%%Page: 81 93
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20676
TeXDict begin 81 92 bop Black 277 51 a Gb(3.3)23 b(Classical)j(Natural)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20677
d(Deduction)2373 b(81)p 277 88 3691 4 v Black 277 365
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20678
a Gg(Since)21 b(the)h(reduction)i(\(3.3\))d(cannot)i(be)e(dispensed)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20679
(with)d(in)g(the)h(sequent)h(calculus,)g(we)e(ha)n(v)o(e)g(to)g(e)o(x-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20680
277 478 y(tend)g(the)g(natural)h(deduction)h(calculus)g(with)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20681
(commuting)i(reductions)i(of)c(the)h(kind)g(sho)n(wn)g(abo)o(v)o(e.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20682
3464 445 y F5(2)277 590 y Gg(As)33 b(can)h(be)f(easily)i(sho)n(wn,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20683
(these)e(commuting)h(reductions)i(are)c(harmless)i(with)f(respect)h(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20684
e(the)277 703 y(subformula)27 b(property)-6 b(.)36 b(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20685
b(e)24 b(shall)i(do)f(the)g(e)o(xtension)i(in)e(the)g(ne)o(xt)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20686
(section)h(and)e(establish)i(the)f(cor)n(-)277 816 y(respondence)h(for)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20687
d(classical)i(logic.)277 1117 y Ge(3.3)119 b(Classical)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20688
b(Natural)g(Deduction)277 1341 y Gg(Gentzen)22 b([1935])h(de\002ned)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20689
(the)f(classical)i(natural)g(deduction)g(calculus)g(NK)d(by)h(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20690
(xtending)i(NJ)e(with)277 1454 y(an)j(additional)i(axiom,)e(kno)n(wn)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20691
(as)f(the)h(e)o(xcluded)h(middle)g(axiom,)e(sho)n(wn)h(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20692
b(.)1740 1671 y Ga(B)20 b F6(_)15 b(:)p Ga(B)277 1842
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20693
y Gg(In)33 b(order)i(to)e(obtain)i(a)d(classical)k(natural)f(deduction)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20694
h(calculus,)i(one)33 b(can,)j(alternati)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20695
b(,)39 b(add)33 b(to)277 1955 y(NJ)f(the)g(stability)i(axiom,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20696
F6(::)p Ga(B)5 b F6(\033)n Ga(B)g Gg(,)33 b(an)f(instance)j(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20697
(Peirce')-5 b(s)33 b(la)o(w)-6 b(,)33 b F4(\(\()p Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20698
b F6(\033?)p F4(\))p F6(\033)p Ga(B)g F4(\))p F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20699
Ga(B)g Gg(,)33 b(or)f(an)277 2067 y(additional)40 b(inference)f(rule)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20700
(called)g(the)f(absurdity)j(rule.)69 b(All)36 b(these)i(calculi)h(are)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20701
(adequate)i(for)277 2180 y(classical)26 b(logic)e(in)f(terms)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20702
(pro)o(v)n(ability;)j(ho)n(we)n(v)o(er)l(,)e(from)f(a)g(proof)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20703
(theoretic)g(point)g(of)e(vie)n(w)g(the)o(y)277 2293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20704
y(are)i(not)g(v)o(ery)g(ele)o(gant.)33 b(Let)23 b(us)i(\002rst)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20705
(mention)i(tw)o(o)e(shortcomings)k(remark)o(ed)e(by)f(Gentzen)g([1935,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20706
277 2406 y(P)o(ages)31 b(189\226190])i(and)f(Pra)o(witz)e([1971,)k(P)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20707
(age)c(251]:)45 b(the)31 b(inference)j(rules)e(for)f(ne)o(gation)h(do)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20708
(not)277 2519 y(f)o(all)25 b(into)g(the)f(general)i(pattern)g(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20709
(introduction)k(and)d(elimination)h(rules,)g(and)e(there)h(are)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20710
(problems)277 2632 y(concerning)32 b(the)d(subformula)i(property)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20711
b(.)47 b(F)o(or)28 b(analysing)j(the)f(correspondence)j(question)e(a)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20712
(third)277 2745 y(shortcoming)h(is)e(particularly)j(anno)o(ying:)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20713
b(classical)29 b(sequent)f(proofs)g(can)f(only)g(be)g(encoded)i(into)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20714
277 2858 y(NK)22 b(via)i(a)f(double)i(ne)o(gation)g(translation.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20715
2987 y(Better)38 b(suited)h(for)f(handling)i(classical)g(logic)e(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20716
(much)g(more)g(con)l(v)o(enient)j(for)c(sho)n(wing)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20717
3100 y(the)f(correspondence)41 b(is)35 b(the)h(sequence-conclusi)q(on)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20718
41 b(natural)d(deduction)g(calculus)3038 3067 y F5(3)3113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20719
3100 y Gg(introduced)277 3213 y(by)28 b(Bori)5 b(\020)-35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20720
b(ci)5 b(\264)-35 b(c)28 b([1985].)42 b(W)-7 b(e)27 b(shall)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20721
(reformulate)i(this)e(calculus)h(into)g(\223sequent-style\224.)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20722
b(F)o(or)26 b(this,)j(we)277 3326 y(shall)i(tak)o(e)g(sequents)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20723
(the)g(form)g F4(\000)p 1529 3314 11 41 v 1540 3296 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20724
5 v 97 w(\001)f Gg(where)h F4(\000)f Gg(and)i F4(\001)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20725
Gg(are)i(sets)f(of)g(\(label,formula\))k(pairs,)277 3439
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20726
y(analogous)24 b(to)e(the)f(left-)h(and)g(right-conte)o(xts)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20727
(introduced)e(for)e(sequent)h(calculi.)30 b(Thus)21 b(the)h(labels)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20728
(in)277 3552 y F4(\000)i Gg(are)g(referred)i(to)f(as)f(names)h(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20729
(in)h F4(\001)e Gg(as)h(co-names.)33 b(The)24 b(inference)j(rules)e(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20730
f(Bori)5 b(\020)-35 b(ci)5 b(\264)-35 b(c')-5 b(s)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20731
b(calcu-)277 3665 y(lus,)21 b(as)f(usual)h(for)f(natural)h(deduction,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20732
(come)d(in)g(pairs)h(introducing)i(a)c(connecti)n(v)o(e)k(and)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20733
(eliminating)277 3778 y(a)j(connecti)n(v)o(e.)31 b(F)o(or)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20734
b(instance)j(the)d(rules)i(introducing)i(and)d(eliminating)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20735
F6(^)c Gg(are:)p Black Black 673 3952 a F4(\000)p 750
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20736
3940 11 41 v 760 3921 46 5 v 96 w(\001)p Ga(;)15 b(a)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20737
F4(:)h Ga(B)95 b F4(\000)p 1291 3940 11 41 v 1302 3921
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20738
46 5 v 96 w(\001)p Ga(;)15 b(b)j F4(:)f Ga(C)p 673 3989
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20739
981 4 v 876 4068 a F4(\000)p 953 4056 11 41 v 964 4038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20740
46 5 v 96 w(\001)p Ga(;)e(c)j F4(:)f Ga(B)5 b F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20741
Ga(C)1695 4008 y F6(^)1756 4022 y Gc(I)2268 3951 y F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20742
2345 3939 11 41 v 2355 3920 46 5 v 96 w(\001)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20743
b(a)i F4(:)h Ga(B)2714 3965 y F9(1)2753 3951 y F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20744
Ga(B)2883 3965 y F9(2)p 2268 3988 655 4 v 2363 4068 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20745
F4(\000)p 2440 4056 11 41 v 2450 4038 46 5 v 96 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20746
Ga(;)d(b)i F4(:)h Ga(B)2800 4082 y Gc(i)2964 4001 y F6(^)3024
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20747
4016 y Gc(E)3076 4026 y FZ(i)277 4262 y Gg(Again,)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20748
b(rede)o(x)o(es)f(arise)f(when)f(a)g(connecti)n(v)o(e)j(is)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20749
(introduced)k(and)d(eliminated)h(afterw)o(ards.)69 b(An)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20750
4375 y(e)o(xample)24 b(is)1526 4456 y Ga(\031)1328 4535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20751
y F4(\000)p 1405 4523 11 41 v 1416 4504 46 5 v 96 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20752
Ga(;)15 b(a)j F4(:)f Ga(B)2050 4455 y(\031)2105 4422
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20753
y FX(0)1870 4535 y F4(\000)p 1947 4523 11 41 v 1957 4504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20754
46 5 v 96 w(\001)p Ga(;)e(b)i F4(:)h Ga(C)p 1328 4572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20755
981 4 v 1532 4651 a F4(\000)p 1609 4639 11 41 v 1619
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20756
4621 46 5 v 96 w(\001)p Ga(;)d(c)j F4(:)f Ga(B)5 b F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20757
Ga(C)2351 4590 y F6(^)2411 4605 y Gc(I)p 1532 4689 574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20758
4 v 1594 4768 a F4(\000)p 1671 4756 11 41 v 1681 4738
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20759
46 5 v 96 w(\001)p Ga(;)15 b(d)j F4(:)f Ga(B)2147 4702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20760
y F6(^)2208 4716 y Gc(E)2260 4725 y FV(1)277 4931 y Gg(which)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20761
b(reduces)h(to)f(the)g(proof)1878 4965 y(.)1878 4998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20762
y(.)1878 5032 y(.)1665 5112 y F4(\000)p 1742 5100 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20763
41 v 1752 5082 46 5 v 96 w(\001)p Ga(;)15 b(d)j F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20764
Ga(B)p Black 277 5238 1290 4 v 383 5294 a F3(2)412 5326
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20765
y F2(While)22 b(this)g(rule)h(has)g(not)g(been)g(considered)h(for)f(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20766
(natural)f(deduction)j(calculus,)f(unpublished)g(w)o(ork)f(by)h(Dybjer)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20767
277 5417 y(and)c(Ong)f(suggests)h(that)f(this)f(rule)h(comes)h(into)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20768
(play)g(when)h(considering)g(bicartesian)f(closed)h(cate)o(gories.)383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20769
5478 y F3(3)412 5510 y F2(T)-6 b(o)17 b(be)h(consistent)h(with)e(our)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(natural)h(deduction)i(calculus)e(as)277 5601 y(sequence-succedent)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20772
(natural)d(deduction)i(calculus,)e(b)o(ut)g(we)f(shall)h(follo)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20773
(Bori)t(\020)-29 b(ci)t(\264)g(c')l(s)19 b(original)g(terminology)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(.)p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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%%Page: 82 94
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(Deduction)p -144 88 3691 4 v Black 321 365 a Gg(obtained)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20778
b(from)e Ga(\031)i Gg(by)e(renaming)h Ga(a)e Gg(to)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20779
Ga(d)p Gg(.)62 b(Ho)n(we)n(v)o(er)l(,)37 b(notice)g(that)e(in)g(case)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20780
(of)g(more)g(than)g(one)321 478 y(formula)h(in)g(the)f(succedent)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20781
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20782
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20783
b(a)c(connecti)n(v)o(e)j(and)e(the)g(rule)g(that)g(eliminates)h(it,)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(as)h(indicated)i(belo)n(w)-6 b(.)1326 746 y F4(\000)1383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20785
713 y FX(0)p 1426 734 11 41 v 1437 716 46 5 v 1502 746
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20786
a F4(\001)1578 713 y FX(0)1601 746 y Ga(;)15 b(a)j F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20787
Ga(B)96 b F4(\000)1971 713 y FX(0)p 2014 734 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20788
2025 716 46 5 v 2090 746 a F4(\001)2166 713 y FX(0)2189
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20789
746 y Ga(;)15 b(b)j F4(:)f Ga(C)p 1326 784 1074 4 v 1553
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20790
870 a F4(\000)1610 837 y FX(0)p 1653 858 11 41 v 1663
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20791
840 46 5 v 1729 870 a F4(\001)1805 837 y FX(0)1828 870
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20792
y Ga(;)e(c)j F4(:)f Ga(B)5 b F6(^)p Ga(C)2441 802 y F6(^)2502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20793
816 y Gc(I)1851 914 y Gg(.)1851 947 y(.)1851 980 y(.)1851
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20794
1013 y(.)1576 1093 y F4(\000)p 1653 1081 11 41 v 1663
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20795
1063 46 5 v 96 w(\001)p Ga(;)15 b(c)j F4(:)f Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20796
F6(^)o Ga(C)p 1576 1131 574 4 v 1638 1210 a F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20797
1715 1198 11 41 v 1725 1180 46 5 v 96 w(\001)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20798
b(d)j F4(:)f Ga(B)2191 1144 y F6(^)2252 1158 y Gc(E)2304
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20799
1167 y FV(1)321 1388 y Gg(In)i(order)h(to)f(apply)h(the)f(reduction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20800
(rule)e(eliminating)i(the)e(rede)o(x,)i(we)d(ha)n(v)o(e)h(to)g(mo)o(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20801
(e)f(the)h(elimination)321 1501 y(rule)32 b(ne)o(xt)g(to)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20802
(introduction)j(rule.)53 b(This)32 b(means)g(already)h(in)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20803
(fragment)h(containing)i(only)321 1614 y(the)e F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20804
Gg(-connecti)n(v)o(e)i(we)c(need)i(commuting)g(reductions.)57
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20805
b(Cellucci)34 b([1992])f(ga)n(v)o(e)f(the)h(\002rst)e(set)i(of)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20806
1727 y(reduction)28 b(rules)e(for)f(Bori)5 b(\020)-35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20807
b(ci)5 b(\264)-35 b(c')-5 b(s)26 b(calculus.)35 b(These)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20808
b(reductions)j(are)d(Gentzen-lik)o(e,)j(in)c(the)i(sense)g(of)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20809
1840 y(re)n(writing)20 b(neighbouring)j(inference)e(rules)f(only)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20810
b(.)28 b(As)18 b(with)g(sequent)j(calculi,)g(a)d(set)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20811
(Gentzen-lik)o(e)321 1953 y(reduction)28 b(rules)e(that)f(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20812
(strongly)i(normalising)g(is)e(rather)h(dif)n(\002cult)g(to)f(\002nd)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20813
(\(we)h(illustrated)i(some)321 2065 y(of)j(the)g(dif)n(\002culties)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20814
(with)d(an)h(LK-proof)h(in)e(Example)h(2.1.3\).)48 b(Unlik)o(e)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20815
b(his)f(rules,)i(which)e(are)g F7(not)321 2178 y Gg(strongly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20816
(normalising,)i(our)c(reduction)j(rules)e(will)e(be)h(similar)h(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20817
(the)g(\223global\224)i(reduction)h(rules)321 2291 y(introduced)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20818
b(for)c F4(\()p FY(T)t Ga(;)1049 2254 y Gc(cut)1018 2291
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20819
y F6(\000)-31 b(\000)f(!)p F4(\))35 b Gg(and)h(in)f(consequence)k(will)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20820
c(be)g(strongly)j(normalising.)67 b(Restricted)321 2404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20821
y(v)o(ersions)27 b(of)d(our)h(rules)g(were)g(studied)h(pre)n(viously)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20822
(in)e(connection)j(with)c Ga(\025\026)f Gg([P)o(arigot,)i(1992].)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20823
b(F)o(or)321 2517 y(space)28 b(reasons)h(we)d(shall)i(gi)n(v)o(e)f(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20824
h(calculus)h(for)e(only)h(the)f(propositional)k(fragment)d(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20825
(classical)321 2630 y(logic.)i(Note)19 b(ho)n(we)n(v)o(er)g(that)h(all)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20826
f(results)i(gi)n(v)o(en)f(belo)n(w)f(can)h(be)f(easily)i(e)o(xtended)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20827
(to)e(more)h(e)o(xpressi)n(v)o(e)321 2743 y(fragments.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20828
2872 y(W)-7 b(e)20 b(alluded)j(in)d(Section)i(2.5)e(to)h(the)g(man)o(y)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20829
f(v)n(ariations)k(of)c(de\002ning)i(sequent)h(calculi;)g(similarly)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20830
2985 y(there)31 b(are)g(man)o(y)f(v)n(ariants)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(sequence-conclusio)q(n)36 b(natural)c(deduction)h(calculi.)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20832
b(T)-7 b(o)29 b(sho)n(w)h(the)321 3098 y(correspondence)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20833
b(we)34 b(shall)i(use)f(the)g(calculus)i(gi)n(v)o(en)f(in)f(Figure)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(3.6.)63 b(The)34 b(notion)j(of)d(implicit)321 3211 y(contractions)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20835
(and)d(the)g(conte)o(xt)g(con)l(v)o(ention)j(can)d(be)f(adapted)i(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20836
(this)f(calculus)j(in)d(the)h(ob)o(vious)321 3324 y(w)o(ay)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20837
b(\(see)h(P)o(age)e(19\).)28 b(Whilst)21 b(our)f(introduction)k(rules)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20838
(are)f(rather)h(standard,)i(our)d(elimination)j(rules)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3437 y(are)28 b(some)n(what)f(non-standard.)44 b(The)o(y)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20840
b(are)g(implicit)h(in)f(the)h(w)o(ork)f(by)g(Bori)5 b(\020)-35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(ci)5 b(\264)-35 b(c)28 b([1985],)i(and)d(some)321 3550
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20842
y(of)e(them)g(appear)h(in)f F7(fr)m(ee)g(deduction)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20843
Gg(introduced)g(by)d(P)o(arigot)g([1991].)34 b(Ho)n(we)n(v)o(er)l(,)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20844
b(we)f(should)j(lik)o(e)321 3663 y(to)i(stress)h(the)f(point)g(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20845
(the)f(reasons)h(for)f(choosing)i(this)e(particular)i(set)e(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20846
(inference)i(rules)e(are)321 3776 y(entirely)24 b(pragmatic:)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20847
b(the)o(y)22 b(simplify)h(considerably)i(the)d(translations)i(between)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20848
(natural)g(deduction)321 3889 y(proofs)34 b(and)e(sequent)i(proofs.)56
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20849
b(Other)32 b(sets)h(of)f(inference)i(rules)f(can)f(also)h(be)f(used)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20850
(to)f(study)h(the)321 4001 y(correspondence,)28 b(although)e(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20851
(machinery)h(required)h(is)d(more)h(complicated.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20852
4131 y(Recall)35 b(the)g(translation)j F6(j)p 1318 4131
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20853
28 4 v 1336 4131 V 1353 4131 V 65 w(j)1406 4098 y Fu(n)1480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20854
4131 y Gg(from)d(intuitionistic)j(sequent)f(proofs)f(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20855
(intuitionistic)39 b(natu-)321 4244 y(ral)29 b(deduction)j(proofs)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20856
(\(Figure)g(3.4\).)44 b(There,)30 b(as)f(illustrated)j(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20857
b(,)30 b(we)e(applied)j(a)d(substitution)321 4357 y(operation)e(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20858
(translate)i(cuts.)1187 4554 y F6(j)p FL(Cut)p F4(\()1385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20859
4542 y FX(h)1412 4554 y Ga(a)1460 4542 y FX(i)1488 4554
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20860
y Ga(M)10 b(;)1626 4542 y F9(\()1654 4554 y Ga(x)1706
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20861
4542 y F9(\))1733 4554 y Ga(N)g F4(\))p F6(j)1876 4516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20862
y Fu(n)1958 4554 y F4(=)40 b F6(j)p Ga(N)10 b F6(j)2202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20863
4516 y Fu(n)2243 4554 y F4([)p Ga(x)25 b F4(:=)h F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20864
Ga(M)10 b F6(j)2615 4516 y Fu(n)2656 4554 y F4(])321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20865
4732 y Gg(In)24 b(consequence,)k(we)23 b(obtained)j(a)d(rather)j(weak)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20866
(correspondence:)35 b(whene)n(v)o(er)25 b Ga(x)e Gg(is)h(not)g(free)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20867
(in)e Ga(N)321 4844 y Gg(and)h(therefore)i(in)e F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20868
Ga(N)10 b F6(j)1053 4811 y Fu(n)1094 4844 y Gg(,)23 b(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20869
(cut-reductions)j(that)e Ga(M)33 b Gg(can)24 b(perform)g(are)g(mapped)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20870
(onto)f(identities.)321 4957 y(Consider)36 b(an)f(analogous)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20871
(translation)g(for)e(classical)h(sequent)h(proofs)f(\(denoted)g(by)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20872
F6(j)p 3184 4957 V 3202 4957 V 3219 4957 V 65 w(j)3272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20873
4924 y Fu(N)3326 4957 y Gg(\),)h(and)321 5070 y(suppose)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20874
b(we)d(are)g(gi)n(v)o(en)h(the)g(term)g(corresponding)j(to)d(Lafont')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20875
b(s)25 b(e)o(xample;)f(that)g(is)1614 5313 y FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20876
F4(\()1787 5301 y FX(h)1815 5313 y Ga(a)1863 5301 y FX(i)1890
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20877
5313 y Ga(M)10 b(;)2028 5301 y F9(\()2056 5313 y Ga(x)2108
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20878
5301 y F9(\))2135 5313 y Ga(N)g F4(\))p Black 1119 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20879
Gg(\(3.6\))p Black 321 5474 a(where)27 b Ga(a)f Gg(is)g(not)h(free)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20880
(in)f Ga(M)36 b Gg(and)27 b Ga(x)e Gg(not)i(free)g(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20881
Ga(N)10 b Gg(.)37 b(Clearly)-6 b(,)27 b(this)g(term)g(can)f(reduce)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20882
(to)f(either)g Ga(M)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20883
%%Page: 83 95
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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TeXDict begin 83 94 bop Black 277 51 a Gb(3.3)23 b(Classical)j(Natural)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20885
d(Deduction)2373 b(83)p 277 88 3691 4 v Black Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20886
277 V 277 2068 4 1792 v 1773 369 676 4 v 1773 447 a Ga(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20887
b F4(:)h Ga(B)5 b(;)15 b F4(\000)p 2076 435 11 41 v 2086
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20888
417 46 5 v 96 w(\001)p Ga(;)g(a)i F4(:)h Ga(B)615 709
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20889
y(x)f F4(:)h Ga(B)5 b(;)15 b F4(\000)p 918 697 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20890
v 928 679 46 5 v 96 w(\001)p Ga(;)g(a)i F4(:)h Ga(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20891
615 746 674 4 v 660 826 a F4(\000)p 737 814 11 41 v 748
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20892
796 46 5 v 96 w(\001)p Ga(;)d(b)j F4(:)f Ga(B)5 b F6(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20893
Ga(C)1331 764 y F6(\033)1401 778 y Gc(I)1946 710 y F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20894
2023 698 11 41 v 2033 680 46 5 v 96 w(\001)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20895
b(a)i F4(:)h Ga(B)5 b F6(\033)o Ga(C)97 b F4(\000)p 2706
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20896
698 11 41 v 2717 680 46 5 v 96 w(\001)p Ga(;)15 b(b)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20897
F4(:)f Ga(B)96 b(x)17 b F4(:)g Ga(C)q(;)e F4(\000)p 3457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20898
698 11 41 v 3467 680 46 5 v 97 w(\001)p 1946 747 1664
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20899
4 v 2663 826 a(\000)p 2740 814 11 41 v 2750 796 46 5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20900
v 96 w(\001)3650 765 y F6(\033)3721 779 y Gc(E)442 1088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20901
y F4(\000)p 519 1076 11 41 v 529 1058 46 5 v 96 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20902
Ga(;)g(a)i F4(:)h Ga(B)145 b F4(\000)p 1110 1076 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20903
v 1121 1058 46 5 v 97 w(\001)p Ga(;)15 b(b)i F4(:)g Ga(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20904
442 1126 1032 4 v 670 1205 a F4(\000)p 747 1193 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20905
v 758 1175 46 5 v 96 w(\001)p Ga(;)e(c)j F4(:)g Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20906
b F6(^)o Ga(C)1514 1144 y F6(^)1575 1158 y Gc(I)2147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20907
1088 y F4(\000)p 2224 1076 11 41 v 2234 1058 46 5 v 96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20908
w(\001)p Ga(;)15 b(a)i F4(:)h Ga(B)2593 1102 y F9(1)2632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20909
1088 y F6(^)p Ga(B)2762 1102 y F9(2)2917 1088 y Ga(x)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20910
F4(:)h Ga(B)3098 1102 y Gc(i)3126 1088 y Ga(;)d F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20911
3243 1076 11 41 v 3254 1058 46 5 v 97 w(\001)p 2147 1126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20912
1249 4 v 2657 1205 a(\000)p 2734 1193 11 41 v 2744 1175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20913
46 5 v 96 w(\001)3437 1139 y F6(^)3497 1153 y Gc(E)3549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20914
1163 y FZ(i)710 1466 y F4(\000)p 787 1454 11 41 v 797
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20915
1436 46 5 v 96 w(\001)p Ga(;)g(a)i F4(:)h Ga(B)1156 1480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20916
y Gc(i)p 624 1504 646 4 v 624 1583 a F4(\000)p 701 1571
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20917
11 41 v 711 1553 46 5 v 96 w(\001)p Ga(;)d(b)i F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20918
Ga(B)1061 1597 y F9(1)1100 1583 y F6(_)p Ga(B)1230 1597
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20919
y F9(2)1311 1517 y F6(_)1371 1531 y Gc(I)1402 1541 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20920
FZ(i)1951 1467 y F4(\000)p 2028 1455 11 41 v 2039 1437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20921
46 5 v 96 w(\001)p Ga(;)d(a)j F4(:)f Ga(B)5 b F6(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20922
Ga(C)98 b(x)17 b F4(:)g Ga(B)5 b(;)15 b F4(\000)p 2927
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20923
1455 11 41 v 2938 1437 46 5 v 96 w(\001)91 b Ga(y)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20924
b F4(:)e Ga(C)q(;)d F4(\000)p 3461 1455 11 41 v 3472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20925
1437 46 5 v 97 w(\001)p 1951 1505 1662 4 v 2668 1583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20926
a(\000)p 2745 1571 11 41 v 2755 1553 46 5 v 96 w(\001)3655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20927
1523 y F6(_)3715 1537 y Gc(E)730 1846 y Ga(x)i F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20928
Ga(B)5 b(;)15 b F4(\000)p 1033 1834 11 41 v 1043 1816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20929
46 5 v 96 w(\001)p 702 1883 512 4 v 702 1962 a(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20930
779 1950 11 41 v 789 1932 46 5 v 96 w(\001)p Ga(;)g(a)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20931
F4(:)h F6(:)p Ga(B)1254 1896 y F6(:)1315 1910 y Gc(I)2261
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20932
1846 y F4(\000)p 2338 1834 11 41 v 2348 1816 46 5 v 96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20933
w(\001)p Ga(;)d(a)i F4(:)h F6(:)p Ga(B)95 b F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20934
2940 1834 11 41 v 2950 1816 46 5 v 96 w(\001)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20935
b(b)i F4(:)h Ga(B)p 2261 1883 1044 4 v 2668 1962 a F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20936
2745 1950 11 41 v 2755 1932 46 5 v 96 w(\001)3346 1896
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20937
y F6(:)3407 1910 y Gc(E)p 3965 2068 4 1792 v 277 2071
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20938
3691 4 v Black 845 2225 a Gg(Figure)25 b(3.6:)k(Natural)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20939
b(deduction)i(calculus)g(for)e(classical)h(logic.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20940
Black 277 2663 a(or)f Ga(N)10 b Gg(.)31 b(T)m(ranslating)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20941
b(this)f(cut-instance)j(to)c(a)g(natural)i(deduction)h(proof)e(using)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20942
(an)o(y)e(kind)h(of)g(substi-)277 2776 y(tution)h(operation)h(w)o(ould)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20943
e(be)g(disastrous:)33 b(it)25 b(w)o(ould)g(result)g(into)g(a)g(f)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20944
(ailure)h(of)e(the)h(correspondence)277 2889 y(between)k(normalisation)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20945
i(and)d(cut-elimination.)46 b(Gi)n(v)o(en)27 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20946
2184 2889 28 4 v 2202 2889 V 2220 2889 V 65 w(j)2272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20947
2856 y Fu(N)2353 2889 y Gg(we)g(w)o(ould)h(translate)i(this)f(term)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20948
(as)277 3002 y(follo)n(ws)1071 3262 y F6(j)p FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20949
F4(\()1269 3250 y FX(h)1297 3262 y Ga(a)1345 3250 y FX(i)1372
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20950
3262 y Ga(M)10 b(;)1510 3250 y F9(\()1538 3262 y Ga(x)1590
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20951
3250 y F9(\))1618 3262 y Ga(N)g F4(\))p F6(j)1761 3225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20952
y Fu(N)1856 3262 y F4(=)40 b F6(j)p Ga(N)10 b F6(j)2100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20953
3225 y Fu(N)2154 3262 y F4([)p Ga(x)26 b F4(:=)2378 3250
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20954
y FX(h)2405 3262 y Ga(a)2453 3250 y FX(i)2481 3262 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20955
F6(j)p Ga(M)10 b F6(j)2629 3229 y Fu(N)2683 3262 y F4(])277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20956
3522 y Gg(where,)23 b(because)i Ga(x)d Gg(not)h(free)g(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20957
F6(j)p Ga(N)10 b F6(j)1455 3489 y Fu(N)1509 3522 y Gg(,)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20958
b(the)h(substitution)j(v)n(anishes;)g(that)d(is)g(the)g(term)f(on)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20959
(the)g(right-)277 3635 y(hand)35 b(side)g(is)g(equi)n(v)n(alent)h(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20960
F6(j)p Ga(N)10 b F6(j)1407 3602 y Fu(N)1461 3635 y Gg(.)60
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20961
b(In)35 b(consequence,)40 b(we)34 b(w)o(ould)h(\223lose\224)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20962
(cut-reductions)277 3747 y(which)29 b(reduce)i(\(3.6\))e(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20963
Ga(M)10 b Gg(.)44 b(This)29 b(means)g(we)f(cannot)j(de\002ne)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20964
F6(j)p 2396 3747 V 2414 3747 V 2432 3747 V 65 w(j)2484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20965
3714 y Fu(N)2566 3747 y Gg(using)h(a)f(substitution)j(oper)n(-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20966
3860 y(ation.)48 b(W)-7 b(e)29 b(shall)h(remedy)h(this)f(problem)h(by)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20967
(introducing)j(the)d(follo)n(wing)h(\(symmetric\))g(e)o(xplicit)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20968
3973 y(substitution)c(rule)1191 4191 y F4(\000)1248 4205
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20969
y F9(1)p 1308 4179 11 41 v 1318 4161 46 5 v 1384 4191
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20970
a F4(\001)1460 4205 y F9(1)1499 4191 y Ga(;)15 b(a)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20971
F4(:)h Ga(B)95 b(x)17 b F4(:)h Ga(B)5 b(;)15 b F4(\000)2094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20972
4205 y F9(2)p 2153 4179 11 41 v 2164 4161 46 5 v 2229
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20973
4191 a F4(\001)2305 4205 y F9(2)p 1191 4229 1154 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20974
1468 4307 a F4(\000)1525 4321 y F9(1)1564 4307 y Ga(;)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20975
F4(\000)1661 4321 y F9(2)p 1721 4295 11 41 v 1731 4277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20976
46 5 v 1797 4307 a F4(\001)1873 4321 y F9(1)1912 4307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20977
y Ga(;)g F4(\001)2028 4321 y F9(2)2386 4259 y Gg(Subst)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20978
4601 y(which)22 b(allo)n(ws)f(us)g(to)g(de\002ne)h(a)f(translation)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20979
F6(j)p 1702 4601 28 4 v 1720 4601 V 1737 4601 V 65 w(j)1790
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20980
4568 y Fu(N)1864 4601 y Gg(and)e(reduction)h(rules)g(so)e(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20981
F6(j)p Gg(\(3.6\))r F6(j)3059 4568 y Fu(N)3133 4601 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20982
Gg(can)h(match)277 4714 y F7(both)j Gg(reductions)i(of)d(\(3.6\).)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20983
b(Thus,)25 b(we)e(add)i(this)f(inference)j(rule)e(to)f(our)g(natural)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20984
(deduction)h(calcu-)277 4827 y(lus.)418 4966 y(In)g(the)h(follo)n(wing)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20985
h(we)d(shall)i(de)n(v)o(elop)h(reduction)h(rules)e(for)g(our)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20986
(classical)j(natural)f(deduction)277 5079 y(calculus.)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20987
b(As)29 b(with)g(our)h(cut-elimination)k(rules,)e(these)e(reduction)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20988
(rules)d(will)g(be)g(de\002ned)g(using)277 5192 y(terms)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20989
b(annotated)i(to)d(proofs.)31 b(The)23 b(ra)o(w)f(terms)i(are)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20990
(de\002ned)g(belo)n(w)-6 b(.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20991
%%Page: 84 96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20992
TeXDict begin 84 95 bop Black -144 51 a Gb(84)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20993
b(Deduction)p -144 88 3691 4 v Black Black 321 365 a(De\002nition)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20994
(3.3.1)g Gg(\(Ra)o(w)g(T)-6 b(erms\))p Gb(:)p Black 33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20995
w Gg(The)24 b(set)f(of)h(ra)o(w)e(terms,)i F4(R)p Gg(,)e(is)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20996
(de\002ned)g(by)g(the)f(grammar)p Black Black 346 539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20997
a Ga(M)5 b(;)15 b(N)5 b(;)15 b(P)113 b F4(::=)100 b FL(Id)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20998
F4(\()p Ga(x;)15 b(a)p F4(\))1220 b Gg(identity)816 652
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 20999
y F6(j)148 b FL(And)1144 666 y Gc(I)1184 652 y F4(\()1219
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21000
640 y FX(h)1246 652 y Ga(a)r F4(:)r Ga(B)1397 640 y FX(i)1425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21001
652 y Ga(M)10 b(;)1563 640 y FX(h)1591 652 y Ga(b)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21002
Ga(C)1731 640 y FX(i)1758 652 y Ga(N)g(;)15 b(c)p F4(\))536
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21003
b Gg(and-introduction)816 766 y F6(j)148 b FL(And)1144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21004
729 y Gc(i)1144 789 y(E)1203 766 y F4(\()1238 754 y FX(h)1266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21005
766 y Ga(a)r F4(:)r Ga(B)1412 780 y F9(1)1452 766 y F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21006
Ga(B)1581 780 y F9(2)1621 754 y FX(i)1648 766 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21007
b(;)1786 754 y F9(\()1814 766 y Ga(x)17 b F4(:)h Ga(B)1995
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21008
780 y Gc(i)2023 754 y F9(\))2050 766 y Ga(N)10 b F4(\))323
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21009
b Gg(and)2621 733 y Gc(i)2650 766 y Gg(-elimination)147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21010
b Ga(i)26 b F4(=)e(1)p Ga(;)15 b F4(2)816 881 y F6(j)148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21011
b FL(Or)1088 844 y Gc(i)1088 904 y(I)1128 881 y F4(\()1163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21012
869 y FX(h)1191 881 y Ga(a)1239 895 y Gc(i)1269 881 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21013
F4(:)r Ga(B)1365 895 y Gc(i)1394 869 y FX(i)1421 881
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21014
y Ga(M)10 b(;)15 b(b)p F4(\))858 b Gg(or)2566 848 y Gc(i)2595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21015
881 y Gg(-introduction)173 b Ga(i)26 b F4(=)e(1)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21016
b F4(2)816 994 y F6(j)148 b FL(Or)1088 1008 y Gc(E)1148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21017
994 y F4(\()1183 982 y FX(h)1211 994 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21018
Ga(B)5 b F6(_)o Ga(C)1494 982 y FX(i)1521 994 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21019
b(;)1659 982 y F9(\()1687 994 y Ga(x)r F4(:)r Ga(B)1842
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21020
982 y F9(\))1869 994 y Ga(N)g(;)1992 982 y F9(\()2020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21021
994 y Ga(y)5 b F4(:)r Ga(C)2169 982 y F9(\))2196 994
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21022
y Ga(P)13 b F4(\))189 b Gg(or)n(-elimination)816 1107
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21023
y F6(j)148 b FL(Imp)1134 1129 y Gc(I)1173 1107 y F4(\()1208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21024
1095 y F9(\()1236 1107 y Ga(x)r F4(:)r Ga(B)1391 1095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21025
y F9(\))p FX(h)1446 1107 y Ga(a)r F4(:)r Ga(C)1595 1095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21026
y FX(i)1622 1107 y Ga(M)10 b(;)15 b(b)p F4(\))657 b Gg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21027
(implication-introducti)q(on)816 1220 y F6(j)148 b FL(Imp)1134
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21028
1241 y Gc(E)1193 1220 y F4(\()1228 1208 y FX(h)1256 1220
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21029
y Ga(a)r F4(:)r Ga(B)5 b F6(\033)p Ga(C)1550 1208 y FX(i)1576
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21030
1220 y Ga(M)11 b(;)1715 1208 y FX(h)1742 1220 y Ga(b)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21031
F4(:)r Ga(C)1882 1208 y FX(i)1910 1220 y Ga(N)f(;)2033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21032
1208 y F9(\()2060 1220 y Ga(x)r F4(:)r Ga(B)2215 1208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21033
y F9(\))2243 1220 y Ga(P)j F4(\))142 b Gg(implication-elimination)816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21034
1333 y F6(j)148 b FL(Not)1132 1347 y Gc(I)1172 1333 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21035
F4(\()1207 1321 y F9(\()1234 1333 y Ga(x)r F4(:)r Ga(B)1389
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21036
1321 y F9(\))1417 1333 y Ga(M)10 b(;)15 b(b)p F4(\))862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21037
b Gg(ne)o(gation-introduction)816 1445 y F6(j)148 b FL(Not)1132
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21038
1459 y Gc(E)1191 1445 y F4(\()1226 1433 y FX(h)1254 1445
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21039
y Ga(a)r F4(:)r F6(:)p Ga(B)1466 1433 y FX(i)1493 1445
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21040
y Ga(M)10 b(;)1631 1433 y FX(h)1659 1445 y Ga(b)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21041
Ga(B)1801 1433 y FX(i)1828 1445 y Ga(N)g F4(\))545 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21042
Gg(ne)o(gation-elimination)816 1558 y F6(j)148 b FL(Subst)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21043
F4(\()1236 1546 y FX(h)1264 1558 y Ga(a)r F4(:)r Ga(B)1415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21044
1546 y FX(i)1442 1558 y Ga(M)10 b(;)1580 1546 y F9(\()1608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21045
1558 y Ga(x)r F4(:)r Ga(B)1763 1546 y F9(\))1791 1558
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21046
y Ga(N)f F4(\))583 b Gg(symmetric)24 b(substitution)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21047
1804 y(Henceforth)j(we)d(shall)i(omit)f(the)g(types)h(on)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21048
(binders,)h(pro)o(vided)f(the)o(y)g(are)f(clear)h(from)f(the)g(con-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21049
1917 y(te)o(xt.)k(Con)l(v)o(ention)24 b(2.2.4)e(concerning)j(the)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21050
(Barendre)o(gt-style)j(naming)e(con)l(v)o(ention)i(e)o(xtends)e(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21051
(the)321 2029 y(terms)k(de\002ned)h(abo)o(v)o(e.)36 b(Gi)n(v)o(en)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21052
b(the)h(notions)i(of)d(free)h(names)h(and)f(co-names)h(for)f(terms)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21053
(of)f FY(R)p Gg(,)g(it)h(is)321 2142 y(routine)g(to)d(de\002ne)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21054
(analogous)i(notions)g(for)d(the)h(terms)g(of)f F4(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21055
Gg(.)462 2272 y(T)-7 b(o)27 b(annotate)j(terms)f(to)f(our)g(natural)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21056
(deduction)i(proofs)f(we)d(shall)i(replace)g(e)n(v)o(ery)g(sequent)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21057
(of)321 2385 y(the)h(form)f F4(\000)p 745 2373 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21058
v 755 2354 46 5 v 96 w(\001)f Gg(with)h(a)f(typing)j(judgement)g(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21059
(the)g(form)g F4(\000)2394 2373 y Gc(.)2449 2385 y Ga(M)2572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21060
2373 y Gc(.)2627 2385 y F4(\001)f Gg(in)h(which)h F4(\000)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21061
Gg(and)h F4(\001)f Gg(are)321 2498 y(conte)o(xts)24 b(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21062
Ga(M)31 b Gg(is)22 b(a)f(term.)28 b(As)21 b(usual,)i(we)f(are)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21063
(interested)i(in)e(only)h(well-typed)h(terms,)e(i.e.,)f(those)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21064
2611 y(for)j(which)f(there)h(is)f(a)g(conte)o(xt)h F4(\000)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21065
Gg(and)g(a)g(conte)o(xt)h F4(\001)f Gg(such)h(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21066
F4(\000)2444 2599 y Gc(.)2499 2611 y Ga(M)2623 2599 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21067
Gc(.)2677 2611 y F4(\001)g Gg(is)f(deri)n(v)n(able)k(gi)n(v)o(en)d(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21068
321 2723 y(rules)i(sho)n(wn)f(in)f(Figure)h(3.7.)29 b(The)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21069
b(set)g(of)h(well-typed)h(terms)f(is)g(de\002ned)g(as)f(follo)n(ws.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21070
Black Black 380 2945 a F4(K)550 2893 y F5(def)558 2945
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21071
y F4(=)735 2844 y FK(n)821 2945 y Ga(M)966 2840 y FK(\014)966
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21072
2895 y(\014)966 2949 y(\014)1044 2945 y Ga(M)35 b F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21073
b F4(R)e Gg(and)h(well-typed)h(by)f(the)g(inference)h(rules)g(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21074
(en)f(in)f(Figure)i(3.7)3427 2844 y FK(o)462 3161 y Gg(Our)f(inference)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21075
i(rules)e(are)g(de\002ned)g(such)h(that)f(only)h(axioms)f(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21076
(introduction)j(rules)e(are)f(con-)321 3274 y(cerned)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21077
b(with)f(introducing)i(a)d(co-name;)33 b(elimination)e(rules,)f(on)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21078
(the)f(other)i(hand,)g(are)f(concerned)321 3387 y(with)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21079
b(only)g(eliminating)i(a)d(co-name.)30 b(This)23 b(is)h(e)o(xpressed)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21080
(in)f(the)g(follo)n(wing)h(de\002nition.)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21081
3575 a Gb(De\002nition)e(3.3.2:)p Black Black 458 3768
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21082
a F6(\017)p Black 46 w Gg(A)f(term)h Ga(M)33 b F7(intr)l(oduces)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21083
b Gg(the)c(co-name)i Ga(c)p Gg(,)e(if)g(and)h(only)h(if)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21084
Ga(M)33 b Gg(is)23 b(of)g(the)h(form:)p Black Black 576
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21085
3965 a FL(Id)q F4(\()p Ga(x;)15 b(c)p F4(\))p 899 3998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21086
4 115 v 101 w FL(And)1105 3979 y Gc(I)1145 3965 y F4(\()1180
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21087
3953 y FX(h)1208 3965 y Ga(a)1256 3953 y FX(i)1283 3965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21088
y Ga(S)5 b(;)1384 3953 y FX(h)1412 3965 y Ga(b)1451 3953
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21089
y FX(i)1479 3965 y Ga(T)12 b(;)j(c)p F4(\))p 1708 3998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21090
V 101 w FL(Or)1858 3928 y Gc(i)1858 3988 y(I)1898 3965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21091
y F4(\()1933 3953 y FX(h)1961 3965 y Ga(a)2009 3953 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21092
FX(i)2037 3965 y Ga(S)5 b(;)15 b(c)p F4(\))p 2261 3998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21093
V 100 w FL(Imp)2457 3986 y Gc(I)2497 3965 y F4(\()2532
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21094
3953 y F9(\()2560 3965 y Ga(x)2612 3953 y F9(\))p FX(h)2666
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21095
3965 y Ga(a)2714 3953 y FX(i)2742 3965 y Ga(S)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21096
b(c)p F4(\))p 2966 3998 V 101 w FL(Not)3160 3979 y Gc(I)3200
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21097
3965 y F4(\()3235 3953 y F9(\()3263 3965 y Ga(x)3315
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21098
3953 y F9(\))3342 3965 y Ga(S)5 b(;)15 b(c)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21099
Black 458 4160 a F6(\017)p Black 46 w Gg(A)23 b(term)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21100
Ga(M)34 b F7(fr)m(eshly)26 b Gg(introduces)i(a)c(co-name,)i(if)f(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21101
(only)h(if)e(none)i(of)f(its)g(proper)h(subterms)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21102
4273 y(introduces)g(this)e(co-name.)p Black 458 4451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21103
a F6(\017)p Black 46 w Gg(A)e(term)h Ga(M)33 b F7(eliminates)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21104
b Gg(the)d(co-name)i Ga(c)p Gg(,)e(if)g(and)h(only)h(if)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21105
Ga(M)33 b Gg(is)23 b(of)g(the)h(form:)p Black Black 390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21106
4648 a FL(And)545 4611 y Gc(i)545 4671 y(E)604 4648 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21107
F4(\()639 4636 y FX(h)667 4648 y Ga(c)706 4636 y FX(i)734
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21108
4648 y Ga(S)5 b(;)835 4636 y F9(\()863 4648 y Ga(x)915
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21109
4636 y F9(\))942 4648 y Ga(T)13 b F4(\))p 1076 4681 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21110
69 w FL(Or)1212 4662 y Gc(E)1271 4648 y F4(\()1306 4636
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21111
y FX(h)1334 4648 y Ga(c)1373 4636 y FX(i)1401 4648 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21112
Ga(S)5 b(;)1502 4636 y F9(\()1530 4648 y Ga(x)1582 4636
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21113
y F9(\))1609 4648 y Ga(T)13 b(;)1715 4636 y F9(\()1743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21114
4648 y Ga(y)1791 4636 y F9(\))1818 4648 y Ga(P)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21115
1957 4681 V 70 w FL(Imp)2138 4669 y Gc(E)2198 4648 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21116
F4(\()2233 4636 y FX(h)2260 4648 y Ga(c)2299 4636 y FX(i)2327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21117
4648 y Ga(S)5 b(;)2428 4636 y FX(h)2456 4648 y Ga(b)2495
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21118
4636 y FX(i)2523 4648 y Ga(T)12 b(;)2628 4636 y F9(\()2656
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21119
4648 y Ga(x)2708 4636 y F9(\))2736 4648 y Ga(P)h F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21120
2875 4681 V 69 w FL(Not)3054 4662 y Gc(E)3114 4648 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21121
F4(\()3149 4636 y FX(h)3176 4648 y Ga(c)3215 4636 y FX(i)3243
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21122
4648 y Ga(S)5 b(;)3344 4636 y FX(h)3372 4648 y Ga(b)3411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21123
4636 y FX(i)3438 4648 y Ga(T)13 b F4(\))462 4910 y Gg(Ne)o(xt)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21124
b(we)f(shall)i(gi)n(v)o(e)f(the)g(translation)j F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21125
1765 4910 28 4 v 1783 4910 V 1801 4910 V 65 w(j)1853
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21126
4877 y Fu(N)1907 4910 y Gg(,)e(which)f(maps)g(terms)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21127
FY(T)f Gg(to)h(terms)g(of)g F4(K)p Gg(,)g(and)321 5023
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21128
y(the)24 b(translation)i F6(j)p 888 5023 V 906 5023 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21129
923 5023 V 64 w(j)975 4990 y Fu(S)1041 5023 y Gg(for)e(the)f(other)h(w)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21130
o(ay)f(around,)h(that)g(is)f(from)g(terms)g(of)g F4(K)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21131
Gg(to)h(terms)g(of)g FY(T)t Gg(.)h(The)o(y)321 5136 y(are)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21132
b(gi)n(v)o(en)f(in)g(Figures)h(3.8)f(and)g(3.9,)i(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21133
b(.)57 b(An)32 b(important)h(property)i(of)d(these)h(transla-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21134
5249 y(tions)f(is)e(that)h(the)o(y)g(respect)h(typing)g(judgements.)53
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21135
b(The)30 b(proof)h(of)g(this)g(property)i(coincides)g(with)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21136
5361 y(sho)n(wing)28 b(the)g(soundness)i(and)d(completeness)j(of)d(our)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21137
g(natural)i(deduction)h(calculus)f(with)e(respect)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21138
5474 y(to)d(pro)o(v)n(ability)i(in)d(classical)j(logic.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21139
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21140
%%Page: 85 97
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21141
TeXDict begin 85 96 bop Black 277 51 a Gb(3.3)23 b(Classical)j(Natural)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21142
d(Deduction)2373 b(85)p 277 88 3691 4 v Black Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21143
1280 V 277 4182 4 2902 v 1599 1372 1024 4 v 1599 1457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21144
a Ga(x)18 b F4(:)f Ga(B)5 b(;)15 b F4(\000)1907 1445
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21145
y Gc(.)1962 1457 y FL(Id)p F4(\()p Ga(x;)g(a)p F4(\))2271
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21146
1445 y Gc(.)2325 1457 y F4(\001)p Ga(;)g(a)j F4(:)f Ga(B)574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21147
1707 y F4(\000)656 1695 y Gc(.)711 1707 y Ga(M)834 1695
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21148
y Gc(.)889 1707 y F4(\001)p Ga(;)e(a)i F4(:)h Ga(B)186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21149
b F4(\000)1450 1695 y Gc(.)1505 1707 y Ga(N)1613 1695
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21150
y Gc(.)1668 1707 y F4(\001)p Ga(;)15 b(b)j F4(:)f Ga(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21151
564 1745 1402 4 v 564 1830 a F4(\000)646 1818 y Gc(.)701
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21152
1830 y FL(And)855 1844 y Gc(I)895 1830 y F4(\()930 1818
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21153
y FX(h)958 1830 y Ga(a)1006 1818 y FX(i)1033 1830 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21154
b(;)1171 1818 y FX(h)1199 1830 y Ga(b)1238 1818 y FX(i)1266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21155
1830 y Ga(N)g(;)15 b(c)p F4(\))1489 1818 y Gc(.)1544
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21156
1830 y F4(\001)p Ga(;)g(c)j F4(:)f Ga(B)5 b F6(^)o Ga(C)2006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21157
1763 y F6(^)2067 1777 y Gc(I)2577 1695 y F4(\000)2660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21158
1683 y Gc(.)2714 1695 y Ga(M)2838 1683 y Gc(.)2893 1695
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21159
y F4(\001)p Ga(;)15 b(a)i F4(:)h Ga(B)3186 1709 y Gc(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21160
2296 1732 1200 4 v 2296 1830 a F4(\000)2378 1818 y Gc(.)2433
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21161
1830 y FL(Or)2532 1793 y Gc(i)2532 1853 y(I)2572 1830
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21162
y F4(\()2607 1818 y FX(h)2635 1830 y Ga(a)2683 1818 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21163
FX(i)2710 1830 y Ga(M)10 b(;)15 b(b)p F4(\))2948 1818
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21164
y Gc(.)3003 1830 y F4(\001)p Ga(;)g(b)j F4(:)f Ga(B)3287
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21165
1844 y F9(1)3326 1830 y F6(_)p Ga(B)3456 1844 y F9(2)3537
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21166
1745 y F6(_)3598 1759 y Gc(I)3629 1769 y FZ(i)841 2079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21167
y Ga(x)g F4(:)h Ga(B)5 b(;)15 b F4(\000)1149 2067 y Gc(.)1204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21168
2079 y Ga(M)1327 2067 y Gc(.)1382 2079 y F4(\001)p Ga(;)g(a)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21169
F4(:)h Ga(C)p 614 2117 1290 4 v 614 2202 a F4(\000)696
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21170
2190 y Gc(.)751 2202 y FL(Imp)896 2223 y Gc(I)935 2202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21171
y F4(\()970 2190 y F9(\()998 2202 y Ga(x)1050 2190 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21172
F9(\))q FX(h)1105 2202 y Ga(a)1153 2190 y FX(i)1181 2202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21173
y Ga(M)10 b(;)15 b(b)p F4(\))1419 2190 y Gc(.)1474 2202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21174
y F4(\001)p Ga(;)g(b)i F4(:)g Ga(B)5 b F6(\033)p Ga(C)1946
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21175
2134 y F6(\033)2016 2148 y Gc(I)2598 2079 y Ga(x)17 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21176
F4(:)h Ga(B)5 b(;)15 b F4(\000)2905 2067 y Gc(.)2960
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21177
2079 y Ga(M)3084 2067 y Gc(.)3139 2079 y F4(\001)p 2345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21178
2117 1122 4 v 2345 2202 a(\000)2427 2190 y Gc(.)2482
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21179
2202 y FL(Not)2625 2216 y Gc(I)2665 2202 y F4(\()2700
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21180
2190 y F9(\()2728 2202 y Ga(x)2780 2190 y F9(\))2807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21181
2202 y Ga(M)10 b(;)15 b(a)p F4(\))3054 2190 y Gc(.)3109
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21182
2202 y F4(\001)p Ga(;)g(a)i F4(:)h F6(:)p Ga(B)3509 2129
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21183
y F6(:)3570 2143 y Gc(I)1207 2451 y F4(\000)1289 2439
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21184
y Gc(.)1344 2451 y Ga(M)1468 2439 y Gc(.)1523 2451 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21185
F4(\001)p Ga(;)d(a)i F4(:)g Ga(B)1815 2465 y F9(1)1855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21186
2451 y F6(^)o Ga(B)1984 2465 y F9(2)2206 2451 y Ga(x)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21187
F4(:)g Ga(B)2386 2465 y Gc(i)2414 2451 y Ga(;)e F4(\000)2537
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21188
2439 y Gc(.)2592 2451 y Ga(N)2700 2439 y Gc(.)2755 2451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21189
y F4(\001)p 1207 2489 1624 4 v 1514 2586 a(\000)1597
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21190
2574 y Gc(.)1651 2586 y FL(And)1806 2549 y Gc(i)1806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21191
2609 y(E)1866 2586 y F4(\()1901 2574 y FX(h)1928 2586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21192
y Ga(a)1976 2574 y FX(i)2004 2586 y Ga(M)10 b(;)2142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21193
2574 y F9(\()2170 2586 y Ga(x)2222 2574 y F9(\))2249
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21194
2586 y Ga(N)g F4(\))2393 2574 y Gc(.)2448 2586 y F4(\001)2872
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21195
2502 y F6(^)2933 2516 y Gc(E)2985 2526 y FZ(i)886 2836
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21196
y F4(\000)968 2824 y Gc(.)1023 2836 y Ga(M)1147 2824
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21197
y Gc(.)1202 2836 y F4(\001)p Ga(;)15 b(a)i F4(:)g Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21198
b F6(_)p Ga(C)188 b(x)17 b F4(:)g Ga(B)5 b(;)15 b F4(\000)2120
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21199
2824 y Gc(.)2175 2836 y Ga(N)2284 2824 y Gc(.)2338 2836
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21200
y F4(\001)182 b Ga(y)20 b F4(:)d Ga(C)q(;)e F4(\000)2892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21201
2824 y Gc(.)2947 2836 y Ga(P)3043 2824 y Gc(.)3098 2836
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21202
y F4(\001)p 886 2873 2288 4 v 1446 2958 a(\000)1528 2946
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21203
y Gc(.)1583 2958 y FL(Or)1683 2972 y Gc(E)1742 2958 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21204
F4(\()1777 2946 y FX(h)1805 2958 y Ga(a)1853 2946 y FX(i)1880
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21205
2958 y Ga(M)c(;)2019 2946 y F9(\()2046 2958 y Ga(x)2098
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21206
2946 y F9(\))2126 2958 y Ga(N)f(;)2249 2946 y F9(\()2277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21207
2958 y Ga(y)2325 2946 y F9(\))2352 2958 y Ga(P)j F4(\))2484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21208
2946 y Gc(.)2538 2958 y F4(\001)3216 2891 y F6(_)3276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21209
2905 y Gc(E)881 3208 y F4(\000)963 3196 y Gc(.)1018 3208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21210
y Ga(M)1141 3196 y Gc(.)1196 3208 y F4(\001)p Ga(;)i(a)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21211
F4(:)h Ga(B)5 b F6(\033)o Ga(C)188 b F4(\000)1899 3196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21212
y Gc(.)1954 3208 y Ga(N)2062 3196 y Gc(.)2117 3208 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21213
F4(\001)p Ga(;)15 b(b)j F4(:)f Ga(B)186 b(x)18 b F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21214
Ga(C)q(;)e F4(\000)2888 3196 y Gc(.)2943 3208 y Ga(P)3039
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21215
3196 y Gc(.)3094 3208 y F4(\001)p 881 3246 2290 4 v 1423
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21216
3331 a(\000)1505 3319 y Gc(.)1560 3331 y FL(Imp)1705
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21217
3353 y Gc(E)1764 3331 y F4(\()1799 3319 y FX(h)1827 3331
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21218
y Ga(a)1875 3319 y FX(i)1902 3331 y Ga(M)c(;)2041 3319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21219
y FX(h)2068 3331 y Ga(b)2107 3319 y FX(i)2135 3331 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21220
Ga(N)f(;)2258 3319 y F9(\()2286 3331 y Ga(x)2338 3319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21221
y F9(\))2365 3331 y Ga(P)j F4(\))2497 3319 y Gc(.)2552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21222
3331 y F4(\001)3211 3264 y F6(\033)3282 3278 y Gc(E)1309
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21223
3581 y F4(\000)1391 3569 y Gc(.)1446 3581 y Ga(M)1569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21224
3569 y Gc(.)1624 3581 y F4(\001)p Ga(;)i(a)i F4(:)h F6(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21225
Ga(B)186 b F4(\000)2246 3569 y Gc(.)2301 3581 y Ga(N)2409
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21226
3569 y Gc(.)2464 3581 y F4(\001)p Ga(;)15 b(b)i F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21227
Ga(B)p 1309 3619 1444 4 v 1538 3704 a F4(\000)1620 3692
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21228
y Gc(.)1675 3704 y FL(Not)1818 3718 y Gc(E)1877 3704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21229
y F4(\()1912 3692 y FX(h)1940 3704 y Ga(a)1988 3692 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21230
FX(i)2016 3704 y Ga(M)10 b(;)2154 3692 y FX(h)2182 3704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21231
y Ga(b)2221 3692 y FX(i)2248 3704 y Ga(N)g F4(\))2392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21232
3692 y Gc(.)2447 3704 y F4(\001)2793 3632 y F6(:)2854
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21233
3646 y Gc(E)1213 3953 y F4(\000)1270 3967 y F9(1)1334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21234
3941 y Gc(.)1389 3953 y Ga(M)1512 3941 y Gc(.)1567 3953
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21235
y F4(\001)1643 3967 y F9(1)1682 3953 y Ga(;)15 b(a)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21236
F4(:)g Ga(B)186 b(x)17 b F4(:)g Ga(B)5 b(;)15 b F4(\000)2368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21237
3967 y F9(2)2433 3941 y Gc(.)2488 3953 y Ga(N)2596 3941
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21238
y Gc(.)2651 3953 y F4(\001)2727 3967 y F9(2)p 1213 3991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21239
1554 4 v 1300 4076 a F4(\000)1357 4090 y F9(1)1397 4076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21240
y Ga(;)g F4(\000)1494 4090 y F9(2)1558 4064 y Gc(.)1613
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21241
4076 y FL(Subst)o F4(\()1860 4064 y FX(h)1888 4076 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21242
Ga(a)1936 4064 y FX(i)1964 4076 y Ga(M)10 b(;)2102 4064
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21243
y F9(\()2130 4076 y Ga(x)2182 4064 y F9(\))2209 4076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21244
y Ga(N)g F4(\))2353 4064 y Gc(.)2408 4076 y F4(\001)2484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21245
4090 y F9(1)2523 4076 y Ga(;)15 b F4(\001)2639 4090 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21246
F9(2)2808 4021 y Gg(Subst)p 3965 4182 4 2902 v 277 4185
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21247
3691 4 v Black 668 4338 a(Figure)24 b(3.7:)29 b(T)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21248
b(erm)23 b(assignment)i(for)f(classical)i(natural)f(deduction)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21249
(proofs.)p Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21250
%%Page: 86 98
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21251
TeXDict begin 86 97 bop Black -144 51 a Gb(86)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21252
b(Deduction)p -144 88 3691 4 v Black Black -144 1278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21253
V -144 4183 4 2905 v -41 1457 a Gg(\(1\))1043 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21254
FL(Id)p F4(\()p Ga(x;)15 b(a)p F4(\))p F6(j)1439 1424
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21255
y Fu(S)1584 1405 y F5(def)1591 1457 y F4(=)106 b FL(Ax)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21256
F4(\()p Ga(x;)15 b(a)p F4(\))-41 1606 y Gg(\(2\))612
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21257
b F6(j)p FL(Subst)o F4(\()948 1594 y FX(h)976 1606 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21258
Ga(a)1024 1594 y FX(i)1052 1606 y Ga(M)10 b(;)1190 1594
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21259
y F9(\()1218 1606 y Ga(x)1270 1594 y F9(\))1297 1606
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21260
y Ga(N)g F4(\))p F6(j)1440 1573 y Fu(S)1584 1555 y F5(def)1591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21261
1606 y F4(=)106 b FL(Cut)p F4(\()1941 1594 y FX(h)1969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21262
1606 y Ga(a)2017 1594 y FX(i)2059 1606 y F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21263
b F6(j)2207 1573 y Fu(S)2251 1606 y Ga(;)2291 1594 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21264
F9(\()2319 1606 y Ga(x)2371 1594 y F9(\))2414 1606 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21265
F6(j)p Ga(N)g F6(j)2547 1573 y Fu(S)2591 1606 y F4(\))-41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21266
1804 y Gg(\(3\))563 b F6(j)p FL(And)807 1818 y Gc(I)847
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21267
1804 y F4(\()882 1792 y FX(h)909 1804 y Ga(a)957 1792
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21268
y FX(i)985 1804 y Ga(M)10 b(;)1123 1792 y FX(h)1151 1804
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21269
y Ga(b)1190 1792 y FX(i)1217 1804 y Ga(N)g(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21270
F4(\))p F6(j)1439 1771 y Fu(S)1584 1752 y F5(def)1591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21271
1804 y F4(=)106 b FL(And)1923 1818 y Gc(R)1980 1804 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21272
F4(\()2015 1792 y FX(h)2043 1804 y Ga(a)2091 1792 y FX(i)2134
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21273
1804 y F6(j)p Ga(M)10 b F6(j)2282 1771 y Fu(S)2326 1804
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21274
y Ga(;)2366 1792 y FX(h)2394 1804 y Ga(b)2433 1792 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21275
FX(i)2475 1804 y F6(j)p Ga(N)g F6(j)2608 1771 y Fu(S)2652
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21276
1804 y Ga(;)15 b(c)p F4(\))-41 1954 y Gg(\(4\))836 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21277
F6(j)p FL(Or)1024 1917 y Gc(i)1024 1977 y(I)1064 1954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21278
y F4(\()1099 1942 y FX(h)1127 1954 y Ga(a)1175 1942 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21279
FX(i)1202 1954 y Ga(M)11 b(;)k(b)p F4(\))p F6(j)1440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21280
1921 y Fu(S)1584 1902 y F5(def)1591 1954 y F4(=)106 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21281
FL(Or)1867 1917 y Gc(i)1867 1977 y(R)1925 1954 y F4(\()1960
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21282
1942 y FX(h)1988 1954 y Ga(a)2036 1942 y FX(i)2078 1954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21283
y F6(j)p Ga(M)10 b F6(j)2226 1921 y Fu(S)2270 1954 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21284
Ga(;)15 b(b)p F4(\))-41 2104 y Gg(\(5\))684 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21285
FL(Imp)917 2125 y Gc(I)957 2104 y F4(\()992 2092 y F9(\()1020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21286
2104 y Ga(x)1072 2092 y F9(\))q FX(h)1127 2104 y Ga(a)1175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21287
2092 y FX(i)1202 2104 y Ga(M)11 b(;)k(b)p F4(\))p F6(j)1440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21288
2071 y Fu(S)1584 2052 y F5(def)1591 2104 y F4(=)106 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21289
FL(Imp)1912 2125 y Gc(R)1970 2104 y F4(\()2005 2092 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21290
F9(\()2033 2104 y Ga(x)2085 2092 y F9(\))p FX(h)2140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21291
2104 y Ga(a)2188 2092 y FX(i)2230 2104 y F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21292
b F6(j)2378 2071 y Fu(S)2422 2104 y Ga(;)15 b(b)p F4(\))-41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21293
2254 y Gg(\(6\))779 b F6(j)p FL(Not)1011 2268 y Gc(I)1051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21294
2254 y F4(\()1086 2242 y F9(\()1114 2254 y Ga(x)1166
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21295
2242 y F9(\))1193 2254 y Ga(M)11 b(;)k(a)p F4(\))p F6(j)1440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21296
2221 y Fu(S)1584 2202 y F5(def)1591 2254 y F4(=)106 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21297
FL(Not)1911 2268 y Gc(R)1968 2254 y F4(\()2003 2242 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21298
F9(\()2031 2254 y Ga(x)2083 2242 y F9(\))2126 2254 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21299
F6(j)p Ga(M)10 b F6(j)2274 2221 y Fu(S)2318 2254 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21300
b(a)p F4(\))-41 2451 y Gg(\(7\))430 b F6(j)p FL(And)674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21301
2414 y Gc(i)674 2474 y(E)734 2451 y F4(\()769 2439 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21302
FX(h)797 2451 y Ga(a)845 2439 y FX(i)872 2451 y FL(Id)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21303
F4(\()p Ga(z)t(;)15 b(a)p F4(\))r Ga(;)1190 2439 y F9(\()1218
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21304
2451 y Ga(x)1270 2439 y F9(\))1297 2451 y Ga(N)10 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21305
F6(j)1440 2418 y Fu(S)1584 2399 y F5(def)1591 2451 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21306
F4(=)106 b FL(And)1923 2414 y Gc(i)1923 2474 y(L)1975
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21307
2451 y F4(\()2010 2439 y F9(\()2038 2451 y Ga(x)2090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21308
2439 y F9(\))2132 2451 y F6(j)p Ga(N)10 b F6(j)2265 2418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21309
y Fu(S)2309 2451 y Ga(;)15 b(z)t F4(\))-41 2601 y Gg(\(8\))272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21310
b F6(j)p FL(Or)460 2615 y Gc(E)520 2601 y F4(\()555 2589
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21311
y FX(h)583 2601 y Ga(a)631 2589 y FX(i)658 2601 y FL(Id)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21312
F4(\()p Ga(z)t(;)15 b(a)p F4(\))r Ga(;)976 2589 y F9(\()1004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21313
2601 y Ga(x)1056 2589 y F9(\))1083 2601 y Ga(N)10 b(;)1206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21314
2589 y F9(\()1234 2601 y Ga(y)1282 2589 y F9(\))1309
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21315
2601 y Ga(P)j F4(\))p F6(j)1440 2568 y Fu(S)1584 2549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21316
y F5(def)1591 2601 y F4(=)106 b FL(Or)1867 2615 y Gc(L)1919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21317
2601 y F4(\()1954 2589 y F9(\()1982 2601 y Ga(x)2034
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21318
2589 y F9(\))2077 2601 y F6(j)p Ga(N)10 b F6(j)2210 2568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21319
y Fu(S)2254 2601 y Ga(;)2294 2589 y F9(\()2321 2601 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21320
Ga(y)2369 2589 y F9(\))2412 2601 y F6(j)p Ga(P)j F6(j)2533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21321
2568 y Fu(S)2577 2601 y Ga(;)i(z)t F4(\))-41 2751 y Gg(\(9\))235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21322
b F6(j)p FL(Imp)469 2772 y Gc(E)529 2751 y F4(\()564
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21323
2739 y FX(h)591 2751 y Ga(a)639 2739 y FX(i)667 2751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21324
y FL(Id)p F4(\()p Ga(z)t(;)15 b(a)p F4(\))q Ga(;)984
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21325
2739 y FX(h)1012 2751 y Ga(b)1051 2739 y FX(i)1079 2751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21326
y Ga(N)10 b(;)1202 2739 y F9(\()1230 2751 y Ga(x)1282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21327
2739 y F9(\))1309 2751 y Ga(P)j F4(\))p F6(j)1440 2718
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21328
y Fu(S)1584 2699 y F5(def)1591 2751 y F4(=)106 b FL(Imp)1912
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21329
2772 y Gc(L)1965 2751 y F4(\()2000 2739 y FX(h)2027 2751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21330
y Ga(b)2066 2739 y FX(i)2109 2751 y F6(j)p Ga(N)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21331
F6(j)2242 2718 y Fu(S)2286 2751 y Ga(;)2326 2739 y F9(\()2354
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21332
2751 y Ga(x)2406 2739 y F9(\))2448 2751 y F6(j)p Ga(P)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21333
F6(j)2569 2718 y Fu(S)2613 2751 y Ga(;)i(z)t F4(\))-41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21334
2901 y Gg(\(10\))410 b F6(j)p FL(Not)687 2915 y Gc(E)747
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21335
2901 y F4(\()782 2889 y FX(h)810 2901 y Ga(a)858 2889
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21336
y FX(i)885 2901 y FL(Id)p F4(\()p Ga(z)t(;)15 b(a)p F4(\))r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21337
Ga(;)1203 2889 y FX(h)1231 2901 y Ga(b)1270 2889 y FX(i)1297
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21338
2901 y Ga(N)10 b F4(\))p F6(j)1440 2868 y Fu(S)1584 2849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21339
y F5(def)1591 2901 y F4(=)106 b FL(Not)1911 2915 y Gc(L)1963
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21340
2901 y F4(\()1998 2889 y FX(h)2026 2901 y Ga(b)2065 2889
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21341
y FX(i)2107 2901 y F6(j)p Ga(N)10 b F6(j)2240 2868 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21342
Fu(S)2284 2901 y Ga(;)15 b(z)t F4(\))-41 3095 y Gb(Otherwise:)-41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21343
3221 y Gg(\(11\))565 b F6(j)p FL(And)854 3184 y Gc(i)854
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21344
3244 y(E)913 3221 y F4(\()948 3209 y FX(h)976 3221 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21345
Ga(a)1024 3209 y FX(i)1052 3221 y Ga(M)10 b(;)1190 3209
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21346
y F9(\()1218 3221 y Ga(x)1270 3209 y F9(\))1297 3221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21347
y Ga(N)g F4(\))p F6(j)1440 3188 y Fu(S)1584 3169 y F5(def)1591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21348
3221 y F4(=)106 b FL(Cut)p F4(\()1941 3209 y FX(h)1969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21349
3221 y Ga(a)2017 3209 y FX(i)2059 3221 y F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21350
b F6(j)2207 3188 y Fu(S)2251 3221 y Ga(;)2291 3209 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21351
F9(\()2319 3221 y Ga(y)2367 3209 y F9(\))2394 3221 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21352
FL(And)2549 3184 y Gc(i)2549 3244 y(L)2601 3221 y F4(\()2636
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21353
3209 y F9(\()2664 3221 y Ga(x)2716 3209 y F9(\))2758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21354
3221 y F6(j)p Ga(N)g F6(j)2891 3188 y Fu(S)2935 3221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21355
y Ga(;)15 b(y)s F4(\))q(\))-41 3371 y Gg(\(12\))406 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21356
F6(j)p FL(Or)640 3385 y Gc(E)699 3371 y F4(\()734 3359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21357
y FX(h)762 3371 y Ga(a)810 3359 y FX(i)838 3371 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21358
b(;)976 3359 y F9(\()1004 3371 y Ga(x)1056 3359 y F9(\))1083
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21359
3371 y Ga(N)g(;)1206 3359 y F9(\()1234 3371 y Ga(y)1282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21360
3359 y F9(\))1309 3371 y Ga(P)j F4(\))p F6(j)1440 3338
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21361
y Fu(S)1584 3319 y F5(def)1591 3371 y F4(=)106 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21362
F4(\()1941 3359 y FX(h)1969 3371 y Ga(a)2017 3359 y FX(i)2059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21363
3371 y F6(j)p Ga(M)10 b F6(j)2207 3338 y Fu(S)2251 3371
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21364
y Ga(;)2291 3359 y F9(\()2319 3371 y Ga(z)2365 3359 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21365
F9(\))2393 3371 y FL(Or)2492 3385 y Gc(L)2544 3371 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21366
F4(\()2579 3359 y F9(\()2607 3371 y Ga(x)2659 3359 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21367
F9(\))2702 3371 y F6(j)p Ga(N)g F6(j)2835 3338 y Fu(S)2878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21368
3371 y Ga(;)2918 3359 y F9(\()2946 3371 y Ga(y)2994 3359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21369
y F9(\))3037 3371 y F6(j)p Ga(P)j F6(j)3158 3338 y Fu(S)3202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21370
3371 y Ga(;)i(z)t F4(\))q(\))-41 3521 y Gg(\(13\))370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21371
b F6(j)p FL(Imp)648 3543 y Gc(E)708 3521 y F4(\()743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21372
3509 y FX(h)771 3521 y Ga(a)819 3509 y FX(i)846 3521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21373
y Ga(M)10 b(;)984 3509 y FX(h)1012 3521 y Ga(b)1051 3509
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21374
y FX(i)1079 3521 y Ga(N)g(;)1202 3509 y F9(\()1230 3521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21375
y Ga(x)1282 3509 y F9(\))1309 3521 y Ga(P)j F4(\))p F6(j)1440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21376
3488 y Fu(S)1584 3469 y F5(def)1591 3521 y F4(=)106 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21377
FL(Cut)p F4(\()1941 3509 y FX(h)1969 3521 y Ga(a)2017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21378
3509 y FX(i)2059 3521 y F6(j)p Ga(M)10 b F6(j)2207 3488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21379
y Fu(S)2251 3521 y Ga(;)2291 3509 y F9(\()2319 3521 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21380
Ga(y)2367 3509 y F9(\))2394 3521 y FL(Imp)2539 3543 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21381
Gc(L)2591 3521 y F4(\()2626 3509 y FX(h)2654 3521 y Ga(b)2693
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21382
3509 y FX(i)2735 3521 y F6(j)p Ga(N)g F6(j)2868 3488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21383
y Fu(S)2912 3521 y Ga(;)2952 3509 y F9(\()2980 3521 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21384
Ga(x)3032 3509 y F9(\))3075 3521 y F6(j)p Ga(P)j F6(j)3196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21385
3488 y Fu(S)3239 3521 y Ga(;)i(y)s F4(\))q(\))-41 3671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21386
y Gg(\(14\))590 b F6(j)p FL(Not)867 3685 y Gc(E)926 3671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21387
y F4(\()961 3659 y FX(h)989 3671 y Ga(a)1037 3659 y FX(i)1065
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21388
3671 y Ga(M)10 b(;)1203 3659 y FX(h)1231 3671 y Ga(b)1270
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21389
3659 y FX(i)1297 3671 y Ga(N)g F4(\))p F6(j)1440 3638
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21390
y Fu(S)1584 3619 y F5(def)1591 3671 y F4(=)106 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21391
F4(\()1941 3659 y FX(h)1969 3671 y Ga(a)2017 3659 y FX(i)2059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21392
3671 y F6(j)p Ga(M)10 b F6(j)2207 3638 y Fu(S)2251 3671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21393
y Ga(;)2291 3659 y F9(\()2319 3671 y Ga(y)2367 3659 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21394
F9(\))2394 3671 y FL(Not)2537 3685 y Gc(L)2589 3671 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21395
F4(\()2624 3659 y FX(h)2652 3671 y Ga(b)2691 3659 y FX(i)2734
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21396
3671 y F6(j)p Ga(N)g F6(j)2867 3638 y Fu(S)2910 3671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21397
y Ga(;)15 b(y)s F4(\))q(\))268 3912 y Gg(where)24 b(in)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21398
b(\(11\))h Ga(y)28 b F6(62)c Ga(F)13 b(N)d F4(\()1177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21399
3900 y F9(\()1205 3912 y Ga(x)1257 3900 y F9(\))1285
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21400
3912 y F6(j)p Ga(N)g F6(j)1418 3879 y Fu(S)1462 3912
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21401
y F4(\))464 b Gg(\(12\))49 b Ga(z)30 b F6(62)24 b Ga(F)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21402
b(N)d F4(\()2506 3900 y F9(\()2534 3912 y Ga(x)2586 3900
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21403
y F9(\))2614 3912 y F6(j)p Ga(N)g F6(j)2747 3879 y Fu(S)2791
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21404
3912 y Ga(;)2831 3900 y F9(\()2858 3912 y Ga(y)2906 3900
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21405
y F9(\))2934 3912 y F6(j)p Ga(P)j F6(j)3055 3879 y Fu(S)3099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21406
3912 y F4(\))631 4059 y Gg(\(13\))49 b Ga(y)28 b F6(62)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21407
Ga(F)13 b(N)d F4(\()1177 4047 y FX(h)1205 4059 y Ga(b)1244
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21408
4047 y FX(i)1272 4059 y F6(j)p Ga(N)g F6(j)1405 4026
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21409
y Fu(S)1449 4059 y Ga(;)1489 4047 y F9(\()1516 4059 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21410
Ga(x)1568 4047 y F9(\))1596 4059 y F6(j)p Ga(P)j F6(j)1717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21411
4026 y Fu(S)1761 4059 y F4(\))165 b Gg(\(14\))49 b Ga(y)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21412
b F6(62)d Ga(F)13 b(N)d F4(\()2508 4047 y FX(h)2536 4059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21413
y Ga(b)2575 4047 y FX(i)2602 4059 y F6(j)p Ga(N)g F6(j)2735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21414
4026 y Fu(S)2779 4059 y F4(\))p 3543 4183 V -144 4186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21415
3691 4 v Black 624 4340 a Gg(Figure)24 b(3.8:)29 b(T)m(ranslation)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21416
(from)f(natural)h(deduction)h(proofs)f(to)f(sequent)h(proofs.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21417
Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21418
%%Page: 87 99
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21419
TeXDict begin 87 98 bop Black 277 51 a Gb(3.3)23 b(Classical)j(Natural)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21420
d(Deduction)2373 b(87)p 277 88 3691 4 v Black Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21421
1275 V 277 4186 4 2911 v 442 1454 a Gg(\(1\))1231 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21422
FL(Ax)o F4(\()p Ga(x;)15 b(a)p F4(\))p F6(j)2140 1421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21423
y Fu(N)2295 1402 y F5(def)2302 1454 y F4(=)107 b FL(Id)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21424
F4(\()p Ga(x;)15 b(a)p F4(\))442 1641 y Gg(\(2\))763
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21425
b F6(j)p FL(And)1490 1655 y Gc(R)1548 1641 y F4(\()1583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21426
1629 y FX(h)1611 1641 y Ga(a)1659 1629 y FX(i)1686 1641
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21427
y Ga(M)10 b(;)1824 1629 y FX(h)1852 1641 y Ga(b)1891
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21428
1629 y FX(i)1919 1641 y Ga(N)f(;)15 b(c)p F4(\))p F6(j)2140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21429
1608 y Fu(N)2295 1590 y F5(def)2302 1641 y F4(=)107 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21430
FL(And)2634 1655 y Gc(I)2674 1641 y F4(\()2709 1629 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21431
FX(h)2737 1641 y Ga(a)2785 1629 y FX(i)2828 1641 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21432
Ga(M)10 b F6(j)2976 1608 y Fu(N)3030 1641 y Ga(;)3070
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21433
1629 y FX(h)3098 1641 y Ga(b)3137 1629 y FX(i)3180 1641
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21434
y F6(j)p Ga(N)g F6(j)3313 1608 y Fu(N)3367 1641 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21435
b(c)p F4(\))442 1791 y Gg(\(3\))1036 b F6(j)p FL(Or)1708
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21436
1754 y Gc(i)1708 1814 y(R)1765 1791 y F4(\()1800 1779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21437
y FX(h)1828 1791 y Ga(a)1876 1779 y FX(i)1904 1791 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21438
Ga(M)10 b(;)15 b(b)p F4(\))p F6(j)2141 1758 y Fu(N)2295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21439
1740 y F5(def)2302 1791 y F4(=)107 b FL(Or)2579 1754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21440
y Gc(i)2579 1814 y(I)2619 1791 y F4(\()2654 1779 y FX(h)2682
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21441
1791 y Ga(a)2730 1779 y FX(i)2772 1791 y F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21442
b F6(j)2920 1758 y Fu(N)2975 1791 y Ga(;)15 b(b)p F4(\))442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21443
1941 y Gg(\(4\))884 b F6(j)p FL(Imp)1601 1963 y Gc(R)1659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21444
1941 y F4(\()1694 1929 y F9(\()1721 1941 y Ga(x)1773
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21445
1929 y F9(\))q FX(h)1828 1941 y Ga(a)1876 1929 y FX(i)1904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21446
1941 y Ga(M)10 b(;)15 b(b)p F4(\))p F6(j)2141 1908 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21447
Fu(N)2295 1890 y F5(def)2302 1941 y F4(=)107 b FL(Imp)2624
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21448
1963 y Gc(I)2664 1941 y F4(\()2699 1929 y F9(\()2727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21449
1941 y Ga(x)2779 1929 y F9(\))p FX(h)2834 1941 y Ga(a)2882
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21450
1929 y FX(i)2925 1941 y F6(j)p Ga(M)10 b F6(j)3073 1908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21451
y Fu(N)3127 1941 y Ga(;)15 b(b)p F4(\))442 2091 y Gg(\(5\))980
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21452
b F6(j)p FL(Not)1695 2105 y Gc(R)1752 2091 y F4(\()1787
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21453
2079 y F9(\()1815 2091 y Ga(x)1867 2079 y F9(\))1895
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21454
2091 y Ga(M)10 b(;)15 b(a)p F4(\))p F6(j)2141 2058 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21455
Fu(N)2295 2040 y F5(def)2302 2091 y F4(=)107 b FL(Not)2623
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21456
2105 y Gc(I)2662 2091 y F4(\()2697 2079 y F9(\()2725
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21457
2091 y Ga(x)2777 2079 y F9(\))2820 2091 y F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21458
b F6(j)2968 2058 y Fu(N)3022 2091 y Ga(;)15 b(a)p F4(\))442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21459
2288 y Gg(\(6\))974 b F6(j)p FL(And)1701 2252 y Gc(i)1701
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21460
2311 y(L)1753 2288 y F4(\()1788 2276 y F9(\()1815 2288
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21461
y Ga(x)1867 2276 y F9(\))1895 2288 y Ga(M)10 b(;)15 b(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21462
F4(\))p F6(j)2141 2255 y Fu(N)2295 2237 y F5(def)2302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21463
2288 y F4(=)107 b FL(And)2634 2252 y Gc(i)2634 2311 y(E)2694
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21464
2288 y F4(\()2729 2276 y FX(h)2757 2288 y Ga(a)2805 2276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21465
y FX(i)2832 2288 y FL(Id)q F4(\()p Ga(y)s(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21466
F4(\))q Ga(;)3152 2276 y F9(\()3179 2288 y Ga(x)3231
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21467
2276 y F9(\))3274 2288 y F6(j)p Ga(M)10 b F6(j)3422 2255
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21468
y Fu(N)3476 2288 y F4(\))442 2438 y Gg(\(7\))805 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21469
FL(Or)1476 2452 y Gc(L)1528 2438 y F4(\()1563 2426 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21470
F9(\()1591 2438 y Ga(x)1643 2426 y F9(\))1670 2438 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21471
Ga(M)10 b(;)1808 2426 y F9(\()1836 2438 y Ga(y)1884 2426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21472
y F9(\))1912 2438 y Ga(N)g(;)15 b(z)t F4(\))p F6(j)2141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21473
2405 y Fu(N)2295 2387 y F5(def)2302 2438 y F4(=)107 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21474
FL(Or)2579 2452 y Gc(E)2639 2438 y F4(\()2674 2426 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21475
FX(h)2701 2438 y Ga(a)2749 2426 y FX(i)2777 2438 y FL(Id)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21476
F4(\()p Ga(z)t(;)15 b(a)p F4(\))r Ga(;)3095 2426 y F9(\()3122
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21477
2438 y Ga(x)3174 2426 y F9(\))3217 2438 y F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21478
b F6(j)3365 2405 y Fu(N)3420 2438 y Ga(;)3460 2426 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21479
F9(\()3487 2438 y Ga(y)3535 2426 y F9(\))3578 2438 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21480
F6(j)p Ga(N)g F6(j)3711 2405 y Fu(N)3765 2438 y F4(\))442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21481
2588 y Gg(\(8\))758 b F6(j)p FL(Imp)1474 2610 y Gc(L)1526
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21482
2588 y F4(\()1561 2576 y FX(h)1589 2588 y Ga(a)1637 2576
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21483
y FX(i)1665 2588 y Ga(M)10 b(;)1803 2576 y F9(\()1831
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21484
2588 y Ga(x)1883 2576 y F9(\))1910 2588 y Ga(N)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21485
b(y)s F4(\))p F6(j)2141 2555 y Fu(N)2295 2537 y F5(def)2302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21486
2588 y F4(=)107 b FL(Imp)2624 2610 y Gc(E)2684 2588 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21487
F4(\()2719 2576 y FX(h)2747 2588 y Ga(b)2786 2576 y FX(i)2813
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21488
2588 y FL(Id)p F4(\()p Ga(y)s(;)15 b(b)p F4(\))q Ga(;)3123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21489
2576 y FX(h)3151 2588 y Ga(a)3199 2576 y FX(i)3242 2588
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21490
y F6(j)p Ga(M)10 b F6(j)3390 2555 y Fu(N)3444 2588 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21491
Ga(;)3484 2576 y F9(\()3512 2588 y Ga(x)3564 2576 y F9(\))3607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21492
2588 y F6(j)p Ga(N)g F6(j)3740 2555 y Fu(N)3794 2588
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21493
y F4(\))442 2738 y Gg(\(9\))985 b F6(j)p FL(Not)1700
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21494
2752 y Gc(L)1752 2738 y F4(\()1787 2726 y FX(h)1815 2738
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21495
y Ga(a)1863 2726 y FX(i)1891 2738 y Ga(M)10 b(;)15 b(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21496
F4(\))p F6(j)2141 2705 y Fu(N)2295 2687 y F5(def)2302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21497
2738 y F4(=)107 b FL(Not)2623 2752 y Gc(E)2682 2738 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21498
F4(\()2717 2726 y FX(h)2745 2738 y Ga(b)2784 2726 y FX(i)2811
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21499
2738 y FL(Id)q F4(\()p Ga(x;)15 b(b)p F4(\))q Ga(;)3126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21500
2726 y FX(h)3154 2738 y Ga(a)3202 2726 y FX(i)3244 2738
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21501
y F6(j)p Ga(M)10 b F6(j)3392 2705 y Fu(N)3447 2738 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21502
F4(\))442 2935 y Gg(\(10\))387 b F6(j)p FL(Cut)p F4(\()1177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21503
2923 y FX(h)1205 2935 y Ga(a)1253 2923 y FX(i)1280 2935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21504
y Ga(M)11 b(;)1419 2923 y F9(\()1446 2935 y Ga(x)1498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21505
2923 y F9(\))1526 2935 y FL(And)1680 2899 y Gc(i)1680
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21506
2959 y(L)1732 2935 y F4(\()1767 2923 y F9(\()1795 2935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21507
y Ga(y)1843 2923 y F9(\))1871 2935 y Ga(N)f(;)15 b(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21508
F4(\)\))p F6(j)2141 2902 y Fu(N)2295 2884 y F5(def)2302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21509
2935 y F4(=)107 b FL(And)2634 2899 y Gc(i)2634 2959 y(E)2694
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21510
2935 y F4(\()2729 2923 y FX(h)2757 2935 y Ga(a)2805 2923
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21511
y FX(i)2832 2935 y F6(j)p Ga(M)10 b F6(j)2980 2902 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21512
Fu(N)3035 2935 y Ga(;)3075 2923 y F9(\()3103 2935 y Ga(y)3151
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21513
2923 y F9(\))3178 2935 y F6(j)p Ga(N)g F6(j)3311 2902
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21514
y Fu(N)3365 2935 y F4(\))442 3085 y Gg(\(11\))230 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21515
FL(Cut)p F4(\()1020 3073 y FX(h)1048 3085 y Ga(a)1096
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21516
3073 y FX(i)1123 3085 y Ga(M)10 b(;)1261 3073 y F9(\()1289
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21517
3085 y Ga(x)1341 3073 y F9(\))1369 3085 y FL(Or)1468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21518
3099 y Gc(L)1520 3085 y F4(\()1555 3073 y F9(\()1583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21519
3085 y Ga(y)1631 3073 y F9(\))1658 3085 y Ga(N)g(;)1781
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21520
3073 y F9(\()1809 3085 y Ga(z)1855 3073 y F9(\))1883
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21521
3085 y Ga(P)j(;)i(x)p F4(\)\))p F6(j)2141 3052 y Fu(N)2295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21522
3034 y F5(def)2302 3085 y F4(=)107 b FL(Or)2579 3099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21523
y Gc(E)2639 3085 y F4(\()2674 3073 y FX(h)2701 3085 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21524
Ga(a)2749 3073 y FX(i)2777 3085 y F6(j)p Ga(M)10 b F6(j)2925
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21525
3052 y Fu(N)2980 3085 y Ga(;)3020 3073 y F9(\()3047 3085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21526
y Ga(y)3095 3073 y F9(\))3123 3085 y F6(j)p Ga(N)g F6(j)3256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21527
3052 y Fu(N)3310 3085 y Ga(;)3350 3073 y F9(\()3378 3085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21528
y Ga(z)3424 3073 y F9(\))3452 3085 y F6(j)p Ga(P)j F6(j)3573
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21529
3052 y Fu(N)3627 3085 y F4(\))442 3235 y Gg(\(12\))191
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21530
b F6(j)p FL(Cut)p F4(\()981 3223 y FX(h)1009 3235 y Ga(a)1057
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21531
3223 y FX(i)1084 3235 y Ga(M)10 b(;)1222 3223 y F9(\()1250
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21532
3235 y Ga(x)1302 3223 y F9(\))1330 3235 y FL(Imp)1474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21533
3257 y Gc(L)1526 3235 y F4(\()1561 3223 y FX(h)1589 3235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21534
y Ga(b)1628 3223 y FX(i)1656 3235 y Ga(Q)o(;)1767 3223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21535
y F9(\()1795 3235 y Ga(y)1843 3223 y F9(\))1871 3235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21536
y Ga(N)g(;)15 b(x)p F4(\)\))p F6(j)2141 3202 y Fu(N)2295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21537
3184 y F5(def)2302 3235 y F4(=)107 b FL(Imp)2624 3257
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21538
y Gc(E)2684 3235 y F4(\()2719 3223 y FX(h)2747 3235 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21539
Ga(a)2795 3223 y FX(i)2822 3235 y F6(j)p Ga(M)10 b F6(j)2970
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21540
3202 y Fu(N)3025 3235 y Ga(;)3065 3223 y FX(h)3093 3235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21541
y Ga(b)3132 3223 y FX(i)3159 3235 y F6(j)p Ga(Q)p F6(j)3281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21542
3202 y Fu(N)3335 3235 y Ga(;)3375 3223 y F9(\()3403 3235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21543
y Ga(y)3451 3223 y F9(\))3479 3235 y F6(j)p Ga(N)g F6(j)3612
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21544
3202 y Fu(N)3666 3235 y F4(\))442 3385 y Gg(\(13\))419
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21545
b F6(j)p FL(Cut)p F4(\()1209 3373 y FX(h)1237 3385 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21546
Ga(a)1285 3373 y FX(i)1312 3385 y Ga(M)10 b(;)1450 3373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21547
y F9(\()1478 3385 y Ga(x)1530 3373 y F9(\))1558 3385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21548
y FL(Not)1700 3399 y Gc(L)1752 3385 y F4(\()1787 3373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21549
y FX(h)1815 3385 y Ga(b)1854 3373 y FX(i)1882 3385 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21550
Ga(Q;)15 b(x)p F4(\)\))p F6(j)2141 3352 y Fu(N)2295 3334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21551
y F5(def)2302 3385 y F4(=)107 b FL(Not)2623 3399 y Gc(E)2682
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21552
3385 y F4(\()2717 3373 y FX(h)2745 3385 y Ga(a)2793 3373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21553
y FX(i)2820 3385 y F6(j)p Ga(M)10 b F6(j)2968 3352 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21554
Fu(N)3023 3385 y Ga(;)3063 3373 y FX(h)3091 3385 y Ga(b)3130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21555
3373 y FX(i)3157 3385 y F6(j)p Ga(Q)p F6(j)3279 3352
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21556
y Fu(N)3334 3385 y F4(\))442 3579 y Gb(Otherwise:)442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21557
3706 y Gg(\(14\))859 b F6(j)p FL(Cut)q F4(\()1650 3694
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21558
y FX(h)1677 3706 y Ga(a)1725 3694 y FX(i)1753 3706 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21559
Ga(M)10 b(;)1891 3694 y F9(\()1919 3706 y Ga(x)1971 3694
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21560
y F9(\))1998 3706 y Ga(N)g F4(\))p F6(j)2141 3673 y Fu(N)2295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21561
3654 y F5(def)2302 3706 y F4(=)107 b FL(Subst)o F4(\()2727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21562
3694 y FX(h)2755 3706 y Ga(a)2803 3694 y FX(i)2830 3706
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21563
y F6(j)p Ga(M)10 b F6(j)2978 3673 y Fu(N)3033 3706 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21564
Ga(;)3073 3694 y F9(\()3101 3706 y Ga(x)3153 3694 y F9(\))3180
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21565
3706 y F6(j)p Ga(N)g F6(j)3313 3673 y Fu(N)3368 3706
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21566
y F4(\))840 3923 y Gg(where)24 b(in)48 b(\(10\))74 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21567
Ga(x)25 b F6(62)g Ga(F)13 b(N)d F4(\()1779 3911 y F9(\()1807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21568
3923 y Ga(y)1855 3911 y F9(\))1882 3923 y Ga(N)g F4(\))325
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21569
b Gg(\(11\))74 b Ga(x)25 b F6(62)g Ga(F)13 b(N)d F4(\()2901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21570
3911 y F9(\()2929 3923 y Ga(y)2977 3911 y F9(\))3004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21571
3923 y Ga(N)g(;)3127 3911 y F9(\()3155 3923 y Ga(z)3201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21572
3911 y F9(\))3228 3923 y Ga(P)j F4(\))1203 4070 y Gg(\(12\))74
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21573
b Ga(x)25 b F6(62)g Ga(F)13 b(N)d F4(\()1779 4058 y F9(\()1807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21574
4070 y Ga(y)1855 4058 y F9(\))1882 4070 y Ga(N)g(;)2005
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21575
4058 y FX(h)2033 4070 y Ga(b)2072 4058 y FX(i)2099 4070
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21576
y Ga(Q)p F4(\))119 b Gg(\(13\))74 b Ga(x)25 b F6(62)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21577
Ga(F)13 b(N)d F4(\()2901 4058 y FX(h)2929 4070 y Ga(b)2968
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21578
4058 y FX(i)2995 4070 y Ga(Q)p F4(\))p 3965 4186 V 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21579
4189 3691 4 v Black 579 4343 a Gg(Figure)24 b(3.9:)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21580
b(T)m(ranslation)25 b(from)f(sequent)h(proofs)g(to)f(natural)h
Christian Urban <urbanc@in.tum.de>
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(deduction)h(proofs.)p Black Black Black Black eop end
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%%Page: 88 100
Christian Urban <urbanc@in.tum.de>
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TeXDict begin 88 99 bop Black -144 51 a Gb(88)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
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b(Deduction)p -144 88 3691 4 v Black Black 321 365 a(Theor)n(em)h
Christian Urban <urbanc@in.tum.de>
parents:
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(3.3.3)f Gg(\(Soundness)j(and)e(Completeness\))p Gb(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21586
Black Black 458 575 a F6(\017)p Black 46 w Gg(If)j Ga(M)44
Christian Urban <urbanc@in.tum.de>
parents:
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b F6(2)33 b F4(K)27 b Gg(with)h(the)g(typing)h(judgement)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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F4(\000)2040 563 y Gc(.)2094 575 y Ga(M)2218 563 y Gc(.)2273
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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575 y F4(\001)o Gg(,)e(then)h F6(j)p Ga(M)10 b F6(j)2731
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21590
542 y Fu(S)2809 575 y F6(2)33 b FY(T)27 b Gg(with)h(the)g(typing)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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688 y(judgement)d F4(\000)1037 676 y Gc(.)1092 688 y
Christian Urban <urbanc@in.tum.de>
parents:
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F6(j)p Ga(M)10 b F6(j)1240 655 y Fu(S)1309 676 y Gc(.)1364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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688 y F4(\001)p Gg(.)p Black 458 873 a F6(\017)p Black
Christian Urban <urbanc@in.tum.de>
parents:
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46 w Gg(If)27 b Ga(M)42 b F6(2)32 b FY(T)27 b Gg(with)g(the)h(typing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(judgement)g F4(\000)2020 861 y Gc(.)2075 873 y Ga(M)2199
Christian Urban <urbanc@in.tum.de>
parents:
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861 y Gc(.)2253 873 y F4(\001)p Gg(,)f(then)g F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Ga(M)10 b F6(j)2711 840 y Fu(N)2798 873 y F6(2)32 b F4(K)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b Gg(with)h(the)h(typing)549 985 y(judgement)d F4(\000)1037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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973 y Gc(.)1092 985 y F6(j)p Ga(M)10 b F6(j)1240 953
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y Fu(N)1320 973 y Gc(.)1375 985 y F4(\001)p Gg(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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321 1223 a F7(Pr)l(oof.)p Black 46 w Gg(Both)24 b(by)g(routine)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(induction)h(on)e(the)g(structure)h(of)f Ga(M)10 b Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
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3480 1223 4 62 v 3484 1165 55 4 v 3484 1223 V 3538 1223
Christian Urban <urbanc@in.tum.de>
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4 62 v 462 1426 a(Before)19 b(we)e(can)i(attack)g(the)g(correspondence)
Christian Urban <urbanc@in.tum.de>
parents:
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j(question)f(with)d(our)g(cut-elimination)k(procedure,)321
Christian Urban <urbanc@in.tum.de>
parents:
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1539 y(we)g(ha)n(v)o(e)g(to)g(de\002ne)h(a)e(normalisation)26
Christian Urban <urbanc@in.tum.de>
parents:
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b(procedure)e(for)f(the)f(natural)i(deduction)h(calculus.)30
Christian Urban <urbanc@in.tum.de>
parents:
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b(Although)321 1651 y(we)25 b(shall)h(de\002ne)g(this)g(procedure)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(using)e(terms,)g(the)g(corresponding)j(notion)e(of)f(a)e(normal)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(term)e(is)321 1764 y(best)c(e)o(xplained)h(on)e(the)g(le)n(v)o(el)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(of)g(proofs:)28 b(unlik)o(e)22 b(in)d(our)i(sequent)g(calculi)h(where)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21612
e(it)f(is)h(immediately)321 1877 y(clear)32 b(what)f(is)g(meant)g(by)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(a)e(normal)i(term)f(\(absence)i(of)e(the)g(cut-term)i(constructor\),)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(things)c(are)321 1990 y(more)27 b(delicate)i(in)f(natural)g(deduction)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21615
i(calculi,)f(as)e(already)i(seen)f(in)f(NJ.)f(T)-7 b(o)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21616
b(formally)j(de\002ne)e(the)321 2103 y(notion)e(of)f(a)f(normal)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21617
(natural)h(deduction)i(proof,)d(the)g(follo)n(wing)h(de\002nition)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21618
(will)e(be)h(useful.)p Black 321 2291 a Gb(De\002nition)e(3.3.4)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21619
Gg(\(Maximal)h(Se)o(gment\))p Gb(:)p Black 35 w Gg(A)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21620
F7(maximal)i(se)l(gment)h Gg(in)e(a)g(natural)i(deduction)h(proof)f(is)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21621
321 2404 y(a)k(sequence)i(of)d(occurrences)k(of)d(a)f(labelled)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(formula)e Ga(a)d F4(:)g Ga(B)5 b Gg(,)27 b(say)h F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Ga(a)e F4(:)f Ga(B)5 b F4(\))2774 2418 y F9(1)2813 2404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21624
y Ga(;)15 b(:)g(:)g(:)i(;)e F4(\()p Ga(a)25 b F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Ga(B)5 b F4(\))3282 2418 y Gc(n)3329 2404 y Gg(,)28 b(such)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21626
2517 y(that)p Black 458 2714 a F6(\017)p Black 46 w Gg(the)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(\002rst)g(occurrence,)40 b F4(\()p Ga(a)23 b F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21628
Ga(B)5 b F4(\))1588 2728 y F9(1)1628 2714 y Gg(,)36 b(is)e(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21629
(labelled)i(formula)g(which)f(is)f(eliminated)i(and)f(not)549
Christian Urban <urbanc@in.tum.de>
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diff changeset
 21630
2827 y(introduced)26 b(by)e(an)f(axiom.)p Black 458 2975
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21631
a F6(\017)p Black 46 w F4(\()p Ga(a)35 b F4(:)g Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21632
b F4(\))836 2989 y Gc(i)p F9(+1)986 2975 y Gg(follo)n(ws)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21633
b F4(\()p Ga(a)h F4(:)g Ga(B)5 b F4(\))1576 2989 y Gc(i)1604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21634
2975 y Gg(,)34 b(if)f Ga(a)i F4(:)g Ga(B)i Gg(occurs)d(in)f(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(conclusion)j(and)e(premise)g(of)f(an)549 3088 y(inference)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21636
b(rule.)p Black 458 3237 a F6(\017)p Black 46 w Gg(the)h(last)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21637
(occurrence)i(of)e Ga(a)22 b F4(:)h Ga(B)30 b Gg(either)d(appears)h(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21638
f(an)f(axiom)h(or)f(is)g(freshly)i(introduced)h(by)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21639
3350 y(an)23 b(introduction)k(rule.)321 3568 y(No)n(w)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21640
b(the)i(notion)h(of)f(a)f(normal)h(natural)h(deduction)i(proof)d(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(as)f(follo)n(ws.)p Black 321 3756 a Gb(De\002nition)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(3.3.5)h Gg(\(Normal)f(Natural)i(Deduction)g(Proof\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Gb(:)p Black 35 w Gg(A)d F7(normal)j Gg(natural)g(deduction)h(proof)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21644
3869 y(contains)26 b(neither)f(a)e(maximal)h(se)o(gment)h(nor)e(an)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(instance)i(of)d(Subst.)462 4123 y(As)30 b(mentioned)j(earlier)l(,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(we)c(can)h(easily)h(pro)o(v)o(e)g(that)f(normal)g(natural)i(deduction)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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g(proofs)f(re-)321 4236 y(spect)k(the)f(subformula)i(property)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(.)65 b(F)o(or)34 b(this)i(notice)g(that)f(all)g(elimination)i(rules)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(must)f(ha)n(v)o(e)h(as)321 4348 y(leftmost)26 b(premise)f(an)g(axiom)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(which)g(introduces)i(the)e(formula)g(that)g(is)g(eliminated.)33
Christian Urban <urbanc@in.tum.de>
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diff changeset
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b(F)o(or)24 b(e)o(xample)321 4461 y(the)g F6(_)516 4475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21652
y Gc(E)575 4461 y Gg(-rule)h(in)e(a)g(normal)i(natural)g(deduction)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(proof)f(must)e(be)h(of)f(the)h(form)p 476 4615 930 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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v 476 4694 a Ga(z)e F4(:)17 b Ga(B)5 b F6(_)o Ga(C)q(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21655
b F4(\000)p 900 4682 11 41 v 911 4664 46 5 v 97 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Ga(;)g(a)j F4(:)f Ga(B)5 b F6(_)o Ga(C)98 b(x)17 b F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Ga(B)5 b(;)15 b(z)22 b F4(:)17 b Ga(B)5 b F6(_)o Ga(C)q(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21658
b F4(\000)p 2146 4682 11 41 v 2157 4664 46 5 v 97 w(\001)91
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21659
b Ga(y)20 b F4(:)d Ga(C)q(;)e(z)23 b F4(:)17 b Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b F6(_)o Ga(C)q(;)15 b F4(\000)p 3027 4682 11 41 v 3038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4664 46 5 v 97 w(\001)p 476 4732 2703 4 v 1540 4811 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Ga(z)21 b F4(:)d Ga(B)5 b F6(_)o Ga(C)q(;)15 b F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1964 4799 11 41 v 1974 4780 46 5 v 97 w(\001)3220 4750
Christian Urban <urbanc@in.tum.de>
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diff changeset
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y F6(_)3281 4764 y Gc(E)3366 4750 y Ga(:)321 5023 y Gg(Clearly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(,)33 b(this)f(instance)g(of)f F6(_)1293 5037 y Gc(E)1382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5023 y Gg(satis\002es)h(the)f(subformula)i(property)-6
Christian Urban <urbanc@in.tum.de>
parents:
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b(.)52 b(Let)30 b(us)h(analyse)i(the)e(three)321 5136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y(other)i(types)g(of)f(instances)i(of)e(this)g(rule)h(and)f(sho)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(in)f(each)i(case)f(that)h(the)o(y)f(cannot)h(occur)g(in)f(a)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5249 y(normal)g(natural)h(deduction)h(proof.)53 b(First,)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(assume)g(the)g(leftmost)g(premise)g(is)f(an)h(axiom)f(which)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21672
5361 y(does)24 b(not)f(introduce)i Ga(a)p Gg(.)j(Then)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21673
b(we)e(ha)n(v)o(e)j(a)e(maximal)h(se)o(gment)h(of)e(length)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21674
F4(1)f Gg(\(\002rst)f(and)i(third)f(clause)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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eop end
Christian Urban <urbanc@in.tum.de>
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%%Page: 89 101
Christian Urban <urbanc@in.tum.de>
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TeXDict begin 89 100 bop Black 277 51 a Gb(3.3)23 b(Classical)j
Christian Urban <urbanc@in.tum.de>
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(Natural)d(Deduction)2373 b(89)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
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277 365 a Gg(of)28 b(De\002nition)g(3.3.4\).)41 b(Second,)29
Christian Urban <urbanc@in.tum.de>
parents:
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b(assume)g(the)f(leftmost)g(premise)h(is)e(an)h(introduction)j(rule.)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(Then)277 478 y(we)24 b(either)j(ha)n(v)o(e)e(a)g(maximal)g(se)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21682
(gment)h(of)f(length)h F4(1)p Gg(,)f(pro)o(vided)i(the)e(introduction)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(rule)c(introduces)277 590 y(freshly)36 b(the)f(formula)h(that)f(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21684
(eliminated,)k(or)c(a)f(maximal)h(se)o(gment)g(of)g(length)h(greater)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21685
(than)g F4(1)277 703 y Gg(\(second)31 b(and)f(third)g(clause)h(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21686
(De\002nition)h(3.3.4\).)47 b(Third,)30 b(if)f(the)h(leftmost)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21687
(premise)f(is)f(an)g(elim-)277 816 y(ination)35 b(rule,)g(then)f(we)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21688
(ha)n(v)o(e)h(a)f(maximal)h(se)o(gment)h(of)f(length)h(greater)g(than)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21689
F4(1)e Gg(\(second)j(clause)277 929 y(of)d(De\002nition)h(3.3.4\).)55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21690
b(So)32 b(in)g(all)g(three)h(cases)g(the)g(proof)g(w)o(ould)g(be)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21691
(non-normal.)57 b(Analogous)277 1042 y(remarks)24 b(apply)h(to)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21692
(other)h(elimination)i(rules.)j(Because)24 b(in)f(normal)h(natural)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21693
(deduction)h(proofs)277 1155 y(all)21 b(instances)j(of)c(introduction)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21694
25 b(rules)d(satisfy)g(the)g(subformula)h(property)g(\(ob)o(vious)g(by)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21695
e(inspection\))277 1268 y(and,)h(as)f(sho)n(wn)h(abo)o(v)o(e,)g(so)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21696
(do)h(all)f(instances)j(of)d(elimination)j(rules,)e(we)f(thus)h(ha)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21697
o(e)g(sho)n(wn)f(that)h(the)277 1381 y(subformula)k(property)g(holds)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21698
(for)g(all)g(normal)g(natural)h(deduction)h(proofs.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21699
1510 y(Ne)o(xt,)21 b(we)g(shall)h(introduce)h(reduction)h(rules)e(that)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21700
g(transform)h(stepwise)f(a)f(non-normal)j(natural)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21701
1623 y(deduction)34 b(proof)f(into)f(a)f(normal)h(one.)52
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21702
b(W)-7 b(e)31 b(be)o(gin)h(with)f(the)h(reduction)i(rules)e(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21703
(eliminate)h(a)277 1736 y(maximal)24 b(se)o(gment)g(of)g(length)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21704
F4(1)p Gg(.)j(Consider)d(the)f(proof)958 1921 y F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21705
1035 1909 11 41 v 1046 1891 46 5 v 96 w(\001)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21706
b(a)j F4(:)f Ga(B)p 897 1959 574 4 v 897 2038 a F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21707
974 2026 11 41 v 984 2008 46 5 v 96 w(\001)p Ga(;)e(b)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21708
F4(:)h Ga(B)5 b F6(_)o Ga(C)1512 1972 y F6(_)1572 1986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21709
y Gc(I)1603 1995 y FV(1)1732 2038 y Ga(x)18 b F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21710
Ga(B)5 b(;)15 b F4(\000)p 2035 2026 11 41 v 2045 2008
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21711
46 5 v 96 w(\001)91 b Ga(y)20 b F4(:)d Ga(C)q(;)e F4(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21712
2569 2026 11 41 v 2580 2008 46 5 v 97 w(\001)p 897 2076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21713
1825 4 v 1694 2155 a(\000)p 1771 2143 11 41 v 1782 2125
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21714
46 5 v 96 w(\001)2762 2094 y F6(_)2823 2108 y Gc(E)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21715
2363 y Gg(where)26 b Ga(b)c F4(:)g Ga(B)5 b F6(_)o Ga(C)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21716
b Gg(forms)c(a)e(maximal)h(se)o(gment)h(of)f(length)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21717
F4(1)p Gg(,)f(pro)o(vided)i Ga(b)22 b F4(:)g Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21718
F6(_)o Ga(C)31 b Gg(does)c(not)f(occur)277 2476 y(in)e(the)f(premise)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21719
(of)e F6(_)974 2490 y Gc(I)1005 2499 y FV(1)1043 2476
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21720
y Gg(.)28 b(This)c(proof)g(will)f(be)h(reduced)h(to)1245
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21721
2664 y F4(\000)p 1322 2652 11 41 v 1332 2634 46 5 v 96
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21722
w(\001)p Ga(;)15 b(a)i F4(:)h Ga(B)95 b(x)17 b F4(:)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21723
Ga(B)5 b(;)15 b F4(\000)p 2089 2652 11 41 v 2099 2634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21724
46 5 v 96 w(\001)p 1245 2702 996 4 v 1628 2780 a(\000)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21725
1705 2768 11 41 v 1716 2750 46 5 v 96 w(\001)2282 2732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21726
y Gg(Subst)26 b Ga(:)277 2988 y Gg(The)35 b(reduction)k(rules,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21727
(denoted)f(by)1581 2951 y Gc(\014)1518 2988 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21728
b(\000)h(!)p Gg(,)38 b(eliminating)g(this)e(type)g(of)g(maximal)g(se)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21729
(gments)h(are)277 3101 y(de\002ned)24 b(belo)n(w)g(using)h(terms.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21730
Black 277 3289 a Gb(De\002nition)e(3.3.6)g Gg(\(Beta-Reductions\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21731
Gb(:)p Black Black Black 285 3472 a FL(And)439 3435 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21732
Gc(i)439 3495 y(E)499 3472 y F4(\()534 3460 y FX(h)562
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21733
3472 y Ga(c)601 3460 y FX(i)629 3472 y FL(And)783 3486
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21734
y Gc(I)823 3472 y F4(\()858 3460 y FX(h)886 3472 y Ga(a)934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21735
3486 y F9(1)973 3460 y FX(i)1001 3472 y Ga(M)1089 3486
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21736
y F9(1)1128 3472 y Ga(;)1168 3460 y FX(h)1196 3472 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21737
Ga(a)1244 3486 y F9(2)1284 3460 y FX(i)1311 3472 y Ga(M)1399
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21738
3486 y F9(2)1439 3472 y Ga(;)15 b(c)p F4(\))q Ga(;)1594
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21739
3460 y F9(\()1622 3472 y Ga(x)1674 3460 y F9(\))1701
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21740
3472 y Ga(N)10 b F4(\))1907 3435 y Gc(\014)1843 3472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21741
y F6(\000)-31 b(\000)f(!)24 b FL(Subst)o F4(\()2284 3460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21742
y FX(h)2312 3472 y Ga(a)2360 3486 y Gc(i)2388 3460 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21743
FX(i)2416 3472 y Ga(M)2504 3486 y Gc(i)2532 3472 y Ga(;)2572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21744
3460 y F9(\()2600 3472 y Ga(x)2652 3460 y F9(\))2679
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21745
3472 y Ga(N)10 b F4(\))1752 3620 y Gg(if)23 b FL(And)1985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21746
3634 y Gc(I)2025 3620 y F4(\()2060 3608 y FX(h)2088 3620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21747
y Ga(a)2136 3634 y F9(1)2175 3608 y FX(i)2203 3620 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21748
Ga(M)2291 3634 y F9(1)2330 3620 y Ga(;)2370 3608 y FX(h)2398
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21749
3620 y Ga(a)2446 3634 y F9(2)2485 3608 y FX(i)2513 3620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21750
y Ga(M)2601 3634 y F9(2)2641 3620 y Ga(;)15 b(c)p F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21751
b Gg(freshly)j(introduces)g Ga(c)407 3820 y FL(Or)507
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21752
3834 y Gc(E)566 3820 y F4(\()601 3808 y FX(h)629 3820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21753
y Ga(b)668 3808 y FX(i)696 3820 y FL(Or)795 3783 y Gc(i)795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21754
3843 y(I)835 3820 y F4(\()870 3808 y FX(h)897 3820 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21755
Ga(a)945 3808 y FX(i)973 3820 y Ga(M)10 b(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21756
F4(\))q Ga(;)1226 3808 y F9(\()1254 3820 y Ga(x)1306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21757
3834 y F9(1)1345 3808 y(\))1373 3820 y Ga(N)1446 3834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21758
y F9(1)1485 3820 y Ga(;)1525 3808 y F9(\()1553 3820 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21759
Ga(x)1605 3834 y F9(2)1644 3808 y(\))1672 3820 y Ga(N)1745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21760
3834 y F9(2)1784 3820 y F4(\))1907 3783 y Gc(\014)1843
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21761
3820 y F6(\000)-31 b(\000)f(!)24 b FL(Subst)o F4(\()2284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21762
3808 y FX(h)2312 3820 y Ga(a)2360 3808 y FX(i)2387 3820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21763
y Ga(M)11 b(;)2526 3808 y F9(\()2553 3820 y Ga(x)2605
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21764
3834 y Gc(i)2634 3808 y F9(\))2661 3820 y Ga(N)2734 3834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21765
y Gc(i)2762 3820 y F4(\))1752 3970 y Gg(if)23 b FL(Or)1930
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21766
3933 y Gc(i)1930 3993 y(I)1969 3970 y F4(\()2004 3958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21767
y FX(h)2032 3970 y Ga(a)2080 3958 y FX(i)2108 3970 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21768
Ga(M)10 b(;)15 b(b)p F4(\))23 b Gg(freshly)j(introduces)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21769
Ga(b)377 4168 y FL(Imp)521 4189 y Gc(E)581 4168 y F4(\()616
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21770
4156 y FX(h)644 4168 y Ga(b)683 4156 y FX(i)710 4168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21771
y FL(Imp)855 4189 y Gc(I)894 4168 y F4(\()929 4156 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21772
F9(\()957 4168 y Ga(x)1009 4156 y F9(\))q FX(h)1064 4168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21773
y Ga(a)1112 4156 y FX(i)1140 4168 y Ga(M)10 b(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21774
F4(\))p Ga(;)1392 4156 y FX(h)1420 4168 y Ga(c)1459 4156
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21775
y FX(i)1487 4168 y Ga(N)10 b(;)1610 4156 y F9(\()1638
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21776
4168 y Ga(y)1686 4156 y F9(\))1713 4168 y Ga(P)j F4(\))1907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21777
4131 y Gc(\014)1843 4168 y F6(\000)-31 b(\000)f(!)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21778
b FL(Subst)o F4(\()2284 4156 y FX(h)2312 4168 y Ga(c)2351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21779
4156 y FX(i)2379 4168 y Ga(N)10 b(;)2502 4156 y F9(\()2530
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21780
4168 y Ga(x)2582 4156 y F9(\))2609 4168 y FL(Subst)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21781
F4(\()2856 4156 y FX(h)2884 4168 y Ga(a)2932 4156 y FX(i)2959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21782
4168 y Ga(M)g(;)3097 4156 y F9(\()3125 4168 y Ga(y)3173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21783
4156 y F9(\))3201 4168 y Ga(P)j F4(\)\))48 b Gg(or)1907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21784
4243 y Gc(\014)1843 4281 y F6(\000)-31 b(\000)f(!)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21785
b FL(Subst)o F4(\()2284 4269 y FX(h)2312 4281 y Ga(a)2360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21786
4269 y FX(i)2387 4281 y FL(Subst)p F4(\()2635 4269 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21787
FX(h)2662 4281 y Ga(c)2701 4269 y FX(i)2729 4281 y Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21788
b(;)2852 4269 y F9(\()2880 4281 y Ga(x)2932 4269 y F9(\))2959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21789
4281 y Ga(M)g F4(\))q Ga(;)3133 4269 y F9(\()3161 4281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21790
y Ga(y)3209 4269 y F9(\))3236 4281 y Ga(P)j F4(\))1752
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21791
4429 y Gg(if)23 b FL(Imp)1975 4451 y Gc(I)2015 4429 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21792
F4(\()2050 4417 y F9(\()2077 4429 y Ga(x)2129 4417 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21793
F9(\))q FX(h)2184 4429 y Ga(a)2232 4417 y FX(i)2260 4429
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21794
y Ga(M)10 b(;)15 b(b)p F4(\))23 b Gg(freshly)j(introduces)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21795
Ga(b)680 4627 y FL(Not)822 4641 y Gc(E)882 4627 y F4(\()917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21796
4615 y FX(h)945 4627 y Ga(a)993 4615 y FX(i)1020 4627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21797
y FL(Not)1163 4641 y Gc(I)1203 4627 y F4(\()1238 4615
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21798
y F9(\()1266 4627 y Ga(x)1318 4615 y F9(\))1345 4627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21799
y Ga(M)10 b(;)15 b(a)p F4(\))q Ga(;)1607 4615 y FX(h)1635
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21800
4627 y Ga(b)1674 4615 y FX(i)1701 4627 y Ga(N)10 b F4(\))1907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21801
4590 y Gc(\014)1843 4627 y F6(\000)-31 b(\000)f(!)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21802
b FL(Subst)o F4(\()2284 4615 y FX(h)2312 4627 y Ga(b)2351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21803
4615 y FX(i)2378 4627 y Ga(N)10 b(;)2501 4615 y F9(\()2529
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21804
4627 y Ga(x)2581 4615 y F9(\))2609 4627 y Ga(M)g F4(\))1752
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21805
4775 y Gg(if)23 b FL(Not)1973 4789 y Gc(I)2013 4775 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21806
F4(\()2048 4763 y F9(\()2076 4775 y Ga(x)2128 4763 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21807
F9(\))2155 4775 y Ga(M)10 b(;)15 b(a)p F4(\))24 b Gg(freshly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21808
(introduces)h Ga(a)p Black 277 5079 a Gb(Remark)33 b(3.3.7:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21809
Black 34 w Gg(There)g(are)g(a)g(fe)n(w)f(subtleties)k(in)d(the)h(third)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21810
g(rule.)58 b(As)32 b(with)h(the)g(cut-reduction)277 5192
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21811
y F6(\033)348 5206 y Gc(R)406 5192 y Ga(=)p F6(\033)522
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21812
5206 y Gc(L)574 5192 y Gg(,)24 b(there)i(are)f(tw)o(o)g(w)o(ays)g(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21813
(eliminate)i(an)e(introduction-eliminat)q(ion)31 b(pair)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21814
b F6(\033)3049 5206 y Gc(E)3109 5192 y Ga(=)p F6(\033)3225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21815
5206 y Gc(I)3265 5192 y Gg(.)32 b(Con-)277 5305 y(sequently)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21816
b(,)24 b(we)c(ha)n(v)o(e)h(tw)o(o)g(reduction)i(rules.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21817
b(W)-7 b(e)20 b(need,)i(ho)n(we)n(v)o(er)l(,)f(to)g(observ)o(e)i(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21818
(side-conditions)277 5418 y Ga(x)k Gg(not)h(free)g(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21819
Ga(F)13 b(N)d F4(\()945 5406 y F9(\()973 5418 y Ga(y)1021
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21820
5406 y F9(\))1048 5418 y Ga(P)j F4(\))25 b Gg(\(\002rst)g(rule\))i(and)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21821
f Ga(a)e Gg(not)i(free)g(in)g Ga(F)13 b(C)7 b F4(\()2381
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21822
5406 y FX(h)2408 5418 y Ga(c)2447 5406 y FX(i)2475 5418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21823
y Ga(N)j F4(\))25 b Gg(\(second)i(rule\),)g(so)e(that)h(no)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21824
5531 y(name)20 b(or)f(co-name)i(capture)g(occurs.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21825
b(This)19 b(can)h(al)o(w)o(ays)h(be)e(achie)n(v)o(ed)i(by)f(renaming)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21826
(some)f(binders.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21827
%%Page: 90 102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21828
TeXDict begin 90 101 bop Black -144 51 a Gb(90)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21829
b(Deduction)p -144 88 3691 4 v Black 462 365 a Gg(T)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21830
b(o)29 b(eliminate)j(all)e(other)g(maximal)h(se)o(gments,)h(we)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21831
(introduce)j(commuting)g(reductions.)50 b(Let)321 478
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21832
y(us)24 b(\002rst)f(gi)n(v)o(e)h(an)f(e)o(xample.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21833
b(Consider)25 b(the)f(proof)957 659 y F4(\000)1014 626
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21834
y FX(0)p 1057 647 11 41 v 1068 629 46 5 v 1133 659 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21835
F4(\001)1209 626 y FX(0)1232 659 y Ga(;)15 b(a)j F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21836
Ga(B)96 b F4(\000)1602 626 y FX(0)p 1645 647 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21837
1656 629 46 5 v 1721 659 a F4(\001)1797 626 y FX(0)1820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21838
659 y Ga(;)15 b(b)j F4(:)f Ga(C)p 957 696 1074 4 v 1184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21839
783 a F4(\000)1241 750 y FX(0)p 1284 771 11 41 v 1294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21840
753 46 5 v 1360 783 a F4(\001)1436 750 y FX(0)1459 783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21841
y Ga(;)e(c)j F4(:)f Ga(B)5 b F6(^)p Ga(C)2072 715 y F6(^)2133
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21842
729 y Gc(I)1482 826 y Gg(.)1482 860 y(.)1482 893 y(.)1482
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21843
926 y(.)1207 1006 y F4(\000)p 1284 994 11 41 v 1294 976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21844
46 5 v 96 w(\001)p Ga(;)15 b(c)j F4(:)f Ga(B)5 b F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21845
Ga(C)490 b(x)17 b F4(:)g Ga(B)5 b(;)15 b F4(\000)p 2566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21846
994 11 41 v 2577 976 46 5 v 96 w(\001)p 1207 1043 1512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21847
4 v 1848 1122 a(\000)p 1925 1110 11 41 v 1936 1092 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21848
5 v 96 w(\001)2760 1057 y F6(^)2820 1071 y Gc(E)2872
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21849
1080 y FV(1)321 1318 y Gg(where)24 b(the)g F6(^)761 1332
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21850
y Gc(E)813 1341 y FV(1)851 1318 y Gg(-rule)g(needs)h(to)f(be)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21851
(permuted)i(so)f(as)f(to)h(yield)g(the)g(proof)885 1497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21852
y F4(\000)942 1464 y FX(\017)p 1002 1485 11 41 v 1012
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21853
1466 46 5 v 1078 1497 a F4(\001)1154 1464 y FX(\017)1193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21854
1497 y Ga(;)15 b(a)j F4(:)f Ga(B)95 b F4(\000)1562 1464
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21855
y FX(\017)p 1622 1485 11 41 v 1632 1466 46 5 v 1698 1497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21856
a F4(\001)1774 1464 y FX(\017)1813 1497 y Ga(;)15 b(b)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21857
F4(:)f Ga(C)p 885 1534 1139 4 v 1128 1614 a F4(\000)1185
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21858
1581 y FX(\017)p 1245 1602 11 41 v 1255 1584 46 5 v 1321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21859
1614 a F4(\001)1397 1581 y FX(\017)1436 1614 y Ga(;)e(c)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21860
F4(:)f Ga(B)5 b F6(^)o Ga(C)2065 1552 y F6(^)2126 1566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21861
y Gc(I)2257 1614 y Ga(x)17 b F4(:)g Ga(B)5 b(;)15 b F4(\000)2539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21862
1581 y FX(\017)p 2599 1602 11 41 v 2609 1584 46 5 v 2675
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21863
1614 a F4(\001)2751 1581 y FX(\017)p 1128 1652 1662 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21864
v 1805 1732 a F4(\000)1862 1699 y FX(\017)p 1922 1720
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21865
11 41 v 1932 1701 46 5 v 1998 1732 a F4(\001)2074 1699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21866
y FX(\017)2831 1665 y F6(^)2892 1679 y Gc(E)2944 1688
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21867
y FV(1)1948 1757 y Gg(.)1948 1791 y(.)1948 1824 y(.)1948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21868
1857 y(.)1845 1937 y F4(\000)p 1922 1925 11 41 v 1932
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21869
1907 46 5 v 96 w(\001)321 2133 y Gg(Here)29 b F4(\000)583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21870
2100 y FX(\017)650 2133 y Gg(stands)h(for)f(the)g(conte)o(xt)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21871
F4(\000)24 b F6([)g F4(\000)1695 2100 y FX(0)1745 2133
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21872
y Gg(\(similarly)31 b F4(\001)2203 2100 y FX(\017)2242
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21873
2133 y Gg(\).)44 b(W)-7 b(e)27 b(realise)k(this)e(kind)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21874
(operation)321 2246 y(using)c(a)e(proof)i(substitution)i(of)c(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21875
(form:)p Black Black 778 2446 a Ga(M)h Fs(^)q FL(And)1078
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21876
2410 y Gc(i)1078 2469 y(E)1138 2446 y F4(\()1173 2434
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21877
y FX(h)1201 2446 y Ga(a)1249 2434 y FX(i)p 1278 2446
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21878
28 4 v 1296 2446 V 1313 2446 V 1341 2446 a Ga(;)1381
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21879
2434 y F9(\()1409 2446 y Ga(x)1461 2434 y F9(\))1488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21880
2446 y Ga(N)10 b F4(\))p Fs(_)396 b Ga(M)25 b Fs(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21881
FL(Imp)2324 2468 y Gc(E)2383 2446 y F4(\()2418 2434 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21882
FX(h)2446 2446 y Ga(a)2494 2434 y FX(i)p 2524 2446 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21883
2541 2446 V 2559 2446 V 2586 2446 a Ga(;)2626 2434 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21884
FX(h)2654 2446 y Ga(b)2693 2434 y FX(i)2721 2446 y Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21885
b(;)2844 2434 y F9(\()2871 2446 y Ga(x)2923 2434 y F9(\))2951
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21886
2446 y Ga(P)j F4(\))p Fs(_)778 2559 y Ga(M)25 b Fs(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21887
FL(Or)1023 2573 y Gc(E)1083 2559 y F4(\()1118 2547 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21888
FX(h)1145 2559 y Ga(a)1193 2547 y FX(i)p 1223 2559 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21889
1241 2559 V 1258 2559 V 1285 2559 a Ga(;)1325 2547 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21890
F9(\()1353 2559 y Ga(x)1405 2547 y F9(\))1433 2559 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21891
Ga(N)10 b(;)1556 2547 y F9(\()1583 2559 y Ga(y)1631 2547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21892
y F9(\))1659 2559 y Ga(P)j F4(\))p Fs(_)237 b Ga(M)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21893
b Fs(^)p FL(Not)2322 2573 y Gc(E)2382 2559 y F4(\()2417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21894
2547 y FX(h)2445 2559 y Ga(a)2493 2547 y FX(i)p 2522
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21895
2559 V 2540 2559 V 2557 2559 V 2585 2559 a Ga(;)2625
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21896
2547 y FX(h)2652 2559 y Ga(b)2691 2547 y FX(i)2719 2559
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21897
y Ga(N)10 b F4(\))p Fs(_)321 2755 y Gg(The)32 b(idea)h(behind)h(this)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21898
(proof)g(substitution)j(is)c(to)g(mo)o(v)o(e)g(the)g(elimination)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21899
(rule)e(inside)g(the)g(term)321 2868 y Ga(M)42 b Gg(until)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21900
b(either)h(a)d(corresponding)37 b(introduction)f(rule)d(or)f(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21901
(corresponding)k(axiom)d(is)f(reached.)321 2981 y(In)c(these)g(tw)o(o)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21902
(cases)g(the)g(term)f Ga(M)37 b Gg(is)28 b(\223plugged\224)i(into)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21903
(the)g(place)h(of)e(the)h(bar)-5 b(.)41 b(If)27 b(there)i(is)e(no)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21904
(cor)n(-)321 3094 y(responding)i(introduction)g(rule)d(nor)g(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21905
(corresponding)31 b(axiom,)c(then)h(the)e(proof)i(substitution)i(just)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21906
321 3207 y(v)n(anishes.)g(Figure)22 b(3.10)g(gi)n(v)o(es)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21907
(complete)i(de\002nition)g(of)f Ga(M)10 b Fs(^)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21908
Fs(_)q Gg(.)27 b(Our)21 b(commuting)i(reductions,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21909
3320 y(denoted)j(by)810 3283 y Gc(\015)745 3320 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21910
b(\000)g(!)p Gg(,)22 b(are)i(then)g(de\002ned)h(as)e(follo)n(ws.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21911
Black 321 3507 a Gb(De\002nition)g(3.3.8)g Gg(\(Commuting)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21912
(Reductions\))p Gb(:)p Black 572 3683 a FL(And)727 3646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21913
y Gc(i)727 3706 y(E)786 3683 y F4(\()821 3671 y FX(h)849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21914
3683 y Ga(a)897 3671 y FX(i)925 3683 y Ga(M)10 b(;)1063
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21915
3671 y F9(\()1091 3683 y Ga(x)1143 3671 y F9(\))1170
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21916
3683 y Ga(N)g F4(\))1436 3646 y Gc(\015)1371 3683 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21917
b(\000)g(!)83 b Ga(M)10 b Fs(^)p FL(And)1910 3646 y Gc(i)1910
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21918
3706 y(E)1969 3683 y F4(\()2004 3671 y FX(h)2032 3683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21919
y Ga(a)2080 3671 y FX(i)p 2110 3683 V 2127 3683 V 2145
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21920
3683 V 2172 3683 a Ga(;)2212 3671 y F9(\()2240 3683 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21921
Ga(x)2292 3671 y F9(\))2319 3683 y Ga(N)g F4(\))p Fs(_)413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21922
3795 y FL(Or)513 3809 y Gc(E)572 3795 y F4(\()607 3783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21923
y FX(h)635 3795 y Ga(a)683 3783 y FX(i)711 3795 y Ga(M)g(;)849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21924
3783 y F9(\()877 3795 y Ga(x)929 3783 y F9(\))956 3795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21925
y Ga(N)g(;)1079 3783 y F9(\()1107 3795 y Ga(y)1155 3783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21926
y F9(\))1182 3795 y Ga(P)j F4(\))1436 3758 y Gc(\015)1371
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21927
3795 y F6(\000)-31 b(\000)g(!)83 b Ga(M)10 b Fs(^)p FL(Or)1854
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21928
3809 y Gc(E)1914 3795 y F4(\()1949 3783 y FX(h)1977 3795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21929
y Ga(a)2025 3783 y FX(i)p 2054 3795 V 2072 3795 V 2090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21930
3795 V 2117 3795 a Ga(;)2157 3783 y F9(\()2185 3795 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21931
Ga(x)2237 3783 y F9(\))2264 3795 y Ga(N)g(;)2387 3783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21932
y F9(\()2415 3795 y Ga(y)2463 3783 y F9(\))2490 3795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21933
y Ga(P)j F4(\))p Fs(_)377 3906 y FL(Imp)522 3928 y Gc(E)581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21934
3906 y F4(\()616 3894 y FX(h)644 3906 y Ga(a)692 3894
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21935
y FX(i)719 3906 y Ga(M)e(;)858 3894 y FX(h)885 3906 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21936
Ga(b)924 3894 y FX(i)952 3906 y Ga(N)f(;)1075 3894 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21937
F9(\()1103 3906 y Ga(x)1155 3894 y F9(\))1182 3906 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21938
Ga(P)j F4(\))1436 3869 y Gc(\015)1371 3906 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21939
b(\000)g(!)83 b Ga(M)10 b Fs(^)p FL(Imp)1900 3928 y Gc(E)1959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21940
3906 y F4(\()1994 3894 y FX(h)2022 3906 y Ga(a)2070 3894
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21941
y FX(i)p 2100 3906 V 2117 3906 V 2135 3906 V 2162 3906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21942
a Ga(;)2202 3894 y FX(h)2230 3906 y Ga(b)2269 3894 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21943
FX(i)2296 3906 y Ga(N)g(;)2419 3894 y F9(\()2447 3906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21944
y Ga(x)2499 3894 y F9(\))2527 3906 y Ga(P)j F4(\))p Fs(_)597
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21945
4018 y FL(Not)740 4032 y Gc(E)799 4018 y F4(\()834 4006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21946
y FX(h)862 4018 y Ga(a)910 4006 y FX(i)938 4018 y Ga(M)d(;)1076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21947
4006 y FX(h)1104 4018 y Ga(b)1143 4006 y FX(i)1170 4018
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21948
y Ga(N)g F4(\))1436 3981 y Gc(\015)1371 4018 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21949
b(\000)g(!)83 b Ga(M)10 b Fs(^)p FL(Not)1898 4032 y Gc(E)1958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21950
4018 y F4(\()1993 4006 y FX(h)2020 4018 y Ga(a)2068 4006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21951
y FX(i)p 2098 4018 V 2115 4018 V 2133 4018 V 2160 4018
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21952
a Ga(;)2200 4006 y FX(h)2228 4018 y Ga(b)2267 4006 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21953
FX(i)2295 4018 y Ga(N)g F4(\))p Fs(_)2665 3615 y FK(9)2665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21954
3696 y(>)2665 3724 y(>)2665 3751 y(=)2665 3915 y(>)2665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21955
3942 y(>)2665 3969 y(;)2811 3799 y Gg(if)23 b Ga(M)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21956
b Gg(does)25 b(not)2811 3912 y(freshly)g(introduce)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21957
Ga(a)462 4275 y Gg(What)j(remains)h(is)e(to)h(de\002ne)g(reduction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21958
(rules)e(eliminating)i(instances)g(of)e(Subst.)44 b(This)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21959
b(term)321 4388 y(constructor)e(w)o(as)c(introduced)k(in)c(order)h(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21960
(match)g(all)f(reductions)k(of)c(commuting)i(cuts,)f(in)f(partic-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21961
4501 y(ular)e(the)g(ones)g(in)g(Lafont')-5 b(s)21 b(e)o(xample.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21962
b(Therefore,)22 b(we)e(shall)h(de\002ne)g(the)f(reduction)j(rules)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21963
(for)e(Subst)321 4614 y(such)29 b(that)f(we)e(can)i(capture)i(the)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21964
(non-deterministic)33 b(beha)n(viour)d(of)d(commuting)i(cuts.)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21965
b(T)-7 b(o)26 b(do)i(so,)321 4727 y(we)23 b(introduce)j(another)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21966
(proof)g(substitution,)i(written)d(as)p Black Black 1238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21967
4931 a Ga(M)10 b Fs(\()-7 b Ga(a)26 b F4(:=)1556 4919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21968
y F9(\()1583 4931 y Ga(x)1635 4919 y F9(\))1663 4931
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21969
y Ga(N)r Fs(\))115 b Gg(or)f Ga(N)10 b Fs(\()-7 b Ga(x)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21970
b F4(:=)2381 4919 y FX(h)2408 4931 y Ga(a)2456 4919 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21971
FX(i)2484 4931 y Ga(M)s Fs(\))p Gg(.)321 5136 y(Figure)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21972
b(3.11)f(gi)n(v)o(es)g(its)g(complete)h(de\002nition,)h(which)e(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21973
(formulated)h(so)f(that,)h(gi)n(v)o(en)f(the)h(translation)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21974
5249 y F6(j)p 348 5249 V 366 5249 V 384 5249 V 65 w(j)436
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21975
5216 y Fu(N)490 5249 y Gg(,)k(there)i(is)e(a)h(perfect)h(match)f(with)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21976
(proof)g(substitutions)k(of)c(the)g(form)f F4([)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21977
F4(])p Gg(.)32 b(Consequently)-6 b(,)28 b(we)321 5361
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21978
y(ha)n(v)o(e)c(the)g(follo)n(wing)h(reduction)h(rule)e(for)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21979
FL(Subst)o Gg(.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21980
%%Page: 91 103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21981
TeXDict begin 91 102 bop Black 277 51 a Gb(3.3)23 b(Classical)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21982
(Natural)d(Deduction)2373 b(91)p 277 88 3691 4 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21983
Black 277 476 V 277 4986 4 4510 v Black Black 489 631
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21984
a(Let)23 b Fs(^)q FO(\033)s Fs(_)g Gb(be)g(of)h(the)e(f)n(orm)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21985
Fs(^)p Fo(And)1566 594 y Fy(i)1566 659 y(E)1630 631 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21986
FD(\()1671 619 y Fz(h)1703 631 y FO(c)1750 619 y Fz(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21987
1781 631 28 4 v 1796 631 V 1811 631 V 1839 631 a FO(;)1885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21988
619 y Fx(\()1917 631 y FO(x)1977 619 y Fx(\))2008 631
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21989
y FO(P)15 b FD(\))p Fs(_)p Gb(,)22 b(then:)1222 818 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21990
FL(Id)p F4(\()p Ga(y)s(;)15 b(c)p F4(\))p Fs(^)r Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21991
Fs(_)1660 767 y F5(def)1667 818 y F4(=)54 b FL(And)1946
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21992
781 y Gc(i)1946 841 y(E)2006 818 y F4(\()2041 806 y FX(h)2069
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21993
818 y Ga(c)2108 806 y FX(i)2135 818 y FL(Id)q F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21994
Ga(y)s(;)15 b(c)p F4(\))q Ga(;)2446 806 y F9(\()2474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21995
818 y Ga(x)2526 806 y F9(\))2553 818 y Ga(P)e F4(\))970
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21996
b Gg(\(1\))729 1005 y FL(And)884 1019 y Gc(I)924 1005
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21997
y F4(\()959 993 y FX(h)986 1005 y Ga(a)1034 993 y FX(i)1062
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21998
1005 y Ga(M)10 b(;)1200 993 y FX(h)1228 1005 y Ga(b)1267
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 21999
993 y FX(i)1294 1005 y Ga(N)g(;)15 b(c)p F4(\))p Fs(^)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22000
Ga(\033)s Fs(_)1660 954 y F5(def)1667 1005 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22001
b FL(And)1946 968 y Gc(i)1946 1028 y(E)2006 1005 y F4(\()2041
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22002
993 y FX(h)2069 1005 y Ga(c)2108 993 y FX(i)2135 1005
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22003
y FL(And)2290 1019 y Gc(I)2330 1005 y F4(\()2365 993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22004
y FX(h)2393 1005 y Ga(a)2441 993 y FX(i)2468 1005 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22005
b Fs(^)q Ga(\033)s Fs(_)q Ga(;)2727 993 y FX(h)2754 1005
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22006
y Ga(b)2793 993 y FX(i)2821 1005 y Ga(N)g Fs(^)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22007
Fs(_)q Ga(;)15 b(c)p F4(\))q Ga(;)3179 993 y F9(\()3207
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22008
1005 y Ga(x)3259 993 y F9(\))3286 1005 y Ga(P)e F4(\))237
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22009
b Gg(\(2\))489 1236 y Gb(Let)23 b Fs(^)q FO(\033)s Fs(_)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22010
Gb(be)g(of)h(the)e(f)n(orm)i Fs(^)p Fo(Or)1505 1250 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22011
Fy(E)1569 1236 y FD(\()1610 1224 y Fz(h)1641 1236 y FO(c)1688
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22012
1224 y Fz(i)p 1720 1236 V 1735 1236 V 1750 1236 V 1777
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22013
1236 a FO(;)1824 1224 y Fx(\()1856 1236 y FO(x)1916 1224
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22014
y Fx(\))1947 1236 y FO(P)14 b(;)2074 1224 y Fx(\()2105
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22015
1236 y FO(y)2162 1224 y Fx(\))2194 1236 y FO(Q)p FD(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22016
Fs(_)p Gb(,)23 b(then:)1222 1423 y FL(Id)p F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22017
b(c)p F4(\))p Fs(^)r Ga(\033)s Fs(_)1660 1372 y F5(def)1667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22018
1423 y F4(=)54 b FL(Or)1891 1437 y Gc(E)1951 1423 y F4(\()1986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22019
1411 y FX(h)2013 1423 y Ga(c)2052 1411 y FX(i)2080 1423
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22020
y FL(Id)p F4(\()p Ga(y)s(;)15 b(c)p F4(\))r Ga(;)2391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22021
1411 y F9(\()2419 1423 y Ga(x)2471 1411 y F9(\))2498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22022
1423 y Ga(P)e(;)2609 1411 y F9(\()2637 1423 y Ga(y)2685
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22023
1411 y F9(\))2712 1423 y Ga(Q)p F4(\))810 b Gg(\(3\))1002
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22024
1610 y FL(Or)1101 1573 y Gc(i)1101 1633 y(I)1141 1610
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22025
y F4(\()1176 1598 y FX(h)1204 1610 y Ga(a)1252 1598 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22026
FX(i)1279 1610 y Ga(M)10 b(;)15 b(c)p F4(\))p Fs(^)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22027
Ga(\033)s Fs(_)1660 1559 y F5(def)1667 1610 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22028
b FL(Or)1891 1624 y Gc(E)1951 1610 y F4(\()1986 1598
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22029
y FX(h)2013 1610 y Ga(c)2052 1598 y FX(i)2080 1610 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22030
FL(Or)2179 1573 y Gc(i)2179 1633 y(I)2219 1610 y F4(\()2254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22031
1598 y FX(h)2282 1610 y Ga(a)2330 1598 y FX(i)2358 1610
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22032
y Ga(M)10 b Fs(^)p Ga(\033)s Fs(_)q Ga(;)15 b(c)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22033
Ga(;)2731 1598 y F9(\()2759 1610 y Ga(x)2811 1598 y F9(\))2838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22034
1610 y Ga(P)e(;)2949 1598 y F9(\()2977 1610 y Ga(y)3025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22035
1598 y F9(\))3052 1610 y Ga(Q)p F4(\))470 b Gg(\(4\))489
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22036
1841 y Gb(Let)23 b Fs(^)q FO(\033)s Fs(_)g Gb(be)g(of)h(the)e(f)n(orm)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22037
Fs(^)p Fo(Imp)1557 1863 y Fy(E)1622 1841 y FD(\()1663
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22038
1829 y Fz(h)1694 1841 y FO(c)1741 1829 y Fz(i)p 1772
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22039
1841 V 1787 1841 V 1803 1841 V 1830 1841 a FO(;)1876
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22040
1829 y Fz(h)1908 1841 y FO(b)1955 1829 y Fz(i)1987 1841
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22041
y FO(P)14 b(;)2114 1829 y Fx(\()2145 1841 y FO(x)2205
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22042
1829 y Fx(\))2237 1841 y FO(Q)p FD(\))p Fs(_)p Gb(,)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22043
b(then:)1222 2028 y FL(Id)p F4(\()p Ga(y)s(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22044
F4(\))p Fs(^)r Ga(\033)s Fs(_)1660 1977 y F5(def)1667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22045
2028 y F4(=)54 b FL(Imp)1936 2050 y Gc(E)1996 2028 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22046
F4(\()2031 2016 y FX(h)2059 2028 y Ga(c)2098 2016 y FX(i)2125
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22047
2028 y FL(Id)q F4(\()p Ga(y)s(;)15 b(c)p F4(\))q Ga(;)2436
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22048
2016 y FX(h)2464 2028 y Ga(b)2503 2016 y FX(i)2530 2028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22049
y Ga(P)e(;)2641 2016 y F9(\()2669 2028 y Ga(x)2721 2016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22050
y F9(\))2748 2028 y Ga(Q)p F4(\))774 b Gg(\(5\))850 2215
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22051
y FL(Imp)994 2237 y Gc(I)1034 2215 y F4(\()1069 2203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22052
y F9(\()1097 2215 y Ga(x)1149 2203 y F9(\))p FX(h)1204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22053
2215 y Ga(a)1252 2203 y FX(i)1279 2215 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22054
b(c)p F4(\))p Fs(^)r Ga(\033)s Fs(_)1660 2164 y F5(def)1667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22055
2215 y F4(=)54 b FL(Imp)1936 2237 y Gc(E)1996 2215 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22056
F4(\()2031 2203 y FX(h)2059 2215 y Ga(c)2098 2203 y FX(i)2125
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22057
2215 y FL(Imp)2270 2237 y Gc(I)2310 2215 y F4(\()2345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22058
2203 y F9(\()2373 2215 y Ga(x)2425 2203 y F9(\))p FX(h)2479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22059
2215 y Ga(a)2527 2203 y FX(i)2555 2215 y Ga(M)10 b Fs(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22060
Ga(\033)s Fs(_)q Ga(;)15 b(c)p F4(\))q Ga(;)2928 2203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22061
y FX(h)2956 2215 y Ga(b)2995 2203 y FX(i)3023 2215 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22062
Ga(P)e(;)3134 2203 y F9(\()3161 2215 y Ga(x)3213 2203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22063
y F9(\))3241 2215 y Ga(Q)p F4(\))281 b Gg(\(6\))489 2446
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22064
y Gb(Let)23 b Fs(^)q FO(\033)s Fs(_)g Gb(be)g(of)h(the)e(f)n(orm)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22065
Fs(^)p Fo(Not)1556 2460 y Fy(E)1621 2446 y FD(\()1662
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22066
2434 y Fz(h)1693 2446 y FO(c)1740 2434 y Fz(i)p 1771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22067
2446 V 1786 2446 V 1802 2446 V 1829 2446 a FO(;)1875
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22068
2434 y Fz(h)1907 2446 y FO(b)1954 2434 y Fz(i)1986 2446
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22069
y FO(P)14 b FD(\))p Fs(_)p Gb(,)23 b(then:)1222 2633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22070
y FL(Id)p F4(\()p Ga(y)s(;)15 b(c)p F4(\))p Fs(^)r Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22071
Fs(_)1660 2582 y F5(def)1667 2633 y F4(=)54 b FL(Not)1934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22072
2647 y Gc(E)1994 2633 y F4(\()2029 2621 y FX(h)2057 2633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22073
y Ga(c)2096 2621 y FX(i)2124 2633 y FL(Id)p F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22074
b(c)p F4(\))q Ga(;)2434 2621 y FX(h)2462 2633 y Ga(b)2501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22075
2621 y FX(i)2528 2633 y Ga(P)e F4(\))995 b Gg(\(7\))954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22076
2820 y FL(Not)1097 2834 y Gc(I)1137 2820 y F4(\()1172
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22077
2808 y F9(\()1200 2820 y Ga(x)1252 2808 y F9(\))1279
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22078
2820 y Ga(M)10 b(;)15 b(c)p F4(\))p Fs(^)r Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22079
Fs(_)1660 2769 y F5(def)1667 2820 y F4(=)54 b FL(Not)1934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22080
2834 y Gc(E)1994 2820 y F4(\()2029 2808 y FX(h)2057 2820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22081
y Ga(c)2096 2808 y FX(i)2124 2820 y FL(Not)2266 2834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22082
y Gc(I)2306 2820 y F4(\()2341 2808 y F9(\()2369 2820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22083
y Ga(x)2421 2808 y F9(\))2448 2820 y Ga(M)10 b Fs(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22084
Ga(\033)s Fs(_)q Ga(;)15 b(c)p F4(\))q Ga(;)2822 2808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22085
y FX(h)2850 2820 y Ga(b)2889 2808 y FX(i)2916 2820 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22086
Ga(P)e F4(\))607 b Gg(\(8\))489 3028 y Gb(Otherwise:)1213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22087
3168 y FL(Id)p F4(\()p Ga(y)s(;)15 b(a)p F4(\))p Fs(^)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22088
Ga(\033)s Fs(_)1660 3116 y F5(def)1667 3168 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22089
b FL(Id)p F4(\()p Ga(x;)15 b(a)p F4(\))729 3355 y FL(And)884
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22090
3369 y Gc(I)924 3355 y F4(\()959 3343 y FX(h)986 3355
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22091
y Ga(a)1034 3343 y FX(i)1062 3355 y Ga(M)10 b(;)1200
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22092
3343 y FX(h)1228 3355 y Ga(b)1267 3343 y FX(i)1294 3355
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22093
y Ga(N)g(;)15 b(c)p F4(\))p Fs(^)r Ga(\033)s Fs(_)1660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22094
3303 y F5(def)1667 3355 y F4(=)54 b FL(And)1946 3369
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22095
y Gc(I)1986 3355 y F4(\()2021 3343 y FX(h)2049 3355 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22096
Ga(a)2097 3343 y FX(i)2140 3355 y Ga(M)10 b Fs(^)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22097
Fs(_)q Ga(;)2398 3343 y FX(h)2426 3355 y Ga(b)2465 3343
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22098
y FX(i)2507 3355 y Ga(N)g Fs(^)q Ga(\033)s Fs(_)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22099
b(c)p F4(\))776 3542 y FL(And)931 3505 y Gc(i)931 3565
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22100
y(E)990 3542 y F4(\()1025 3530 y FX(h)1053 3542 y Ga(a)1101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22101
3530 y FX(i)1129 3542 y Ga(M)10 b(;)1267 3530 y F9(\()1295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22102
3542 y Ga(x)1347 3530 y F9(\))1374 3542 y Ga(N)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22103
Fs(^)q Ga(\033)s Fs(_)1660 3490 y F5(def)1667 3542 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22104
F4(=)54 b FL(And)1946 3505 y Gc(i)1946 3565 y(E)2006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22105
3542 y F4(\()2041 3530 y FX(h)2069 3542 y Ga(a)2117 3530
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22106
y FX(i)2159 3542 y Ga(M)10 b Fs(^)q Ga(\033)s Fs(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22107
Ga(;)2418 3530 y F9(\()2445 3542 y Ga(x)2497 3530 y F9(\))2540
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22108
3542 y Ga(N)g Fs(^)p Ga(\033)s Fs(_)q F4(\))1002 3729
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22109
y FL(Or)1101 3692 y Gc(i)1101 3752 y(I)1141 3729 y F4(\()1176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22110
3717 y FX(h)1204 3729 y Ga(a)1252 3717 y FX(i)1279 3729
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22111
y Ga(M)h(;)k(b)p F4(\))p Fs(^)q Ga(\033)s Fs(_)1660 3677
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22112
y F5(def)1667 3729 y F4(=)54 b FL(Or)1891 3692 y Gc(i)1891
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22113
3752 y(I)1931 3729 y F4(\()1966 3717 y FX(h)1994 3729
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22114
y Ga(a)2042 3717 y FX(i)2084 3729 y Ga(M)10 b Fs(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22115
Ga(\033)s Fs(_)q Ga(;)15 b(b)p F4(\))617 3916 y FL(Or)717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22116
3930 y Gc(E)776 3916 y F4(\()811 3904 y FX(h)839 3916
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22117
y Ga(a)887 3904 y FX(i)915 3916 y Ga(M)10 b(;)1053 3904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22118
y F9(\()1081 3916 y Ga(x)1133 3904 y F9(\))1160 3916
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22119
y Ga(N)g(;)1283 3904 y F9(\()1311 3916 y Ga(y)1359 3904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22120
y F9(\))1386 3916 y Ga(P)j F4(\))p Fs(^)q Ga(\033)s Fs(_)1660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22121
3864 y F5(def)1667 3916 y F4(=)54 b FL(Or)1891 3930 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22122
Gc(E)1951 3916 y F4(\()1986 3904 y FX(h)2013 3916 y Ga(a)2061
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22123
3904 y FX(i)2104 3916 y Ga(M)10 b Fs(^)q Ga(\033)s Fs(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22124
Ga(;)2362 3904 y F9(\()2390 3916 y Ga(x)2442 3904 y F9(\))2485
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22125
3916 y Ga(N)g Fs(^)p Ga(\033)s Fs(_)q Ga(;)2728 3904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22126
y F9(\()2756 3916 y Ga(y)2804 3904 y F9(\))2846 3916
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22127
y Ga(P)j Fs(^)q Ga(\033)s Fs(_)p F4(\))850 4103 y FL(Imp)994
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22128
4125 y Gc(I)1034 4103 y F4(\()1069 4091 y F9(\()1097
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22129
4103 y Ga(x)1149 4091 y F9(\))q FX(h)1204 4103 y Ga(a)1252
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22130
4091 y FX(i)1279 4103 y Ga(M)e(;)k(b)p F4(\))p Fs(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22131
Ga(\033)s Fs(_)1660 4051 y F5(def)1667 4103 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22132
b FL(Imp)1936 4125 y Gc(I)1976 4103 y F4(\()2011 4091
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22133
y F9(\()2039 4103 y Ga(x)2091 4091 y F9(\))p FX(h)2146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22134
4103 y Ga(a)2194 4091 y FX(i)2236 4103 y Ga(M)10 b Fs(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22135
Ga(\033)s Fs(_)q Ga(;)15 b(b)p F4(\))581 4290 y FL(Imp)725
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22136
4312 y Gc(E)785 4290 y F4(\()820 4278 y FX(h)848 4290
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22137
y Ga(a)896 4278 y FX(i)923 4290 y Ga(M)10 b(;)1061 4278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22138
y FX(h)1089 4290 y Ga(b)1128 4278 y FX(i)1156 4290 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22139
Ga(N)g(;)1279 4278 y F9(\()1307 4290 y Ga(x)1359 4278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22140
y F9(\))1386 4290 y Ga(P)j F4(\))p Fs(^)q Ga(\033)s Fs(_)1660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22141
4238 y F5(def)1667 4290 y F4(=)54 b FL(Imp)1936 4312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22142
y Gc(E)1996 4290 y F4(\()2031 4278 y FX(h)2059 4290 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22143
Ga(a)2107 4278 y FX(i)2149 4290 y Ga(M)10 b Fs(^)q Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22144
Fs(_)p Ga(;)2407 4278 y FX(h)2435 4290 y Ga(b)2474 4278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22145
y FX(i)2517 4290 y Ga(N)g Fs(^)p Ga(\033)s Fs(_)q Ga(;)2760
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22146
4278 y F9(\()2788 4290 y Ga(x)2840 4278 y F9(\))2882
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22147
4290 y Ga(P)j Fs(^)q Ga(\033)s Fs(_)q F4(\))955 4477
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22148
y FL(Not)1097 4491 y Gc(I)1137 4477 y F4(\()1172 4465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22149
y F9(\()1200 4477 y Ga(x)1252 4465 y F9(\))1279 4477
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22150
y Ga(M)e(;)k(b)p F4(\))p Fs(^)q Ga(\033)s Fs(_)1660 4425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22151
y F5(def)1667 4477 y F4(=)54 b FL(Not)1934 4491 y Gc(I)1974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22152
4477 y F4(\()2009 4465 y F9(\()2037 4477 y Ga(x)2089
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22153
4465 y F9(\))2132 4477 y Ga(M)10 b Fs(^)p Ga(\033)s Fs(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22154
Ga(;)15 b(b)p F4(\))801 4664 y FL(Not)944 4678 y Gc(E)1003
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22155
4664 y F4(\()1038 4652 y FX(h)1066 4664 y Ga(a)1114 4652
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22156
y FX(i)1142 4664 y Ga(M)10 b(;)1280 4652 y FX(h)1308
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22157
4664 y Ga(b)1347 4652 y FX(i)1374 4664 y Ga(N)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22158
Fs(^)q Ga(\033)s Fs(_)1660 4612 y F5(def)1667 4664 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22159
F4(=)54 b FL(Not)1934 4678 y Gc(E)1994 4664 y F4(\()2029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22160
4652 y FX(h)2057 4664 y Ga(a)2105 4652 y FX(i)2147 4664
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22161
y Ga(M)10 b Fs(^)q Ga(\033)s Fs(_)q Ga(;)2406 4652 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22162
FX(h)2434 4664 y Ga(b)2473 4652 y FX(i)2515 4664 y Ga(N)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22163
Fs(^)q Ga(\033)s Fs(_)p F4(\))778 4851 y FL(Subst)o F4(\()1025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22164
4839 y FX(h)1053 4851 y Ga(a)1101 4839 y FX(i)1129 4851
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22165
y Ga(M)g(;)1267 4839 y F9(\()1295 4851 y Ga(x)1347 4839
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22166
y F9(\))1374 4851 y Ga(N)g F4(\))p Fs(^)q Ga(\033)s Fs(_)1660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22167
4799 y F5(def)1667 4851 y F4(=)54 b FL(Subst)o F4(\()2039
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22168
4839 y FX(h)2067 4851 y Ga(a)2115 4839 y FX(i)2157 4851
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22169
y Ga(M)10 b Fs(^)q Ga(\033)s Fs(_)q Ga(;)2416 4839 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22170
F9(\()2443 4851 y Ga(x)2495 4839 y F9(\))2538 4851 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22171
Ga(N)g Fs(^)p Ga(\033)s Fs(_)q F4(\))p 3965 4986 4 4510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22172
v 277 4989 3691 4 v Black 916 5142 a Gg(Figure)24 b(3.10:)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22173
b(Proof)23 b(substitution)k(in)d(natural)h(deduction)h(I.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22174
Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22175
%%Page: 92 104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22176
TeXDict begin 92 103 bop Black -144 51 a Gb(92)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22177
b(Deduction)p -144 88 3691 4 v Black Black -144 934 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22178
-144 4528 4 3594 v 142 1085 a(Let)g Fs(\()p FO(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22179
Fs(\))h Gb(be)f(of)g(the)g(f)n(orm)h Fs(\()-7 b FO(x)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22180
b FD(:=)1271 1073 y Fz(h)1303 1085 y FO(c)1350 1073 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22181
Fz(i)1381 1085 y FO(P)7 b Fs(\))q Gb(,)22 b(then:)767
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22182
1272 y FL(Id)q F4(\()p Ga(x;)15 b(a)p F4(\))p Fs(\()q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22183
Ga(\033)s Fs(\))1218 1221 y F5(def)1225 1272 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22184
b Ga(P)13 b F4([)p Ga(c)d F6(7!)g Ga(a)p F4(])1465 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22185
Gg(\(1\))142 1480 y Gb(Let)23 b Fs(\()p FO(\033)s Fs(\))h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22186
Gb(be)f(of)g(the)g(f)n(orm)h Fs(\()-7 b FO(c)29 b FD(:=)1258
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22187
1468 y Fx(\()1290 1480 y FO(x)1350 1468 y Fx(\))1381
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22188
1480 y FO(P)7 b Fs(\))q Gb(,)22 b(then:)776 1667 y FL(Id)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22189
F4(\()p Ga(x;)15 b(c)p F4(\))p Fs(\()r Ga(\033)s Fs(\))1218
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22190
1615 y F5(def)1225 1667 y F4(=)54 b Ga(P)13 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22191
Ga(y)g F6(7!)d Ga(x)p F4(])1452 b Gg(\(2\))287 1854 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22192
FL(And)442 1868 y Gc(I)482 1854 y F4(\()517 1842 y FX(h)545
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22193
1854 y Ga(a)593 1842 y FX(i)620 1854 y Ga(M)10 b(;)758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22194
1842 y FX(h)786 1854 y Ga(b)825 1842 y FX(i)853 1854
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22195
y Ga(N)g(;)15 b(c)p F4(\))p Fs(\()q Ga(\033)s Fs(\))1218
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22196
1802 y F5(def)1225 1854 y F4(=)54 b FL(Subst)o F4(\()1597
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22197
1842 y FX(h)1625 1854 y Ga(c)1664 1842 y FX(i)1692 1854
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22198
y FL(And)1846 1868 y Gc(I)1886 1854 y F4(\()1921 1842
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22199
y FX(h)1949 1854 y Ga(a)1997 1842 y FX(i)2025 1854 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22200
Ga(M)10 b Fs(\()p Ga(\033)s Fs(\))q Ga(;)2283 1842 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22201
FX(h)2311 1854 y Ga(b)2350 1842 y FX(i)2377 1854 y Ga(N)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22202
Fs(\()q Ga(\033)s Fs(\))p Ga(;)15 b(c)p F4(\))q Ga(;)2735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22203
1842 y F9(\()2763 1854 y Ga(y)2811 1842 y F9(\))2838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22204
1854 y Ga(P)e F4(\))190 b Gg(\(3\))560 2041 y FL(Or)659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22205
2004 y Gc(i)659 2064 y(I)699 2041 y F4(\()734 2029 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22206
FX(h)762 2041 y Ga(a)810 2029 y FX(i)837 2041 y Ga(M)11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22207
b(;)k(c)p F4(\))p Fs(\()q Ga(\033)s Fs(\))1218 1989 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22208
F5(def)1225 2041 y F4(=)54 b FL(Subst)o F4(\()1597 2029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22209
y FX(h)1625 2041 y Ga(c)1664 2029 y FX(i)1692 2041 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22210
FL(Or)1791 2004 y Gc(i)1791 2064 y(I)1831 2041 y F4(\()1866
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22211
2029 y FX(h)1894 2041 y Ga(a)1942 2029 y FX(i)1969 2041
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22212
y Ga(M)10 b Fs(\()q Ga(\033)s Fs(\))q Ga(;)15 b(c)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22213
Ga(;)2343 2029 y F9(\()2370 2041 y Ga(y)2418 2029 y F9(\))2446
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22214
2041 y Ga(P)e F4(\))582 b Gg(\(4\))408 2228 y FL(Imp)552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22215
2250 y Gc(I)592 2228 y F4(\()627 2216 y F9(\()655 2228
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22216
y Ga(x)707 2216 y F9(\))q FX(h)762 2228 y Ga(a)810 2216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22217
y FX(i)837 2228 y Ga(M)11 b(;)k(c)p F4(\))p Fs(\()q Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22218
Fs(\))1218 2176 y F5(def)1225 2228 y F4(=)54 b FL(Subst)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22219
F4(\()1597 2216 y FX(h)1625 2228 y Ga(c)1664 2216 y FX(i)1692
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22220
2228 y FL(Imp)1836 2250 y Gc(I)1876 2228 y F4(\()1911
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22221
2216 y F9(\()1939 2228 y Ga(x)1991 2216 y F9(\))p FX(h)2046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22222
2228 y Ga(a)2094 2216 y FX(i)2121 2228 y Ga(M)10 b Fs(\()q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22223
Ga(\033)s Fs(\))q Ga(;)15 b(c)p F4(\))q Ga(;)2495 2216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22224
y F9(\()2523 2228 y Ga(y)2571 2216 y F9(\))2598 2228
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22225
y Ga(P)e F4(\))430 b Gg(\(5\))513 2415 y FL(Not)655 2429
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22226
y Gc(I)695 2415 y F4(\()730 2403 y F9(\()758 2415 y Ga(x)810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22227
2403 y F9(\))837 2415 y Ga(M)11 b(;)k(c)p F4(\))p Fs(\()q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22228
Ga(\033)s Fs(\))1218 2363 y F5(def)1225 2415 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22229
b FL(Subst)o F4(\()1597 2403 y FX(h)1625 2415 y Ga(c)1664
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22230
2403 y FX(i)1692 2415 y FL(Not)1835 2429 y Gc(I)1874
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22231
2415 y F4(\()1909 2403 y F9(\()1937 2415 y Ga(x)1989
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22232
2403 y F9(\))2017 2415 y Ga(M)10 b Fs(\()p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22233
Fs(\))q Ga(;)15 b(c)p F4(\))q Ga(;)2390 2403 y F9(\()2418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22234
2415 y Ga(y)2466 2403 y F9(\))2493 2415 y Ga(P)e F4(\))535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22235
b Gg(\(6\))142 2599 y Gb(Otherwise:)771 2738 y FL(Id)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22236
F4(\()p Ga(y)s(;)15 b(a)p F4(\))p Fs(\()q Ga(\033)s Fs(\))1218
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22237
2687 y F5(def)1225 2738 y F4(=)54 b FL(Id)p F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22238
b(a)p F4(\))287 2925 y FL(And)442 2939 y Gc(I)482 2925
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22239
y F4(\()517 2913 y FX(h)545 2925 y Ga(a)593 2913 y FX(i)620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22240
2925 y Ga(M)10 b(;)758 2913 y FX(h)786 2925 y Ga(b)825
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22241
2913 y FX(i)853 2925 y Ga(N)g(;)15 b(c)p F4(\))p Fs(\()q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22242
Ga(\033)s Fs(\))1218 2874 y F5(def)1225 2925 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22243
b FL(And)1505 2939 y Gc(I)1545 2925 y F4(\()1580 2913
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22244
y FX(h)1607 2925 y Ga(a)1655 2913 y FX(i)1698 2925 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22245
Ga(M)10 b Fs(\()q Ga(\033)s Fs(\))p Ga(;)1956 2913 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22246
FX(h)1984 2925 y Ga(b)2023 2913 y FX(i)2066 2925 y Ga(N)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22247
Fs(\()p Ga(\033)s Fs(\))q Ga(;)15 b(c)p F4(\))335 3112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22248
y FL(And)489 3076 y Gc(i)489 3136 y(E)549 3112 y F4(\()584
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22249
3100 y FX(h)612 3112 y Ga(a)660 3100 y FX(i)687 3112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22250
y Ga(M)10 b(;)825 3100 y F9(\()853 3112 y Ga(x)905 3100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22251
y F9(\))932 3112 y Ga(N)g F4(\))p Fs(\()q Ga(\033)s Fs(\))1218
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22252
3061 y F5(def)1225 3112 y F4(=)54 b FL(And)1505 3076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22253
y Gc(i)1505 3136 y(E)1564 3112 y F4(\()1599 3100 y FX(h)1627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22254
3112 y Ga(a)1675 3100 y FX(i)1718 3112 y Ga(M)10 b Fs(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22255
Ga(\033)s Fs(\))q Ga(;)1976 3100 y F9(\()2004 3112 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22256
Ga(x)2056 3100 y F9(\))2098 3112 y Ga(N)g Fs(\()q Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22257
Fs(\))p F4(\))560 3300 y FL(Or)660 3263 y Gc(i)660 3323
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22258
y(I)700 3300 y F4(\()735 3288 y FX(h)762 3300 y Ga(a)810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22259
3288 y FX(i)838 3300 y Ga(M)g(;)15 b(b)p F4(\))p Fs(\()q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22260
Ga(\033)s Fs(\))1218 3248 y F5(def)1225 3300 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22261
b FL(Or)1449 3263 y Gc(i)1449 3323 y(I)1489 3300 y F4(\()1524
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22262
3288 y FX(h)1552 3300 y Ga(a)1600 3288 y FX(i)1643 3300
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22263
y Ga(M)10 b Fs(\()p Ga(\033)s Fs(\))q Ga(;)15 b(b)p F4(\))176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22264
3487 y FL(Or)275 3501 y Gc(E)335 3487 y F4(\()370 3475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22265
y FX(h)397 3487 y Ga(a)445 3475 y FX(i)473 3487 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22266
b(;)611 3475 y F9(\()639 3487 y Ga(x)691 3475 y F9(\))718
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22267
3487 y Ga(N)g(;)841 3475 y F9(\()869 3487 y Ga(y)917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22268
3475 y F9(\))944 3487 y Ga(P)j F4(\))p Fs(\()q Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22269
Fs(\))1218 3435 y F5(def)1225 3487 y F4(=)54 b FL(Or)1449
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22270
3501 y Gc(E)1509 3487 y F4(\()1544 3475 y FX(h)1572 3487
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22271
y Ga(a)1620 3475 y FX(i)1662 3487 y Ga(M)10 b Fs(\()q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22272
Ga(\033)s Fs(\))q Ga(;)1921 3475 y F9(\()1949 3487 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22273
Ga(x)2001 3475 y F9(\))2043 3487 y Ga(N)g Fs(\()p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22274
Fs(\))q Ga(;)2286 3475 y F9(\()2314 3487 y Ga(y)2362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22275
3475 y F9(\))2404 3487 y Ga(P)j Fs(\()q Ga(\033)s Fs(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22276
F4(\))408 3674 y FL(Imp)553 3695 y Gc(I)593 3674 y F4(\()628
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22277
3662 y F9(\()655 3674 y Ga(x)707 3662 y F9(\))q FX(h)762
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22278
3674 y Ga(a)810 3662 y FX(i)838 3674 y Ga(M)d(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22279
F4(\))p Fs(\()q Ga(\033)s Fs(\))1218 3622 y F5(def)1225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22280
3674 y F4(=)54 b FL(Imp)1495 3695 y Gc(I)1535 3674 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22281
F4(\()1570 3662 y F9(\()1597 3674 y Ga(x)1649 3662 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22282
F9(\))q FX(h)1704 3674 y Ga(a)1752 3662 y FX(i)1795 3674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22283
y Ga(M)10 b Fs(\()p Ga(\033)s Fs(\))q Ga(;)15 b(b)p F4(\))139
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22284
3861 y FL(Imp)284 3882 y Gc(E)343 3861 y F4(\()378 3849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22285
y FX(h)406 3861 y Ga(a)454 3849 y FX(i)482 3861 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22286
b(;)620 3849 y FX(h)648 3861 y Ga(b)687 3849 y FX(i)714
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22287
3861 y Ga(N)g(;)837 3849 y F9(\()865 3861 y Ga(x)917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22288
3849 y F9(\))944 3861 y Ga(P)j F4(\))p Fs(\()q Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22289
Fs(\))1218 3809 y F5(def)1225 3861 y F4(=)54 b FL(Imp)1495
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22290
3882 y Gc(E)1554 3861 y F4(\()1589 3849 y FX(h)1617 3861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22291
y Ga(a)1665 3849 y FX(i)1708 3861 y Ga(M)10 b Fs(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22292
Ga(\033)s Fs(\))q Ga(;)1966 3849 y FX(h)1994 3861 y Ga(b)2033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22293
3849 y FX(i)2075 3861 y Ga(N)g Fs(\()q Ga(\033)s Fs(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22294
Ga(;)2318 3849 y F9(\()2346 3861 y Ga(x)2398 3849 y F9(\))2441
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22295
3861 y Ga(P)j Fs(\()p Ga(\033)s Fs(\))q F4(\))513 4048
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22296
y FL(Not)656 4062 y Gc(I)696 4048 y F4(\()731 4036 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22297
F9(\()758 4048 y Ga(x)810 4036 y F9(\))838 4048 y Ga(M)d(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22298
b(b)p F4(\))p Fs(\()q Ga(\033)s Fs(\))1218 3996 y F5(def)1225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22299
4048 y F4(=)54 b FL(Not)1493 4062 y Gc(I)1533 4048 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22300
F4(\()1568 4036 y F9(\()1596 4048 y Ga(x)1648 4036 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22301
F9(\))1690 4048 y Ga(M)10 b Fs(\()q Ga(\033)s Fs(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22302
Ga(;)15 b(b)p F4(\))359 4235 y FL(Not)502 4249 y Gc(E)562
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22303
4235 y F4(\()597 4223 y FX(h)624 4235 y Ga(a)672 4223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22304
y FX(i)700 4235 y Ga(M)10 b(;)838 4223 y FX(h)866 4235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22305
y Ga(b)905 4223 y FX(i)932 4235 y Ga(N)g F4(\))p Fs(\()q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22306
Ga(\033)s Fs(\))1218 4183 y F5(def)1225 4235 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22307
b FL(Not)1493 4249 y Gc(E)1552 4235 y F4(\()1587 4223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22308
y FX(h)1615 4235 y Ga(a)1663 4223 y FX(i)1706 4235 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22309
Ga(M)10 b Fs(\()p Ga(\033)s Fs(\))q Ga(;)1964 4223 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22310
FX(h)1992 4235 y Ga(b)2031 4223 y FX(i)2074 4235 y Ga(N)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22311
Fs(\()p Ga(\033)s Fs(\))q F4(\))337 4422 y FL(Subst)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22312
F4(\()584 4410 y FX(h)612 4422 y Ga(a)660 4410 y FX(i)687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22313
4422 y Ga(M)g(;)825 4410 y F9(\()853 4422 y Ga(x)905
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22314
4410 y F9(\))932 4422 y Ga(N)g F4(\))p Fs(\()q Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22315
Fs(\))1218 4370 y F5(def)1225 4422 y F4(=)54 b FL(Subst)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22316
F4(\()1597 4410 y FX(h)1625 4422 y Ga(a)1673 4410 y FX(i)1716
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22317
4422 y Ga(M)10 b Fs(\()p Ga(\033)s Fs(\))q Ga(;)1974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22318
4410 y F9(\()2002 4422 y Ga(x)2054 4410 y F9(\))2096
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22319
4422 y Ga(N)g Fs(\()q Ga(\033)s Fs(\))p F4(\))p 3543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22320
4528 V -144 4531 3691 4 v Black 945 4684 a Gg(Figure)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22321
b(3.11:)30 b(Proof)23 b(substitution)k(in)d(natural)h(deduction)h(II.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22322
Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22323
%%Page: 93 105
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22324
TeXDict begin 93 104 bop Black 277 51 a Gb(3.4)23 b(The)g(Corr)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22325
(espondence)h(between)f(Cut-Elimination)g(and)f(Normalisation)k(II)814
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22326
b(93)p 277 88 3691 4 v Black Black 277 365 a(De\002nition)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22327
b(3.3.9)g Gg(\(Substitution)k(Elimination\))p Gb(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22328
Black Black Black 463 555 a FL(Subst)p F4(\()711 543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22329
y FX(h)738 555 y Ga(a)786 543 y FX(i)814 555 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22330
b(;)952 543 y F9(\()980 555 y Ga(x)1032 543 y F9(\))1059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22331
555 y Ga(N)g F4(\))1341 518 y Gc(\033)1277 555 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22332
b(\000)f(!)100 b Ga(M)10 b Fs(\()-6 b Ga(a)25 b F4(:=)1865
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22333
543 y F9(\()1892 555 y Ga(x)1944 543 y F9(\))1972 555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22334
y Ga(M)s Fs(\))48 b Gg(if)24 b Ga(M)32 b Gg(does)25 b(not)f(freshly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22335
(introduce)h Ga(a)p Gg(,)c(or)1341 655 y Gc(\033)1277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22336
692 y F6(\000)-31 b(\000)f(!)115 b Ga(N)10 b Fs(\()-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22337
b Ga(x)26 b F4(:=)1869 680 y FX(h)1896 692 y Ga(a)1944
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22338
680 y FX(i)1972 692 y Ga(M)s Fs(\))418 937 y Gg(T)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22339
b(o)23 b(sum)g(up)g(the)h(discussion)i(about)f(the)f(reduction)i(rules)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22340
e(let)f(us)h(combine)g(them)g(in)f(the)h(de\002ni-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22341
1050 y(tion)g(of)g(a)f(normalisation)k(procedure)f(for)d(classical)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22342
(natural)f(deduction)i(proofs.)p Black 277 1237 a Gb(De\002nition)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22343
(3.3.10)h Gg(\(Normalisation)i(for)e(Classical)h(Natural)f(Deduction)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22344
(Proofs\))p Gb(:)p Black 277 1350 a Gg(The)d(normalisation)j(procedure)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22345
f(for)e(classical)i(natural)g(deduction)g(proofs,)g F4(\(K)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22346
Ga(;)2942 1313 y Gc(\024)2877 1350 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22347
F4(\))p Gg(,)22 b(is)g(a)f(reduc-)277 1463 y(tion)j(system)h(where:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22348
Black 414 1666 a F6(\017)p Black 45 w F4(K)e Gg(is)g(the)h(set)g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22349
(well-typed)j(terms,)d(and)p Black 414 1853 a F6(\017)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22350
Black 569 1816 a Gc(\024)504 1853 y F6(\000)-31 b(\000)g(!)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22351
b Gg(consists)k(of)d(beta-reductions,)25 b(commuting)c(reductions)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22352
(and)e(reductions)i(to)e(eliminate)504 1966 y(substitutions;)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22353
b(that)c(is)1511 2056 y Gc(\024)1446 2093 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22354
b(\000)f(!)1642 2042 y F5(def)1649 2093 y F4(=)1815 2056
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22355
y Gc(\014)1752 2093 y F6(\000)g(\000)h(!)25 b([)2098
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22356
2056 y Gc(\015)2033 2093 y F6(\000)-31 b(\000)f(!)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22357
b([)2378 2056 y Gc(\033)2314 2093 y F6(\000)-31 b(\000)g(!)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22358
b Ga(:)277 2347 y Gg(W)-7 b(e)23 b(should)i(lik)o(e)f(to)g(state)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22359
(that)1349 2310 y Gc(\024)1284 2347 y F6(\000)-31 b(\000)f(!)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22360
b Gg(respects)j(the)d(subject)j(reduction)g(property)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22361
b(.)p Black 277 2535 a Gb(Pr)n(oposition)33 b(3.3.11)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22362
Gg(\(Subject)h(Reduction\))p Gb(:)p Black 60 w Gg(Suppose)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22363
Ga(M)50 b F6(2)40 b F4(K)31 b Gg(with)h(the)g(typing)h(judgement)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22364
2648 y F4(\000)359 2636 y Gc(.)414 2648 y Ga(M)537 2636
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22365
y Gc(.)592 2648 y F4(\001)23 b Gg(and)h Ga(M)1033 2611
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22366
y Gc(\024)968 2648 y F6(\000)-31 b(\000)g(!)25 b Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22367
b Gg(,)22 b(then)i Ga(N)36 b F6(2)24 b F4(K)f Gg(with)g(the)h(typing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22368
(judgement)h F4(\000)2821 2636 y Gc(.)2876 2648 y Ga(N)2984
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22369
2636 y Gc(.)3039 2648 y F4(\001)o Gg(.)p Black 277 2885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22370
a F7(Pr)l(oof.)p Black 46 w Gg(Analogous)g(to)d(proof)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22371
(Proposition)j(2.2.10.)p 3436 2885 4 62 v 3440 2827 55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22372
4 v 3440 2885 V 3494 2885 4 62 v 277 3089 a(As)19 b(mentioned)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22373
(earlier)l(,)h(the)e(reduction)1627 3052 y Gc(\024)1562
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22374
3089 y F6(\000)-31 b(\000)f(!)19 b Gg(is)g(complete)i(in)e(the)h(sense)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22375
h(that)f(to)f(e)n(v)o(ery)h(non-normal)277 3202 y(natural)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22376
b(deduction)i(proof)e(a)e(reduction)j(rule)e(can)g(be)g(applied.)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22377
b(T)-7 b(o)25 b(sho)n(w)i(that)g(e)n(v)o(ery)g(non-normal)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22378
3315 y(natural)33 b(deduction)h(proof)f(can)f(be)f(reduced)j(to)d(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22379
(normal)h(one,)i(we)d(ha)n(v)o(e)h(to)f(sho)n(w)h(that)3305
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22380
3278 y Gc(\024)3241 3315 y F6(\000)-32 b(\000)h(!)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22381
b Gg(is)277 3428 y(terminating.)i(This)24 b(is)g(what)g(we)f(shall)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22382
(do)f(ne)o(xt)g(by)h(sho)n(wing)g(the)f(correspondence)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22383
b(between)3397 3391 y Gc(\024)3332 3428 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22384
b(\000)f(!)277 3541 y Gg(and)462 3503 y Gc(cut)431 3541
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22385
y F6(\000)h(\000)f(!)p Gg(.)277 3849 y Ge(3.4)119 b(The)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22386
b(Corr)n(espondence)i(between)f(Cut-Elimination)g(and)g(Nor)l(-)546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22387
3999 y(malisation)29 b(II)277 4222 y Gg(In)h(this)g(section)i(we)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22388
(shall)h(sho)n(w)g(the)g(correspondence)35 b(between)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22389
b(normalisation)j(and)e(cut-elimi-)277 4335 y(nation)f(in)d(classical)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22390
(logic.)43 b(A)27 b(byproduct,)32 b(we)27 b(shall)i(sho)n(w)e(that)2500
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22391
4298 y Gc(\024)2435 4335 y F6(\000)-31 b(\000)f(!)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22392
b Gg(is)h(strongly)i(normalising.)277 4448 y(The)d(lengthy)i(part)f(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22393
f(this)h(proof)h(is)e(to)g(establish)j(the)e(correspondence)k(between)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22394
(the)g(proof)h(sub-)277 4561 y(stitutions)d Fs(^)p 668
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22395
4561 28 4 v 685 4561 V 703 4561 V 65 w(_)q Gg(,)c Fs(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22396
843 4561 V 860 4561 V 878 4561 V 65 w(\))h Gg(and)h F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22397
1141 4561 V 1159 4561 V 1177 4561 V 65 w(])p Gg(.)k(This)c(is)f(what)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22398
(we)e(shall)j(do)e(\002rst.)p Black 277 4749 a Gb(Lemma)g(3.4.1:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22399
Black 34 w Gg(F)o(or)g(all)g Ga(M)5 b(;)15 b(N)36 b F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22400
b F4(K)p Gg(,)57 b F6(j)p Ga(M)10 b F6(j)1731 4716 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22401
Fu(S)1775 4749 y F4([)p Ga(a)26 b F4(:=)1995 4737 y F9(\()2023
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22402
4749 y Ga(x)2075 4737 y F9(\))2102 4749 y F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22403
b F6(j)2235 4716 y Fu(S)2279 4749 y F4(])25 b F6(\021)g(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22404
Ga(M)10 b Fs(\()-6 b Ga(a)25 b F4(:=)2769 4737 y F9(\()2796
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22405
4749 y Ga(x)2848 4737 y F9(\))2875 4749 y Ga(N)s Fs(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22406
F6(j)3009 4716 y Fu(S)3052 4749 y Gg(.)p Black 277 4961
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22407
a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)e(induction)j(on)d(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22408
(structure)i(of)d Ga(M)33 b Gg(\(see)24 b(P)o(age)f(161\).)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22409
3436 4961 4 62 v 3440 4903 55 4 v 3440 4961 V 3494 4961
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22410
4 62 v Black 277 5149 a Gb(Lemma)g(3.4.2:)p Black 34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22411
w Gg(F)o(or)g(all)g Ga(M)5 b(;)15 b(N)36 b F6(2)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22412
F4(K)p Gg(,)57 b F6(j)p Ga(M)10 b F6(j)1731 5116 y Fu(S)1775
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22413
5149 y F4([)p Ga(x)26 b F4(:=)1999 5137 y FX(h)2027 5149
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22414
y Ga(a)2075 5137 y FX(i)2102 5149 y F6(j)p Ga(N)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22415
F6(j)2235 5116 y Fu(S)2279 5149 y F4(])2360 5112 y Gc(cut)2329
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22416
5149 y F6(\000)-31 b(\000)g(!)2500 5116 y FX(\003)2564
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22417
5149 y F6(j)p Ga(M)10 b Fs(\()-6 b Ga(x)25 b F4(:=)2911
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22418
5137 y FX(h)2939 5149 y Ga(a)2987 5137 y FX(i)3014 5149
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22419
y Ga(N)s Fs(\))q F6(j)3148 5116 y Fu(S)3191 5149 y Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22420
Black 277 5361 a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)e(induction)j(on)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22421
(the)h(structure)i(of)d Ga(M)33 b Gg(\(see)24 b(P)o(age)f(161\).)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22422
3436 5361 V 3440 5303 55 4 v 3440 5361 V 3494 5361 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22423
62 v Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22424
%%Page: 94 106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22425
TeXDict begin 94 105 bop Black -144 51 a Gb(94)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22426
b(Deduction)p -144 88 3691 4 v Black Black 321 365 a(Lemma)f(3.4.3:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22427
Black 34 w Gg(F)o(or)f(all)h Ga(M)5 b(;)15 b(N)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22428
b(P)s(;)g(Q)27 b F6(2)d F4(K)d Gg(with)h Ga(z)30 b F6(62)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22429
b Ga(F)13 b(N)d F4(\()2302 353 y F9(\()2330 365 y Ga(x)2382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22430
353 y F9(\))2409 365 y F6(j)p Ga(N)g F6(j)2542 332 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22431
Fu(S)2586 365 y Ga(;)2626 353 y F9(\()2654 365 y Ga(y)2702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22432
353 y F9(\))2729 365 y F6(j)p Ga(P)j F6(j)2850 332 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22433
Fu(S)2894 365 y Ga(;)2934 353 y FX(h)2962 365 y Ga(b)3001
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22434
353 y FX(i)3028 365 y F6(j)p Ga(Q)p F6(j)3150 332 y Fu(S)3194
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22435
365 y F4(\))22 b Gg(we)f(ha)n(v)o(e)p Black Black 840
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22436
637 a F6(j)p Ga(M)10 b F6(j)988 604 y Fu(S)1032 637 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22437
F4([)p Ga(a)26 b F4(:=)1252 625 y F9(\()1279 637 y Ga(z)1325
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22438
625 y F9(\))1353 637 y FL(And)1508 600 y Gc(i)1508 660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22439
y(L)1560 637 y F4(\()1595 625 y F9(\()1623 637 y Ga(x)1675
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22440
625 y F9(\))1717 637 y F6(j)p Ga(N)10 b F6(j)1850 604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22441
y Fu(S)1894 637 y Ga(;)15 b(z)t F4(\))q(])43 b F6(\021)f(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22442
Ga(M)10 b Fs(^)q FL(And)2507 600 y Gc(i)2507 660 y(E)2567
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22443
637 y F4(\()2602 625 y FX(h)2630 637 y Ga(a)2678 625
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22444
y FX(i)p 2707 637 28 4 v 2725 637 V 2743 637 V 2770 637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22445
a Ga(;)2810 625 y F9(\()2838 637 y Ga(x)2890 625 y F9(\))2917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22446
637 y Ga(N)g F4(\))p Fs(_)q F6(j)3093 604 y Fu(S)572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22447
796 y F6(j)p Ga(M)g F6(j)720 763 y Fu(S)764 796 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22448
Ga(a)26 b F4(:=)984 784 y F9(\()1012 796 y Ga(z)1058
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22449
784 y F9(\))1085 796 y FL(Or)1185 810 y Gc(L)1237 796
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22450
y F4(\()1272 784 y F9(\()1299 796 y Ga(x)1351 784 y F9(\))1394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22451
796 y F6(j)p Ga(N)10 b F6(j)1527 763 y Fu(S)1571 796
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22452
y Ga(;)1611 784 y F9(\()1639 796 y Ga(y)1687 784 y F9(\))1729
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22453
796 y F6(j)p Ga(P)j F6(j)1850 763 y Fu(S)1894 796 y Ga(;)i(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22454
F4(\))q(])43 b F6(\021)f(j)p Ga(M)10 b Fs(^)q FL(Or)2452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22455
810 y Gc(E)2512 796 y F4(\()2547 784 y FX(h)2574 796
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22456
y Ga(a)2622 784 y FX(i)p 2652 796 V 2670 796 V 2687 796
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22457
V 2714 796 a Ga(;)2754 784 y F9(\()2782 796 y Ga(x)2834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22458
784 y F9(\))2862 796 y Ga(N)g(;)2985 784 y F9(\()3013
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22459
796 y Ga(y)3061 784 y F9(\))3088 796 y Ga(P)j F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22460
Fs(_)q F6(j)3252 763 y Fu(S)551 956 y F6(j)p Ga(M)d F6(j)699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22461
923 y Fu(S)743 956 y F4([)p Ga(a)26 b F4(:=)963 944 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22462
F9(\()990 956 y Ga(z)1036 944 y F9(\))1064 956 y FL(Imp)1209
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22463
977 y Gc(L)1261 956 y F4(\()1296 944 y FX(h)1324 956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22464
y Ga(b)1363 944 y FX(i)1405 956 y F6(j)p Ga(Q)p F6(j)1527
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22465
923 y Fu(S)1571 956 y Ga(;)1611 944 y F9(\()1639 956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22466
y Ga(y)1687 944 y F9(\))1729 956 y F6(j)p Ga(P)13 b F6(j)1850
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22467
923 y Fu(S)1894 956 y Ga(;)i(z)t F4(\))q(])43 b F6(\021)f(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22468
Ga(M)10 b Fs(^)q FL(Imp)2497 977 y Gc(E)2557 956 y F4(\()2592
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22469
944 y FX(h)2620 956 y Ga(a)2668 944 y FX(i)p 2697 956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22470
V 2715 956 V 2732 956 V 2760 956 a Ga(;)2800 944 y FX(h)2828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22471
956 y Ga(b)2867 944 y FX(i)2894 956 y Ga(Q;)3006 944
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22472
y F9(\()3034 956 y Ga(y)3082 944 y F9(\))3109 956 y Ga(P)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22473
F4(\))p Fs(_)q F6(j)3273 923 y Fu(S)876 1115 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22474
Ga(M)d F6(j)1024 1082 y Fu(S)1068 1115 y F4([)p Ga(a)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22475
b F4(:=)1288 1103 y F9(\()1315 1115 y Ga(z)1361 1103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22476
y F9(\))1389 1115 y FL(Not)1532 1129 y Gc(L)1584 1115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22477
y F4(\()1619 1103 y FX(h)1647 1115 y Ga(b)1686 1103 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22478
FX(i)1728 1115 y F6(j)p Ga(Q)p F6(j)1850 1082 y Fu(S)1894
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22479
1115 y Ga(;)15 b(z)t F4(\))q(])43 b F6(\021)f(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22480
b Fs(^)q FL(Not)2496 1129 y Gc(E)2555 1115 y F4(\()2590
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22481
1103 y FX(h)2618 1115 y Ga(a)2666 1103 y FX(i)p 2696
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22482
1115 V 2713 1115 V 2731 1115 V 2758 1115 a Ga(;)2798
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22483
1103 y FX(h)2826 1115 y Ga(b)2865 1103 y FX(i)2892 1115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22484
y Ga(Q)p F4(\))p Fs(_)q F6(j)3057 1082 y Fu(S)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22485
321 1325 a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)23 b(induction)j(on)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22486
(the)f(structure)j(of)e Ga(M)32 b Gg(\(see)25 b(P)o(age)e(162\).)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22487
3480 1325 4 62 v 3484 1267 55 4 v 3484 1325 V 3538 1325
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22488
4 62 v 321 1543 a(No)n(w)31 b(we)g(are)h(in)f(a)h(position)i(to)d(sho)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22489
(w)h(that)g(under)h F6(j)p 2057 1543 28 4 v 2075 1543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22490
V 2093 1543 V 65 w(j)2145 1510 y Fu(S)2220 1543 y Gg(e)n(v)o(ery)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22491
(normalisation)j(reduction)f(maps)321 1656 y(onto)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22492
b(a)e(series)h(of)g(cut-reductions.)p Black 321 1844
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22493
a Gb(Theor)n(em)g(3.4.4:)p Black 34 w Gg(F)o(or)f(all)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22494
Ga(M)5 b(;)15 b(N)36 b F6(2)25 b F4(K)p Gg(,)d(if)h Ga(M)1913
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22495
1807 y Gc(\024)1848 1844 y F6(\000)-31 b(\000)f(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22496
b Ga(N)10 b Gg(,)23 b(then)h F6(j)p Ga(M)10 b F6(j)2499
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22497
1811 y Fu(S)2599 1807 y Gc(cut)2568 1844 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22498
b(\000)f(!)2738 1811 y F9(+)2823 1844 y F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22499
b F6(j)2956 1811 y Fu(S)2999 1844 y Gg(.)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22500
2057 a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)23 b(induction)j(on)e(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22501
(de\002nition)j(of)1856 2019 y Gc(\024)1791 2057 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22502
b(\000)f(!)23 b Gg(\(see)h(P)o(age)f(163\).)p 3480 2057
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22503
4 62 v 3484 1999 55 4 v 3484 2057 V 3538 2057 4 62 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22504
321 2275 a(Since)e(we)e(ha)n(v)o(e)i(sho)n(wn)f(that)1331
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22505
2238 y Gc(\024)1266 2275 y F6(\000)-31 b(\000)f(!)p Gg(-reductions)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22506
b(can)c(be)g(simulated)i(by)2629 2238 y Gc(cut)2599 2275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22507
y F6(\000)-32 b(\000)h(!)p Gg(,)20 b(we)f(e)o(xpect)i(that)g(e)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22508
(ery)386 2350 y Gc(\024)321 2388 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22509
Gg(-reduction)26 b(sequence)g(terminates.)k(This)24 b(is)f(indeed)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22510
(the)f(case.)p Black 321 2575 a Gb(Theor)n(em)g(3.4.5:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22511
Black 34 w Gg(The)f(reduction)j(system)f F4(\(K)p Ga(;)1958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22512
2538 y Gc(\024)1893 2575 y F6(\000)-31 b(\000)f(!)p F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22513
b Gg(is)h(strongly)h(normalising.)p Black 321 2788 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22514
F7(Pr)l(oof)o(.)p Black 34 w Gg(By)i(Theorem)i(3.4.4)f(we)g(kno)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22515
(that)h(e)n(v)o(ery)g(reduction)h(sequence)h(of)2927
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22516
2751 y Gc(\024)2862 2788 y F6(\000)-31 b(\000)g(!)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22517
b Gg(maps)h(to)h(a)e(re-)321 2901 y(duction)f(sequence)g(of)1102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22518
2864 y Gc(cut)1071 2901 y F6(\000)-31 b(\000)g(!)o Gg(.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22519
b(Because)24 b F4(\()p FY(T)t Ga(;)1782 2864 y Gc(cut)1751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22520
2901 y F6(\000)-31 b(\000)f(!)p F4(\))23 b Gg(is)h(strongly)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22521
(normalising,)f F4(\(K)p Ga(;)3075 2864 y Gc(\024)3010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22522
2901 y F6(\000)-31 b(\000)f(!)p F4(\))23 b Gg(must)h(be,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22523
3014 y(too.)p 3480 3014 V 3484 2955 55 4 v 3484 3014
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22524
V 3538 3014 4 62 v 462 3232 a(W)-7 b(e)19 b(proceed)i(with)e(sho)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22525
(wing)h(the)g(correspondence)k(in)19 b(the)h(opposite)h(direction.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22526
b(First)19 b(we)g(sho)n(w)321 3344 y(the)31 b(correspondence)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22527
b(between)31 b(the)g(proof)h(substitutions\227again)k(this)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22528
b(in)l(v)n(olv)o(es)i(rather)f(lengthy)321 3457 y(calculations.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22529
Black 321 3645 a Gb(Lemma)23 b(3.4.6:)p Black 34 w Gg(F)o(or)g(all)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22530
Ga(M)5 b(;)15 b(N)35 b F6(2)25 b FY(T)t Gg(,)54 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22531
Ga(M)10 b F6(j)1762 3612 y Fu(N)1817 3645 y Fs(\()-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22532
b Ga(x)25 b F4(:=)2040 3633 y FX(h)2068 3645 y Ga(a)2116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22533
3633 y FX(i)2143 3645 y F6(j)p Ga(N)10 b F6(j)2276 3612
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22534
y Fu(N)2324 3645 y Fs(\))25 b F6(\021)g(j)p Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22535
F4([)p Ga(x)26 b F4(:=)2824 3633 y FX(h)2852 3645 y Ga(a)2900
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22536
3633 y FX(i)2927 3645 y Ga(N)10 b F4(])p F6(j)3060 3612
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22537
y Fu(N)3115 3645 y Gg(.)p Black 321 3858 a F7(Pr)l(oof)o(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22538
Black 34 w Gg(By)23 b(induction)j(on)e(the)f(structure)j(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22539
Ga(M)32 b Gg(\(see)25 b(P)o(age)e(163\).)p 3480 3858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22540
V 3484 3799 55 4 v 3484 3858 V 3538 3858 4 62 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22541
321 4045 a Gb(Lemma)g(3.4.7:)p Black 34 w Gg(F)o(or)g(all)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22542
Ga(M)5 b(;)15 b(N)35 b F6(2)25 b FY(T)t Gg(,)19 b(if)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22543
Ga(N)32 b Gg(freshly)26 b(introduces)g Ga(x)d Gg(and)h(is)f(not)h(an)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22544
(axiom,)f(then)p Black Black 1213 4293 a F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22545
b F6(j)1361 4260 y Fu(N)1416 4293 y Fs(^)p Ga(N)1521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22546
4308 y Fu(N)1575 4293 y Fs(_)1772 4256 y Gc(\024)1708
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22547
4293 y F6(\000)-32 b(\000)h(!)1878 4260 y FX(\003)2017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22548
4293 y F6(j)p Ga(M)10 b F4([)p Ga(a)26 b F4(:=)2360 4281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22549
y F9(\()2387 4293 y Ga(x)2439 4281 y F9(\))2467 4293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22550
y Ga(N)10 b F4(])p F6(j)2600 4260 y Fu(N)321 4755 y Gg(where)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22551
b Ga(N)639 4770 y Fu(N)718 4755 y F6(\021)814 4514 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22552
FK(8)814 4596 y(>)814 4623 y(>)814 4651 y(<)814 4814
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22553
y(>)814 4841 y(>)814 4869 y(:)937 4586 y FL(And)1091
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22554
4549 y Gc(i)1091 4609 y(E)1151 4586 y F4(\()1186 4574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22555
y FX(h)1214 4586 y Ga(a)1262 4574 y FX(i)p 1291 4586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22556
28 4 v 1309 4586 V 1326 4586 V 1354 4586 a Ga(;)1394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22557
4574 y F9(\()1421 4586 y Ga(y)1469 4574 y F9(\))1497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22558
4586 y F6(j)p Ga(P)13 b F6(j)1618 4553 y Fu(N)1672 4586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22559
y F4(\))937 4699 y FL(Or)1036 4713 y Gc(E)1096 4699 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22560
F4(\()1131 4687 y FX(h)1158 4699 y Ga(a)1206 4687 y FX(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22561
1236 4699 V 1253 4699 V 1271 4699 V 1298 4699 a Ga(;)1338
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22562
4687 y F9(\()1366 4699 y Ga(y)1414 4687 y F9(\))1441
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22563
4699 y F6(j)p Ga(P)g F6(j)1562 4666 y Fu(N)1617 4699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22564
y Ga(;)1657 4687 y F9(\()1685 4699 y Ga(z)1731 4687 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22565
F9(\))1758 4699 y F6(j)p Ga(Q)p F6(j)1880 4666 y Fu(N)1935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22566
4699 y F4(\))937 4812 y FL(Imp)1081 4834 y Gc(E)1141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22567
4812 y F4(\()1176 4800 y FX(h)1204 4812 y Ga(a)1252 4800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22568
y FX(i)p 1281 4812 V 1299 4812 V 1316 4812 V 1344 4812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22569
a Ga(;)1384 4800 y FX(h)1411 4812 y Ga(b)1450 4800 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22570
FX(i)1478 4812 y F6(j)p Ga(P)g F6(j)1599 4779 y Fu(N)1653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22571
4812 y Ga(;)1693 4800 y F9(\()1721 4812 y Ga(y)1769 4800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22572
y F9(\))1796 4812 y F6(j)p Ga(Q)p F6(j)1918 4779 y Fu(N)1973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22573
4812 y F4(\))937 4925 y FL(Not)1079 4939 y Gc(E)1139
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22574
4925 y F4(\()1174 4913 y FX(h)1202 4925 y Ga(a)1250 4913
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22575
y FX(i)p 1279 4925 V 1297 4925 V 1314 4925 V 1342 4925
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22576
a Ga(;)1382 4913 y FX(h)1410 4925 y Ga(b)1449 4913 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22577
FX(i)1476 4925 y F6(j)p Ga(P)g F6(j)1597 4892 y Fu(N)1652
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22578
4925 y F4(\))2082 4755 y Gg(pro)o(vided)26 b Ga(N)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22579
b F6(\021)2631 4514 y FK(8)2631 4596 y(>)2631 4623 y(>)2631
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22580
4651 y(<)2631 4814 y(>)2631 4841 y(>)2631 4869 y(:)2753
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22581
4586 y FL(And)2908 4550 y Gc(i)2908 4609 y(L)2960 4586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22582
y F4(\()2995 4574 y F9(\()3023 4586 y Ga(y)3071 4574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22583
y F9(\))3098 4586 y Ga(P)13 b(;)i(x)p F4(\))2753 4699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22584
y FL(Or)2853 4713 y Gc(L)2905 4699 y F4(\()2940 4687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22585
y F9(\()2968 4699 y Ga(y)3016 4687 y F9(\))3043 4699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22586
y Ga(P)e(;)3154 4687 y F9(\()3182 4699 y Ga(z)3228 4687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22587
y F9(\))3255 4699 y Ga(Q;)i(x)p F4(\))2753 4812 y FL(Imp)2898
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22588
4834 y Gc(L)2950 4812 y F4(\()2985 4800 y FX(h)3013 4812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22589
y Ga(b)3052 4800 y FX(i)3079 4812 y Ga(P)e(;)3190 4800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22590
y F9(\()3218 4812 y Ga(y)3266 4800 y F9(\))3293 4812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22591
y Ga(Q;)i(x)p F4(\))2753 4925 y FL(Not)2896 4939 y Gc(L)2948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22592
4925 y F4(\()2983 4913 y FX(h)3011 4925 y Ga(b)3050 4913
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22593
y FX(i)3077 4925 y Ga(P)e(;)i(x)p F4(\))p Black 321 5154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22594
a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)23 b(induction)j(on)e(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22595
(structure)j(of)e Ga(M)32 b Gg(\(see)25 b(P)o(age)e(164\).)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22596
3480 5154 4 62 v 3484 5096 55 4 v 3484 5154 V 3538 5154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22597
4 62 v Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22598
%%Page: 95 107
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22599
TeXDict begin 95 106 bop Black 277 51 a Gb(3.5)23 b(Notes)3248
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22600
b(95)p 277 88 3691 4 v Black Black 277 365 a(Lemma)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22601
b(3.4.8:)p Black 34 w Gg(F)o(or)h(all)g Ga(M)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22602
b(N)35 b F6(2)25 b FY(T)t Gg(,)16 b(pro)o(vided)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22603
Ga(N)31 b Gg(does)21 b(not)h(freshly)g(introduce)i Ga(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22604
Gg(,)c(e)o(xcept)i(where)277 478 y Ga(N)33 b Gg(is)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22605
b(an)h(axiom,)f(then)p Black Black 1115 672 a F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22606
Ga(M)10 b F4([)p Ga(a)26 b F4(:=)1458 660 y F9(\()1486
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22607
672 y Ga(x)1538 660 y F9(\))1565 672 y Ga(N)10 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22608
F6(j)1698 639 y Fu(N)1778 672 y F6(\021)25 b(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22609
b F6(j)2022 639 y Fu(N)2076 672 y Fs(\()-6 b Ga(a)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22610
b F4(:=)2296 660 y F9(\()2324 672 y Ga(x)2376 660 y F9(\))2403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22611
672 y F6(j)p Ga(N)10 b F6(j)2536 639 y Fu(N)2584 672
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22612
y Fs(\))25 b Gg(.)p Black 277 884 a F7(Pr)l(oof)o(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22613
Black 34 w Gg(By)e(induction)j(on)d(the)h(structure)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22614
Ga(M)33 b Gg(\(see)24 b(P)o(age)f(165\).)p 3436 884 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22615
62 v 3440 826 55 4 v 3440 884 V 3494 884 4 62 v 277 1088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22616
a(The)g(correspondence)28 b(between)d(the)f(reduction)i(rules)e(is)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22617
(no)n(w)f(a)g(matter)h(of)f(simple)h(calculations.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22618
Black 277 1276 a Gb(Theor)n(em)g(3.4.9:)p Black 34 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22619
Gg(F)o(or)f(all)g Ga(M)5 b(;)15 b(N)36 b F6(2)25 b FY(T)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22620
Gg(,)18 b(if)24 b Ga(M)1821 1239 y Gc(cut)1790 1276 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22621
F6(\000)-31 b(\000)f(!)25 b Ga(N)10 b Gg(,)23 b(then)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22622
F6(j)p Ga(M)10 b F6(j)2441 1243 y Fu(N)2586 1239 y Gc(\024)2521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22623
1276 y F6(\000)-31 b(\000)f(!)2691 1243 y FX(\003)2756
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22624
1276 y F6(j)p Ga(N)10 b F6(j)2889 1243 y Fu(N)2943 1276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22625
y Gg(.)p Black 277 1489 a F7(Pr)l(oof)o(.)p Black 34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22626
w Gg(By)23 b(induction)j(on)d(the)h(de\002nition)h(of)1778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22627
1452 y Gc(cut)1747 1489 y F6(\000)-31 b(\000)f(!)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22628
Gg(\(see)h(P)o(age)f(165\).)p 3436 1489 V 3440 1431 55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22629
4 v 3440 1489 V 3494 1489 4 62 v 418 1693 a(T)-7 b(o)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22630
b(sum)g(up,)h(we)e(ha)n(v)o(e)j(sho)n(wn)f(in)f(this)h(section)i(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22631
(close)g(correspondence)29 b(between)24 b(normal-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22632
1806 y(isation)k(and)f(cut-elimination)j(in)c(propositional)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22633
(classical)e(logic;)h(by)d(Theorems)h(3.4.4)g(and)f(3.4.9)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22634
1919 y(we)d(ha)n(v)o(e)h(the)g(correspondences)k(listed)d(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22635
b(.)p Black Black 1182 2113 a Ga(S)1411 2076 y Gc(\024)1310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22636
2113 y F6(\000)-21 b(\000)h(\000)f(!)64 b Ga(T)1171 2477
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22637
y(S)1232 2439 y FX(0)1146 2295 y Fu(S)1191 2351 y F6(#)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22638
1211 2288 4 142 v 1353 2440 a Gc(cut)1281 2477 y F6(\000)-21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22639
b(\000)g(\000)g(!)1522 2439 y F9(+)1605 2477 y Ga(T)1671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22640
2439 y FX(0)1675 2295 y Fu(S)1628 2351 y F6(#)p 1648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22641
2288 V 2116 2113 a Ga(M)2335 2076 y Gc(cut)2263 2113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22642
y F6(\000)g(\000)g(\000)g(!)56 b Ga(N)2104 2477 y(M)2202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22643
2439 y FX(0)2088 2295 y Fu(N)2144 2351 y F6(#)p 2164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22644
2288 V 2358 2440 a Gc(\024)2258 2477 y F6(\000)-21 b(\000)h(\000)e(!)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22645
2494 2439 y FX(\003)2561 2477 y Ga(N)2644 2439 y FX(0)2627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22646
2295 y Fu(N)2580 2351 y F6(#)p 2600 2288 V 277 2674 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22647
Gg(These)23 b(correspondences)28 b(can)23 b(be)g(e)o(xtended)i(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22648
(\002rst-order)i(classical)g(logic)f(and)f(to)g(other)h(v)n(ariants)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22649
2787 y(of)g(sequence-conclusion)30 b(natural)25 b(deduction)h(calculi,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22650
f(b)n(ut)f(the)g(details)h(are)f(omitted.)277 3094 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22651
Ge(3.5)119 b(Notes)277 3318 y Gg(In)22 b(this)g(section,)h(we)e(shall)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22652
(\002rst)e(compare)i(our)f(w)o(ork)g(with)f(earlier)i(w)o(ork)f(by)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22653
(Ungar)g([1992],)h(which)277 3431 y(also)29 b(studies)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22654
(correspondence)k(between)c(cut-elimination)j(and)d(normalisation,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22655
(and)d(second)277 3544 y(describe)f(brie\003y)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22656
Ga(\025\026)p Gg(-calculus)h(de)n(v)o(eloped)h(by)e(P)o(arigot)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22657
([1992].)36 b(W)-7 b(e)25 b(shall)h(sho)n(w)g(that)g(strong)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22658
3657 y(normalisation)39 b(for)c(\(propositional\))40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22659
b Ga(\025\026)35 b Gg(can)h(be)f(inferred)j(by)d(a)g(simple)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22660
(translation)j(from)c(our)277 3769 y(strong)25 b(normalisation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22661
(result,)d(b)n(ut)g(not)g F7(vice)g(ver)o(sa)p Gg(.)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22662
4033 y Fq(3.5.1)99 b(Comparison)22 b(with)h(Ungar')l(s)g(A)n(ppr)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22663
(oach)g(to)g(the)g(Corr)n(espondence)i(Question)277 4224
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22664
y Gg(While)e(Zuck)o(er)g([1974])h(ar)n(gued)g(that)f(in)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22665
F4(\()p F6(^)p Ga(;)15 b F6(\033)p Ga(;)g F6(_)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22666
Gg(-fragment)24 b(of)e(intuitionistic)27 b(logic)c(the)g(lack)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22667
4337 y(of)i(strong)h(normalisation)i(of)d(cut-elimination)j(is)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22668
(responsible)j(for)c(the)h(f)o(ailure)i(of)d(the)h(correspon-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22669
4450 y(dence,)h(we)e(ha)n(v)o(e)i(sho)n(wn)f(that)h(in)f(f)o(act)g(the)
Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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277 4676 y(dence)38 b(using)f(NJ\).)f(Ungar)g(made)h(the)g(same)f
Christian Urban <urbanc@in.tum.de>
parents:
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(observ)n(ation)2229 4643 y F5(4)2306 4676 y Gg(and)h(already)h(e)o
Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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(an)277 5270 y(application)d(of)e Fr(9)p F2(-)h(or)f
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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(41])p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
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%%Page: 96 108
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
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1426 y Ga(:)321 1727 y Gg(Here)21 b Ga(B)k Gg(and)c Ga(C)26
Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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b(inferences)j(with)d(tw)o(o)f(separate)j(conclusions)321
Christian Urban <urbanc@in.tum.de>
parents:
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1840 y(has)k(the)g(profound)i(ef)n(fect)f(that,)f(in)g(general,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(natural)g(deduction)h(calculus.)p Black Black 1811 2201
Christian Urban <urbanc@in.tum.de>
parents:
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a Ga(B)p 1714 2222 275 4 v 1714 2298 a(C)131 b(D)p 1714
Christian Urban <urbanc@in.tum.de>
parents:
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2318 73 4 v 1910 2318 244 4 v 1714 2394 a(E)h(F)111 b(G)p
Christian Urban <urbanc@in.tum.de>
parents:
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1714 2415 275 4 v 2082 2415 72 4 v 1807 2490 a(H)212
Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22730
Ga(B)k Gg(and)c Ga(C)321 3187 y Gg(must)g(not)g(ha)n(v)o(e)g(an)o(y)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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4 v 1735 3504 a(B)95 b(C)2012 3444 y F6(_)2073 3458 y
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Gc(E)p 1735 3524 V 1750 3603 a Ga(B)5 b F6(^)o Ga(C)2012
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y(.)2003 4366 y(.)2003 4399 y(.)2003 4432 y(.)321 4633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b Gg(,)22 b(and)i(the)g(proof)1933 4772 y(.)1933 4805
Christian Urban <urbanc@in.tum.de>
parents:
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y(.)1933 4838 y(.)1841 4918 y Ga(B)5 b F6(_)o Ga(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1826 4938 V 1826 5016 a(B)95 b(C)2103 4956 y F6(_)2164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4970 y Gc(E)321 5249 y Gg(ending)27 b(with)e(tw)o(o)g(conclusions,)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
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b(the)e(latter)g(proof)h(for)e(the)h(assumptions)p Black
Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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730 y Ga(B)5 b F6(_)o Ga(C)p 1773 768 233 4 v 1773 843
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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a(B)92 b(C)1799 891 y Gg(.)1799 925 y(.)1799 958 y(.)1959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22762
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22764
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22769
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22773
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22775
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22779
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2602 y F9(\()p 1392 2614 28 4 v 1409 2614 V 1427 2614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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V 65 w(\))1510 2614 y Gg(and)1670 2602 y FX(h)p 1699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2614 V 1717 2614 V 1734 2614 V 65 w(i)1789 2614 y Gg(,)g(and)g(record)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22789
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22790
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22792
i(calculus)e(w)o(as)e(gi)n(v)o(en)h(by)g(P)o(arigot)g([1992].)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22793
b(The)24 b(corre-)277 2953 y(sponding)i(term)e(calculus)h(w)o(as)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(named)h Ga(\025)q(\026)p Gg(-calculus,)h(which)f(we)f(shall)h(study)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22795
(brie\003y)f(ne)o(xt.)277 3238 y Fq(3.5.2)99 b(P)o(arigot')l(s)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b Fn(\025)o(\026)p Fq(-Calculus)277 3436 y Gg(The)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22797
b Ga(\025\026)p Gg(-calculus)i(can)f(be)f(seen)h(as)f(a)f(simple)i(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22798
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22799
3549 y(culus.)64 b(The)34 b(e)o(xtension)j(is)e(such)h(that)f(via)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22800
(the)h(Curry-Ho)n(w)o(ard)g(correspondence)j Ga(\025\026)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22801
Gg(-terms)c(no)277 3662 y(longer)21 b(correspond)h(to)d(proofs)h(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22802
(Gentzen')-5 b(s)21 b(NJ-calculus,)h(b)n(ut)e(to)f(proofs)h(in)f(Bori)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22803
b(\020)-35 b(ci)5 b(\264)-35 b(c')-5 b(s)21 b(sequence-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22804
3774 y(conclusion)34 b(natural)e(deduction)h(calculus)g(\(see)e(P)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22805
(age)f(81\).)50 b(In)30 b(ef)n(fect,)j(P)o(arigot)e(ga)n(v)o(e)g(an)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22806
(ele)o(gant)277 3887 y(solution)j(to)d(the)h(problem)g(of)g(ho)n(w)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22807
(to)i(modify)g(simply-typed)i(lambda)f(terms)e(so)h(that)g(their)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22808
(sin-)277 4000 y(gle)f(type)h(is)f(generalised)j(to)d(a)f(sequence)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22809
(of)d(types.)49 b(The)29 b(technical)k(crux)d(of)g(P)o(arigot')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22810
b(s)31 b(solution)277 4113 y(is)f(the)g(addition)h(of)f(tw)o(o)f(ne)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22811
(w)g(term)h(constructors)j(to)d(the)f(simply-typed)k(lambda)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22812
(calculus.)49 b(The)277 4226 y(corresponding)28 b(ra)o(w)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22813
b Ga(\025\026)p Gg(-terms)g(are)h(gi)n(v)o(en)g(by)g(the)g(grammar)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22814
Black Black 1187 4438 a Ga(M)5 b(;)15 b(N)110 b F4(::=)100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22815
b Ga(x)412 b Gg(v)n(ariable)1551 4560 y F6(j)148 b Ga(\025x:M)246
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22816
b Gg(abstraction)1551 4682 y F6(j)148 b Ga(M)25 b(N)278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22817
b Gg(application)1551 4805 y F6(j)148 b F4([)p Ga(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22818
F4(])p Ga(M)278 b Gg(passi)n(v)n(ate)1551 4927 y F6(j)148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22819
b Ga(\026a:M)248 b Gg(acti)n(v)n(ate)277 5136 y(where)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22820
b Ga(x)e Gg(is)h(tak)o(en)h(from)f(a)g(set)h(of)f(v)n(ariables)i(and)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22821
Ga(a)e Gg(from)h(a)g(set)g(of)g Ga(\026)p Gg(-v)n(ariables.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22822
b(As)18 b(a)h(consequence,)277 5249 y(he)k(could)i(consider)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22823
(sequence)h(of)f(conclusions)i(of)d(proofs)i(in)e(Bori)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22824
b(\020)-35 b(ci)5 b(\264)-35 b(c')-5 b(s)25 b(calculus)g(as)e(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22825
(whole,)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22826
%%Page: 98 110
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22827
TeXDict begin 98 109 bop Black -144 51 a Gb(98)2876 b(Natural)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22828
b(Deduction)p -144 88 3691 4 v Black 321 365 a Gg(b)n(ut)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22829
b(distinguishing)k(one)c(of)g(them)f(as)h F7(active)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22830
Gg(\227in)h(what)f(follo)n(ws)g(annotated)i(with)e(a)f(disc\227and)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22831
478 y(the)26 b(others)h(as)f F7(passive)p Gg(.)37 b(The)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22832
b(latter)h(are)g(signi\002ed)h(by)f(being)h(labelled)g(with)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22833
Ga(\026)p Gg(-v)n(ariables.)37 b(T)-7 b(yping)321 590
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22834
y(judgements)26 b(in)e Ga(\025\026)e Gg(ha)n(v)o(e)i(thus)h(the)e(form)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22835
1672 822 y F4(\000)1754 810 y Gc(.)1809 822 y Ga(M)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22836
F4(:)18 b Ga(B)2041 789 y FX(\017)2080 822 y Ga(;)d F4(\001)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22837
1033 y Gg(where)24 b F4(\000)f Gg(is)g(a)g(set)h(of)f(\(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22838
(ariable,formula\))28 b(pairs)c(and)g F4(\001)f Gg(a)g(set)g(of)h(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22839
Ga(\026)p Gg(-v)n(ariable,formula\))j(pairs.)462 1163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22840
y(Since)d Ga(\025\026)e Gg(is)h(a)g(natural)i(deduction)h(calculus,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22841
(its)f(inference)h(rules)g(are)e(concerned)j(with)d(intro-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22842
1276 y(ducing)g(or)e(eliminating)j(formulae)f(in)e(the)g(succedent.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22843
b(Ho)n(we)n(v)o(er)l(,)21 b(the)o(y)h(are)f(restricted)j(so)d(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22844
(the)o(y)321 1389 y(act)g(on)h(acti)n(v)o(e)f(formulae)h(only)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22845
b(,)23 b(as)f(can)h(be)f(seen)g(in)g(the)g(term)g(formation)i(rules)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22846
(for)f(abstraction)j(and)321 1502 y(application.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22847
Black 516 1728 a Ga(x)17 b F4(:)g Ga(B)5 b(;)15 b F4(\000)823
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22848
1716 y Gc(.)878 1728 y Ga(M)28 b F4(:)17 b Ga(C)1108
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22849
1695 y FX(\017)1147 1728 y Ga(;)e F4(\001)p 456 1766
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22850
867 4 v 456 1851 a(\000)538 1839 y Gc(.)593 1851 y Ga(\025x:M)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22851
b F4(:)17 b(\()p Ga(B)5 b F6(\033)p Ga(C)i F4(\))1168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22852
1818 y FX(\017)1207 1851 y Ga(;)15 b F4(\001)1364 1783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22853
y F6(\033)1435 1797 y Gc(I)1853 1728 y F4(\000)1935 1716
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22854
y Gc(.)1990 1728 y Ga(M)27 b F4(:)18 b(\()p Ga(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22855
F6(\033)o Ga(C)i F4(\))2434 1695 y FX(\017)2474 1728
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22856
y Ga(;)15 b F4(\001)91 b(\000)2763 1716 y Gc(.)2818 1728
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22857
y Ga(N)27 b F4(:)17 b Ga(B)3034 1695 y FX(\017)3073 1728
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22858
y Ga(;)e F4(\001)p 1853 1771 1337 4 v 2211 1851 a(\000)2293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22859
1839 y Gc(.)2348 1851 y Ga(M)25 b(N)j F4(:)17 b Ga(C)2676
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22860
1818 y FX(\017)2715 1851 y Ga(;)e F4(\001)3231 1788 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22861
F6(\033)3302 1802 y Gc(E)3386 1788 y Ga(:)321 2085 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22862
Gg(The)32 b(formation)i(rules)f(for)f(the)g(ne)n(w)g(term)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22863
(constructors)36 b(are)c(called,)j(using)f(terminology)g(intro-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22864
2198 y(duced)25 b(by)f(Bierman)g([1998],)g F7(passivate)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22865
Gg(and)e F7(activate)p Gg(.)31 b(The)o(y)23 b(are)h(as)g(follo)n(ws.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22866
2885 2165 y F5(5)p Black Black 1112 2420 a F4(\000)1194
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22867
2408 y Gc(.)1249 2420 y Ga(M)j F4(:)17 b Ga(B)1480 2387
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22868
y FX(\017)1519 2420 y Ga(;)e F4(\001)p 953 2458 842 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22869
v 953 2542 a(\000)1035 2530 y Gc(.)1090 2542 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22870
Ga(a)p F4(])p Ga(M)28 b F4(:)17 b F6(?)1417 2509 y FX(\017)1456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22871
2542 y Ga(;)e(a)j F4(:)f Ga(B)5 b(;)15 b F4(\001)2172
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22872
2425 y(\000)2254 2413 y Gc(.)2309 2425 y Ga(M)27 b F4(:)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22873
b F6(?)2538 2392 y FX(\017)2577 2425 y Ga(;)d(a)j F4(:)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22874
Ga(B)5 b(;)15 b F4(\001)p 2172 2463 743 4 v 2217 2542
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22875
a(\000)2299 2530 y Gc(.)2354 2542 y Ga(\026a:M)28 b F4(:)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22876
b Ga(B)2714 2509 y FX(\017)2753 2542 y Ga(;)e F4(\001)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22877
2777 y Gg(The)28 b(\002rst)g(rule)h(changes)i(the)e(status)g(of)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22878
(acti)n(v)o(e)h(formula)h(to)e(become)i(passi)n(v)o(e.)44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22879
b(The)28 b(resulting)321 2890 y(term)i(has)h(then)g(the)f(acti)n(v)o(e)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22880
h(type)g F6(?)p Gg(,)g(a)e(distinguished)35 b(atomic)c(formula.)49
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22881
b(The)30 b(other)h(rule)g(w)o(orks)321 3003 y(similar)l(,)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22882
b(b)n(ut)f(in)f(the)h(re)n(v)o(erse)h(direction.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22883
3133 y(Before)f(passing)h(to)d(the)i(translations)i(between)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22884
Ga(\025\026)p Gg(-terms)f(and)g(terms)g(belonging)j(to)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22885
F4(K)p Gg(,)f(let)h(us)321 3245 y(brie\003y)30 b(comment)g(on)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22886
(syntactic)j(con)l(v)o(entions)g(in)e(both)f(systems.)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22887
b(It)29 b(seems)g(the)h(dif)n(ferences)321 3358 y(are)e(chie\003y)h(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22888
(matter)h(of)f(taste:)39 b(whereas)29 b(in)e Ga(\025\026)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22889
Gg(a)g(typing)i(judgement)h(has)e(only)h(a)e(single)i(acti)n(v)o(e)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22890
3471 y(formula,)34 b(and)d(therefore)i(it)e(does)h(not)f(need)h(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22891
(be)h(recorded)i(in)e(the)g(term,)h(e)o(xcept)g(of)f(course)i(in)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22892
3584 y(the)24 b(cases)g(where)g(another)h(formula)f(becomes)h(acti)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22893
o(e;)f(in)f(our)h(term)f(calculus,)j(on)d(the)h(other)g(hand,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22894
3697 y(all)k(formulae)h(in)f(the)g(succedent)i(are)e(acti)n(v)o(e,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22895
(so)e(to)h(speak,)i(and)e(therefore)i(it)d(has)h(to)g(be)f(recorded)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22896
3810 y(e)o(xplicitly)36 b(which)e(formulae)h(are)f(dischar)n(ged)j(and)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22897
d(introduced.)62 b(Clearly)-6 b(,)36 b(both)f(syntactic)h(con-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22898
3923 y(v)o(entions)29 b(can)e(be)f(\223simulated\224)j(by)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22899
(other)-5 b(.)39 b(On)26 b(the)h(other)g(hand,)h(considering)i(the)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22900
(amount)h(of)321 4036 y(prolixity)-6 b(,)33 b(the)d(syntax)h(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22901
Ga(\025\026)f Gg(is)i(more)f(concise)i(in)e(the)h(a)n(v)o(erage)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22902
(case,)g(b)n(ut)f(less)g(so)f(in)h(the)f(w)o(orst)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22903
4149 y(case.)h(F)o(or)22 b(e)o(xample)j(in)e Ga(\025\026)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22904
Gg(we)f(can)i(form)g(the)g(term)1681 4380 y Ga(\026a:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22905
F4([)p Ga(a)p F4(])p Ga(\026a:)p F4([)p Ga(a)p F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22906
Ga(x)321 4615 y Gg(which)g(corresponds)j(to)d(the)f(term)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22907
FL(Id)p F4(\()p Ga(x;)15 b(a)p F4(\))24 b Gg(in)f(our)h(calculus.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22908
4744 y(Our)d(natural)j(deduction)g(calculus)f F4(K)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22909
Gg(is)g(a)g(slightly)j(modi\002ed)e(v)n(ariant)h(of)e(Bori)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22910
b(\020)-35 b(ci)5 b(\264)-35 b(c')-5 b(s)23 b(sequence-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22911
4857 y(conclusion)36 b(natural)f(deduction)h(calculus)f(\(see)f(Figure)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22912
f(3.7\).)58 b(W)-7 b(e)32 b(made)h(the)g(modi\002cations)j(in)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22913
4970 y(order)24 b(to)e(f)o(acilitate)j(the)d(translation)k(with)c(our)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22914
(sequent)h(calculus)h FY(T)t Gg(.)e(It)f(may)h(be)f(therefore)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22915
(surpris-)321 5083 y(ing,)30 b(that)e(there)h(are)g(also)g(relati)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22916
(ely)h(simple)f(translations)i(for)d F4(K)f Gg(to)h(and)h(from)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22917
Ga(\025\026)p Gg(-calculus.)p Black 321 5161 1290 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22918
427 5217 a F3(5)456 5249 y F2(W)-6 b(e)18 b(use)i(the)e(con)m(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22919
(ention)j(introduced)f(by)g(Bierman)f([1998])h(where)f(all)f(terms)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22920
(ha)o(v)o(e)g(an)g(acti)n(v)o(e)g(type.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22921
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22922
%%Page: 99 111
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22923
TeXDict begin 99 110 bop Black 277 51 a Gb(3.5)23 b(Notes)3248
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22924
b(99)p 277 88 3691 4 v Black 277 388 a Gg(The)23 b(translations)k(for)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22925
(both)g(directions)i(are)e(listed)h(belo)n(w)-6 b(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22926
Black Black 305 606 a F6(j)p 332 606 28 4 v 350 606 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22927
367 606 V 64 w(j)419 573 y Gc(\026)491 606 y F4(:)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22928
b(K)f F6(!)g Ga(\025\026)1219 734 y F6(j)p FL(Id)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22929
Ga(x;)15 b(a)p F4(\))p F6(j)1551 701 y Gc(\026)1699 683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22930
y F5(def)1706 734 y F4(=)106 b([)p Ga(a)p F4(])p Ga(x)860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22931
874 y F6(j)p FL(Imp)1029 896 y Gc(I)1069 874 y F4(\()1104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22932
862 y F9(\()1132 874 y Ga(x)1184 862 y F9(\))p FX(h)1239
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22933
874 y Ga(a)1287 862 y FX(i)1314 874 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22934
b(b)p F4(\))p F6(j)1551 841 y Gc(\026)1699 822 y F5(def)1706
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22935
874 y F4(=)106 b([)p Ga(b)p F4(]\()p Ga(\025)q(x:\026a:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22936
F6(j)p Ga(M)10 b F6(j)2414 841 y Gc(\026)2462 874 y F4(\))591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22937
1014 y F6(j)p FL(Imp)760 1035 y Gc(E)820 1014 y F4(\()855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22938
1002 y FX(h)883 1014 y Ga(a)931 1002 y FX(i)958 1014
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22939
y Ga(M)g(;)1096 1002 y FX(h)1124 1014 y Ga(b)1163 1002
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22940
y FX(i)1191 1014 y Ga(N)g(;)1314 1002 y F9(\()1341 1014
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22941
y Ga(x)1393 1002 y F9(\))1421 1014 y Ga(P)j F4(\))p F6(j)1552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22942
981 y Gc(\026)1699 962 y F5(def)1706 1014 y F4(=)106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22943
b F6(j)p Ga(P)13 b F6(j)2004 981 y Gc(\026)2051 1014
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22944
y F4([)p Ga(x)26 b F4(:=)f(\()p Ga(\026a:)p F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22945
b F6(j)2586 981 y Gc(\026)2634 1014 y F4(\))15 b(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22946
Ga(\026b:)p F6(j)p Ga(N)10 b F6(j)2971 981 y Gc(\026)3018
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22947
1014 y F4(\)])788 1153 y F6(j)p FL(Subst)o F4(\()1060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22948
1141 y FX(h)1088 1153 y Ga(a)1136 1141 y FX(i)1164 1153
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22949
y Ga(M)g(;)1302 1141 y F9(\()1329 1153 y Ga(x)1381 1141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22950
y F9(\))1409 1153 y Ga(N)g F4(\))p F6(j)1552 1120 y Gc(\026)1699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22951
1102 y F5(def)1706 1153 y F4(=)106 b F6(j)p Ga(N)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22952
F6(j)2016 1120 y Gc(\026)2063 1153 y F4([)p Ga(x)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22953
F4(:=)f Ga(\026a:)p F6(j)p Ga(M)10 b F6(j)2563 1120 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22954
Gc(\026)2610 1153 y F4(])305 1314 y F6(j)p 332 1314 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22955
350 1314 V 367 1314 V 64 w(j)419 1281 y Gc(\024)490 1314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22956
y F4(:)25 b Ga(\025\026)g F6(!)g F4(K)1452 1442 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22957
Ga(x)p F6(j)1554 1409 y Gc(\024)1554 1464 y(c)1699 1390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22958
y F5(def)1706 1442 y F4(=)106 b FL(Id)p F4(\()p Ga(x;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22959
b(c)p F4(\))1275 1581 y F6(j)p Ga(\025)q(x:M)10 b F6(j)1554
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22960
1548 y Gc(\024)1554 1604 y(c)1699 1530 y F5(def)1706
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22961
1581 y F4(=)106 b FL(Imp)2028 1603 y Gc(I)2067 1581 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22962
F4(\()2102 1569 y F9(\()2130 1581 y Ga(x)2182 1569 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22963
F9(\))q FX(h)2237 1581 y Ga(a)2285 1569 y FX(i)2313 1581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22964
y F6(j)p Ga(M)10 b F6(j)2461 1548 y Gc(\024)2461 1604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22965
y(a)2506 1581 y Ga(;)15 b(c)p F4(\))1323 1721 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22966
Ga(M)10 b(N)g F6(j)1554 1688 y Gc(\024)1554 1744 y(c)1699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22967
1670 y F5(def)1706 1721 y F4(=)106 b FL(Imp)2028 1743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22968
y Gc(E)2087 1721 y F4(\()2122 1709 y FX(h)2150 1721 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22969
Ga(a)2198 1709 y FX(i)2225 1721 y F6(j)p Ga(M)10 b F6(j)2373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22970
1688 y Gc(\024)2373 1744 y(a)2419 1721 y Ga(;)2459 1709
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22971
y FX(h)2487 1721 y Ga(b)2526 1709 y FX(i)2553 1721 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22972
F6(j)p Ga(N)g F6(j)2686 1688 y Gc(\024)2686 1749 y(b)2731
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22973
1721 y Ga(;)2771 1709 y F9(\()2799 1721 y Ga(x)2851 1709
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22974
y F9(\))2879 1721 y FL(Id)p F4(\()p Ga(x;)15 b(c)p F4(\))q(\))1316
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22975
1861 y F6(j)p F4([)p Ga(c)p F4(])p Ga(M)10 b F6(j)1553
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22976
1828 y Gc(\024)1699 1809 y F5(def)1706 1861 y F4(=)106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22977
b F6(j)p Ga(M)10 b F6(j)2031 1828 y Gc(\024)2031 1883
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22978
y(c)1278 2001 y F6(j)p Ga(\026a:M)g F6(j)1554 1968 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22979
Gc(\024)1554 2023 y(c)1699 1949 y F5(def)1706 2001 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22980
F4(=)106 b F6(j)p Ga(M)10 b F4([)p Ga(a)g F6(7!)g Ga(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22981
F4(])p F6(j)2279 1968 y Gc(\024)277 2280 y Gg(Note)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22982
b(that)h(in)f(the)g(case)h(where)f(the)g Ga(\025\026)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22983
Gg(-term)g(has)h(an)f(acti)n(v)o(e)g(type)h(dif)n(ferent)h(from)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22984
F6(?)p Gg(,)f(the)i(clauses)277 2393 y(of)e F6(j)p 404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22985
2393 V 422 2393 V 440 2393 V 65 w(j)492 2360 y Gc(\024)561
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22986
2393 y Gg(are)g(inde)o(x)o(ed)i(with)d(a)h(label.)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22987
b(It)25 b(is)g(easy)g(to)g(v)o(erify)h(that)g(these)g(translations)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22988
(respect)f(typing)277 2506 y(judgements.)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22989
2694 a Gb(Pr)n(oposition)e(3.5.1:)p Black Black 414 2921
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22990
a F6(\017)p Black 45 w Gg(If)g Ga(M)38 b F6(2)28 b F4(K)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22991
Gg(with)h(the)g(typing)i(judgement)g F4(\000)1967 2909
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22992
y Gc(.)2022 2921 y Ga(M)2145 2909 y Gc(.)2200 2921 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22993
F4(\001)p Gg(,)e(then)g F6(j)p Ga(M)10 b F6(j)2652 2888
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22994
y Gc(\026)2728 2921 y F6(2)27 b Ga(\025\026)e Gg(with)f(the)i(typing)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22995
504 3034 y(judgement)g F4(\000)993 3022 y Gc(.)1048 3034
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22996
y F6(j)p Ga(M)10 b F6(j)1196 3001 y Gc(\026)1268 3022
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22997
y(.)1323 3034 y F6(?)1394 3001 y FX(\017)1433 3034 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22998
Ga(;)15 b F4(\001)q Gg(.)p Black 414 3232 a F6(\017)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 22999
Black 45 w Gg(Assume)38 b Ga(c)g Gg(is)f(a)g(unique)j(label,)i(not)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23000
(occurring)i(an)o(ywhere)f(in)f Ga(M)10 b Gg(.)70 b(If)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23001
b Ga(M)62 b F6(2)51 b Ga(\025\026)37 b Gg(with)504 3345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23002
y(the)30 b(typing)h(judgement)h F4(\000)1400 3333 y Gc(.)1455
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23003
3345 y Ga(M)1578 3333 y Gc(.)1633 3345 y Ga(B)1707 3312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23004
y FX(\017)1746 3345 y Ga(;)15 b F4(\001)p Gg(,)30 b(then)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23005
F6(j)p Ga(M)10 b F6(j)2249 3312 y Gc(\024)2249 3368 y(c)2330
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23006
3345 y F6(2)36 b F4(K)29 b Gg(with)g(the)h(typing)h(judgement)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23007
3458 y F4(\000)586 3446 y Gc(.)641 3458 y F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23008
b F6(j)789 3425 y Gc(\024)789 3481 y(c)860 3446 y(.)915
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23009
3458 y Ga(c)17 b F4(:)h Ga(B)5 b(;)15 b F4(\001)p Gg(.)49
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23010
b(If)31 b Ga(M)49 b F6(2)38 b Ga(\025\026)30 b Gg(with)g(the)h(typing)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23011
(judgement)g F4(\000)2832 3446 y Gc(.)2887 3458 y Ga(M)3010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23012
3446 y Gc(.)3065 3458 y F6(?)3136 3425 y FX(\017)3175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23013
3458 y Ga(;)15 b F4(\001)p Gg(,)32 b(then)504 3571 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23014
F6(j)p Ga(M)10 b F6(j)652 3538 y Gc(\024)723 3571 y F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23015
b F4(K)d Gg(with)i(the)g(typing)h(judgement)g F4(\000)1964
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23016
3559 y Gc(.)2019 3571 y F6(j)p Ga(M)10 b F6(j)2167 3538
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23017
y Gc(\024)2238 3559 y(.)2293 3571 y F4(\001)o Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23018
Black 277 3809 a F7(Pr)l(oof.)p Black 46 w Gg(Both)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23019
b(cases)g(by)g(induction)i(on)e(the)g(structure)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23020
Ga(M)10 b Gg(.)p 3436 3809 4 62 v 3440 3751 55 4 v 3440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23021
3809 V 3494 3809 4 62 v 418 4034 a(There)36 b(are)f(a)f(number)i(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23022
(reduction)j(rules)e(associated)i(with)d(the)g Ga(\025\026)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23023
Gg(-calculus,)40 b(which)35 b(we)277 4147 y(ho)n(we)n(v)o(er)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23024
b(omit)f(and)h(instead)h(refer)f(the)g(reader)h(to)e([P)o(arigot,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23025
(1992])g(and)g([Bierman,)g(1998].)29 b(P)o(arigot)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23026
4259 y(pro)o(v)o(ed)c(for)e(the)h(corresponding)k(reduction)e(system)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23027
(the)g(follo)n(wing)h(tw)o(o)e(properties.)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23028
4447 a Gb(Theor)n(em)h(3.5.2)f Gg(\(P)o(arigot\))p Gb(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23029
Black 36 w Gg(The)g(reduction)j(system)e(of)f(the)h Ga(\025\026)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23030
Gg(-calculus)i(is:)p Black Black 579 4656 a(\(i\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23031
b(strongly)26 b(normalising,)f(and)554 4792 y(\(ii\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23032
b(con\003uent.)277 5046 y(Strong)26 b(normalisation)j(can,)e(for)f(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23033
(xample,)g(be)g(obtain)h(using)g F6(j)p 2341 5046 28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23034
4 v 2359 5046 V 2377 5046 V 65 w(j)2429 5013 y Gc(\024)2499
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23035
5046 y Gg(and)f(appealing)i(to)e(our)g(strong)277 5159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23036
y(normalisation)40 b(result)d(for)g F4(\(K)p Ga(;)1403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23037
5122 y Gc(\024)1338 5159 y F6(\000)-31 b(\000)g(!)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23038
Gg(.)66 b(Unfortunately)40 b(this)d(method)g(does)g(not)g(e)o(xtend)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23039
(to)e(the)277 5272 y(second-order)29 b(v)o(ersion)e(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23040
Ga(\025\026)p Gg(,)f(for)h(which)h(P)o(arigot)f(sho)n(wed)h(strong)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23041
(normalisation)j(using)d(a)f(can-)277 5385 y(didates)g(method.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23042
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23043
%%Page: 100 112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23044
TeXDict begin 100 111 bop Black -144 51 a Gb(100)2831
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23045
b(Natural)23 b(Deduction)p -144 88 3691 4 v Black 462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23046
388 a Gg(T)-7 b(o)26 b(conclude)k(the)d(discussion)j(about)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23047
Ga(\025\026)p Gg(,)e(let)i(us)e(consider)k(the)d(con\003uence)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23048
(property)g(of)e Ga(\025\026)p Gg(.)321 501 y(At)21 b(\002rst)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23049
(glance,)j(it)e(may)g(be)g(surprising)j(that)e Ga(\025\026)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23050
Gg(satis\002es)i(this)g(property)-6 b(,)24 b(b)n(ut)e(at)f(a)f(second)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23051
(it)e(is)g(less)321 614 y(so,)27 b(if)g(we)f(tak)o(e)h(into)g(account)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23052
(that)e(the)g(e)o(xplicit)h(substitution)i(operator)f(in)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23053
F4(K)g Gg(w)o(as)g(a)g(prerequisite)321 727 y(in)h(order)i(to)e(obtain)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23054
h(a)f(correspondence)32 b(with)27 b(our)g(\(non-con\003uent\))k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23055
(cut-elimination)g(procedure.)321 840 y(As)23 b(remark)o(ed)i(earlier)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23056
(,)g(translating)i(this)d(substitution)j(operator)e(as)1084
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23057
1080 y F6(j)p FL(Subst)p F4(\()1357 1068 y FX(h)1385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23058
1080 y Ga(a)1433 1068 y FX(i)1460 1080 y Ga(M)10 b(;)1598
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23059
1068 y F9(\()1626 1080 y Ga(x)1678 1068 y F9(\))1705
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23060
1080 y Ga(N)g F4(\))p F6(j)1848 1043 y Gc(\026)1921 1029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23061
y F5(def)1928 1080 y F4(=)32 b F6(j)p Ga(N)10 b F6(j)2164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23062
1043 y Gc(\026)2211 1080 y F4([)p Ga(x)25 b F4(:=)h Ga(\026a:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23063
F6(j)p Ga(M)10 b F6(j)2711 1043 y Gc(\026)2758 1080 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23064
F4(])321 1301 y Gg(then)26 b(all)f(reductions)j(that)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23065
Ga(M)34 b Gg(can)25 b(perform)h(are)g(\223lost\224)g(pro)o(vided)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23066
Ga(x)d Gg(is)h(not)g(free)h(in)f Ga(N)10 b Gg(.)32 b(Therefore)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23067
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 23207
/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23208
A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23209
get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23210
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23211
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23212
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23213
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23214
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23215
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23216
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23217
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23218
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23219
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23220
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23221
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23222
1000 div/DVImag X/IE 256 array N 2 string 0 1 255{IE S A 360 add 36 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23223
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23224
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23225
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23226
(LaserWriter 16/600)]{A length product length le{A length product exch 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23227
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23228
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23229
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23230
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23231
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23232
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23233
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23234
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23235
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23236
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23237
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23238
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23239
%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23240
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23241
% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23242
% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23243
% For copying restrictions, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23244
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23245
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23246
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23247
/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23248
/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23249
/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23250
/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23251
/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23252
/L /lineto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23253
/T /translate load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23254
/TMatrix { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23255
/RAngle { 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23256
/Atan { /atan load stopped { pop pop 0 } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23257
/Div { dup 0 eq { pop } { div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23258
/NET { neg exch neg exch T } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23259
/Pyth { dup mul exch dup mul add sqrt } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23260
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23261
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23262
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23263
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23264
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23265
/pathforall load stopped { pop pop pop pop } if z } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23266
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23267
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23268
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23269
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23270
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23271
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23272
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23273
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23274
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23275
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23276
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23277
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23278
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23279
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23280
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23281
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23282
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23283
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23284
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23285
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23286
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23287
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23288
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23289
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23290
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23291
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23292
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23293
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23294
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23295
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23296
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23297
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23298
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23299
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23300
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23301
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23302
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23303
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23304
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23305
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23306
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23307
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23308
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23309
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23310
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23311
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23312
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23313
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23314
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23315
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23316
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23317
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23318
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23319
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23320
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23321
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23322
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23323
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23324
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23325
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23326
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23327
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23328
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23329
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23330
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23331
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23332
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23333
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23334
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23335
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23336
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23337
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23338
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23339
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23340
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23341
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23342
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23343
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23344
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23345
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23346
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23347
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23348
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23349
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23350
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23351
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23352
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23353
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23354
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23355
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23356
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23357
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23358
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23359
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23360
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23361
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23362
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23363
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23364
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23365
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23366
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23367
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23368
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23369
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23370
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23371
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23372
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23373
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23374
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23375
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23376
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23377
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23378
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23379
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23380
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23381
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23382
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23383
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23384
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23385
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23386
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23387
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23388
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23389
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23390
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23391
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23392
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23393
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23394
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23395
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23396
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23397
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23398
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23399
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23400
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23401
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23402
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23403
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23404
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23405
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23406
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23407
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23408
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23409
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23410
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23411
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23412
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23413
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23414
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23415
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23416
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23417
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23418
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23419
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23420
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23421
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23422
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23423
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23424
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23425
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23426
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23427
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23428
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23429
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23430
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23431
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23432
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23433
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23434
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23435
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23436
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23437
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23438
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23439
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23440
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23441
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23442
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23443
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23444
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23445
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23446
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23447
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23448
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23449
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23450
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23451
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23452
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23453
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23454
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23455
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23456
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23457
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23458
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23459
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23460
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23461
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23462
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23463
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23464
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23466
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23467
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23468
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23469
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23470
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23471
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23472
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23473
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23474
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23475
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23476
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23477
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23478
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23479
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23480
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23481
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23482
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23483
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23484
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23485
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23486
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23487
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23488
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23489
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23490
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23491
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23492
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23493
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23494
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23495
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23496
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23497
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23498
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23499
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23500
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23501
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23502
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23503
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23504
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23505
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23506
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23507
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23508
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23509
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23510
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23511
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23512
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23513
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23514
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23515
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23516
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23517
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23518
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23519
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23520
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23521
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23522
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23523
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23524
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23525
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23526
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23527
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23528
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23529
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23530
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23531
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23532
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23533
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23534
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23535
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23536
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23537
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23538
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23539
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23540
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23541
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23542
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23543
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23544
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23545
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23546
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23547
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23548
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23549
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23550
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23551
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23552
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23553
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23554
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23555
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23556
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23557
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23558
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23559
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23560
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23561
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23562
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23563
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23564
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23565
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23566
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23567
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23568
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23569
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23570
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23571
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23572
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23573
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23574
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23575
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23576
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23577
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23578
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23579
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23580
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23581
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23582
% For copying restrictions, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23583
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23584
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23585
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23586
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23587
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23588
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23589
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23590
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23591
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23592
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23593
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23594
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23595
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23596
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23597
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23598
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23599
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23600
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23601
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23602
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23603
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23604
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23605
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23606
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23607
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23608
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23609
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23610
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23611
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23612
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23613
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23614
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23615
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23616
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23617
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23618
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23619
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23620
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23621
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23622
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23623
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23624
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23625
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23626
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23627
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23628
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23629
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23630
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23631
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23632
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23633
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23634
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23635
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23636
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23637
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23638
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23639
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23640
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23641
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23642
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23643
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23644
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23645
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23646
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23647
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23648
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23649
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23650
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23651
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23652
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23653
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23654
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23655
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23656
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23657
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23658
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23659
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23660
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23661
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23662
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23663
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23664
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23665
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23666
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23667
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23668
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23669
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23670
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23671
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23672
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23673
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23674
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23675
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23676
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23677
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23678
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23679
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23680
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23681
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23682
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23683
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23684
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23685
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23686
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23687
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23688
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23689
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23690
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23691
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23692
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23693
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23694
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23695
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23696
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23697
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23698
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23699
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23700
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23701
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23702
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23703
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23704
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23705
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23706
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23707
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23708
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23709
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23710
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23711
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23712
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23713
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23714
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23715
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23716
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23717
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23718
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23719
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23720
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23721
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23722
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23723
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23724
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23725
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23726
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23727
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23728
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23729
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23730
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23731
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23732
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23733
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23734
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23735
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23736
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23737
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23738
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23739
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23740
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23741
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23742
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23743
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23744
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23745
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23746
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23747
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23748
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23749
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23750
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23751
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23752
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23753
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23754
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23755
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23756
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23757
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23758
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23759
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23760
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23761
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23762
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23763
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23764
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23765
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23766
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23767
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23768
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23769
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23770
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23771
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23772
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23773
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23774
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23775
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23776
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23777
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23778
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23779
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23780
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23781
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23782
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23783
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23784
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23785
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23786
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23787
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23788
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23789
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23790
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23791
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23792
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23793
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23794
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23795
% For copying restrictions, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23796
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23797
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23798
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23799
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23800
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23801
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23802
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23803
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23804
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23805
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23806
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23807
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23808
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23809
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23810
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23811
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23812
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23813
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23814
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23815
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23817
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23818
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23819
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23820
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23821
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23822
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23823
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23824
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23825
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23826
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23827
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23828
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23829
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23830
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23831
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23832
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23833
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23834
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23835
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23836
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23837
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23838
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23839
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23840
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23841
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23842
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23843
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23844
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23845
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23846
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23847
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23848
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23850
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23851
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23852
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23853
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23854
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23855
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23856
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23857
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23858
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23859
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23861
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23862
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23863
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23864
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23865
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23866
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23867
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23868
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23869
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23870
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23871
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23872
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23873
    Char end stringwidth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23874
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23875
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23876
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23877
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23878
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23879
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23880
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23881
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23882
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23883
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23884
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23885
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23887
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23888
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23889
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23890
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23891
%%BeginProcSet: special.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23892
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23893
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23894
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23895
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23896
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23897
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23898
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23899
/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23900
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23901
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23902
maxlength ge{/md md dup length 20 add dict copy def}if end md begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23903
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23904
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 23905
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
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TeXDict begin 103 114 bop Black 277 51 a Gb(4.1)23 b(Implementation)
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2798 b(103)p 277 88 3691 4 v Black 277 388 a Gg(ministic)27
Christian Urban <urbanc@in.tum.de>
parents:
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b(e)n(v)n(aluation)g(function,)h(which,)e(gi)n(v)o(en)g(a)f(proof,)i
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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b Gg(without)h(the)g(need)g(of)g(backtracking.)418 631
Christian Urban <urbanc@in.tum.de>
parents:
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 24210
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24218
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24222
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24223
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24224
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24225
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24227
1309 y F6(\000)-32 b(\000)h(!)p Gg(,)20 b(we)g(shall)i(\002rst)e(\002x)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24228
h(some)f(terminology)k(and)d(pro)o(v)o(e)h(a)e(lemma)h(sho)n(wing)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24229
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24231
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24232
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24233
1609 y F6(\000)-31 b(\000)f(!)p Gg(,)23 b(if)h(and)g(only)h(if)f(it)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24235
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24236
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24237
b(\000)h(!)p Gg(.)41 b(This)27 b(means)i(the)f(reduction)i(violates)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24238
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24239
V 411 1798 a Gc(aux)391 1835 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24240
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24241
34 w Gg(F)o(or)g(e)n(v)o(ery)h(reduction)i(sequence)g(of)e(the)g(form)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24242
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24243
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24244
1998 V 2888 2035 a Gc(aux)2868 2073 y F6(\000)-31 b(\000)f(!)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24246
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24247
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24248
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24249
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24250
2152 y FX(\003)2224 2185 y Ga(M)2312 2199 y F9(3)2352
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24251
2185 y Gg(.)p Black 277 2398 a F7(Pr)l(oof)o(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24252
34 w Gg(By)23 b(induction)j(on)d(the)h(structure)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24253
Ga(M)1804 2412 y F9(1)1867 2398 y Gg(\(details)i(on)f(P)o(age)f(167\).)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24254
p 3436 2398 4 62 v 3440 2340 55 4 v 3440 2398 V 3494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24255
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24256
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24257
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24258
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24259
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24260
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24261
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24262
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24263
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 24264
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24265
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24266
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24267
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24268
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24269
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24270
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24272
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24277
3640 y F6(\000)-31 b(\000)g(!)p Gg(,)23 b(does)h(not)h(restrict)g(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24278
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24279
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24281
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24282
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24285
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24286
(apply)g(to)f FL(Cut)p F4(\()2186 4199 y FX(h)2214 4211
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24287
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24288
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24289
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24290
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24291
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24292
F4(])24 b Gg(and)g(the)g(other)h FL(Ax)o F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24293
b(b)p F4(\)[)p Ga(y)e F6(7!)d Ga(x)p F4(])p Gg(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24294
348 4514 a(\(ii\))p Black 46 w(The)21 b(cut-reduction)k(for)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24295
(commuting)h(cuts)g(may)f(reduce)h(the)f(term)g FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24296
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24297
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24298
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24299
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24300
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24301
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24302
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24303
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24304
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24305
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24306
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24307
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24308
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24309
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24310
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24311
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24312
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24313
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24314
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24315
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24316
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24317
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24318
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24319
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24320
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24321
F4(\()855 5350 y FX(h)883 5362 y Ga(a)931 5350 y FX(i)959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24322
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24323
5350 y FX(i)1226 5362 y Ga(N)10 b(;)1349 5350 y F9(\()1377
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24324
5362 y Ga(x)1429 5350 y F9(\))1456 5362 y Ga(M)g F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24325
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24326
5362 y Ga(P)j F4(\))101 b Gg(or)h FL(Cut)p F4(\()2290
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24327
5350 y FX(h)2318 5362 y Ga(c)2357 5350 y FX(i)2384 5362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24328
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24329
5350 y F9(\))2615 5362 y FL(Cut)o F4(\()2787 5350 y FX(h)2815
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24330
5362 y Ga(a)2863 5350 y FX(i)2891 5362 y Ga(M)g(;)3029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24331
5350 y F9(\()3057 5362 y Ga(y)3105 5350 y F9(\))3132
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24332
5362 y Ga(P)j F4(\)\))26 b Ga(:)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24333
%%Page: 104 116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24334
TeXDict begin 104 115 bop Black -144 51 a Gb(104)2310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24335
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24336
3691 4 v Black Black -144 300 V -144 1259 4 959 v 1112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24337
429 a FU(M)1244 394 y FS(aux)1224 429 y FT(\000)-24 b(\000)f(!)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24338
b FU(N)55 b Gd(on)20 b(the)g(outermost)f(le)n(v)o(el)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24339
1112 450 1159 4 v 1503 535 a FU(M)p 1635 466 113 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24340
1635 500 a FS(aux)1616 535 y FT(\000)-25 b(\000)g(!)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24341
b FU(N)587 717 y(M)p 719 648 V 719 682 a FS(aux)699 717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24342
y FT(\000)-24 b(\000)f(!)23 b FU(M)976 687 y FQ(0)p -46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24343
737 1678 4 v -46 822 a FF(And)95 834 y FS(R)150 822 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24344
FG(\()182 810 y FQ(h)209 822 y FU(a)253 810 y FQ(i)279
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24345
822 y FU(M)9 b(;)406 810 y FQ(h)433 822 y FU(b)469 810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24346
y FQ(i)495 822 y FU(N)g(;)14 b(c)p FG(\))p 719 753 113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24347
3 v 719 787 a FS(aux)699 822 y FT(\000)-24 b(\000)f(!)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24348
b FF(And)1027 834 y FS(R)1082 822 y FG(\()1114 810 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24349
FQ(h)1141 822 y FU(a)1185 810 y FQ(i)1212 822 y FU(M)1302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24350
792 y FQ(0)1324 822 y FU(;)1361 810 y FQ(h)1388 822 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24351
FU(b)1424 810 y FQ(i)1451 822 y FU(N)8 b(;)14 b(c)p FG(\))2073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24352
716 y FU(M)32 b Gd(is)21 b(normal)220 b FU(N)p 2837 647
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24353
V 2837 681 a FS(aux)2818 716 y FT(\000)-25 b(\000)g(!)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24354
b FU(N)3080 686 y FQ(0)p 1750 737 1678 4 v 1750 822 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24355
FF(And)1891 834 y FS(R)1945 822 y FG(\()1977 810 y FQ(h)2004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24356
822 y FU(a)2048 810 y FQ(i)2075 822 y FU(M)9 b(;)2202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24357
810 y FQ(h)2229 822 y FU(b)2265 810 y FQ(i)2291 822 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24358
FU(N)g(;)14 b(c)p FG(\))p 2515 753 113 3 v 2515 787 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24359
FS(aux)2495 822 y FT(\000)-25 b(\000)h(!)23 b FF(And)2823
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24360
834 y FS(R)2877 822 y FG(\()2909 810 y FQ(h)2936 822
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24361
y FU(a)2980 810 y FQ(i)3007 822 y FU(M)9 b(;)3134 810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24362
y FQ(h)3161 822 y FU(b)3197 810 y FQ(i)3223 822 y FU(N)3299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24363
792 y FQ(0)3322 822 y FU(;)14 b(c)p FG(\))1484 1027 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24364
FU(M)p 1617 958 V 1617 992 a FS(aux)1597 1027 y FT(\000)-24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24365
b(\000)f(!)23 b FU(M)1874 997 y FQ(0)p 1049 1047 1283
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24366
4 v 1049 1139 a FF(And)1190 1102 y FS(i)1190 1160 y(L)1240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24367
1139 y FG(\()1272 1127 y FJ(\()1298 1139 y FU(x)1345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24368
1127 y FJ(\))1372 1139 y FU(M)8 b(;)14 b(y)s FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24369
1617 1070 113 3 v 1617 1104 a FS(aux)1597 1139 y FT(\000)-24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24370
b(\000)f(!)23 b FF(And)1925 1102 y FS(i)1925 1160 y(L)1975
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24371
1139 y FG(\()2007 1127 y FJ(\()2033 1139 y FU(x)2080
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24372
1127 y FJ(\))2106 1139 y FU(M)2196 1109 y FQ(0)2219 1139
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24373
y FU(;)14 b(y)s FG(\))p 3543 1259 4 959 v -144 1262 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24374
4 v 321 1416 a Gg(Figure)23 b(4.1:)29 b(Excerpt)23 b(from)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24375
(reduction)i(system)e(for)f(the)h(leftmost-outermost)k(reduction)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24376
(strat-)321 1529 y(e)o(gy)f(of)593 1492 y Gc(aux)572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24377
1529 y F6(\000)-31 b(\000)g(!)p Gg(.)p Black 321 1937
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24378
a(In)28 b(the)g(\002rst)g(case)g(it)f(is)h(not)g(dif)n(\002cult)h(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24379
(see)g(that)g(the)g(choice)h(does)g(not)f(matter:)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24380
b(both)29 b(rules)g(yield)321 2050 y(the)g(normal)h(form)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24381
FL(Ax)o F4(\()p Ga(x;)15 b(b)p F4(\))p Gg(.)46 b(In)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24382
b(the)i(second)g(we)e(recei)n(v)o(e,)j(in)e(general,)j(dif)n(ferent)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24383
(normal)e(forms)321 2163 y(depending)36 b(on)d(which)h(choice)g(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24384
(made.)57 b(So)32 b(both)i(choices)h(need)f(to)f(be)g(e)o(xplored)i(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24385
e(order)h(to)321 2276 y(compute)k(all)f(normal)h(forms)f(reachable)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24386
(from)e(a)f(proof.)70 b(Also)37 b(in)g(the)g(third)g(case)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24387
(choice)321 2388 y(matters)26 b(with)f(respect)i(to)e(the)h(reachable)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24388
(normal)f(forms,)g(as)f(illustrated)j(in)d(Figure)h(4.2.)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24389
b(If)25 b(we)f(do)321 2501 y(not)g(e)o(xplore)h(both)g(choices,)g(we)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24390
(may)i(\223lose\224)h(some)e(normal)h(forms.)462 2631
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24391
y(T)-7 b(o)30 b(a)n(v)n(oid)j(backtracking)i(o)o(v)o(er)c(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24392
(choices)i(in)e(Cases)h(\(ii\))f(and)g(\(iii\))h(we)e(add)i(the)f(ne)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24393
(w)g(term)321 2744 y(constructor)1719 2857 y FL(Mix)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24394
F4(\()p Ga(M)5 b(;)15 b(N)10 b F4(\))321 3015 y Gg(to)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24395
b(the)g(grammar)f(of)h(terms)g(and,)f(correspondingly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24396
b(,)29 b(add)24 b(the)g(rule)1330 3181 y F4(\000)1387
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24397
3195 y F9(1)1451 3169 y Gc(.)1506 3181 y Ga(M)1629 3169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24398
y Gc(.)1684 3181 y F4(\001)1760 3195 y F9(1)1890 3181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24399
y F4(\000)1947 3195 y F9(2)2012 3169 y Gc(.)2067 3181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24400
y Ga(N)2175 3169 y Gc(.)2230 3181 y F4(\001)2306 3195
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24401
y F9(2)p 1290 3214 1095 4 v 1290 3299 a F4(\000)1347
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24402
3313 y F9(1)1386 3299 y Ga(;)15 b F4(\000)1483 3313 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24403
F9(2)1548 3287 y Gc(.)1603 3299 y FL(Mix)o F4(\()p Ga(M)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24404
b(;)15 b(N)10 b F4(\))2059 3287 y Gc(.)2114 3299 y F4(\001)2190
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24405
3313 y F9(1)2229 3299 y Ga(;)15 b F4(\001)2345 3313 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24406
F9(2)2426 3245 y Gg(Mix)321 3487 y(to)26 b(the)f(set)h(of)f(rules)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24407
(for)f(forming)i(typing)g(judgements.)36 b(W)-7 b(e)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24408
b(shall)i(denote)h(the)f(corresponding)j(set)321 3600
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24409
y(of)23 b(well-typed)i(terms)e(by)h FY(T)1229 3567 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24410
Gc(m)1291 3600 y Gg(.)k(There)23 b(are)g(no)g(reduction)j(rules)e(that)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24411
f(apply)i(to)e(Mix,)f(b)n(ut)i(using)g(this)321 3713
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24412
y(ne)n(w)g(term)g(constructor)k(we)c(can)h(reformulate)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24413
(reductions)j(that)d(apply)g(in)g(Cases)g(\(ii\))f(and)h(\(iii\).)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24414
3826 y(The)e(ne)n(w)g(reductions)k(are)c(listed)i(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24415
b(.)p Black Black 371 4104 a FL(Cut)p F4(\()544 4092
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24416
y FX(h)572 4104 y Ga(a)620 4092 y FX(i)647 4104 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24417
b(;)785 4092 y F9(\()813 4104 y Ga(x)865 4092 y F9(\))893
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24418
4104 y Ga(N)g F4(\))502 4227 y Gc(c)533 4203 y FC(000)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24419
4264 y F6(\000)-31 b(\000)f(!)100 b Ga(M)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24420
b Ga(a)26 b F4(:=)1058 4252 y F9(\()1086 4264 y Ga(x)1138
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24421
4252 y F9(\))1165 4264 y Ga(N)q F6(g)142 b Gg(if)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24422
Ga(N)33 b Gg(freshly)25 b(introduces)h Ga(x)p Gg(,)d(b)n(ut)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24423
Ga(M)33 b Gg(does)24 b(not)g Ga(a)p Gg(,)53 b(or)502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24424
4375 y Gc(c)533 4352 y FC(000)462 4412 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24425
b(\000)f(!)100 b Ga(N)5 b F6(f)-7 b Ga(x)26 b F4(:=)1047
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24426
4400 y FX(h)1074 4412 y Ga(a)1122 4400 y FX(i)1150 4412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24427
y Ga(M)q F6(g)142 b Gg(if)23 b Ga(M)33 b Gg(freshly)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24428
b(introduces)h Ga(a)p Gg(,)d(b)n(ut)h Ga(N)33 b Gg(does)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24429
b(not)g Ga(x)p Gg(,)53 b(or)502 4524 y Gc(c)533 4500
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24430
y FC(000)462 4561 y F6(\000)-31 b(\000)f(!)100 b FL(Mix)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24431
F4(\()p Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)1237 4549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24432
y F9(\()1264 4561 y Ga(x)1316 4549 y F9(\))1344 4561
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24433
y Ga(N)p F6(g)15 b Ga(;)31 b(N)5 b F6(f)-7 b Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24434
b F4(:=)1848 4549 y FX(h)1876 4561 y Ga(a)1924 4549 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24435
FX(i)1951 4561 y Ga(M)q F6(g)p F4(\))1426 4697 y Gg(if)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24436
Ga(M)33 b Gg(and)24 b Ga(N)32 b Gg(do)24 b(not,)g(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24437
b(,)26 b(freshly)f(introduce)h Ga(a)d Gg(and)h Ga(x)371
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24438
4928 y FL(Cut)p F4(\()544 4916 y FX(h)572 4928 y Ga(b)611
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24439
4916 y FX(i)638 4928 y FL(Imp)783 4950 y Gc(R)840 4928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24440
y F4(\()875 4916 y F9(\()903 4928 y Ga(x)955 4916 y F9(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24441
FX(h)1010 4928 y Ga(a)1058 4916 y FX(i)1086 4928 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24442
b(;)15 b(b)p F4(\))p Ga(;)1338 4916 y F9(\()1366 4928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24443
y Ga(z)1412 4916 y F9(\))1440 4928 y FL(Imp)1584 4950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24444
y Gc(L)1637 4928 y F4(\()1672 4916 y FX(h)1699 4928 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24445
Ga(c)1738 4916 y FX(i)1766 4928 y Ga(N)10 b(;)1889 4916
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24446
y F9(\()1917 4928 y Ga(y)1965 4916 y F9(\))1992 4928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24447
y Ga(P)j(;)i(z)t F4(\))q(\))516 5040 y Gc(l)538 5016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24448
y FC(00)462 5077 y F6(\000)-31 b(\000)f(!)100 b FL(Mix)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24449
F4(\()p FL(Cut)p F4(\()1083 5065 y FX(h)1111 5077 y Ga(a)1159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24450
5065 y FX(i)1187 5077 y FL(Cut)o F4(\()1359 5065 y FX(h)1387
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24451
5077 y Ga(c)1426 5065 y FX(i)1454 5077 y Ga(N)10 b(;)1577
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24452
5065 y F9(\()1605 5077 y Ga(x)1657 5065 y F9(\))1684
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24453
5077 y Ga(M)g F4(\))q Ga(;)1858 5065 y F9(\()1886 5077
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24454
y Ga(y)1934 5065 y F9(\))1961 5077 y Ga(P)j F4(\))i Ga(;)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24455
b FL(Cut)p F4(\()2311 5065 y FX(h)2339 5077 y Ga(c)2378
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24456
5065 y FX(i)2405 5077 y Ga(N)10 b(;)2528 5065 y F9(\()2556
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24457
5077 y Ga(x)2608 5065 y F9(\))2636 5077 y FL(Cut)o F4(\()2808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24458
5065 y FX(h)2836 5077 y Ga(a)2884 5065 y FX(i)2912 5077
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24459
y Ga(M)g(;)3050 5065 y F9(\()3078 5077 y Ga(y)3126 5065
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24460
y F9(\))3153 5077 y Ga(P)j F4(\)\)\))1426 5213 y Gg(if)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24461
b FL(Imp)1648 5235 y Gc(R)1706 5213 y F4(\()1741 5201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24462
y F9(\()1769 5213 y Ga(x)1821 5201 y F9(\))p FX(h)1876
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24463
5213 y Ga(a)1924 5201 y FX(i)1951 5213 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24464
b(b)p F4(\))24 b Gg(freshly)h(introduces)i Ga(b)22 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24465
Gg(and)1426 5326 y(if)h FL(Imp)1648 5348 y Gc(L)1701
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24466
5326 y F4(\()1736 5314 y FX(h)1763 5326 y Ga(c)1802 5314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24467
y FX(i)1830 5326 y Ga(N)10 b(;)1953 5314 y F9(\()1981
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24468
5326 y Ga(y)2029 5314 y F9(\))2056 5326 y Ga(P)j(;)i(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24469
F4(\))24 b Gg(freshly)h(introduces)i Ga(z)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24470
eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24471
%%Page: 105 117
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24472
TeXDict begin 105 116 bop Black 277 51 a Gb(4.1)23 b(Implementation)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24473
2798 b(105)p 277 88 3691 4 v Black Black 277 1079 V 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24474
3771 4 2693 v Black Black 1270 1171 322 4 v 1270 1244
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24475
a FU(A;)14 b(B)p 1455 1232 10 38 v 1465 1216 42 4 v 93
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24476
w(B)p 1211 1280 440 4 v 1211 1354 a(A)p FT(^)q FU(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24477
1514 1342 10 38 v 1523 1325 42 4 v 92 w(B)1692 1293 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24478
FT(^)1747 1305 y FS(L)1793 1313 y FP(2)p 1198 1390 467
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24479
4 v 1198 1463 a FU(A)p FT(^)p FU(A)p 1396 1451 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24480
v 1406 1434 42 4 v 88 w(B)t FT(\033)p FU(B)1706 1406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24481
y FT(\033)1770 1418 y FS(R)p 1974 1203 223 4 v 1974 1276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24482
a FU(B)p 2059 1264 10 38 v 2069 1248 42 4 v 92 w(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24483
1912 1296 345 4 v 1912 1370 a(B)t FT(^)q FU(B)p 2121
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24484
1358 10 38 v 2130 1341 42 4 v 92 w(B)2299 1309 y FT(^)2354
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24485
1322 y FS(L)2400 1330 y FP(1)p 2580 1203 223 4 v 2580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24486
1276 a FU(B)p 2666 1264 10 38 v 2676 1248 42 4 v 93 w(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24487
2519 1296 345 4 v 2519 1370 a(B)p 2605 1358 10 38 v 2614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24488
1341 42 4 v 92 w(B)t FT(_)q FU(B)2905 1309 y FT(_)2961
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24489
1322 y FS(R)3011 1330 y FP(1)p 1912 1390 952 4 v 2037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24490
1463 a FU(B)t FT(\033)o FU(B)t(;)g(B)t FT(^)q FU(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24491
2481 1451 10 38 v 2490 1434 42 4 v 92 w(B)t FT(_)q FU(B)2905
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24492
1406 y FT(\033)2970 1418 y FS(L)p 1198 1499 1543 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24493
1627 1572 a FU(A)p FT(^)p FU(A;)g(B)t FT(^)q FU(B)p 2052
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24494
1560 10 38 v 2061 1544 42 4 v 92 w(B)t FT(_)q FU(B)2781
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24495
1527 y Gd(Cut)1247 1785 y FN(.)1511 b(&)p 400 2050 223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24496
4 v 400 2123 a FU(B)p 485 2111 10 38 v 495 2095 42 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24497
v 92 w(B)p 338 2143 345 4 v 338 2217 a(B)t FT(^)q FU(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24498
546 2205 10 38 v 556 2188 42 4 v 92 w(B)711 2156 y FT(^)766
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24499
2169 y FS(L)812 2177 y FP(1)p 948 2034 322 4 v 948 2107
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24500
a FU(A;)14 b(B)p 1133 2095 10 38 v 1143 2079 42 4 v 93
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24501
w(B)p 890 2143 440 4 v 890 2217 a(A)p FT(^)p FU(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24502
1192 2205 10 38 v 1202 2188 42 4 v 93 w(B)1356 2156 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24503
FT(^)1412 2169 y FS(L)1458 2177 y FP(2)p 338 2253 991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24504
4 v 553 2326 a FU(A)p FT(^)p FU(A;)g(B)t FT(^)q FU(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24505
978 2314 10 38 v 987 2298 42 4 v 92 w(B)1356 2281 y Gd(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24506
1610 2160 223 4 v 1610 2233 a FU(B)p 1696 2221 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24507
v 1706 2204 42 4 v 92 w(B)p 1549 2253 345 4 v 1549 2326
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24508
a(B)p 1635 2314 10 38 v 1644 2298 42 4 v 92 w(B)t FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24509
FU(B)1922 2265 y FT(_)1977 2278 y FS(R)2027 2286 y FP(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24510
553 2362 1342 4 v 881 2436 a FU(A)p FT(^)q FU(A;)g(B)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24511
FT(^)q FU(B)p 1306 2424 10 38 v 1316 2407 42 4 v 92 w(B)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24512
FT(_)p FU(B)1922 2390 y Gd(Cut)p 2243 2160 223 4 v 2243
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24513
2233 a FU(B)p 2328 2221 10 38 v 2338 2204 42 4 v 92 w(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24514
2182 2253 345 4 v 2182 2326 a(B)t FT(^)p FU(B)p 2390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24515
2314 10 38 v 2399 2298 42 4 v 92 w(B)2554 2265 y FT(^)2609
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24516
2278 y FS(L)2655 2286 y FP(1)p 2806 2034 322 4 v 2806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24517
2107 a FU(A;)g(B)p 2990 2095 10 38 v 3000 2079 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24518
92 w(B)p 2747 2143 440 4 v 2747 2217 a(A)p FT(^)p FU(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24519
3049 2205 10 38 v 3059 2188 42 4 v 93 w(B)3213 2156 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24520
FT(^)3269 2169 y FS(L)3315 2177 y FP(2)p 3454 2050 223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24521
4 v 3454 2123 a FU(B)p 3539 2111 10 38 v 3549 2095 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24522
4 v 92 w(B)p 3392 2143 345 4 v 3392 2217 a(B)p 3478 2205
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24523
10 38 v 3488 2188 42 4 v 92 w(B)t FT(_)q FU(B)3765 2156
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24524
y FT(_)3820 2169 y FS(R)3870 2177 y FP(1)p 2747 2253
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24525
991 4 v 2961 2326 a FU(B)t(;)g(A)p FT(^)q FU(A)p 3264
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24526
2314 10 38 v 3273 2298 42 4 v 88 w(B)t FT(_)q FU(B)3765
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24527
2281 y Gd(Cut)p 2182 2362 1342 4 v 2510 2436 a FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24528
FT(^)q FU(A;)g(B)t FT(^)p FU(B)p 2935 2424 10 38 v 2945
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24529
2407 42 4 v 92 w(B)t FT(_)q FU(B)3550 2390 y Gd(Cut)1171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24530
2648 y FN(#)1783 b(#)1186 2740 y FM(.)1186 2773 y(.)1186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24531
2806 y(.)3029 2740 y(.)3029 2773 y(.)3029 2806 y(.)1171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24532
2919 y FN(#)g(#)p 948 3168 322 4 v 948 3242 a FU(A;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24533
b(B)p 1133 3230 10 38 v 1143 3213 42 4 v 93 w(B)p 890
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24534
3278 440 4 v 890 3351 a(A)p FT(^)p FU(A;)g(B)p 1192 3339
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24535
10 38 v 1202 3322 42 4 v 93 w(B)1370 3290 y FT(^)1426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24536
3303 y FS(L)1472 3311 y FP(2)p 828 3387 562 4 v 828 3460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24537
a FU(A)p FT(^)q FU(A;)g(B)t FT(^)q FU(B)p 1253 3448 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24538
38 v 1263 3432 42 4 v 92 w(B)1431 3400 y FT(^)1487 3412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24539
y FS(L)1533 3420 y FP(1)p 767 3496 684 4 v 767 3570 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24540
FU(A)p FT(^)q FU(A;)g(B)t FT(^)p FU(B)p 1192 3558 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24541
38 v 1202 3541 42 4 v 93 w(B)t FT(_)p FU(B)1493 3509
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24542
y FT(_)1548 3522 y FS(R)1598 3530 y FP(1)p 2794 3168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24543
322 4 v 2794 3242 a FU(A;)g(B)p 2979 3230 10 38 v 2988
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24544
3213 42 4 v 92 w(B)p 2735 3278 440 4 v 2735 3351 a(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24545
FT(^)q FU(A;)g(B)p 3038 3339 10 38 v 3047 3322 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24546
92 w(B)3216 3290 y FT(^)3271 3303 y FS(L)3317 3311 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24547
FP(2)p 2674 3387 562 4 v 2674 3460 a FU(A)p FT(^)q FU(A;)g(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24548
2976 3448 10 38 v 2986 3432 42 4 v 92 w(B)t FT(_)p FU(B)3277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24549
3400 y FT(_)3332 3412 y FS(R)3382 3420 y FP(1)p 2613
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24550
3496 684 4 v 2613 3570 a FU(A)p FT(^)p FU(A;)g(B)t FT(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24551
FU(B)p 3038 3558 10 38 v 3047 3541 42 4 v 92 w(B)t FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24552
FU(B)3338 3509 y FT(^)3393 3522 y FS(L)3439 3530 y FP(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24553
3965 3771 4 2693 v 277 3774 3691 4 v 277 3928 a Gg(Figure)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24554
b(4.2:)29 b(The)21 b(cut)h(in)g(the)g(upper)g(proof)h(is)f(replaced)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24555
(by)e(tw)o(o)f(cuts)h(on)g(smaller)h(formulae.)29 b(There)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24556
4041 y(are)22 b(tw)o(o)f(possibilites)k(of)d(ho)n(w)f(to)h(nest)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24557
(resulting)i(cuts,)f(as)e(is)h(sho)n(wn)g(in)f(the)h(second)h(line.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24558
b(From)277 4154 y(these)d(proofs)g(the)f(normal)h(forms)f(gi)n(v)o(en)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24559
(belo)n(w)g(can)g(be)g(reached.)35 b(The)24 b(normal)i(form)e(on)h(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24560
g(left,)277 4267 y(ho)n(we)n(v)o(er)l(,)e(cannot)g(be)f(reached)h(from)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24561
f(the)g(proof)h(in)f(the)g(second)h(line)g(on)e(the)i(right,)f(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24562
(the)f(normal)277 4380 y(form)f(on)h(the)f(right)i(not)e(from)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24563
(proof)i(in)e(the)h(second)h(line)e(on)h(the)g(left.)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24564
b(Thus)21 b(the)h(choice)h(of)e(ho)n(w)277 4493 y(to)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24565
b(nest)h(the)g(cuts)g(when)g(applying)h(a)e(logical)i(cut)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24566
F6(\033)1985 4507 y Gc(R)2042 4493 y Ga(=)p F6(\033)2159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24567
4507 y Gc(L)2238 4493 y Gg(matters)g(with)g(respect)h(to)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24568
(normal)277 4606 y(forms)24 b(that)g(can)g(be)f(reached.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24569
Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24570
%%Page: 106 118
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24571
TeXDict begin 106 117 bop Black -144 51 a Gb(106)2310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24572
b(A)n(pplications)23 b(of)h(Cut-Elimination)p -144 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24573
3691 4 v Black 321 388 a Gg(It)d(is)g(not)h(hard)g(to)f(see)g(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24574
(the)f(reformulation)k(does)d(not)f(af)n(fect)h(the)g(beha)n(viour)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24575
(of)d(cut-elimination)321 501 y(with)g(respect)i(to)f(strong)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24576
(normalisation.)31 b(The)21 b(corresponding)k(ne)n(w)c(strate)o(gy)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24577
(is)e(denoted)i(by)p 3366 427 119 3 v 3366 464 a Gc(aux)3496
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24578
441 y FC(0)3351 501 y F6(\000)-31 b(\000)f(!)p Gg(;)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24579
614 y(the)32 b(ne)n(w)f(reduction)j(system)e(is)g(denoted)h(by)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24580
F4(\()p FY(T)1928 581 y Gc(m)1991 614 y Ga(;)p 2046 539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24581
V 2046 577 a Gc(aux)2176 554 y FC(0)2031 614 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24582
b(\000)g(!)o F4(\))p Gg(.)53 b(T)-7 b(o)30 b(ensure)j(that)f(we)f(do)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24583
(not)f(ha)n(v)o(e)i(to)321 727 y(e)o(xplore)25 b(an)o(y)f(choices)h(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24584
f F4(\()p FY(T)1253 694 y Gc(m)1316 727 y Ga(;)p 1371
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24585
652 V 1371 690 a Gc(aux)1501 666 y FC(0)1356 727 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24586
b(\000)f(!)p F4(\))p Gg(,)23 b(we)g(sho)n(w)g(that)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24587
2122 652 V 2122 690 a Gc(aux)2252 666 y FC(0)2107 727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24588
y F6(\000)-31 b(\000)g(!)22 b Gg(is)i(con\003uent.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24589
Black 321 915 a Gb(Pr)n(oposition)h(4.1.4:)p Black 34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24590
w Gg(The)e(reduction)p 1582 840 V 1582 878 a Gc(aux)1712
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24591
854 y FC(0)1567 915 y F6(\000)-31 b(\000)g(!)22 b Gg(is)i(con\003uent.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24592
p Black 321 1127 a F7(Pr)l(oof)o(.)p Black 34 w Gg(By)f(inspection)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24593
(of)e(the)f(reduction)k(rules:)j(the)24 b(only)g(critical)h(pair)f(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24594
(of)h(the)g(form)1399 1392 y FL(Cut)p F4(\()1572 1380
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24595
y FX(h)1600 1392 y Ga(a)1648 1380 y FX(i)1676 1392 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24596
FL(Ax)o F4(\()p Ga(x;)15 b(a)p F4(\))q Ga(;)2029 1380
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24597
y F9(\()2057 1392 y Ga(y)2105 1380 y F9(\))2132 1392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24598
y FL(Ax)p F4(\()p Ga(y)s(;)g(b)p F4(\))q(\))321 1637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24599
y Gg(which)24 b(in)g(both)g(cases)h(reduces)g(to)e FL(Ax)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24600
F4(\()p Ga(x;)15 b(b)p F4(\))p Gg(.)p 3480 1637 4 62
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24601
v 3484 1579 55 4 v 3484 1637 V 3538 1637 4 62 v 321 1848
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24602
a(Consequently)-6 b(,)24 b(it)c(does)h(not)g(matter)f(which)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24603
1758 1773 119 3 v 1758 1811 a Gc(aux)1887 1787 y FC(0)1743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24604
1848 y F6(\000)-32 b(\000)h(!)p Gg(-reduction)23 b(we)c(apply:)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24605
b(we)19 b(recei)n(v)o(e)i(only)g(a)f(single)321 1961
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24606
y(normal)i(form,)g(which)f(of)h(course)h(in)e(general)i(contains)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24607
(instances)h(of)d FL(Mix)p Gg(.)27 b(This)21 b(normal)h(form)f(can)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24608
2074 y(be)28 b(seen)h(as)f(a)f(compact)i(encoding)i(of)c(the)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24609
(collection)h(of)e(all)g(normal)h(forms)f(reachable)i(by)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24610
3374 1999 V 3374 2037 a Gc(aux)3353 2074 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24611
b(\000)g(!)p Gg(,)321 2187 y(and)35 b(thus)h(also)f(by)997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24612
2149 y Gc(aux)976 2187 y F6(\000)-31 b(\000)g(!)33 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24613
Gg(and)1376 2149 y Gc(cut)1345 2187 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24614
Gg(.)61 b(In)34 b(order)i(to)e(distinguish)k(them)d(from)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24615
(normal)g(forms)321 2299 y(reachable,)30 b(for)c(e)o(xample,)i(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24616
F4(\()p FY(T)t Ga(;)p 1455 2225 V 1455 2262 a Gc(aux)1435
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24617
2299 y F6(\000)-31 b(\000)g(!)p F4(\))p Gg(,)26 b(we)g(shall)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24618
(henceforth)i(refer)f(to)e(the)h(normal)g(forms)g(with)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24619
2412 y(instances)h(of)c FL(Mix)g Gg(as)g F7(mix-normal)j(forms)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24620
Gg(.)33 b(An)24 b(\223unwinding\224)j(function)g(will)e(allo)n(w)g(us)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24621
(to)f(e)o(xtract)321 2525 y(the)g(collection)i(of)e(\223real\224)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24622
(normal)f(forms)g(from)f(a)g(mix-normal)i(form.)462 2661
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24623
y(Notice)j(that)f(the)g(\223trick\224)h(with)f(the)g(mix)g(term)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24624
(constructor)k(cannot)e(be)f(used)h(in)e F4(\()p FY(T)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24625
Ga(;)3245 2624 y Gc(aux)3225 2661 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24626
F4(\))p Gg(:)35 b(if)321 2774 y(we)28 b(replace)803 2737
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24627
y Gc(c)834 2713 y FC(0)745 2774 y F6(\000)-32 b(\000)h(!)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24628
b Gg(with)1172 2737 y Gc(c)1203 2713 y FC(000)1131 2774
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24629
y F6(\000)-31 b(\000)g(!)27 b Gg(in)h F4(\()p FY(T)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24630
Ga(;)1580 2737 y Gc(aux)1560 2774 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24631
F4(\))p Gg(,)29 b(we)e(do)h F7(not)h Gg(obtain)h(a)e(con\003uent)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24632
(reduction)i(system,)321 2887 y(as)24 b(demonstrated)i(in)e(the)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24633
(xample)g(belo)n(w)-6 b(.)p Black 321 3074 a Gb(Example)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24634
b(4.1.5:)p Black 34 w Gg(Suppose)i Ga(M)10 b Gg(,)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24635
b Ga(N)35 b Gg(and)27 b Ga(P)38 b Gg(are)27 b(cut-free)h(terms)e(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24636
(assume:)35 b(\(i\))26 b Ga(a)k F6(62)g Ga(F)13 b(C)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24637
b F4(\()p Ga(M)j F4(\))p Gg(,)321 3187 y(\(ii\))27 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24638
Ga(M)36 b Gg(freshly)28 b(introduces)h Ga(b)p Gg(,)d(\(iii\))h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24639
Ga(y)33 b F6(62)d Ga(F)13 b(N)d F4(\()p Ga(P)j F4(\))27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24640
b Gg(and)f(\(i)n(v\))h Ga(N)35 b Gg(freshly)28 b(introduces)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24641
Ga(x)p Gg(.)36 b(Consider)321 3300 y(then)25 b(the)e(follo)n(wing)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24642
(term)1330 3565 y FL(Cut)p F4(\()1503 3553 y FX(h)1531
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24643
3565 y Ga(b)1570 3553 y FX(i)1597 3565 y FL(Cut)p F4(\()1770
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24644
3553 y FX(h)1798 3565 y Ga(a)1846 3553 y FX(i)1874 3565
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24645
y Ga(M)10 b(;)2012 3553 y F9(\()2040 3565 y Ga(x)2092
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24646
3553 y F9(\))2119 3565 y Ga(N)g F4(\))p Ga(;)2277 3553
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24647
y F9(\()2305 3565 y Ga(y)2353 3553 y F9(\))2380 3565
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24648
y Ga(P)j F4(\))26 b Ga(:)321 3810 y Gg(Reducing)f(\002rst)f(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24649
(outer)i(cut,)f(we)e(recei)n(v)o(e)j(the)f(follo)n(wing)h(reduction)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24650
(sequence.)p Black Black 982 4064 a FL(Cut)p F4(\()1155
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24651
4052 y FX(h)1183 4064 y Ga(b)1222 4052 y FX(i)1249 4064
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24652
y FL(Cut)p F4(\()1422 4052 y FX(h)1450 4064 y Ga(a)1498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24653
4052 y FX(i)1526 4064 y Ga(M)10 b(;)1664 4052 y F9(\()1692
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24654
4064 y Ga(x)1744 4052 y F9(\))1771 4064 y Ga(N)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24655
Ga(;)1929 4052 y F9(\()1957 4064 y Ga(y)2005 4052 y F9(\))2032
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24656
4064 y Ga(P)j F4(\))733 4140 y Gc(aux)712 4177 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24657
b(\000)g(!)99 b FL(Mix)p F4(\()p FL(Cut)p F4(\()1334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24658
4165 y FX(h)1362 4177 y Ga(a)1410 4165 y FX(i)1437 4177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24659
y FL(Cut)p F4(\()1610 4165 y FX(h)1638 4177 y Ga(b)1677
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24660
4165 y FX(i)1704 4177 y Ga(M)10 b(;)1842 4165 y F9(\()1870
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24661
4177 y Ga(y)1918 4165 y F9(\))1945 4177 y Ga(P)j F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24662
Ga(;)2092 4165 y F9(\()2120 4177 y Ga(x)2172 4165 y F9(\))2199
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24663
4177 y Ga(N)d F4(\))15 b Ga(;)31 b(P)13 b F4(\))123 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24664
Gg(by)24 b(\(ii\))g(and)g(\(iii\))733 4253 y Gc(aux)712
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24665
4290 y F6(\000)-31 b(\000)g(!)99 b FL(Mix)p F4(\()p FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24666
F4(\()1334 4278 y FX(h)1362 4290 y Ga(a)1410 4278 y FX(i)1437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24667
4290 y Ga(P)13 b(;)1548 4278 y F9(\()1576 4290 y Ga(x)1628
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24668
4278 y F9(\))1655 4290 y Ga(N)d F4(\))15 b Ga(;)31 b(P)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24669
b F4(\))667 b Gg(by)24 b(\(ii\))g(and)g(\(iii\))733 4366
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24670
y Gc(aux)712 4403 y F6(\000)-31 b(\000)g(!)99 b FL(Mix)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24671
F4(\()p Ga(P)28 b(;)j(P)13 b F4(\))1208 b Gg(by)24 b(\(i\))g(and)g(\(i)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24672
n(v\))321 4654 y(Reducing)h(the)f(inner)h(cut)f(\002rst,)f(we)f(recei)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24673
(v)o(e)j(a)e(slightly)i(dif)n(ferent)h(reduction)g(sequence.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24674
Black Black 982 4908 a FL(Cut)p F4(\()1155 4896 y FX(h)1183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24675
4908 y Ga(b)1222 4896 y FX(i)1249 4908 y FL(Cut)p F4(\()1422
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24676
4896 y FX(h)1450 4908 y Ga(a)1498 4896 y FX(i)1526 4908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24677
y Ga(M)10 b(;)1664 4896 y F9(\()1692 4908 y Ga(x)1744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24678
4896 y F9(\))1771 4908 y Ga(N)g F4(\))p Ga(;)1929 4896
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24679
y F9(\()1957 4908 y Ga(y)2005 4896 y F9(\))2032 4908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24680
y Ga(P)j F4(\))733 4984 y Gc(aux)712 5021 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24681
b(\000)g(!)99 b FL(Cut)p F4(\()1155 5009 y FX(h)1183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24682
5021 y Ga(b)1222 5009 y FX(i)1249 5021 y Ga(M)11 b(;)1388
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24683
5009 y F9(\()1415 5021 y Ga(y)1463 5009 y F9(\))1491
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24684
5021 y Ga(P)i F4(\))1020 b Gg(by)24 b(\(i\))g(and)g(\(i)n(v\))733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24685
5097 y Gc(aux)712 5134 y F6(\000)-31 b(\000)g(!)99 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24686
Ga(P)1577 b Gg(by)24 b(\(ii\))g(and)g(\(iii\))321 5385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24687
y(So)f(the)h(restrictions)j(imposed)d(by)g(the)g(strate)o(gy)h(are)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24688
(necessary)i(for)e(con\003uence.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24689
%%Page: 107 119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24690
TeXDict begin 107 118 bop Black 277 51 a Gb(4.1)23 b(Implementation)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24691
2798 b(107)p 277 88 3691 4 v Black 277 388 a Fq(4.1.1)99
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24692
b(The)26 b(Code)277 596 y Gg(Let)j(us)h(no)n(w)g(gi)n(v)o(e)g(some)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24693
(details)i(of)d(the)i(code.)49 b(Figure)30 b(4.3)g(sho)n(ws)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24694
(main)g(datatypes)j(imple-)277 709 y(menting)25 b(the)f(follo)n(wing)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24695
(design)g(decisions:)p Black 414 981 a F6(\017)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24696
45 w Gg(names,)f(co-names)h(and)f(v)n(ariables)i(are)e(implemented)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24697
(as)e(strings,)p Black 414 1202 a F6(\017)p Black 45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24698
w Gg(in)30 b(e)n(v)o(ery)g(cut)f(we)g(record)i(e)o(xplicitly)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24699
(cut-formula)i(\(a)d(typing)i(algorithm)g(for)f(terms)g(is)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24700
1315 y(then)25 b(rather)f(simple,)g(b)n(ut)g(be)o(yond)h(the)f(scope)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24701
(of)e(this)h(thesis\),)h(and)p Black 414 1537 a F6(\017)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24702
Black 45 w Gg(there)k(are)f(four)g(syntactic)i(cate)o(gories)h(of)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24703
(terms:)38 b(terms,)29 b(named)f(terms,)h(co-named)g(terms)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24704
1650 y(and)24 b(double-bounded)29 b(terms.)277 1922 y(The)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24705
b(double-bounded)29 b(terms)c(implement)g(the)f(special)i(binding)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24706
(operation)h(in)d(terms)h(of)f(the)g(form)277 2034 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24707
FL(Imp)422 2056 y Gc(R)479 2034 y F4(\()514 2022 y F9(\()542
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24708
2034 y Ga(x)594 2022 y F9(\))p FX(h)649 2034 y Ga(a)697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24709
2022 y FX(i)724 2034 y Ga(M)11 b(;)k(b)p F4(\))p Gg(.)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24710
b(Some)19 b(e)o(xamples)h(of)f(ho)n(w)g(these)i(datatypes)h(relate)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24711
(to)g(our)f(formal)h(de\002nitions)277 2147 y(are)k(gi)n(v)o(en)g(belo)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24712
n(w)-6 b(.)p 277 2335 V 277 3745 4 1410 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24713
354 2496 a(Example)923 b(Code)354 2678 y(e)o(xpression)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24714
b FL(f)6 b F4(\()p FL(g)q F4(\()p FL(x)p F4(\)\))572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24715
b Fm(F\("f",[F\("g",[Var\("x"\)]\)]\))354 2814 y Gg(formula)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24716
b F6(8)p FL(x)p Ga(:)p F4(\()p Ga(A)15 b F6(^)g Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24717
b FL(x)p F4(\))471 b Fm(Forall\("x",And\(Atm\("A"\),Pre\("B",[Var\()o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24718
("x"\)])o(\)\)\))354 2949 y Gg(term)24 b FL(And)698 2913
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24719
y Gc(i)698 2973 y(L)750 2949 y F4(\()785 2937 y F9(\()813
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24720
2949 y Ga(x)865 2937 y F9(\))893 2949 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24721
b(y)s F4(\))485 b Fm(AndL\(t,m,"y"\))97 b Gg(where)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24722
b Fm(t)15 b F6(2)25 b(f)p Fm(I)p Gg(,)p Fm(II)p F6(g)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24723
Gg(and)2346 3085 y Fm(m)g Gg(is)g(the)h(code)h(of)2929
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24724
3073 y F9(\()2956 3085 y Ga(x)3008 3073 y F9(\))3036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24725
3085 y Ga(M)354 3220 y Gg(term)f FL(Imp)688 3242 y Gc(R)746
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24726
3220 y F4(\()781 3208 y F9(\()809 3220 y Ga(x)861 3208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24727
y F9(\))p FX(h)916 3220 y Ga(a)964 3208 y FX(i)991 3220
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24728
y Ga(M)10 b(;)15 b(b)p F4(\))396 b Fm(ImpR\(m,"b"\))197
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24729
b Gg(where)24 b Fm(m)f Gg(is)g(the)h(code)g(of)3174 3208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24730
y F9(\()3201 3220 y Ga(x)3253 3208 y F9(\))q FX(h)3308
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24731
3220 y Ga(a)3356 3208 y FX(i)3384 3220 y Ga(M)364 3356
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24732
y Gg(named)g(term)819 3344 y F9(\()846 3356 y Ga(x)898
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24733
3344 y F9(\))926 3356 y Ga(M)579 b Fm(\("x",m\))364 3491
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24734
y Gg(co-named)25 b(term)935 3479 y FX(h)962 3491 y Ga(a)1010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24735
3479 y FX(i)1038 3491 y Ga(M)467 b Fm(\("a",m\))364 3627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24736
y Gg(double-bounded)29 b(term)1167 3615 y F9(\()1195
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24737
3627 y Ga(x)1247 3615 y F9(\))p FX(h)1302 3627 y Ga(a)1350
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24738
3615 y FX(i)1377 3627 y Ga(M)128 b Fm(\("x","a",m\))2216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24739
3273 y FK(9)2216 3355 y(>)2216 3382 y(=)2216 3546 y(>)2216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24740
3573 y(;)2360 3487 y Gg(where)23 b Fm(m)g Gg(is)g(the)h(code)h(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24741
Ga(M)p 3965 3745 V 277 3748 3691 4 v 277 3975 a Gg(Ne)o(xt)h(we)g(gi)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24742
(v)o(e)h(the)g(details)h(for)e(the)h(cut-reductions)k(and)c(the)g(e)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24743
(v)n(aluation)i(function.)34 b(The)o(y)24 b(call)h(six)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24744
4088 y(subsidiary)i(functions,)e F7(viz.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 24757
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24785
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24786
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24787
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24788
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24789
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24790
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24793
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24794
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24795
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24796
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24797
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24798
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24799
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24800
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24801
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24802
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24804
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24805
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24806
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24807
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24808
4980 a Gg(Figure)24 b(4.3:)29 b(Code)24 b(for)g(main)f(datatypes.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24809
Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
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diff changeset
 24810
%%Page: 109 121
Christian Urban <urbanc@in.tum.de>
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diff changeset
 24811
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 24812
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24813
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24814
(cut-reductions.)p Black Black Black Black 277 604 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24815
Fl(let)44 b(rec)h(aux_redu)e(term)h(=)367 695 y(match)g(term)g(with)456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24816
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24817
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24818
1243 y(|)g(Cut\(Or\(f1,f2\),\(c,OrR\(t,\(a,m\),b\)\),\(z,OrL\(\(x)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24819
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24820
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24821
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24822
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24823
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24824
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24825
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24826
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24827
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24828
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24829
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24830
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24831
g F6(_)3264 2773 y Gc(R)3321 2759 y Ga(=)p F6(_)3427
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24832
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24833
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24834
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24835
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24836
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24837
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24838
i F4(\()p FY(T)t Ga(;)2285 3192 y Gc(aux)2265 3229 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24839
F6(\000)-31 b(\000)g(!)o F4(\))23 b Gg(w)o(as)g(to)f(obtain)j(a)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24840
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24841
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24842
(tail-recursi)n(v)o(e)g(function.)p Black Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24843
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24844
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24845
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24846
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24847
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24848
4105 y(|)i(AndR\(\(a,m\),\(b,n\),c\))d(->)i(AndR\(\(a,eval)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24849
(m\),\(b,eval)g(n\),c\))456 4196 y(|)i(...)456 4288 y(|)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24850
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24851
4379 y(|)i(Mix\(m,n\))537 b(->)44 b(Mix\(eval)g(m,eval)g(n\))277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24852
4470 y(;;)277 4708 y Gg(This)32 b(e)n(v)n(aluation)j(function)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24853
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24854
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24855
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24856
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24857
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24858
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24859
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24860
v 30 w(up)p 1712 5046 V 29 w(lists)p Gg(,)21 b(which)j(returns)h(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24861
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24862
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24863
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24864
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24865
Black Black eop end
Christian Urban <urbanc@in.tum.de>
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diff changeset
 24866
%%Page: 110 122
Christian Urban <urbanc@in.tum.de>
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diff changeset
 24867
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 24868
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24869
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24870
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24871
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24872
(NotL\(\(a,m\),x\))1083 754 y(->)h(map)f(\(fun)g(m)h(->)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24873
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24874
(AndR\(\(a,m\),\(b,n\),c\))1083 936 y(->)h(let)f(plist)g(=)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24875
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24876
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24877
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24878
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24879
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24880
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24881
1726 y Gg(The)25 b(ne)o(xt)g(function)i(calculates)h(the)d(number)h(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24882
e(normal)i(forms)f(encoded)i(in)e(a)f(mix-normal)j(form)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24883
1839 y(without)d(actually)g(e)o(xtracting)h(the)e(normal)g(forms.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24884
b(This)22 b(function)j(is)d(useful)i(for)e(analysing)k(e)o(xam-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24885
1952 y(ples.)p Black Black Black Black 321 2173 a Fl(let)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24886
b(rec)f(count)g(term)89 b(=)411 2264 y(match)44 b(term)g(with)590
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24887
2355 y(Ax\(x,a\))582 b(->)44 b(1)501 2447 y(|)g(NotL\(\(a,m\),x\))312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24888
b(->)44 b(\(count)g(m\))501 2538 y(|)g(AndR\(\(a,m\),\(b,n\),c\))e(->)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24889
(\(count)g(m\))2070 2551 y(*)2159 2538 y(\(count)g(n\))501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24890
2629 y(|)g(...)501 2720 y(|)g(ImpR\(\(x,a,m\),b\))222
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24891
b(->)44 b(\(count)g(m\))501 2812 y(|)g(Mix\(m,n\))537
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24892
b(->)44 b(\(count)g(m\))h(+)f(\(count)g(n\))321 2903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24893
y(;;)462 3145 y Gg(W)-7 b(e)25 b(should)i(no)n(w)d(lik)o(e)i(to)f(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24894
(xamine)h(an)f(e)o(xample.)35 b(Consider)26 b(the)g(proof)g(in)f(which)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24895
h(for)f(bre)n(vity)321 3258 y(all)f(names)g(and)g(co-names)h(are)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24896
(omitted.)p 584 3494 293 4 v 584 3568 a FU(A)p FF(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24897
704 3556 10 38 v 714 3539 42 4 v 88 w FU(A)p FF(a)p 959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24898
3493 299 4 v 83 w FU(A)p FF(b)p 1082 3556 10 38 v 1092
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24899
3539 42 4 v 88 w FU(A)p FF(b)p 584 3588 674 4 v 623 3662
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24900
a FU(A)p FF(a)p FT(_)p FU(A)p FF(b)p 904 3650 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24901
914 3633 42 4 v 89 w FU(A)p FF(a)p FU(;)14 b(A)p FF(b)1257
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24902
3604 y FT(_)1312 3617 y FS(L)p 537 3698 768 4 v 537 3777
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24903
a FT(8)p FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q FT(_)p FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24904
FF(b)p FG(\))p 990 3765 10 38 v 1000 3748 42 4 v 89 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24905
FU(A)p FF(a)p FU(;)g(A)p FF(b)1304 3720 y FT(8)1351 3732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24906
y FS(L)p 508 3817 825 4 v 508 3896 a FT(8)p FF(x)p FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24907
FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q FT(_)q FU(A)p FF(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24908
FG(\))p 1019 3884 10 38 v 1028 3868 42 4 v 89 w FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24909
FF(a)p FU(;)g(A)p FF(b)1332 3840 y FT(8)1379 3852 y FS(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24910
455 3937 931 4 v 455 4016 a FT(8)p FF(x)p FU(:)p FF(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24911
FU(:)p FG(\()p FU(A)p FF(y)q FT(_)r FU(A)p FF(x)p FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24912
966 4004 10 38 v 975 3987 42 4 v 88 w FU(A)p FF(b)p FU(;)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24913
FT(8)p FF(x)p FU(:A)p FF(x)1385 3960 y FT(8)1432 3972
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24914
y FS(R)p 403 4056 1036 4 v 403 4135 a FT(8)p FF(x)p FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24915
FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q FT(_)q FU(A)p FF(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24916
FG(\))p 913 4123 10 38 v 923 4107 42 4 v 89 w FT(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24917
FF(x)p FU(:A)p FF(x)p FU(;)g FT(8)p FF(y)q FU(:A)p FF(y)1438
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24918
4079 y FT(8)1485 4091 y FS(R)p 1637 3747 287 4 v 1637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24919
3821 a FU(A)p FF(c)p 1754 3809 10 38 v 1764 3792 42 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24920
v 88 w FU(A)p FF(c)p 2006 3746 299 4 v 83 w FU(A)p FF(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24921
2130 3809 10 38 v 2139 3792 42 4 v 88 w FU(A)p FF(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24922
1637 3840 668 4 v 1676 3915 a FU(A)p FF(c)p FU(;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24923
FF(d)p 1936 3903 10 38 v 1945 3886 42 4 v 88 w FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24924
FF(c)p FT(^)q FU(A)p FF(d)2304 3857 y FT(^)2360 3870
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24925
y FS(R)p 1622 3951 698 4 v 1622 4025 a FU(A)p FF(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24926
FU(;)g FT(8)p FF(x)p FU(:A)p FF(x)p 1990 4013 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24927
2000 3996 42 4 v 88 w FU(A)p FF(c)p FT(^)p FU(A)p FF(d)2319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24928
3973 y FT(8)2366 3985 y FS(L)p 1622 4061 698 4 v 1693
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24929
4135 a FT(8)p FF(x)p FU(:A)p FF(x)p 1919 4123 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24930
1929 4107 42 4 v 88 w FU(A)p FF(c)p FT(^)p FU(A)p FF(d)2319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24931
4080 y FT(8)2388 4050 y Fy(?)2366 4103 y FS(L)p 403 4176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24932
1846 4 v 782 4255 a FT(8)p FF(x)p FU(:)p FF(y)q FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24933
FG(\()p FU(A)p FF(y)q FT(_)q FU(A)p FF(x)p FG(\))p 1293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24934
4243 10 38 v 1302 4226 42 4 v 89 w FT(8)p FF(y)q FU(:A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24935
FF(y)q FU(;)g(A)p FF(c)p FT(^)q FU(A)p FF(d)2248 4204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24936
y Gd(Cut)p 2532 3867 287 4 v 2532 3940 a FU(A)p FF(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24937
2650 3928 10 38 v 2659 3911 42 4 v 88 w FU(A)p FF(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24938
2901 3866 275 4 v 83 w FU(A)p FF(f)p 3013 3928 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24939
v 3023 3911 42 4 v 95 w FU(A)p FF(f)p 2532 3960 645 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24940
v 2571 4034 a FU(A)p FF(e)p FU(;)g(A)p FF(f)p 2819 4022
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24941
10 38 v 2829 4006 42 4 v 95 w FU(A)p FF(e)p FT(^)p FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24942
FF(f)3176 3977 y FT(^)3232 3989 y FS(R)p 2516 4070 677
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24943
4 v 2516 4145 a FU(A)p FF(f)6 b FU(;)14 b FT(8)p FF(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24944
FU(:A)p FF(y)p 2875 4133 10 38 v 2884 4116 42 4 v 89
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24945
w FU(A)p FF(e)p FT(^)q FU(A)p FF(f)3192 4093 y FT(8)3239
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24946
4105 y FS(L)p 2516 4181 677 4 v 2581 4255 a FT(8)p FF(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24947
FU(:A)p FF(y)p 2810 4243 10 38 v 2819 4226 42 4 v 89
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24948
w FU(A)p FF(e)p FT(^)q FU(A)p FF(f)3192 4200 y FT(8)3261
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24949
4170 y Fy(?)3239 4223 y FS(L)p 782 4296 2346 4 v 1392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24950
4374 a FT(8)p FF(x)p FU(:)p FF(y)q FU(:)p FG(\()p FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24951
FF(y)q FT(_)r FU(A)p FF(x)p FG(\))p 1903 4362 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24952
1912 4346 42 4 v 88 w FU(A)p FF(c)p FT(^)q FU(A)p FF(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24953
FU(;)g(A)p FF(e)p FT(^)p FU(A)p FF(f)3127 4323 y Gd(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24954
Black 3387 4374 a(\(4.1\))p Black 321 4708 a Gg(One)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24955
b(of)h(its)f(normal)i(forms)e(is)h(gi)n(v)o(en)g(in)f(Figure)h(4.4.)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24956
b(Ho)n(we)n(v)o(er)26 b(let)h(us)f(analyse)j(the)d(collection)k(of)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24957
4821 y(all)d(normal)g(forms)g(reachable)i(from)e(this)g(proof.)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24958
b(As)26 b(can)h(be)g(seen,)g(the)g(proof)h(contains)h(tw)o(o)d(cuts)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24959
4933 y(and)i(tw)o(o)f(implicit)i(contractions)i(\(in)c(the)h(inference)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24960
i(rules)e(mark)o(ed)g(with)g(a)e(star\).)41 b(If)28 b(we)e(feed)i(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24961
321 5046 y(corresponding)33 b(term)28 b(into)h(our)f(e)n(v)n(aluation)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24962
(function)f(and)f(count)g(all)g(normal)g(forms)f(that)h(can)g(be)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24963
5159 y(e)o(xtracted)34 b(from)d(the)h(corresponding)k(mix-normal)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24964
(form,)g(we)e(recei)n(v)o(e)h(the)g(number)g(1,618,912!)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24965
5272 y(Although)h(the)f(normal)g(forms)g(are)g(all)f(moderately)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24966
(small,)f(we)e(are)g(confronted)k(with)c(a)g(serious)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24967
5385 y(problem:)58 b(the)37 b(collection)j(of)d(normal)h(forms)f(does)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24968
(not)f(\002t)g(into)g(the)h(memory)f(of)g(an)g(a)n(v)o(erage)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24969
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24970
%%Page: 111 123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24971
TeXDict begin 111 122 bop Black 277 51 a Gb(4.1)23 b(Implementation)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24972
2798 b(111)p 277 88 3691 4 v Black 277 388 a Gg(computer)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24973
b(.)636 355 y F5(1)726 388 y Gg(If)30 b(we)g(w)o(ant)h(to)g(analyse)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24974
(the)e(normal)g(forms)g(with)g(re)o(gard)h(to)e(their)i(computational)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24975
277 501 y(content,)c(we)c(ha)n(v)o(e)i(to)g(alle)n(viate)h(this)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24976
(problem.)36 b(There)25 b(are)h(tw)o(o)f(reasons)i(for)f(this)g(\223e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24977
(xplosion\224)i(in)277 614 y(the)c(number)g(of)g(normal)g(forms:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24978
Black 414 836 a F6(\017)p Black 45 w Gg(First,)34 b(in)e(case)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24979
(commuting)h(cuts)g(whose)f(cut-formula)i(is)e(not)g(freshly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24980
(introduced)i(on)504 949 y(either)i(side,)h(the)d(reduction)1533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24981
911 y Gc(c)1564 888 y FC(0)1475 949 y F6(\000)-32 b(\000)h(!)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24982
b Gg(leads,)k(in)d(general,)40 b(to)34 b(dif)n(ferent)j(normal)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24983
(forms,)i(and)504 1061 y(thus)29 b(the)f(reduct)h(of)1218
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24984
1024 y Gc(c)1249 1001 y FC(000)1178 1061 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24985
b(\000)f(!)27 b Gg(includes)j(an)e(instance)i(of)e FL(Mix)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24986
Gg(.)40 b(In)28 b(some)g(cases,)i(ho)n(we)n(v)o(er)l(,)f(this)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24987
1174 y(reduction)37 b(does)e(not)g(generate)h(dif)n(ferent)g(normal)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24988
(forms.)62 b(Consider)l(,)38 b(for)d(e)o(xample,)i(the)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24989
1287 y(follo)n(wing)25 b(proof.)p 1075 1448 315 4 v 1075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24990
1521 a FU(A)p 1155 1509 10 38 v 1165 1493 42 4 v 88 w(A;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24991
b(C)p 1472 1448 311 4 v 89 w(C)q(;)g(A)p 1651 1509 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24992
38 v 1660 1493 42 4 v 89 w(A)p 1075 1557 708 4 v 1322
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24993
1631 a(A)p 1403 1619 10 38 v 1413 1602 42 4 v 89 w(A)1824
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24994
1585 y Gd(Cut)p 2027 1448 331 4 v 2027 1521 a FU(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24995
2113 1509 10 38 v 2122 1493 42 4 v 92 w(B)t(;)g(D)p 2440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24996
1448 331 4 v 85 w(D)r(;)g(B)p 2634 1509 10 38 v 2643
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24997
1493 42 4 v 92 w(B)p 2027 1557 744 4 v 2288 1631 a(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24998
2373 1619 10 38 v 2383 1602 42 4 v 92 w(B)2812 1585 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 24999
Gd(Cut)p 1322 1651 1188 4 v 1697 1724 a FU(A;)g(B)p 1881
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25000
1712 10 38 v 1891 1695 42 4 v 92 w(A)p FT(^)p FU(B)2551
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25001
1668 y FT(^)2607 1680 y FS(R)504 2063 y Gg(Our)32 b(e)n(v)n(aluation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25002
(function)g(calculates)g(a)e(mix-normal)h(form)f(for)g(this)g(proof)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25003
(from)f(which)504 2176 y(four)25 b(normal)f(forms)g(can)g(be)f(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25004
(xtracted.)31 b(But)23 b(all)h(of)f(them)h(stand)g(for)g(the)g(proof)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25005
1646 2336 213 4 v 1646 2410 a FU(A)p 1726 2398 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25006
v 1736 2381 42 4 v 88 w(A)p 1941 2336 223 4 v 83 w(B)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25007
2027 2398 10 38 v 2036 2381 42 4 v 92 w(B)p 1646 2430
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25008
518 4 v 1685 2503 a(A;)14 b(B)p 1870 2491 10 38 v 1879
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25009
2474 42 4 v 92 w(A)p FT(^)q FU(B)2205 2446 y FT(^)2260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25010
2459 y FS(R)2338 2446 y FU(:)504 2842 y Gg(W)-7 b(e)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25011
b(could)i(alle)n(viate)h(our)e(problem)h(by)e(letting)2051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25012
2805 y Gc(c)2082 2781 y FC(000)2011 2842 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25013
b(\000)h(!)19 b Gg(only)h(reduce)i(to)d FL(Mix)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25014
Ga(M)5 b(;)15 b(N)10 b F4(\))20 b Gg(pro)o(vided)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25015
2955 y Ga(M)42 b F6(6\021)31 b Ga(N)10 b Gg(.)38 b(Ho)n(we)n(v)o(er)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25016
(,)28 b(in)e(the)i(e)o(xamples)g(we)e(analysed)j(this)f(did)f(not)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25017
(reduce)i(substantially)504 3067 y(the)24 b(number)h(of)e(normal)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25018
(forms.)p Black 414 3253 a F6(\017)p Black 45 w Gg(The)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25019
b(second)i(reason)f(is)f(more)g(subtle.)65 b(T)-7 b(o)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25020
b(e)o(xplain)i(the)g(issues)g(we)e(gi)n(v)o(e)i(the)f(reader)h(an)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25021
3365 y(e)o(xample.)30 b(Consider)25 b(the)f(proof)p 1535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25022
3526 213 4 v 1535 3600 a FU(A)p 1616 3588 10 38 v 1625
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25023
3571 42 4 v 88 w(A)p 1474 3620 335 4 v 1474 3693 a(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25024
FT(^)p FU(B)p 1677 3681 10 38 v 1686 3664 42 4 v 92 w(A)1850
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25025
3642 y FT(^)1906 3612 y FJ(1)1906 3665 y FS(L)p 2099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25026
3526 213 4 v 2099 3600 a FU(A)p 2179 3588 10 38 v 2189
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25027
3571 42 4 v 88 w(A)p 2038 3620 334 4 v 2038 3693 a(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25028
2119 3681 10 38 v 2129 3664 42 4 v 88 w(A)p FT(_)q FU(C)2413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25029
3642 y FT(_)2468 3612 y FJ(1)2468 3665 y FS(R)p 1474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25030
3713 898 4 v 1695 3786 a FU(A)p FT(^)p FU(B)p 1898 3774
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25031
10 38 v 1907 3757 42 4 v 92 w(A)p FT(_)q FU(C)2413 3741
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25032
y Gd(Cut)504 4125 y Gg(and)33 b(let)e(us)h(calculate)i(the)e(normal)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25033
(forms)f(reachable)i(from)d(this)i(proof\227we)f(recei)n(v)o(e)h(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25034
504 4238 y(follo)n(wing)25 b(tw)o(o)f(proofs)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25035
Black 1355 4414 213 4 v 1355 4487 a FU(A)p 1436 4475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25036
10 38 v 1445 4459 42 4 v 88 w(A)p 1295 4507 334 4 v 1295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25037
4580 a(A)p 1375 4568 10 38 v 1385 4552 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25038
FT(_)q FU(C)1669 4529 y FT(_)1725 4499 y FJ(1)1725 4552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25039
y FS(R)p 1233 4600 456 4 v 1233 4674 a FU(A)p FT(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25040
FU(B)p 1437 4662 10 38 v 1446 4645 42 4 v 92 w(A)p FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25041
FU(C)1731 4623 y FT(^)1786 4593 y FJ(1)1786 4646 y FS(L)1935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25042
4603 y Gg(and)p 2288 4414 213 4 v 2288 4487 a FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25043
2368 4475 10 38 v 2378 4459 42 4 v 88 w(A)p 2226 4507
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25044
335 4 v 2226 4580 a(A)p FT(^)q FU(B)p 2430 4568 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25045
v 2439 4552 42 4 v 92 w(A)2603 4529 y FT(^)2658 4499
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25046
y FJ(1)2658 4552 y FS(L)p 2166 4600 456 4 v 2166 4674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25047
a FU(A)p FT(^)q FU(B)p 2369 4662 10 38 v 2379 4645 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25048
4 v 92 w(A)p FT(_)p FU(C)2663 4623 y FT(_)2719 4593 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25049
FJ(1)2719 4646 y FS(R)504 4902 y Gg(which)30 b(dif)n(fer)h(only)g(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25050
(the)h(order)h(of)f(their)g(inferences)j(and)d(can)g(be)g(identi\002ed)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25051
h(by)f(simple)504 5015 y(permutation)d(rules)d(introduced)j(by)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25052
(Kleene)g([1952b].)32 b(In)23 b(what)h(follo)n(ws,)g(we)f(shall)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25053
(restrict)504 5128 y(the)32 b(reduction)1061 5091 y Gc(c)1092
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25054
5067 y FC(000)1021 5128 y F6(\000)-32 b(\000)h(!)31 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25055
Gg(such)h(that)h(in)e(the)h(e)o(xamples)h(of)f(the)g(kind)h(abo)o(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25056
(e)f(only)g(one)h(normal)p Black 277 5206 1290 4 v 383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25057
5262 a F3(1)412 5294 y F2(If)19 b(we)h(assume)h(that)f(e)n(v)o(ery)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25058
(normal)g(form)f(occupies)i(100)f(Bytes)f(of)g(memory)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25059
b(,)21 b(which)g(is)e(an)i(optimistic)f(estimation)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25060
5385 y(for)f(the)g(smallest)g(of)f(these)i(normal)f(forms,)g(we)g(need)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25061
g(a)g(computer)h(with)f(more)g(than)g(150)h(MByte)f(of)g(RAM.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25062
Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25063
%%Page: 112 124
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25064
TeXDict begin 112 123 bop Black -144 51 a Gb(112)2310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25065
b(A)n(pplications)23 b(of)h(Cut-Elimination)p -144 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25066
3691 4 v Black Black 321 569 3226 4 v 321 4940 4 4371
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25067
v 1914 641 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25068
 gsave currentpoint currentpoint translate -90 neg rotate neg exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25069
neg exch translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25070
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25071
FU(A)p FF(c)p 2264 -284 10 38 v 2273 -301 42 4 v 88 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25072
FU(A)p FF(c)p 2515 -345 287 4 v 83 w FU(A)p FF(e)p 2633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25073
-284 10 38 v 2642 -301 42 4 v 88 w FU(A)p FF(e)p 2146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25074
-252 656 4 v 2185 -179 a FU(A)p FF(c)p FT(_)q FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25075
FF(e)p 2457 -191 10 38 v 2467 -207 42 4 v 88 w FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25076
FF(c)p FU(;)14 b(A)p FF(e)2802 -235 y FT(_)2857 -223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25077
y FS(L)p 2097 -143 753 4 v 2097 -64 a FT(8)p FF(y)q FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25078
FG(\()p FU(A)p FF(y)q FT(_)q FU(A)p FF(e)p FG(\))p 2545
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25079
-76 10 38 v 2555 -92 42 4 v 89 w FU(A)p FF(c)p FU(;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25080
FF(e)2850 -120 y FT(8)2897 -108 y FS(L)p 2066 -23 816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25081
4 v 2066 56 a FT(8)p FF(x)p FU(:)p FF(y)q FU(:)p FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25082
FU(A)p FF(y)q FT(_)q FU(A)p FF(x)p FG(\))p 2577 44 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25083
38 v 2586 27 42 4 v 89 w FU(A)p FF(c)p FU(;)g(A)p FF(e)2881
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25084
-1 y FT(8)2928 11 y FS(L)p 3137 -347 299 4 v 3137 -273
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25085
a FU(A)p FF(d)p 3261 -285 10 38 v 3270 -302 42 4 v 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25086
w FU(A)p FF(d)p 3518 -346 287 4 v 83 w FU(A)p FF(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25087
3636 -285 10 38 v 3646 -302 42 4 v 89 w FU(A)p FF(e)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25088
3137 -253 668 4 v 3176 -179 a FU(A)p FF(d)p FT(_)q FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25089
FF(e)p 3455 -191 10 38 v 3464 -207 42 4 v 88 w FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25090
FF(d)p FU(;)g(A)p FF(e)3805 -236 y FT(_)3860 -224 y FS(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25091
3092 -143 759 4 v 3092 -64 a FT(8)p FF(y)q FU(:)p FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25092
FU(A)p FF(y)q FT(_)q FU(A)p FF(e)p FG(\))p 3539 -76 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25093
38 v 3549 -92 42 4 v 88 w FU(A)p FF(d)p FU(;)g(A)p FF(e)3850
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25094
-120 y FT(8)3897 -108 y FS(L)p 3060 -23 822 4 v 3060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25095
56 a FT(8)p FF(x)p FU(:)p FF(y)q FU(:)p FG(\()p FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25096
FF(y)q FT(_)r FU(A)p FF(x)p FG(\))p 3571 44 10 38 v 3580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25097
27 42 4 v 88 w FU(A)p FF(d)p FU(;)g(A)p FF(e)3882 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25098
y FT(8)3929 11 y FS(L)p 2066 96 1816 4 v 2418 175 a FT(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25099
FF(x)p FU(:)p FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25100
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25101
v 89 w FU(A)p FF(e)p FU(;)g(A)p FF(e)p FU(;)g(A)p FF(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25102
FT(^)q FU(A)p FF(d)3882 113 y FT(^)3937 126 y FS(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25103
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25104
10 38 v 4284 -302 42 4 v 88 w FU(A)p FF(c)p 4526 -347
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25105
275 4 v 83 w FU(A)p FF(f)p 4638 -285 10 38 v 4648 -302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25106
42 4 v 95 w FU(A)p FF(f)p 4157 -253 645 4 v 4197 -179
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25107
a FU(A)p FF(c)p FT(_)p FU(A)p FF(f)p 4463 -191 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25108
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25109
-236 y FT(_)4857 -224 y FS(L)p 4109 -143 742 4 v 4109
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25110
-64 a FT(8)p FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25111
FU(A)p FF(f)6 b FG(\))p 4551 -76 10 38 v 4560 -92 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25112
4 v 88 w FU(A)p FF(c)p FU(;)14 b(A)p FF(f)4850 -120 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25113
FT(8)4897 -108 y FS(L)p 4074 -23 810 4 v 4074 56 a FT(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25114
FF(x)p FU(:)p FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q FT(_)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25115
FU(A)p FF(x)p FG(\))p 4585 44 10 38 v 4595 27 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25116
88 w FU(A)p FF(c)p FU(;)g(A)p FF(f)4884 -1 y FT(8)4931
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25117
11 y FS(L)p 5143 -347 299 4 v 5143 -273 a FU(A)p FF(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25118
5266 -285 10 38 v 5276 -302 42 4 v 88 w FU(A)p FF(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25119
5524 -347 275 4 v 83 w FU(A)p FF(f)p 5636 -285 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25120
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25121
v 5182 -179 a FU(A)p FF(d)p FT(_)p FU(A)p FF(f)p 5454
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25122
-191 10 38 v 5464 -207 42 4 v 95 w FU(A)p FF(d)p FU(;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25123
FF(f)5799 -236 y FT(_)5854 -224 y FS(L)p 5097 -143 748
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25124
4 v 5097 -64 a FT(8)p FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25125
FT(_)q FU(A)p FF(f)6 b FG(\))p 5539 -76 10 38 v 5549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25126
-92 42 4 v 89 w FU(A)p FF(d)p FU(;)14 b(A)p FF(f)5844
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25127
-120 y FT(8)5891 -108 y FS(L)p 5063 -23 816 4 v 5063
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25128
56 a FT(8)p FF(x)p FU(:)p FF(y)q FU(:)p FG(\()p FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25129
FF(y)q FT(_)q FU(A)p FF(x)p FG(\))p 5573 44 10 38 v 5583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25130
27 42 4 v 89 w FU(A)p FF(d)p FU(;)g(A)p FF(f)5878 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25131
y FT(8)5925 11 y FS(L)p 4074 96 1804 4 v 4426 175 a FT(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25132
FF(x)p FU(:)p FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25133
FU(A)p FF(x)p FG(\))p 4937 163 10 38 v 4946 147 42 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25134
v 89 w FU(A)p FF(f)6 b FU(;)14 b(A)p FF(f)6 b FU(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25135
b(A)p FF(c)p FT(^)q FU(A)p FF(d)5878 113 y FT(^)5934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25136
126 y FS(R)p 2418 216 3109 4 v 3277 295 a FT(8)p FF(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25137
FU(:)p FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q FT(_)q FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25138
FF(x)p FG(\))p 3787 283 10 38 v 3797 266 42 4 v 89 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25139
FU(A)p FF(e)p FU(;)g(A)p FF(f)6 b FU(;)14 b(A)p FF(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25140
FT(^)q FU(A)p FF(d)p FU(;)g(A)p FF(e)p FT(^)p FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25141
FF(f)5527 233 y FT(^)5582 245 y FS(R)2066 350 y FI(|)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25142
2103 350 1887 10 v 1887 w({z)p 4064 350 V 1887 w(})3998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25143
429 y FS(X)p 2044 938 287 4 v 2044 1012 a FU(A)p FF(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25144
2161 1000 10 38 v 2171 983 42 4 v 88 w FU(A)p FF(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25145
2413 938 287 4 v 83 w FU(A)p FF(e)p 2531 1000 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25146
2540 983 42 4 v 88 w FU(A)p FF(e)p 2044 1032 656 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25147
2083 1105 a FU(A)p FF(c)p FT(_)q FU(A)p FF(e)p 2355 1093
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25148
10 38 v 2365 1076 42 4 v 88 w FU(A)p FF(c)p FU(;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25149
FF(e)2699 1049 y FT(_)2755 1061 y FS(L)p 1995 1141 753
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25150
4 v 1995 1220 a FT(8)p FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25151
FT(_)q FU(A)p FF(e)p FG(\))p 2443 1208 10 38 v 2453 1191
Christian Urban <urbanc@in.tum.de>
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 25152
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Christian Urban <urbanc@in.tum.de>
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 25153
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Christian Urban <urbanc@in.tum.de>
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 25154
FT(8)p FF(x)p FU(:)p FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q
Christian Urban <urbanc@in.tum.de>
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 25155
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Christian Urban <urbanc@in.tum.de>
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 25156
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Christian Urban <urbanc@in.tum.de>
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 25157
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Christian Urban <urbanc@in.tum.de>
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 25158
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Christian Urban <urbanc@in.tum.de>
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 25159
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25163
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Christian Urban <urbanc@in.tum.de>
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 25164
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Christian Urban <urbanc@in.tum.de>
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 25165
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Christian Urban <urbanc@in.tum.de>
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 25166
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Christian Urban <urbanc@in.tum.de>
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 25167
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Christian Urban <urbanc@in.tum.de>
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 25168
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Christian Urban <urbanc@in.tum.de>
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 25169
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Christian Urban <urbanc@in.tum.de>
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 25170
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Christian Urban <urbanc@in.tum.de>
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 25171
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Christian Urban <urbanc@in.tum.de>
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a FT(8)p FF(x)p FU(:)p FF(y)q FU(:)p FG(\()p FU(A)p FF(y)q
Christian Urban <urbanc@in.tum.de>
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 25173
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25188
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25199
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Christian Urban <urbanc@in.tum.de>
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 25200
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25201
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25203
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Christian Urban <urbanc@in.tum.de>
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 25204
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25208
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25224
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Christian Urban <urbanc@in.tum.de>
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 25225
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Christian Urban <urbanc@in.tum.de>
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%%Page: 113 125
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25231
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Christian Urban <urbanc@in.tum.de>
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 25232
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Christian Urban <urbanc@in.tum.de>
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 25233
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Christian Urban <urbanc@in.tum.de>
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 25234
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Christian Urban <urbanc@in.tum.de>
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 25235
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Christian Urban <urbanc@in.tum.de>
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 25236
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Christian Urban <urbanc@in.tum.de>
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 25237
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Christian Urban <urbanc@in.tum.de>
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 25238
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25241
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Christian Urban <urbanc@in.tum.de>
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 25242
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Christian Urban <urbanc@in.tum.de>
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 25243
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25246
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25247
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25248
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Christian Urban <urbanc@in.tum.de>
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 25249
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Christian Urban <urbanc@in.tum.de>
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 25250
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Christian Urban <urbanc@in.tum.de>
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 25251
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Christian Urban <urbanc@in.tum.de>
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 25252
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Christian Urban <urbanc@in.tum.de>
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 25253
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25255
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Christian Urban <urbanc@in.tum.de>
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 25256
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25258
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25260
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Christian Urban <urbanc@in.tum.de>
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 25261
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25263
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Christian Urban <urbanc@in.tum.de>
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 25264
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Christian Urban <urbanc@in.tum.de>
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 25265
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Christian Urban <urbanc@in.tum.de>
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 25266
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25267
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25268
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Christian Urban <urbanc@in.tum.de>
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 25269
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25270
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25271
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Christian Urban <urbanc@in.tum.de>
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 25272
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25273
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25274
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25275
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25276
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25277
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25278
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25279
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25280
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Christian Urban <urbanc@in.tum.de>
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 25281
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Christian Urban <urbanc@in.tum.de>
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 25282
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25283
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Christian Urban <urbanc@in.tum.de>
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 25284
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Christian Urban <urbanc@in.tum.de>
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 25285
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25289
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Christian Urban <urbanc@in.tum.de>
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 25290
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Christian Urban <urbanc@in.tum.de>
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 25291
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Christian Urban <urbanc@in.tum.de>
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 25292
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Christian Urban <urbanc@in.tum.de>
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 25293
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25294
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25297
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25303
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Christian Urban <urbanc@in.tum.de>
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 25304
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25308
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25310
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Christian Urban <urbanc@in.tum.de>
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 25311
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Christian Urban <urbanc@in.tum.de>
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 25312
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Christian Urban <urbanc@in.tum.de>
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 25313
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25316
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Christian Urban <urbanc@in.tum.de>
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 25317
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25321
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Christian Urban <urbanc@in.tum.de>
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 25322
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Christian Urban <urbanc@in.tum.de>
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 25323
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25325
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25328
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Christian Urban <urbanc@in.tum.de>
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 25329
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25334
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25335
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Christian Urban <urbanc@in.tum.de>
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 25336
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25338
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25339
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25340
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25341
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25342
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25343
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25344
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25345
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25346
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25347
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25348
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Christian Urban <urbanc@in.tum.de>
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 25349
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25350
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Christian Urban <urbanc@in.tum.de>
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 25351
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25352
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25353
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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 25355
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25358
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Christian Urban <urbanc@in.tum.de>
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 25359
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25360
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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 25362
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Christian Urban <urbanc@in.tum.de>
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 25363
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25364
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25365
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25369
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25371
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25372
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25375
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Christian Urban <urbanc@in.tum.de>
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 25376
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Christian Urban <urbanc@in.tum.de>
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 25377
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25378
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Christian Urban <urbanc@in.tum.de>
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 25379
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25380
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25381
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25382
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Christian Urban <urbanc@in.tum.de>
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 25383
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Christian Urban <urbanc@in.tum.de>
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 25384
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Christian Urban <urbanc@in.tum.de>
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 25385
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Christian Urban <urbanc@in.tum.de>
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 25386
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25388
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Christian Urban <urbanc@in.tum.de>
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 25389
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Christian Urban <urbanc@in.tum.de>
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 25390
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Christian Urban <urbanc@in.tum.de>
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 25391
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25394
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25396
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Christian Urban <urbanc@in.tum.de>
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 25397
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Christian Urban <urbanc@in.tum.de>
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 25398
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25402
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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 25404
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25405
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Christian Urban <urbanc@in.tum.de>
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 25406
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25407
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25408
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25409
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25413
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25414
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25415
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25416
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25417
b F4(=)f FL(i)91 b F4(\000)p 1865 1340 11 41 v 1875 1322
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25418
46 5 v 96 w(\001)p Ga(;)15 b FL(fx)25 b F4(=)g FL(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25419
1214 1389 1057 4 v 1476 1469 a F4(\000)p 1553 1457 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25420
41 v 1564 1439 46 5 v 96 w(\001)p Ga(;)15 b FL(fx)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25421
b F4(=)g FL(fy)2311 1419 y Gg(T)m(rans)h Ga(:)277 1640
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25422
y Gg(Ho)n(we)n(v)o(er)k(we)g(shall)i(refrain)g(from)f(adding)i(either)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25423
(of)e(them,)j(because)g(it)d(w)o(ould)i(require)g(a)e(fresh)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25424
1753 y(cut-elimination)24 b(proof)e(\(cut-elimination)i(in)c(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25425
(presence)h(of)f(non-logical)i(axioms)e(or)g(additional)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25426
1866 y(inference)28 b(rules)e(is)g(non-tri)n(vial\).)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25427
b(Instead,)28 b(we)c(shall)j(represent)g(principles)i(of)c(arithmetic)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25428
(as)f(for)n(-)277 1979 y(mulae,)f(which)g(we)f(add)h(to)g(our)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25429
(assumptions.)35 b(T)m(ransiti)n(vity)-6 b(,)27 b(for)e(e)o(xample,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25430
(is)g(represented)i(as)e(the)277 2092 y(formula)1237
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25431
2221 y F6(8)p FL(i)p Ga(:)p FL(x)p Ga(:)p FL(y)q Ga(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25432
F4(\(\()p FL(fy)i F4(=)e FL(i)15 b F6(^)g FL(fx)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25433
F4(=)f FL(i)p F4(\))15 b F6(\033)h FL(fx)25 b F4(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25434
FL(fy)q F4(\))i Ga(:)p Black 786 w Gg(\(4.5\))p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25435
277 2396 a(Other)i(principles)h(we)e(need)h(concern)h(the)e(predicates)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25436
j Ga(<)c Gg(and)i F6(\024)p Gg(.)39 b(The)26 b(corresponding)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25437
b(formulae)277 2509 y(are)1570 2664 y F6(8)p FL(x)p Ga(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25438
FL(y)q Ga(:)p F4(\()p FL(sx)25 b F6(\024)g FL(y)17 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25439
F6(\033)e FL(x)25 b Ga(<)g FL(y)q F4(\))p Black 953 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25440
Gg(\(4.6\))p Black 1570 2802 a F6(8)p FL(y)q Ga(:)p FL(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25441
Ga(:)p F4(\()p FL(x)h F6(\024)f FL(m)2027 2764 y Fu(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25442
Gc(;)p Fu(y)2117 2802 y F4(\))p Black 1176 w Gg(\(4.7\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25443
Black 1570 2939 a F6(8)p FL(y)q Ga(:)p FL(x)p Ga(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25444
F4(\()p FL(y)i F6(\024)e FL(m)2028 2902 y Fu(x)p Gc(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25445
Fu(y)2118 2939 y F4(\))p Black 1175 w Gg(\(4.8\))p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25446
277 3110 a(in)d(which)g FL(s)e Gg(denotes)k(the)e(successor)i(function)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25447
g(and)e FL(m)2015 3077 y Fu(x)p Gc(;)p Fu(y)2126 3110
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25448
y Gg(denotes)i(the)e(maximum)f(function)j(of)e(tw)o(o)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25449
3223 y(inte)o(gers,)33 b FL(x)d Gg(and)h FL(y)q Gg(.)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25450
b(F)o(or)29 b(instance)k FL(m)1518 3190 y Fu(s0)p Gc(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25451
Fu(0)1669 3223 y Gg(is)45 b FL(s0)p Gg(.)j(In)31 b(the)f(sequel,)j(we)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25452
(shall)h(abbre)n(viate)48 b FL(s0)p Gg(,)e FL(ss0)p Gg(,)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25453
3336 y FL(sss0)23 b Gg(as)g FL(1)p Gg(,)g FL(2)g Gg(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25454
FL(3)p Gg(,)f(respecti)n(v)o(ely)-6 b(.)32 b(A)22 b(proof)j(of)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25455
(statement)41 b FL(0)10 b Ga(<)g FL(1)23 b Gg(is)g(then)i(as)e(follo)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25456
(ws.)924 3808 y Ga(\031)1016 3439 y FI(8)1016 3513 y(>)1016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25457
3538 y(>)1016 3563 y(>)1016 3588 y(>)1016 3613 y(>)1016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25458
3638 y(>)1016 3663 y(>)1016 3688 y(>)1016 3713 y(<)1016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25459
3862 y(>)1016 3887 y(>)1016 3912 y(>)1016 3937 y(>)1016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25460
3962 y(>)1016 3987 y(>)1016 4012 y(>)1016 4036 y(>)1016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25461
4061 y(:)p 1420 3450 421 4 v 1420 3521 a FF(1)9 b FT(\024)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25462
FF(1)p 1604 3509 10 38 v 1614 3492 42 4 v 87 w(1)g FT(\024)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25463
FF(1)p 1289 3552 683 4 v 1275 3637 a FT(8)p FF(x)p FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25464
FG(\()p FF(x)g FT(\024)g FF(m)1602 3607 y Ft(x)p FS(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25465
Ft(0)1685 3637 y FG(\))p 1736 3625 10 38 v 1745 3608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25466
42 4 v 88 w FF(1)g FT(\024)g FF(1)1985 3575 y FT(8)2032
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25467
3587 y FS(L)p 1237 3678 786 4 v 1237 3757 a FT(8)p FF(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25468
FU(:)p FF(x)p FU(:)p FG(\()p FF(x)24 b FT(\024)f FF(m)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25469
3726 y Ft(x)p FS(;)p Ft(y)1736 3757 y FG(\))p 1787 3745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25470
10 38 v 1797 3728 42 4 v 88 w FF(1)9 b FT(\024)g FF(1)2036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25471
3700 y FT(8)2083 3712 y FS(L)p 2215 3686 421 4 v 2215
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25472
3757 a FF(0)g FU(<)g FF(1)p 2400 3745 10 38 v 2409 3728
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25473
42 4 v 87 w(0)g FU(<)g FF(1)p 1237 3797 1398 4 v 1327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25474
3876 a(1)g FT(\024)g FF(1)p FT(\033)o FF(0)g FU(<)g FF(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25475
FU(;)14 b FT(8)p FF(y)q FU(:)p FF(x)p FU(:)o FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25476
FF(x)23 b FT(\024)g FF(m)2178 3846 y Ft(x)p FS(;)p Ft(y)2259
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25477
3876 y FG(\))p 2310 3864 10 38 v 2320 3848 42 4 v 89
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25478
w FF(0)9 b FU(<)g FF(1)2649 3813 y FT(\033)2714 3826
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25479
y FS(L)p 1249 3917 1374 4 v 1236 3996 a FT(8)p FF(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25480
FU(:)p FG(\()p FF(1)g FT(\024)g FF(y)q FT(\033)o FF(0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25481
FU(<)g FF(y)q FG(\))p FU(;)14 b FT(8)p FF(y)q FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25482
FF(x)p FU(:)p FG(\()p FF(x)24 b FT(\024)e FF(m)2255 3966
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25483
y Ft(x)p FS(;)p Ft(y)2337 3996 y FG(\))p 2388 3984 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25484
38 v 2397 3967 42 4 v 88 w FF(0)9 b FU(<)g FF(1)2637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25485
3939 y FT(8)2684 3951 y FS(L)p 1158 4036 1558 4 v 1158
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25486
4115 a FT(8)p FF(x)p FU(:)p FF(y)q FU(:)p FG(\()p FF(sx)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25487
b FT(\024)g FF(y)15 b FT(\033)e FF(x)24 b FU(<)e FF(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25488
FG(\))p FU(;)14 b FT(8)p FF(y)q FU(:)p FF(x)p FU(:)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25489
FG(\()p FF(x)24 b FT(\024)e FF(m)2347 4085 y Ft(x)p FS(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25490
Ft(y)2429 4115 y FG(\))p 2480 4103 10 38 v 2489 4087
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25491
42 4 v 88 w FF(0)9 b FU(<)g FF(1)2729 4059 y FT(8)2776
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25492
4071 y FS(L)277 4318 y Gg(There)30 b(are)g(se)n(v)o(eral)h(w)o(ays)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25493
(of)e(ho)n(w)h(to)g(formalise)h(a)f(proof)h(of)f FL(0)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25494
Ga(<)f FL(1)p Gg(;)33 b(it)c(will,)i(ho)n(we)n(v)o(er)l(,)h(become)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25495
4431 y(clear)d(later)g(on)f(why)g(we)f(ha)n(v)o(e)i(chosen)g(this)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25496
(particular)i(formalisation.)45 b(Before)29 b(we)e(proceed,)k(let)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25497
4544 y(us)24 b(introduce)i(some)d(abbre)n(viations)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25498
b(for)23 b(the)h(formulae)h(gi)n(v)o(en)f(in)g(\(4.3\))g(to)f(\(4.8\).)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25499
p Black Black 700 4744 a Ga(A)833 4693 y F5(def)840 4744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25500
y F4(=)66 b F6(8)p FL(x)p Ga(:)p F4(\()p FL(fx)25 b F4(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25501
FL(0)15 b F6(_)h FL(fx)24 b F4(=)h FL(1)p F4(\))693 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25502
Gg(assumption)103 b(\(4.3\))699 4884 y Ga(P)833 4832
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25503
y F5(def)840 4884 y F4(=)66 b F6(9)p FL(n)p Ga(:)p FL(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25504
Ga(:)p F4(\()p FL(n)26 b Ga(<)f FL(m)15 b F6(^)g FL(fn)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25505
b F4(=)g FL(fm)p F4(\))526 b Gg(proposition)104 b(\(4.4\))701
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25506
5024 y Ga(T)833 4972 y F5(def)840 5024 y F4(=)66 b F6(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25507
FL(i)p Ga(:)p FL(x)p Ga(:)p FL(y)q Ga(:)p F4(\(\()p FL(fy)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25508
b F4(=)e FL(i)15 b F6(^)g FL(fx)25 b F4(=)g FL(i)p F4(\))15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25509
b F6(\033)g FL(fx)25 b F4(=)g FL(fy)q F4(\))189 b Gg(transiti)n(vity)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25510
126 b(\(4.5\))704 5164 y Ga(S)833 5112 y F5(def)840 5164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25511
y F4(=)66 b F6(8)p FL(x)p Ga(:)p FL(y)q Ga(:)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25512
FL(sx)26 b F6(\024)f FL(y)16 b F6(\033)f FL(x)25 b Ga(<)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25513
FL(y)q F4(\))639 b Gg(successor)169 b(\(4.6\))670 5303
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25514
y Ga(M)758 5317 y F9(1)833 5252 y F5(def)840 5303 y F4(=)66
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25515
b F6(8)p FL(y)q Ga(:)p FL(x)p Ga(:)p F4(\()p FL(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25516
b F6(\024)f FL(m)1434 5270 y Fu(x)p Gc(;)p Fu(y)1524
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25517
5303 y F4(\))862 b Gg(maximum)2789 5317 y F9(1)2935 5303
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25518
y Gg(\(4.7\))670 5443 y Ga(M)758 5457 y F9(2)833 5392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25519
y F5(def)840 5443 y F4(=)66 b F6(8)p FL(y)q Ga(:)p FL(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25520
Ga(:)p F4(\()p FL(y)27 b F6(\024)e FL(m)1435 5410 y Fu(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25521
Gc(;)p Fu(y)1525 5443 y F4(\))861 b Gg(maximum)2789 5457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25522
y F9(2)2935 5443 y Gg(\(4.8\))p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25523
%%Page: 116 128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25524
TeXDict begin 116 127 bop Black -144 51 a Gb(116)2310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25525
b(A)n(pplications)23 b(of)h(Cut-Elimination)p -144 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25526
3691 4 v Black 321 388 a Gg(The)f(sequent)j(that)e(we)f(are)g(going)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25527
(to)f(pro)o(v)o(e)g(is)p Black Black 421 561 a FT(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25528
FF(y)q FU(:)p FF(x)p FU(:)p FG(\()p FF(x)f FT(\024)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25529
FF(m)838 530 y Ft(x)p FS(;)p Ft(y)920 561 y FG(\))p Gd(,)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25530
FT(8)p FF(y)q FU(:)p FF(x)p FU(:)p FG(\()p FF(y)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25531
FT(\024)e FF(m)1412 530 y Ft(x)p FS(;)p Ft(y)1493 561
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25532
y FG(\))p Gd(,)e FT(8)p FF(x)p FU(:)p FF(y)q FU(:)p FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25533
FF(sx)i FT(\024)g FF(y)15 b FT(\033)f FF(x)23 b FU(<)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25534
FF(y)q FG(\))p FU(;)421 660 y FT(8)p FF(i)p FU(:)p FF(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25535
FU(:)p FF(y)q FU(:)p FG(\(\()p FF(fy)i FG(=)e FF(i)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25536
b FT(^)f FF(fx)24 b FG(=)f FF(i)p FG(\))14 b FT(\033)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25537
FF(fx)24 b FG(=)f FF(fy)q FG(\))p Gd(,)421 760 y FT(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25538
FF(x)p FU(:)o FG(\()p FF(fx)h FG(=)f FF(0)14 b FT(_)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25539
FF(fx)24 b FG(=)f FF(1)p FG(\))p 2434 649 20 78 v 2454
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25540
614 87 8 v 2579 637 a F6(9)p FL(n)p Ga(:)p FL(m)p Ga(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25541
F4(\()p FL(n)i Ga(<)g FL(m)16 b F6(^)e FL(fn)25 b F4(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25542
FL(fm)p F4(\))321 935 y Gg(or)f(using)g(our)g(abbre)n(viations)1524
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25543
1047 y Ga(M)1612 1061 y F9(1)1652 1047 y Ga(;)15 b(M)1780
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25544
1061 y F9(2)1820 1047 y Ga(;)g(S;)g(T)8 b(;)15 b(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25545
2146 1035 11 41 v 2156 1017 46 5 v 97 w(P)38 b(:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25546
1029 w Gg(\(4.9\))p Black 462 1221 a(W)-7 b(e)26 b(turn)h(no)n(w)f(to)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25547
(the)g(question)h(of)f(ho)n(w)f(one)h(might)g(pro)o(v)o(e)g(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25548
(sequent)h(just)f(gi)n(v)o(en.)38 b(As)26 b(one)321 1334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25549
y(may)d(e)o(xpect,)i(there)f(are)f(se)n(v)o(eral)i(possibilities.)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25550
b(W)-7 b(e)23 b(\002rst)g(consider)i(proofs)g(that)f(\223perform\224)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25551
(a)e(case)321 1447 y(analysis;)37 b(by)31 b(this)h(we)e(mean)h(the)o(y)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25552
g(consist)i(of)e(a)f(number)i(of)f(tests,)i(which)f(check)g(whether)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25553
(tw)o(o)321 1560 y(cells)25 b(on)e(a)g(gi)n(v)o(en)h(tape)h(ha)n(v)o(e)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25554
f(the)g(same)f(content.)31 b(T)-7 b(ak)o(e)23 b(for)h(e)o(xample)g(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25555
g(follo)n(wing)h(tape.)p Black Black 1539 1689 790 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25556
v 1537 1802 4 113 v 1589 1768 a FL(1)p 1683 1802 V 100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25557
w(0)p 1828 1802 V 100 w(0)p 1973 1802 V 100 w(1)p 2118
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25558
1802 V 100 w Gg(.)14 b(.)g(.)p 1539 1806 790 4 v 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25559
1978 a(Here)25 b(the)g(test)g FL(f1)g F4(=)g FL(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25560
F4(=)e FL(0)q Gg(,)g(or)g FL(f0)i F4(=)f FL(f3)g F4(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25561
FL(1)q Gg(,)e(is)i(suf)n(\002cient)h(to)e(sho)n(w)g Ga(P)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25562
b Gg(for)25 b(this)g(particular)i(tape.)321 2091 y(Ho)n(we)n(v)o(er)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25563
(,)c(both)i(tests)f(f)o(ail)g(for)g(the)g(tape)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25564
Black 1539 2220 V 1537 2333 4 113 v 1589 2299 a FL(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25565
1683 2333 V 100 w(1)p 1828 2333 V 100 w(0)p 1973 2333
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25566
V 100 w(0)p 2118 2333 V 100 w Gg(.)14 b(.)g(.)p 1539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25567
2337 790 4 v 321 2509 a(Since)30 b(we)e(w)o(ant)h(to)g(sho)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25568
Ga(P)42 b Gg(for)29 b(all)g(tapes,)i(we)e(ha)n(v)o(e)h(to)f(do)g(some)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25569
(more)g(tests.)47 b(It)29 b(is)g(left)g(to)g(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25570
2622 y(reader)c(to)f(v)o(erify)g(that)g(the)g(follo)n(wing)h(tests)1340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25571
2838 y FL(f0)g F4(=)g FL(f1)h F4(=)f FL(0)1340 2951 y(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25572
F4(=)g FL(f2)h F4(=)f FL(0)1340 3064 y(f1)g F4(=)g FL(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25573
F4(=)f FL(0)2093 2838 y(f0)h F4(=)f FL(f1)g F4(=)g FL(1)2093
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25574
2951 y(f0)h F4(=)f FL(f2)g F4(=)g FL(1)2093 3064 y(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25575
F4(=)f FL(f2)g F4(=)g FL(1)p Black 3327 2951 a Gg(\(4.10\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25576
Black 321 3230 a(guarantee)f(that)e(there)g(is)f(an)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25577
FL(n)f Gg(and)i FL(m)f Gg(such)h(that)f FL(n)26 b Ga(<)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25578
FL(m)20 b Gg(and)i FL(fn)j F4(=)g FL(fm)20 b Gg(for)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25579
F7(e)o(very)h Gg(tape.)29 b(Ob)o(viously)321 3343 y(the)e(collection)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25580
(of)c(tests)i(gi)n(v)o(en)f(abo)o(v)o(e)g(is)g(not)g(the)f(only)i(one)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25581
(to)g(establish)i(the)e(proposition)j(for)c(all)321 3456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25582
y(tapes.)j(W)-7 b(e)20 b(could,)i(for)e(e)o(xample,)i(test)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25583
(13th,)g(101th)h(and)f(999th)h(cell,)f(instead)i(of)d(the)h(\002rst)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25584
(three.)321 3569 y(Also,)j(we)g(might)h(consider)l(,)i(as)d(we)g(shall)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25585
i(see)e(later)l(,)i(more)e(than)i(just)f(three)g(cells.)462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25586
3698 y(Let)g(us)f(sho)n(w)h(ho)n(w)f(a)g(test)i(translates)h(into)e(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25587
(sequent)j(proof.)k(Belo)n(w)24 b(we)e(gi)n(v)o(e)i(a)g(proof,)g(which)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25588
321 3811 y(corresponds)j(to)d(the)g(test)39 b FL(f0)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25589
b F4(=)g FL(f1)h F4(=)f FL(0)p Gg(.)181 4806 y FU(\031)-12
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25590
4880 y(M)69 4892 y FJ(1)106 4880 y FU(;)14 b(S)p 217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25591
4868 10 38 v 226 4851 42 4 v 92 w FF(0)-5 b FU(<)g FF(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25592
419 3998 471 4 v 419 4073 a(f1)10 b FG(=)f FF(0)p 629
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25593
4061 10 38 v 639 4044 42 4 v 87 w(f1)g FG(=)g FF(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25594
973 3998 471 4 v 82 w(f1)g FG(=)g FF(1)p 1183 4061 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25595
38 v 1193 4044 42 4 v 88 w(f1)g FG(=)g FF(1)p 419 4093
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25596
1025 4 v 459 4167 a(f1)g FG(=)g FF(0)p FT(_)p FF(f1)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25597
FG(=)g FF(1)p 915 4155 10 38 v 925 4138 42 4 v 87 w(f1)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25598
FG(=)g FF(0)p FU(;)14 b FF(f1)8 b FG(=)h FF(1)1444 4112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25599
y FT(_)1499 4082 y Fz(\016)1499 4135 y FS(L)p 383 4203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25600
1098 4 v 369 4282 a FT(8)p FF(x)p FU(:)p FG(\()p FF(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25601
FG(=)f FF(0)p FT(_)o FF(fx)h FG(=)f FF(1)p FG(\))p 991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25602
4270 10 38 v 1001 4253 42 4 v 88 w FF(f1)g FG(=)g FF(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25603
FU(;)14 b FF(f1)8 b FG(=)h FF(1)1481 4223 y FT(8)1528
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25604
4193 y Fz(\016)1528 4246 y FS(L)p 1696 3998 471 4 v 1696
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25605
4073 a FF(f0)g FG(=)g FF(0)p 1906 4061 10 38 v 1915 4044
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25606
42 4 v 87 w(f0)g FG(=)g FF(0)p 2250 3998 471 4 v 83 w(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25607
FG(=)g FF(1)p 2460 4061 10 38 v 2469 4044 42 4 v 87 w(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25608
FG(=)g FF(1)p 1696 4093 1025 4 v 1735 4167 a(f0)g FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25609
FF(0)p FT(_)p FF(f0)g FG(=)g FF(1)p 2192 4155 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25610
2201 4138 42 4 v 87 w(f0)g FG(=)g FF(0)p FU(;)14 b FF(f0)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25611
b FG(=)g FF(1)2720 4112 y FT(_)2776 4082 y Fz(\016)2776
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25612
4135 y FS(L)p 1659 4203 1098 4 v 1646 4282 a FT(8)p FF(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25613
FU(:)p FG(\()p FF(fx)g FG(=)g FF(0)p FT(_)p FF(fx)h FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25614
FF(1)p FG(\))p 2268 4270 10 38 v 2277 4253 42 4 v 87
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25615
w FF(f0)g FG(=)g FF(0)p FU(;)14 b FF(f0)9 b FG(=)g FF(1)2757
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25616
4223 y FT(8)2804 4193 y Fz(\016)2804 4246 y FS(L)p 383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25617
4322 2375 4 v 770 4401 a FT(8)p FF(x)p FU(:)p FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25618
FF(fx)g FG(=)g FF(0)p FT(_)p FF(fx)h FG(=)f FF(1)p FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25619
1392 4389 10 38 v 1402 4373 42 4 v 87 w FF(f1)h FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25620
FF(0)p FT(^)o FF(f0)g FG(=)g FF(0)p FU(;)14 b FF(f1)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25621
b FG(=)g FF(1)p FU(;)14 b FF(f0)8 b FG(=)h FF(1)2743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25622
4342 y FT(^)2798 4312 y Fz(\017)2798 4365 y FS(R)p 2936
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25623
4327 522 4 v 2936 4401 a FF(f0)g FG(=)g FF(f1)p 3171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25624
4389 10 38 v 3181 4373 42 4 v 88 w(f0)g FG(=)g FF(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25625
784 4442 2674 4 v 0.75 TeXcolorgray 0.75 TeXcolorgray
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25626
1034 4542 785 84 v 0.75 TeXcolorgray Black 1034 4521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25627
a FG(\()p FF(f1)h FG(=)f FF(0)p FT(^)o FF(f0)h FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25628
FF(0)p FG(\))p FT(\033)o FF(f0)g FG(=)g FF(f1)p 0.75
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25629
TeXcolorgray Black FU(;)p FT(8)p FF(x)p FU(:)p FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25630
FF(fx)g FG(=)g FF(0)p FT(_)p FF(fx)h FG(=)f FF(1)p FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25631
2464 4509 10 38 v 2473 4492 42 4 v 87 w FF(f0)g FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25632
FF(f1)p FU(;)14 b FF(f1)9 b FG(=)g FF(1)p FU(;)14 b FF(f0)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25633
b FG(=)h FF(1)3444 4462 y FT(\033)3508 4432 y Fz(\017)3508
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25634
4485 y FS(L)p 957 4562 2328 4 v 943 4640 a FT(8)p FF(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25635
FU(:)p FG(\(\()p FF(fy)i FG(=)e FF(0)p FT(^)p FF(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25636
FG(=)g FF(0)p FG(\))p FT(\033)o FF(f0)g FG(=)g FF(fy)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25637
FG(\))p FU(;)p FT(8)p FF(x)p FU(:)p FG(\()p FF(fx)h FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25638
FF(0)p FT(_)o FF(fx)h FG(=)f FF(1)p FG(\))p 2541 4628
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25639
10 38 v 2551 4612 42 4 v 88 w FF(f0)g FG(=)g FF(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25640
FU(;)14 b FF(f1)9 b FG(=)g FF(1)p FU(;)14 b FF(f0)8 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25641
FG(=)h FF(1)3284 4581 y FT(8)3331 4551 y Fz(\017)3331
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25642
4604 y FS(L)p 929 4681 2383 4 v 916 4760 a FT(8)p FF(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25643
FU(:)p FF(y)q FU(:)p FG(\(\()p FF(fy)i FG(=)e FF(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25644
FT(^)p FF(fx)h FG(=)f FF(0)p FG(\))p FT(\033)o FF(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25645
FG(=)f FF(fy)q FG(\))p FU(;)p FT(8)p FF(x)p FU(:)p FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25646
FF(fx)h FG(=)f FF(0)p FT(_)p FF(fx)h FG(=)f FF(1)p FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25647
2569 4748 10 38 v 2578 4731 42 4 v 87 w FF(f0)g FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25648
FF(f1)p FU(;)14 b FF(f1)9 b FG(=)g FF(1)p FU(;)14 b FF(f0)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25649
b FG(=)h FF(1)3312 4701 y FT(8)3359 4671 y Fz(\017)3359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25650
4724 y FS(L)p 929 4801 2383 4 v 1385 4880 a FU(T)e(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25651
b FT(8)p FF(x)p FU(:)p FG(\()p FF(fx)9 b FG(=)g FF(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25652
FT(_)p FF(fx)g FG(=)g FF(1)p FG(\))p 2099 4868 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25653
v 2109 4851 42 4 v 88 w FF(f0)g FG(=)g FF(f1)p FU(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25654
b FF(f1)9 b FG(=)g FF(1)p FU(;)14 b FF(f0)8 b FG(=)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25655
FF(1)3312 4820 y FT(8)3359 4790 y Fz(\017)3359 4843 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25656
FS(L)p -12 4920 2855 4 v 427 4999 a FU(M)508 5011 y FJ(1)544
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25657
4999 y FU(;)14 b(S;)g(T)7 b(;)14 b FT(8)p FF(x)p FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25658
FG(\()p FF(fx)9 b FG(=)g FF(0)14 b FT(_)g FF(fx)9 b FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25659
FF(1)p FG(\))p 1411 4987 10 38 v 1421 4970 42 4 v 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25660
w FF(0)-5 b FU(<)g FF(1)14 b FT(^)f FF(f0)d FG(=)f FF(f1)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25661
FU(;)28 b FF(f1)9 b FG(=)g FF(1)p FU(;)27 b FF(f0)9 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25662
FG(=)g FF(1)2842 4941 y FT(^)2898 4911 y Fk(?)2898 4964
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25663
y FS(R)p 302 5040 2226 4 v 302 5119 a FU(M)383 5131 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25664
FJ(1)420 5119 y FU(;)14 b(S;)g(T)7 b(;)14 b FT(8)p FF(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25665
FU(:)p FG(\()p FF(fx)9 b FG(=)g FF(0)14 b FT(_)f FF(fx)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25666
FG(=)f FF(1)p FG(\))p 1287 5107 10 38 v 1297 5090 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25667
4 v 88 w FT(9)p FF(m)p FU(:)p FG(\()p FF(0)-5 b FU(<)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25668
FF(m)14 b FT(^)g FF(f0)9 b FG(=)g FF(fm)p FG(\))p FU(;)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25669
b FF(f1)9 b FG(=)g FF(1)p FU(;)27 b FF(f0)9 b FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25670
FF(1)2528 5061 y FT(9)2574 5031 y Fk(?)2574 5084 y FS(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25671
254 5159 2322 4 v 254 5238 a FU(M)335 5250 y FJ(1)372
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25672
5238 y FU(;)14 b(S;)g(T)7 b(;)14 b FT(8)p FF(x)p FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25673
FG(\()p FF(fx)9 b FG(=)g FF(0)14 b FT(_)f FF(fx)d FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25674
FF(1)p FG(\))590 5294 y FI(|)p 627 5294 241 10 v 241
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25675
w({z)p 942 5294 V 241 w(})874 5389 y FU(A)p 1253 5226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25676
10 38 v 1262 5210 42 4 v 1336 5238 a FT(9)p FF(n)p FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25677
FF(m)p FU(:)p FG(\()p FF(n)-5 b FU(<)g FF(m)14 b FT(^)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25678
FF(fn)9 b FG(=)g FF(fm)p FG(\))1336 5294 y FI(|)p 1373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25679
5294 304 10 v 304 w({z)p 1751 5294 V 304 w(})1682 5389
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25680
y FU(P)2092 5238 y(;)27 b FF(f1)10 b FG(=)f FF(1)p FU(;)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25681
b FF(f0)9 b FG(=)g FF(1)2576 5181 y FT(9)2622 5150 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25682
Fk(?)2622 5203 y FS(R)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25683
%%Page: 117 129
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25684
TeXDict begin 117 128 bop Black 277 51 a Gb(4.2)23 b(Non-Deterministic)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25685
i(Computation:)k(Case)23 b(Study)1667 b(117)p 277 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25686
3691 4 v Black 277 388 a Gg(In)27 b(this)h(proof,)i(the)d(inference)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25687
(rules)e(mark)o(ed)h(with)e(a)g(star)h(are)f(concerned)j(with)e(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25688
(proposition,)277 501 y Ga(P)13 b Gg(,)31 b(the)f(inference)i(rules)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25689
(mark)o(ed)g(with)f(a)f(circle)j(are)e(concerned)i(with)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25690
(assumption,)k Ga(A)p Gg(,)d(and)277 614 y(the)f(inference)j(rules)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25691
(mark)o(ed)g(with)f(a)f(disk)i(concern)h(the)e(formula)h(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25692
(transiti)n(vity)-6 b(,)35 b Ga(T)13 b Gg(.)47 b(Clearly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25693
b(,)277 727 y(the)29 b(order)h(of)f(the)g(inference)i(rules)e(is)g(not)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25694
g(important)i(with)d(respect)j(to)d(which)h(test)h(is)e(made:)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25695
b(the)277 840 y(inference)29 b(rules)f(can)f(be)g(easily)h(rearranged.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25696
41 b(Ho)n(we)n(v)o(er)26 b(it)h(is)f(relati)n(v)o(ely)j(simple)e(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25697
(determine)h(in)277 953 y(an)o(y)d(proof)h(of)f(this)g(kind)h(which)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25698
(test)g(is)g(made:)32 b(one)25 b(only)h(has)f(to)g(\002nd)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25699
(place)h(where)f(the)g(proof)277 1066 y(deals)31 b(with)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25700
(transiti)n(vity)i(\(shaded)f(formula\).)48 b(So)29 b(in)g(the)h(proof)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25701
g(gi)n(v)o(en)g(abo)o(v)o(e,)h(the)f(inference)277 1179
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25702
y(rule)p Black Black 1090 1360 a FU(:)14 b(:)g(:)p 1219
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25703
1348 10 38 v 1228 1331 42 4 v 101 w FF(f1)23 b FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25704
FF(0)14 b FT(^)f FF(f0)23 b FG(=)g FF(0)p FU(;)14 b(:)g(:)g(:)174
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25705
b FF(f1)9 b FG(=)g FF(f0)p 2354 1348 10 38 v 2363 1331
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25706
42 4 v 102 w FU(:)14 b(:)g(:)p 1090 1396 1445 4 v 1177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25707
1474 a FG(\()p FF(f1)24 b FG(=)e FF(0)14 b FT(^)g FF(f0)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25708
b FG(=)f FF(0)p FG(\))14 b FT(\033)f FF(f0)23 b FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25709
FF(f1)p FU(;)14 b(:)g(:)g(:)p 2266 1462 10 38 v 2275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25710
1446 42 4 v 115 w(:)g(:)g(:)2575 1412 y FT(\033)2640
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25711
1424 y FS(L)277 1701 y Gg(is)29 b(an)h(indicator)i(for)e(the)f(test)46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25712
b FL(f0)25 b F4(=)g FL(f1)g F4(=)g FL(0)30 b Gg(being)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25713
(\223performed\224.)50 b(Note)29 b(that)h(the)g(tw)o(o)f(formulae)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25714
1814 y FL(f1)d F4(=)f FL(1)h Gg(and)g FL(f0)g F4(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25715
FL(0)p Gg(,)h(which)h(appear)h(in)e(the)h(endsequent,)j(are)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25716
F7(not)h Gg(part)g(of)f(the)h(test)42 b FL(f0)26 b F4(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25717
FL(f1)g F4(=)g FL(0)p Gg(,)277 1927 y(b)n(ut)30 b(ha)n(v)o(e)f(to)g(be)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25718
g(used)h(in)f(other)h(tests.)46 b(T)-7 b(w)o(o)28 b(proofs)i(for)g(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25719
f(sequent)i Ga(M)2682 1941 y F9(1)2722 1927 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25720
b(M)2850 1941 y F9(2)2890 1927 y Ga(;)g(S;)g(T)8 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25721
b(A)p 3216 1915 11 41 v 3226 1897 46 5 v 97 w(P)41 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25722
Gg(are)277 2040 y(gi)n(v)o(en)34 b(in)f(Appendix)h(A.)56
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25723
b(Each)33 b(of)g(them)g(corresponds)j(to)d(a)g(particular)i(collection)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25724
h(of)d(tests,)i(as)277 2153 y(indicated)26 b(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25725
b(.)p Black Black 672 2355 a(Proof)24 b(on)f(P)o(ages)h(132\226134)597
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25726
2516 y FL(f0)i F4(=)f FL(f1)g F4(=)g FL(0)597 2628 y(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25727
F4(=)f FL(f2)g F4(=)g FL(0)597 2741 y(f0)h F4(=)f FL(f2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25728
F4(=)g FL(0)597 2854 y(f1)h F4(=)f FL(f3)g F4(=)g FL(0)1202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25729
2572 y(f0)h F4(=)f FL(f1)g F4(=)g FL(1)1202 2685 y(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25730
F4(=)f FL(f3)g F4(=)g FL(1)1202 2798 y(f1)h F4(=)f FL(f2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25731
F4(=)g FL(1)2217 2355 y Gg(Proof)f(on)g(P)o(ages)f(135\226140)2143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25732
2572 y FL(f0)i F4(=)g FL(f1)h F4(=)f FL(0)2143 2685 y(f2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25733
F4(=)g FL(f3)h F4(=)f FL(0)2143 2798 y(f1)g F4(=)g FL(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25734
F4(=)f FL(0)2748 2516 y(f0)g F4(=)g FL(f1)h F4(=)f FL(1)2748
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25735
2628 y(f1)g F4(=)g FL(f2)h F4(=)f FL(1)2748 2741 y(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25736
F4(=)g FL(f2)h F4(=)f FL(1)2748 2854 y(f1)g F4(=)g FL(f3)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25737
F4(=)f FL(1)418 3072 y Gg(No)n(w)d(we)h(return)i(to)e(the)g(question)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25738
(whether)f(cut-elimination)i(in)c(classical)i(logic)g(corresponds)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25739
3185 y(to)i(non-deterministic)k(computation:)39 b(the)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25740
b(preceding)i(discussion)h(suggests)f(the)e(follo)n(wing)h(intu-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25741
3298 y(iti)n(v)o(e)c(criterion)i(for)d(deciding)j(whether)f(tw)o(o)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25742
(proofs)i(are)f(dif)n(ferent.)p Black Black 504 3525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25743
a Gb(Criterion:)45 b Gg(Gi)n(v)o(en)30 b(tw)o(o)g(cut-free)j(proofs)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25744
(of)f(the)g(sequent)h Ga(M)2571 3539 y F9(1)2611 3525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25745
y Ga(;)15 b(M)2739 3539 y F9(2)2779 3525 y Ga(;)g(S;)g(T)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25746
b(;)15 b(A)p 3105 3513 11 41 v 3116 3495 46 5 v 97 w(P)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25747
Gg(.)504 3638 y(These)24 b(proofs)h(are)f(said)g(to)g(be)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25748
F7(dif)n(fer)m(ent)p Gg(,)j(if)d(the)o(y)h(consist)h(of)f(dif)n(ferent)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25749
h(tests.)277 3865 y(According)j(to)d(this)i(criterion,)h(the)e(tw)o(o)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25750
(proofs)i(gi)n(v)o(en)g(in)e(Appendix)j(A)c(are)i(dif)n(ferent,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25751
(since)f(the)o(y)277 3978 y(consist)32 b(of)e(dif)n(ferent)i(tests.)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25752
b(Let)30 b(us)g(point)h(out)g(that)g(our)f(criterion)j(is)d(suf)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25753
(\002cient)i(for)e(tw)o(o)g(proofs)277 4091 y(being)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25754
b(dif)n(ferent,)g(b)n(ut)f(it)f(is)h(by)f(no)h(means)g(a)f(necessary)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25755
(one.)418 4220 y(In)g(the)g(rest)g(of)g(this)h(section)g(we)e(shall)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25756
(gi)n(v)o(e)f(a)f(proof)i(of)f(Sequent)h(\(4.9\),)g(whose)f(collection)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25757
i(of)277 4333 y(normal)21 b(forms)g(includes)i(the)d(tw)o(o)g(proofs)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25758
(gi)n(v)o(en)f(in)g(Appendix)h(A.)k(This)21 b(proof)g(contains)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25759
(instances)277 4446 y(of)h(the)f(cut-rule)j(and)e(is)f(based)i(upon)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25760
(the)g(three)h(observ)n(ations)i(listed)d(belo)n(w:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25761
Black 373 4673 a(\(i\))p Black 46 w(From)31 b(the)g(assumption,)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25762
Ga(A)p Gg(,)e(one)e(can)h(pro)o(v)o(e)f(that)h(there)g(are)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25763
(in\002nitely)g(man)o(y)f(`)p FL(0)p Gg(')-5 b(s)32 b(or)f(in-)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25764
4786 y(\002nitely)25 b(man)o(y)e(`)p FL(1)p Gg(')-5 b(s)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25765
b(on)g(the)g(tape.)p Black 348 4972 a(\(ii\))p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25766
46 w(If)h(on)h(the)f(tape)h(there)h(are)e(in\002nitely)i(man)o(y)e(`)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25767
FL(0)p Gg(')-5 b(s,)26 b(then)g(there)g(are)g(tw)o(o)f(cells)h(with)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25768
(the)g(same)504 5085 y(content.)p Black 323 5272 a(\(iii\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25769
Black 46 w(Similarly)-6 b(,)24 b(if)g(on)f(the)h(tape)g(there)g(are)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25770
(in\002nitely)h(man)o(y)e(`)p FL(1)p Gg(')-5 b(s,)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25771
b(then)g(there)h(are)e(tw)o(o)g(cells)i(with)504 5385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25772
y(the)f(same)g(content.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25773
%%Page: 118 130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25774
TeXDict begin 118 129 bop Black -144 51 a Gb(118)2310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25775
b(A)n(pplications)23 b(of)h(Cut-Elimination)p -144 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25776
3691 4 v Black 321 388 a Gg(The)f(formulae)h(that)g(represent)h(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25777
(statements)i(\223there)f(are)g(in\002nitely)g(man)o(y)f(`)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25778
FL(0)p Gg(')-5 b(s\224)24 b(and)f(\223there)h(are)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25779
501 y(in\002nitely)h(man)o(y)f(`)p FL(1)p Gg(')-5 b(s\224)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25780
b(are)p Black Black 574 736 a F6(1)665 750 y F9(0)745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25781
685 y F5(def)752 737 y F4(=)870 736 y F6(8)p FL(n)p Ga(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25782
F6(9)p FL(k)p Ga(:)p F4(\()p FL(n)i F6(\024)e FL(k)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25783
b F6(^)h FL(fk)25 b F4(=)g FL(0)q F4(\))118 b Gg(and)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25784
F6(1)2185 750 y F9(1)2264 685 y F5(def)2271 737 y F4(=)2390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25785
736 y F6(8)p FL(n)p Ga(:)o F6(9)p FL(k)p Ga(:)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25786
FL(n)26 b F6(\024)f FL(k)15 b F6(^)g FL(fk)26 b F4(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25787
FL(1)p F4(\))h Gg(.)321 946 y(The)d(sequent)j(that)e(corresponds)j(to)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25788
(the)h(\002rst)f(observ)n(ation)k(is)1533 1179 y Ga(M)1621
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25789
1193 y F9(1)1661 1179 y Ga(;)15 b(M)1789 1193 y F9(2)1829
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25790
1179 y Ga(;)g(A)p 1957 1167 11 41 v 1968 1149 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25791
97 w F6(1)2125 1193 y F9(0)2164 1179 y Ga(;)g F6(1)2295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25792
1193 y F9(1)321 1394 y Gg(for)27 b(which)f(a)g(proof,)h(denoted)h(by)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25793
Ga(\034)10 b Gg(,)25 b(is)h(gi)n(v)o(en)h(in)f(Figure)h(4.5.)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25794
b(The)25 b(sequent)j(that)f(corresponds)j(to)321 1506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25795
y(the)24 b(observ)n(ations)j(stated)e(in)e(\(ii\))h(and)g(\(iii\))g(is)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25796
1689 1737 y F6(1)1780 1751 y Gc(i)1808 1737 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25797
b(S;)g(T)p 2031 1725 11 41 v 2042 1706 46 5 v 110 w(P)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25798
1951 y Gg(where)23 b Ga(i)i F6(2)g(f)p F4(0)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25799
b F4(1)p F6(g)p Gg(.)30 b(A)21 b(proof)i(of)g(this)f(sequent,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25800
(denoted)f(by)e Ga(")2321 1965 y Gc(i)2350 1951 y Gg(,)f(is)h(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25801
(en)h(in)f(Figure)h(4.5,)f(too.)29 b(No)n(w)321 2063
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25802
y(we)23 b(can)h(use)g(the)g(cut-rule)h(for)f(forming)g(the)g(follo)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25803
(wing)h(proof)g(for)f(Sequent)g(4.9.)486 2690 y Ga(\033)582
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25804
2286 y FK(8)582 2368 y(>)582 2395 y(>)582 2422 y(>)582
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25805
2449 y(>)582 2477 y(>)582 2504 y(>)582 2531 y(>)582 2559
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25806
y(>)582 2586 y(<)582 2749 y(>)582 2777 y(>)582 2804 y(>)582
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25807
2831 y(>)582 2859 y(>)582 2886 y(>)582 2913 y(>)582 2940
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25808
y(>)582 2968 y(:)867 2710 y @beginspecial 179 @llx 477
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25809
@lly 318 @urx 605 @ury 639 @rwi @clip @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25810
%%BeginDocument: pics/tau.ps
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25811
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25812
%%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25813
%%Title: tau.dvi
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25814
%%Pages: 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25815
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 25816
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25817
%%EndComments
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25818
%DVIPSWebPage: (www.radicaleye.com)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25819
%DVIPSCommandLine: dvips -o tau.ps tau.dvi
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25820
%DVIPSParameters: dpi=600, compressed
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25821
%DVIPSSource:  TeX output 2000.03.10:0010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25822
%%BeginProcSet: texc.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25823
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25824
/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25825
N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25826
mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25827
0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25828
landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25829
mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25830
matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25831
exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25832
statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25833
N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25834
/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25835
/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25836
array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25837
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25838
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25839
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25840
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25841
1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25842
1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25843
0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25844
sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25845
rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25846
gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25847
/chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25848
/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25849
A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25850
get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25851
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25852
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25853
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25854
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25855
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25856
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25857
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25858
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25859
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25860
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25861
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25862
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25863
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25864
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25865
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25866
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25867
(LaserWriter 16/600)]{A length product length le{A length product exch 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25868
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25869
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25870
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25871
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25872
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25873
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25874
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25875
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25876
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25877
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25879
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25880
%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25881
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25882
% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25883
% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25884
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25885
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25886
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25887
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25888
/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25889
/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25890
/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25891
/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25892
/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25893
/L /lineto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25894
/T /translate load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25895
/TMatrix { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25896
/RAngle { 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25897
/Atan { /atan load stopped { pop pop 0 } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25898
/Div { dup 0 eq { pop } { div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25899
/NET { neg exch neg exch T } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25900
/Pyth { dup mul exch dup mul add sqrt } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25901
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25902
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25903
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25904
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25905
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25906
/pathforall load stopped { pop pop pop pop } if z } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25907
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25908
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25909
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25910
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25911
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25912
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25913
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25914
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25915
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25916
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25917
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25918
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25919
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25920
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25921
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25922
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25923
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25924
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25925
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25926
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25927
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25928
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25929
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25930
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25931
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25932
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25933
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25934
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25935
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25936
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25937
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25938
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25939
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25940
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25941
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25942
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25943
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25944
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25945
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25946
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25947
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25948
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25949
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25950
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25951
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25952
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25953
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25954
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25955
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25956
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25957
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25958
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25959
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25960
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25961
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25962
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25963
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25964
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25965
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25966
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25967
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25968
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25969
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25970
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25971
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25972
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25973
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25974
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25975
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25976
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25977
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25978
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25979
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25980
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25981
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25982
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25983
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25984
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25985
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25986
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25987
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25988
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25989
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25990
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25991
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25992
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25993
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25994
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25995
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25996
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25997
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25998
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 25999
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26000
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26001
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26002
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26003
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26004
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26005
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26006
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26007
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26008
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26009
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26010
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26011
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26012
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26013
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26014
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26015
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26016
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26017
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26018
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26019
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26020
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26021
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26022
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26023
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26024
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26025
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26026
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26027
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26028
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26029
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26030
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26031
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26032
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26033
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26034
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26035
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26036
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26037
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26038
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26039
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26040
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26041
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26042
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26043
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26044
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26045
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26046
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26047
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26048
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26049
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26050
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26051
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26052
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26053
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26054
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26055
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26056
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26057
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26058
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26059
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26060
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26061
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26062
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26063
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26064
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26065
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26066
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26067
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26068
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26069
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26070
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26071
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26072
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26073
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26074
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26075
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26076
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26077
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26078
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26079
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26080
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26081
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26082
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26083
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26084
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26085
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26086
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26087
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26088
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26089
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26090
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26091
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26092
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26093
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26094
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26095
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26096
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26097
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26098
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26099
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26100
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26101
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26102
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26103
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26104
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26105
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26107
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26108
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26109
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26110
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26111
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26112
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26113
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26114
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26115
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26116
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26117
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26118
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26119
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26120
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26121
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26122
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26123
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26124
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26125
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26126
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26127
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26128
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26129
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26130
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26131
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26132
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26133
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26134
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26135
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26136
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26137
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26138
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26139
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26140
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26141
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26142
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26143
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26144
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26145
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26146
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26147
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26148
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26149
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26150
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26151
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26152
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26153
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26154
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26155
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26156
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26157
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26158
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26159
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26160
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26161
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26162
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26163
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26164
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26165
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26166
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26167
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26168
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26169
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26170
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26171
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26172
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26173
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26174
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26175
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26176
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26177
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26178
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26179
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26180
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26181
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26182
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26183
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26184
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26185
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26186
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26187
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26188
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26189
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26190
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26191
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26192
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26193
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26194
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26195
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26196
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26197
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26198
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26199
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26200
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26201
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26202
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26203
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26204
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26205
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26206
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26207
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26208
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26209
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26210
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26211
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26212
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26213
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26214
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26215
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26216
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26218
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26219
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26220
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26221
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26222
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26223
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26224
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26225
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26226
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26227
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26228
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26229
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26230
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26231
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26232
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26233
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26234
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26235
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26236
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26237
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26238
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26239
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26240
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26241
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26242
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26243
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26244
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26245
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26246
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26247
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26248
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26249
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26250
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26251
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26252
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26253
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26254
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26255
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26256
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26257
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26258
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26259
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26260
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26261
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26262
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26263
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26264
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26265
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26266
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26267
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26268
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26269
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26270
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26271
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26272
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26273
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26274
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26275
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26276
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26277
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26278
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26279
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26280
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26281
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26282
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26283
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26284
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26285
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26286
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26287
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26288
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26289
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26290
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26291
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26292
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26293
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26294
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26295
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26296
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26297
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26298
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26299
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26300
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26301
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26302
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26303
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26304
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26305
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26306
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26307
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26308
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26309
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26310
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26311
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26312
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26313
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26314
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26315
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26316
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26317
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26318
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26319
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26320
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26321
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26322
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26323
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26324
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26325
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26326
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26327
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26328
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26329
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26330
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26331
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26332
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26333
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26334
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26335
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26336
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26337
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26338
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26339
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26340
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26341
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26342
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26343
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26344
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26345
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26346
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26347
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26348
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26349
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26350
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26351
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26352
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26353
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26354
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26355
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26356
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26357
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26358
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26359
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26360
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26361
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26362
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26363
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26364
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26365
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26366
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26367
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26368
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26369
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26370
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26371
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26372
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26373
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26374
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26375
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26376
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26377
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26378
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26379
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26380
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26381
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26382
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26383
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26384
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26385
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26386
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26387
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26388
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26389
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26390
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26391
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26392
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26393
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26394
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26395
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26396
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26397
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26398
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26399
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26400
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26401
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26402
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26403
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26404
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26405
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26406
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26407
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26408
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26409
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26410
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26411
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26412
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26413
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26414
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26415
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26416
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26417
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26418
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26419
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26420
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26421
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26422
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26423
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26424
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26425
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26426
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26427
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26428
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26429
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26430
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26431
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26432
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26433
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26434
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26435
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26436
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26438
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26439
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26441
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26442
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26443
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26444
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26445
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26446
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26447
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26448
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26449
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26450
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26451
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26452
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26453
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26454
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26455
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26456
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26458
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26459
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26460
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26461
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26463
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26464
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26465
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26466
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26467
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26468
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26469
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26470
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26471
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26472
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26473
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26474
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26475
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26476
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26477
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26478
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26479
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26480
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26481
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26482
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26483
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26484
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26485
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26486
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26487
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26488
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26489
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26491
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
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 26655
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26656
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26657
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26659
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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139.41826  1. 0.1 0.  /c ED /b ED /a ED false OpenCurve  gsave 2.0
Christian Urban <urbanc@in.tum.de>
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diff changeset
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SLW 0  setgray 0 setlinecap stroke  grestore end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26671
@endspecial 841 3165 a
Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26674
a Fb(\034)p Black 1918 5251 a Fa(1)p Black eop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26675
%%Trailer
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26676
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26677
userdict /end-hook known{end-hook}if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26678
%%EOF
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26679
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26680
%%EndDocument
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26681
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26682
y Ga(;)15 b(M)988 2802 y F9(2)1028 2788 y Ga(;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26683
1157 2776 11 41 v 1167 2758 46 5 v 97 w F6(1)1324 2802
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26684
y F9(0)1363 2788 y Ga(;)g F6(1)1494 2802 y F9(1)1625
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26685
2710 y @beginspecial 365 @llx 477 @lly 519 @urx 605 @ury
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26686
708 @rwi @clip @setspecial
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26687
%%BeginDocument: pics/eps0.ps
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26688
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26689
%%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26690
%%Title: eps0.dvi
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26691
%%Pages: 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26692
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26693
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26694
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26695
%DVIPSWebPage: (www.radicaleye.com)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26696
%DVIPSCommandLine: dvips -o eps0.ps eps0.dvi
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26697
%DVIPSParameters: dpi=600, compressed
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26698
%DVIPSSource:  TeX output 2000.03.10:0010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26699
%%BeginProcSet: texc.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26700
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26701
/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26702
N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26703
mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26704
0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26705
landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26706
mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26707
matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26708
exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26709
statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26710
N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26711
/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26712
/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26713
array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26714
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26715
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26716
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26717
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26718
1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26719
1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26720
0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26721
sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26722
rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26723
gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26724
/chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26725
/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26726
A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26727
get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26728
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26729
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26730
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26731
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26732
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26733
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26734
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26735
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26736
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26737
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26738
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26739
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26740
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26741
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26742
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26743
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26744
(LaserWriter 16/600)]{A length product length le{A length product exch 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26745
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26746
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26747
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26748
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26749
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26750
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26751
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26752
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26753
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26754
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26756
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26757
%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26758
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26759
% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26760
% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26761
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26762
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26763
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26764
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26765
/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26766
/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26767
/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26768
/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26769
/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26770
/L /lineto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26771
/T /translate load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26772
/TMatrix { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26773
/RAngle { 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26774
/Atan { /atan load stopped { pop pop 0 } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26775
/Div { dup 0 eq { pop } { div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26776
/NET { neg exch neg exch T } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26777
/Pyth { dup mul exch dup mul add sqrt } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26778
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26779
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26780
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26781
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26782
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26783
/pathforall load stopped { pop pop pop pop } if z } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26784
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26785
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26786
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26787
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26788
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26789
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26790
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26791
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26792
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26793
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26794
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26795
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26796
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26797
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26798
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26799
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26800
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26801
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26802
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26803
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26804
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26805
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26806
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26807
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26808
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26809
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26810
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26811
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26812
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26813
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26814
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26815
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26816
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26817
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26818
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26819
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26820
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26821
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26822
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26823
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26824
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26825
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26826
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26827
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26828
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26829
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26830
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26831
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26832
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26833
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26834
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26835
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26836
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26837
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26838
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26839
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26840
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26841
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26842
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26843
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26844
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26845
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26846
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26847
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26848
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26849
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26850
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26851
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26852
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26853
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26854
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26855
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26856
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26857
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26858
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26859
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26860
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26861
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26862
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26863
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26864
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26865
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26866
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26867
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26868
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26869
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26870
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26871
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26872
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26873
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26874
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26875
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26876
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26877
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26878
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26879
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26880
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26881
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26882
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26883
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26884
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26885
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26886
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26887
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26888
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26889
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26890
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26891
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26892
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26893
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26894
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26895
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26896
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26897
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26898
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26899
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26900
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26901
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26902
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26903
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26904
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26905
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26906
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26907
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26908
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26909
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26910
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26911
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26912
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26913
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26914
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26915
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26916
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26917
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26918
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26919
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26920
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26921
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26922
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26923
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26924
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26925
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26926
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26927
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26928
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26929
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26930
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26931
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26932
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26933
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26934
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26935
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26936
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26937
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26938
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26939
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26940
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26941
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26942
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26943
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26944
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26945
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26946
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26947
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26948
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26949
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26950
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26951
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26952
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26953
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26954
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26955
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26956
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26957
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26958
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26959
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26960
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26961
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26962
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26963
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26964
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26965
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26966
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26967
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26968
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26969
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26970
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26971
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26972
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26973
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26974
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26975
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26976
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26977
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26978
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26979
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26980
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26981
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26982
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26983
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26984
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26985
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26986
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26987
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26988
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26989
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26990
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26991
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26992
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26993
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26994
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26995
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26996
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26997
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26998
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 26999
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27000
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27001
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27002
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27003
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27004
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27005
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27006
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27007
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27008
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27009
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27010
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27011
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27012
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27013
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27014
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27015
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27016
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27017
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27018
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27019
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27020
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27021
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27022
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27023
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27024
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27025
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27026
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27027
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27028
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27029
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27030
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27031
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27032
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27033
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27034
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27035
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27036
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27037
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27038
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27039
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27040
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27041
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27042
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27043
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27044
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27045
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27046
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27047
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27048
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27049
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27050
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27051
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27052
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27053
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27054
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27055
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27056
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27057
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27058
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27059
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27060
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27061
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27062
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27063
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27064
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27065
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27066
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27067
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27068
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27069
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27070
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27071
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27072
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27073
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27074
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27075
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27076
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27077
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27078
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27079
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27080
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27081
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27082
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27083
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27084
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27085
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27086
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27087
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27088
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27089
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27090
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27091
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27092
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27093
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27095
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27096
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27097
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27098
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27099
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27100
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27101
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27102
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27103
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27104
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27105
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27106
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27107
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27108
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27109
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27110
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27111
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27112
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27113
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27114
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27115
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27116
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27117
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27118
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27119
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27120
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27121
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27122
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27123
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27124
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27125
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27126
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27127
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27128
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27129
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27130
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27131
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27132
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27133
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27134
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27135
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27136
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27137
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27138
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27139
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27140
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27141
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27142
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27143
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27144
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27145
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27146
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27147
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27148
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27149
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27150
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27151
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27152
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27153
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27154
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27155
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27156
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27157
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27158
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27159
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27160
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27161
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27162
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27163
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27164
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27165
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27166
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27167
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27168
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27169
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27170
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27171
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27172
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27173
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27174
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27175
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27176
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27177
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27178
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27179
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27180
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27181
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27182
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27183
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27184
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27185
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27186
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27187
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27188
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27189
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27190
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27191
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27192
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27193
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27194
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27195
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27196
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27197
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27198
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27199
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27200
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27201
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27202
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27203
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27204
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27205
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27206
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27207
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27208
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27209
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27210
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27211
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27212
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27213
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27214
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27215
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27216
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27217
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27218
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27219
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27220
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27221
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27222
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27223
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27224
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27225
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27226
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27227
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27228
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27229
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27230
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27231
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27232
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27233
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27234
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27235
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27236
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27237
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27238
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27239
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27240
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27241
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27242
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27243
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27244
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27245
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27246
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27247
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27248
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27249
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27250
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27251
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27252
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27253
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27254
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27255
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27256
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27257
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27258
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27259
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27260
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27261
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27262
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27263
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27264
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27265
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27266
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27267
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27268
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27269
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27270
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27271
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27272
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27273
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27274
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27275
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27276
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27277
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27278
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27279
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27280
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27281
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27282
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27283
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27284
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27285
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27286
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27287
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27288
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27289
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27290
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27291
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27292
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27293
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27294
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27295
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27296
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27297
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27298
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27299
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27300
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27301
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27302
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27303
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27304
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27305
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27306
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27308
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27309
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27310
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27311
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27312
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27313
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27315
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27316
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27317
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27318
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27319
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27320
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27321
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27322
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27323
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27324
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27325
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27326
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27327
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27328
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27329
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27330
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27331
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27332
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27333
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27335
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27336
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27337
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27338
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27339
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27340
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27341
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27342
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27343
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27344
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27345
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27346
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27347
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27348
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27349
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27350
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27351
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27352
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27353
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27354
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27355
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27356
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27357
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27358
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27359
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27360
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27361
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27362
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27363
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27364
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27365
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27366
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27367
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27368
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27369
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27370
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27371
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27372
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27373
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27374
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27375
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27376
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27377
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27378
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27379
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27380
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27381
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27382
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27383
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27384
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27385
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27386
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27387
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27388
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27389
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27390
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27391
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27392
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27393
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27394
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27395
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27396
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27397
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27398
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27399
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27400
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27401
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27402
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27403
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27405
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27406
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27407
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27408
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27409
%%BeginProcSet: special.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27410
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27411
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27412
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27413
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27414
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27415
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27416
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27417
/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27418
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27419
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27420
maxlength ge{/md md dup length 20 add dict copy def}if end md begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27421
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27422
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27423
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27424
transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27425
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27427
if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27428
-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27429
get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27430
yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27431
neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27432
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27433
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27434
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27435
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27436
2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
Christian Urban <urbanc@in.tum.de>
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diff changeset
 27437
-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27438
TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27439
Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27440
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27441
save N userdict maxlength dict begin/magscale true def normalscale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27442
currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27443
/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27444
psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27445
psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27446
TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27447
psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27448
roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27449
moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27450
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27451
@SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27452
CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27453
closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27454
sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27455
}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27456
CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27457
lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27458
/@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27459
repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27460
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27461
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27462
moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27463
/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27464
1 startangle endangle arc savematrix setmatrix}N end
Christian Urban <urbanc@in.tum.de>
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 27465
Christian Urban <urbanc@in.tum.de>
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diff changeset
 27466
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27467
%%BeginProcSet: color.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27468
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27469
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27470
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27471
}repeat setrgbcolor pop}ifelse}B}ifelse/TeXcolorcmyk{setcmykcolor}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27472
/TeXcolorrgb{setrgbcolor}def/TeXcolorgrey{setgray}def/TeXcolorgray{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27473
setgray}def/TeXcolorhsb{sethsbcolor}def/currentcmykcolor where{pop}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27474
/currentcmykcolor{currentrgbcolor 10}B}ifelse/DC{exch dup userdict exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27475
known{pop pop}{X}ifelse}B/GreenYellow{0.15 0 0.69 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27476
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27477
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27478
setcmykcolor}DC/Peach{0 0.50 0.70 0 setcmykcolor}DC/Melon{0 0.46 0.50 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27479
setcmykcolor}DC/YellowOrange{0 0.42 1 0 setcmykcolor}DC/Orange{0 0.61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27480
0.87 0 setcmykcolor}DC/BurntOrange{0 0.51 1 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27481
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27482
setcmykcolor}DC/Mahogany{0 0.85 0.87 0.35 setcmykcolor}DC/Maroon{0 0.87
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27483
0.68 0.32 setcmykcolor}DC/BrickRed{0 0.89 0.94 0.28 setcmykcolor}DC/Red{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27484
0 1 1 0 setcmykcolor}DC/OrangeRed{0 1 0.50 0 setcmykcolor}DC/RubineRed{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27485
0 1 0.13 0 setcmykcolor}DC/WildStrawberry{0 0.96 0.39 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27486
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27487
setcmykcolor}DC/Magenta{0 1 0 0 setcmykcolor}DC/VioletRed{0 0.81 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27488
setcmykcolor}DC/Rhodamine{0 0.82 0 0 setcmykcolor}DC/Mulberry{0.34 0.90
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27489
0 0.02 setcmykcolor}DC/RedViolet{0.07 0.90 0 0.34 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27490
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27491
setcmykcolor}DC/Thistle{0.12 0.59 0 0 setcmykcolor}DC/Orchid{0.32 0.64 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27492
0 setcmykcolor}DC/DarkOrchid{0.40 0.80 0.20 0 setcmykcolor}DC/Purple{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27493
0.45 0.86 0 0 setcmykcolor}DC/Plum{0.50 1 0 0 setcmykcolor}DC/Violet{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27494
0.79 0.88 0 0 setcmykcolor}DC/RoyalPurple{0.75 0.90 0 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27495
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27496
setcmykcolor}DC/CadetBlue{0.62 0.57 0.23 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27497
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27498
0.43 setcmykcolor}DC/NavyBlue{0.94 0.54 0 0 setcmykcolor}DC/RoyalBlue{1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27499
0.50 0 0 setcmykcolor}DC/Blue{1 1 0 0 setcmykcolor}DC/Cerulean{0.94 0.11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27500
0 0 setcmykcolor}DC/Cyan{1 0 0 0 setcmykcolor}DC/ProcessBlue{0.96 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27501
setcmykcolor}DC/SkyBlue{0.62 0 0.12 0 setcmykcolor}DC/Turquoise{0.85 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27502
0.20 0 setcmykcolor}DC/TealBlue{0.86 0 0.34 0.02 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27503
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27504
setcmykcolor}DC/Emerald{1 0 0.50 0 setcmykcolor}DC/JungleGreen{0.99 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27505
0.52 0 setcmykcolor}DC/SeaGreen{0.69 0 0.50 0 setcmykcolor}DC/Green{1 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27506
1 0 setcmykcolor}DC/ForestGreen{0.91 0 0.88 0.12 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27507
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27508
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27673
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27674
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27675
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27676
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27677
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27678
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27679
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27680
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27681
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27682
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27683
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27684
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27685
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27686
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27687
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27688
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27689
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27690
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27691
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27692
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27693
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27694
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27695
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27696
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27697
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27698
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27699
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27700
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27701
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27702
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27703
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27704
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27705
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27706
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27707
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27708
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27709
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27710
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27711
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27712
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27713
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27714
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27715
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27716
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27717
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27718
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27719
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27720
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27721
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27722
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27723
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27724
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27725
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27726
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27727
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27728
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27729
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27730
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27731
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27732
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27733
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27734
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27735
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27736
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27737
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27738
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27739
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27740
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27741
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27742
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27743
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27744
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27745
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27746
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27747
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27748
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27749
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27750
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27751
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27752
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27753
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27754
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27755
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27756
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27757
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27758
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27759
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27760
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27761
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27762
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27763
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27764
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27765
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27766
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27767
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27768
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27769
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27770
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27771
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27772
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27773
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27774
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27775
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27776
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27777
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27778
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27779
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27780
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27781
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27782
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27783
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27784
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27785
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27786
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27787
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27788
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27789
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27790
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27791
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27792
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27793
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27794
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27795
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27796
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27797
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27798
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27799
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27800
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27801
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27802
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27803
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27804
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27805
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27806
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27807
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27808
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27809
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27810
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27811
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27812
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27813
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27814
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27815
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27816
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27817
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27818
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27819
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27820
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27821
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27822
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27823
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27824
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27825
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27826
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27827
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27828
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27829
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27830
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27831
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27832
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27833
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27834
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27835
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27836
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27837
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27838
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27839
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27840
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27841
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27842
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27843
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27844
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27845
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27846
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27847
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27848
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27849
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27850
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27851
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27852
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27853
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27854
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27855
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27856
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27857
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27858
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27859
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27860
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27861
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27862
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27863
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27864
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27865
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27866
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27867
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27868
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27869
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27870
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27871
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27872
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27873
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27874
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27875
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27876
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27877
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27878
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27879
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27880
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27881
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27882
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27883
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27884
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27885
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27886
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27887
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27888
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27889
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27890
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27891
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27892
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27893
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27894
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27895
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27896
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27897
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27898
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27899
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27900
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27901
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27902
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27903
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27904
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27905
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27906
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27907
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27908
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27909
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27910
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27911
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27912
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27913
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27914
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27915
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27916
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27917
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27918
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27919
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27920
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27921
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27922
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27923
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27924
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27925
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27926
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27927
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27928
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27929
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27930
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27931
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27932
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27933
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27934
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27935
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27936
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27937
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27938
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27939
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27940
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27941
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27942
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27943
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27944
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27945
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27946
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27947
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27948
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27949
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27950
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27951
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27952
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27953
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27954
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27955
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27956
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27957
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27958
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27959
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27960
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27961
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27962
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27963
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27964
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27965
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27966
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27967
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27968
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27969
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27970
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27971
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27972
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27973
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27974
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27975
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27976
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27977
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27978
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27979
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27980
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27981
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27982
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27983
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27984
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27985
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27986
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27987
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27988
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27989
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27990
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27991
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27992
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27993
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27994
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27995
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27996
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27997
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27998
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 27999
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28000
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28001
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28002
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28003
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28004
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28005
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28006
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28007
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28008
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28009
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28010
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28011
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28012
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28013
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28014
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28015
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28016
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28017
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28018
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28019
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28020
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28021
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28022
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28023
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28024
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28025
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28026
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28027
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28028
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28029
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28030
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28031
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28032
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28033
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28034
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28035
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28036
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28037
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28038
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28039
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28040
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28041
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28042
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28043
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28044
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28045
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28046
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28047
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28048
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28049
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28050
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28051
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28052
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28053
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28054
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28055
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28056
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28057
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28058
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28059
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28060
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28061
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28062
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28063
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28064
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28065
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28066
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28067
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28068
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28069
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28070
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28071
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28072
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28073
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28074
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28075
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28076
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28077
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28078
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28079
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28080
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28081
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28082
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28083
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28084
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28085
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28086
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28087
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28088
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28089
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28090
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28091
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28092
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28093
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28094
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28095
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28096
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28097
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28098
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28099
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28100
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28101
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28102
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28103
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28104
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28105
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28106
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28107
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28108
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28109
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28110
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28111
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28112
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28113
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28114
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28115
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28116
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28117
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28118
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28119
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28120
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28121
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28122
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28123
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28124
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28125
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28126
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28127
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28128
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28129
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28130
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28131
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28132
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28133
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28134
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28135
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28136
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28137
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28138
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28139
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28140
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28141
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28142
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28143
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28144
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28145
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28146
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28147
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28148
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28149
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28150
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28151
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28152
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28153
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28154
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28155
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28156
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28157
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28158
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28159
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28160
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28161
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28162
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28163
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28164
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28165
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28166
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28167
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28168
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28169
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28170
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28171
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28172
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28173
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28174
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28175
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28176
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28177
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28178
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28179
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28180
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28181
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28182
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28183
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28184
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28185
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28186
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28187
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28188
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28189
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28190
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28191
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28192
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28193
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28194
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28195
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28197
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28198
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28199
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28200
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28201
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28202
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28204
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28205
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28207
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28208
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28209
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28210
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28211
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28212
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28213
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28214
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28215
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28216
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28217
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28218
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28219
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28220
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28222
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28224
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28225
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28226
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28227
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28228
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28229
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28230
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28231
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28232
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28233
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28234
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28235
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28236
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28237
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28238
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28239
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28240
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28241
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28242
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28243
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28244
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28245
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28246
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28247
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28248
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28249
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28250
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28251
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28252
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28253
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28254
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28255
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28257
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28258
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28259
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28260
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28261
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28262
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28263
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28264
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28265
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28266
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28267
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28268
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28269
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28270
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28271
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28272
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28273
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28274
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28275
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28277
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28278
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28279
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28280
    Char end stringwidth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28281
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28282
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28283
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28284
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28285
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28286
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28287
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28288
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28289
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28290
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28291
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28292
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28294
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28295
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28296
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28297
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28298
%%BeginProcSet: special.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28299
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28300
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28301
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28302
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28303
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28304
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28305
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28306
/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28307
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28308
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28309
maxlength ge{/md md dup length 20 add dict copy def}if end md begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28310
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28311
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28312
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28313
transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28314
curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28315
pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28316
if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28317
-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28318
get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28319
yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28320
neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28321
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28322
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28323
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28324
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28325
2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28326
-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28327
TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28328
Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28329
}if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28330
save N userdict maxlength dict begin/magscale true def normalscale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28331
currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28332
/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28333
psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28334
psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28335
TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28336
psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28337
roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28338
moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28339
begin/SpecialSave save N gsave normalscale currentpoint TR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28340
@SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28341
CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28342
closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28343
sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28344
}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28345
CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28346
lineto closepath clip}if/showpage{}N/erasepage{}N/copypage{}N newpath}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28347
/@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{end}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28348
repeat grestore SpecialSave restore end}N/@defspecial{SDict begin}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28349
/@fedspecial{end}B/li{lineto}B/rl{rlineto}B/rc{rcurveto}B/np{/SaveX
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28350
currentpoint/SaveY X N 1 setlinecap newpath}N/st{stroke SaveX SaveY
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28351
moveto}N/fil{fill SaveX SaveY moveto}N/ellipse{/endangle X/startangle X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28352
/yrad X/xrad X/savematrix matrix currentmatrix N TR xrad yrad scale 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28353
1 startangle endangle arc savematrix setmatrix}N end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28354
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28355
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28356
%%BeginProcSet: color.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28357
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28358
TeXDict begin/setcmykcolor where{pop}{/setcmykcolor{dup 10 eq{pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28359
setrgbcolor}{1 sub 4 1 roll 3{3 index add neg dup 0 lt{pop 0}if 3 1 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28360
}repeat setrgbcolor pop}ifelse}B}ifelse/TeXcolorcmyk{setcmykcolor}def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28361
/TeXcolorrgb{setrgbcolor}def/TeXcolorgrey{setgray}def/TeXcolorgray{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28362
setgray}def/TeXcolorhsb{sethsbcolor}def/currentcmykcolor where{pop}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28363
/currentcmykcolor{currentrgbcolor 10}B}ifelse/DC{exch dup userdict exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28364
known{pop pop}{X}ifelse}B/GreenYellow{0.15 0 0.69 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28365
/Yellow{0 0 1 0 setcmykcolor}DC/Goldenrod{0 0.10 0.84 0 setcmykcolor}DC
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28366
/Dandelion{0 0.29 0.84 0 setcmykcolor}DC/Apricot{0 0.32 0.52 0
Christian Urban <urbanc@in.tum.de>
parents:
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diff changeset
 28514
1362 y FT(8)1187 1374 y FS(L)p 1387 943 665 4 v 1387
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28515
1038 a FF(n)g FT(\024)g FF(m)1579 1007 y Ft(n)p FS(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28516
Ft(n)1663 982 y Fh(0)p 1694 1026 10 38 v 1704 1009 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28517
4 v 1764 1038 a FF(n)g FT(\024)g FF(m)1956 1007 y Ft(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28518
FS(;)p Ft(n)2040 982 y Fh(0)p 1292 1069 856 4 v 1292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28519
1164 a FT(8)p FF(x)p FU(:)p FG(\()p FF(x)23 b FT(\024)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28520
FF(m)1646 1133 y Ft(x)p FS(;)p Ft(n)1726 1108 y Fh(0)1739
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28521
1164 y FG(\))p 1790 1152 10 38 v 1799 1135 42 4 v 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28522
w FF(n)9 b FT(\024)g FF(m)2051 1133 y Ft(n)p FS(;)p Ft(n)2135
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28523
1108 y Fh(0)2147 1091 y FT(8)2194 1103 y FS(L)p 1292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28524
1204 856 4 v 1473 1299 a FU(M)1554 1311 y FJ(1)p 1609
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28525
1287 10 38 v 1618 1270 42 4 v 1678 1299 a FF(n)g FT(\024)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28526
FF(m)1870 1269 y Ft(n)p FS(;)p Ft(n)1954 1244 y Fh(0)2147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28527
1227 y FT(8)2194 1239 y FS(L)p 2299 959 713 4 v 2299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28528
1053 a FF(fm)2390 1023 y Ft(n)p FS(;)p Ft(n)2474 998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28529
y Fh(0)2495 1053 y FG(=)g FF(0)p 2629 1041 10 38 v 2639
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28530
1025 42 4 v 88 w(fm)2790 1023 y Ft(n)p FS(;)p Ft(n)2874
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28531
998 y Fh(0)2896 1053 y FG(=)g FF(0)p 3094 959 713 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28532
82 w(fm)3185 1023 y Ft(n)p FS(;)p Ft(n)3269 998 y Fh(0)3291
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28533
1053 y FG(=)g FF(1)p 3424 1041 10 38 v 3434 1025 42 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28534
v 87 w(fm)3585 1023 y Ft(n)p FS(;)p Ft(n)3669 998 y Fh(0)3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28535
1053 y FG(=)g FF(1)p 2299 1073 1508 4 v 2338 1168 a(fm)2429
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28536
1138 y Ft(n)p FS(;)p Ft(n)2513 1113 y Fh(0)2535 1168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28537
y FG(=)g FF(0)p FT(_)o FF(fm)2796 1138 y Ft(n)p FS(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28538
Ft(n)2880 1113 y Fh(0)2902 1168 y FG(=)g FF(1)p 3036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28539
1156 10 38 v 3046 1140 42 4 v 88 w(fm)3197 1138 y Ft(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28540
FS(;)p Ft(n)3281 1113 y Fh(0)3302 1168 y FG(=)g FF(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28541
FU(;)14 b FF(fm)3546 1138 y Ft(n)p FS(;)p Ft(n)3630 1113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28542
y Fh(0)3651 1168 y FG(=)9 b FF(1)3806 1090 y FT(_)3861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28543
1103 y FS(L)p 2338 1204 1429 4 v 2376 1299 a FT(8)p FF(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28544
FU(:)p FG(\()p FF(fx)h FG(=)f FF(0)p FT(_)o FF(fx)h FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28545
FF(1)p FG(\))p 2998 1287 10 38 v 3008 1270 42 4 v 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28546
w FF(fm)3159 1269 y Ft(n)p FS(;)p Ft(n)3243 1244 y Fh(0)3264
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28547
1299 y FG(=)g FF(0)p FU(;)14 b FF(fm)3508 1269 y Ft(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28548
FS(;)p Ft(n)3592 1244 y Fh(0)3613 1299 y FG(=)9 b FF(1)3767
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28549
1227 y FT(8)3814 1239 y FS(L)p 1473 1340 2257 4 v 1675
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28550
1435 a FU(M)1756 1447 y FJ(1)1793 1435 y FU(;)14 b FT(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28551
FF(x)p FU(:)p FG(\()p FF(fx)c FG(=)f FF(0)p FT(_)o FF(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28552
FG(=)f FF(1)p FG(\))p 2452 1423 10 38 v 2462 1406 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28553
4 v 88 w FF(n)g FT(\024)g FF(m)2714 1405 y Ft(n)p FS(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28554
Ft(n)2798 1379 y Fh(0)2810 1435 y FT(^)p FF(fm)2956 1405
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28555
y Ft(n)p FS(;)p Ft(n)3040 1379 y Fh(0)3062 1435 y FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28556
FF(0)p FU(;)14 b FF(fm)3306 1405 y Ft(n)p FS(;)p Ft(n)3390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28557
1379 y Fh(0)3411 1435 y FG(=)9 b FF(1)3729 1357 y FT(^)3784
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28558
1369 y FS(R)p 474 1475 3052 4 v 814 1570 a FU(M)895 1582
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28559
y FJ(1)932 1570 y FU(;)14 b(M)1050 1582 y FJ(2)1087 1570
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28560
y FU(;)g FT(8)p FF(x)p FU(:)p FG(\()p FF(fx)9 b FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28561
FF(0)p FT(_)p FF(fx)h FG(=)f FF(1)p FG(\))p 1746 1558
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28562
10 38 v 1755 1542 42 4 v 87 w FF(n)g FT(\024)g FF(m)2007
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28563
1540 y Ft(n)p FS(;)p Ft(n)2091 1515 y Fh(0)2103 1570
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28564
y FT(^)q FF(fm)2250 1540 y Ft(n)p FS(;)p Ft(n)2334 1515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28565
y Fh(0)2355 1570 y FG(=)g FF(0)p FU(;)14 b FF(n)2551
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28566
1540 y FQ(0)2583 1570 y FT(\024)9 b FF(m)2723 1540 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28567
Ft(n)p FS(;)p Ft(n)2807 1515 y Fh(0)2819 1570 y FT(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28568
FF(fm)2965 1540 y Ft(n)p FS(;)p Ft(n)3049 1515 y Fh(0)3071
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28569
1570 y FG(=)g FF(1)3526 1492 y FT(^)3582 1505 y FS(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28570
814 1611 2373 4 v 849 1706 a FU(M)930 1718 y FJ(1)966
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28571
1706 y FU(;)14 b(M)1084 1718 y FJ(2)1121 1706 y FU(;)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28572
FT(8)p FF(x)p FU(:)p FG(\()p FF(fx)c FG(=)f FF(0)p FT(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28573
FF(fx)h FG(=)f FF(1)p FG(\))p 1780 1694 10 38 v 1790
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28574
1677 42 4 v 88 w FT(9)p FF(k)p FU(:)p FG(\()p FF(n)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28575
FT(\024)g FF(k)p FT(^)p FF(fk)g FG(=)g FF(0)p FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28576
FU(;)14 b FF(n)2517 1676 y FQ(0)2549 1706 y FT(\024)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28577
b FF(m)2689 1676 y Ft(n)p FS(;)p Ft(n)2773 1651 y Fh(0)2785
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28578
1706 y FT(^)p FF(fm)2931 1676 y Ft(n)p FS(;)p Ft(n)3015
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28579
1651 y Fh(0)3037 1706 y FG(=)g FF(1)3187 1633 y FT(9)3233
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28580
1645 y FS(R)p 848 1746 2305 4 v 848 1825 a FU(M)929 1837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28581
y FJ(1)966 1825 y FU(;)14 b(M)1084 1837 y FJ(2)1121 1825
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28582
y FU(;)g FT(8)p FF(x)p FU(:)p FG(\()p FF(fx)9 b FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28583
FF(0)p FT(_)p FF(fx)h FG(=)f FF(1)p FG(\))p 1780 1813
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28584
10 38 v 1789 1797 42 4 v 87 w FT(9)p FF(k)p FU(:)p FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28585
FF(n)g FT(\024)g FF(k)p FT(^)p FF(fk)g FG(=)g FF(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28586
FG(\))p FU(;)14 b FT(9)p FF(k)2560 1795 y FQ(0)2583 1825
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28587
y FU(:)p FG(\()p FF(n)2681 1795 y FQ(0)2713 1825 y FT(\024)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28588
b FF(k)2828 1795 y FQ(0)2851 1825 y FT(^)q FF(fk)2973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28589
1795 y FQ(0)3005 1825 y FG(=)g FF(1)p FG(\))3153 1769
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28590
y FT(9)3199 1781 y FS(R)p 792 1866 2417 4 v 792 1945
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28591
a FU(M)873 1957 y FJ(1)910 1945 y FU(;)14 b(M)1028 1957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28592
y FJ(2)1064 1945 y FU(;)g FT(8)p FF(x)p FU(:)p FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28593
FF(fx)c FG(=)f FF(0)p FT(_)p FF(fx)h FG(=)f FF(1)p FG(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28594
1724 1933 10 38 v 1733 1916 42 4 v 87 w FT(8)p FF(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28595
FU(:)p FT(9)p FF(k)p FU(:)p FG(\()p FF(n)g FT(\024)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28596
FF(k)p FT(^)o FF(fk)g FG(=)g FF(0)p FG(\))p FU(;)14 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28597
FT(9)p FF(k)2616 1915 y FQ(0)2639 1945 y FU(:)p FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28598
FF(n)2737 1915 y FQ(0)2770 1945 y FT(\024)9 b FF(k)2885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28599
1915 y FQ(0)2907 1945 y FT(^)q FF(fk)3029 1915 y FQ(0)3061
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28600
1945 y FG(=)g FF(1)p FG(\))3209 1889 y FT(8)3256 1901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28601
y FS(R)p 704 1986 2594 4 v 704 2069 a FU(M)785 2081 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28602
FJ(1)821 2069 y FU(;)14 b(M)939 2081 y FJ(2)976 2069
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28603
y FU(;)g FT(8)p FF(x)p FU(:)p FG(\()p FF(fx)c FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28604
FF(0)p FT(_)o FF(fx)h FG(=)f FF(1)p FG(\))1013 2125 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28605
FI(|)p 1050 2125 228 10 v 228 w({z)p 1352 2125 V 228
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28606
w(})1284 2220 y FU(A)p 1649 2057 10 38 v 1659 2040 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28607
4 v 1732 2069 a FT(8)p FF(n)p FU(:)p FT(9)p FF(k)p FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28608
FG(\()p FF(n)g FT(\024)g FF(k)p FT(^)o FF(fk)g FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28609
FF(0)p FG(\))1732 2125 y FI(|)p 1769 2125 275 10 v 275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28610
w({z)p 2118 2125 V 275 w(})2022 2199 y FT(1)2105 2211
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28611
y FJ(0)2431 2069 y FU(;)28 b FT(8)p FF(n)2572 2035 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28612
FQ(0)2594 2069 y FU(:)p FT(9)p FF(k)2704 2035 y FQ(0)2727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28613
2069 y FU(:)p FG(\()p FF(n)2825 2035 y FQ(0)2858 2069
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28614
y FT(\024)9 b FF(k)2973 2035 y FQ(0)2996 2069 y FT(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28615
FF(fk)3117 2035 y FQ(0)3149 2069 y FG(=)g FF(1)p FG(\))2482
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28616
2125 y FI(|)p 2519 2125 333 10 v 333 w({z)p 2926 2125
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28617
V 333 w(})2829 2199 y FT(1)2912 2211 y FJ(1)3297 2008
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28618
y FT(8)3344 2020 y FS(R)366 2588 y Fj(")429 2606 y FH(i)467
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28619
2539 y F5(def)472 2588 y Fi(=)p 504 2876 536 4 v 504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28620
2945 a FF(sn)g FT(\024)g FF(m)p 746 2933 10 38 v 756
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28621
2917 42 4 v 88 w(sn)g FT(\024)g FF(m)p 1123 2880 472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28622
4 v 83 w(n)g FU(<)g FF(m)p 1333 2933 10 38 v 1342 2917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28623
42 4 v 87 w(n)g FU(<)g FF(m)p 504 2977 1091 4 v 539 3046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28624
a(sn)g FT(\024)g FF(m)p FT(\033)o FF(n)g FU(<)g FF(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28625
FU(;)14 b FF(sn)9 b FT(\024)g FF(m)p 1298 3034 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28626
v 1308 3017 42 4 v 88 w(n)g FU(<)g FF(m)1608 2993 y FT(\033)1673
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28627
3005 y FS(L)p 479 3082 1141 4 v 479 3161 a FT(8)p FF(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28628
FU(:)p FG(\()p FF(sn)g FT(\024)g FF(y)q FT(\033)p FF(n)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28629
FU(<)g FF(y)q FG(\))p FU(;)14 b FF(sn)9 b FT(\024)g FF(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28630
1358 3149 10 38 v 1368 3132 42 4 v 88 w(n)g FU(<)g FF(m)1633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28631
3105 y FT(8)1680 3117 y FS(L)p 479 3202 1141 4 v 751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28632
3275 a FF(sn)g FT(\024)g FF(m)p FU(;)14 b(S)p 1086 3263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28633
10 38 v 1095 3246 42 4 v 92 w FF(n)9 b FU(<)g FF(m)1633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28634
3224 y FT(8)1680 3236 y FS(L)p 1812 2638 459 4 v 1812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28635
2712 a FF(fn)h FG(=)f Fg(i)p 2016 2700 10 38 v 2025 2683
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28636
42 4 v 96 w FF(fn)h FG(=)f Fg(i)p 2354 2638 505 4 v 92
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28637
w FF(fm)g FG(=)g Fg(i)p 2580 2700 10 38 v 2590 2683 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28638
4 v 97 w FF(fm)g FG(=)g Fg(i)p 1812 2732 1046 4 v 1852
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28639
2806 a FF(fm)g FG(=)g Fg(i)f FU(;)14 b FF(fn)c FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28640
Fg(i)p 2300 2794 10 38 v 2310 2777 42 4 v 97 w FF(fn)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28641
FG(=)g Fg(i)f FT(^)r FF(fm)h FG(=)g Fg(i)2872 2749 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28642
FT(^)2927 2761 y FS(R)p 3065 2732 574 4 v 3065 2806 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28643
FF(fn)g FG(=)g FF(fm)p 3326 2794 10 38 v 3335 2777 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28644
4 v 88 w(fn)h FG(=)f FF(fm)p 1852 2842 1787 4 v 1936
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28645
2921 a FG(\()p FF(fn)g FG(=)g Fg(i)f FT(^)q FF(fm)i FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28646
Fg(i)f FG(\))p FT(\033)q FF(fn)h FG(=)g FF(fm)p FU(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28647
b FF(fm)c FG(=)f Fg(i)f FU(;)14 b FF(fn)c FG(=)f Fg(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28648
3242 2909 10 38 v 3251 2892 42 4 v 96 w FF(fn)h FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28649
FF(fm)3652 2858 y FT(\033)3716 2871 y FS(L)p 1876 2962
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28650
1739 4 v 1876 3041 a FT(8)p FF(y)q FU(:)p FG(\(\()p FF(fn)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28651
FG(=)g Fg(i)f FT(^)q FF(fy)j FG(=)e Fg(i)f FG(\))p FT(\033)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28652
FF(fn)h FG(=)g FF(fy)r FG(\))p FU(;)14 b FF(fm)c FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28653
Fg(i)f FU(;)14 b FF(fn)c FG(=)f Fg(i)p 3302 3029 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28654
v 3312 3012 42 4 v 96 w FF(fn)h FG(=)f FF(fm)3628 2984
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28655
y FT(8)3675 2996 y FS(L)p 1849 3081 1791 4 v 1849 3160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28656
a FT(8)p FF(x)p FU(:)p FF(y)q FU(:)p FG(\(\()p FF(fx)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28657
FG(=)e Fg(i)f FT(^)q FF(fy)j FG(=)e Fg(i)f FG(\))p FT(\033)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28658
FF(fx)h FG(=)g FF(fy)r FG(\))p FU(;)14 b FF(fm)c FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28659
Fg(i)f FU(;)14 b FF(fn)c FG(=)f Fg(i)p 3328 3148 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28660
v 3338 3131 42 4 v 97 w FF(fn)g FG(=)g FF(fm)3654 3104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28661
y FT(8)3701 3116 y FS(L)p 1849 3201 1791 4 v 2316 3275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28662
a FF(fm)g FG(=)g Fg(i)f FU(;)14 b FF(fn)d FG(=)e Fg(i)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28663
FU(;)14 b(T)p 2862 3263 10 38 v 2871 3246 42 4 v 99 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28664
FF(fn)c FG(=)f FF(fm)3654 3223 y FT(8)3701 3235 y FS(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28665
751 3311 2423 4 v 1236 3385 a FF(fm)g FG(=)g Fg(i)f FU(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28666
b FF(sn)c FT(\024)f FF(m)p FU(;)14 b FF(fn)9 b FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28667
Fg(i)f FU(;)14 b(S;)g(T)p 2130 3373 10 38 v 2140 3357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28668
42 4 v 99 w FF(n)9 b FU(<)g FF(m)p FT(^)q FF(fn)g FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28669
FF(fm)3188 3328 y FT(^)3243 3340 y FS(R)p 1226 3421 1473
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28670
4 v 1226 3496 a FF(fn)h FG(=)f Fg(i)f FU(;)14 b FF(sn)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28671
b FT(\024)g FF(m)p FT(^)q FF(fm)g FG(=)g Fg(i)f FU(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28672
b(S;)g(T)p 2139 3484 10 38 v 2149 3467 42 4 v 100 w FF(n)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28673
b FU(<)g FF(m)p FT(^)p FF(fn)g FG(=)g FF(fm)2712 3441
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28674
y FT(^)2768 3411 y FQ(0)2768 3464 y FS(L)p 1115 3532
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28675
1695 4 v 1115 3606 a FF(sn)g FT(\024)g FF(m)p FT(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28676
FF(fm)h FG(=)f Fg(i)f FU(;)14 b FF(0)9 b FT(\024)g FF(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28677
FT(^)p FF(fn)g FG(=)g Fg(i)f FU(;)14 b(S;)g(T)p 2251
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28678
3594 10 38 v 2260 3577 42 4 v 100 w FF(n)9 b FU(<)g FF(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28679
FT(^)p FF(fn)h FG(=)f FF(fm)2824 3544 y FT(^)2879 3557
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28680
y FS(L)2925 3565 y FP(2)p 1015 3642 1895 4 v 1015 3721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28681
a FF(sn)g FT(\024)g FF(m)p FT(^)p FF(fm)h FG(=)f Fg(i)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28682
FU(;)14 b FF(0)9 b FT(\024)g FF(n)p FT(^)p FF(fn)h FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28683
Fg(i)f FU(;)14 b(S;)g(T)p 2151 3709 10 38 v 2160 3692
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28684
42 4 v 99 w FT(9)p FF(m)p FU(:)p FG(\()p FF(n)9 b FU(<)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28685
FF(m)p FT(^)q FF(fn)g FG(=)g FF(fm)p FG(\))2924 3665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28686
y FT(9)2970 3677 y FS(R)p 982 3762 1961 4 v 982 3840
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28687
a FF(sn)g FT(\024)g FF(m)p FT(^)p FF(fm)h FG(=)f Fg(i)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28688
FU(;)14 b FF(0)9 b FT(\024)g FF(n)p FT(^)p FF(fn)h FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28689
Fg(i)f FU(;)14 b(S;)g(T)p 2118 3828 10 38 v 2127 3812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28690
42 4 v 99 w FT(9)p FF(n)p FU(:)p FF(m)p FU(:)p FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28691
FF(n)9 b FU(<)g FF(m)p FT(^)q FF(fn)g FG(=)g FF(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28692
FG(\))2957 3784 y FT(9)3003 3796 y FS(R)p 920 3881 2085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28693
4 v 920 3960 a FT(9)p FF(k)p FU(:)p FG(\()p FF(sn)g FT(\024)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28694
FF(k)p FT(^)p FF(fk)g FG(=)g Fg(i)f FG(\))p FU(;)14 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28695
FF(0)9 b FT(\024)g FF(n)p FT(^)q FF(fn)g FG(=)g Fg(i)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28696
FU(;)14 b(S;)g(T)p 2180 3948 10 38 v 2189 3931 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28697
100 w FT(9)p FF(n)p FU(:)p FF(m)p FU(:)p FG(\()p FF(n)9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28698
b FU(<)g FF(m)p FT(^)q FF(fn)g FG(=)g FF(fm)p FG(\))3018
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28699
3904 y FT(9)3064 3916 y FS(L)p 880 4001 2165 4 v 880
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28700
4080 a FF(0)g FT(\024)g FF(n)p FT(^)p FF(fn)h FG(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28701
Fg(i)f FU(;)14 b FT(8)p FF(n)p FU(:)p FT(9)p FF(k)p FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28702
FG(\()p FF(n)9 b FT(\024)g FF(k)p FT(^)o FF(fk)g FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28703
Fg(i)f FG(\))p FU(;)14 b(S;)g(T)p 2220 4068 10 38 v 2229
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28704
4051 42 4 v 100 w FT(9)p FF(n)p FU(:)p FF(m)p FU(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28705
FG(\()p FF(n)9 b FU(<)g FF(m)p FT(^)q FF(fn)g FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28706
FF(fm)p FG(\))3059 4023 y FT(8)3106 4035 y FS(L)p 795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28707
4120 2335 4 v 795 4199 a FT(9)p FF(k)p FU(:)p FG(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28708
FF(0)g FT(\024)g FF(k)p FT(^)p FF(fk)g FG(=)g Fg(i)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28709
FG(\))p FU(;)14 b FT(8)p FF(n)p FU(:)p FT(9)p FF(k)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28710
FU(:)p FG(\()p FF(n)9 b FT(\024)g FF(k)p FT(^)p FF(fk)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28711
FG(=)g Fg(i)f FG(\))p FU(;)14 b(S;)g(T)p 2305 4187 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28712
38 v 2314 4171 42 4 v 100 w FT(9)p FF(n)p FU(:)p FF(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28713
FU(:)p FG(\()p FF(n)9 b FU(<)g FF(m)p FT(^)q FF(fn)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28714
FG(=)g FF(fm)p FG(\))3143 4143 y FT(9)3189 4155 y FS(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28715
795 4240 2335 4 v 1096 4319 a FT(8)p FF(n)p FU(:)p FT(9)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28716
FF(k)p FU(:)p FG(\()p FF(n)g FT(\024)g FF(k)p FT(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28717
FF(fk)g FG(=)g Fg(i)g FG(\))1096 4374 y FI(|)p 1133 4374
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28718
271 10 v 271 w({z)p 1478 4374 V 271 w(})1386 4449 y FT(1)1469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28719
4461 y FS(i)1787 4319 y FU(;)14 b(S;)g(T)p 1990 4307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28720
10 38 v 2000 4290 42 4 v 113 w FT(9)p FF(n)p FU(:)p FF(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28721
FU(:)p FG(\()p FF(n)9 b FU(<)g FF(m)p FT(^)p FF(fn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28722
FG(=)f FF(fm)p FG(\))2074 4374 y FI(|)p 2111 4374 304
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28723
10 v 304 w({z)p 2489 4374 V 304 w(})2419 4470 y FU(P)3143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28724
4262 y FT(8)3190 4274 y FS(L)p Black Black 1844 4623
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28725
a Gd(where)20 b FU(i)i FT(2)i(f)p FF(0)p FU(;)14 b FF(1)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28726
FT(g)p 3965 4791 4 4073 v 277 4794 3691 4 v Black 1156
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28727
4948 a Gg(Figure)24 b(4.5:)29 b(The)24 b(sequent)h(proofs)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28728
Ga(\034)33 b Gg(and)24 b Ga(")2572 4962 y Gc(i)2600 4948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28729
y Gg(.)p Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28730
%%Page: 120 132
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28731
TeXDict begin 120 131 bop Black -144 51 a Gb(120)2310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28732
b(A)n(pplications)23 b(of)h(Cut-Elimination)p -144 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28733
3691 4 v Black 321 388 a Gg(of)31 b(the)g F6(\033)p Gg(-fragment)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28734
(classical)i(logic)f(and)f(whose)h(computations)i(can)d(be)g(simulated)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28735
i(by)e(cut-)321 501 y(elimination.)462 632 y(As)f(we)g(metioned)i(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28736
(the)g(introduction,)k(the)c(simply-typed)j(lambda)d(calculus)i(can)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28737
(be)g(seen)321 745 y(as)k(a)g(prototypical)k(programming)e(language)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28738
(in)d(which)g(program)i(e)o(x)o(ecution)g(\(beta-reduction\))321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28739
858 y(is)30 b(a)f(form)h(of)g(sequential)i(computation.)51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28740
b(In)30 b(Section)g(3.1,)h(we)e(sho)n(wed)i(that)f(this)g(programming)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28741
321 971 y(language)37 b(gi)n(v)o(es)e(a)f(computational)k(meaning)e(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28742
e(the)h F6(\033)p Gg(-fragment)h(of)e(intuitionistic)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28743
b(logic)c(\(we)321 1084 y(de\002ned)24 b(translations)h(between)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28744
(simply-typed)h(lambda)f(terms)e(and)h(proofs)h(in)e(this)h(fragment,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28745
(and)321 1197 y(sho)n(wed)k(that)f(beta-reduction)k(and)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28746
(cut-elimination)j(can)d(simulate)h(each)f(other\).)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28747
b(Consequently)-6 b(,)321 1310 y(we)36 b(kno)n(w)h(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28748
(cut-elimination)k(captures)e(features)g(of)d(sequential)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28749
(computation.)71 b(The)37 b(main)321 1423 y(result)d(of)e(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28750
(preceding)i(section)f(w)o(as)e(that)g(cut-elimination)k(in)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28751
(classical)h(logic)f(captures)i(also)321 1535 y(features)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28752
b(of)e(non-deterministic)31 b(computation.)39 b(In)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28753
b(this)h(section)g(we)f(shall)h(combine)g(both)g(results)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28754
1648 y(by)e(e)o(xtending)h(the)e(simply-typed)k(lambda)d(calculus)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28755
(with)e(an)g(erratic)h(choice)h(operator)g(and)f(sho)n(w)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28756
1761 y(an)h(analogous)i(result)e(to)f(that)h(gi)n(v)o(en)g(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28757
(Proposition)j(3.2.4.)34 b(As)24 b(will)h(be)g(seen)h(shortly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28758
b(,)28 b(this)d(result)321 1874 y(requires)h(classical)g(logic:)k(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28759
24 b(erratic)h(choice)g(operator)h(cannot)f(be)e(represented)k(in)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28760
(intuitionistic)321 1987 y(logic.)462 2118 y(Hereafter)l(,)f(we)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28761
(shall)i(refer)f(to)g(the)f(simply-typed)k(lambda)e(calculus)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28762
(xtended)h(with)d(an)h(erratic)321 2231 y(choice)27 b(operator)g(as)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28763
Ga(\025)1063 2245 y F9(+)1122 2231 y Gg(-calculus,)j(or)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28764
Ga(\025)1651 2245 y F9(+)1735 2231 y Gg(for)g(short.)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28765
b(Intuiti)n(v)o(ely)-6 b(,)28 b(a)c(term)h(of)g Ga(\025)2929
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28766
2245 y F9(+)3012 2231 y Gg(corresponds)k(to)321 2344
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28767
y(a)36 b(sequential)i(program,)i(which)c(e)o(x)o(ecuted)h(may)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28768
(produce)j(dif)n(ferent)f(results)h(\(see)e(for)g(e)o(xample)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28769
2457 y([Hennessy)26 b(and)e(Ashcroft,)g(1980]\).)31 b(The)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28770
b(ra)o(w)f(terms)i(of)f Ga(\025)2237 2471 y F9(+)2318
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28771
2457 y Gg(are)h(gi)n(v)o(en)g(by)g(the)g(grammar)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28772
Black 1179 2689 a Ga(M)5 b(;)15 b(N)110 b F4(::=)100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28773
b Ga(x)429 b Gg(v)n(ariable)1543 2802 y F6(j)148 b Ga(\025x:M)263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28774
b Gg(abstraction)1543 2914 y F6(j)148 b Ga(M)25 b(N)295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28775
b Gg(application)1543 3027 y F6(j)148 b Ga(M)30 b F4(+)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28776
b Ga(N)199 b Gg(erratic)25 b(choice)321 3257 y(As)e(usual,)h(we)f(are)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28777
(interested)i(in)d(only)i(the)f(set)f(of)h(well-typed)h(terms)f(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28778
Ga(\025)2743 3271 y F9(+)2802 3257 y Gg(;)f(that)h(is)f(in)h(the)f(set)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28779
p Black Black 499 3516 a F4(\003)562 3530 y F9(+)720
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28780
3464 y F5(def)728 3515 y F4(=)905 3414 y FK(n)991 3516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28781
y Ga(M)1136 3411 y FK(\014)1136 3465 y(\014)1136 3520
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28782
y(\014)1214 3516 y F4(\000)1296 3504 y Gc(.)1351 3516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28783
y Ga(M)k F4(:)17 b Ga(B)28 b Gg(is)23 b(deri)n(v)n(able)i(using)g(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28784
(rules)g(gi)n(v)o(en)g(in)g(Figure)g(4.6)3308 3414 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28785
FK(o)321 3770 y Gg(There)35 b(are)g(tw)o(o)g(reduction)i(rules)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28786
(associated)h(with)e Ga(\025)2126 3784 y F9(+)2185 3770
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28787
y Gg(:)51 b(one)35 b(is)g(the)g(beta-reduction)k(rule)c(\(see)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28788
3883 y(P)o(age)23 b(72\))h(and)g(the)g(other)h(is)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28789
Black 1427 4093 a Ga(M)30 b F4(+)20 b Ga(N)1880 4044
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28790
y FV(+)1818 4093 y F6(\000)-31 b(\000)g(!)99 b Ga(M)58
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28791
b Gg(or)49 b Ga(N)321 4303 y Gg(The)26 b(reduction)k(system)d(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28792
Ga(\025)1290 4317 y F9(+)1375 4303 y Gg(is)h(thus)g(de\002ned)g(as)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28793
F4(\(\003)2132 4317 y F9(+)2191 4303 y Ga(;)2272 4266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28794
y Gc(\014)2315 4254 y FV(+)2232 4303 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28795
F4(\))p Gg(,)27 b(where)2775 4266 y Gc(\014)2818 4254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28796
y FV(+)2735 4303 y F6(\000)-32 b(\000)h(!)26 b Gg(consists)i(of)3409
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28797
4266 y Gc(\014)3346 4303 y F6(\000)-31 b(\000)f(!)p Gg(-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28798
4416 y(reductions)29 b(and)942 4367 y FV(+)880 4416 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28799
F6(\000)-31 b(\000)g(!)p Gg(-reductions,)28 b(which)e(both)g(are)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28800
(assumed)h(to)e(be)g(closed)i(under)g(term)e(forma-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28801
4529 y(tion.)462 4660 y(Recall)20 b(Lafont')-5 b(s)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28802
b(e)o(xample)g(gi)n(v)o(en)f(on)f(P)o(age)g(4,)h(and)g(consider)i(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28803
(term)f Ga(M)d F4(+)5 b Ga(N)10 b Gg(.)26 b(The)19 b(dynamic)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28804
4773 y(beha)n(viour)24 b(of)d(this)g(term)g(corresponds)j(e)o(xactly)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28805
(to)d(the)h(beha)n(viour)j(of)d(cut-elimination)k(in)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28806
b(Lafont')-5 b(s)321 4886 y(e)o(xample.)30 b(Thus)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28807
b(the)h(translation)j F6(j)p 1451 4886 28 4 v 1469 4886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28808
V 1486 4886 V 65 w(j)1539 4853 y Fu(s)p FV(+)1642 4886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28809
y F4(:)e(\003)1755 4900 y F9(+)1840 4886 y F6(!)g FY(T)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28810
Gg(is)g(as)h(follo)n(ws.)p Black Black 890 5147 a F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28811
Ga(x)p F6(j)992 5114 y Fu(s)p FV(+)992 5169 y Gc(c)1169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28812
5095 y F5(def)1176 5147 y F4(=)107 b FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28813
Ga(x;)15 b(c)p F4(\))713 5287 y F6(j)p Ga(\025x:M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28814
b F6(j)991 5254 y Fu(s)p FV(+)991 5309 y Gc(c)1169 5235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28815
y F5(def)1176 5287 y F4(=)107 b FL(Imp)1498 5309 y Gc(R)1556
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28816
5287 y F4(\()1591 5275 y F9(\()1619 5287 y Ga(x)1671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28817
5275 y F9(\))p FX(h)1726 5287 y Ga(a)1774 5275 y FX(i)1816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28818
5287 y F6(j)p Ga(M)10 b F6(j)1964 5254 y Fu(s)p FV(+)1964
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28819
5309 y Gc(a)2043 5287 y Ga(;)15 b(c)p F4(\))735 5426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28820
y F6(j)p Ga(M)36 b(N)10 b F6(j)992 5393 y Fu(s)p FV(+)992
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28821
5449 y Gc(c)1169 5375 y F5(def)1176 5426 y F4(=)107 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28822
F6(j)p Ga(M)10 b F6(j)1502 5393 y Fu(s)p FV(+)1502 5449
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28823
y Gc(a)1580 5426 y F4([)p Ga(a)26 b F4(:=)1800 5414 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28824
F9(\()1827 5426 y Ga(y)1875 5414 y F9(\))1903 5426 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28825
FL(Imp)2047 5448 y Gc(L)2099 5426 y F4(\()2134 5414 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28826
FX(h)2162 5426 y Ga(b)2201 5414 y FX(i)2229 5426 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28827
Ga(N)10 b F6(j)2362 5388 y Fu(s)p FV(+)2362 5456 y Gc(b)2440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28828
5426 y Ga(;)2480 5414 y F9(\()2508 5426 y Ga(z)2554 5414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28829
y F9(\))2581 5426 y FL(Ax)p F4(\()p Ga(z)t(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28830
F4(\))r Ga(;)g(y)s F4(\)])3124 5393 y F9(\(1\))649 5566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28831
y F6(j)p Ga(M)31 b F4(+)20 b Ga(N)10 b F6(j)992 5533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28832
y Fu(s)p FV(+)992 5589 y Gc(c)1169 5515 y F5(def)1176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28833
5566 y F4(=)107 b FL(Cut)p F4(\()1527 5554 y FX(h)1555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28834
5566 y Ga(a)1603 5554 y FX(i)1630 5566 y F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28835
b F6(j)1778 5533 y Fu(s)p FV(+)1778 5589 y Gc(c)1856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28836
5566 y Ga(;)1896 5554 y F9(\()1924 5566 y Ga(x)1976 5554
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28837
y F9(\))2004 5566 y F6(j)p Ga(N)g F6(j)2137 5533 y Fu(s)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28838
FV(+)2137 5589 y Gc(c)2215 5566 y F4(\))3124 5533 y F9(\(2\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28839
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28840
%%Page: 121 133
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28841
TeXDict begin 121 132 bop Black 277 51 a Gb(4.3)23 b(A)g(Simple,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28842
(Non-Deterministic)j(Pr)n(ogramming)f(Language)1327 b(121)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28843
277 88 3691 4 v Black Black 277 300 3226 4 v 277 1198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28844
4 898 v 1439 392 548 4 v 1439 471 a Ga(x)17 b F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28845
Ga(B)5 b(;)15 b F4(\000)1746 459 y Gc(.)1801 471 y Ga(x)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28846
F4(:)h Ga(B)746 662 y(x)g F4(:)f Ga(B)5 b(;)15 b F4(\000)1054
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28847
650 y Gc(.)1109 662 y Ga(M)27 b F4(:)18 b Ga(C)p 722
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28848
699 641 4 v 722 779 a F4(\000)804 767 y Gc(.)859 779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28849
y Ga(\025x:M)27 b F4(:)18 b Ga(B)5 b F6(\033)o Ga(C)1404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28850
730 y Gg(Abs)1905 677 y F4(\000)1987 665 y Gc(.)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28851
677 y Ga(M)27 b F4(:)18 b Ga(B)5 b F6(\033)o Ga(C)97
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28852
b F4(\000)2588 665 y Gc(.)2643 677 y Ga(N)27 b F4(:)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28853
b Ga(B)p 1905 700 955 4 v 2150 779 a F4(\000)2232 767
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28854
y Gc(.)2287 779 y Ga(M)25 b(N)i F4(:)18 b Ga(C)2901 721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28855
y Gg(App)1427 993 y F4(\000)1510 981 y Gc(.)1564 993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28856
y Ga(M)28 b F4(:)17 b Ga(B)96 b F4(\000)1969 981 y Gc(.)2024
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28857
993 y Ga(N)27 b F4(:)17 b Ga(B)p 1427 1013 813 4 v 1552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28858
1092 a F4(\000)1634 1080 y Gc(.)1689 1092 y Ga(M)31 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28859
F4(+)20 b Ga(N)27 b F4(:)17 b Ga(B)2281 1036 y F4(+)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28860
3499 1198 4 898 v 277 1201 3226 4 v Black 1306 1369 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28861
Gg(Figure)24 b(4.6:)29 b(T)-7 b(yping)24 b(rules)g(for)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28862
Ga(\025)2392 1332 y F9(+)2451 1369 y Gg(.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28863
277 1789 a(where)j(in)g F4(\(1\))h Ga(y)34 b F6(62)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28864
Ga(F)13 b(N)d F4(\()p F6(j)p Ga(N)g F6(j)1259 1751 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28865
Fu(s)p FV(+)1259 1818 y Gc(b)1338 1789 y F4(\))p Gg(,)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28866
b(and)g(in)g F4(\(2\))h Ga(a)j F6(62)h Ga(F)13 b(C)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28867
b F4(\()p F6(j)p Ga(M)j F6(j)2318 1756 y Fu(s)p FV(+)2318
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28868
1811 y Gc(a)2396 1789 y F4(\))27 b Gg(and)g Ga(x)32 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28869
F6(62)f Ga(F)13 b(N)d F4(\()p F6(j)p Ga(M)g F6(j)3128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28870
1756 y Fu(s)p FV(+)3128 1811 y Gc(a)3207 1789 y F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28871
Gg(.)39 b(Ne)o(xt,)277 1902 y(we)23 b(w)o(ould)h(lik)o(e)g(to)g(sho)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28872
(w)f(that)h(this)g(reduction)i(respects)g(typing)f(judgements.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28873
Black 277 2089 a Gb(Pr)n(oposition)g(4.3.1:)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28874
2202 a Gg(If)37 b Ga(M)60 b F6(2)49 b F4(\003)695 2216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28875
y F9(+)790 2202 y Gg(with)37 b(the)g(typing)h(judgement)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28876
F4(\000)1905 2190 y Gc(.)1960 2202 y Ga(M)27 b F4(:)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28877
b Ga(B)t Gg(,)39 b(then)f F6(j)p Ga(M)10 b F6(j)2594
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28878
2169 y Fu(s)p FV(+)2594 2225 y Gc(a)2722 2202 y F6(2)49
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28879
b FY(T)36 b Gg(with)h(the)g(typing)277 2315 y(judgement)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28880
b F4(\000)766 2303 y Gc(.)821 2315 y F6(j)p Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28881
F6(j)969 2282 y Fu(s)p FV(+)969 2338 y Gc(a)1072 2303
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28882
y(.)1127 2315 y Ga(a)17 b F4(:)h Ga(B)t Gg(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28883
277 2553 a F7(Pr)l(oof.)p Black 46 w Gg(By)23 b(routine)i(induction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28884
(on)c(the)h(structure)i(of)d Ga(M)10 b Gg(.)p 3436 2553
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28885
4 62 v 3440 2495 55 4 v 3440 2553 V 3494 2553 4 62 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28886
277 2760 a(Finally)-6 b(,)24 b(we)f(can)h(pro)o(v)o(e)g(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28887
(cut-elimination)j(in)d FY(T)f Gg(can)g(simulate)i(normalisation)i(in)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28888
Ga(\025)3138 2774 y F9(+)3197 2760 y Gg(.)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28889
2947 a Gb(Theor)n(em)h(4.3.2:)p Black Black Black 504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28890
3166 a Gg(If)g Ga(M)5 b(;)15 b(N)35 b F6(2)25 b F4(\003)978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28891
3180 y F9(+)1060 3166 y Gg(and)f Ga(M)1378 3129 y Gc(\014)1421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28892
3117 y FV(+)1337 3166 y F6(\000)-31 b(\000)g(!)25 b Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28893
b Gg(,)22 b(then)j(for)e(all)h(co-names)h Ga(a)p Gg(,)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28894
F6(j)p Ga(M)10 b F6(j)2696 3133 y Fu(s)p FV(+)2696 3188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28895
y Gc(a)2830 3129 y(cut)2799 3166 y F6(\000)-31 b(\000)g(!)2969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28896
3133 y F9(+)3054 3166 y F6(j)p Ga(N)10 b F6(j)3187 3133
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28897
y Fu(s)p FV(+)3187 3188 y Gc(a)3265 3166 y Gg(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28898
277 3403 a F7(Pr)l(oof.)p Black 46 w Gg(Analogous)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28899
b(to)d(proof)i(of)e(Proposition)j(3.2.4.)p 3436 3403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28900
V 3440 3345 55 4 v 3440 3403 V 3494 3403 4 62 v 418 3610
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28901
a(On)k(P)o(age)h(14)g(we)f(introduced)k(three)e(criteria)g(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28902
(designing)j(a)c(cut-elimination)35 b(procedure.)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28903
3723 y(Let)23 b(us)h(analyse)h(these)g(criteria)g(with)e(respect)i(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28904
(the)g(simulation)h(result)g(just)f(gi)n(v)o(en.)29 b(In)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28905
b(particular)l(,)277 3836 y(let)k(us)g(analyse)h(whether)g(we)e(w)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28906
(ould)h(ha)n(v)o(e,)h(gi)n(v)o(en)f(analogous)i(translations,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28907
(obtained)e(the)e(simu-)277 3949 y(lation)d(result)g(with)f(an)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28908
(xisting)h(cut-elimination)j(procedure)e(for)e(classical)i(logic.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28909
b(W)-7 b(e)23 b(analyse)i(in)277 4062 y(turn)f(three)h(procedures)h(by)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28910
e(Gentzen)h([1935],)f(Dragalin)h([1988])g(and)f(Danos)g(et)f(al.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28911
([1997].)p Black 414 4293 a F6(\017)p Black 45 w Gg(Gentzen')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28912
b(s)39 b(original)f(cut-elimination)j(procedure)e(is)d(de\002ned)i(as)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28913
(innermost)i(reductions)504 4406 y(strate)o(gy)-6 b(.)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28914
b(Consequently)-6 b(,)28 b(we)c(w)o(ould)i(ha)n(v)o(e)g(obtained)h(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28915
e(weak)o(er)h(result)g(saying)g(that)g(cut-)504 4519
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28916
y(elimination)38 b(can)d(simulate)i(only)e(a)g(reduction)j(strate)o(gy)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28917
e(in)f Ga(\025)2582 4533 y F9(+)2641 4519 y Gg(.)63 b(Ho)n(we)n(v)o(er)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28918
l(,)37 b(as)e(soon)h(as)504 4632 y(we)25 b(impose)h(an)f(innermost)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28919
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28920
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28921
4744 y(forms;)d(by)f(this)h(we)e(mean)h(that)g(gi)n(v)o(en)g(a)g(term)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28922
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28923
y Gg(some)d(of)g(its)g(normal)g(forms)g(are)g(no)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28924
4857 y(longer)j(reachable.)p Black 414 5046 a F6(\017)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28925
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28926
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28927
5159 y(hope)g(that)g(this)g(procedure)i(is)d(able)g(to)g(simulate)i(e)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28928
(v)o(ery)e(reduction)j(in)d Ga(\025)2880 5173 y F9(+)2939
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28929
5159 y Gg(;)j(unfortunately)-6 b(,)504 5272 y(Dragalin)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28930
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28931
(permuted)h(with)e(other)i(cuts.)504 5385 y(Ho)n(we)n(v)o(er)l(,)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28932
b(this,)h(as)g(remark)o(ed)h(in)e(Section)i(3.2,)e(is)g(necessary)j(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28933
e(simulate)h(beta-reduction.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28934
%%Page: 122 134
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28936
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28937
3691 4 v Black Black 458 388 a F6(\017)p Black 46 w Gg(The)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28938
(cut-elimination)32 b(procedure)e(de)n(v)o(eloped)g(by)e(Danos)g(et.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28939
(al.)f(for)h(LK)2950 355 y Gc(tq)3040 388 y Gg(allo)n(ws)g(cuts)h(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28940
549 501 y(be)20 b(permuted)i(with)e(other)h(cuts)g(and)f(is)g(strongly)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28941
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28942
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28943
b(commuting)j(cuts)e(are)g(reduced.)30 b(Notice,)21 b(ho)n(w-)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28944
727 y(e)n(v)o(er)l(,)k(that)h(there)g(is)g(no)f(possibility)j(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28945
(predetermining)i(the)e(beha)n(viour)i(of)d(a)g(term)g(with)h(an)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28946
840 y(erratic)e(choice.)31 b(So)22 b(also)j(in)e(this)h(case)g(an)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28947
(analogous)i(simulation)g(result)f(w)o(ould)f(f)o(ail.)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28948
1019 y(As)29 b(f)o(ar)g(as)g(I)g(ha)n(v)o(e)h(been)g(able)g(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28949
(ascertain,)k(no)c(other)h(cut-elimination)j(procedure)f(for)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28950
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28951
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28952
1262 y(Clearly)-6 b(,)36 b(the)c(fragment)i(of)e(classical)i(logic)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28953
(into)f(which)f(we)g(translated)j F4(\003)2955 1276 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28954
F9(+)3045 1262 y Gg(is)e(v)o(ery)f(weak:)321 1374 y(no)e(ne)n(w)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28955
(sequents)k(are)e(deri)n(v)n(able)h(in)e(comparison)j(with)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28956
(intuitionistic)k(logic.)47 b(Important)31 b(future)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28957
1487 y(w)o(ork)h(is)f(to)g(e)o(xtend)i(suitably)h(the)d(programming)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28958
(language)g Ga(\025)2400 1501 y F9(+)2489 1487 y Gg(so)e(that)g(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28959
(simulation)i(result)g(is)321 1600 y(obtained)e(for)e(lar)n(ger)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28960
(fragments)g(of)e(classical)j(logic.)45 b(Good)28 b(candidates)k(for)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28961
(e)o(xtensions)k(are)c(the)321 1713 y(control)j(operators)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28962
F6(A)d Gg(and)h F6(C)5 b Gg(,)30 b(which)f(were)g(studied)h(pre)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28963
(viously)i(in)d(the)g(conte)o(xt)h(of)f Ga(\025\026)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28964
Gg(\(see)h(for)321 1826 y(e)o(xample)c([Bierman,)e(1998]\).)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28965
2127 y Ge(4.4)119 b(Notes)321 2351 y Gg(In)19 b(order)g(to)f(sho)n(w)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28966
(that)h(the)g(tw)o(o)f(normal)h(forms)f(gi)n(v)o(en)h(in)f(Appendix)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28967
(A)e(dif)n(fer)h(in)f(essential)j(features)321 2463 y(we)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28968
b(had)i(to)f(e)o(xtract)h(some)f(kind)h(of)f(data)h(from)e(them.)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28969
b(Kreisel)27 b([1958,)i(P)o(age)e(163])h(demonstrated)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28970
2576 y(that)f(e)o(xtracting)i(data)f(from)e(classical)j(proofs)f(is)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28971
F7(not)h Gg(possible)i(in)e(general.)39 b(But)26 b(it)h(is)f(kno)n(wn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28972
(that)321 2689 y(data)c(can)f(al)o(w)o(ays)h(be)e(e)o(xtracted)j(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28973
(certain)h(fragments)h(of)e(classical)i(logic,)f(as)e(sho)n(wn)h(for)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28974
(instance)321 2802 y(by)h(Friedman)g([1978].)30 b(Our)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28975
b(e)o(xample)i(f)o(alls)h(into)f(such)g(a)f(fragment)h(of)f(classical)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28976
(logic,)e(and)g(so)f(the)321 2915 y(non-determinism,)k(which)e(we)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28977
(inferred)j(from)e(the)g(e)o(xistence)i(of)e(tw)o(o)f(dif)n(ferent)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28978
(normal)f(forms,)f(is)321 3028 y(independent)33 b(from)c(the)g(result)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28979
(by)f(Kreisel.)46 b(Let)28 b(us)h(conclude)i(this)f(chapter)g(with)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28980
(sho)n(wing)h(that)321 3141 y(the)24 b(colour)h(annotation)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28981
(introduced)f(in)e(LK)1744 3108 y Gc(tq)1829 3141 y Gg(completely)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28982
(hides)e(this)h(non-determinism.)321 3398 y Fq(4.4.1)99
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28983
b(A)25 b(Beautiful)h(Theory)g(f)n(or)e(the)i(Colours)f(in)g(LK)2392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28984
3362 y Fy(tq)2466 3398 y Fq(,)g(Slain)g(by)g(A)g(F)n(act)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28985
3589 y Gg(The)c(cut-elimination)k(procedure)e(in)e(LK)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28986
3556 y Gc(tq)1738 3589 y Gg(has)h(three)g(remarkable)h(properties:)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28987
b(it)21 b(is)f(strongly)k(nor)n(-)321 3702 y(malising,)g(con\003uent,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28988
(and)f(there)h(is)f(a)f(strong)j(connection)g(to)e(linear)h(logic)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28989
(since)g(there)g(is)e(a)h(simple)321 3815 y(translation)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28990
b(between)e(LK)1192 3782 y Gc(tq)1255 3815 y Gg(-proofs)h(and)e(proof)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28991
(nets.)54 b(Furthermore,)36 b(cut-elimination)g(in)c(LK)3484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28992
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28993
(as)e(cut-elimination)k(in)c(proof)h(nets)f([Danos)h(et)e(al.,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28994
(1997].)321 4041 y(P)o(articularly)j(interesting)h(is)d(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28995
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28996
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28997
4154 y(tenc)o(y\224)h(described)i(in)d(the)g(introduction.)67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 28998
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 28999
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29000
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 29001
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 29002
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 29003
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 29004
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 29005
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 29006
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 29007
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 29008
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 29009
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Christian Urban <urbanc@in.tum.de>
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diff changeset
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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diff changeset
 29053
get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29054
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29055
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29056
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29057
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29058
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29059
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29060
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29061
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29062
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29063
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29064
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29065
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29066
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29067
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29068
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29069
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29070
(LaserWriter 16/600)]{A length product length le{A length product exch 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29071
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29072
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29073
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29074
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29075
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29076
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29077
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29078
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29079
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29080
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29081
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29082
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29083
%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29084
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29085
% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29086
% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29087
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29088
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29089
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29090
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29091
/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29092
/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29093
/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29094
/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29095
/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29096
/L /lineto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29097
/T /translate load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29098
/TMatrix { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29099
/RAngle { 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29100
/Atan { /atan load stopped { pop pop 0 } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29101
/Div { dup 0 eq { pop } { div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29102
/NET { neg exch neg exch T } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29103
/Pyth { dup mul exch dup mul add sqrt } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29104
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29105
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29106
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29107
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29108
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29109
/pathforall load stopped { pop pop pop pop } if z } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29110
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29111
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29112
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29113
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29114
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29115
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29116
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29117
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29118
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29119
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29120
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29121
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29122
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29123
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29124
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29125
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29126
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29127
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29128
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29129
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29130
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29131
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29132
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29133
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29134
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29135
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29136
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29137
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29138
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29139
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29140
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29141
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29142
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29143
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29144
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29145
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29146
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29147
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29148
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29149
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29150
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29151
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29152
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29153
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29154
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29155
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29156
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29157
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29158
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29159
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29160
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29161
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29162
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29163
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29164
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29165
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29166
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29167
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29168
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29169
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29170
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29171
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29172
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29173
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29174
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29175
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29176
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29177
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29178
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29179
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29180
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29181
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29182
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29183
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29184
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29185
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29186
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29187
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29188
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29189
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29190
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29191
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29192
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29193
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29194
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29195
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29196
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29197
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29198
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29199
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29200
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29201
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29202
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29203
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29204
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29205
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29206
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29207
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29208
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29209
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29210
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29211
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29212
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29213
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29214
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29215
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29216
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29217
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29218
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29219
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29220
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29221
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29222
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29223
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29224
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29225
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29226
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29227
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29228
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29229
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29230
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29231
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29232
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29233
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29234
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29235
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29236
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29237
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29238
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29239
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29240
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29241
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29242
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29243
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29244
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29245
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29246
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29247
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29248
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29249
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29250
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29251
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29252
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29253
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29254
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29255
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29256
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29257
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29258
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29259
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29260
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29261
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29262
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29263
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29264
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29265
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29266
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29267
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29268
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29269
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29270
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29271
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29272
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29273
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29274
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29275
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29276
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29277
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29278
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29279
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29280
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29281
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29282
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29283
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29284
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29285
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29286
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29287
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29288
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29289
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29290
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29291
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29292
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29293
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29294
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29295
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29296
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29297
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29298
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29299
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29300
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29301
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29302
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29303
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29304
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29305
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29306
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29307
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29308
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29309
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29310
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29311
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29312
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29313
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29314
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29315
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29316
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29317
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29318
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29319
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29320
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29321
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29322
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29323
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29324
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29325
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29326
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29327
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29328
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29329
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29330
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29331
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29332
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29333
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29334
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29335
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29336
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29337
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29338
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29339
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29340
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29341
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29342
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29343
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29344
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29345
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29346
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29347
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29348
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29349
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29350
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29351
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29352
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29353
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29354
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29355
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29356
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29357
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29358
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29359
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29360
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29361
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29362
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29363
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29364
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29365
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29366
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29367
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29368
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29369
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29370
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29371
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29372
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29373
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29374
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29375
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29376
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29377
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29378
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29379
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29380
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29381
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29382
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29383
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29384
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29385
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29386
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29387
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29388
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29389
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29390
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29391
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29392
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29393
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29394
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29395
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29396
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29397
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29398
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29399
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29400
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29401
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29402
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29403
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29404
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29405
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29406
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29407
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29408
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29409
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29410
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29411
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29412
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29413
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29414
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29415
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29416
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29417
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29418
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29419
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29420
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29421
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29422
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29423
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29424
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29425
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29426
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29427
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29428
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29429
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29430
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29431
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29432
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29433
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29434
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29435
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29436
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29437
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29438
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29439
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29440
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29441
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29442
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29443
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29444
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29445
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29446
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29447
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29448
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29449
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29450
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29451
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29452
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29453
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29454
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29455
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29456
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29457
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29458
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29459
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29460
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29461
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29462
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29463
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29464
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29465
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29466
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29467
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29468
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29469
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29470
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29471
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29472
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29473
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29474
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29475
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29476
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29477
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29478
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29479
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29480
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29481
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29482
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29483
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29484
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29485
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29486
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29487
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29488
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29489
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29490
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29491
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29492
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29493
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29494
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29495
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29496
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29497
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29498
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29499
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29500
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29501
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29502
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29503
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29504
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29505
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29506
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29507
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29508
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29509
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29510
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29511
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29512
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29513
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29514
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29515
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29516
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29517
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29518
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29519
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29520
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29521
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29522
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29523
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29524
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29525
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29526
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29527
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29528
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29529
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29530
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29531
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29532
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29533
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29534
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29535
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29536
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29537
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29538
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29539
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29540
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29541
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29542
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29543
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29544
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29545
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29546
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29547
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29548
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29549
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29550
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29551
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29552
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29553
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29554
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29555
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29556
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29557
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29558
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29559
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29560
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29561
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29562
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29563
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29564
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29565
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29566
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29567
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29568
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29569
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29570
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29571
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29572
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29573
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29574
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29575
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29576
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29577
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29578
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29579
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29580
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29581
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29582
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29583
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29584
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29585
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29586
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29587
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29588
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29589
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29590
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29591
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29592
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29593
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29594
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29595
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29596
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29597
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29598
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29599
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29600
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29601
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29602
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29603
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29604
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29605
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29606
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29607
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29608
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29609
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29610
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29611
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29612
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29613
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29614
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29615
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29616
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29617
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29618
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29619
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29620
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29621
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29622
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29623
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29624
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29625
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29626
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29627
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29628
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29629
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29630
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29631
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29632
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29634
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29635
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29636
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29637
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29638
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29639
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29640
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29641
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29642
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29643
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29644
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29645
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29646
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29647
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29648
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29649
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29650
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29651
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29652
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29653
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29654
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29655
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29656
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29657
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29659
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29661
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29662
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29663
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29664
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29666
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29668
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29669
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29670
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29671
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29672
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29673
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29674
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29675
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29676
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29677
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29678
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29679
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29680
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29681
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29682
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29683
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29684
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29685
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29686
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29687
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29688
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29689
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29690
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29691
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29692
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29693
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29694
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29695
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29696
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29697
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29698
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29699
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29700
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29701
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29702
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29703
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29705
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29706
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29707
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29708
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29709
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29710
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29711
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29712
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29713
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29714
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29715
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29716
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29717
    Char end stringwidth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29718
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29719
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29720
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29721
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29722
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29723
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29724
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29725
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29726
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29727
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29728
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29729
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29730
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29731
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29732
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29734
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29735
%%BeginProcSet: special.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29736
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29737
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29738
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29739
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29740
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29741
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29742
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29743
/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29744
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29745
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29746
maxlength ge{/md md dup length 20 add dict copy def}if end md begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29747
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29748
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29749
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29750
transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29751
curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 29752
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30067
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
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%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
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%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30072
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30073
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30074
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30075
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30078
/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30079
/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30080
/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30081
/L /lineto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30082
/T /translate load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30083
/TMatrix { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30084
/RAngle { 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30085
/Atan { /atan load stopped { pop pop 0 } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30086
/Div { dup 0 eq { pop } { div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30087
/NET { neg exch neg exch T } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30088
/Pyth { dup mul exch dup mul add sqrt } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30089
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30090
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30091
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30092
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30093
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30094
/pathforall load stopped { pop pop pop pop } if z } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30095
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30096
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30097
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30098
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30099
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30100
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30101
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30102
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30103
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30104
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30105
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30106
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30107
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30108
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30109
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30110
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30111
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30112
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30113
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30114
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30115
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30116
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30117
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30118
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30119
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30120
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30121
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30122
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30123
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30124
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30125
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30126
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30127
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30128
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30129
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30130
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30131
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30132
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30133
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30134
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30135
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30136
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30137
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30138
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30139
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30140
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30141
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30142
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30143
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30144
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30145
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30146
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30147
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30148
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30149
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30150
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30151
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30152
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30153
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30154
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30155
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30156
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30157
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30158
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30159
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30160
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30161
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30162
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30163
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30164
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30165
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30166
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30167
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30168
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30169
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30170
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30171
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30172
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30173
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30174
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30175
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30176
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30177
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30178
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30179
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30180
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30181
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30182
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30183
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30184
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30185
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30186
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30187
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30188
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30189
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30190
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30191
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30192
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30193
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30194
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30195
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30196
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30197
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30198
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30199
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30200
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30201
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30202
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30203
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30204
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30205
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30206
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30207
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30208
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30209
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30210
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30211
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30212
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30213
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30214
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30215
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30216
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30217
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30218
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30219
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30220
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30221
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30222
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30223
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30224
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30225
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30226
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30227
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30228
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30229
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30230
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30231
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30232
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30233
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30234
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30235
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30236
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30237
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30238
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30239
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30240
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30241
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30242
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30243
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30244
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30245
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30246
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30247
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30248
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30249
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30250
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30251
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30252
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30253
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30254
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30255
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30256
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30257
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30258
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30259
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30260
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30261
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30262
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30263
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30264
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30265
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30266
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30267
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30268
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30269
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30270
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30271
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30272
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30273
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30274
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30275
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30276
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30277
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30278
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30279
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30280
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30281
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30282
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30283
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30284
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30285
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30286
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30287
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30288
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30289
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30290
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30291
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30292
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30293
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30295
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30296
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30297
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30298
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30299
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30300
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30301
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30302
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30303
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30304
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30305
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30306
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30307
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30308
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30309
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30310
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30311
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30312
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30313
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30314
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30315
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30316
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30317
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30318
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30319
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30320
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30321
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30322
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30323
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30324
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30325
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30326
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30327
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30328
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30329
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30330
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30331
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30332
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30333
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30334
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30335
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30336
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30337
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30338
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30339
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30340
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30341
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30342
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30343
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30344
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30345
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30346
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30347
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30348
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30349
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30350
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30351
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30352
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30353
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30354
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30355
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30356
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30357
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30358
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30359
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30360
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30361
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30362
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30363
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30364
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30365
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30366
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30367
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30368
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30369
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30370
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30371
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30372
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30373
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30374
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30375
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30376
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30377
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30378
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30379
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30380
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30381
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30382
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30383
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30384
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30385
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30386
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30387
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30388
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30389
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30390
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30391
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30392
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30393
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30394
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30395
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30396
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30397
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30398
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30399
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30400
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30401
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30402
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30403
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30404
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30405
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30406
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30407
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30408
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30409
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30410
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30411
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30412
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30413
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30414
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30415
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30416
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30417
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30418
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30419
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30420
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30421
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30422
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30423
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30424
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30425
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30426
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30427
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30428
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30429
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30430
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30431
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30432
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30433
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30434
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30435
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30436
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30437
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30438
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30439
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30440
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30441
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30442
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30443
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30444
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30445
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30446
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30447
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30448
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30449
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30450
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30451
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30452
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30453
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30454
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30455
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30456
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30457
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30458
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30459
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30460
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30461
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30462
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30463
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30464
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30465
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30466
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30467
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30468
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30469
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30470
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30471
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30472
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30473
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30474
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30475
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30476
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30477
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30478
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30479
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30480
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30481
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30482
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30483
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30484
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30485
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30486
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30487
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30488
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30489
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30490
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30491
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30492
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30493
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30494
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30495
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30496
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30497
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30498
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30499
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30500
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30501
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30502
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30503
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30504
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30505
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30506
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30507
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30508
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30509
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30510
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30511
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30512
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30513
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30514
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30515
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30516
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30517
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30518
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30519
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30520
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30521
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30522
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30523
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30524
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30525
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30526
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30527
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30528
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30529
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30530
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30531
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30532
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30533
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30534
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30535
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30536
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30537
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30538
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30539
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30540
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30541
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30542
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30543
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30544
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30545
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30546
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30547
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30548
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30549
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30550
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30551
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30552
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30553
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30554
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30555
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30556
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30557
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30558
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30559
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30560
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30561
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30562
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30563
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30564
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30565
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30566
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30567
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30568
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30569
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30570
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30571
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30572
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30573
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30574
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30575
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30576
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30577
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30578
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30579
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30580
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30581
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30582
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30583
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30584
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30585
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30586
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30587
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30588
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30589
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30590
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30591
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30592
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30593
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30594
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30595
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30596
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30597
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30598
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30599
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30600
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30601
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30602
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30603
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30604
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30605
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30606
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30607
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30608
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30609
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30610
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30611
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30612
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30613
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30614
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30615
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30616
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30617
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30618
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30619
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30620
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30621
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30622
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30623
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30624
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30625
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30626
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30627
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30628
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30629
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30630
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30631
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30632
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30633
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30634
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30635
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30636
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30637
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30638
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30639
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30640
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30641
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30642
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30643
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30644
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30645
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30646
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30647
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30648
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30649
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30650
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30651
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30652
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30653
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30654
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30655
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30656
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30657
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30658
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30659
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30660
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30661
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30662
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30663
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30664
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30665
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30666
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30667
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30668
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30669
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30670
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30671
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30672
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30673
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30674
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30675
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30676
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30677
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30678
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30679
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30680
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30681
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30682
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30683
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30684
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30685
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30686
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30687
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30688
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30689
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30690
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30691
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30692
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30693
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30694
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30695
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30696
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30697
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30698
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30699
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30700
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30701
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30702
    Char end stringwidth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30703
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30704
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30705
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30706
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30707
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30708
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30709
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30710
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30711
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30712
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30713
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30714
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30715
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30716
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30717
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30718
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30719
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30720
%%BeginProcSet: special.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30721
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30722
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30723
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30724
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30725
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30726
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30727
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30728
/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30729
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30730
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30731
maxlength ge{/md md dup length 20 add dict copy def}if end md begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30732
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30733
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30734
itransform lineto}}{6 -2 roll transform 6 -2 roll transform 6 -2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30735
transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30736
curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30737
pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30738
if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30739
-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30740
get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30741
yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30742
neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30743
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30744
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30745
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30746
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30747
2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30748
-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30749
TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30750
Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30751
}if 0 setgray}N/psfts{S 65781.76 div N}N/startTexFig{/psf$SavedState
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30752
save N userdict maxlength dict begin/magscale true def normalscale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30753
currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30754
/psf$y psfts/psf$x psfts currentpoint/psf$cy X/psf$cx X/psf$sx psf$x
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30755
psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30756
psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30757
TR/showpage{}N/erasepage{}N/copypage{}N/p 3 def @MacSetUp}N/doclip{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30758
psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30759
roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30760
moveto}N/endTexFig{end psf$SavedState restore}N/@beginspecial{SDict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30761
begin/SpecialSave save N gsave normalscale currentpoint TR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30762
@SpecialDefaults count/ocount X/dcount countdictstack N}N/@setspecial{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30763
CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30764
closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30765
sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30766
}{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse
Christian Urban <urbanc@in.tum.de>
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diff changeset
 30922
1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30923
0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30924
sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30925
rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30926
gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30927
/chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30928
/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30929
A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30930
get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30931
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30932
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30933
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30934
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30935
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30936
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30937
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30938
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30939
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30940
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30941
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30942
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30943
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30944
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30945
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30946
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30947
(LaserWriter 16/600)]{A length product length le{A length product exch 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30948
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30949
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30950
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30951
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30952
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30953
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30954
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30955
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30956
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30957
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30959
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30960
%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30961
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30962
% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30963
% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30964
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30965
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30966
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30967
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30968
/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30969
/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30970
/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30971
/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30972
/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30973
/L /lineto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30974
/T /translate load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30975
/TMatrix { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30976
/RAngle { 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30977
/Atan { /atan load stopped { pop pop 0 } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30978
/Div { dup 0 eq { pop } { div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30979
/NET { neg exch neg exch T } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30980
/Pyth { dup mul exch dup mul add sqrt } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30981
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30982
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30983
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30984
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30985
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30986
/pathforall load stopped { pop pop pop pop } if z } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30987
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30988
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30989
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30990
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30991
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30992
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30993
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30994
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30995
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30996
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30997
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30998
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 30999
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31000
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31001
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31002
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31003
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31004
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31005
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31006
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31007
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31008
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31009
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31010
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31011
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31012
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31013
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31014
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31015
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31016
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31017
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31018
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31019
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31020
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31021
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31022
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31023
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31024
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31025
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31026
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31027
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31028
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31029
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31030
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31031
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31032
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31033
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31034
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31035
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31036
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31037
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31038
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31039
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31040
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31041
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31042
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31043
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31044
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31045
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31046
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31047
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31048
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31049
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31050
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31051
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31052
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31053
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31054
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31055
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31056
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31057
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31058
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31059
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31060
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31061
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31062
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31063
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31064
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31065
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31066
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31067
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31068
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31069
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31070
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31071
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31072
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31073
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31074
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31075
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31076
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31077
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31078
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31079
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31080
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31081
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31082
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31083
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31084
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31085
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31086
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31087
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31088
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31089
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31090
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31091
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31092
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31093
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31094
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31095
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31096
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31097
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31098
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31099
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31100
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31101
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31102
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31103
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31104
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31105
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31106
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31107
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31108
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31109
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31110
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31111
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31112
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31113
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31114
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31115
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31116
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31117
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31118
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31119
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31120
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31121
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31122
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31123
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31124
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31125
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31126
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31127
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31128
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31129
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31130
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31131
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31132
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31133
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31134
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31135
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31136
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31137
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31138
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31139
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31140
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31141
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31142
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31143
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31144
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31145
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31146
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31147
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31148
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31149
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31150
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31151
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31152
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31153
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31154
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31155
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31156
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31157
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31158
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31159
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31160
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31161
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31162
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31163
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31164
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31165
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31166
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31167
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31168
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31169
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31170
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31171
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31172
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31173
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31174
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31175
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31176
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31177
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31178
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31179
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31180
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31181
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31182
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31183
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31184
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31185
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31187
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31188
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31189
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31190
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31191
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31192
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31193
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31194
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31195
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31196
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31197
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31198
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31199
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31200
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31201
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31202
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31203
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31204
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31205
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31206
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31207
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31208
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31209
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31210
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31211
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31212
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31213
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31214
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31215
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31216
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31217
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31218
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31219
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31220
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31221
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31222
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31223
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31224
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31225
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31226
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31227
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31228
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31229
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31230
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31231
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31232
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31233
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31234
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31235
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31236
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31237
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31238
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31239
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31240
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31241
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31242
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31243
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31244
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31245
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31246
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31247
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31248
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31249
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31250
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31251
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31252
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31253
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31254
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31255
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31256
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31257
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31258
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31259
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31260
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31261
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31262
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31263
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31264
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31265
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31266
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31267
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31268
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31269
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31270
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31271
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31272
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31273
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31274
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31275
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31276
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31277
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31278
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31279
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31280
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31281
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31282
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31283
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31284
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31285
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31286
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31287
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31288
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31289
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31290
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31291
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31292
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31293
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31294
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31295
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31296
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31297
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31298
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31299
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31300
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31301
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31302
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31303
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31304
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31305
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31306
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31307
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31308
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31309
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31310
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31311
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31312
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31313
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31314
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31315
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31316
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31317
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31318
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31319
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31320
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31321
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31322
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31323
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31324
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31325
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31326
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31327
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31328
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31329
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31330
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31331
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31332
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31333
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31334
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31335
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31336
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31337
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31338
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31339
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31340
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31341
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31342
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31343
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31344
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31345
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31346
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31347
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31348
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31349
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31350
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31351
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31352
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31353
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31354
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31355
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31356
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31357
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31358
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31359
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31360
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31361
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31362
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31363
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31364
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31365
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31366
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31367
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31368
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31369
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31370
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31371
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31372
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31373
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31374
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31375
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31376
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31377
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31378
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31379
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31380
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31381
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31382
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31383
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31384
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31385
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31386
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31387
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31388
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31389
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31390
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31391
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31392
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31393
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31394
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31395
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31396
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31397
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31398
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31399
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31400
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31401
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31402
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31403
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31404
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31405
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31406
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31407
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31408
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31409
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31410
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31411
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31412
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31413
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31414
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31415
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31416
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31417
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31418
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31419
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31420
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31421
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31422
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31423
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31424
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31425
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31426
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31427
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31428
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31429
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31430
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31431
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31432
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31433
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31434
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31435
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31436
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31437
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31438
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31439
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31440
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31441
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31442
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31443
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31444
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31445
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31446
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31447
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31448
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31449
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31450
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31451
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31452
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31453
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31454
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31455
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31456
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31457
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31458
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31459
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31460
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31461
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31462
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31463
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31464
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31465
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31466
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31467
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31468
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31469
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31470
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31471
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31472
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31473
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31474
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31475
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31476
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31477
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31478
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31479
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31480
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31481
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31482
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31483
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31484
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31485
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31486
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31487
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31488
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31489
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31490
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31491
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31492
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31493
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31494
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31495
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31496
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31497
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31498
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31499
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31500
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31501
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31502
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31503
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31504
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31505
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31506
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31507
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31508
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31509
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31511
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31512
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31513
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31514
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31515
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31516
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31517
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31518
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31519
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31520
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31521
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31522
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31523
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31524
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31525
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31526
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31527
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31528
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31529
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31530
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31531
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31532
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31533
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31534
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31536
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31537
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31538
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31539
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31540
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31541
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31542
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31543
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31544
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31545
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31546
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31547
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31548
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31549
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31550
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31551
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31552
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31553
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31554
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31555
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31556
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31557
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31558
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31559
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31560
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31561
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31562
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31563
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31564
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31565
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31566
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31567
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31568
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31569
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31570
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31571
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31572
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31573
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31574
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31575
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31576
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31577
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31578
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31579
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31580
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31582
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31583
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31584
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31585
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31586
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31587
  translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31588
  Angle rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31589
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31590
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31591
/ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31592
{ /Char exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31593
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31594
    Char end stringwidth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31595
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31596
    2 div /Sy exch def 2 div /Sx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31597
    currentpoint
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31598
    Voffset sub Sy add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31599
    Hoffset sub Sx add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31600
    Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31601
    Sx neg Sy neg moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31602
    Char end tx@TextPathSavedShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31603
    tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31604
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31605
  Sx 2 mul Sy 2 mul rmoveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31606
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31608
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31609
% END pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31610
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31611
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31612
%%BeginProcSet: special.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31613
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31614
TeXDict begin/SDict 200 dict N SDict begin/@SpecialDefaults{/hs 612 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31615
/vs 792 N/ho 0 N/vo 0 N/hsc 1 N/vsc 1 N/ang 0 N/CLIP 0 N/rwiSeen false N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31616
/rhiSeen false N/letter{}N/note{}N/a4{}N/legal{}N}B/@scaleunit 100 N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31617
/@hscale{@scaleunit div/hsc X}B/@vscale{@scaleunit div/vsc X}B/@hsize{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31618
/hs X/CLIP 1 N}B/@vsize{/vs X/CLIP 1 N}B/@clip{/CLIP 2 N}B/@hoffset{/ho
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31619
X}B/@voffset{/vo X}B/@angle{/ang X}B/@rwi{10 div/rwi X/rwiSeen true N}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31620
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31777
%DVIPSWebPage: (www.radicaleye.com)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31778
%DVIPSCommandLine: dvips -o sigma.ps sigma.dvi
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31779
%DVIPSParameters: dpi=600, compressed
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31780
%DVIPSSource:  TeX output 2000.03.10:0010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31781
%%BeginProcSet: texc.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31782
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31783
/TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31784
N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31785
mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31786
0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31787
landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31788
mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31789
matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31790
exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31791
statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31792
N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31793
/FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31794
/BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31795
array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31796
df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31797
definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31798
}B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31799
B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31800
1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31801
1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31802
0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31803
sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31804
rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31805
gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31806
/chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31807
/cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31808
A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31809
get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31810
ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31811
fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31812
{2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31813
chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31814
1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31815
forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31816
/BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31817
}if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31818
bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31819
mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31820
SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31821
userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31822
1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31823
index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31824
/p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31825
/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31826
(LaserWriter 16/600)]{A length product length le{A length product exch 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31827
exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31828
end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31829
grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31830
imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31831
exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31832
fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31833
delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31834
B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31835
p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31836
rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31838
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31839
%%BeginProcSet: pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31840
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31841
% PostScript prologue for pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31842
% Version 97 patch 3, 98/06/01
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31843
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31844
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31845
/tx@Dict 200 dict def tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31846
/ADict 25 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31847
/CM { matrix currentmatrix } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31848
/SLW /setlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31849
/CLW /currentlinewidth load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31850
/CP /currentpoint load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31851
/ED { exch def } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31852
/L /lineto load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31853
/T /translate load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31854
/TMatrix { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31855
/RAngle { 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31856
/Atan { /atan load stopped { pop pop 0 } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31857
/Div { dup 0 eq { pop } { div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31858
/NET { neg exch neg exch T } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31859
/Pyth { dup mul exch dup mul add sqrt } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31860
/PtoC { 2 copy cos mul 3 1 roll sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31861
/PathLength@ { /z z y y1 sub x x1 sub Pyth add def /y1 y def /x1 x def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31862
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31863
/PathLength { flattenpath /z 0 def { /y1 ED /x1 ED /y2 y1 def /x2 x1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31864
} { /y ED /x ED PathLength@ } {} { /y y2 def /x x2 def PathLength@ }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31865
/pathforall load stopped { pop pop pop pop } if z } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31866
/STP { .996264 dup scale } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31867
/STV { SDict begin normalscale end STP  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31868
/DashLine { dup 0 gt { /a .5 def PathLength exch div } { pop /a 1 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31869
PathLength } ifelse /b ED /x ED /y ED /z y x add def b a .5 sub 2 mul y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31870
mul sub z Div round z mul a .5 sub 2 mul y mul add b exch Div dup y mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31871
/y ED x mul /x ED x 0 gt y 0 gt and { [ y x ] 1 a sub y mul } { [ 1 0 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31872
0 } ifelse setdash stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31873
/DotLine { /b PathLength def /a ED /z ED /y CLW def /z y z add def a 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31874
{ /b b a div def } { a 0 eq { /b b y sub def } { a -3 eq { /b b y add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31875
def } if } ifelse } ifelse [ 0 b b z Div round Div dup 0 le { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31876
] a 0 gt { 0 } { y 2 div a -2 gt { neg } if } ifelse setdash 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31877
setlinecap stroke } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31878
/LineFill { gsave abs CLW add /a ED a 0 dtransform round exch round exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31879
2 copy idtransform exch Atan rotate idtransform pop /a ED .25 .25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31880
% DG/SR modification begin - Dec. 12, 1997 - Patch 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31881
%itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31882
itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31883
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31884
Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31885
/setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31886
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31887
% a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31888
% def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31889
a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31890
pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31891
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31892
/BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31893
exch 3 index sub exch Atan rotate newpath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31894
/EndArrow { @mtrx setmatrix CP grestore end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31895
/Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31896
scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31897
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31898
/Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31899
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31900
/Bracket { CLW mul add dup CLW sub 2 div /x ED mul CLW add /y ED /z CLW 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31901
div def x neg y moveto x neg CLW 2 div L x CLW 2 div L x y L stroke 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31902
CLW moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31903
/RoundBracket { CLW mul add dup 2 div /x ED mul /y ED /mtrx CM def 0 CLW
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31904
2 div T x y mul 0 ne { x y scale } if 1 1 moveto .85 .5 .35 0 0 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31905
curveto -.35 0 -.85 .5 -1 1 curveto mtrx setmatrix stroke 0 CLW moveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31906
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31907
/SD { 0 360 arc fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31908
/EndDot { { /z DS def } { /z 0 def } ifelse /b ED 0 z DS SD b { 0 z DS
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31909
CLW sub SD } if 0 DS z add CLW 4 div sub moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31910
/Shadow { [ { /moveto load } { /lineto load } { /curveto load } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31911
/closepath load } /pathforall load stopped { pop pop pop pop CP /moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31912
load } if ] cvx newpath 3 1 roll T exec } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31913
/NArray { aload length 2 div dup dup cvi eq not { exch pop } if /n exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31914
cvi def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31915
/NArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31916
f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31917
/Line { NArray n 0 eq not { n 1 eq { 0 0 /n 2 def } if ArrowA /n n 2 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31918
def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31919
/Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31920
ifelse { pop } repeat a } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31921
/CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31922
and { pop pop /n n 1 sub def } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31923
/Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31924
repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31925
ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31926
moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31927
Lineto pop pop closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31928
/Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31929
neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31930
def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31931
div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31932
setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31933
% DG modification begin - Jan. 15, 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31934
%/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31935
%pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31936
%div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31937
%d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31938
%setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31939
/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31940
CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31941
div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31942
d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31943
% DG/SR modification begin - Jun.  1, 1998 - Patch 3 (from Michael Vulis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31944
% setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31945
setmatrix pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31946
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31947
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31948
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31949
/CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31950
def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31951
/CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31952
def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31953
mul dy1 l0 c exp mul add def /m dx0 dy0 Atan dx1 dy1 Atan sub 2 div cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31954
abs b exp a mul dx dy Pyth Div 2 div def /x2 x l0 dx mul m mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31955
/y2 y l0 dy mul m mul sub def /dx l1 dx mul m mul neg def /dy l1 dy mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31956
m mul neg def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31957
/IC { /c c 1 add def c 0 lt { /c 0 def } { c 3 gt { /c 3 def } if }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31958
ifelse /a a 2 mul 3 div 45 cos b exp div def CCA /dx 0 def /dy 0 def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31959
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31960
/BOC { IC CC x2 y2 x1 y1 ArrowA CP 4 2 roll x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31961
/NC { CC x1 y1 x2 y2 x y curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31962
/EOC { x dx sub y dy sub 4 2 roll ArrowB 2 copy curveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31963
/BAC { IC CC x y moveto CC x1 y1 CP ArrowA } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31964
/NAC { x2 y2 x y curveto CC x1 y1 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31965
/EAC { x2 y2 x y ArrowB curveto pop pop } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31966
/OpenCurve { NArray n 3 lt { n { pop pop } repeat } { BOC /n n 3 sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31967
n { NC } repeat EOC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31968
/AltCurve { { false NArray n 2 mul 2 roll [ n 2 mul 3 sub 1 roll ] aload
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31969
/Points ED n 2 mul -2 roll } { false NArray } ifelse n 4 lt { n { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31970
pop } repeat } { BAC /n n 4 sub def n { NAC } repeat EAC } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31971
/ClosedCurve { NArray n 3 lt { n { pop pop } repeat } { n 3 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31972
CheckClosed } if 6 copy n 2 mul 6 add 6 roll IC CC x y moveto n { NC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31973
repeat closepath pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31974
/SQ { /r ED r r moveto r r neg L r neg r neg L r neg r L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31975
/ST { /y ED /x ED x y moveto x neg y L 0 x L fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31976
/SP { /r ED gsave 0 r moveto 4 { 72 rotate 0 r L } repeat fill grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31977
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31978
/FontDot { DS 2 mul dup matrix scale matrix concatmatrix exch matrix
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31979
rotate matrix concatmatrix exch findfont exch makefont setfont } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31980
/Rect { x1 y1 y2 add 2 div moveto x1 y2 lineto x2 y2 lineto x2 y1 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31981
x1 y1 lineto closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31982
/OvalFrame { x1 x2 eq y1 y2 eq or { pop pop x1 y1 moveto x2 y2 L } { y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31983
y2 sub abs x1 x2 sub abs 2 copy gt { exch pop } { pop } ifelse 2 div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31984
exch { dup 3 1 roll mul exch } if 2 copy lt { pop } { exch pop } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31985
/b ED x1 y1 y2 add 2 div moveto x1 y2 x2 y2 b arcto x2 y2 x2 y1 b arcto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31986
x2 y1 x1 y1 b arcto x1 y1 x1 y2 b arcto 16 { pop } repeat closepath }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31987
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31988
/Frame { CLW mul /a ED 3 -1 roll 2 copy gt { exch } if a sub /y2 ED a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31989
/y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31990
pop pop Rect } { OvalFrame } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31991
/BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31992
} if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31993
aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31994
/OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31995
2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31996
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31997
/ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31998
6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 31999
/BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32000
sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32001
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32002
/Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32003
sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32004
x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32005
y0 x0 2 mul x1 sub y1 ] def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32006
/Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32007
if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32008
/dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32009
round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32010
round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32011
0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32012
/Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32013
if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32014
{ /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32015
exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32016
div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32017
dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32018
dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32019
pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32020
stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg}
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32021
{z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32022
w add def } repeat grestore gsave n 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32023
% DG/SR modification begin - Nov. 7, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32024
%{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32025
{ 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32026
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32027
{ 2 setlinecap } ifelse /i y1 def /f x1 dx mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32028
n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32029
div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32030
1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32031
dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32032
{z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32033
exch t L stroke /i i h add def } repeat grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32034
/ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32035
1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32036
exch r a PtoC y add exch x add exch b pop pop pop pop a e d CLW 8 div c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32037
mul neg d } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32038
/Ellipse { /mtrx CM def T scale 0 0 1 5 3 roll arc mtrx setmatrix } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32039
/Rot { CP CP translate 3 -1 roll neg rotate NET  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32040
/RotBegin { tx@Dict /TMatrix known not { /TMatrix { } def /RAngle { 0 }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32041
def } if /TMatrix [ TMatrix CM ] cvx def /a ED a Rot /RAngle [ RAngle
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32042
dup a add ] cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32043
/RotEnd { /TMatrix [ TMatrix setmatrix ] cvx def /RAngle [ RAngle pop ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32044
cvx def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32045
/PutCoor { gsave CP T CM STV exch exec moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32046
/PutBegin { /TMatrix [ TMatrix CM ] cvx def CP 4 2 roll T moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32047
/PutEnd { CP /TMatrix [ TMatrix setmatrix ] cvx def moveto } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32048
/Uput { /a ED add 2 div /h ED 2 div /w ED /s a sin def /c a cos def /b s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32049
abs c abs 2 copy gt dup /q ED { pop } { exch pop } ifelse def /w1 c b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32050
div w mul def /h1 s b div h mul def q { w1 abs w sub dup c mul abs } {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32051
h1 abs h sub dup s mul abs } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32052
/UUput { /z ED abs /y ED /x ED q { x s div c mul abs y gt } { x c div s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32053
mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32054
{ x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32055
exch } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32056
/BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32057
/IfVisible true def } if } { IfVisible { Invisible /IfVisible false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32058
} if } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32059
/InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32060
/Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32061
neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32062
/IfVisible true def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32063
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32064
% END pstricks.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32065
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32066
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32067
%%BeginProcSet: pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32068
%!PS-Adobe-2.0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32069
%%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32070
%%Creator: Timothy Van Zandt <tvz@Princeton.EDU>
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32071
%%Creation Date: May 7, 1993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32072
10 dict dup begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32073
  /FontType 3 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32074
  /FontMatrix [ .001 0 0 .001 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32075
  /FontBBox [ 0 0 0 0 ] def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32076
  /Encoding 256 array def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32077
  0 1 255 { Encoding exch /.notdef put } for
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32078
  Encoding
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32079
    dup (b) 0 get /Bullet put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32080
    dup (c) 0 get /Circle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32081
    dup (C) 0 get /BoldCircle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32082
    dup (u) 0 get /SolidTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32083
    dup (t) 0 get /Triangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32084
    dup (T) 0 get /BoldTriangle put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32085
    dup (r) 0 get /SolidSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32086
    dup (s) 0 get /Square put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32087
    dup (S) 0 get /BoldSquare put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32088
    dup (q) 0 get /SolidPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32089
    dup (p) 0 get /Pentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32090
    (P) 0 get /BoldPentagon put
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32091
  /Metrics 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32092
  Metrics begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32093
    /Bullet 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32094
    /Circle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32095
    /BoldCircle 1000 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32096
    /SolidTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32097
    /Triangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32098
    /BoldTriangle 1344 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32099
    /SolidSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32100
    /Square 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32101
    /BoldSquare 886 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32102
    /SolidPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32103
    /Pentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32104
    /BoldPentagon 1093.2 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32105
    /.notdef 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32106
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32107
  /BBoxes 13 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32108
  BBoxes begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32109
    /Circle { -550 -550 550 550 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32110
    /BoldCircle /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32111
    /Bullet /Circle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32112
    /Triangle { -571.5 -330 571.5 660 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32113
    /BoldTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32114
    /SolidTriangle /Triangle load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32115
    /Square { -450 -450 450 450 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32116
    /BoldSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32117
    /SolidSquare /Square load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32118
    /Pentagon { -546.6 -465 546.6 574.7 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32119
    /BoldPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32120
    /SolidPentagon /Pentagon load def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32121
    /.notdef { 0 0 0 0 } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32122
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32123
  /CharProcs 20 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32124
  CharProcs begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32125
    /Adjust {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32126
      2 copy dtransform floor .5 add exch floor .5 add exch idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32127
      3 -1 roll div 3 1 roll exch div exch scale
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32128
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32129
    /CirclePath { 0 0 500 0 360 arc closepath } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32130
    /Bullet { 500 500 Adjust CirclePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32131
    /Circle { 500 500 Adjust CirclePath .9 .9 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32132
    /BoldCircle { 500 500 Adjust CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32133
    /BoldCircle { CirclePath .8 .8 scale CirclePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32134
    /TrianglePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32135
      0  660 moveto -571.5 -330 lineto 571.5 -330 lineto closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32136
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32137
    /SolidTriangle { TrianglePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32138
    /Triangle { TrianglePath .85 .85 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32139
    /BoldTriangle { TrianglePath .7 .7 scale TrianglePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32140
    /SquarePath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32141
      -450 450 moveto 450 450 lineto 450 -450 lineto -450 -450 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32142
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32143
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32144
    /SolidSquare { SquarePath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32145
    /Square { SquarePath .89 .89 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32146
    /BoldSquare { SquarePath .78 .78 scale SquarePath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32147
    /PentagonPath {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32148
      -337.8 -465   moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32149
       337.8 -465   lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32150
       546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32151
         0    574.7 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32152
      -546.6  177.6 lineto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32153
      closepath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32154
    } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32155
    /SolidPentagon { PentagonPath fill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32156
    /Pentagon { PentagonPath .89 .89 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32157
    /BoldPentagon { PentagonPath .78 .78 scale PentagonPath eofill } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32158
    /.notdef { } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32159
  end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32160
  /BuildGlyph {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32161
    exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32162
    begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32163
      Metrics 1 index get exec 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32164
      BBoxes 3 index get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32165
      setcachedevice
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32166
      CharProcs begin load exec end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32167
    end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32168
  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32169
  /BuildChar {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32170
    1 index /Encoding get exch get
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32171
    1 index /BuildGlyph get exec
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32172
  } bind def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32173
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32174
/PSTricksDotFont exch definefont pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32175
% END pst-dots.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32177
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32178
%%BeginProcSet: pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32179
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32180
% PostScript prologue for pst-node.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32181
% Version 97 patch 1, 97/05/09.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32182
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32183
%
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32184
/tx@NodeDict 400 dict def tx@NodeDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32185
tx@Dict begin /T /translate load def end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32186
/NewNode { gsave /next ED dict dup 3 1 roll def exch { dup 3 1 roll def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32187
if begin tx@Dict begin STV CP T exec end /NodeMtrx CM def next end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32188
grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32189
/InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32190
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32191
/InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32192
Sin mul } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32193
/GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32194
} ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32195
ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32196
dup Sin mul Cos Div } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32197
/InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32198
/u ED /NodePos { GetRnodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32199
/DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32200
Cos mul exch Sin mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32201
/TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32202
mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32203
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32204
/InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32205
ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32206
sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32207
sin def /NodePos { TriNodePos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32208
/OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32209
hh mul Atan dup cos ww mul exch sin hh mul } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32210
/GetCenter { begin X Y NodeMtrx transform CM itransform end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32211
/XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32212
Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32213
{ 0 Dist Sin mul } ifelse } ifelse Do } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32214
/GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32215
exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32216
dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32217
XYPos } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32218
/AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32219
roll sin mul sub exch } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32220
/GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32221
AddOffset yA add /yA1 ED xA add /xA1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32222
/GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32223
AddOffset yB add /yB1 ED xB add /xB1 ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32224
/GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32225
AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32226
ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32227
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32228
/GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32229
AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32230
ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32231
} ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32232
/InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32233
/NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32234
known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32235
ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32236
/LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32237
mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32238
/LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32239
{ pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32240
} repeat LPutLine cleartomark } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32241
/BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32242
/t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32243
mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32244
sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32245
add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32246
y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32247
mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32248
/HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32249
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32250
/HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32251
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32252
/HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32253
/HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32254
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32255
} { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd 
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32256
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32257
/VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32258
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32259
/VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32260
/NAngle yyB yyA sub xxB xxA sub Atan def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32261
/VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32262
/VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32263
} ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32264
exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32265
VPosEnd  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32266
/HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32267
roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32268
def grestore exec /LPutVar /SaveLPutVar load def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32269
/NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32270
GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32271
LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32272
LPutVar VPutLine } def LPutVar } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32273
/NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32274
end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32275
/NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32276
Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32277
/AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32278
yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32279
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32280
ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32281
/NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32282
roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32283
GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32284
begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32285
cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32286
} HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32287
/NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32288
def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32289
itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32290
yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32291
yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32292
/HPutPos { HPutLines } def /VPutPos { VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32293
/NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32294
yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32295
/y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32296
tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32297
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32298
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32299
/NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32300
xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32301
transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32302
{ /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32303
0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32304
begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32305
def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32306
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32307
/NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32308
xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32309
/LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32310
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32311
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32312
/NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32313
GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32314
false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32315
LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32316
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32317
/NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32318
def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32319
mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32320
mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32321
xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32322
Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ]
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32323
cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32324
VPutLines } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32325
% DG/SR modification begin - May 9, 1997 - Patch 1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32326
%/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32327
%r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32328
%exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32329
%mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32330
/NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32331
r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32332
exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32333
mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32334
% DG/SR modification end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32335
} def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32336
AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32337
Div def /y ED /x ED } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32338
/NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32339
180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32340
mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32341
/LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32342
add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32343
{ xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32344
tx@Dict begin false Polygon end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32345
/NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32346
def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32347
mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32348
/x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32349
def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32350
GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32351
sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32352
eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32353
{ x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32354
0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32355
roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32356
0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32357
roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32358
ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32359
{ LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32360
r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32361
{ t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32362
add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32363
mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32364
sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32365
y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos {
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32366
LPutPos } def } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32367
/Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32368
sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32369
ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32370
w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32371
{ LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32372
{ pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32373
def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32374
/ArrowA { moveto } def /ArrowB { } def false Line closepath end } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32375
/LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32376
exch X sub neg exch moveto setmatrix CP grestore } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32377
/LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32378
def } ifelse LPutCoor  } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32379
/HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32380
if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32381
ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32382
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32383
/VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32384
r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32385
ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32386
def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32387
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32388
% END pst-node.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32389
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32390
%%EndProcSet
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32391
%%BeginProcSet: pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32392
%!
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32393
% PostScript header file pst-text.pro
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32394
% Version 97, 94/04/20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32395
% For distribution, see pstricks.tex.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32396
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32397
/tx@TextPathDict 40 dict def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32398
tx@TextPathDict begin
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32399
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32400
% Syntax:  <dist> PathPosition -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32401
% Function: Searches for position of currentpath distance <dist> from
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32402
%           beginning. Sets (X,Y)=position, and Angle=tangent.
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32403
/PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32404
{ /targetdist exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32405
  /pathdist 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32406
  /continue true def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32407
  /X { newx } def /Y { newy } def /Angle 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32408
  gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32409
    flattenpath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32410
    { movetoproc }  { linetoproc } { } { firstx firsty linetoproc }
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32411
    /pathforall load stopped { pop pop pop pop /X 0 def /Y 0 def } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32412
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32413
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32415
/movetoproc { continue { @movetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32416
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32417
/@movetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32418
{ /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32419
  /firstx newx def /firsty newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32420
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32422
/linetoproc { continue { @linetoproc } { pop pop } ifelse } def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32423
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32424
/@linetoproc
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32425
{
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32426
  /oldx newx def /oldy newy def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32427
  /newy exch def /newx exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32428
  /dx newx oldx sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32429
  /dy newy oldy sub def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32430
  /dist dx dup mul dy dup mul add sqrt def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32431
  /pathdist pathdist dist add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32432
  pathdist targetdist ge
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32433
  { pathdist targetdist sub dist div dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32434
    dy mul neg newy add /Y exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32435
    dx mul neg newx add /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32436
    /Angle dy dx atan def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32437
    /continue false def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32438
  } if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32439
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32441
/TextPathShow
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32442
{ /String exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32443
  /CharCount 0 def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32444
  String length
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32445
  { String CharCount 1 getinterval ShowChar
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32446
    /CharCount CharCount 1 add def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32447
  } repeat
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32448
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32449
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32450
% Syntax: <pathlength> <position> InitTextPath -
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32451
/InitTextPath
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32452
{ gsave
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32453
    currentpoint /Y exch def /X exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32454
    exch X Hoffset sub sub mul
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32455
    Voffset Hoffset sub add
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32456
    neg X add /Hoffset exch def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32457
    /Voffset Y def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32458
  grestore
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32459
} def
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32461
/Transform
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32462
{ PathPosition
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32463
  dup
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32464
  Angle cos mul Y add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32465
  Angle sin mul neg X add exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32466
  translate
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-5 b(s)28 b(e)o(xample)f(to)f(the)h(impli-)321 1760 y(cational)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(fragment)e(of)f(intuitionistic)k(logic,)e(then)f(this)g(fragment)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(can)g(simulate)g(reduction)i(in)d(the)321 1873 y(simply-typed)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(lambda)d(calculus)i(enriched)g(with)d(the)h(erratic)h(choice)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(operator)-5 b(.)462 2002 y(It)38 b(is)g(hoped)h(that)f(the)g(reader)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32769
(will)d(no)n(w)g(be)h(con)l(vinced)j(of)d(the)g(point)h(of)f(vie)n(w)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(that)i(non-)321 2115 y(determinism)31 b F7(is)e Gg(an)f(intrinsic)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(feature)g(of)e(classical)i(logic.)45 b(This)29 b(point)h(of)f(vie)n(w)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32772
f(w)o(as)h(\002rst)f(tak)o(en)321 2228 y(by)f(Barbanera)h(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(Berardi)h([1994],)i(and)d(later)h(analysed)i(from)d(a)g(programming)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(point)g(of)e(vie)n(w)321 2341 y(by)21 b(Barbanera)h(et)e(al.)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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([1997];)1282 2308 y F5(1)1346 2341 y Gg(it)h(w)o(as)g(also)h(hinted)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(at)f(by)h(Coquand)g([1995])h(and)f(Herbelin)g([1995],)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2454 y(and)i(more)f(recently)h(suggested)i(by)d(Girard)g([2001,)i(P)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(age)d(88])h(and)h(adv)n(ocated)i(by)d(Hyland)g([2000].)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32779
2761 y Ge(5.1)119 b(Further)31 b(W)-9 b(ork)321 2985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y Gg(There)25 b(are)f(man)o(y)g(areas)h(for)f(further)i(w)o(ork.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(W)-7 b(e)23 b(list)h(some)h(of)f(them)g(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(,)24 b(roughly)i(in)e(the)h(order)321 3098 y(as)f(the)o(y)g(appear)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(in)e(the)h(thesis.)462 3227 y(Pleasant)35 b(though)g(our)f(strong)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(normalisation)i(result)d(for)g(cut-elimination)j(is,)f(it)d(is)g(not)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(the)321 3340 y(strongest)f(result)e(we)e(could)i(hoped)h(for:)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(our)30 b(strong)i(normalisation)h(theorems)e(say)f(little)h(about)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3453 y(strong)c(normalisation)g(of)e(cut-elimination)k(in)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(second-order)29 b(classical)e(logic.)33 b(Clearly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(,)25 b(it)g(w)o(ould)321 3566 y(be)i(nice)g(to)f(ha)n(v)o(e)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(stronger)j(result:)36 b(for)26 b(e)o(xample)h(it)f(can)h(sa)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(lengthy)h(strong)f(normalisation)321 3679 y(proofs)h(from)e(\002rst)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(principles)j(for)d(other)h(term)f(re)n(write)g(systems)i(\(see)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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([Benton,)h(1995]\).)38 b(T)-7 b(o)25 b(gi)n(v)o(e)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3792 y(a)36 b(proof)g(of)g(the)g(stronger)i(result,)i(one)c(has)g(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(e)o(xtend)g(the)g(notion)i(of)d(symmetric)i(reducibility)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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3904 y(candidates)c(using)d(what)f(is)g(by)h(Gallier)g([1990])g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(referred)i(to)d(as)g(\223Girard')-5 b(s)31 b(trick\224.)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(W)-7 b(e)28 b(ha)n(v)o(e)i(not)321 4017 y(check)o(ed)f(whether)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(this)g(e)o(xtension)h(is)e(possible)i(gi)n(v)o(en)f(the)f(\002x)o(ed)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(point)h(operation)h(we)e(emplo)o(yed)321 4130 y(in)d(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(propositional)j(and)d(\002rst-order)h(case.)462 4260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y(There)h(are)g(se)n(v)o(eral)g(possibilities)j(of)c(ho)n(w)g(to)h(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(xtend)g(our)g(sequent)h(calculus)h(in)d(order)i(to)e(cap-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4373 y(ture,)f(for)g(e)o(xample,)g(features)h(of)f(recursi)n(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(computation.)32 b(One)23 b(is)g(to)h(add)g(the)f(induction)k(rule)1150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4566 y Ga(B)5 b F4([)p FL(x)25 b F4(:=)g FL(y)q F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Ga(;)15 b F4(\000)p 1623 4554 11 41 v 1634 4536 46 5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32808
v 97 w(\001)p Ga(;)g(B)5 b F4([)p FL(x)25 b F4(:=)h FL(y)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b F4(+)f FL(1)p F4(])p 1150 4608 1178 4 v 1232 4693 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Ga(B)5 b F4([)p FL(x)25 b F4(:=)h FL(0)p F4(])p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b F4(\000)p 1708 4681 11 41 v 1718 4663 46 5 v 97 w(\001)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32812
Ga(;)g(B)5 b F4([)p FL(x)25 b F4(:=)g FL(t)p F4(])2369
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4639 y Gg(Induction)321 4903 y(where)i FL(y)g Gg(is)g(an)f(eigen)l(v)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(ariable)31 b(not)c(occurring)i(else)n(where)f(in)f(the)f(premise,)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(and)e FL(t)e Gg(is)i(an)f(arbitrary)321 5016 y(e)o(xpression.)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(Another)30 b(is)f(to)g(allo)n(w)g(non-logical)j(axioms)e(of)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(form)p 2699 4937 246 4 v 29 w Ga(C)p 2790 5004 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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v 2801 4986 46 5 v 103 w(D)r Gg(.)45 b(In)28 b(general,)k(the)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5129 y(latter)21 b(means)e(that)h(some)f(cuts)h(cannot)h(be)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(eliminated.)29 b(T)-7 b(o)18 b(a)n(v)n(oid)j(this)f(problem)g(one)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(has)g(to)f(impose)p Black 321 5206 1290 4 v 427 5262
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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a F3(1)456 5294 y F2(The)k(main)g(technical)g(no)o(v)o(elty)g(in)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(this)g(thesis)f(is)h(that)f(we)h(sho)n(wed)h(the)f(non-determinism)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(using)f(a)g(standard)h(for)o(-)321 5385 y(mulation)c(of)f(classical)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(logic;)h(for)g(this)g(we)g(had)g(to)g(e)o(xtend)h(their)e(results)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(in)g(order)g(to)g(deal)g(with)g(all)f(connecti)n(v)o(es.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
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%%Page: 127 139
Christian Urban <urbanc@in.tum.de>
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TeXDict begin 127 138 bop Black 277 51 a Gb(5.1)23 b(Further)g(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
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b(ork)2867 b(127)p 277 88 3691 4 v Black 277 388 a Gg(suitable)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(constraints)h(on)e(the)f(form)g(of)g(the)g(non-logical)k(axioms.)52
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(Both)31 b(e)o(xtensions)j(ha)n(v)o(e)e(been)277 501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y(in)l(v)o(estigated)23 b(in)c(an)g(intuitionistic)k(setting)e(by)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(McDo)n(well)f(and)h(Miller)g([2000],)h(and)f(the)o(y)g(seem)f(not)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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614 y(to)g(pose)g(an)o(y)g(dif)n(\002culties)h(with)f(respect)h(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(our)g(strong)h(normalisation)h(proof)f(based)g(on)f(candidates.)418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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751 y(W)-7 b(e)34 b(designed)i(our)f(cut-elimination)j(procedures)f(so)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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d(that)h(restrictions)i(imposed)f(in)e(earlier)277 864
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y(cut-elimination)h(procedures)f(are)d(remo)o(v)o(ed,)i(b)n(ut)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(without)h(breaking)i(the)d(strong)h(normalisation)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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977 y(property)-6 b(.)46 b(F)o(or)27 b(e)o(xample,)j(we)e(sho)n(wed)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(that)g(the)g(colour)h(annotations)h(of)e(LK)2827 944
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y Gc(tq)2917 977 y Gg(are)f(not)h(required)277 1090 y(for)c(strong)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32844
(normalisation.)35 b(But,)24 b(we)f(k)o(ept)i(one)g(restriction)i(of)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32845
(LK)2465 1057 y Gc(tq)2528 1090 y Gg(:)30 b(commuting)c(cuts)f(can)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32846
(only)277 1202 y(be)j(permuted)i(into)e(one)h(direction)h(and)f(not)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32847
(into)h(alternating)i(ones.)43 b(One)27 b(side)i(ef)n(fect)g(of)f(this)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32848
g(re-)277 1315 y(striction)d(is)d(that)h(some)f(normal)h(forms)g(are)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32849
(not)f(reachable,)j(which)e(w)o(ould)g(ho)n(we)n(v)o(er)f(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(reachable)277 1428 y(using)28 b(a)e(completely)i(unrestricted)i(\(and)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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d(thus)g(not)g(strongly)h(normalising\))i(cut-elimination)g(pro-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32852
1541 y(cedure.)36 b(Consider)27 b(again)g(the)f(in\002nite)g(reduction)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32853
i(sequence)g(outlined)g(in)d(Example)h(2.1.3.)35 b(Start-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32854
1654 y(ing)30 b(from)f(the)g(\002rst)g(proof,)j(one)d(can)h(reach)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32855
(the)g(normal)g(form)f(gi)n(v)o(en)h(in)f(Figure)h(5.1,)g(b)n(ut)g(we)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(are)277 1767 y(not)33 b(a)o(w)o(are)f(of)g(an)o(y)g(strongly)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(normalising)f(cut-elimination)i(procedure)f(with)d(which)h(we)e(could)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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277 1880 y(reach)36 b(it.)581 1847 y F5(2)679 1880 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Gg(The)e(only)h(normal)g(forms)g(that)g(can)f(be)h(reached)h(using)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32860
(our)g(strongly)i(normalising)277 1993 y(cut-elimination)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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b(procedures)g(are)p Black Black 289 2193 213 4 v 289
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2267 a FU(A)p 369 2255 10 38 v 379 2238 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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584 2193 213 4 v 83 w(A)p 665 2255 10 38 v 674 2238 42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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4 v 88 w(A)p 289 2287 509 4 v 328 2360 a(A)p FT(_)q FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32865
526 2348 10 38 v 536 2331 42 4 v 88 w(A;)14 b(A)811 2303
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y FT(_)866 2316 y FS(L)p 328 2396 430 4 v 378 2469 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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FU(A)p FT(_)p FU(A)p 576 2457 10 38 v 586 2441 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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88 w(A)799 2418 y Gd(Contr)989 2430 y FS(R)p 1098 2193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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213 4 v 1098 2267 a FU(A)p 1178 2255 10 38 v 1188 2238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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42 4 v 88 w(A)p 1393 2193 213 4 v 83 w(A)p 1474 2255
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32871
10 38 v 1484 2238 42 4 v 89 w(A)p 1098 2287 509 4 v 1137
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32872
2360 a(A)p FT(_)q FU(A)p 1335 2348 10 38 v 1345 2331
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32873
42 4 v 88 w(A;)g(A)1620 2303 y FT(_)1675 2316 y FS(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32874
1137 2396 430 4 v 1187 2469 a FU(A)p FT(_)p FU(A)p 1385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32875
2457 10 38 v 1395 2441 42 4 v 89 w(A)1608 2418 y Gd(Contr)1798
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32876
2430 y FS(R)p 378 2489 1140 4 v 615 2563 a FU(A)p FT(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32877
FU(A;)g(A)p FT(_)p FU(A)p 1030 2551 10 38 v 1040 2534
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32878
42 4 v 89 w(A)p FT(^)p FU(A)1531 2506 y FT(^)1586 2519
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32879
y FS(R)p 615 2599 665 4 v 723 2672 a FU(A)p FT(_)q FU(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32880
922 2660 10 38 v 931 2643 42 4 v 88 w(A)p FT(^)q FU(A)1321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32881
2621 y Gd(Contr)1511 2633 y FS(L)p 1937 2193 213 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32882
1937 2267 a FU(A)p 2018 2255 10 38 v 2028 2238 42 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32883
89 w(A)p 2233 2193 213 4 v 83 w(A)p 2314 2255 10 38 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32884
2323 2238 42 4 v 88 w(A)p 1937 2287 509 4 v 1977 2360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32885
a(A;)g(A)p 2157 2348 10 38 v 2166 2331 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32886
FT(^)q FU(A)2459 2303 y FT(^)2514 2316 y FS(R)p 1977
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32887
2396 430 4 v 2026 2469 a FU(A)p 2107 2457 10 38 v 2117
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32888
2441 42 4 v 88 w(A)p FT(^)q FU(A)2447 2418 y Gd(Contr)2637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32889
2430 y FS(L)p 2742 2193 213 4 v 2742 2267 a FU(A)p 2822
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32890
2255 10 38 v 2832 2238 42 4 v 88 w(A)p 3037 2193 213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32891
4 v 83 w(A)p 3118 2255 10 38 v 3127 2238 42 4 v 88 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32892
2742 2287 509 4 v 2781 2360 a(A;)g(A)p 2961 2348 10 38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32893
v 2970 2331 42 4 v 88 w(A)p FT(^)q FU(A)3263 2303 y FT(^)3319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32894
2316 y FS(R)p 2781 2396 430 4 v 2831 2469 a FU(A)p 2911
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32895
2457 10 38 v 2921 2441 42 4 v 88 w(A)p FT(^)p FU(A)3252
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2418 y Gd(Contr)3442 2430 y FS(L)p 2026 2489 1135 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2261 2563 a FU(A)p FT(_)q FU(A)p 2460 2551 10 38 v 2469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32898
2534 42 4 v 88 w(A)p FT(^)q FU(A;)g(A)p FT(^)p FU(A)3175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32899
2506 y FT(_)3230 2519 y FS(L)p 2261 2599 665 4 v 2370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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2672 a FU(A)p FT(_)p FU(A)p 2568 2660 10 38 v 2577 2643
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32901
42 4 v 88 w(A)p FT(^)q FU(A)2967 2621 y Gd(Contr)3157
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32902
2633 y FS(R)277 2925 y Gg(W)-7 b(e)23 b(\002nd)h(it)f(is)h(highly)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32903
(desirable)g(to)e(ha)n(v)o(e)g(a)g(con)l(vincing)j(e)o(xplanation)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32904
(for)d(what)g(it)g(means)g(compu-)277 3038 y(tationally)i(that)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32905
(normal)g(form)f(gi)n(v)o(en)h(in)f(Figure)h(5.1)f(is)g(not)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(reachable)h(using)g(our)f(procedures.)277 3151 y(Because)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32907
b(there)f(are)g(some)f(similarities)j(between)e(the)g(tw)o(o)f(normal)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32908
(forms)g(gi)n(v)o(en)g(abo)o(v)o(e)g(and)g(the)277 3264
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32909
y(one)c(in)g(Figure)h(5.1,)f(an)f(appealing)k(e)o(xplanation)g(w)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(ould)d(be)g(that)h(using)g(our)f(cut-elimination)k(pro-)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32911
3376 y(cedure)f(only)f(\223essential\224)i(normal)e(forms)g(can)f(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32912
(reached.)39 b(Ho)n(we)n(v)o(er)25 b(at)h(present,)j(this)e(is)f(only)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32913
(a)277 3489 y(v)n(ague)e(hope.)418 3626 y(In)39 b(Section)h(4.1)f(we)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32914
(de)n(v)o(eloped)i(an)e(implementation)j(of)d(the)h(cut-elimination)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32915
(procedure)277 3739 y F4(\()p FY(T)t Ga(;)430 3702 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32916
Gc(aux)410 3739 y F6(\000)-31 b(\000)f(!)p F4(\))36 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32917
Gg(in)h(OCaml.)66 b(A)36 b(more)g(suitable)j(programming)f(language)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(for)e(this)g(kind)g(of)g(imple-)277 3852 y(mentations)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32919
b(has)f(been)g(recently)h(proposed)h(by)d(Pitts)g(and)g(Gabbay)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32920
([2000].)31 b(In)23 b(this)g(programming)277 3965 y(language,)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32921
b(named)e(FreshML,)f(one)h(does)g(not)g(ha)n(v)o(e)g(to)g(deal)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(xplicitly)i(with)d(alpha-con)l(v)o(ersions,)277 4078
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32923
y(and)j(thus)g(one)g(can)g(write)g(much)f(clearer)i(code)f(reminiscent)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32924
i(of)e(the)f(simple)i(de\002nitions)g(we)e(pre-)277 4191
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32925
y(sented)f(in)f(this)g(thesis.)36 b(As)25 b(soon)i(as)e(there)i(is)e(a)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32926
g(w)o(orking)i(prototype)i(a)n(v)n(ailable)e(for)f(FreshML,)f(we)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32927
4304 y(hope)g(to)e(pro)o(vide)i(a)e(simpli\002ed)i(implementation)h(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32928
e F4(\()p FY(T)t Ga(;)2164 4267 y Gc(aux)2143 4304 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32929
F6(\000)-31 b(\000)g(!)p F4(\))p Gg(.)418 4440 y(Grif)n(\002n)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32930
b([1990])j(noted)f(a)e(correspondence)39 b(between)d(classical)h(logic)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32931
f(and)f(functional)j(pro-)277 4553 y(gramming)21 b(languages)j(with)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32932
(control)i(operators.)30 b(This)20 b(observ)n(ation)k(has)c(been)i(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32933
(xploited)g(in)f(man)o(y)277 4666 y(w)o(orks,)34 b(for)e(e)o(xample)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32934
([Murthy,)g(1990,)g(Ber)n(ger)h(et)e(al.,)g(2000],)k(where)c(programs)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(are)d(e)o(xtracted)277 4779 y(from)h(classical)j(proofs)f(via)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32936
(double)i(ne)o(gation)g(translations.)59 b(An)31 b(interesting)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32937
b(twist)c(of)g(this)h(ap-)277 4892 y(proach)25 b(has)f(been)g(studied)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32938
(by)e(Coquand)i([1995])g(and)e(Herbelin)i([1995].)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32939
b(The)o(y)23 b(analyse)i(a)e(v)n(ariant)277 5005 y(of)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32940
b(the)g(classical)j(proof)e(gi)n(v)o(en)f(in)g(Section)h(4.2)f(and)h(e)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32941
o(xtract)g(tw)o(o)e(dif)n(ferent)j(programs)g(by)e(using)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32942
Black 277 5115 1290 4 v 383 5171 a F3(2)412 5202 y F2(The)e(strongly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32943
(normalising)g(cut-elimination)g(procedure)h(seems)f(not)g(to)f(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32944
(meaningful,)j(where)c(one)i(can)e(do)277 5294 y(\002nitely)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32945
b(man)o(y)g(steps)h(\223ho)n(we)n(v)o(er)g(one)g(w)o(ants\224,)g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32946
(then)f(annotate)h(colours)g(and)g(reduce)g(the)f(proof)g(according)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32947
(to)e(the)277 5385 y(cut-elimination)c(procedure)i(of)e(LK)1251
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32948
5353 y FZ(tq)1308 5385 y F2(.)p Black Black Black eop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32949
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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%%Page: 128 140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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TeXDict begin 128 139 bop Black -144 51 a Gb(128)3121
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32953
4 v 321 1438 4 1139 v 646 607 233 4 v 646 686 a Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32954
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32955
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Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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646 706 557 4 v 689 785 a(A;)15 b(A)p 886 773 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32957
896 754 46 5 v 97 w(A)p F6(^)p Ga(A)1243 724 y F6(^)1304
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32958
738 y Gc(R)p 689 822 471 4 v 743 901 a Ga(A)p 831 889
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32959
11 41 v 842 871 46 5 v 97 w(A)p F6(^)o Ga(A)1200 846
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32960
y Gg(Contr)1406 860 y Gc(L)p 1601 607 233 4 v 1601 686
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32961
a Ga(A)p 1689 674 11 41 v 1700 656 46 5 v 97 w(A)p 1925
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32962
392 233 4 v 1925 471 a(A)p 2013 459 11 41 v 2023 441
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32963
46 5 v 96 w(A)p 2248 392 233 4 v 91 w(A)p 2337 459 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32964
41 v 2347 441 46 5 v 97 w(A)p 1925 491 557 4 v 1968 570
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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a(A;)g(A)p 2165 558 11 41 v 2175 539 46 5 v 97 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32966
F6(^)p Ga(A)2522 509 y F6(^)2583 523 y Gc(R)p 1968 607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32967
471 4 v 2022 686 a Ga(A)p 2110 674 11 41 v 2121 656 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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5 v 97 w(A)p F6(^)o Ga(A)2479 631 y Gg(Contr)2685 645
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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y Gc(L)p 1601 706 783 4 v 1693 785 a Ga(A)p F6(_)p Ga(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32970
1910 773 11 41 v 1920 754 46 5 v 96 w(A;)g(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32971
Ga(A)2425 724 y F6(_)2486 738 y Gc(L)p 2829 706 233 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32972
v 2829 785 a Ga(A)p 2918 773 11 41 v 2928 754 46 5 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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97 w(A)p 1693 822 1370 4 v 1959 901 a(A;)g(A)p F6(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32974
Ga(A)p 2285 889 11 41 v 2296 871 46 5 v 96 w(A)p F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32975
Ga(A;)g(A)p F6(^)p Ga(A)3104 840 y F6(^)3164 855 y Gc(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32976
743 939 2053 4 v 1168 1017 a Ga(A)p F6(_)p Ga(A;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32977
F6(_)p Ga(A)p 1623 1005 11 41 v 1633 987 46 5 v 97 w(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32978
F6(^)p Ga(A;)g(A)p F6(^)p Ga(A;)g(A)p F6(^)p Ga(A)2837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32979
957 y F6(_)2898 971 y Gc(L)p 1168 1055 1203 4 v 1287
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32980
1134 a Ga(A)p F6(_)p Ga(A)p 1504 1122 11 41 v 1514 1103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32981
46 5 v 96 w(A)p F6(^)p Ga(A;)g(A)p F6(^)p Ga(A;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32982
F6(^)q Ga(A)2412 1079 y Gg(Contr)2618 1093 y Gc(L)p 1287
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32983
1171 966 4 v 1406 1250 a Ga(A)p F6(_)o Ga(A)p 1623 1238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32984
11 41 v 1633 1220 46 5 v 97 w(A)p F6(^)p Ga(A;)g(A)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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F6(^)p Ga(A)2293 1195 y Gg(Contr)2499 1209 y Gc(R)p 1406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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1288 728 4 v 1524 1366 a Ga(A)p F6(_)p Ga(A)p 1741 1354
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32987
11 41 v 1752 1336 46 5 v 96 w(A)p F6(^)p Ga(A)2175 1312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32988
y Gg(Contr)2381 1326 y Gc(R)p 3543 1438 4 1139 v 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32989
1441 3226 4 v 321 1595 a Gg(Figure)21 b(5.1:)28 b(A)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32990
b(normal)i(form)f(of)h(the)f(LK-proof)h(gi)n(v)o(en)g(in)f(Example)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
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(2.1.3,)g(P)o(age)f(15.)28 b(This)20 b(normal)321 1708
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32992
y(form)25 b(is)f(reachable)j(only)f(if)e(commuting)i(cuts)f(can)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32993
(\223change\224)i(the)e(direction)i(into)e(which)g(the)o(y)g(are)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32994
1821 y(permuted.)p Black 321 2226 a(tw)o(o)30 b(dif)n(ferent)i(double)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32995
(ne)o(gation)g(translations.)52 b(W)-7 b(e)29 b(conjecture)k(that)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32996
(double)h(ne)o(gation)g(transla-)321 2339 y(tions)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32997
b(correspond)i(to)d(particular)j(strate)o(gies)f(of)e(cut-elimination.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32998
32 b(This)19 b(conjecture)k(does)e(not)g(seem)321 2452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 32999
y(dif)n(\002cult)k(to)e(v)o(erify)-6 b(,)24 b(b)n(ut)g(the)g(details)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33000
(remain)g(future)f(w)o(ork.)462 2581 y(Important)36 b(future)f(w)o(ork)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33001
f(is)g(to)g(reconsider)j(the)d(proof)h(theory)h(of)e(classical)i(logic)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33002
f(in)f(a)f(set-)321 2694 y(ting)23 b(where)f(proofs)h(are)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33003
(identi\002ed)h(up)f(to)f F7(Kleene)i(permutations)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33004
Gg([Kleene,)f(1952b],)g(analogous)i(to)321 2807 y(proof)j(nets)f(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33005
(linear)h(logic)h([Girard,)e(1987a].)39 b(One)26 b(practical)i(reason)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33006
(for)f(this)f(identi\002cation)k(is)321 2920 y(the)d(huge)h(number)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33007
(of)f(normal)i(forms)f(we)e(found)j(by)f(feeding)h(the)f(classical)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33008
(proof)e(gi)n(v)o(en)g(in)g(Sec-)321 3033 y(tion)22 b(4.2)e(into)h(our)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33009
g(implementation.)31 b(W)-7 b(e)20 b(conjecture)j(man)o(y)e(of)f(these)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33010
i(normal)f(forms)g(are)g(equal)g(up)321 3146 y(to)i(Kleene)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33011
(permutations.)31 b(In)23 b(our)h(implementation)i(we)c(introduced)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33012
(some)d F7(ad)h(hoc)f Gg(assumptions,)321 3259 y(which)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33013
b(reduced)g(the)g(number)f(of)g(normal)h(forms)f(some)n(what,)h(b)n(ut)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33014
g(we)e(felt)h(that)g(we)f(did)i(not)f(ha)n(v)o(e)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33015
3372 y(much)f(control)h(o)o(v)o(er)e(the)h(cut-elimination)j(process.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33016
39 b(Identi\002cation)29 b(up)d(to)h(Kleene)g(permutations)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33017
3485 y(might)e(alle)n(viate)i(this)e(problem.)34 b(Unfortunately)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33018
b(,)28 b(syntactic)f(methods)f(\(e.g.)e(additional)k(reduction)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33019
3597 y(rules\))g(to)f(achie)n(v)o(e)g(this)h(identi\002cation)h(are)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33020
(rather)h(dif)n(\002cult)f(to)g(obtain)h([Schwichtenber)n(g,)i(1999].)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33021
321 3710 y(So)g(one)g(might)h(ha)n(v)o(e)g(to)f(resort)i(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33022
(semantical)i(methods,)h(as)d(proposed)j(by)d(Hyland)h([2000],)i(in-)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33023
321 3823 y(stead.)462 3953 y(The)d(most)f(interesting)k(future)d(w)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33024
(ork)g(concerns)i(the)e(computational)j(interpretation)g(of)d(clas-)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33025
4066 y(sical)h(proofs.)48 b(In)30 b(Section)g(4.3)g(we)f(ga)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33026
(an)h(encoding)i(into)e(classical)i(logic)f(of)e(a)h(simply-typed)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33027
4179 y(lambda)g(calculus)h(enriched)g(with)d(an)h(erratic)h(choice)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33028
(operator)-5 b(.)46 b(This)29 b(result)h(should)g(be)f(seen)h(as)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33029
4291 y(proof)d(of)e(concept:)36 b(it)25 b(barely)i(scratches)h(at)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33030
(the)h(surf)o(ace)h(of)f(what)f(can)h(be)g(encoded)h(into)f(classical)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33031
321 4404 y(logic.)k(F)o(or)22 b(e)o(xample,)h(we)f(did)h(not)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33032
(consider)h(an)d(encoding)k(of)c(a)h(\(weak\))g(form)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33033
(communication)321 4517 y(into)h(classical)i(logic)f(using)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33034
(quanti\002ers.)31 b(F)o(or)23 b(e)o(xample)h(the)g(cut-instance)1140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33035
4695 y Ga(:)15 b(:)g(:)p 1281 4683 11 41 v 1292 4664
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33036
46 5 v 128 w(:)g(:)g(:)h(;)f(B)5 b FL(a)p 1082 4732 628
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33037
4 v 1082 4812 a Ga(:)15 b(:)g(:)p 1223 4800 11 41 v 1234
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33038
4782 46 5 v 128 w(:)g(:)g(:)h(;)f F6(8)p FL(x)p Ga(:B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33039
b FL(x)1751 4757 y F6(8)1802 4771 y Gc(R)2064 4695 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33040
Ga(B)g FL(t)p Ga(;)15 b(:)g(:)g(:)p 2352 4683 11 41 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33041
2363 4664 46 5 v 127 w(:)g(:)g(:)p 2001 4732 613 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33042
2001 4812 a F6(8)p FL(x)p Ga(:B)5 b FL(x)p Ga(;)15 b(:)g(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33043
2415 4800 11 41 v 2426 4782 46 5 v 127 w(:)g(:)g(:)2654
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33044
4757 y F6(8)2705 4771 y Gc(L)p 1082 4850 1531 4 v 1678
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33045
4919 a Ga(:)g(:)g(:)p 1819 4907 11 41 v 1830 4889 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33046
5 v 128 w(:)g(:)g(:)2654 4880 y Gg(Cut)321 5119 y(which)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33047
b(reduces)i(to)1284 5278 y Ga(:)15 b(:)g(:)p 1425 5266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33048
11 41 v 1435 5248 46 5 v 127 w(:)g(:)g(:)i(;)e(B)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33049
FL(t)141 b Ga(B)5 b FL(t)p Ga(;)15 b(:)g(:)g(:)p 2214
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33050
5266 11 41 v 2224 5248 46 5 v 127 w(:)g(:)g(:)p 1284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33051
5316 1128 4 v 1678 5385 a(:)g(:)g(:)p 1819 5373 11 41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33052
v 1830 5355 46 5 v 128 w(:)g(:)g(:)2453 5346 y Gg(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33053
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33054
%%Page: 129 141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33055
TeXDict begin 129 140 bop Black 277 51 a Gb(5.1)23 b(Further)g(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33056
b(ork)2867 b(129)p 277 88 3691 4 v Black 277 388 a Gg(can)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33057
b(be)f(understood)k(as)c(communicating)j(the)e(e)o(xpression)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33058
FL(t)c Gg(from)h(the)h(proof)g(on)g(the)f(right-hand)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33059
501 y(side)34 b(to)f(the)g(one)h(on)f(the)g(left-hand)i(side.)58
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33060
b(Using)34 b(multiple)g(quanti\002ers,)j(one)d(can)f(also)h(encode)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33061
614 y(certain)25 b(dependencies)j(between)c(communications,)j(as)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33062
(illustrated)k(with)c(the)h(cut-instance)948 813 y Ga(:)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33063
b(:)g(:)p 1090 801 11 41 v 1100 783 46 5 v 128 w(:)g(:)g(:)h(;)f(B)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33064
b FL(a)15 b(t)1523 827 y Fu(1)p 890 851 730 4 v 890 930
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33065
a Ga(:)g(:)g(:)p 1032 918 11 41 v 1042 900 46 5 v 128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33066
w(:)g(:)g(:)h(;)f F6(8)p FL(x)p Ga(:B)20 b FL(x)15 b(t)1581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33067
944 y Fu(1)1662 875 y F6(8)1713 889 y Gc(R)p 845 968
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33068
820 4 v 845 1048 a Ga(:)g(:)g(:)p 987 1036 11 41 v 997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33069
1018 46 5 v 128 w(:)g(:)g(:)h(;)f F6(9)p FL(y)q Ga(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33070
F6(8)p FL(x)p Ga(:B)20 b FL(x)15 b(y)1707 993 y F6(9)1758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33071
1007 y Gc(R)2058 813 y Ga(B)k FL(t)2179 827 y Fu(2)2234
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33072
813 y FL(b)p Ga(;)c(:)g(:)g(:)p 2462 801 11 41 v 2473
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33073
783 46 5 v 128 w(:)g(:)g(:)p 2014 851 690 4 v 2014 930
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33074
a F6(8)p FL(x)p Ga(:B)k FL(x)c(b)p Ga(;)g(:)g(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33075
2506 918 11 41 v 2517 900 46 5 v 128 w(:)g(:)g(:)2745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33076
875 y F6(8)2796 889 y Gc(L)p 1956 968 805 4 v 1956 1048
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33077
a F6(9)p FL(y)q Ga(:)p F6(8)p FL(x)p Ga(:B)20 b FL(x)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33078
b(y)q Ga(;)g(:)g(:)g(:)p 2564 1036 11 41 v 2574 1018
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33079
46 5 v 128 w(:)g(:)g(:)2803 993 y F6(9)2854 1007 y Gc(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33080
845 1085 1916 4 v 1634 1155 a Ga(:)g(:)g(:)p 1775 1143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33081
11 41 v 1786 1124 46 5 v 128 w(:)g(:)g(:)2803 1116 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33082
Gg(Cut)277 1375 y(which)20 b(reduces)i(so)e(that)g(\002rst)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33083
FL(t)1264 1389 y Fu(1)1322 1375 y Gg(is)g(communicated)i(to)e(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33084
(proof)h(on)f(the)g(right-hand)j(side)d(and)h(then)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33085
1488 y FL(t)310 1502 y Fu(2)367 1488 y Gg(to)e(the)g(one)h(on)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33086
(left-hand)i(side.)28 b(Our)18 b(cut-elimination)23 b(procedures)f(pro)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33087
o(vide)e(a)f(frame)n(w)o(ork)g(to)277 1601 y(study)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33088
b(this)e(idea)h(further)-5 b(.)30 b(In)22 b(terms)h(of)f(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33089
(computational)j(interpretation,)h(most)d(unclear)l(,)h(ho)n(we)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33090
(er)l(,)277 1714 y(are)g(cut-instances)j(whose)d(cut-formula)i(is)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33091
(contracted)k(on)c(both)i(sides.)978 1944 y Ga(:)15 b(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33092
1120 1932 11 41 v 1130 1913 46 5 v 128 w(:)g(:)g(:)h(;)f(C)q(;)g(C)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33093
978 1981 573 4 v 1032 2060 a(:)g(:)g(:)p 1173 2048 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33094
41 v 1183 2030 46 5 v 127 w(:)g(:)g(:)h(;)f(C)1592 2005
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33095
y Gg(Contr)1798 2019 y Gc(R)1948 1944 y Ga(C)q(;)g(C)q(;)g(:)g(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33096
2303 1932 11 41 v 2314 1913 46 5 v 129 w(:)g(:)g(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33097
1948 1981 553 4 v 2001 2060 a(C)q(;)g(:)g(:)g(:)p 2250
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33098
2048 11 41 v 2260 2030 46 5 v 129 w(:)g(:)g(:)2542 2005
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33099
y Gg(Contr)2748 2019 y Gc(L)p 1032 2097 1416 4 v 1570
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33100
2167 a Ga(:)g(:)g(:)p 1711 2155 11 41 v 1722 2137 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33101
5 v 128 w(:)g(:)g(:)2488 2128 y Gg(Cut)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33102
eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33103
%%Page: 130 142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33104
TeXDict begin 130 141 bop Black Black Black Black eop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33105
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33106
%%Page: 131 143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33107
TeXDict begin 131 142 bop Black Black 277 1027 a F8(A)l(ppendix)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33108
b(A)277 1459 y Gf(Experimental)52 b(Data)277 1920 y Gg(In)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33109
b(this)g(appendix)j(we)c(shall)i(gi)n(v)o(e)f(tw)o(o)f(normal)i(forms)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33110
(for)g(the)g(proof,)i Ga(\033)s Gg(,)d(gi)n(v)o(en)i(on)f(P)o(age)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33111
(118.)28 b(Both)277 2020 y(normal)c(forms)g(pro)o(v)o(e,)f(by)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33112
(analysing)j(some)e(cases,)g(Sequent)g(\(4.9\))g(sho)n(wn)f(on)h(P)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33113
(age)e(116.)30 b(T)-7 b(o)22 b(\002nd)277 2119 y(which)i(cases)h(are)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33114
(analysed,)j(one)e(has)g(to)f(look)i(for)e(proof)i(fragments)g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33115
(the)f(follo)n(wing)i(form.)1811 2218 y Gd(.)1811 2251
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33116
y(.)1811 2284 y(.)1811 2317 y(.)1113 2397 y FU(:)14 b(:)g(:)g(;)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33117
FG(\()p FF(f1)23 b FG(=)f FF(1)14 b FT(^)g FF(f0)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33118
FG(=)f FF(1)p FG(\))14 b FT(\033)f FF(f0)23 b FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33119
FF(f1)p FU(;)14 b(:)g(:)g(:)p 2349 2385 10 38 v 2358
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33120
2369 42 4 v 115 w(:)g(:)g(:)p 1028 2438 1586 4 v 1028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33121
2517 a(:)g(:)g(:)g(;)g FT(8)p FF(y)q FU(:)p FG(\(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33122
FF(fy)25 b FG(=)e FF(1)14 b FT(^)f FF(f1)23 b FG(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33123
FF(1)p FG(\))14 b FT(\033)f FF(f0)23 b FG(=)f FF(fy)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33124
FG(\))p FU(;)14 b(:)g(:)g(:)p 2433 2505 10 38 v 2443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33125
2488 42 4 v 116 w(:)g(:)g(:)2655 2461 y FT(8)2702 2473
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33126
y FS(L)p 1001 2558 1641 4 v 1001 2636 a FU(:)g(:)g(:)g(;)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33127
FT(8)p FF(x)p FU(:)p FF(y)p FU(:)p FG(\(\()p FF(fy)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33128
b FG(=)c FF(1)14 b FT(^)g FF(fx)24 b FG(=)e FF(1)p FG(\))14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33129
b FT(\033)f FF(fx)24 b FG(=)f FF(fy)r FG(\))p FU(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33130
b(:)g(:)g(:)p 2461 2624 10 38 v 2470 2608 42 4 v 115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33131
w(:)g(:)g(:)2683 2580 y FT(8)2730 2592 y FS(L)p 1001
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33132
2677 1641 4 v 1001 2756 a FU(:)g(:)g(:)g(;)g FT(8)p FF(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33133
FU(:)p FF(x)p FU(:)p FF(y)p FU(:)p FG(\(\()p FF(fy)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33134
b FG(=)d FF(i)14 b FT(^)f FF(fx)24 b FG(=)f FF(i)p FG(\))14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33135
b FT(\033)f FF(fx)24 b FG(=)f FF(fy)q FG(\))p FU(;)14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33136
b(:)g(:)g(:)p 2461 2744 10 38 v 2470 2727 42 4 v 116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33137
w(:)g(:)g(:)2683 2700 y FT(8)2730 2712 y FS(L)277 3044
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33138
y Gg(This)27 b(particular)i(proof)e(fragment)h(indicates)h(that)f(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33139
(case)g FL(f0)k F4(=)g FL(f1)g F4(=)g(1)26 b Gg(is)g(analysed.)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33140
b(The)26 b(\002rst)277 3157 y(normal)e(form)g(\(P)o(ages)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33141
(132\226134\))i(analyses)f(the)f(cases)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33142
803 3335 a FL(f0)i F4(=)f FL(f1)g F4(=)g FL(0)100 b(f1)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33143
b F4(=)f FL(f2)g F4(=)g FL(0)237 b(f0)25 b F4(=)g FL(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33144
F4(=)f FL(1)100 b(f2)25 b F4(=)g FL(f3)h F4(=)f FL(1)803
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33145
3448 y(f0)h F4(=)f FL(f2)g F4(=)g FL(0)100 b(f1)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33146
F4(=)f FL(f3)g F4(=)g FL(0)237 b(f1)25 b F4(=)g FL(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33147
F4(=)f FL(1)277 3623 y Gg(and)f(the)g(second)h(\(P)o(ages)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33148
(135\226140\))i(the)e(cases)p Black Black 803 3801 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33149
FL(f0)i F4(=)f FL(f1)g F4(=)g FL(0)100 b(f2)26 b F4(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33150
FL(f3)g F4(=)g FL(0)237 b(f0)25 b F4(=)g FL(f1)h F4(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33151
FL(1)100 b(f1)25 b F4(=)g FL(f2)h F4(=)f FL(1)803 3914
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33152
y(f1)h F4(=)f FL(f2)g F4(=)g FL(0)771 b(f0)25 b F4(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33153
FL(f2)h F4(=)f FL(1)100 b(f1)25 b F4(=)g FL(f3)h F4(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33154
FL(1)277 4090 y Gg(W)-7 b(e)23 b(tak)o(e)i(this)f(as)g(e)n(vidence)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33155
(that)f(both)g(normal)f(forms)h(are)f(\223essentially\224)k(dif)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33156
(ferent)e(\(see)e(the)g(crite-)277 4203 y(rion)h(gi)n(v)o(en)g(on)f(P)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33157
(age)g(117\).)31 b(Because)25 b(of)g(their)g(size,)f(the)h(tw)o(o)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33158
(normal)i(forms)g(are)f(spread)i(o)o(v)o(er)e(the)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33159
4316 y(follo)n(wing)h(nine)f(\002gures.)414 4462 y(1)164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33160
b(First)23 b(normal)h(form)g(and)g(Subproof)h Ga(X)1882
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33161
4476 y F9(6)1922 4462 y Gg(.)59 b(.)46 b(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33162
g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black 43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33163
w(132)p Black 414 4592 a(2)164 b(Subproofs)25 b Ga(X)1094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33164
4606 y F9(1)1157 4592 y Gg(and)f Ga(X)1386 4606 y F9(2)1425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33165
4592 y Gg(.)79 b(.)45 b(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33166
(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33167
Black 43 w(133)p Black 414 4721 a(3)164 b(Subproofs)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33168
b Ga(X)1094 4735 y F9(3)1134 4721 y Gg(,)d Ga(X)1254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33169
4735 y F9(4)1317 4721 y Gg(and)i Ga(X)1546 4735 y F9(5)1585
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33170
4721 y Gg(.)55 b(.)46 b(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33171
(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33172
43 w(134)p Black 414 4851 a(4)164 b(Second)24 b(normal)g(form)g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33173
(Subproof)h Ga(Y)1961 4865 y F9(8)2000 4851 y Gg(.)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33174
b(.)45 b(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33175
h(.)f(.)p Black 43 w(135)p Black 414 4980 a(5)164 b(Subproofs)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33176
b Ga(Y)1072 4994 y F9(6)1134 4980 y Gg(and)f Ga(Y)1341
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33177
4994 y F9(7)1380 4980 y Gg(.)56 b(.)45 b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33178
h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33179
(.)f(.)p Black 43 w(136)p Black 414 5110 a(6)164 b(Subproofs)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33180
b Ga(Y)1072 5124 y F9(4)1134 5110 y Gg(and)f Ga(Y)1341
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33181
5124 y F9(5)1380 5110 y Gg(.)56 b(.)45 b(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33182
h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33183
(.)f(.)p Black 43 w(137)p Black 414 5239 a(7)164 b(Subproofs)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33184
b Ga(Y)1072 5253 y F9(1)1111 5239 y Gg(,)e Ga(Y)1210
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33185
5253 y F9(2)1272 5239 y Gg(and)h Ga(Y)1479 5253 y F9(3)1518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33186
5239 y Gg(.)54 b(.)45 b(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33187
(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33188
Black 43 w(138)p Black 414 5369 a(8)164 b(Subproofs)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33189
b Ga(Y)1072 5383 y F9(12)1169 5369 y Gg(and)f Ga(Y)1376
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33190
5383 y F9(13)1451 5369 y Gg(.)53 b(.)45 b(.)g(.)h(.)f(.)g(.)g(.)g(.)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33191
(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33192
f(.)p Black 43 w(139)p Black 414 5498 a(9)164 b(Subproofs)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33193
b Ga(Y)1072 5512 y F9(9)1111 5498 y Gg(,)e Ga(Y)1210
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33194
5512 y F9(10)1307 5498 y Gg(and)h Ga(Y)1514 5512 y F9(11)1588
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33195
5498 y Gg(.)52 b(.)46 b(.)f(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33196
(.)g(.)g(.)g(.)h(.)f(.)g(.)g(.)g(.)g(.)h(.)f(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33197
43 w(140)p Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33198
%%Page: 132 144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33199
TeXDict begin 132 143 bop Black -144 51 a Gb(132)2816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33200
b(Experimental)24 b(Data)p -144 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33201
321 393 3226 4 v 321 5116 4 4724 v 1927 465 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33202
 gsave currentpoint currentpoint translate -90 neg rotate neg exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33203
neg exch translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33204
 1927 465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33205
a 2583 -821 247 4 v 2583 -769 a Ff(3)l FC(\024)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33206
2691 -781 6 23 v 2697 -791 25 3 v 61 w(3)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33207
Ff(3)p 2490 -739 433 4 v 2479 -675 a FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33208
FV(\()p Ff(3)l FC(\024)l Ff(m)2689 -696 y Fe(x)p Fd(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33209
Fe(3)2746 -675 y FV(\))p 2784 -687 6 23 v 2790 -697 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33210
3 v 59 w Ff(3)l FC(\024)l Ff(3)2922 -728 y FC(8)2957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33211
-717 y Fd(L)p 2490 -642 433 4 v 2585 -591 a FZ(M)2643
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33212
-581 y FP(2)p 2689 -603 6 23 v 2695 -613 25 3 v 2734
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33213
-591 a Ff(3)l FC(\024)l Ff(3)2922 -631 y FC(8)2957 -620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33214
y Fd(L)p 3050 -641 247 4 v 3050 -591 a Ff(2)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33215
Ff(3)p 3158 -603 6 23 v 3164 -613 25 3 v 61 w(2)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33216
Ff(3)p 2585 -561 713 4 v 2687 -509 a(3)l FC(\024)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33217
FC(\033)q Ff(2)l FZ(<)l Ff(3)p FZ(;)13 b(M)3010 -499
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33218
y FP(2)p 3056 -521 6 23 v 3062 -531 25 3 v 3101 -509
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33219
a Ff(2)l FZ(<)l Ff(3)3296 -552 y FC(\033)3344 -540 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33220
Fd(L)p 2630 -479 622 4 v 2619 -425 a FC(8)p Ff(y)q FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33221
FV(\()p Ff(3)l FC(\024)l Ff(y)q FC(\033)r Ff(2)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33222
Ff(y)q FV(\))p FZ(;)f(M)3067 -415 y FP(2)p 3113 -437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33223
6 23 v 3119 -447 25 3 v 3158 -425 a Ff(2)l FZ(<)l Ff(3)3251
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33224
-467 y FC(8)3286 -456 y Fd(L)p 2630 -393 622 4 v 2784
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33225
-342 a FZ(M)2842 -332 y FP(2)2875 -342 y FZ(;)e(S)p 2959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33226
-354 6 23 v 2964 -364 25 3 v 61 w Ff(2)l FZ(<)l Ff(3)3251
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33227
-381 y FC(8)3286 -370 y Fd(L)4423 -835 y F3(.)4423 -802
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33228
y(.)4423 -769 y(.)4423 -736 y(.)4388 -684 y FZ(X)4438
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33229
-674 y FP(2)p 4538 -736 351 4 v 4538 -684 a Ff(f2)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33230
Ff(f3)p 4698 -696 6 23 v 4704 -706 25 3 v 59 w(f2)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33231
Ff(f3)p 3514 -655 2232 4 v 3514 -601 a FV(\()p Ff(f3)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33232
FV(=)f Ff(1)p FC(^)o Ff(f2)h FV(=)f Ff(1)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33233
Ff(f2)h FV(=)f Ff(f3)p FZ(;)k(M)4145 -591 y FP(1)4177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33234
-601 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33235
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33236
4774 -613 6 23 v 4780 -623 25 3 v 68 w Ff(f2)d FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33237
Ff(f3)q FZ(;)j Ff(1)l FZ(<)l Ff(3)p FC(^)r Ff(f1)d FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33238
Ff(f3)q FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33239
Ff(f2)p FZ(;)k Ff(f1)d FV(=)f Ff(1)5745 -646 y FC(\033)5793
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33240
-635 y Fd(L)p 3457 -569 2346 4 v 3447 -515 a FC(8)p Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33241
FZ(:)p FV(\(\()p Ff(fy)h FV(=)f Ff(1)p FC(^)p Ff(f2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33242
FV(=)g Ff(1)p FV(\))p FC(\033)p Ff(f2)h FV(=)f Ff(fy)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33243
FV(\))p FZ(;)k(M)4202 -505 y FP(1)4234 -515 y FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33244
FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33245
Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4831
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33246
-527 6 23 v 4837 -537 25 3 v 68 w Ff(f2)d FV(=)g Ff(f3)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33247
FZ(;)j Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)e FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33248
FZ(;)k Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33249
FZ(;)k Ff(f1)d FV(=)f Ff(1)5802 -557 y FC(8)5837 -546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33250
y Fd(L)p 3438 -482 2385 4 v 3427 -428 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33251
FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i FV(=)e Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33252
FC(^)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FC(\033)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33253
FV(=)f Ff(fy)q FV(\))p FZ(;)j(M)4221 -418 y FP(1)4253
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33254
-428 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33255
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33256
4850 -440 6 23 v 4856 -450 25 3 v 67 w Ff(f2)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33257
Ff(f3)p FZ(;)k Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33258
Ff(f3)p FZ(;)k Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33259
Ff(f2)p FZ(;)j Ff(f1)e FV(=)f Ff(1)5821 -471 y FC(8)5856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33260
-460 y Fd(L)p 3438 -396 2385 4 v 3800 -342 a FZ(M)3858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33261
-332 y FP(1)3891 -342 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33262
Ff(fx)g FV(=)g Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33263
FZ(;)j(S)o(;)h(T)p 4488 -354 6 23 v 4493 -364 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33264
67 w Ff(f2)d FV(=)f Ff(f3)p FZ(;)k Ff(1)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33265
FC(^)q Ff(f1)d FV(=)f Ff(f3)p FZ(;)k Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33266
FC(^)q Ff(f1)d FV(=)f Ff(f2)p FZ(;)k Ff(f1)c FV(=)g Ff(1)5821
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33267
-384 y FC(8)5856 -373 y Fd(L)p 2784 -310 2676 4 v 3165
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33268
-256 a FZ(M)3223 -246 y FP(1)3255 -256 y FZ(;)k(M)3343
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33269
-246 y FP(2)3375 -256 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33270
Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33271
FZ(;)k(S)o(;)f(T)p 3972 -268 6 23 v 3978 -277 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33272
68 w Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)e FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33273
FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)e FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33274
FZ(;)j Ff(2)l FZ(<)l Ff(3)p FC(^)r Ff(f2)d FV(=)g Ff(f3)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33275
FZ(;)j Ff(f1)e FV(=)f Ff(1)5459 -301 y FC(^)5500 -290
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33276
y Fd(R)p 3083 -223 2079 4 v 3083 -169 a FZ(M)3141 -159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33277
y FP(1)3173 -169 y FZ(;)j(M)3260 -159 y FP(2)3293 -169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33278
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33279
Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33280
3890 -181 6 23 v 3895 -191 25 3 v 57 w FC(9)p Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33281
FZ(:)p FV(\()p Ff(1)l FZ(<)l Ff(m)p FC(^)q Ff(f1)c FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33282
Ff(fm)p FV(\))p FZ(;)k Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33283
FV(=)f Ff(f2)p FZ(;)k Ff(2)l FZ(<)l Ff(3)p FC(^)q Ff(f2)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33284
FV(=)f Ff(f3)p FZ(;)j Ff(f1)e FV(=)f Ff(1)5160 -212 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33285
FC(9)5195 -201 y Fd(R)p 3059 -137 2127 4 v 3059 -83 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33286
FZ(M)3117 -73 y FP(1)3149 -83 y FZ(;)j(M)3236 -73 y FP(2)3269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33287
-83 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33288
Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33289
3866 -95 6 23 v 3871 -105 25 3 v 67 w Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33290
FC(^)r Ff(f1)c FV(=)g Ff(f2)q FZ(;)j Ff(2)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33291
FC(^)r Ff(f2)d FV(=)g Ff(f3)q FZ(;)j Ff(f1)e FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33292
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33293
FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33294
-125 y FC(9)5219 -114 y Fd(R)p 2976 -51 2292 4 v 2976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33295
3 a FZ(M)3034 13 y FP(1)3067 3 y FZ(;)j(M)3154 13 y FP(2)3186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33296
3 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33297
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33298
3783 -9 6 23 v 3789 -18 25 3 v 57 w FC(9)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33299
FV(\()p Ff(1)l FZ(<)l Ff(m)p FC(^)p Ff(f1)d FV(=)f Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33300
FV(\))p FZ(;)j Ff(2)l FZ(<)l Ff(3)p FC(^)r Ff(f2)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33301
Ff(f3)p FZ(;)j Ff(f1)e FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33302
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33303
Ff(fn)h FV(=)f Ff(fm)o FV(\))5267 -39 y FC(9)5302 -28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33304
y Fd(R)p 2976 36 2292 4 v 3214 90 a FZ(M)3272 100 y FP(1)3304
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33305
90 y FZ(;)k(M)3392 100 y FP(2)3424 90 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33306
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33307
FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4021 78 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33308
v 4027 68 25 3 v 68 w Ff(2)l FZ(<)l Ff(3)p FC(^)q Ff(f2)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33309
FV(=)f Ff(f3)p FZ(;)k Ff(f1)c FV(=)g Ff(1)p FZ(;)p FC(9)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33310
Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33311
FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5267 48 y FC(9)5302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33312
59 y Fd(R)p 3132 122 1981 4 v 3132 176 a FZ(M)3190 186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33313
y FP(1)3222 176 y FZ(;)k(M)3310 186 y FP(2)3342 176 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33314
FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33315
FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33316
3939 164 6 23 v 3945 154 25 3 v 57 w FC(9)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33317
FV(\()p Ff(2)l FZ(<)l Ff(m)p FC(^)q Ff(f2)e FV(=)f Ff(fm)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33318
FV(\))p FZ(;)k Ff(f1)d FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33319
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33320
Ff(fn)g FV(=)g Ff(fm)p FV(\))5111 134 y FC(9)5146 145
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33321
y Fd(R)p 3132 209 1981 4 v 3369 262 a FZ(M)3427 272 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33322
FP(1)3460 262 y FZ(;)j(M)3547 272 y FP(2)3579 262 y FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33323
FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33324
Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 4176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33325
250 6 23 v 4182 241 25 3 v 67 w Ff(f1)d FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33326
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33327
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)o FV(\))5111
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33328
220 y FC(9)5146 231 y Fd(R)6003 112 y F3(.)6003 145 y(.)6003
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33329
178 y(.)6003 211 y(.)5968 262 y FZ(X)6018 272 y FP(5)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33330
3369 295 2699 4 v 3880 349 a FZ(M)3938 359 y FP(1)3971
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33331
349 y FZ(;)j(M)4058 359 y FP(2)4090 349 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33332
Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33333
FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 4687 337 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33334
v 4693 327 25 3 v 67 w Ff(f1)d FV(=)f Ff(1)p FC(^)p Ff(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33335
FV(=)g Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33336
FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33337
FV(\))6067 303 y FC(^)6108 314 y Fd(R)2490 393 y Fc(|)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33338
2514 393 1785 8 v 1785 w({z)p 4347 393 V 1785 w(})4283
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33339
455 y Fd(X)4329 467 y FP(6)p 2567 627 247 4 v 2567 678
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33340
a Ff(1)l FC(\024)l Ff(1)p 2675 666 6 23 v 2680 656 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33341
3 v 61 w(1)l FC(\024)l Ff(1)p 2474 708 433 4 v 2463 773
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33342
a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(1)l FC(\024)l Ff(m)2673
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33343
752 y Fe(x)p Fd(;)p Fe(1)2730 773 y FV(\))p 2768 761
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33344
6 23 v 2774 751 25 3 v 59 w Ff(1)l FC(\024)l Ff(1)2906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33345
720 y FC(8)2941 731 y Fd(L)p 2474 805 433 4 v 2569 857
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33346
a FZ(M)2627 867 y FP(2)p 2673 845 6 23 v 2679 835 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33347
3 v 2718 857 a Ff(1)l FC(\024)l Ff(1)2906 817 y FC(8)2941
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33348
828 y Fd(L)p 3034 806 247 4 v 3034 857 a Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33349
Ff(1)p 3142 845 6 23 v 3148 835 25 3 v 61 w(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33350
Ff(1)p 2569 887 713 4 v 2671 939 a(1)l FC(\024)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33351
FC(\033)q Ff(0)l FZ(<)l Ff(1)p FZ(;)13 b(M)2994 949 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33352
FP(2)p 3040 927 6 23 v 3046 917 25 3 v 3085 939 a Ff(0)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33353
FZ(<)l Ff(1)3280 896 y FC(\033)3328 907 y Fd(L)p 2614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33354
969 622 4 v 2603 1023 a FC(8)p Ff(y)q FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33355
Ff(1)l FC(\024)l Ff(y)q FC(\033)r Ff(0)l FZ(<)l Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33356
FV(\))p FZ(;)f(M)3051 1033 y FP(2)p 3097 1011 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33357
3103 1001 25 3 v 3142 1023 a Ff(0)l FZ(<)l Ff(1)3235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33358
980 y FC(8)3270 991 y Fd(L)p 2614 1055 622 4 v 2768 1106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33359
a FZ(M)2826 1116 y FP(2)2858 1106 y FZ(;)f(S)p 2942 1094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33360
6 23 v 2948 1084 25 3 v 61 w Ff(0)l FZ(<)l Ff(1)3235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33361
1067 y FC(8)3270 1078 y Fd(L)4338 612 y F3(.)4338 646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33362
y(.)4338 679 y(.)4338 712 y(.)4303 763 y FZ(X)4353 773
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33363
y FP(6)p 4453 712 351 4 v 4453 763 a Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33364
4613 751 6 23 v 4619 741 25 3 v 59 w(f0)h FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33365
3498 793 2094 4 v 3498 847 a FV(\()p Ff(f1)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33366
FC(^)o Ff(f0)h FV(=)f Ff(1)p FV(\))p FC(\033)p Ff(f0)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33367
FV(=)f Ff(f1)p FZ(;)k(M)4129 857 y FP(1)4161 847 y FZ(;)f(M)4248
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33368
857 y FP(2)4281 847 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33369
Ff(fx)d FV(=)g Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33370
FZ(;)j(S)o(;)h(T)p 4878 835 6 23 v 4883 825 25 3 v 67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33371
w Ff(f0)d FV(=)f Ff(f1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33372
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33373
FV(=)f Ff(fm)p FV(\))5591 802 y FC(\033)5639 813 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33374
3441 879 2208 4 v 3430 933 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33375
Ff(fy)i FV(=)e Ff(1)p FC(^)o Ff(f0)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33376
FC(\033)p Ff(f0)h FV(=)f Ff(fy)q FV(\))p FZ(;)j(M)4185
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33377
943 y FP(1)4218 933 y FZ(;)g(M)4305 943 y FP(2)4337 933
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33378
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33379
Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33380
4934 921 6 23 v 4940 911 25 3 v 67 w Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33381
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33382
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33383
891 y FC(8)5683 902 y Fd(L)p 3421 965 2247 4 v 3411 1019
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33384
a FC(8)p Ff(x)p FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33385
FV(=)e Ff(1)p FC(^)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33386
Ff(fx)h FV(=)f Ff(fy)p FV(\))p FZ(;)k(M)4205 1029 y FP(1)4237
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33387
1019 y FZ(;)g(M)4325 1029 y FP(2)4357 1019 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33388
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33389
FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4954 1007 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33390
23 v 4960 998 25 3 v 68 w Ff(f0)e FV(=)f Ff(f1)p FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33391
FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33392
Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5667 977 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33393
FC(8)5702 988 y Fd(L)p 3421 1052 2247 4 v 3784 1106 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33394
FZ(M)3842 1116 y FP(1)3874 1106 y FZ(;)k(M)3962 1116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33395
y FP(2)3994 1106 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33396
Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33397
FZ(;)k(S)o(;)g(T)p 4591 1094 6 23 v 4597 1084 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33398
67 w Ff(f0)d FV(=)f Ff(f1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33399
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33400
FV(=)f Ff(fm)o FV(\))5667 1064 y FC(8)5702 1075 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33401
2768 1138 2538 4 v 3209 1192 a FZ(M)3267 1202 y FP(1)3299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33402
1192 y FZ(;)j(M)3386 1202 y FP(2)3419 1192 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33403
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)g Ff(0)p FC(_)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33404
FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 4016 1180 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33405
23 v 4021 1170 25 3 v 67 w Ff(0)l FZ(<)l Ff(1)p FC(^)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33406
Ff(f0)c FV(=)g Ff(f1)q FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33407
FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33408
Ff(fm)p FV(\))5304 1146 y FC(^)5346 1158 y Fd(R)p 3126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33409
1225 1821 4 v 3126 1278 a FZ(M)3184 1288 y FP(1)3217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33410
1278 y FZ(;)j(M)3304 1288 y FP(2)3336 1278 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33411
Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33412
FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p 3933 1266 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33413
23 v 3939 1257 25 3 v 57 w FC(9)p Ff(m)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33414
Ff(0)l FZ(<)l Ff(m)p FC(^)p Ff(f0)d FV(=)f Ff(fm)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33415
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33416
FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))4946
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33417
1236 y FC(9)4981 1247 y Fd(R)p 3126 1311 1821 4 v 3364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33418
1365 a FZ(M)3422 1375 y FP(1)3454 1365 y FZ(;)k(M)3542
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33419
1375 y FP(2)3574 1365 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33420
Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33421
FZ(;)k(S)o(;)f(T)p 4171 1353 6 23 v 4177 1343 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33422
57 w FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33423
FZ(<)l Ff(m)p FC(^)q Ff(fn)e FV(=)f Ff(fm)p FV(\))4946
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33424
1323 y FC(9)4981 1334 y Fd(R)2026 1612 y Gc(M)2094 1621
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33425
y FV(1)2148 1612 y F9(=)20 b FX(8)p Fu(y)q Gc(:)p Fu(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33426
Gc(:)m F9(\()p Fu(x)8 b FX(\024)g Fu(m)2555 1588 y Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33427
FZ(;)p Ff(y)2626 1612 y F9(\))101 b Gc(T)30 b F9(=)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33428
b FX(8)p Fu(i)p Gc(:)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33429
Fu(fy)i F9(=)e Fu(i)p FX(^)p Fu(fx)g F9(=)g Fu(i)p F9(\))12
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33430
b FX(\033)g Fu(fx)19 b F9(=)g Fu(fy)p F9(\))2026 1738
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33431
y Gc(M)2094 1747 y FV(2)2148 1738 y F9(=)h FX(8)p Fu(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33432
Gc(:)p Fu(x)p Gc(:)m F9(\()p Fu(y)9 b FX(\024)f Fu(m)2556
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33433
1714 y Ff(x)p FZ(;)p Ff(y)2627 1738 y F9(\))100 b Gc(S)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33434
b F9(=)d FX(8)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\()p Fu(sx)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33435
b FX(\024)g Fu(y)k FX(\033)g Fu(x)c Gc(<)g Fu(y)q F9(\))6506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33436
465 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33437
 currentpoint grestore moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33438
 6506 465 a 3543 5116 4 4724 v 321 5119 3226 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33439
Black 1093 5273 a Gg(Figure)25 b(1:)j(First)c(normal)g(form)g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33440
(Subproof)h Ga(X)2712 5287 y F9(6)2751 5273 y Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33441
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33442
%%Page: 133 145
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33443
TeXDict begin 133 144 bop Black 3831 51 a Gb(133)p 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33444
88 3691 4 v Black Black 277 393 3226 4 v 277 5116 4 4724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33445
v 1882 465 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33446
 gsave currentpoint currentpoint translate -90 neg rotate neg exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33447
neg exch translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33448
 1882 465 a 2677 -559 247 4 v 2677 -507 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33449
Ff(2)l FC(\024)l Ff(2)p 2785 -519 6 23 v 2791 -529 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33450
3 v 61 w(2)l FC(\024)l Ff(2)p 2585 -477 431 4 v 2574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33451
-412 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(x)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33452
Ff(m)2782 -433 y Fe(x)p Fd(;)p Fe(0)2839 -412 y FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33453
2877 -424 6 23 v 2883 -434 25 3 v 59 w Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33454
Ff(2)3015 -466 y FC(8)3050 -455 y Fd(L)p 2585 -380 431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33455
4 v 2679 -329 a FZ(M)2737 -319 y FP(1)p 2783 -341 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33456
v 2789 -350 25 3 v 2828 -329 a Ff(2)l FC(\024)l Ff(2)3015
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33457
-368 y FC(8)3050 -357 y Fd(L)p 3143 -379 247 4 v 3143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33458
-329 a Ff(1)l FZ(<)l Ff(2)p 3251 -341 6 23 v 3257 -350
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33459
25 3 v 61 w(1)l FZ(<)l Ff(2)p 2679 -299 712 4 v 2780
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33460
-247 a(2)l FC(\024)l Ff(2)p FC(\033)r Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33461
FZ(;)12 b(M)3103 -237 y FP(1)p 3150 -259 6 23 v 3155
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33462
-269 25 3 v 3195 -247 a Ff(1)l FZ(<)l Ff(2)3389 -289
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33463
y FC(\033)3437 -278 y Fd(L)p 2723 -217 622 4 v 2713 -163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33464
a FC(8)p Ff(y)q FZ(:)p FV(\()p Ff(2)l FC(\024)l Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33465
FC(\033)q Ff(1)l FZ(<)l Ff(y)q FV(\))p FZ(;)g(M)3160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33466
-153 y FP(1)p 3207 -175 6 23 v 3212 -185 25 3 v 3251
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33467
-163 a Ff(1)l FZ(<)l Ff(2)3344 -205 y FC(8)3379 -194
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33468
y Fd(L)p 2723 -131 622 4 v 2878 -80 a FZ(M)2936 -70 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33469
FP(1)2968 -80 y FZ(;)f(S)p 3052 -92 6 23 v 3058 -102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33470
25 3 v 61 w Ff(1)l FZ(<)l Ff(2)3344 -119 y FC(8)3379
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33471
-108 y Fd(L)p 3567 -717 319 4 v 3567 -666 a Ff(f2)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33472
Ff(0)p 3711 -678 6 23 v 3717 -688 25 3 v 59 w(f2)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33473
Ff(0)p 3936 -717 319 4 v 50 w(f2)h FV(=)f Ff(1)p 4080
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33474
-678 6 23 v 4086 -688 25 3 v 59 w(f2)g FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33475
3567 -646 687 4 v 3586 -595 a(f2)h FV(=)f Ff(0)p FC(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33476
Ff(f2)g FV(=)g Ff(1)p 3902 -607 6 23 v 3907 -617 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33477
v 59 w(f2)h FV(=)f Ff(0)p FZ(;)k Ff(f2)c FV(=)g Ff(1)4254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33478
-638 y FC(_)4295 -627 y Fd(L)p 3531 -565 761 4 v 3520
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33479
-511 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33480
FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p 3957 -523 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33481
3963 -533 25 3 v 59 w Ff(f2)h FV(=)f Ff(0)p FZ(;)k Ff(f2)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33482
FV(=)g Ff(1)4290 -554 y FC(8)4325 -543 y Fd(L)p 4456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33483
-717 319 4 v 4456 -666 a Ff(f1)h FV(=)f Ff(0)p 4600 -678
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33484
6 23 v 4606 -688 25 3 v 59 w(f1)h FV(=)f Ff(0)p 4824
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33485
-717 319 4 v 49 w(f1)h FV(=)f Ff(1)p 4968 -678 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33486
4974 -688 25 3 v 59 w(f1)h FV(=)f Ff(1)p 4456 -646 687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33487
4 v 4475 -595 a(f1)h FV(=)f Ff(0)p FC(_)o Ff(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33488
Ff(1)p 4790 -607 6 23 v 4796 -617 25 3 v 59 w(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33489
Ff(0)p FZ(;)j Ff(f1)e FV(=)f Ff(1)5142 -638 y FC(_)5184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33490
-627 y Fd(L)p 4419 -565 761 4 v 4409 -511 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33491
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33492
Ff(1)p FV(\))p 4846 -523 6 23 v 4852 -533 25 3 v 59 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33493
Ff(f1)h FV(=)f Ff(0)p FZ(;)j Ff(f1)e FV(=)f Ff(1)5179
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33494
-554 y FC(8)5214 -543 y Fd(L)p 3531 -479 1650 4 v 3799
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33495
-425 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33496
FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p 4236 -437 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33497
4242 -447 25 3 v 59 w Ff(f2)h FV(=)f Ff(0)p FC(^)p Ff(f1)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33498
FV(=)g Ff(0)p FZ(;)k Ff(f2)d FV(=)f Ff(1)p FZ(;)j Ff(f1)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33499
FV(=)f Ff(1)5179 -471 y FC(^)5221 -460 y Fd(R)p 5317
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33500
-476 351 4 v 5317 -425 a Ff(f1)h FV(=)f Ff(f2)p 5478
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33501
-437 6 23 v 5483 -447 25 3 v 59 w(f1)h FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33502
3810 -393 1859 4 v 3984 -339 a FV(\()p Ff(f2)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33503
Ff(0)p FC(^)p Ff(f1)g FV(=)g Ff(0)p FV(\))p FC(\033)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33504
Ff(f1)g FV(=)g Ff(f2)q FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33505
Ff(fx)g FV(=)g Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33506
4984 -351 6 23 v 4990 -361 25 3 v 59 w Ff(f1)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33507
Ff(f2)q FZ(;)j Ff(f2)e FV(=)f Ff(1)p FZ(;)k Ff(f1)c FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33508
Ff(1)5667 -384 y FC(\033)5715 -373 y Fd(L)p 3928 -306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33509
1623 4 v 3917 -252 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33510
FV(=)f Ff(0)p FC(^)p Ff(f1)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33511
Ff(f1)g FV(=)g Ff(fy)q FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33512
FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33513
FV(\))p 5041 -264 6 23 v 5047 -274 25 3 v 59 w Ff(f1)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33514
FV(=)g Ff(f2)q FZ(;)j Ff(f2)e FV(=)f Ff(1)p FZ(;)j Ff(f1)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33515
FV(=)f Ff(1)5549 -295 y FC(8)5584 -284 y Fd(L)p 3908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33516
-220 1662 4 v 3897 -166 a FC(8)p Ff(x)p FZ(:)p Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33517
FZ(:)p FV(\(\()p Ff(fy)i FV(=)e Ff(0)p FC(^)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33518
FV(=)g Ff(0)p FV(\))p FC(\033)p Ff(fx)h FV(=)f Ff(fy)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33519
FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33520
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p 5060 -178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33521
6 23 v 5066 -188 25 3 v 59 w Ff(f1)h FV(=)f Ff(f2)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33522
Ff(f2)c FV(=)g Ff(1)p FZ(;)k Ff(f1)d FV(=)f Ff(1)5569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33523
-208 y FC(8)5604 -197 y Fd(L)p 3908 -134 1662 4 v 4223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33524
-80 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33525
FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(T)p 4735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33526
-92 6 23 v 4740 -102 25 3 v 67 w Ff(f1)d FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33527
FZ(;)k Ff(f2)d FV(=)f Ff(1)p FZ(;)j Ff(f1)e FV(=)f Ff(1)5569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33528
-122 y FC(8)5604 -111 y Fd(L)p 2878 -47 2367 4 v 3395
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33529
7 a FZ(M)3453 17 y FP(1)3485 7 y FZ(;)p FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33530
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33531
Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p 4082 -5 6 23 v 4088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33532
-15 25 3 v 67 w Ff(f2)d FV(=)f Ff(1)p FZ(;)j Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33533
Ff(2)p FC(^)r Ff(f1)d FV(=)g Ff(f2)q FZ(;)j Ff(f1)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33534
Ff(1)5243 -39 y FC(^)5285 -28 y Fd(R)2585 51 y Fc(|)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33535
2609 51 1540 8 v 1540 w({z)p 4197 51 V 1540 w(})4133
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33536
113 y Fd(X)4179 125 y FP(1)p 2549 485 247 4 v 2549 537
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33537
a Ff(2)l FC(\024)l Ff(3)p 2657 525 6 23 v 2663 515 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33538
3 v 61 w(2)l FC(\024)l Ff(3)p 2457 567 431 4 v 2447 632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33539
a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(x)l FC(\024)l Ff(m)2655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33540
611 y Fe(x)p Fd(;)p Fe(3)2711 632 y FV(\))p 2749 620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33541
6 23 v 2755 610 25 3 v 59 w Ff(2)l FC(\024)l Ff(3)2887
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33542
578 y FC(8)2922 589 y Fd(L)p 2457 664 431 4 v 2551 715
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33543
a FZ(M)2609 725 y FP(1)p 2656 703 6 23 v 2661 694 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33544
3 v 2700 715 a Ff(2)l FC(\024)l Ff(3)2887 676 y FC(8)2922
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33545
687 y Fd(L)p 3016 665 247 4 v 3016 715 a Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33546
Ff(3)p 3124 703 6 23 v 3130 694 25 3 v 61 w(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33547
Ff(3)p 2551 745 712 4 v 2653 797 a(2)l FC(\024)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33548
FC(\033)r Ff(1)l FZ(<)l Ff(3)p FZ(;)12 b(M)2976 807 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33549
FP(1)p 3022 785 6 23 v 3028 775 25 3 v 3067 797 a Ff(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33550
FZ(<)l Ff(3)3261 755 y FC(\033)3309 766 y Fd(L)p 2596
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33551
827 622 4 v 2585 881 a FC(8)p Ff(y)q FZ(:)p FV(\()p Ff(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33552
FC(\024)l Ff(y)q FC(\033)r Ff(1)l FZ(<)l Ff(y)q FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33553
FZ(;)g(M)3033 891 y FP(1)p 3079 869 6 23 v 3085 859 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33554
3 v 3124 881 a Ff(1)l FZ(<)l Ff(3)3217 839 y FC(8)3252
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33555
850 y Fd(L)p 2596 913 622 4 v 2750 964 a FZ(M)2808 974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33556
y FP(1)2841 964 y FZ(;)e(S)p 2925 952 6 23 v 2930 942
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33557
25 3 v 61 w Ff(1)l FZ(<)l Ff(3)3217 925 y FC(8)3252 936
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33558
y Fd(L)p 3440 327 319 4 v 3440 378 a Ff(f3)e FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33559
3584 366 6 23 v 3590 356 25 3 v 59 w(f3)g FV(=)g Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33560
3808 327 319 4 v 50 w(f3)h FV(=)f Ff(1)p 3952 366 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33561
v 3958 356 25 3 v 59 w(f3)h FV(=)f Ff(1)p 3440 398 687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33562
4 v 3459 449 a(f3)h FV(=)f Ff(0)p FC(_)o Ff(f3)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33563
Ff(1)p 3774 437 6 23 v 3780 427 25 3 v 59 w(f3)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33564
Ff(0)p FZ(;)j Ff(f3)e FV(=)f Ff(1)4126 406 y FC(_)4168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33565
417 y Fd(L)p 3403 479 761 4 v 3392 532 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33566
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33567
Ff(1)p FV(\))p 3830 520 6 23 v 3836 511 25 3 v 59 w Ff(f3)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33568
FV(=)g Ff(0)p FZ(;)k Ff(f3)d FV(=)f Ff(1)4163 490 y FC(8)4198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33569
501 y Fd(L)p 4329 327 319 4 v 4329 378 a Ff(f1)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33570
Ff(0)p 4473 366 6 23 v 4478 356 25 3 v 59 w(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33571
Ff(0)p 4697 327 319 4 v 50 w(f1)h FV(=)f Ff(1)p 4841
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33572
366 6 23 v 4847 356 25 3 v 59 w(f1)g FV(=)g Ff(1)p 4329
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33573
398 687 4 v 4347 449 a(f1)h FV(=)f Ff(0)p FC(_)p Ff(f1)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33574
FV(=)g Ff(1)p 4663 437 6 23 v 4668 427 25 3 v 60 w(f1)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33575
FV(=)g Ff(0)p FZ(;)k Ff(f1)d FV(=)f Ff(1)5015 406 y FC(_)5056
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33576
417 y Fd(L)p 4292 479 761 4 v 4281 532 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33577
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33578
Ff(1)p FV(\))p 4719 520 6 23 v 4724 511 25 3 v 59 w Ff(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33579
FV(=)f Ff(0)p FZ(;)k Ff(f1)c FV(=)g Ff(1)5052 490 y FC(8)5087
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33580
501 y Fd(L)p 3403 565 1650 4 v 3671 619 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33581
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33582
Ff(1)p FV(\))p 4109 607 6 23 v 4115 597 25 3 v 59 w Ff(f3)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33583
FV(=)g Ff(0)p FC(^)p Ff(f1)h FV(=)f Ff(0)p FZ(;)j Ff(f3)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33584
FV(=)f Ff(1)p FZ(;)k Ff(f1)c FV(=)g Ff(1)5052 573 y FC(^)5093
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33585
584 y Fd(R)p 5190 568 351 4 v 5190 619 a Ff(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33586
Ff(f3)p 5350 607 6 23 v 5356 597 25 3 v 59 w(f1)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33587
Ff(f3)p 3682 651 1859 4 v 3857 705 a FV(\()p Ff(f3)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33588
FV(=)f Ff(0)p FC(^)o Ff(f1)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33589
Ff(f1)h FV(=)f Ff(f3)p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33590
Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33591
4856 693 6 23 v 4862 683 25 3 v 59 w Ff(f1)h FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33592
FZ(;)k Ff(f3)c FV(=)g Ff(1)p FZ(;)k Ff(f1)d FV(=)f Ff(1)5540
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33593
660 y FC(\033)5588 671 y Fd(L)p 3800 738 1623 4 v 3789
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33594
792 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i FV(=)e Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33595
FC(^)o Ff(f1)h FV(=)f Ff(0)p FV(\))p FC(\033)p Ff(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33596
FV(=)f Ff(fy)q FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33597
Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33598
4913 780 6 23 v 4919 770 25 3 v 59 w Ff(f1)h FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33599
FZ(;)k Ff(f3)c FV(=)g Ff(1)p FZ(;)k Ff(f1)d FV(=)f Ff(1)5422
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33600
749 y FC(8)5457 760 y Fd(L)p 3780 824 1662 4 v 3770 878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33601
a FC(8)p Ff(x)p FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33602
FV(=)e Ff(0)p FC(^)o Ff(fx)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33603
Ff(fx)h FV(=)f Ff(fy)p FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33604
FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33605
FV(\))p 4933 866 6 23 v 4939 856 25 3 v 59 w Ff(f1)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33606
FV(=)g Ff(f3)q FZ(;)j Ff(f3)e FV(=)f Ff(1)p FZ(;)k Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33607
FV(=)g Ff(1)5441 836 y FC(8)5476 847 y Fd(L)p 3780 910
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33608
1662 4 v 4096 964 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33609
FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33610
4607 952 6 23 v 4613 942 25 3 v 68 w Ff(f1)e FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33611
FZ(;)j Ff(f3)e FV(=)f Ff(1)p FZ(;)k Ff(f1)c FV(=)g Ff(1)5441
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33612
922 y FC(8)5476 933 y Fd(L)p 2750 997 2367 4 v 3267 1051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33613
a FZ(M)3325 1061 y FP(1)3358 1051 y FZ(;)p FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33614
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33615
Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 3955 1039 6 23 v 3960
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33616
1029 25 3 v 67 w Ff(f3)d FV(=)f Ff(1)p FZ(;)k Ff(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33617
FZ(<)l Ff(3)p FC(^)q Ff(f1)d FV(=)f Ff(f3)p FZ(;)k Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33618
FV(=)g Ff(1)5116 1005 y FC(^)5157 1016 y Fd(R)5734 900
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33619
y F3(.)5734 933 y(.)5734 966 y(.)5734 999 y(.)5699 1051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33620
y FZ(X)5749 1061 y FP(1)p 3267 1083 2532 4 v 3626 1137
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33621
a FZ(M)3684 1147 y FP(1)3716 1137 y FZ(;)p FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33622
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33623
Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 4313 1125 6 23 v 4319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33624
1115 25 3 v 67 w Ff(f3)d FV(=)f Ff(1)p FC(^)p Ff(f2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33625
FV(=)g Ff(1)p FZ(;)k Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33626
FV(=)f Ff(f3)p FZ(;)k Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33627
FV(=)f Ff(f2)p FZ(;)k Ff(f1)c FV(=)g Ff(1)5798 1091 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33628
FC(^)5840 1102 y Fd(R)2457 1182 y Fc(|)p 2481 1182 1667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33629
8 v 1667 w({z)p 4196 1182 V 1667 w(})4133 1243 y Fd(X)4179
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33630
1255 y FP(2)1982 1492 y Gc(M)2050 1501 y FV(1)2104 1492
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33631
y F9(=)19 b FX(8)p Fu(y)q Gc(:)p Fu(x)p Gc(:)n F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33632
Fu(x)8 b FX(\024)g Fu(m)2511 1469 y Ff(x)p FZ(;)p Ff(y)2582
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33633
1492 y F9(\))101 b Gc(T)30 b F9(=)19 b FX(8)p Fu(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33634
Gc(:)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\(\()p Fu(fy)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33635
F9(=)g Fu(i)p FX(^)p Fu(fx)e F9(=)i Fu(i)p F9(\))12 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33636
FX(\033)g Fu(fx)18 b F9(=)i Fu(fy)p F9(\))1982 1618 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33637
Gc(M)2050 1627 y FV(2)2104 1618 y F9(=)f FX(8)p Fu(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33638
Gc(:)p Fu(x)p Gc(:)n F9(\()p Fu(y)9 b FX(\024)f Fu(m)2512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33639
1595 y Ff(x)p FZ(;)p Ff(y)2583 1618 y F9(\))100 b Gc(S)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33640
b F9(=)d FX(8)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\()p Fu(sx)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33641
b FX(\024)g Fu(y)k FX(\033)g Fu(x)c Gc(<)g Fu(y)q F9(\))6462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33642
465 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33643
 currentpoint grestore moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33644
 6462 465 a 3499 5116 4 4724 v 277 5119 3226 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33645
Black 1297 5273 a Gg(Figure)25 b(2:)j(Subproofs)e Ga(X)2128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33646
5287 y F9(1)2190 5273 y Gg(and)e Ga(X)2419 5287 y F9(2)2459
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33647
5273 y Gg(.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33648
%%Page: 134 146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33649
TeXDict begin 134 145 bop Black -144 51 a Gb(134)2816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33650
b(Experimental)24 b(Data)p -144 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33651
321 383 V 321 5012 4 4630 v 2159 408 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33652
 gsave currentpoint currentpoint translate -90 neg rotate neg exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33653
neg exch translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33654
 2159 408 a 3367
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33655
-1151 247 4 v 3367 -1100 a Ff(1)l FC(\024)l Ff(1)p 3475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33656
-1112 6 23 v 3481 -1122 25 3 v 61 w(1)l FC(\024)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33657
3275 -1070 431 4 v 3265 -1005 a FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33658
FV(\()p Ff(x)l FC(\024)l Ff(m)3473 -1026 y Fe(x)p Fd(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33659
Fe(0)3529 -1005 y FV(\))p 3567 -1017 6 23 v 3573 -1027
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33660
25 3 v 59 w Ff(1)l FC(\024)l Ff(1)3705 -1058 y FC(8)3740
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33661
-1047 y Fd(L)p 3275 -973 431 4 v 3369 -921 a FZ(M)3427
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33662
-911 y FP(1)p 3474 -933 6 23 v 3479 -943 25 3 v 3518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33663
-921 a Ff(1)l FC(\024)l Ff(1)3705 -961 y FC(8)3740 -950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33664
y Fd(L)p 3834 -972 247 4 v 3834 -921 a Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33665
Ff(1)p 3942 -933 6 23 v 3947 -943 25 3 v 60 w(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33666
Ff(1)p 3369 -891 712 4 v 3471 -840 a(1)l FC(\024)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33667
FC(\033)q Ff(0)l FZ(<)l Ff(1)p FZ(;)13 b(M)3794 -830
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33668
y FP(1)p 3840 -852 6 23 v 3846 -861 25 3 v 3885 -840
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33669
a Ff(0)l FZ(<)l Ff(1)4079 -882 y FC(\033)4127 -871 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33670
Fd(L)p 3414 -810 622 4 v 3403 -756 a FC(8)p Ff(y)q FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33671
FV(\()p Ff(1)l FC(\024)l Ff(y)q FC(\033)r Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33672
Ff(y)q FV(\))p FZ(;)f(M)3851 -746 y FP(1)p 3897 -768
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33673
6 23 v 3903 -777 25 3 v 3942 -756 a Ff(0)l FZ(<)l Ff(1)4035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33674
-798 y FC(8)4070 -787 y Fd(L)p 3414 -723 622 4 v 3568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33675
-672 a FZ(M)3626 -662 y FP(1)3659 -672 y FZ(;)e(S)p 3743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33676
-684 6 23 v 3748 -694 25 3 v 61 w Ff(0)l FZ(<)l Ff(1)4035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33677
-712 y FC(8)4070 -701 y Fd(L)p 4258 -1310 319 4 v 4258
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33678
-1259 a Ff(f1)e FV(=)f Ff(0)p 4402 -1271 6 23 v 4408
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33679
-1281 25 3 v 59 w(f1)g FV(=)g Ff(0)p 4626 -1310 319 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33680
v 50 w(f1)h FV(=)f Ff(1)p 4770 -1271 6 23 v 4776 -1281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33681
25 3 v 59 w(f1)h FV(=)f Ff(1)p 4258 -1239 687 4 v 4277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33682
-1188 a(f1)g FV(=)g Ff(0)p FC(_)p Ff(f1)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33683
4592 -1200 6 23 v 4598 -1210 25 3 v 59 w(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33684
Ff(0)p FZ(;)j Ff(f1)e FV(=)f Ff(1)4944 -1231 y FC(_)4985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33685
-1219 y Fd(L)p 4221 -1158 761 4 v 4210 -1104 a FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33686
Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33687
FV(=)f Ff(1)p FV(\))p 4648 -1116 6 23 v 4654 -1126 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33688
3 v 59 w Ff(f1)g FV(=)g Ff(0)p FZ(;)k Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33689
Ff(1)4981 -1146 y FC(8)5016 -1135 y Fd(L)p 5146 -1310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33690
319 4 v 5146 -1259 a Ff(f0)h FV(=)f Ff(0)p 5290 -1271
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33691
6 23 v 5296 -1281 25 3 v 59 w(f0)h FV(=)f Ff(0)p 5515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33692
-1310 319 4 v 50 w(f0)h FV(=)f Ff(1)p 5659 -1271 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33693
v 5665 -1281 25 3 v 59 w(f0)g FV(=)g Ff(1)p 5146 -1239
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33694
687 4 v 5165 -1188 a(f0)h FV(=)f Ff(0)p FC(_)p Ff(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33695
FV(=)g Ff(1)p 5481 -1200 6 23 v 5486 -1210 25 3 v 59
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33696
w(f0)h FV(=)f Ff(0)p FZ(;)k Ff(f0)c FV(=)g Ff(1)5833
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33697
-1231 y FC(_)5874 -1219 y Fd(L)p 5110 -1158 761 4 v 5099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33698
-1104 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33699
FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p 5536 -1116 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33700
v 5542 -1126 25 3 v 59 w Ff(f0)h FV(=)f Ff(0)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33701
Ff(f0)c FV(=)g Ff(1)5869 -1146 y FC(8)5904 -1135 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33702
4221 -1072 1650 4 v 4489 -1018 a FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33703
FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33704
FV(\))p 4927 -1030 6 23 v 4933 -1040 25 3 v 59 w Ff(f1)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33705
FV(=)g Ff(0)p FC(^)p Ff(f0)h FV(=)f Ff(0)p FZ(;)j Ff(f1)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33706
FV(=)f Ff(1)p FZ(;)k Ff(f0)c FV(=)g Ff(1)5869 -1063 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33707
FC(^)5911 -1052 y Fd(R)p 6008 -1069 351 4 v 6008 -1018
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33708
a Ff(f0)g FV(=)g Ff(f1)p 6168 -1030 6 23 v 6174 -1040
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33709
25 3 v 60 w(f0)g FV(=)g Ff(f1)p 4500 -985 1859 4 v 4675
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33710
-931 a FV(\()p Ff(f1)h FV(=)f Ff(0)p FC(^)o Ff(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33711
Ff(0)p FV(\))p FC(\033)p Ff(f0)h FV(=)f Ff(f1)p FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33712
FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33713
Ff(fx)h FV(=)f Ff(1)p FV(\))p 5674 -943 6 23 v 5680 -953
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33714
25 3 v 59 w Ff(f0)h FV(=)f Ff(f1)p FZ(;)k Ff(f1)c FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33715
Ff(1)p FZ(;)k Ff(f0)d FV(=)f Ff(1)6358 -976 y FC(\033)6406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33716
-965 y Fd(L)p 4618 -899 1623 4 v 4607 -845 a FC(8)p Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33717
FZ(:)p FV(\(\()p Ff(fy)i FV(=)e Ff(0)p FC(^)o Ff(f0)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33718
FV(=)f Ff(0)p FV(\))p FC(\033)p Ff(f0)h FV(=)f Ff(fy)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33719
FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33720
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p 5731 -857
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33721
6 23 v 5737 -867 25 3 v 59 w Ff(f0)h FV(=)f Ff(f1)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33722
Ff(f1)c FV(=)g Ff(1)p FZ(;)k Ff(f0)d FV(=)f Ff(1)6240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33723
-887 y FC(8)6275 -876 y Fd(L)p 4598 -813 1662 4 v 4588
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33724
-759 a FC(8)p Ff(x)p FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33725
FV(=)f Ff(0)p FC(^)p Ff(fx)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33726
Ff(fx)g FV(=)g Ff(fy)q FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33727
FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33728
FV(\))p 5751 -771 6 23 v 5757 -781 25 3 v 59 w Ff(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33729
FV(=)g Ff(f1)q FZ(;)j Ff(f1)e FV(=)f Ff(1)p FZ(;)j Ff(f0)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33730
FV(=)f Ff(1)6259 -801 y FC(8)6294 -790 y Fd(L)p 4598
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33731
-726 1662 4 v 4914 -672 a FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33732
Ff(fx)g FV(=)g Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33733
FZ(;)j(T)p 5425 -684 6 23 v 5431 -694 25 3 v 68 w Ff(f0)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33734
FV(=)f Ff(f1)p FZ(;)j Ff(f1)e FV(=)f Ff(1)p FZ(;)k Ff(f0)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33735
FV(=)g Ff(1)6259 -715 y FC(8)6294 -704 y Fd(L)p 3568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33736
-640 2367 4 v 4085 -586 a FZ(M)4143 -576 y FP(1)4176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33737
-586 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33738
Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33739
4773 -598 6 23 v 4778 -608 25 3 v 67 w Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33740
Ff(1)p FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33741
Ff(f1)p FZ(;)k Ff(f0)c FV(=)g Ff(1)5934 -632 y FC(^)5975
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33742
-621 y Fd(R)3275 -541 y Fc(|)p 3299 -541 1540 8 v 1540
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33743
w({z)p 4887 -541 V 1540 w(})4823 -480 y Fd(X)4869 -468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33744
y FP(3)p 2885 -224 247 4 v 2885 -172 a Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33745
Ff(2)p 2993 -184 6 23 v 2999 -194 25 3 v 61 w(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33746
Ff(2)p 2793 -142 431 4 v 2783 -77 a FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33747
FV(\()p Ff(x)l FC(\024)l Ff(m)2991 -98 y Fe(x)p Fd(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33748
Fe(2)3047 -77 y FV(\))p 3086 -89 6 23 v 3091 -99 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33749
v 59 w Ff(1)l FC(\024)l Ff(2)3223 -130 y FC(8)3258 -119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33750
y Fd(L)p 2793 -45 431 4 v 2887 7 a FZ(M)2945 17 y FP(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33751
2992 -5 6 23 v 2997 -15 25 3 v 3036 7 a Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33752
Ff(2)3223 -33 y FC(8)3258 -22 y Fd(L)p 3352 -44 247 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33753
v 3352 7 a Ff(0)l FZ(<)l Ff(2)p 3460 -5 6 23 v 3466 -15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33754
25 3 v 61 w(0)l FZ(<)l Ff(2)p 2887 37 712 4 v 2989 88
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33755
a(1)l FC(\024)l Ff(2)p FC(\033)r Ff(0)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33756
FZ(;)12 b(M)3312 98 y FP(1)p 3358 76 6 23 v 3364 66 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33757
3 v 3403 88 a Ff(0)l FZ(<)l Ff(2)3598 46 y FC(\033)3646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33758
57 y Fd(L)p 2932 118 622 4 v 2921 172 a FC(8)p Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33759
FZ(:)p FV(\()p Ff(1)l FC(\024)l Ff(y)q FC(\033)r Ff(0)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33760
FZ(<)l Ff(y)q FV(\))p FZ(;)g(M)3369 182 y FP(1)p 3415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33761
160 6 23 v 3421 150 25 3 v 3460 172 a Ff(0)l FZ(<)l Ff(2)3553
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33762
130 y FC(8)3588 141 y Fd(L)p 2932 205 622 4 v 3086 255
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33763
a FZ(M)3144 265 y FP(1)3177 255 y FZ(;)e(S)p 3261 243
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33764
6 23 v 3266 233 25 3 v 61 w Ff(0)l FZ(<)l Ff(2)3553 216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33765
y FC(8)3588 227 y Fd(L)p 3776 -382 319 4 v 3776 -331
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33766
a Ff(f2)e FV(=)f Ff(0)p 3920 -343 6 23 v 3926 -353 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33767
3 v 59 w(f2)h FV(=)f Ff(0)p 4144 -382 319 4 v 49 w(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33768
FV(=)f Ff(1)p 4288 -343 6 23 v 4294 -353 25 3 v 59 w(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33769
FV(=)f Ff(1)p 3776 -311 687 4 v 3795 -260 a(f2)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33770
Ff(0)p FC(_)o Ff(f2)h FV(=)f Ff(1)p 4110 -272 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33771
4116 -282 25 3 v 59 w(f2)h FV(=)f Ff(0)p FZ(;)j Ff(f2)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33772
FV(=)f Ff(1)4462 -303 y FC(_)4504 -292 y Fd(L)p 3739
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33773
-230 761 4 v 3729 -176 a FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33774
Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33775
4166 -188 6 23 v 4172 -198 25 3 v 59 w Ff(f2)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33776
Ff(0)p FZ(;)j Ff(f2)e FV(=)f Ff(1)4499 -219 y FC(8)4534
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33777
-208 y Fd(L)p 4665 -382 319 4 v 4665 -331 a Ff(f0)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33778
Ff(0)p 4809 -343 6 23 v 4814 -353 25 3 v 59 w(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33779
Ff(0)p 5033 -382 319 4 v 50 w(f0)h FV(=)f Ff(1)p 5177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33780
-343 6 23 v 5183 -353 25 3 v 59 w(f0)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33781
4665 -311 687 4 v 4684 -260 a(f0)g FV(=)g Ff(0)p FC(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33782
Ff(f0)h FV(=)f Ff(1)p 4999 -272 6 23 v 5005 -282 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33783
v 59 w(f0)g FV(=)g Ff(0)p FZ(;)k Ff(f0)d FV(=)f Ff(1)5351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33784
-303 y FC(_)5392 -292 y Fd(L)p 4628 -230 761 4 v 4617
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33785
-176 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33786
FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p 5055 -188 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33787
5060 -198 25 3 v 59 w Ff(f0)h FV(=)f Ff(0)p FZ(;)k Ff(f0)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33788
FV(=)g Ff(1)5388 -219 y FC(8)5423 -208 y Fd(L)p 3739
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33789
-144 1650 4 v 4007 -90 a FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33790
Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33791
4445 -102 6 23 v 4451 -112 25 3 v 59 w Ff(f2)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33792
Ff(0)p FC(^)p Ff(f0)h FV(=)f Ff(0)p FZ(;)j Ff(f2)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33793
Ff(1)p FZ(;)k Ff(f0)c FV(=)g Ff(1)5388 -136 y FC(^)5429
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33794
-125 y Fd(R)p 5526 -141 351 4 v 5526 -90 a Ff(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33795
Ff(f2)p 5686 -102 6 23 v 5692 -112 25 3 v 59 w(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33796
Ff(f2)p 4018 -58 1859 4 v 4193 -4 a FV(\()p Ff(f2)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33797
Ff(0)p FC(^)o Ff(f0)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33798
Ff(f0)h FV(=)f Ff(f2)p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33799
Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33800
5192 -16 6 23 v 5198 -26 25 3 v 59 w Ff(f0)h FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33801
FZ(;)k Ff(f2)c FV(=)g Ff(1)p FZ(;)k Ff(f0)d FV(=)f Ff(1)5876
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33802
-49 y FC(\033)5924 -37 y Fd(L)p 4136 29 1623 4 v 4126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33803
83 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33804
FC(^)p Ff(f0)g FV(=)g Ff(0)p FV(\))p FC(\033)p Ff(f0)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33805
FV(=)f Ff(fy)q FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33806
Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33807
5249 71 6 23 v 5255 61 25 3 v 59 w Ff(f0)h FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33808
FZ(;)k Ff(f2)c FV(=)g Ff(1)p FZ(;)k Ff(f0)d FV(=)f Ff(1)5758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33809
40 y FC(8)5793 51 y Fd(L)p 4117 115 1662 4 v 4106 169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33810
a FC(8)p Ff(x)p FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33811
FV(=)e Ff(0)p FC(^)o Ff(fx)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33812
Ff(fx)h FV(=)f Ff(fy)q FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33813
FV(\()p Ff(fx)g FV(=)g Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33814
FV(\))p 5269 157 6 23 v 5275 147 25 3 v 59 w Ff(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33815
FV(=)g Ff(f2)q FZ(;)j Ff(f2)e FV(=)f Ff(1)p FZ(;)k Ff(f0)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33816
FV(=)g Ff(1)5777 127 y FC(8)5812 138 y Fd(L)p 4117 201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33817
1662 4 v 4432 255 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33818
FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33819
4943 243 6 23 v 4949 233 25 3 v 67 w Ff(f0)d FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33820
FZ(;)j Ff(f2)e FV(=)f Ff(1)p FZ(;)k Ff(f0)d FV(=)f Ff(1)5777
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33821
213 y FC(8)5812 224 y Fd(L)p 3086 288 2367 4 v 3603 342
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33822
a FZ(M)3661 352 y FP(1)3694 342 y FZ(;)p FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33823
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33824
Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4291 330 6 23 v 4297
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33825
320 25 3 v 68 w Ff(f2)d FV(=)g Ff(1)p FZ(;)k Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33826
Ff(2)p FC(^)q Ff(f0)d FV(=)f Ff(f2)p FZ(;)k Ff(f0)c FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33827
Ff(1)5452 296 y FC(^)5493 307 y Fd(R)6070 191 y F3(.)6070
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33828
224 y(.)6070 257 y(.)6070 290 y(.)6035 342 y FZ(X)6085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33829
352 y FP(3)p 3603 374 2532 4 v 3962 428 a FZ(M)4020 438
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33830
y FP(1)4052 428 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33831
Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33832
FZ(;)j(S)o(;)h(T)p 4649 416 6 23 v 4655 406 25 3 v 67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33833
w Ff(f2)d FV(=)f Ff(1)p FC(^)p Ff(f1)g FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33834
FZ(;)k Ff(0)l FZ(<)l Ff(2)p FC(^)q Ff(f0)d FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33835
FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33836
FZ(;)k Ff(f0)c FV(=)g Ff(1)6134 382 y FC(^)6176 394 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33837
Fd(R)2793 473 y Fc(|)p 2817 473 1667 8 v 1667 w({z)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33838
4532 473 V 1667 w(})4469 534 y Fd(X)4515 546 y FP(4)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33839
2660 648 247 4 v 2660 699 a Ff(2)l FC(\024)l Ff(2)p 2768
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33840
687 6 23 v 2773 678 25 3 v 61 w(2)l FC(\024)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33841
2567 729 433 4 v 2556 794 a FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33842
Ff(2)l FC(\024)l Ff(m)2766 773 y Fe(x)p Fd(;)p Fe(2)2823
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33843
794 y FV(\))p 2861 782 6 23 v 2867 772 25 3 v 59 w Ff(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33844
FC(\024)l Ff(2)2999 741 y FC(8)3034 752 y Fd(L)p 2567
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33845
827 433 4 v 2662 878 a FZ(M)2720 888 y FP(2)p 2766 866
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33846
6 23 v 2772 856 25 3 v 2811 878 a Ff(2)l FC(\024)l Ff(2)2999
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33847
838 y FC(8)3034 849 y Fd(L)p 3127 827 247 4 v 3127 878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33848
a Ff(1)l FZ(<)l Ff(2)p 3235 866 6 23 v 3241 856 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33849
v 61 w(1)l FZ(<)l Ff(2)p 2662 908 713 4 v 2764 960 a(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33850
FC(\024)l Ff(2)p FC(\033)q Ff(1)l FZ(<)l Ff(2)p FZ(;)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33851
b(M)3087 970 y FP(2)p 3133 948 6 23 v 3139 938 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33852
3178 960 a Ff(1)l FZ(<)l Ff(2)3373 917 y FC(\033)3421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33853
928 y Fd(L)p 2707 990 622 4 v 2696 1044 a FC(8)p Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33854
FZ(:)p FV(\()p Ff(2)l FC(\024)l Ff(y)q FC(\033)r Ff(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33855
FZ(<)l Ff(y)q FV(\))p FZ(;)f(M)3144 1054 y FP(2)p 3190
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33856
1032 6 23 v 3196 1022 25 3 v 3235 1044 a Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33857
Ff(2)3328 1001 y FC(8)3363 1012 y Fd(L)p 2707 1076 622
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33858
4 v 2861 1127 a FZ(M)2919 1137 y FP(2)2951 1127 y FZ(;)f(S)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33859
3035 1115 6 23 v 3041 1105 25 3 v 61 w Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33860
Ff(2)3328 1088 y FC(8)3363 1099 y Fd(L)4500 634 y F3(.)4500
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33861
667 y(.)4500 700 y(.)4500 733 y(.)4465 784 y FZ(X)4515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33862
794 y FP(4)p 4615 733 351 4 v 4615 784 a Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33863
Ff(f2)p 4775 772 6 23 v 4781 763 25 3 v 59 w(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33864
Ff(f2)p 3591 814 2232 4 v 3591 868 a FV(\()p Ff(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33865
FV(=)f Ff(1)p FC(^)o Ff(f1)h FV(=)f Ff(1)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33866
Ff(f1)h FV(=)f Ff(f2)p FZ(;)k(M)4222 878 y FP(1)4254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33867
868 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33868
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33869
4851 856 6 23 v 4857 846 25 3 v 68 w Ff(f1)d FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33870
FZ(;)j Ff(0)l FZ(<)l Ff(2)p FC(^)q Ff(f0)e FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33871
FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33872
FZ(;)k Ff(f0)d FV(=)f Ff(1)5822 823 y FC(\033)5870 834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33873
y Fd(L)p 3534 900 2346 4 v 3523 954 a FC(8)p Ff(y)q FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33874
FV(\(\()p Ff(fy)i FV(=)e Ff(1)p FC(^)o Ff(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33875
Ff(1)p FV(\))p FC(\033)p Ff(f1)h FV(=)f Ff(fy)q FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33876
FZ(;)j(M)4278 964 y FP(1)4311 954 y FZ(;)p FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33877
FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33878
Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 4908 942 6 23 v 4913
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33879
932 25 3 v 67 w Ff(f1)d FV(=)f Ff(f2)p FZ(;)k Ff(0)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33880
FZ(<)l Ff(2)p FC(^)q Ff(f0)d FV(=)f Ff(f2)p FZ(;)k Ff(0)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33881
FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)k Ff(f0)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33882
FV(=)g Ff(1)5879 912 y FC(8)5914 923 y Fd(L)p 3514 987
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33883
2385 4 v 3504 1041 a FC(8)p Ff(x)p FZ(:)p Ff(y)q FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33884
FV(\(\()p Ff(fy)i FV(=)e Ff(1)p FC(^)o Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33885
Ff(1)p FV(\))p FC(\033)p Ff(fx)h FV(=)f Ff(fy)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33886
FZ(;)k(M)4298 1051 y FP(1)4330 1041 y FZ(;)p FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33887
FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33888
Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p 4927 1029 6 23 v 4933
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33889
1019 25 3 v 67 w Ff(f1)d FV(=)f Ff(f2)p FZ(;)k Ff(0)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33890
FZ(<)l Ff(2)p FC(^)q Ff(f0)d FV(=)f Ff(f2)p FZ(;)j Ff(0)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33891
FZ(<)l Ff(1)p FC(^)r Ff(f0)d FV(=)g Ff(f1)q FZ(;)j Ff(f0)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33892
FV(=)f Ff(1)5898 998 y FC(8)5933 1009 y Fd(L)p 3514 1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33893
2385 4 v 3877 1127 a FZ(M)3935 1137 y FP(1)3967 1127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33894
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33895
Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33896
4565 1115 6 23 v 4570 1105 25 3 v 67 w Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33897
Ff(f2)p FZ(;)k Ff(0)l FZ(<)l Ff(2)p FC(^)q Ff(f0)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33898
Ff(f2)p FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33899
Ff(f1)p FZ(;)k Ff(f0)c FV(=)g Ff(1)5898 1085 y FC(8)5933
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33900
1096 y Fd(L)p 2861 1159 2676 4 v 3242 1213 a FZ(M)3300
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33901
1223 y FP(1)3332 1213 y FZ(;)j(M)3419 1223 y FP(2)3452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33902
1213 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33903
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33904
4049 1201 6 23 v 4054 1191 25 3 v 68 w Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33905
Ff(2)p FC(^)q Ff(f0)d FV(=)f Ff(f2)p FZ(;)j Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33906
Ff(1)p FC(^)r Ff(f0)d FV(=)g Ff(f1)q FZ(;)j Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33907
Ff(2)p FC(^)r Ff(f1)d FV(=)g Ff(f2)q FZ(;)j Ff(f0)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33908
Ff(1)5535 1168 y FC(^)5577 1179 y Fd(R)p 3159 1246 2079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33909
4 v 3159 1300 a FZ(M)3217 1310 y FP(1)3250 1300 y FZ(;)j(M)3337
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33910
1310 y FP(2)3369 1300 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33911
Ff(fx)e FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33912
FZ(;)j(S)o(;)h(T)p 3966 1288 6 23 v 3972 1278 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33913
57 w FC(9)p Ff(m)p FZ(:)p FV(\()p Ff(0)l FZ(<)l Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33914
FC(^)q Ff(f0)c FV(=)g Ff(fm)p FV(\))p FZ(;)k Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33915
Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)j Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33916
Ff(2)p FC(^)r Ff(f1)e FV(=)f Ff(f2)p FZ(;)j Ff(f0)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33917
Ff(1)5237 1257 y FC(9)5272 1268 y Fd(R)p 3135 1332 2127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33918
4 v 3135 1386 a FZ(M)3193 1396 y FP(1)3226 1386 y FZ(;)j(M)3313
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33919
1396 y FP(2)3345 1386 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33920
Ff(fx)e FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33921
FZ(;)j(S)o(;)h(T)p 3943 1374 6 23 v 3948 1364 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33922
67 w Ff(0)l FZ(<)l Ff(1)p FC(^)r Ff(f0)c FV(=)g Ff(f1)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33923
FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)e FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33924
FZ(;)k Ff(f0)d FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33925
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33926
FV(=)g Ff(fm)p FV(\))5261 1344 y FC(9)5296 1355 y Fd(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33927
3053 1418 2292 4 v 3053 1472 a FZ(M)3111 1482 y FP(1)3143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33928
1472 y FZ(;)k(M)3231 1482 y FP(2)3263 1472 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33929
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33930
FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p 3860 1460 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33931
23 v 3866 1450 25 3 v 56 w FC(9)p Ff(m)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33932
Ff(0)l FZ(<)l Ff(m)p FC(^)q Ff(f0)d FV(=)f Ff(fm)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33933
FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)d FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33934
FZ(;)j Ff(f0)e FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33935
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33936
FV(=)f Ff(fm)o FV(\))5343 1430 y FC(9)5378 1441 y Fd(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33937
3053 1505 2292 4 v 3291 1559 a FZ(M)3349 1569 y FP(1)3381
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33938
1559 y FZ(;)k(M)3469 1569 y FP(2)3501 1559 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33939
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33940
FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4098 1547 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33941
23 v 4104 1537 25 3 v 68 w Ff(1)l FZ(<)l Ff(2)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33942
Ff(f1)e FV(=)f Ff(f2)p FZ(;)k Ff(f0)c FV(=)g Ff(1)p FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33943
FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33944
Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5343 1516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33945
y FC(9)5378 1527 y Fd(R)p 3209 1591 1981 4 v 3209 1645
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33946
a FZ(M)3267 1655 y FP(1)3299 1645 y FZ(;)j(M)3386 1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33947
y FP(2)3419 1645 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33948
Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33949
FZ(;)j(S)o(;)h(T)p 4016 1633 6 23 v 4021 1623 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33950
57 w FC(9)p Ff(m)p FZ(:)p FV(\()p Ff(1)l FZ(<)l Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33951
FC(^)q Ff(f1)c FV(=)g Ff(fm)p FV(\))p FZ(;)k Ff(f0)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33952
FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33953
FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33954
FV(\))5188 1603 y FC(9)5223 1614 y Fd(R)p 3209 1677 1981
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33955
4 v 3446 1731 a FZ(M)3504 1741 y FP(1)3537 1731 y FZ(;)j(M)3624
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33956
1741 y FP(2)3656 1731 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33957
Ff(fx)e FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33958
FZ(;)k(S)o(;)g(T)p 4253 1719 6 23 v 4259 1709 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33959
67 w Ff(f0)d FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33960
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33961
FV(=)g Ff(fm)p FV(\))5188 1689 y FC(9)5223 1700 y Fd(R)2567
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33962
1776 y Fc(|)p 2591 1776 1658 8 v 1658 w({z)p 4297 1776
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33963
V 1658 w(})4233 1838 y Fd(X)4279 1850 y FP(5)2259 1970
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33964
y Gc(M)2327 1979 y FV(1)2381 1970 y F9(=)19 b FX(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33965
Fu(y)q Gc(:)p Fu(x)p Gc(:)n F9(\()p Fu(x)8 b FX(\024)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33966
Fu(m)2788 1947 y Ff(x)p FZ(;)p Ff(y)2859 1970 y F9(\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33967
b Gc(T)30 b F9(=)19 b FX(8)p Fu(i)p Gc(:)p Fu(x)p Gc(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33968
Fu(y)q Gc(:)m F9(\(\()p Fu(fy)h F9(=)g Fu(i)p FX(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33969
Fu(fx)f F9(=)g Fu(i)p F9(\))12 b FX(\033)g Fu(fx)18 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33970
F9(=)i Fu(fy)p F9(\))2259 2097 y Gc(M)2327 2106 y FV(2)2381
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33971
2097 y F9(=)f FX(8)p Fu(y)q Gc(:)p Fu(x)p Gc(:)n F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33972
Fu(y)9 b FX(\024)f Fu(m)2789 2073 y Ff(x)p FZ(;)p Ff(y)2860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33973
2097 y F9(\))100 b Gc(S)23 b F9(=)d FX(8)p Fu(x)p Gc(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33974
Fu(y)q Gc(:)m F9(\()p Fu(sx)8 b FX(\024)g Fu(y)k FX(\033)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33975
Fu(x)c Gc(<)g Fu(y)q F9(\))6739 408 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33976
 currentpoint grestore moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33977
 6739 408 a 4009
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33978
5012 4 4630 v 321 5015 3691 4 v Black 1262 5283 a Gg(Figure)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33979
b(3:)29 b(Subproofs)c Ga(X)2092 5297 y F9(3)2132 5283
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33980
y Gg(,)d Ga(X)2252 5297 y F9(4)2315 5283 y Gg(and)i Ga(X)2544
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33981
5297 y F9(5)2583 5283 y Gg(.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33982
%%Page: 135 147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33983
TeXDict begin 135 146 bop Black 3831 51 a Gb(135)p 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33984
88 3691 4 v Black Black 277 393 3226 4 v 277 5116 4 4724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33985
v 1882 465 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33986
 gsave currentpoint currentpoint translate -90 neg rotate neg exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33987
neg exch translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33988
 1882 465 a 2724 -868 247 4 v 2724 -816 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33989
Ff(2)l FC(\024)l Ff(2)p 2832 -828 6 23 v 2837 -838 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33990
3 v 60 w(2)l FC(\024)l Ff(2)p 3020 -867 247 4 v 52 w(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33991
FZ(<)l Ff(2)p 3128 -828 6 23 v 3134 -838 25 3 v 61 w(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33992
FZ(<)l Ff(2)p 2724 -786 543 4 v 2739 -734 a(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33993
Ff(2)p FC(\033)r Ff(1)l FZ(<)l Ff(2)p FZ(;)12 b Ff(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33994
FC(\024)l Ff(2)p 3112 -746 6 23 v 3118 -756 25 3 v 61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33995
w(1)l FZ(<)l Ff(2)3266 -777 y FC(\033)3314 -766 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33996
2682 -704 626 4 v 2672 -650 a FC(8)p Ff(y)q FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33997
Ff(2)l FC(\024)l Ff(y)q FC(\033)q Ff(1)l FZ(<)l Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33998
FV(\))p FZ(;)g Ff(2)l FC(\024)l Ff(2)p 3169 -662 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 33999
v 3175 -672 25 3 v 61 w(1)l FZ(<)l Ff(2)3307 -693 y FC(8)3342
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34000
-682 y Fd(L)p 2682 -618 626 4 v 2837 -566 a Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34001
Ff(2)p FZ(;)g(S)p 3015 -578 6 23 v 3020 -588 25 3 v 61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34002
w Ff(1)l FZ(<)l Ff(2)3307 -606 y FC(8)3342 -595 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34003
3448 -1029 319 4 v 3448 -977 a Ff(f2)c FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34004
3592 -989 6 23 v 3598 -999 25 3 v 59 w(f2)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34005
3817 -1029 319 4 v 50 w(f1)g FV(=)g Ff(0)p 3961 -989
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34006
6 23 v 3966 -999 25 3 v 59 w(f1)h FV(=)f Ff(0)p 3448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34007
-958 687 4 v 3467 -906 a(f2)h FV(=)f Ff(0)p FZ(;)k Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34008
FV(=)g Ff(0)p 3770 -918 6 23 v 3776 -928 25 3 v 59 w(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34009
FV(=)f Ff(0)p FC(^)p Ff(f1)g FV(=)g Ff(0)4134 -949 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34010
FC(^)4176 -938 y Fd(R)p 4273 -958 351 4 v 4273 -906 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34011
Ff(f1)g FV(=)g Ff(f2)p 4433 -918 6 23 v 4439 -928 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34012
3 v 60 w(f1)g FV(=)g Ff(f2)p 3467 -877 1157 4 v 3512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34013
-823 a FV(\()p Ff(f2)h FV(=)f Ff(0)p FC(^)o Ff(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34014
Ff(0)p FV(\))p FC(\033)p Ff(f1)h FV(=)f Ff(f2)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34015
Ff(f2)c FV(=)g Ff(0)p FZ(;)k Ff(f1)d FV(=)f Ff(0)p 4388
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34016
-835 6 23 v 4394 -845 25 3 v 59 w(f1)g FV(=)g Ff(f2)4623
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34017
-868 y FC(\033)4671 -856 y Fd(L)p 3455 -790 1181 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34018
3445 -736 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34019
Ff(0)p FC(^)p Ff(f1)g FV(=)g Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34020
Ff(f1)h FV(=)f Ff(fy)q FV(\))p FZ(;)k Ff(f2)c FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34021
Ff(0)p FZ(;)k Ff(f1)d FV(=)f Ff(0)p 4445 -748 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34022
4451 -758 25 3 v 59 w(f1)g FV(=)g Ff(f2)4635 -779 y FC(8)4670
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34023
-768 y Fd(L)p 3436 -704 1220 4 v 3425 -650 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34024
FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i FV(=)e Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34025
FC(^)o Ff(fx)h FV(=)f Ff(0)p FV(\))p FC(\033)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34026
FV(=)f Ff(fy)q FV(\))p FZ(;)j Ff(f2)e FV(=)f Ff(0)p FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34027
Ff(f1)e FV(=)f Ff(0)p 4464 -662 6 23 v 4470 -672 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34028
v 59 w(f1)h FV(=)f Ff(f2)4654 -692 y FC(8)4689 -681 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34029
Fd(L)p 3436 -618 1220 4 v 3761 -566 a Ff(f2)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34030
Ff(0)p FZ(;)k Ff(f1)c FV(=)g Ff(0)p FZ(;)k(T)p 4139 -578
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34031
6 23 v 4144 -588 25 3 v 67 w Ff(f1)d FV(=)f Ff(f2)4654
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34032
-606 y FC(8)4689 -595 y Fd(L)p 2837 -537 1493 4 v 3136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34033
-485 a Ff(f2)h FV(=)f Ff(0)p FZ(;)k Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34034
Ff(2)p FZ(;)h Ff(f1)c FV(=)f Ff(0)p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34035
3704 -497 6 23 v 3709 -507 25 3 v 68 w Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34036
Ff(2)p FC(^)q Ff(f1)d FV(=)f Ff(f2)4329 -528 y FC(^)4370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34037
-517 y Fd(R)4829 -636 y F3(.)4829 -603 y(.)4829 -569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34038
y(.)4829 -536 y(.)4801 -485 y FZ(Y)4837 -475 y FP(7)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34039
2660 -455 2705 4 v 2660 -401 a Ff(f2)g FV(=)g Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34040
FC(_)p Ff(f2)h FV(=)f Ff(1)p FZ(;)j Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34041
Ff(2)p FZ(;)j Ff(1)l FC(\024)l Ff(2)p FZ(;)f Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34042
FV(=)f Ff(0)p FZ(;)j(M)3454 -391 y FP(1)3487 -401 y FZ(;)g(M)3574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34043
-391 y FP(2)3606 -401 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34044
Ff(fx)e FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34045
FZ(;)j(S)o(;)h(T)p 4203 -413 6 23 v 4209 -423 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34046
67 w Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)c FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34047
FZ(;)j Ff(0)l FZ(<)l Ff(2)p FC(^)q Ff(f0)e FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34048
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34049
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5363
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34050
-447 y FC(_)5404 -435 y Fd(L)p 2660 -369 2705 4 v 2825
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34051
-315 a Ff(2)l FC(\024)l Ff(2)p FZ(;)12 b Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34052
Ff(2)p FZ(;)h Ff(f1)7 b FV(=)g Ff(0)p FZ(;)k(M)3289 -305
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34053
y FP(1)3321 -315 y FZ(;)g(M)3409 -305 y FP(2)3441 -315
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34054
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34055
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34056
4038 -327 6 23 v 4044 -337 25 3 v 68 w Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34057
Ff(2)p FC(^)q Ff(f1)e FV(=)f Ff(f2)p FZ(;)k Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34058
Ff(2)p FC(^)q Ff(f0)d FV(=)f Ff(f2)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34059
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34060
Ff(fn)h FV(=)f Ff(fm)o FV(\))5363 -357 y FC(8)5398 -346
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34061
y Fd(L)p 2733 -282 2558 4 v 2722 -218 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34062
FZ(:)p FV(\()p Ff(x)l FC(\024)l Ff(m)2930 -238 y Fe(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34063
Fd(;)p Fe(1)2987 -218 y FV(\))p FZ(;)j Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34064
Ff(2)p FZ(;)j Ff(f1)7 b FV(=)g Ff(0)p FZ(;)k(M)3381 -208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34065
y FP(1)3413 -218 y FZ(;)g(M)3501 -208 y FP(2)3533 -218
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34066
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34067
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34068
4130 -230 6 23 v 4136 -239 25 3 v 67 w Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34069
Ff(2)p FC(^)q Ff(f1)d FV(=)f Ff(f2)p FZ(;)k Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34070
Ff(2)p FC(^)q Ff(f0)d FV(=)f Ff(f2)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34071
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34072
Ff(fn)h FV(=)f Ff(fm)o FV(\))5290 -271 y FC(8)5325 -260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34073
y Fd(L)p 2733 -185 2558 4 v 2887 -131 a Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34074
Ff(2)p FZ(;)12 b Ff(f1)c FV(=)f Ff(0)p FZ(;)j(M)3227
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34075
-121 y FP(1)3260 -131 y FZ(;)g(M)3347 -121 y FP(2)3379
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34076
-131 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34077
Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34078
3976 -143 6 23 v 3982 -153 25 3 v 67 w Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34079
Ff(2)p FC(^)r Ff(f1)c FV(=)g Ff(f2)q FZ(;)j Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34080
Ff(2)p FC(^)q Ff(f0)e FV(=)f Ff(f2)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34081
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34082
Ff(fn)h FV(=)f Ff(fm)p FV(\))5290 -173 y FC(8)5325 -162
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34083
y Fd(L)p 2794 -99 2437 4 v 2783 -34 a FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34084
FV(\()p Ff(1)l FC(\024)l Ff(m)2993 -55 y Fe(x)p Fd(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34085
Fe(1)3050 -34 y FV(\))p FZ(;)j Ff(f1)e FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34086
FZ(;)j(M)3320 -24 y FP(1)3353 -34 y FZ(;)g(M)3440 -24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34087
y FP(2)3473 -34 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34088
Ff(fx)d FV(=)g Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34089
FZ(;)j(S)o(;)h(T)p 4070 -46 6 23 v 4075 -56 25 3 v 67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34090
w Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)c FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34091
FZ(;)j Ff(0)l FZ(<)l Ff(2)p FC(^)q Ff(f0)e FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34092
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34093
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5229
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34094
-87 y FC(8)5264 -76 y Fd(L)p 2794 -2 2437 4 v 2948 52
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34095
a Ff(f1)h FV(=)f Ff(0)p FZ(;)k(M)3166 62 y FP(1)3198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34096
52 y FZ(;)f(M)3285 62 y FP(2)3318 52 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34097
Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34098
FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 3915 40 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34099
v 3920 30 25 3 v 68 w Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34100
FV(=)f Ff(f2)p FZ(;)j Ff(0)l FZ(<)l Ff(2)p FC(^)r Ff(f0)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34101
FV(=)g Ff(f2)q FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34102
FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34103
FV(\))5229 10 y FC(8)5264 21 y Fd(L)2660 97 y Fc(|)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34104
2684 97 1347 8 v 1347 w({z)p 4079 97 V 1347 w(})4021
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34105
159 y Fd(Y)4055 171 y FP(8)p 2140 531 247 4 v 2140 583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34106
a Ff(1)l FC(\024)l Ff(1)p 2248 571 6 23 v 2253 561 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34107
3 v 60 w(1)l FC(\024)l Ff(1)p 2047 613 433 4 v 2036 678
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34108
a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(1)l FC(\024)l Ff(m)2246
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34109
657 y Fe(x)p Fd(;)p Fe(1)2303 678 y FV(\))p 2341 666
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34110
6 23 v 2347 656 25 3 v 59 w Ff(1)l FC(\024)l Ff(1)2478
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34111
624 y FC(8)2513 635 y Fd(L)p 2047 710 433 4 v 2142 761
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34112
a FZ(M)2200 771 y FP(2)p 2246 749 6 23 v 2252 740 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34113
3 v 2291 761 a Ff(1)l FC(\024)l Ff(1)2478 722 y FC(8)2513
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34114
733 y Fd(L)p 2607 711 247 4 v 2607 761 a Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34115
Ff(1)p 2715 749 6 23 v 2721 740 25 3 v 61 w(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34116
Ff(1)p 2142 791 713 4 v 2244 843 a(1)l FC(\024)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34117
FC(\033)q Ff(0)l FZ(<)l Ff(1)p FZ(;)13 b(M)2567 853 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34118
FP(2)p 2613 831 6 23 v 2619 821 25 3 v 2658 843 a Ff(0)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34119
FZ(<)l Ff(1)2853 801 y FC(\033)2901 812 y Fd(L)p 2187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34120
873 622 4 v 2176 927 a FC(8)p Ff(y)q FZ(:)p FV(\()p Ff(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34121
FC(\024)l Ff(y)q FC(\033)r Ff(0)l FZ(<)l Ff(y)q FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34122
FZ(;)f(M)2624 937 y FP(2)p 2670 915 6 23 v 2676 905 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34123
3 v 2715 927 a Ff(0)l FZ(<)l Ff(1)2808 885 y FC(8)2843
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34124
896 y Fd(L)p 2187 959 622 4 v 2341 1010 a FZ(M)2399 1020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34125
y FP(2)2431 1010 y FZ(;)f(S)p 2515 998 6 23 v 2521 988
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34126
25 3 v 61 w Ff(0)l FZ(<)l Ff(1)2808 971 y FC(8)2843 982
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34127
y Fd(L)4033 258 y F3(.)4033 291 y(.)4033 324 y(.)4033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34128
357 y(.)4005 409 y FZ(Y)4041 419 y FP(8)p 4141 357 319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34129
4 v 4141 409 a Ff(f1)c FV(=)g Ff(1)p 4285 397 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34130
4290 387 25 3 v 59 w(f1)h FV(=)f Ff(1)p 2994 438 2458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34131
4 v 2994 492 a(f1)h FV(=)f Ff(0)p FC(_)p Ff(f1)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34132
Ff(1)p FZ(;)k(M)3383 502 y FP(1)3415 492 y FZ(;)g(M)3503
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34133
502 y FP(2)3535 492 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34134
Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34135
FZ(;)k(S)o(;)g(T)p 4132 480 6 23 v 4138 470 25 3 v 67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34136
w Ff(f1)d FV(=)f Ff(1)p FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34137
Ff(f1)d FV(=)g Ff(f2)q FZ(;)j Ff(0)l FZ(<)l Ff(2)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34138
Ff(f0)e FV(=)f Ff(f2)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34139
FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34140
Ff(fm)p FV(\))5451 446 y FC(_)5492 458 y Fd(L)p 2994
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34141
525 2458 4 v 3160 578 a FZ(M)3218 588 y FP(1)3250 578
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34142
y FZ(;)k(M)3338 588 y FP(2)3370 578 y FZ(;)p FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34143
FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34144
Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 3967 566 6 23 v 3973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34145
557 25 3 v 68 w Ff(f1)d FV(=)g Ff(1)p FZ(;)k Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34146
Ff(2)p FC(^)q Ff(f1)d FV(=)f Ff(f2)p FZ(;)k Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34147
Ff(2)p FC(^)q Ff(f0)d FV(=)f Ff(f2)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34148
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34149
Ff(fn)h FV(=)f Ff(fm)o FV(\))5451 536 y FC(8)5486 547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34150
y Fd(L)5646 428 y F3(.)5646 461 y(.)5646 494 y(.)5646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34151
527 y(.)5604 578 y FZ(Y)5640 588 y FP(13)p 3160 611 2559
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34152
4 v 3290 665 a FZ(M)3348 675 y FP(1)3380 665 y FZ(;)k(M)3468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34153
675 y FP(2)3500 665 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34154
Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34155
FZ(;)k(S)o(;)f(T)p 4097 653 6 23 v 4103 643 25 3 v 68
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34156
w Ff(f1)e FV(=)f Ff(1)p FC(^)o Ff(f0)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34157
FZ(;)k Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34158
FZ(;)j Ff(0)l FZ(<)l Ff(2)p FC(^)r Ff(f0)d FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34159
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34160
FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34161
619 y FC(^)5759 630 y Fd(R)p 5856 614 351 4 v 5856 665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34162
a Ff(f0)g FV(=)g Ff(f1)p 6016 653 6 23 v 6021 643 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34163
3 v 60 w(f0)g FV(=)g Ff(f1)p 3290 697 2917 4 v 3390 751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34164
a FV(\()p Ff(f1)h FV(=)f Ff(1)p FC(^)p Ff(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34165
Ff(1)p FV(\))p FC(\033)p Ff(f0)g FV(=)g Ff(f1)q FZ(;)j(M)4021
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34166
761 y FP(1)4053 751 y FZ(;)h(M)4141 761 y FP(2)4173 751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34167
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34168
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34169
4770 739 6 23 v 4776 729 25 3 v 67 w Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34170
FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)e FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34171
FZ(;)j Ff(0)l FZ(<)l Ff(2)p FC(^)r Ff(f0)d FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34172
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34173
FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))6206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34174
706 y FC(\033)6254 717 y Fd(L)p 3334 784 2830 4 v 3323
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34175
838 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34176
FC(^)p Ff(f0)h FV(=)f Ff(1)p FV(\))p FC(\033)p Ff(f0)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34177
FV(=)f Ff(fy)p FV(\))p FZ(;)k(M)4078 848 y FP(1)4110
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34178
838 y FZ(;)g(M)4198 848 y FP(2)4230 838 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34179
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34180
FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4827 826 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34181
v 4833 816 25 3 v 68 w Ff(f0)e FV(=)f Ff(f1)p FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34182
Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)d FV(=)g Ff(f2)q FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34183
Ff(0)l FZ(<)l Ff(2)p FC(^)r Ff(f0)d FV(=)g Ff(f2)q FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34184
FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34185
Ff(m)p FC(^)p Ff(fn)h FV(=)f Ff(fm)p FV(\))6162 795 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34186
FC(8)6197 806 y Fd(L)p 3314 870 2869 4 v 3303 924 a FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34187
Ff(x)p FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i FV(=)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34188
Ff(1)p FC(^)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34189
Ff(fx)h FV(=)f Ff(fy)q FV(\))p FZ(;)k(M)4098 934 y FP(1)4130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34190
924 y FZ(;)f(M)4217 934 y FP(2)4250 924 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34191
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)g Ff(0)p FC(_)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34192
FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 4847 912 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34193
v 4852 902 25 3 v 67 w Ff(f0)d FV(=)f Ff(f1)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34194
Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d FV(=)f Ff(f2)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34195
Ff(0)l FZ(<)l Ff(2)p FC(^)q Ff(f0)d FV(=)f Ff(f2)p FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34196
FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34197
Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)o FV(\))6182 882 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34198
FC(8)6217 893 y Fd(L)p 3314 956 2869 4 v 3677 1010 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34199
FZ(M)3735 1020 y FP(1)3767 1010 y FZ(;)k(M)3855 1020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34200
y FP(2)3887 1010 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34201
Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34202
FZ(;)k(S)o(;)f(T)p 4484 998 6 23 v 4490 988 25 3 v 68
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34203
w Ff(f0)d FV(=)g Ff(f1)q FZ(;)j Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34204
FC(^)r Ff(f1)d FV(=)g Ff(f2)q FZ(;)j Ff(0)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34205
FC(^)q Ff(f0)e FV(=)f Ff(f2)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34206
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34207
FV(=)f Ff(fm)p FV(\))6182 968 y FC(8)6217 979 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34208
2341 1043 3479 4 v 2941 1097 a FZ(M)2999 1107 y FP(1)3032
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34209
1097 y FZ(;)j(M)3119 1107 y FP(2)3151 1097 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34210
Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34211
FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 3748 1085 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34212
23 v 3754 1075 25 3 v 67 w Ff(1)l FZ(<)l Ff(2)p FC(^)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34213
Ff(f1)c FV(=)g Ff(f2)q FZ(;)j Ff(0)l FZ(<)l Ff(2)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34214
Ff(f0)e FV(=)f Ff(f2)p FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34215
Ff(f0)d FV(=)f Ff(f1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34216
FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34217
Ff(fm)p FV(\))5819 1051 y FC(^)5860 1062 y Fd(R)p 2859
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34218
1129 2443 4 v 2859 1183 a FZ(M)2917 1193 y FP(1)2949
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34219
1183 y FZ(;)k(M)3037 1193 y FP(2)3069 1183 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34220
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34221
FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p 3666 1171 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34222
23 v 3672 1161 25 3 v 56 w FC(9)p Ff(m)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34223
Ff(1)l FZ(<)l Ff(m)p FC(^)q Ff(f1)d FV(=)f Ff(fm)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34224
FZ(;)j Ff(0)l FZ(<)l Ff(2)p FC(^)r Ff(f0)d FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34225
FZ(;)j Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)e FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34226
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34227
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5301
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34228
1141 y FC(9)5336 1152 y Fd(R)p 2859 1215 2443 4 v 3097
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34229
1269 a FZ(M)3155 1279 y FP(1)3187 1269 y FZ(;)k(M)3275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34230
1279 y FP(2)3307 1269 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34231
Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34232
FZ(;)k(S)o(;)f(T)p 3904 1257 6 23 v 3910 1247 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34233
68 w Ff(0)l FZ(<)l Ff(2)p FC(^)q Ff(f0)e FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34234
FZ(;)j Ff(0)l FZ(<)l Ff(1)p FC(^)r Ff(f0)e FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34235
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34236
FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5301
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34237
1227 y FC(9)5336 1238 y Fd(R)p 3015 1302 2132 4 v 3015
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34238
1356 a FZ(M)3073 1366 y FP(1)3105 1356 y FZ(;)j(M)3192
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34239
1366 y FP(2)3225 1356 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34240
Ff(fx)d FV(=)g Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34241
FZ(;)j(S)o(;)h(T)p 3822 1344 6 23 v 3827 1334 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34242
57 w FC(9)p Ff(m)p FZ(:)p FV(\()p Ff(0)l FZ(<)l Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34243
FC(^)q Ff(f0)c FV(=)g Ff(fm)p FV(\))p FZ(;)k Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34244
Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34245
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34246
Ff(fn)h FV(=)f Ff(fm)o FV(\))5145 1313 y FC(9)5180 1324
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34247
y Fd(R)p 3015 1388 2132 4 v 3252 1442 a FZ(M)3310 1452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34248
y FP(1)3342 1442 y FZ(;)k(M)3430 1452 y FP(2)3462 1442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34249
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34250
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34251
4059 1430 6 23 v 4065 1420 25 3 v 67 w Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34252
Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34253
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34254
Ff(fn)h FV(=)f Ff(fm)p FV(\))5145 1400 y FC(9)5180 1411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34255
y Fd(R)p 3170 1474 1821 4 v 3170 1528 a FZ(M)3228 1538
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34256
y FP(1)3260 1528 y FZ(;)k(M)3348 1538 y FP(2)3380 1528
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34257
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34258
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34259
3977 1516 6 23 v 3983 1506 25 3 v 57 w FC(9)p Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34260
FZ(:)p FV(\()p Ff(0)l FZ(<)l Ff(m)p FC(^)q Ff(f0)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34261
Ff(fm)p FV(\))p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34262
FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34263
FV(\))4990 1486 y FC(9)5025 1497 y Fd(R)p 3170 1561 1821
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34264
4 v 3408 1615 a FZ(M)3466 1625 y FP(1)3498 1615 y FZ(;)j(M)3585
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34265
1625 y FP(2)3618 1615 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34266
Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34267
FZ(;)j(S)o(;)h(T)p 4215 1603 6 23 v 4220 1593 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34268
57 w FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34269
FZ(<)l Ff(m)p FC(^)q Ff(fn)c FV(=)g Ff(fm)p FV(\))4990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34270
1572 y FC(9)5025 1583 y Fd(R)1982 1803 y Gc(M)2050 1812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34271
y FV(1)2104 1803 y F9(=)19 b FX(8)p Fu(y)q Gc(:)p Fu(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34272
Gc(:)n F9(\()p Fu(x)8 b FX(\024)g Fu(m)2511 1780 y Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34273
FZ(;)p Ff(y)2582 1803 y F9(\))101 b Gc(T)30 b F9(=)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34274
b FX(8)p Fu(i)p Gc(:)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34275
Fu(fy)h F9(=)g Fu(i)p FX(^)p Fu(fx)e F9(=)i Fu(i)p F9(\))12
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34276
b FX(\033)g Fu(fx)18 b F9(=)i Fu(fy)p F9(\))1982 1930
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34277
y Gc(M)2050 1939 y FV(2)2104 1930 y F9(=)f FX(8)p Fu(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34278
Gc(:)p Fu(x)p Gc(:)n F9(\()p Fu(y)9 b FX(\024)f Fu(m)2512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34279
1906 y Ff(x)p FZ(;)p Ff(y)2583 1930 y F9(\))100 b Gc(S)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34280
b F9(=)d FX(8)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\()p Fu(sx)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34281
b FX(\024)g Fu(y)k FX(\033)g Fu(x)c Gc(<)g Fu(y)q F9(\))6462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34282
465 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34283
 currentpoint grestore moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34284
 6462 465 a 3499 5116 4 4724 v 277 5119 3226 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34285
Black 1010 5273 a Gg(Figure)24 b(4:)29 b(Second)24 b(normal)h(form)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34286
(and)h(Subproof)h Ga(Y)2707 5287 y F9(8)2746 5273 y Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34287
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34288
%%Page: 136 148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34289
TeXDict begin 136 147 bop Black -144 51 a Gb(136)2816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34290
b(Experimental)24 b(Data)p -144 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34291
321 393 3226 4 v 321 5116 4 4724 v 1927 465 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34292
 gsave currentpoint currentpoint translate -90 neg rotate neg exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34293
neg exch translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34294
 1927 465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34295
a 2969 -802 247 4 v 2969 -750 a Ff(1)l FC(\024)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34296
3077 -762 6 23 v 3082 -772 25 3 v 60 w(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34297
Ff(1)p 3265 -801 247 4 v 52 w(0)l FZ(<)l Ff(1)p 3373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34298
-762 6 23 v 3379 -772 25 3 v 61 w(0)l FZ(<)l Ff(1)p 2969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34299
-720 543 4 v 2984 -669 a(1)l FC(\024)l Ff(1)p FC(\033)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34300
Ff(0)l FZ(<)l Ff(1)p FZ(;)12 b Ff(1)l FC(\024)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34301
3357 -681 6 23 v 3363 -691 25 3 v 61 w(0)l FZ(<)l Ff(1)3511
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34302
-711 y FC(\033)3559 -700 y Fd(L)p 2927 -639 626 4 v 2917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34303
-585 a FC(8)p Ff(y)q FZ(:)p FV(\()p Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34304
Ff(y)q FC(\033)q Ff(0)l FZ(<)l Ff(y)q FV(\))p FZ(;)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34305
Ff(1)l FC(\024)l Ff(1)p 3414 -597 6 23 v 3420 -607 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34306
3 v 61 w(0)l FZ(<)l Ff(1)3552 -627 y FC(8)3587 -616 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34307
Fd(L)p 2927 -552 626 4 v 3082 -501 a Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34308
Ff(1)p FZ(;)g(S)p 3260 -513 6 23 v 3265 -523 25 3 v 61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34309
w Ff(0)l FZ(<)l Ff(1)3552 -541 y FC(8)3587 -530 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34310
3693 -963 319 4 v 3693 -912 a Ff(f1)c FV(=)f Ff(0)p 3837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34311
-924 6 23 v 3843 -934 25 3 v 59 w(f1)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34312
4062 -963 319 4 v 50 w(f0)g FV(=)g Ff(0)p 4206 -924 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34313
23 v 4211 -934 25 3 v 59 w(f0)h FV(=)f Ff(0)p 3693 -892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34314
687 4 v 3712 -841 a(f1)h FV(=)f Ff(0)p FZ(;)j Ff(f0)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34315
FV(=)f Ff(0)p 4015 -853 6 23 v 4021 -862 25 3 v 59 w(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34316
FV(=)f Ff(0)p FC(^)p Ff(f0)g FV(=)g Ff(0)4379 -884 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34317
FC(^)4421 -872 y Fd(R)p 4518 -892 351 4 v 4518 -841 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34318
Ff(f0)g FV(=)g Ff(f1)p 4678 -853 6 23 v 4684 -862 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34319
3 v 60 w(f0)g FV(=)g Ff(f1)p 3712 -811 1157 4 v 3757
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34320
-757 a FV(\()p Ff(f1)h FV(=)f Ff(0)p FC(^)o Ff(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34321
Ff(0)p FV(\))p FC(\033)p Ff(f0)h FV(=)f Ff(f1)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34322
Ff(f1)c FV(=)g Ff(0)p FZ(;)k Ff(f0)d FV(=)f Ff(0)p 4633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34323
-769 6 23 v 4639 -779 25 3 v 59 w(f0)g FV(=)g Ff(f1)4868
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34324
-802 y FC(\033)4916 -791 y Fd(L)p 3700 -725 1181 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34325
3690 -671 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34326
Ff(0)p FC(^)p Ff(f0)g FV(=)g Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34327
Ff(f0)h FV(=)f Ff(fy)q FV(\))p FZ(;)j Ff(f1)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34328
Ff(0)p FZ(;)k Ff(f0)d FV(=)f Ff(0)p 4690 -683 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34329
4696 -693 25 3 v 59 w(f0)g FV(=)g Ff(f1)4880 -713 y FC(8)4915
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34330
-702 y Fd(L)p 3681 -638 1220 4 v 3670 -584 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34331
FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i FV(=)e Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34332
FC(^)o Ff(fx)h FV(=)f Ff(0)p FV(\))p FC(\033)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34333
FV(=)f Ff(fy)q FV(\))p FZ(;)j Ff(f1)e FV(=)f Ff(0)p FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34334
Ff(f0)e FV(=)f Ff(0)p 4709 -596 6 23 v 4715 -606 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34335
v 59 w(f0)h FV(=)f Ff(f1)4899 -627 y FC(8)4934 -616 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34336
Fd(L)p 3681 -552 1220 4 v 4006 -501 a Ff(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34337
Ff(0)p FZ(;)k Ff(f0)c FV(=)g Ff(0)p FZ(;)k(T)p 4384 -513
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34338
6 23 v 4389 -523 25 3 v 67 w Ff(f0)d FV(=)f Ff(f1)4899
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34339
-540 y FC(8)4934 -529 y Fd(L)p 3082 -471 1493 4 v 3381
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34340
-419 a Ff(f1)h FV(=)f Ff(0)p FZ(;)k Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34341
Ff(1)p FZ(;)h Ff(f0)c FV(=)f Ff(0)p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34342
3949 -431 6 23 v 3954 -441 25 3 v 67 w Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34343
Ff(1)p FC(^)r Ff(f0)d FV(=)f Ff(f1)4573 -462 y FC(^)4615
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34344
-451 y Fd(R)5074 -570 y F3(.)5074 -537 y(.)5074 -504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34345
y(.)5074 -470 y(.)5046 -419 y FZ(Y)5082 -409 y FP(5)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34346
2887 -389 2741 4 v 2887 -335 a Ff(f1)g FV(=)g Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34347
FC(_)p Ff(f1)h FV(=)f Ff(1)p FZ(;)j Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34348
Ff(1)p FZ(;)j Ff(f0)7 b FV(=)g Ff(0)p FZ(;)k(M)3558 -325
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34349
y FP(1)3590 -335 y FZ(;)g(M)3678 -325 y FP(2)3710 -335
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34350
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34351
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34352
4307 -347 6 23 v 4313 -357 25 3 v 68 w Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34353
Ff(1)p FC(^)q Ff(f0)e FV(=)f Ff(f1)p FZ(;)k Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34354
Ff(2)p FC(^)q Ff(f1)d FV(=)f Ff(f2)p FZ(;)k Ff(f0)c FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34355
Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34356
Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5626
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34357
-381 y FC(_)5667 -370 y Fd(L)p 2887 -303 2741 4 v 3052
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34358
-249 a Ff(1)l FC(\024)l Ff(1)p FZ(;)12 b Ff(f0)c FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34359
Ff(0)p FZ(;)k(M)3393 -239 y FP(1)3425 -249 y FZ(;)f(M)3512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34360
-239 y FP(2)3545 -249 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34361
Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34362
FZ(;)j(S)o(;)h(T)p 4142 -261 6 23 v 4147 -271 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34363
67 w Ff(0)l FZ(<)l Ff(1)p FC(^)r Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34364
FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)d FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34365
FZ(;)j Ff(f0)e FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34366
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34367
FV(=)g Ff(fm)p FV(\))5626 -291 y FC(8)5661 -280 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34368
2960 -216 2594 4 v 2949 -152 a FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34369
Ff(x)l FC(\024)l Ff(m)3157 -173 y Fe(x)p Fd(;)p Fe(0)3214
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34370
-152 y FV(\))p FZ(;)j Ff(f0)e FV(=)f Ff(0)p FZ(;)k(M)3485
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34371
-142 y FP(1)3517 -152 y FZ(;)f(M)3604 -142 y FP(2)3637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34372
-152 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34373
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34374
4234 -164 6 23 v 4240 -174 25 3 v 68 w Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34375
Ff(1)p FC(^)q Ff(f0)e FV(=)f Ff(f1)p FZ(;)j Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34376
Ff(2)p FC(^)r Ff(f1)d FV(=)g Ff(f2)q FZ(;)j Ff(f0)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34377
Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34378
Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)o FV(\))5552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34379
-205 y FC(8)5587 -194 y Fd(L)p 2960 -119 2594 4 v 3114
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34380
-65 a Ff(f0)g FV(=)g Ff(0)p FZ(;)k(M)3331 -55 y FP(1)3363
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34381
-65 y FZ(;)g(M)3451 -55 y FP(2)3483 -65 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34382
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34383
FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4080 -77 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34384
v 4086 -87 25 3 v 68 w Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34385
FV(=)f Ff(f1)p FZ(;)k Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34386
FV(=)f Ff(f2)p FZ(;)k Ff(f0)c FV(=)g Ff(1)p FZ(;)p FC(9)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34387
Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34388
FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5552 -108 y FC(8)5587
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34389
-97 y Fd(L)2887 -21 y Fc(|)p 2911 -21 1365 8 v 1365 w({z)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34390
4324 -21 V 1365 w(})4266 41 y Fd(Y)4300 53 y FP(6)p 2180
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34391
995 247 4 v 2180 1047 a Ff(1)l FC(\024)l Ff(2)p 2288
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34392
1035 6 23 v 2294 1025 25 3 v 61 w(1)l FC(\024)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34393
2476 996 247 4 v 51 w(0)l FZ(<)l Ff(2)p 2584 1035 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34394
v 2590 1025 25 3 v 61 w(0)l FZ(<)l Ff(2)p 2180 1077 543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34395
4 v 2196 1128 a(1)l FC(\024)l Ff(2)p FC(\033)q Ff(0)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34396
FZ(<)l Ff(2)p FZ(;)13 b Ff(1)l FC(\024)l Ff(2)p 2569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34397
1116 6 23 v 2574 1106 25 3 v 60 w(0)l FZ(<)l Ff(2)2722
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34398
1086 y FC(\033)2770 1097 y Fd(L)p 2139 1158 626 4 v 2128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34399
1212 a FC(8)p Ff(y)q FZ(:)p FV(\()p Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34400
Ff(y)q FC(\033)r Ff(0)l FZ(<)l Ff(y)q FV(\))p FZ(;)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34401
Ff(1)l FC(\024)l Ff(2)p 2626 1200 6 23 v 2631 1190 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34402
3 v 60 w(0)l FZ(<)l Ff(2)2763 1170 y FC(8)2798 1181 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34403
Fd(L)p 2139 1244 626 4 v 2293 1296 a Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34404
Ff(2)p FZ(;)h(S)p 2471 1284 6 23 v 2477 1274 25 3 v 61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34405
w Ff(0)l FZ(<)l Ff(2)2763 1256 y FC(8)2798 1267 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34406
2892 813 319 4 v 2892 864 a Ff(f2)8 b FV(=)f Ff(1)p 3036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34407
852 6 23 v 3042 842 25 3 v 59 w(f2)g FV(=)g Ff(1)4299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34408
198 y F3(.)4299 231 y(.)4299 265 y(.)4299 298 y(.)4270
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34409
349 y FZ(Y)4306 359 y FP(6)p 4407 298 319 4 v 4407 349
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34410
a Ff(f0)g FV(=)g Ff(1)p 4551 337 6 23 v 4556 327 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34411
v 59 w(f0)h FV(=)f Ff(1)p 3260 379 2458 4 v 3260 433
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34412
a(f0)h FV(=)f Ff(0)p FC(_)p Ff(f0)g FV(=)g Ff(1)p FZ(;)k(M)3649
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34413
443 y FP(1)3681 433 y FZ(;)g(M)3769 443 y FP(2)3801 433
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34414
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34415
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34416
4398 421 6 23 v 4404 411 25 3 v 68 w Ff(0)l FZ(<)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34417
FC(^)q Ff(f0)e FV(=)f Ff(f1)p FZ(;)k Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34418
FC(^)q Ff(f1)d FV(=)f Ff(f2)p FZ(;)k Ff(f0)c FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34419
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34420
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34421
387 y FC(_)5758 398 y Fd(L)p 3260 465 2458 4 v 3426 519
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34422
a FZ(M)3484 529 y FP(1)3516 519 y FZ(;)j(M)3603 529 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34423
FP(2)3636 519 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34424
FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34425
4233 507 6 23 v 4238 497 25 3 v 67 w Ff(0)l FZ(<)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34426
FC(^)r Ff(f0)d FV(=)f Ff(f1)p FZ(;)j Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34427
FC(^)r Ff(f1)d FV(=)g Ff(f2)q FZ(;)j Ff(f0)e FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34428
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34429
FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34430
477 y FC(8)5752 488 y Fd(L)p 3343 551 2292 4 v 3343 605
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34431
a FZ(M)3401 615 y FP(1)3434 605 y FZ(;)j(M)3521 615 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34432
FP(2)3553 605 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34433
FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34434
4150 593 6 23 v 4156 583 25 3 v 57 w FC(9)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34435
FV(\()p Ff(0)l FZ(<)l Ff(m)p FC(^)q Ff(f0)c FV(=)g Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34436
FV(\))p FZ(;)k Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34437
Ff(f2)p FZ(;)j Ff(f0)e FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34438
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34439
Ff(fn)h FV(=)f Ff(fm)p FV(\))5634 563 y FC(9)5669 574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34440
y Fd(R)p 3343 638 2292 4 v 3581 692 a FZ(M)3639 702 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34441
FP(1)3671 692 y FZ(;)k(M)3759 702 y FP(2)3791 692 y FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34442
FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34443
Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4388
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34444
680 6 23 v 4394 670 25 3 v 68 w Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34445
FC(^)q Ff(f1)e FV(=)f Ff(f2)p FZ(;)k Ff(f0)c FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34446
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34447
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34448
649 y FC(9)5669 660 y Fd(R)p 3499 724 1981 4 v 3499 778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34449
a FZ(M)3557 788 y FP(1)3589 778 y FZ(;)k(M)3677 788 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34450
FP(2)3709 778 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34451
FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34452
4306 766 6 23 v 4312 756 25 3 v 57 w FC(9)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34453
FV(\()p Ff(1)l FZ(<)l Ff(m)p FC(^)q Ff(f1)e FV(=)f Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34454
FV(\))p FZ(;)j Ff(f0)e FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34455
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34456
Ff(fn)g FV(=)g Ff(fm)p FV(\))5478 736 y FC(9)5513 747
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34457
y Fd(R)p 3499 810 1981 4 v 3736 864 a FZ(M)3794 874 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34458
FP(1)3827 864 y FZ(;)j(M)3914 874 y FP(2)3947 864 y FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34459
FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)g Ff(0)p FC(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34460
Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 4544
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34461
852 6 23 v 4549 842 25 3 v 67 w Ff(f0)d FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34462
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34463
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)o FV(\))5478
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34464
822 y FC(9)5513 833 y Fd(R)p 2892 897 2350 4 v 3149 951
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34465
a Ff(f2)h FV(=)f Ff(1)p FZ(;)j(M)3366 961 y FP(1)3398
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34466
951 y FZ(;)h(M)3486 961 y FP(2)3518 951 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34467
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34468
FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p 4115 939 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34469
v 4121 929 25 3 v 67 w Ff(f2)d FV(=)f Ff(1)p FC(^)o Ff(f0)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34470
FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34471
FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34472
FV(\))5241 905 y FC(^)5282 916 y Fd(R)p 5852 899 351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34473
4 v 5852 951 a Ff(f0)h FV(=)f Ff(f2)p 6012 939 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34474
6018 929 25 3 v 59 w(f0)h FV(=)f Ff(f2)p 3149 983 3054
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34475
4 v 3549 1037 a FV(\()p Ff(f2)h FV(=)f Ff(1)p FC(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34476
Ff(f0)g FV(=)g Ff(1)p FV(\))p FC(\033)p Ff(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34477
Ff(f2)p FZ(;)k Ff(f2)d FV(=)f Ff(1)p FZ(;)j(M)4339 1047
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34478
y FP(1)4371 1037 y FZ(;)h(M)4459 1047 y FP(2)4491 1037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34479
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34480
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34481
5088 1025 6 23 v 5094 1015 25 3 v 67 w Ff(f0)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34482
Ff(f2)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34483
Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)o FV(\))6202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34484
992 y FC(\033)6250 1003 y Fd(L)p 3492 1069 2367 4 v 3482
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34485
1123 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34486
Ff(1)p FC(^)p Ff(f0)h FV(=)f Ff(1)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34487
Ff(f0)g FV(=)g Ff(fy)q FV(\))p FZ(;)k Ff(f2)c FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34488
Ff(1)p FZ(;)k(M)4396 1133 y FP(1)4428 1123 y FZ(;)g(M)4516
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34489
1133 y FP(2)4548 1123 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34490
Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34491
FZ(;)k(S)o(;)g(T)p 5145 1111 6 23 v 5151 1101 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34492
67 w Ff(f0)d FV(=)f Ff(f2)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34493
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34494
FV(=)g Ff(fm)p FV(\))5858 1081 y FC(8)5893 1092 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34495
3473 1156 2407 4 v 3462 1210 a FC(8)p Ff(x)p FZ(:)p Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34496
FZ(:)p FV(\(\()p Ff(fy)i FV(=)e Ff(1)p FC(^)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34497
FV(=)g Ff(1)p FV(\))p FC(\033)p Ff(fx)h FV(=)f Ff(fy)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34498
FV(\))p FZ(;)j Ff(f2)e FV(=)f Ff(1)p FZ(;)k(M)4416 1220
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34499
y FP(1)4448 1210 y FZ(;)f(M)4535 1220 y FP(2)4568 1210
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34500
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34501
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34502
5165 1198 6 23 v 5170 1188 25 3 v 68 w Ff(f0)c FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34503
Ff(f2)q FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34504
Ff(n)l FZ(<)l Ff(m)p FC(^)p Ff(fn)h FV(=)f Ff(fm)p FV(\))5878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34505
1167 y FC(8)5913 1178 y Fd(L)p 3473 1242 2407 4 v 3836
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34506
1296 a Ff(f2)g FV(=)g Ff(1)p FZ(;)k(M)4053 1306 y FP(1)4085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34507
1296 y FZ(;)g(M)4173 1306 y FP(2)4205 1296 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34508
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34509
FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4802 1284 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34510
23 v 4808 1274 25 3 v 68 w Ff(f0)d FV(=)g Ff(f2)q FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34511
FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34512
Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5878 1254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34513
y FC(8)5913 1265 y Fd(L)p 2293 1328 3223 4 v 2935 1382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34514
a Ff(f2)h FV(=)f Ff(1)p FZ(;)j Ff(1)l FC(\024)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34515
FZ(;)j(M)3276 1392 y FP(1)3308 1382 y FZ(;)e(M)3396 1392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34516
y FP(2)3428 1382 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34517
Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34518
FZ(;)k(S)o(;)f(T)p 4025 1370 6 23 v 4031 1360 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34519
68 w Ff(0)l FZ(<)l Ff(2)p FC(^)q Ff(f0)e FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34520
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34521
FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34522
1337 y FC(^)5557 1348 y Fd(R)2139 1427 y Fc(|)p 2163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34523
1427 2030 8 v 2030 w({z)p 4241 1427 V 2030 w(})4183 1489
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34524
y Fd(Y)4217 1501 y FP(7)2026 1738 y Gc(M)2094 1747 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34525
FV(1)2148 1738 y F9(=)20 b FX(8)p Fu(y)q Gc(:)p Fu(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34526
Gc(:)m F9(\()p Fu(x)8 b FX(\024)g Fu(m)2555 1714 y Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34527
FZ(;)p Ff(y)2626 1738 y F9(\))101 b Gc(T)30 b F9(=)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34528
b FX(8)p Fu(i)p Gc(:)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34529
Fu(fy)i F9(=)e Fu(i)p FX(^)p Fu(fx)g F9(=)g Fu(i)p F9(\))12
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34530
b FX(\033)g Fu(fx)19 b F9(=)g Fu(fy)p F9(\))2026 1864
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34531
y Gc(M)2094 1873 y FV(2)2148 1864 y F9(=)h FX(8)p Fu(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34532
Gc(:)p Fu(x)p Gc(:)m F9(\()p Fu(y)9 b FX(\024)f Fu(m)2556
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34533
1840 y Ff(x)p FZ(;)p Ff(y)2627 1864 y F9(\))100 b Gc(S)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34534
b F9(=)d FX(8)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\()p Fu(sx)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34535
b FX(\024)g Fu(y)k FX(\033)g Fu(x)c Gc(<)g Fu(y)q F9(\))6506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34536
465 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34537
 currentpoint grestore moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34538
 6506 465 a 3543 5116 4 4724 v 321 5119 3226 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34539
Black 1364 5273 a Gg(Figure)24 b(5:)29 b(Subproofs)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34540
Ga(Y)2173 5287 y F9(6)2235 5273 y Gg(and)e Ga(Y)2442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34541
5287 y F9(7)2481 5273 y Gg(.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34542
%%Page: 137 149
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34543
TeXDict begin 137 148 bop Black 3831 51 a Gb(137)p 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34544
88 3691 4 v Black Black 277 393 3226 4 v 277 5116 4 4724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34545
v 1882 465 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34546
 gsave currentpoint currentpoint translate -90 neg rotate neg exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34547
neg exch translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34548
 1882 465 a 2724 -693 247 4 v 2724 -641 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34549
Ff(1)l FC(\024)l Ff(1)p 2832 -653 6 23 v 2837 -663 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34550
3 v 60 w(1)l FC(\024)l Ff(1)p 3020 -692 247 4 v 52 w(0)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34551
FZ(<)l Ff(1)p 3128 -653 6 23 v 3134 -663 25 3 v 61 w(0)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34552
FZ(<)l Ff(1)p 2724 -611 543 4 v 2739 -559 a(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34553
Ff(1)p FC(\033)r Ff(0)l FZ(<)l Ff(1)p FZ(;)12 b Ff(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34554
FC(\024)l Ff(1)p 3112 -571 6 23 v 3118 -581 25 3 v 61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34555
w(0)l FZ(<)l Ff(1)3266 -602 y FC(\033)3314 -591 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34556
2682 -529 626 4 v 2672 -475 a FC(8)p Ff(y)q FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34557
Ff(1)l FC(\024)l Ff(y)q FC(\033)q Ff(0)l FZ(<)l Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34558
FV(\))p FZ(;)g Ff(1)l FC(\024)l Ff(1)p 3169 -487 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34559
v 3175 -497 25 3 v 61 w(0)l FZ(<)l Ff(1)3307 -518 y FC(8)3342
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34560
-507 y Fd(L)p 2682 -443 626 4 v 2837 -392 a Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34561
Ff(1)p FZ(;)g(S)p 3015 -404 6 23 v 3020 -413 25 3 v 61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34562
w Ff(0)l FZ(<)l Ff(1)3307 -431 y FC(8)3342 -420 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34563
3448 -854 319 4 v 3448 -802 a Ff(f1)c FV(=)f Ff(0)p 3592
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34564
-814 6 23 v 3598 -824 25 3 v 59 w(f1)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34565
3817 -854 319 4 v 50 w(f0)g FV(=)g Ff(0)p 3961 -814 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34566
23 v 3966 -824 25 3 v 59 w(f0)h FV(=)f Ff(0)p 3448 -783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34567
687 4 v 3467 -731 a(f1)h FV(=)f Ff(0)p FZ(;)k Ff(f0)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34568
FV(=)g Ff(0)p 3770 -743 6 23 v 3776 -753 25 3 v 59 w(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34569
FV(=)f Ff(0)p FC(^)p Ff(f0)g FV(=)g Ff(0)4134 -774 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34570
FC(^)4176 -763 y Fd(R)p 4273 -783 351 4 v 4273 -731 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34571
Ff(f0)g FV(=)g Ff(f1)p 4433 -743 6 23 v 4439 -753 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34572
3 v 60 w(f0)g FV(=)g Ff(f1)p 3467 -702 1157 4 v 3512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34573
-648 a FV(\()p Ff(f1)h FV(=)f Ff(0)p FC(^)o Ff(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34574
Ff(0)p FV(\))p FC(\033)p Ff(f0)h FV(=)f Ff(f1)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34575
Ff(f1)c FV(=)g Ff(0)p FZ(;)k Ff(f0)d FV(=)f Ff(0)p 4388
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34576
-660 6 23 v 4394 -670 25 3 v 59 w(f0)g FV(=)g Ff(f1)4623
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34577
-693 y FC(\033)4671 -681 y Fd(L)p 3455 -615 1181 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34578
3445 -561 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34579
Ff(0)p FC(^)p Ff(f0)g FV(=)g Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34580
Ff(f0)h FV(=)f Ff(fy)q FV(\))p FZ(;)k Ff(f1)c FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34581
Ff(0)p FZ(;)k Ff(f0)d FV(=)f Ff(0)p 4445 -573 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34582
4451 -583 25 3 v 59 w(f0)g FV(=)g Ff(f1)4635 -604 y FC(8)4670
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34583
-593 y Fd(L)p 3436 -529 1220 4 v 3425 -475 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34584
FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i FV(=)e Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34585
FC(^)o Ff(fx)h FV(=)f Ff(0)p FV(\))p FC(\033)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34586
FV(=)f Ff(fy)q FV(\))p FZ(;)j Ff(f1)e FV(=)f Ff(0)p FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34587
Ff(f0)e FV(=)f Ff(0)p 4464 -487 6 23 v 4470 -497 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34588
v 59 w(f0)h FV(=)f Ff(f1)4654 -517 y FC(8)4689 -506 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34589
Fd(L)p 3436 -443 1220 4 v 3761 -392 a Ff(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34590
Ff(0)p FZ(;)k Ff(f0)c FV(=)g Ff(0)p FZ(;)k(T)p 4139 -404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34591
6 23 v 4144 -413 25 3 v 67 w Ff(f0)d FV(=)f Ff(f1)4654
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34592
-431 y FC(8)4689 -420 y Fd(L)p 2837 -362 1493 4 v 3136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34593
-310 a Ff(f1)h FV(=)f Ff(0)p FZ(;)k Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34594
Ff(1)p FZ(;)h Ff(f0)c FV(=)f Ff(0)p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34595
3704 -322 6 23 v 3709 -332 25 3 v 68 w Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34596
Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)4329 -353 y FC(^)4370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34597
-342 y Fd(R)4829 -461 y F3(.)4829 -428 y(.)4829 -394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34598
y(.)4829 -361 y(.)4801 -310 y FZ(Y)4837 -300 y FP(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34599
2425 -280 3175 4 v 2425 -226 a Ff(f1)g FV(=)g Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34600
FC(_)p Ff(f1)h FV(=)f Ff(1)p FZ(;)j Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34601
Ff(1)p FZ(;)j Ff(f0)7 b FV(=)g Ff(0)p FZ(;)k Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34602
Ff(2)p FZ(;)h Ff(f1)c FV(=)f Ff(1)p FZ(;)k(M)3379 -216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34603
y FP(1)3411 -226 y FZ(;)f(M)3498 -216 y FP(2)3531 -226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34604
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34605
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34606
4128 -238 6 23 v 4133 -248 25 3 v 67 w Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34607
Ff(1)p FC(^)r Ff(f0)d FV(=)f Ff(f1)p FZ(;)j Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34608
Ff(3)p FC(^)r Ff(f1)d FV(=)g Ff(f3)q FZ(;)j Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34609
Ff(2)p FC(^)r Ff(f1)d FV(=)g Ff(f2)q FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34610
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34611
Ff(fn)h FV(=)f Ff(fm)p FV(\))5598 -272 y FC(_)5639 -261
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34612
y Fd(L)p 2425 -194 3175 4 v 2590 -140 a Ff(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34613
Ff(1)p FZ(;)12 b Ff(f0)c FV(=)f Ff(0)p FZ(;)k Ff(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34614
FC(\024)l Ff(2)p FZ(;)h Ff(f1)c FV(=)f Ff(1)p FZ(;)j(M)3213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34615
-130 y FP(1)3246 -140 y FZ(;)g(M)3333 -130 y FP(2)3365
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34616
-140 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34617
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34618
3962 -152 6 23 v 3968 -162 25 3 v 67 w Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34619
Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)k Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34620
Ff(3)p FC(^)q Ff(f1)d FV(=)f Ff(f3)p FZ(;)k Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34621
Ff(2)p FC(^)q Ff(f1)d FV(=)f Ff(f2)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34622
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34623
Ff(fn)h FV(=)f Ff(fm)o FV(\))5598 -182 y FC(8)5633 -171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34624
y Fd(L)p 2498 -107 3028 4 v 2487 -43 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34625
FZ(:)p FV(\()p Ff(x)l FC(\024)l Ff(m)2695 -63 y Fe(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34626
Fd(;)p Fe(0)2752 -43 y FV(\))p FZ(;)j Ff(f0)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34627
Ff(0)p FZ(;)k Ff(2)l FC(\024)l Ff(2)p FZ(;)h Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34628
FV(=)f Ff(1)p FZ(;)j(M)3305 -33 y FP(1)3338 -43 y FZ(;)g(M)3425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34629
-33 y FP(2)3457 -43 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34630
Ff(fx)e FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34631
FZ(;)j(S)o(;)h(T)p 4054 -55 6 23 v 4060 -64 25 3 v 67
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34632
w Ff(0)l FZ(<)l Ff(1)p FC(^)r Ff(f0)c FV(=)g Ff(f1)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34633
FZ(;)j Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)e FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34634
FZ(;)k Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34635
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34636
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34637
-96 y FC(8)5560 -85 y Fd(L)p 2498 -10 3028 4 v 2652 44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34638
a Ff(f0)g FV(=)g Ff(0)p FZ(;)k Ff(2)l FC(\024)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34639
FZ(;)h Ff(f1)c FV(=)f Ff(1)p FZ(;)k(M)3152 54 y FP(1)3184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34640
44 y FZ(;)f(M)3271 54 y FP(2)3304 44 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34641
Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34642
FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 3901 32 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34643
v 3906 22 25 3 v 67 w Ff(0)l FZ(<)l Ff(1)p FC(^)r Ff(f0)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34644
FV(=)g Ff(f1)q FZ(;)j Ff(1)l FZ(<)l Ff(3)p FC(^)r Ff(f1)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34645
FV(=)g Ff(f3)q FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34646
FV(=)f Ff(f2)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34647
FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34648
FV(\))5525 2 y FC(8)5560 13 y Fd(L)2425 88 y Fc(|)p 2449
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34649
88 1582 8 v 1582 w({z)p 4079 88 V 1582 w(})4021 150 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34650
Fd(Y)4055 162 y FP(4)4057 424 y F3(.)4057 457 y(.)4057
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34651
490 y(.)4057 523 y(.)4029 575 y FZ(Y)4065 585 y FP(4)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34652
4165 523 319 4 v 4165 575 a Ff(f0)h FV(=)f Ff(1)p 4309
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34653
563 6 23 v 4315 553 25 3 v 59 w(f0)h FV(=)f Ff(1)p 2722
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34654
604 3051 4 v 2722 658 a(f0)h FV(=)f Ff(0)p FC(_)p Ff(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34655
FV(=)g Ff(1)p FZ(;)k Ff(2)l FC(\024)l Ff(2)p FZ(;)h Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34656
FV(=)f Ff(1)p FZ(;)k(M)3394 668 y FP(1)3426 658 y FZ(;)g(M)3514
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34657
668 y FP(2)3546 658 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34658
Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34659
FZ(;)k(S)o(;)f(T)p 4143 646 6 23 v 4149 636 25 3 v 68
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34660
w Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)e FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34661
FZ(;)k Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)d FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34662
FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)d FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34663
FZ(;)j Ff(f0)e FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34664
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34665
FV(=)f Ff(fm)o FV(\))5772 612 y FC(_)5814 623 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34666
2722 690 3051 4 v 2888 744 a Ff(2)l FC(\024)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34667
FZ(;)12 b Ff(f1)c FV(=)f Ff(1)p FZ(;)j(M)3228 754 y FP(1)3261
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34668
744 y FZ(;)g(M)3348 754 y FP(2)3381 744 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34669
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)g Ff(0)p FC(_)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34670
FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 3978 732 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34671
v 3983 722 25 3 v 67 w Ff(0)l FZ(<)l Ff(1)p FC(^)r Ff(f0)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34672
FV(=)g Ff(f1)q FZ(;)j Ff(1)l FZ(<)l Ff(3)p FC(^)r Ff(f1)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34673
FV(=)g Ff(f3)q FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34674
FV(=)f Ff(f2)p FZ(;)k Ff(f0)d FV(=)f Ff(1)p FZ(;)p FC(9)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34675
Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34676
FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5772 702 y FC(8)5807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34677
713 y Fd(L)p 2806 777 2885 4 v 2806 831 a Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34678
Ff(2)p FZ(;)12 b Ff(f1)c FV(=)f Ff(1)p FZ(;)j(M)3146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34679
841 y FP(1)3178 831 y FZ(;)h(M)3266 841 y FP(2)3298 831
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34680
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34681
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34682
3895 819 6 23 v 3901 809 25 3 v 56 w FC(9)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34683
FV(\()p Ff(0)l FZ(<)l Ff(m)p FC(^)q Ff(f0)d FV(=)f Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34684
FV(\))p FZ(;)j Ff(1)l FZ(<)l Ff(3)p FC(^)r Ff(f1)d FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34685
Ff(f3)q FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)d FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34686
Ff(f2)q FZ(;)j Ff(f0)e FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34687
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34688
Ff(fn)g FV(=)g Ff(fm)p FV(\))5689 788 y FC(9)5724 799
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34689
y Fd(R)p 2806 863 2885 4 v 3043 917 a Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34690
Ff(2)p FZ(;)13 b Ff(f1)7 b FV(=)g Ff(1)p FZ(;)k(M)3384
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34691
927 y FP(1)3416 917 y FZ(;)g(M)3504 927 y FP(2)3536 917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34692
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34693
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34694
4133 905 6 23 v 4139 895 25 3 v 68 w Ff(1)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34695
FC(^)q Ff(f1)e FV(=)f Ff(f3)p FZ(;)k Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34696
FC(^)q Ff(f1)d FV(=)f Ff(f2)p FZ(;)j Ff(f0)e FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34697
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34698
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5689
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34699
875 y FC(9)5724 886 y Fd(R)p 2961 949 2574 4 v 2961 1003
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34700
a Ff(2)l FC(\024)l Ff(2)p FZ(;)12 b Ff(f1)c FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34701
FZ(;)k(M)3302 1013 y FP(1)3334 1003 y FZ(;)f(M)3421 1013
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34702
y FP(2)3454 1003 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34703
Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34704
FZ(;)j(S)o(;)h(T)p 4051 991 6 23 v 4056 981 25 3 v 57
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34705
w FC(9)p Ff(m)p FZ(:)p FV(\()p Ff(1)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34706
Ff(f1)c FV(=)g Ff(fm)p FV(\))p FZ(;)k Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34707
FC(^)q Ff(f1)d FV(=)f Ff(f2)p FZ(;)k Ff(f0)c FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34708
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34709
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5534
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34710
961 y FC(9)5569 972 y Fd(R)p 2961 1036 2574 4 v 3199
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34711
1090 a Ff(2)l FC(\024)l Ff(2)p FZ(;)12 b Ff(f1)c FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34712
Ff(1)p FZ(;)j(M)3539 1100 y FP(1)3572 1090 y FZ(;)g(M)3659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34713
1100 y FP(2)3691 1090 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34714
Ff(fx)e FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34715
FZ(;)j(S)o(;)h(T)p 4288 1078 6 23 v 4294 1068 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34716
67 w Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)c FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34717
FZ(;)j Ff(f0)e FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34718
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34719
FV(=)g Ff(fm)p FV(\))5534 1047 y FC(9)5569 1058 y Fd(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34720
3106 1122 2285 4 v 3095 1187 a FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34721
Ff(2)l FC(\024)l Ff(m)3305 1166 y Fe(x)p Fd(;)p Fe(2)3362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34722
1187 y FV(\))p FZ(;)j Ff(f1)e FV(=)f Ff(1)p FZ(;)j(M)3632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34723
1197 y FP(1)3665 1187 y FZ(;)g(M)3752 1197 y FP(2)3784
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34724
1187 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34725
Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34726
4382 1175 6 23 v 4387 1165 25 3 v 67 w Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34727
Ff(2)p FC(^)r Ff(f1)c FV(=)g Ff(f2)q FZ(;)j Ff(f0)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34728
Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34729
Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5389
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34730
1134 y FC(8)5424 1145 y Fd(L)p 3106 1219 2285 4 v 3260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34731
1273 a Ff(f1)h FV(=)f Ff(1)p FZ(;)k(M)3478 1283 y FP(1)3510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34732
1273 y FZ(;)f(M)3597 1283 y FP(2)3630 1273 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34733
Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34734
FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 4227 1261 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34735
23 v 4232 1251 25 3 v 68 w Ff(1)l FZ(<)l Ff(2)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34736
Ff(f1)d FV(=)f Ff(f2)p FZ(;)j Ff(f0)e FV(=)f Ff(1)p FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34737
FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34738
Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5389 1231
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34739
y FC(8)5424 1242 y Fd(L)2722 1318 y Fc(|)p 2746 1318
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34740
1520 8 v 1520 w({z)p 4314 1318 V 1520 w(})4257 1379 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34741
Fd(Y)4291 1391 y FP(5)1982 1629 y Gc(M)2050 1638 y FV(1)2104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34742
1629 y F9(=)19 b FX(8)p Fu(y)q Gc(:)p Fu(x)p Gc(:)n F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34743
Fu(x)8 b FX(\024)g Fu(m)2511 1605 y Ff(x)p FZ(;)p Ff(y)2582
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34744
1629 y F9(\))101 b Gc(T)30 b F9(=)19 b FX(8)p Fu(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34745
Gc(:)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\(\()p Fu(fy)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34746
F9(=)g Fu(i)p FX(^)p Fu(fx)e F9(=)i Fu(i)p F9(\))12 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34747
FX(\033)g Fu(fx)18 b F9(=)i Fu(fy)p F9(\))1982 1755 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34748
Gc(M)2050 1764 y FV(2)2104 1755 y F9(=)f FX(8)p Fu(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34749
Gc(:)p Fu(x)p Gc(:)n F9(\()p Fu(y)9 b FX(\024)f Fu(m)2512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34750
1731 y Ff(x)p FZ(;)p Ff(y)2583 1755 y F9(\))100 b Gc(S)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34751
b F9(=)d FX(8)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\()p Fu(sx)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34752
b FX(\024)g Fu(y)k FX(\033)g Fu(x)c Gc(<)g Fu(y)q F9(\))6462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34753
465 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34754
 currentpoint grestore moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34755
 6462 465 a 3499 5116 4 4724 v 277 5119 3226 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34756
Black 1320 5273 a Gg(Figure)24 b(6:)29 b(Subproofs)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34757
Ga(Y)2128 5287 y F9(4)2190 5273 y Gg(and)f Ga(Y)2397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34758
5287 y F9(5)2437 5273 y Gg(.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34759
%%Page: 138 150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34760
TeXDict begin 138 149 bop Black -144 51 a Gb(138)2816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34761
b(Experimental)24 b(Data)p -144 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34762
321 393 3226 4 v 321 5116 4 4724 v 1927 465 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34763
 gsave currentpoint currentpoint translate -90 neg rotate neg exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34764
neg exch translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34765
 1927 465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34766
a 2094 -771 247 4 v 2094 -719 a Ff(2)l FC(\024)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34767
2202 -731 6 23 v 2207 -741 25 3 v 61 w(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34768
Ff(3)p 2390 -770 247 4 v 51 w(1)l FZ(<)l Ff(3)p 2498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34769
-731 6 23 v 2504 -741 25 3 v 61 w(1)l FZ(<)l Ff(3)p 2094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34770
-689 543 4 v 2109 -637 a(2)l FC(\024)l Ff(3)p FC(\033)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34771
Ff(1)l FZ(<)l Ff(3)p FZ(;)13 b Ff(2)l FC(\024)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34772
2483 -649 6 23 v 2488 -659 25 3 v 60 w(1)l FZ(<)l Ff(3)2636
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34773
-680 y FC(\033)2684 -669 y Fd(L)p 2053 -607 626 4 v 2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34774
-554 a FC(8)p Ff(y)q FZ(:)p FV(\()p Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34775
Ff(y)q FC(\033)q Ff(1)l FZ(<)l Ff(y)q FV(\))p FZ(;)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34776
Ff(2)l FC(\024)l Ff(3)p 2539 -566 6 23 v 2545 -575 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34777
3 v 61 w(1)l FZ(<)l Ff(3)2677 -596 y FC(8)2712 -585 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34778
Fd(L)p 2053 -521 626 4 v 2207 -470 a Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34779
Ff(3)p FZ(;)g(S)p 2385 -482 6 23 v 2391 -491 25 3 v 62
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34780
w Ff(1)l FZ(<)l Ff(3)2677 -509 y FC(8)2712 -498 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34781
2819 -932 319 4 v 2819 -881 a Ff(f3)7 b FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34782
2962 -893 6 23 v 2968 -902 25 3 v 59 w(f3)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34783
3187 -932 319 4 v 50 w(f1)h FV(=)f Ff(1)p 3331 -893 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34784
23 v 3337 -902 25 3 v 59 w(f1)g FV(=)g Ff(1)p 2819 -861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34785
687 4 v 2837 -809 a(f3)h FV(=)f Ff(1)p FZ(;)k Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34786
FV(=)g Ff(1)p 3141 -821 6 23 v 3146 -831 25 3 v 59 w(f3)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34787
FV(=)f Ff(1)p FC(^)p Ff(f1)h FV(=)f Ff(1)3505 -852 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34788
FC(^)3546 -841 y Fd(R)p 3643 -861 351 4 v 3643 -809 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34789
Ff(f1)h FV(=)f Ff(f3)p 3803 -821 6 23 v 3809 -831 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34790
3 v 59 w(f1)g FV(=)g Ff(f3)p 2837 -780 1157 4 v 2882
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34791
-726 a FV(\()p Ff(f3)h FV(=)f Ff(1)p FC(^)p Ff(f1)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34792
Ff(1)p FV(\))p FC(\033)p Ff(f1)h FV(=)f Ff(f3)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34793
Ff(f3)d FV(=)f Ff(1)p FZ(;)j Ff(f1)e FV(=)f Ff(1)p 3758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34794
-738 6 23 v 3764 -748 25 3 v 59 w(f1)h FV(=)f Ff(f3)3993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34795
-771 y FC(\033)4041 -760 y Fd(L)p 2825 -693 1181 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34796
2815 -639 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34797
Ff(1)p FC(^)p Ff(f1)g FV(=)g Ff(1)p FV(\))p FC(\033)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34798
Ff(f1)g FV(=)g Ff(fy)q FV(\))p FZ(;)k Ff(f3)c FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34799
Ff(1)p FZ(;)k Ff(f1)d FV(=)f Ff(1)p 3815 -651 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34800
3821 -661 25 3 v 59 w(f1)h FV(=)f Ff(f3)4005 -682 y FC(8)4040
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34801
-671 y Fd(L)p 2806 -607 1220 4 v 2795 -553 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34802
FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i FV(=)e Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34803
FC(^)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FC(\033)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34804
FV(=)f Ff(fy)q FV(\))p FZ(;)j Ff(f3)e FV(=)f Ff(1)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34805
Ff(f1)c FV(=)g Ff(1)p 3835 -565 6 23 v 3840 -575 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34806
v 59 w(f1)h FV(=)f Ff(f3)4024 -595 y FC(8)4059 -584 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34807
Fd(L)p 2806 -521 1220 4 v 3132 -470 a Ff(f3)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34808
Ff(1)p FZ(;)k Ff(f1)d FV(=)f Ff(1)p FZ(;)j(T)p 3509 -482
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34809
6 23 v 3515 -491 25 3 v 68 w Ff(f1)d FV(=)g Ff(f3)4024
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34810
-509 y FC(8)4059 -498 y Fd(L)p 2207 -440 1493 4 v 2507
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34811
-388 a Ff(f3)g FV(=)g Ff(1)p FZ(;)k Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34812
Ff(3)p FZ(;)h Ff(f1)c FV(=)f Ff(1)p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34813
3074 -400 6 23 v 3080 -410 25 3 v 68 w Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34814
Ff(3)p FC(^)q Ff(f1)e FV(=)f Ff(f3)3699 -431 y FC(^)3740
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34815
-420 y Fd(R)2053 -346 y Fc(|)p 2077 -346 978 8 v 978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34816
w({z)p 3103 -346 V 978 w(})3045 -284 y Fd(Y)3079 -272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34817
y FP(1)p 4370 -771 247 4 v 4370 -719 a Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34818
Ff(2)p 4478 -731 6 23 v 4484 -741 25 3 v 61 w(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34819
Ff(2)p 4667 -770 247 4 v 52 w(1)l FZ(<)l Ff(2)p 4775
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34820
-731 6 23 v 4780 -741 25 3 v 60 w(1)l FZ(<)l Ff(2)p 4370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34821
-689 543 4 v 4386 -637 a(2)l FC(\024)l Ff(2)p FC(\033)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34822
Ff(1)l FZ(<)l Ff(2)p FZ(;)12 b Ff(2)l FC(\024)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34823
4759 -649 6 23 v 4765 -659 25 3 v 61 w(1)l FZ(<)l Ff(2)4912
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34824
-680 y FC(\033)4960 -669 y Fd(L)p 4329 -607 626 4 v 4318
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34825
-554 a FC(8)p Ff(y)q FZ(:)p FV(\()p Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34826
Ff(y)q FC(\033)r Ff(1)l FZ(<)l Ff(y)q FV(\))p FZ(;)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34827
Ff(2)l FC(\024)l Ff(2)p 4816 -566 6 23 v 4822 -575 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34828
3 v 61 w(1)l FZ(<)l Ff(2)4954 -596 y FC(8)4989 -585 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34829
Fd(L)p 4329 -521 626 4 v 4484 -470 a Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34830
Ff(2)p FZ(;)g(S)p 4661 -482 6 23 v 4667 -491 25 3 v 61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34831
w Ff(1)l FZ(<)l Ff(2)4954 -509 y FC(8)4989 -498 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34832
5095 -932 319 4 v 5095 -881 a Ff(f2)c FV(=)f Ff(1)p 5239
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34833
-893 6 23 v 5245 -902 25 3 v 59 w(f2)g FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34834
5463 -932 319 4 v 50 w(f1)h FV(=)f Ff(1)p 5607 -893 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34835
23 v 5613 -902 25 3 v 59 w(f1)h FV(=)f Ff(1)p 5095 -861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34836
687 4 v 5114 -809 a(f2)h FV(=)f Ff(1)p FZ(;)j Ff(f1)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34837
FV(=)f Ff(1)p 5417 -821 6 23 v 5423 -831 25 3 v 59 w(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34838
FV(=)f Ff(1)p FC(^)o Ff(f1)h FV(=)f Ff(1)5781 -852 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34839
FC(^)5823 -841 y Fd(R)p 5919 -861 351 4 v 5919 -809 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34840
Ff(f1)h FV(=)f Ff(f2)p 6079 -821 6 23 v 6085 -831 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34841
3 v 59 w(f1)h FV(=)f Ff(f2)p 5114 -780 1157 4 v 5159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34842
-726 a FV(\()p Ff(f2)g FV(=)g Ff(1)p FC(^)p Ff(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34843
Ff(1)p FV(\))p FC(\033)p Ff(f1)g FV(=)g Ff(f2)q FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34844
Ff(f2)e FV(=)f Ff(1)p FZ(;)k Ff(f1)c FV(=)g Ff(1)p 6035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34845
-738 6 23 v 6040 -748 25 3 v 59 w(f1)h FV(=)f Ff(f2)6269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34846
-771 y FC(\033)6317 -760 y Fd(L)p 5102 -693 1181 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34847
5091 -639 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i FV(=)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34848
Ff(1)p FC(^)o Ff(f1)h FV(=)f Ff(1)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34849
Ff(f1)h FV(=)f Ff(fy)q FV(\))p FZ(;)j Ff(f2)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34850
Ff(1)p FZ(;)k Ff(f1)c FV(=)g Ff(1)p 6092 -651 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34851
6097 -661 25 3 v 59 w(f1)h FV(=)f Ff(f2)6281 -682 y FC(8)6316
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34852
-671 y Fd(L)p 5082 -607 1220 4 v 5072 -553 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34853
FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34854
FC(^)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FC(\033)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34855
FV(=)g Ff(fy)q FV(\))p FZ(;)k Ff(f2)c FV(=)g Ff(1)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34856
Ff(f1)d FV(=)f Ff(1)p 6111 -565 6 23 v 6117 -575 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34857
v 59 w(f1)h FV(=)f Ff(f2)6301 -595 y FC(8)6336 -584 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34858
Fd(L)p 5082 -521 1220 4 v 5408 -470 a Ff(f2)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34859
Ff(1)p FZ(;)j Ff(f1)e FV(=)f Ff(1)p FZ(;)k(T)p 5785 -482
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34860
6 23 v 5791 -491 25 3 v 67 w Ff(f1)d FV(=)f Ff(f2)6301
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34861
-509 y FC(8)6336 -498 y Fd(L)p 4484 -440 1493 4 v 4783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34862
-388 a Ff(f2)h FV(=)f Ff(1)p FZ(;)j Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34863
Ff(2)p FZ(;)j Ff(f1)7 b FV(=)g Ff(1)p FZ(;)k(S)o(;)g(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34864
5350 -400 6 23 v 5356 -410 25 3 v 67 w Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34865
Ff(2)p FC(^)r Ff(f1)c FV(=)g Ff(f2)5975 -431 y FC(^)6017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34866
-420 y Fd(R)4329 -346 y Fc(|)p 4353 -346 978 8 v 978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34867
w({z)p 5379 -346 V 978 w(})5321 -284 y Fd(Y)5355 -272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34868
y FP(2)p 2823 91 247 4 v 2823 142 a Ff(3)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34869
Ff(3)p 2931 130 6 23 v 2936 120 25 3 v 60 w(3)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34870
Ff(3)p 3119 92 247 4 v 52 w(2)l FZ(<)l Ff(3)p 3227 130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34871
6 23 v 3233 120 25 3 v 61 w(2)l FZ(<)l Ff(3)p 2823 172
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34872
543 4 v 2838 224 a(3)l FC(\024)l Ff(3)p FC(\033)r Ff(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34873
FZ(<)l Ff(3)p FZ(;)12 b Ff(3)l FC(\024)l Ff(3)p 3211
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34874
212 6 23 v 3217 202 25 3 v 61 w(2)l FZ(<)l Ff(3)3365
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34875
181 y FC(\033)3413 193 y Fd(L)p 2781 254 626 4 v 2771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34876
308 a FC(8)p Ff(y)q FZ(:)p FV(\()p Ff(3)l FC(\024)l Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34877
FC(\033)q Ff(2)l FZ(<)l Ff(y)q FV(\))p FZ(;)g Ff(3)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34878
FC(\024)l Ff(3)p 3268 296 6 23 v 3274 286 25 3 v 61 w(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34879
FZ(<)l Ff(3)3406 266 y FC(8)3441 277 y Fd(L)p 2781 340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34880
626 4 v 2936 392 a Ff(3)l FC(\024)l Ff(3)p FZ(;)g(S)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34881
3114 380 6 23 v 3119 370 25 3 v 61 w Ff(2)l FZ(<)l Ff(3)3406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34882
352 y FC(8)3441 363 y Fd(L)p 3547 -70 319 4 v 3547 -19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34883
a Ff(f3)c FV(=)f Ff(0)p 3691 -31 6 23 v 3697 -41 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34884
v 59 w(f3)h FV(=)f Ff(0)p 3916 -70 319 4 v 50 w(f2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34885
FV(=)g Ff(0)p 4060 -31 6 23 v 4065 -41 25 3 v 59 w(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34886
FV(=)f Ff(0)p 3547 1 687 4 v 3566 52 a(f3)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34887
FZ(;)k Ff(f2)c FV(=)g Ff(0)p 3869 40 6 23 v 3875 30 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34888
3 v 59 w(f3)h FV(=)f Ff(0)p FC(^)p Ff(f2)g FV(=)g Ff(0)4233
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34889
9 y FC(^)4275 20 y Fd(R)p 4372 1 351 4 v 4372 52 a Ff(f2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34890
FV(=)g Ff(f3)p 4532 40 6 23 v 4538 30 25 3 v 60 w(f2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34891
FV(=)g Ff(f3)p 3566 82 1157 4 v 3611 136 a FV(\()p Ff(f3)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34892
FV(=)f Ff(0)p FC(^)o Ff(f2)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34893
Ff(f2)h FV(=)f Ff(f3)p FZ(;)k Ff(f3)c FV(=)g Ff(0)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34894
Ff(f2)d FV(=)f Ff(0)p 4487 124 6 23 v 4493 114 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34895
59 w(f2)g FV(=)g Ff(f3)4722 91 y FC(\033)4770 102 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34896
3554 168 1181 4 v 3544 222 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34897
Ff(fy)h FV(=)f Ff(0)p FC(^)p Ff(f2)g FV(=)g Ff(0)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34898
FC(\033)p Ff(f2)h FV(=)f Ff(fy)q FV(\))p FZ(;)k Ff(f3)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34899
FV(=)g Ff(0)p FZ(;)k Ff(f2)d FV(=)f Ff(0)p 4544 210 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34900
23 v 4550 200 25 3 v 59 w(f2)g FV(=)g Ff(f3)4734 180
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34901
y FC(8)4769 191 y Fd(L)p 3535 254 1220 4 v 3524 308 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34902
FC(8)p Ff(x)p FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34903
FV(=)e Ff(0)p FC(^)o Ff(fx)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34904
Ff(fx)h FV(=)f Ff(fy)q FV(\))p FZ(;)j Ff(f3)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34905
Ff(0)p FZ(;)j Ff(f2)e FV(=)f Ff(0)p 4563 296 6 23 v 4569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34906
286 25 3 v 59 w(f2)h FV(=)f Ff(f3)4753 266 y FC(8)4788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34907
277 y Fd(L)p 3535 341 1220 4 v 3860 392 a Ff(f3)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34908
Ff(0)p FZ(;)k Ff(f2)c FV(=)g Ff(0)p FZ(;)k(T)p 4238 380
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34909
6 23 v 4243 370 25 3 v 67 w Ff(f2)d FV(=)f Ff(f3)4753
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34910
352 y FC(8)4788 363 y Fd(L)p 2936 422 1493 4 v 3235 473
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34911
a Ff(f3)h FV(=)f Ff(0)p FZ(;)k Ff(3)l FC(\024)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34912
FZ(;)h Ff(f2)c FV(=)f Ff(0)p FZ(;)j(S)o(;)h(T)p 3803
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34913
461 6 23 v 3808 451 25 3 v 68 w Ff(2)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34914
FC(^)q Ff(f2)d FV(=)f Ff(f3)4428 430 y FC(^)4469 441
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34915
y Fd(R)4928 323 y F3(.)4928 356 y(.)4928 389 y(.)4928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34916
422 y(.)4900 473 y FZ(Y)4936 483 y FP(1)p 3235 503 1751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34917
4 v 3282 555 a Ff(f3)g FV(=)g Ff(0)p FC(_)p Ff(f3)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34918
Ff(1)p FZ(;)j Ff(3)l FC(\024)l Ff(3)p FZ(;)j Ff(f2)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34919
b FV(=)g Ff(0)p FZ(;)k Ff(2)l FC(\024)l Ff(3)p FZ(;)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34920
Ff(f1)c FV(=)f Ff(1)p FZ(;)k(S)o(;)f(T)p 4303 543 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34921
v 4309 533 25 3 v 68 w Ff(2)l FZ(<)l Ff(3)p FC(^)q Ff(f2)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34922
FV(=)f Ff(f3)p FZ(;)j Ff(1)l FZ(<)l Ff(3)p FC(^)r Ff(f1)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34923
FV(=)f Ff(f3)4985 512 y FC(_)5027 523 y Fd(L)p 3226 585
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34924
1770 4 v 3226 639 a Ff(3)l FC(\024)l Ff(3)p FZ(;)12 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34925
Ff(f2)c FV(=)f Ff(0)p FZ(;)j Ff(2)l FC(\024)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34926
FZ(;)j Ff(f1)7 b FV(=)g Ff(1)p FZ(;)p FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34927
FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34928
FV(\))p FZ(;)j(S)o(;)h(T)p 4359 627 6 23 v 4364 617 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34929
3 v 67 w Ff(2)l FZ(<)l Ff(3)p FC(^)r Ff(f2)c FV(=)g Ff(f3)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34930
FZ(;)j Ff(1)l FZ(<)l Ff(3)p FC(^)r Ff(f1)d FV(=)g Ff(f3)4995
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34931
597 y FC(8)5030 608 y Fd(L)p 3134 671 1954 4 v 3123 736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34932
a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(x)l FC(\024)l Ff(m)3331
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34933
715 y Fe(x)p Fd(;)p Fe(2)3388 736 y FV(\))p FZ(;)j Ff(f2)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34934
FV(=)f Ff(0)p FZ(;)k Ff(2)l FC(\024)l Ff(3)p FZ(;)h Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34935
FV(=)f Ff(1)p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34936
FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34937
4451 724 6 23 v 4456 714 25 3 v 67 w Ff(2)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34938
FC(^)r Ff(f2)d FV(=)f Ff(f3)p FZ(;)j Ff(1)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34939
FC(^)r Ff(f1)d FV(=)g Ff(f3)5087 683 y FC(8)5122 694
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34940
y Fd(L)p 3134 768 1954 4 v 3228 822 a Ff(f2)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34941
Ff(0)p FZ(;)k Ff(2)l FC(\024)l Ff(3)p FZ(;)h Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34942
FV(=)f Ff(1)p FZ(;)k(M)3728 832 y FP(1)3760 822 y FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34943
FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34944
Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34945
810 6 23 v 4363 800 25 3 v 68 w Ff(2)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34946
FC(^)q Ff(f2)e FV(=)f Ff(f3)p FZ(;)j Ff(1)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34947
FC(^)r Ff(f1)e FV(=)f Ff(f3)5087 780 y FC(8)5122 791
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34948
y Fd(L)5261 672 y F3(.)5261 705 y(.)5261 738 y(.)5261
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34949
771 y(.)5233 822 y FZ(Y)5269 832 y FP(2)p 3008 855 2532
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34950
4 v 3008 909 a Ff(f2)h FV(=)f Ff(0)p FC(_)o Ff(f2)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34951
Ff(1)p FZ(;)k Ff(2)l FC(\024)l Ff(3)p FZ(;)h Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34952
FV(=)f Ff(1)p FZ(;)j Ff(2)l FC(\024)l Ff(2)p FZ(;)j Ff(f1)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34953
b FV(=)g Ff(1)p FZ(;)k(M)3962 919 y FP(1)3994 909 y FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34954
FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34955
Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p 4591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34956
897 6 23 v 4597 887 25 3 v 67 w Ff(2)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34957
FC(^)q Ff(f2)d FV(=)f Ff(f3)p FZ(;)k Ff(1)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34958
FC(^)q Ff(f1)d FV(=)f Ff(f3)p FZ(;)k Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34959
FC(^)q Ff(f1)d FV(=)f Ff(f2)5538 863 y FC(_)5580 874
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34960
y Fd(L)p 3008 941 2532 4 v 3173 995 a Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34961
Ff(3)p FZ(;)13 b Ff(f1)7 b FV(=)g Ff(1)p FZ(;)k Ff(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34962
FC(\024)l Ff(2)p FZ(;)h Ff(f1)c FV(=)f Ff(1)p FZ(;)k(M)3797
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34963
1005 y FP(1)3829 995 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34964
Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34965
FZ(;)k(S)o(;)f(T)p 4426 983 6 23 v 4432 973 25 3 v 68
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34966
w Ff(2)l FZ(<)l Ff(3)p FC(^)q Ff(f2)e FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34967
FZ(;)k Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)d FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34968
FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)d FV(=)g Ff(f2)5538
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34969
953 y FC(8)5573 964 y Fd(L)p 3091 1027 2365 4 v 3091
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34970
1081 a Ff(2)l FC(\024)l Ff(3)p FZ(;)12 b Ff(f1)c FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34971
Ff(1)p FZ(;)k Ff(2)l FC(\024)l Ff(2)p FZ(;)h Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34972
FV(=)f Ff(1)p FZ(;)j(M)3714 1091 y FP(1)3747 1081 y FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34973
FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)g Ff(0)p FC(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34974
Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 4344
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34975
1069 6 23 v 4349 1060 25 3 v 57 w FC(9)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34976
FV(\()p Ff(2)l FZ(<)l Ff(m)p FC(^)q Ff(f2)c FV(=)g Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34977
FV(\))p FZ(;)k Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34978
Ff(f3)p FZ(;)k Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34979
Ff(f2)5455 1039 y FC(9)5490 1050 y Fd(R)p 3067 1114 2413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34980
4 v 3067 1168 a Ff(2)l FC(\024)l Ff(3)p FZ(;)12 b Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34981
FV(=)f Ff(1)p FZ(;)k Ff(2)l FC(\024)l Ff(2)p FZ(;)h Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34982
FV(=)f Ff(1)p FZ(;)j(M)3690 1178 y FP(1)3723 1168 y FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34983
FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34984
Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 4320
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34985
1156 6 23 v 4325 1146 25 3 v 67 w Ff(1)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34986
FC(^)r Ff(f1)c FV(=)g Ff(f3)q FZ(;)j Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34987
FC(^)r Ff(f1)d FV(=)g Ff(f2)q FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34988
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34989
FV(=)g Ff(fm)p FV(\))5479 1125 y FC(9)5514 1136 y Fd(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34990
2974 1200 2600 4 v 2963 1265 a FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34991
Ff(2)l FC(\024)l Ff(m)3173 1244 y Fe(x)p Fd(;)p Fe(2)3230
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34992
1265 y FV(\))p FZ(;)j Ff(f1)e FV(=)f Ff(1)p FZ(;)k Ff(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34993
FC(\024)l Ff(2)p FZ(;)h Ff(f1)c FV(=)f Ff(1)p FZ(;)j(M)3783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34994
1275 y FP(1)3816 1265 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34995
Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34996
FZ(;)k(S)o(;)f(T)p 4413 1253 6 23 v 4418 1243 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34997
68 w Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)e FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34998
FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)d FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 34999
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35000
FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35001
1212 y FC(8)5607 1223 y Fd(L)p 2974 1297 2600 4 v 3069
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35002
1351 a Ff(f1)g FV(=)g Ff(1)p FZ(;)k Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35003
Ff(2)p FZ(;)h Ff(f1)c FV(=)f Ff(1)p FZ(;)k(M)3569 1361
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35004
y FP(1)3601 1351 y FZ(;)g(M)3689 1361 y FP(2)3721 1351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35005
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35006
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35007
4318 1339 6 23 v 4324 1329 25 3 v 68 w Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35008
Ff(3)p FC(^)q Ff(f1)e FV(=)f Ff(f3)p FZ(;)k Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35009
Ff(2)p FC(^)q Ff(f1)d FV(=)f Ff(f2)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35010
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35011
Ff(fn)g FV(=)g Ff(fm)p FV(\))5572 1309 y FC(8)5607 1320
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35012
y Fd(L)2781 1396 y Fc(|)p 2805 1396 1387 8 v 1387 w({z)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35013
4240 1396 V 1387 w(})4183 1457 y Fd(Y)4217 1469 y FP(3)2026
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35014
1707 y Gc(M)2094 1716 y FV(1)2148 1707 y F9(=)20 b FX(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35015
Fu(y)q Gc(:)p Fu(x)p Gc(:)m F9(\()p Fu(x)8 b FX(\024)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35016
Fu(m)2555 1683 y Ff(x)p FZ(;)p Ff(y)2626 1707 y F9(\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35017
b Gc(T)30 b F9(=)19 b FX(8)p Fu(i)p Gc(:)p Fu(x)p Gc(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35018
Fu(y)q Gc(:)m F9(\(\()p Fu(fy)i F9(=)e Fu(i)p FX(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35019
Fu(fx)g F9(=)g Fu(i)p F9(\))12 b FX(\033)g Fu(fx)19 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35020
F9(=)g Fu(fy)p F9(\))2026 1833 y Gc(M)2094 1842 y FV(2)2148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35021
1833 y F9(=)h FX(8)p Fu(y)q Gc(:)p Fu(x)p Gc(:)m F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35022
Fu(y)9 b FX(\024)f Fu(m)2556 1809 y Ff(x)p FZ(;)p Ff(y)2627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35023
1833 y F9(\))100 b Gc(S)23 b F9(=)d FX(8)p Fu(x)p Gc(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35024
Fu(y)q Gc(:)m F9(\()p Fu(sx)8 b FX(\024)g Fu(y)k FX(\033)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35025
Fu(x)c Gc(<)g Fu(y)q F9(\))6506 465 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35026
 currentpoint grestore moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35027
 6506 465 a 3543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35028
5116 4 4724 v 321 5119 3226 4 v Black 1295 5273 a Gg(Figure)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35029
b(7:)29 b(Subproofs)d Ga(Y)2104 5287 y F9(1)2143 5273
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35030
y Gg(,)c Ga(Y)2241 5287 y F9(2)2303 5273 y Gg(and)i Ga(Y)2510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35031
5287 y F9(3)2550 5273 y Gg(.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35032
%%Page: 139 151
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35033
TeXDict begin 139 150 bop Black 3831 51 a Gb(139)p 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35034
88 3691 4 v Black Black 277 393 3226 4 v 277 5116 4 4724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35035
v 1882 465 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35036
 gsave currentpoint currentpoint translate -90 neg rotate neg exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35037
neg exch translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35038
 1882 465 a 2799 -418 a F3(.)2799 -384 y(.)2799
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35039
-351 y(.)2799 -318 y(.)2757 -267 y FZ(Y)2793 -257 y FP(11)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35040
3014 -832 247 4 v 3014 -780 a Ff(1)l FC(\024)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35041
3121 -792 6 23 v 3127 -802 25 3 v 60 w(1)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35042
Ff(1)p 2921 -750 431 4 v 2911 -686 a FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35043
FV(\()p Ff(x)l FC(\024)l Ff(m)3119 -707 y Fe(x)p Fd(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35044
Fe(0)3175 -686 y FV(\))p 3214 -698 6 23 v 3219 -708 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35045
3 v 59 w Ff(1)l FC(\024)l Ff(1)3351 -739 y FC(8)3386
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35046
-728 y Fd(L)p 2921 -653 431 4 v 3015 -602 a FZ(M)3073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35047
-592 y FP(1)p 3120 -614 6 23 v 3125 -624 25 3 v 3164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35048
-602 a Ff(1)l FC(\024)l Ff(1)3351 -642 y FC(8)3386 -631
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35049
y Fd(L)p 3480 -653 247 4 v 3480 -602 a Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35050
Ff(1)p 3588 -614 6 23 v 3594 -624 25 3 v 61 w(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35051
Ff(1)p 3015 -572 712 4 v 3117 -520 a(1)l FC(\024)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35052
FC(\033)r Ff(0)l FZ(<)l Ff(1)p FZ(;)12 b(M)3440 -510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35053
y FP(1)p 3486 -532 6 23 v 3492 -542 25 3 v 3531 -520
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35054
a Ff(0)l FZ(<)l Ff(1)3726 -563 y FC(\033)3774 -552 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35055
Fd(L)p 3060 -490 622 4 v 3049 -436 a FC(8)p Ff(y)q FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35056
FV(\()p Ff(1)l FC(\024)l Ff(y)q FC(\033)r Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35057
Ff(y)q FV(\))p FZ(;)g(M)3497 -426 y FP(1)p 3543 -448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35058
6 23 v 3549 -458 25 3 v 3588 -436 a Ff(0)l FZ(<)l Ff(1)3681
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35059
-479 y FC(8)3716 -468 y Fd(L)p 3060 -404 622 4 v 3214
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35060
-353 a FZ(M)3272 -343 y FP(1)3305 -353 y FZ(;)e(S)p 3389
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35061
-365 6 23 v 3394 -375 25 3 v 61 w Ff(0)l FZ(<)l Ff(1)3681
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35062
-392 y FC(8)3716 -381 y Fd(L)p 3904 -991 319 4 v 3904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35063
-939 a Ff(f1)e FV(=)f Ff(0)p 4048 -951 6 23 v 4054 -961
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35064
25 3 v 59 w(f1)h FV(=)f Ff(0)p 4272 -991 319 4 v 49 w(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35065
FV(=)f Ff(1)p 4416 -951 6 23 v 4422 -961 25 3 v 59 w(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35066
FV(=)f Ff(1)p 3904 -920 687 4 v 3923 -868 a(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35067
Ff(0)p FC(_)o Ff(f1)h FV(=)f Ff(1)p 4238 -880 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35068
4244 -890 25 3 v 59 w(f1)h FV(=)f Ff(0)p FZ(;)j Ff(f1)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35069
FV(=)f Ff(1)4590 -911 y FC(_)4632 -900 y Fd(L)p 3867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35070
-839 761 4 v 3857 -785 a FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35071
Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35072
4294 -797 6 23 v 4300 -807 25 3 v 59 w Ff(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35073
Ff(0)p FZ(;)j Ff(f1)e FV(=)f Ff(1)4627 -827 y FC(8)4662
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35074
-816 y Fd(L)p 4793 -991 319 4 v 4793 -939 a Ff(f0)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35075
Ff(0)p 4937 -951 6 23 v 4942 -961 25 3 v 59 w(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35076
Ff(0)p 5161 -991 319 4 v 50 w(f0)h FV(=)f Ff(1)p 5305
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35077
-951 6 23 v 5311 -961 25 3 v 59 w(f0)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35078
4793 -920 687 4 v 4812 -868 a(f0)g FV(=)g Ff(0)p FC(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35079
Ff(f0)h FV(=)f Ff(1)p 5127 -880 6 23 v 5133 -890 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35080
v 59 w(f0)g FV(=)g Ff(0)p FZ(;)k Ff(f0)d FV(=)f Ff(1)5479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35081
-911 y FC(_)5520 -900 y Fd(L)p 4756 -839 761 4 v 4745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35082
-785 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35083
FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p 5183 -797 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35084
5188 -807 25 3 v 59 w Ff(f0)h FV(=)f Ff(0)p FZ(;)k Ff(f0)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35085
FV(=)g Ff(1)5516 -827 y FC(8)5551 -816 y Fd(L)p 3867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35086
-752 1650 4 v 4136 -698 a FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35087
Ff(fx)g FV(=)g Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35088
4573 -710 6 23 v 4579 -720 25 3 v 59 w Ff(f1)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35089
Ff(0)p FC(^)p Ff(f0)h FV(=)f Ff(0)p FZ(;)j Ff(f1)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35090
Ff(1)p FZ(;)k Ff(f0)d FV(=)f Ff(1)5516 -744 y FC(^)5557
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35091
-733 y Fd(R)p 5654 -750 351 4 v 5654 -698 a Ff(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35092
Ff(f1)p 5814 -710 6 23 v 5820 -720 25 3 v 59 w(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35093
Ff(f1)p 4146 -666 1859 4 v 4321 -612 a FV(\()p Ff(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35094
FV(=)f Ff(0)p FC(^)o Ff(f0)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35095
Ff(f0)h FV(=)f Ff(f1)p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35096
Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35097
5321 -624 6 23 v 5326 -634 25 3 v 59 w Ff(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35098
Ff(f1)p FZ(;)k Ff(f1)d FV(=)f Ff(1)p FZ(;)j Ff(f0)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35099
Ff(1)6004 -657 y FC(\033)6052 -646 y Fd(L)p 4264 -580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35100
1623 4 v 4254 -526 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35101
FV(=)f Ff(0)p FC(^)p Ff(f0)g FV(=)g Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35102
Ff(f0)h FV(=)f Ff(fy)q FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35103
FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35104
FV(\))p 5377 -538 6 23 v 5383 -548 25 3 v 59 w Ff(f0)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35105
FV(=)f Ff(f1)p FZ(;)k Ff(f1)c FV(=)g Ff(1)p FZ(;)k Ff(f0)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35106
FV(=)f Ff(1)5886 -568 y FC(8)5921 -557 y Fd(L)p 4245
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35107
-493 1662 4 v 4234 -439 a FC(8)p Ff(x)p FZ(:)p Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35108
FZ(:)p FV(\(\()p Ff(fy)i FV(=)e Ff(0)p FC(^)o Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35109
FV(=)f Ff(0)p FV(\))p FC(\033)p Ff(fx)h FV(=)f Ff(fy)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35110
FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35111
Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p 5397 -451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35112
6 23 v 5403 -461 25 3 v 59 w Ff(f0)g FV(=)g Ff(f1)q FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35113
Ff(f1)e FV(=)f Ff(1)p FZ(;)k Ff(f0)c FV(=)g Ff(1)5906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35114
-482 y FC(8)5941 -471 y Fd(L)p 4245 -407 1662 4 v 4560
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35115
-353 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35116
FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(T)p 5071
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35117
-365 6 23 v 5077 -375 25 3 v 67 w Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35118
FZ(;)j Ff(f1)e FV(=)f Ff(1)p FZ(;)k Ff(f0)d FV(=)f Ff(1)5906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35119
-395 y FC(8)5941 -384 y Fd(L)p 3214 -321 2367 4 v 3732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35120
-267 a FZ(M)3790 -257 y FP(1)3822 -267 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35121
Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35122
FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4419 -279 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35123
23 v 4425 -289 25 3 v 68 w Ff(f1)d FV(=)g Ff(1)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35124
Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35125
Ff(f0)c FV(=)g Ff(1)5580 -312 y FC(^)5621 -301 y Fd(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35126
2624 -234 2556 4 v 2624 -180 a FZ(M)2682 -170 y FP(1)2714
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35127
-180 y FZ(;)j(M)2801 -170 y FP(2)2834 -180 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35128
Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35129
FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 3431 -192 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35130
23 v 3436 -202 25 3 v 67 w Ff(f2)d FV(=)f Ff(1)p FC(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35131
Ff(f1)h FV(=)f Ff(1)p FZ(;)j Ff(2)l FZ(<)l Ff(3)p FC(^)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35132
Ff(f2)d FV(=)g Ff(f3)q FZ(;)j Ff(0)l FZ(<)l Ff(1)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35133
Ff(f0)e FV(=)f Ff(f1)p FZ(;)k Ff(1)l FZ(<)l Ff(3)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35134
Ff(f1)d FV(=)f Ff(f3)p FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35135
Ff(f0)d FV(=)f Ff(f1)p FZ(;)k Ff(f0)c FV(=)g Ff(1)5178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35136
-226 y FC(^)5220 -215 y Fd(R)2624 -136 y Fc(|)p 2648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35137
-136 1689 8 v 1689 w({z)p 4385 -136 V 1689 w(})4312 -74
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35138
y Fd(Y)4346 -62 y FP(12)p 2164 39 247 4 v 2164 91 a Ff(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35139
FC(\024)l Ff(2)p 2272 79 6 23 v 2278 69 25 3 v 61 w(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35140
FC(\024)l Ff(2)p 2071 121 433 4 v 2061 186 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35141
FZ(:)p FV(\()p Ff(2)l FC(\024)l Ff(m)2271 165 y Fe(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35142
Fd(;)p Fe(2)2327 186 y FV(\))p 2365 174 6 23 v 2371 164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35143
25 3 v 59 w Ff(2)l FC(\024)l Ff(2)2503 133 y FC(8)2538
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35144
144 y Fd(L)p 2071 218 433 4 v 2166 270 a FZ(M)2224 280
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35145
y FP(2)p 2270 258 6 23 v 2276 248 25 3 v 2315 270 a Ff(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35146
FC(\024)l Ff(2)2503 230 y FC(8)2538 241 y Fd(L)p 2632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35147
219 247 4 v 2632 270 a Ff(1)l FZ(<)l Ff(2)p 2740 258
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35148
6 23 v 2745 248 25 3 v 60 w(1)l FZ(<)l Ff(2)p 2166 300
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35149
713 4 v 2268 351 a(2)l FC(\024)l Ff(2)p FC(\033)r Ff(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35150
FZ(<)l Ff(2)p FZ(;)12 b(M)2591 361 y FP(2)p 2638 339
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35151
6 23 v 2643 329 25 3 v 2682 351 a Ff(1)l FZ(<)l Ff(2)2877
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35152
309 y FC(\033)2925 320 y Fd(L)p 2211 381 622 4 v 2201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35153
435 a FC(8)p Ff(y)q FZ(:)p FV(\()p Ff(2)l FC(\024)l Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35154
FC(\033)q Ff(1)l FZ(<)l Ff(y)q FV(\))p FZ(;)g(M)2648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35155
445 y FP(2)p 2695 423 6 23 v 2700 413 25 3 v 2739 435
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35156
a Ff(1)l FZ(<)l Ff(2)2832 393 y FC(8)2867 404 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35157
2211 468 622 4 v 2366 519 a FZ(M)2424 529 y FP(2)2456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35158
519 y FZ(;)f(S)p 2540 507 6 23 v 2546 497 25 3 v 61 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35159
Ff(1)l FZ(<)l Ff(2)2832 479 y FC(8)2867 490 y Fd(L)4375
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35160
25 y F3(.)4375 58 y(.)4375 92 y(.)4375 125 y(.)4333 176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35161
y FZ(Y)4369 186 y FP(12)p 4498 125 351 4 v 4498 176 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35162
Ff(f1)c FV(=)g Ff(f2)p 4658 164 6 23 v 4663 154 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35163
v 59 w(f1)h FV(=)f Ff(f2)p 3095 206 2974 4 v 3095 259
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35164
a FV(\()p Ff(f2)h FV(=)f Ff(1)p FC(^)p Ff(f1)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35165
Ff(1)p FV(\))p FC(\033)q Ff(f1)g FV(=)g Ff(f2)q FZ(;)j(M)3726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35166
269 y FP(1)3758 259 y FZ(;)h(M)3846 269 y FP(2)3878 259
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35167
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35168
Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35169
4475 247 6 23 v 4481 238 25 3 v 68 w Ff(f1)e FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35170
FZ(;)j Ff(2)l FZ(<)l Ff(3)p FC(^)r Ff(f2)d FV(=)g Ff(f3)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35171
FZ(;)j Ff(0)l FZ(<)l Ff(1)p FC(^)r Ff(f0)d FV(=)g Ff(f1)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35172
FZ(;)j Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)e FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35173
FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35174
FZ(;)k Ff(f0)d FV(=)f Ff(1)6068 215 y FC(\033)6116 226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35175
y Fd(L)p 3039 292 3087 4 v 3028 346 a FC(8)p Ff(y)q FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35176
FV(\(\()p Ff(fy)h FV(=)f Ff(1)p FC(^)p Ff(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35177
Ff(1)p FV(\))p FC(\033)p Ff(f1)g FV(=)g Ff(fy)q FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35178
FZ(;)k(M)3783 356 y FP(1)3815 346 y FZ(;)g(M)3903 356
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35179
y FP(2)3935 346 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35180
Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35181
FZ(;)k(S)o(;)f(T)p 4532 334 6 23 v 4538 324 25 3 v 68
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35182
w Ff(f1)d FV(=)g Ff(f2)q FZ(;)j Ff(2)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35183
FC(^)r Ff(f2)d FV(=)g Ff(f3)q FZ(;)j Ff(0)l FZ(<)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35184
FC(^)q Ff(f0)e FV(=)f Ff(f1)p FZ(;)k Ff(1)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35185
FC(^)q Ff(f1)d FV(=)f Ff(f3)p FZ(;)k Ff(0)l FZ(<)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35186
FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)k Ff(f0)c FV(=)g Ff(1)6125
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35187
304 y FC(8)6160 315 y Fd(L)p 3019 378 3127 4 v 3008 432
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35188
a FC(8)p Ff(x)p FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35189
FV(=)e Ff(1)p FC(^)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35190
Ff(fx)h FV(=)f Ff(fy)q FV(\))p FZ(;)j(M)3802 442 y FP(1)3835
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35191
432 y FZ(;)g(M)3922 442 y FP(2)3955 432 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35192
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)g Ff(0)p FC(_)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35193
FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 4552 420 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35194
v 4557 410 25 3 v 67 w Ff(f1)d FV(=)f Ff(f2)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35195
Ff(2)l FZ(<)l Ff(3)p FC(^)q Ff(f2)d FV(=)f Ff(f3)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35196
Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35197
Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)d FV(=)f Ff(f3)p FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35198
Ff(0)l FZ(<)l Ff(1)p FC(^)r Ff(f0)d FV(=)g Ff(f1)q FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35199
Ff(f0)e FV(=)f Ff(1)6144 390 y FC(8)6179 401 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35200
3019 465 3127 4 v 3382 519 a FZ(M)3440 529 y FP(1)3472
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35201
519 y FZ(;)j(M)3559 529 y FP(2)3592 519 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35202
Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35203
FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4189 507 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35204
v 4195 497 25 3 v 68 w Ff(f1)d FV(=)g Ff(f2)q FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35205
Ff(2)l FZ(<)l Ff(3)p FC(^)q Ff(f2)e FV(=)f Ff(f3)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35206
Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35207
Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)d FV(=)f Ff(f3)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35208
Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35209
Ff(f0)e FV(=)f Ff(1)6144 476 y FC(8)6179 487 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35210
2366 551 3417 4 v 2806 605 a FZ(M)2864 615 y FP(1)2896
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35211
605 y FZ(;)k(M)2984 615 y FP(2)3016 605 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35212
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35213
FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 3613 593 6 23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35214
v 3619 583 25 3 v 68 w Ff(2)l FZ(<)l Ff(3)p FC(^)q Ff(f2)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35215
FV(=)f Ff(f3)p FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35216
FV(=)f Ff(f1)p FZ(;)k Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35217
FV(=)f Ff(f3)p FZ(;)j Ff(0)l FZ(<)l Ff(1)p FC(^)r Ff(f0)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35218
FV(=)g Ff(f1)q FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35219
FV(=)g Ff(f2)q FZ(;)j Ff(f0)e FV(=)f Ff(1)5781 559 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35220
FC(^)5823 570 y Fd(R)p 2724 637 2701 4 v 2724 691 a FZ(M)2782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35221
701 y FP(1)2814 691 y FZ(;)k(M)2902 701 y FP(2)2934 691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35222
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35223
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35224
3531 679 6 23 v 3537 669 25 3 v 57 w FC(9)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35225
FV(\()p Ff(2)l FZ(<)l Ff(m)p FC(^)q Ff(f2)e FV(=)f Ff(fm)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35226
FV(\))p FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35227
Ff(f1)p FZ(;)k Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35228
Ff(f3)p FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35229
Ff(f1)p FZ(;)k Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35230
Ff(f2)p FZ(;)j Ff(f0)e FV(=)f Ff(1)5423 649 y FC(9)5458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35231
660 y Fd(R)p 2700 724 2749 4 v 2700 778 a FZ(M)2758 788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35232
y FP(1)2790 778 y FZ(;)k(M)2878 788 y FP(2)2910 778 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35233
FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35234
FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35235
3507 766 6 23 v 3513 756 25 3 v 68 w Ff(0)l FZ(<)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35236
FC(^)q Ff(f0)e FV(=)f Ff(f1)p FZ(;)k Ff(1)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35237
FC(^)q Ff(f1)d FV(=)f Ff(f3)p FZ(;)j Ff(0)l FZ(<)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35238
FC(^)r Ff(f0)d FV(=)g Ff(f1)q FZ(;)j Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35239
FC(^)r Ff(f1)d FV(=)g Ff(f2)q FZ(;)j Ff(f0)e FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35240
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35241
FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5447
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35242
735 y FC(9)5482 746 y Fd(R)p 2618 810 2913 4 v 2618 864
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35243
a FZ(M)2676 874 y FP(1)2708 864 y FZ(;)j(M)2795 874 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35244
FP(2)2828 864 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35245
FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35246
3425 852 6 23 v 3430 842 25 3 v 57 w FC(9)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35247
FV(\()p Ff(0)l FZ(<)l Ff(m)p FC(^)q Ff(f0)c FV(=)g Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35248
FV(\))p FZ(;)k Ff(1)l FZ(<)l Ff(3)p FC(^)q Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35249
Ff(f3)p FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35250
Ff(f1)p FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35251
Ff(f2)p FZ(;)j Ff(f0)e FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35252
FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35253
Ff(fn)h FV(=)f Ff(fm)o FV(\))5530 822 y FC(9)5565 833
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35254
y Fd(R)p 2618 896 2913 4 v 2855 950 a FZ(M)2913 960 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35255
FP(1)2946 950 y FZ(;)j(M)3033 960 y FP(2)3065 950 y FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35256
FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35257
Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p 3662
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35258
938 6 23 v 3668 928 25 3 v 67 w Ff(1)l FZ(<)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35259
FC(^)q Ff(f1)d FV(=)f Ff(f3)p FZ(;)k Ff(0)l FZ(<)l Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35260
FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)k Ff(1)l FZ(<)l Ff(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35261
FC(^)q Ff(f1)d FV(=)f Ff(f2)p FZ(;)k Ff(f0)c FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35262
FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35263
FZ(<)l Ff(m)p FC(^)q Ff(fn)h FV(=)f Ff(fm)p FV(\))5530
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35264
908 y FC(9)5565 919 y Fd(R)p 2773 983 2603 4 v 2773 1037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35265
a FZ(M)2831 1047 y FP(1)2863 1037 y FZ(;)k(M)2951 1047
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35266
y FP(2)2983 1037 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35267
Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35268
FZ(;)k(S)o(;)f(T)p 3580 1025 6 23 v 3586 1015 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35269
57 w FC(9)p Ff(m)p FZ(:)p FV(\()p Ff(1)l FZ(<)l Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35270
FC(^)q Ff(f1)e FV(=)f Ff(fm)p FV(\))p FZ(;)j Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35271
Ff(1)p FC(^)r Ff(f0)d FV(=)g Ff(f1)q FZ(;)j Ff(1)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35272
Ff(2)p FC(^)q Ff(f1)e FV(=)f Ff(f2)p FZ(;)k Ff(f0)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35273
Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35274
Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5374
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35275
994 y FC(9)5409 1005 y Fd(R)p 2773 1069 2603 4 v 3011
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35276
1123 a FZ(M)3069 1133 y FP(1)3101 1123 y FZ(;)j(M)3188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35277
1133 y FP(2)3221 1123 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35278
Ff(fx)e FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35279
FZ(;)j(S)o(;)h(T)p 3818 1111 6 23 v 3823 1101 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35280
68 w Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35281
FZ(;)j Ff(1)l FZ(<)l Ff(2)p FC(^)r Ff(f1)d FV(=)g Ff(f2)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35282
FZ(;)j Ff(f0)e FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35283
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35284
FV(=)f Ff(fm)o FV(\))5374 1081 y FC(9)5409 1092 y Fd(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35285
2928 1155 2292 4 v 2928 1209 a FZ(M)2986 1219 y FP(1)3019
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35286
1209 y FZ(;)j(M)3106 1219 y FP(2)3138 1209 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35287
Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f Ff(0)p FC(_)p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35288
FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p 3735 1197 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35289
23 v 3741 1187 25 3 v 57 w FC(9)p Ff(m)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35290
Ff(0)l FZ(<)l Ff(m)p FC(^)q Ff(f0)c FV(=)g Ff(fm)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35291
FZ(;)k Ff(1)l FZ(<)l Ff(2)p FC(^)q Ff(f1)d FV(=)f Ff(f2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35292
FZ(;)j Ff(f0)e FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35293
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35294
FV(=)f Ff(fm)p FV(\))5219 1167 y FC(9)5254 1178 y Fd(R)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35295
2928 1242 2292 4 v 3166 1296 a FZ(M)3224 1306 y FP(1)3256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35296
1296 y FZ(;)k(M)3344 1306 y FP(2)3376 1296 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35297
Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)f Ff(0)p FC(_)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35298
FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p 3973 1284 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35299
23 v 3979 1274 25 3 v 67 w Ff(1)l FZ(<)l Ff(2)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35300
Ff(f1)d FV(=)f Ff(f2)p FZ(;)k Ff(f0)d FV(=)f Ff(1)p FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35301
FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35302
Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p FV(\))5219 1253
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35303
y FC(9)5254 1264 y Fd(R)p 3084 1328 1981 4 v 3084 1382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35304
a FZ(M)3142 1392 y FP(1)3174 1382 y FZ(;)k(M)3262 1392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35305
y FP(2)3294 1382 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35306
Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35307
FZ(;)k(S)o(;)f(T)p 3891 1370 6 23 v 3897 1360 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35308
57 w FC(9)p Ff(m)p FZ(:)p FV(\()p Ff(1)l FZ(<)l Ff(m)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35309
FC(^)q Ff(f1)e FV(=)f Ff(fm)p FV(\))p FZ(;)j Ff(f0)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35310
FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p Ff(m)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35311
FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)g FV(=)g Ff(fm)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35312
FV(\))5063 1340 y FC(9)5098 1351 y Fd(R)p 3084 1414 1981
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35313
4 v 3322 1468 a FZ(M)3380 1478 y FP(1)3412 1468 y FZ(;)j(M)3499
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35314
1478 y FP(2)3532 1468 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35315
Ff(fx)d FV(=)g Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35316
FZ(;)j(S)o(;)h(T)p 4129 1456 6 23 v 4134 1446 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35317
67 w Ff(f0)d FV(=)f Ff(1)p FZ(;)p FC(9)p Ff(n)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35318
Ff(m)p FZ(:)p FV(\()p Ff(n)l FZ(<)l Ff(m)p FC(^)q Ff(fn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35319
FV(=)f Ff(fm)p FV(\))5063 1426 y FC(9)5098 1437 y Fd(R)2071
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35320
1513 y Fc(|)p 2095 1513 2028 8 v 2028 w({z)p 4171 1513
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35321
V 2028 w(})4100 1574 y Fd(Y)4134 1586 y FP(13)1982 1765
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35322
y Gc(M)2050 1774 y FV(1)2104 1765 y F9(=)19 b FX(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35323
Fu(y)q Gc(:)p Fu(x)p Gc(:)n F9(\()p Fu(x)8 b FX(\024)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35324
Fu(m)2511 1742 y Ff(x)p FZ(;)p Ff(y)2582 1765 y F9(\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35325
b Gc(T)30 b F9(=)19 b FX(8)p Fu(i)p Gc(:)p Fu(x)p Gc(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35326
Fu(y)q Gc(:)m F9(\(\()p Fu(fy)h F9(=)g Fu(i)p FX(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35327
Fu(fx)e F9(=)i Fu(i)p F9(\))12 b FX(\033)g Fu(fx)18 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35328
F9(=)i Fu(fy)p F9(\))1982 1892 y Gc(M)2050 1901 y FV(2)2104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35329
1892 y F9(=)f FX(8)p Fu(y)q Gc(:)p Fu(x)p Gc(:)n F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35330
Fu(y)9 b FX(\024)f Fu(m)2512 1868 y Ff(x)p FZ(;)p Ff(y)2583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35331
1892 y F9(\))100 b Gc(S)23 b F9(=)d FX(8)p Fu(x)p Gc(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35332
Fu(y)q Gc(:)m F9(\()p Fu(sx)8 b FX(\024)g Fu(y)k FX(\033)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35333
Fu(x)c Gc(<)g Fu(y)q F9(\))6462 465 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35334
 currentpoint grestore moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35335
 6462 465 a 3499
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35336
5116 4 4724 v 277 5119 3226 4 v Black 1285 5273 a Gg(Figure)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35337
b(8:)29 b(Subproofs)c Ga(Y)2093 5287 y F9(12)2190 5273
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35338
y Gg(and)f Ga(Y)2397 5287 y F9(13)2472 5273 y Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35339
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35340
%%Page: 140 152
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35341
TeXDict begin 140 151 bop Black -144 51 a Gb(140)2816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35342
b(Experimental)24 b(Data)p -144 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35343
321 393 3226 4 v 321 5116 4 4724 v 1927 465 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35344
 gsave currentpoint currentpoint translate -90 neg rotate neg exch
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35345
neg exch translate
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35346
 1927 465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35347
a 2283 -864 247 4 v 2283 -813 a Ff(3)l FC(\024)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35348
2391 -825 6 23 v 2396 -835 25 3 v 61 w(3)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35349
Ff(3)p 2191 -783 431 4 v 2180 -718 a FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35350
FV(\()p Ff(x)l FC(\024)l Ff(m)2388 -739 y Fe(x)p Fd(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35351
Fe(2)2445 -718 y FV(\))p 2483 -730 6 23 v 2489 -740 25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35352
3 v 59 w Ff(3)l FC(\024)l Ff(3)2621 -771 y FC(8)2656
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35353
-760 y Fd(L)p 2191 -686 431 4 v 2285 -634 a FZ(M)2343
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35354
-624 y FP(1)p 2389 -646 6 23 v 2395 -656 25 3 v 2434
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35355
-634 a Ff(3)l FC(\024)l Ff(3)2621 -674 y FC(8)2656 -663
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35356
y Fd(L)p 2749 -685 247 4 v 2749 -634 a Ff(2)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35357
Ff(3)p 2857 -646 6 23 v 2863 -656 25 3 v 61 w(2)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35358
Ff(3)p 2285 -604 712 4 v 2386 -552 a(3)l FC(\024)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35359
FC(\033)r Ff(2)l FZ(<)l Ff(3)p FZ(;)12 b(M)2709 -542
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35360
y FP(1)p 2756 -564 6 23 v 2761 -574 25 3 v 2800 -552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35361
a Ff(2)l FZ(<)l Ff(3)2995 -595 y FC(\033)3043 -584 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35362
Fd(L)p 2329 -522 622 4 v 2319 -468 a FC(8)p Ff(y)q FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35363
FV(\()p Ff(3)l FC(\024)l Ff(y)q FC(\033)q Ff(2)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35364
Ff(y)q FV(\))p FZ(;)g(M)2766 -458 y FP(1)p 2813 -480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35365
6 23 v 2818 -490 25 3 v 2857 -468 a Ff(2)l FZ(<)l Ff(3)2950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35366
-511 y FC(8)2985 -500 y Fd(L)p 2329 -436 622 4 v 2484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35367
-385 a FZ(M)2542 -375 y FP(1)2574 -385 y FZ(;)f(S)p 2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35368
-397 6 23 v 2664 -407 25 3 v 61 w Ff(2)l FZ(<)l Ff(3)2950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35369
-424 y FC(8)2985 -413 y Fd(L)p 3173 -1023 319 4 v 3173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35370
-972 a Ff(f3)d FV(=)f Ff(0)p 3317 -984 6 23 v 3323 -994
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35371
25 3 v 59 w(f3)h FV(=)f Ff(0)p 3542 -1023 319 4 v 50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35372
w(f3)g FV(=)g Ff(1)p 3686 -984 6 23 v 3691 -994 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35373
v 60 w(f3)g FV(=)g Ff(1)p 3173 -952 687 4 v 3192 -900
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35374
a(f3)h FV(=)f Ff(0)p FC(_)p Ff(f3)g FV(=)g Ff(1)p 3508
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35375
-912 6 23 v 3513 -922 25 3 v 59 w(f3)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35376
FZ(;)k Ff(f3)c FV(=)g Ff(1)3859 -943 y FC(_)3901 -932
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35377
y Fd(L)p 3136 -871 761 4 v 3126 -817 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35378
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35379
Ff(1)p FV(\))p 3563 -829 6 23 v 3569 -839 25 3 v 59 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35380
Ff(f3)h FV(=)f Ff(0)p FZ(;)j Ff(f3)e FV(=)f Ff(1)3896
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35381
-859 y FC(8)3931 -848 y Fd(L)p 4062 -1023 319 4 v 4062
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35382
-972 a Ff(f2)h FV(=)f Ff(0)p 4206 -984 6 23 v 4212 -994
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35383
25 3 v 59 w(f2)g FV(=)g Ff(0)p 4430 -1023 319 4 v 50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35384
w(f2)h FV(=)f Ff(1)p 4574 -984 6 23 v 4580 -994 25 3
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35385
v 59 w(f2)h FV(=)f Ff(1)p 4062 -952 687 4 v 4081 -900
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35386
a(f2)h FV(=)f Ff(0)p FC(_)o Ff(f2)h FV(=)f Ff(1)p 4396
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35387
-912 6 23 v 4402 -922 25 3 v 59 w(f2)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35388
FZ(;)j Ff(f2)e FV(=)f Ff(1)4748 -943 y FC(_)4790 -932
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35389
y Fd(L)p 4025 -871 761 4 v 4014 -817 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35390
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35391
Ff(1)p FV(\))p 4452 -829 6 23 v 4458 -839 25 3 v 59 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35392
Ff(f2)g FV(=)g Ff(0)p FZ(;)k Ff(f2)d FV(=)f Ff(1)4785
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35393
-859 y FC(8)4820 -848 y Fd(L)p 3136 -785 1650 4 v 3405
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35394
-731 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35395
FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p 3842 -743 6 23 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35396
3848 -752 25 3 v 59 w Ff(f3)h FV(=)f Ff(0)p FC(^)o Ff(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35397
FV(=)f Ff(0)p FZ(;)k Ff(f3)c FV(=)g Ff(1)p FZ(;)k Ff(f2)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35398
FV(=)f Ff(1)4785 -776 y FC(^)4826 -765 y Fd(R)p 4923
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35399
-782 351 4 v 4923 -731 a Ff(f2)h FV(=)f Ff(f3)p 5083
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35400
-743 6 23 v 5089 -752 25 3 v 59 w(f2)h FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35401
3415 -698 1859 4 v 3590 -644 a FV(\()p Ff(f3)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35402
Ff(0)p FC(^)p Ff(f2)g FV(=)g Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35403
Ff(f2)h FV(=)f Ff(f3)p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35404
Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35405
4590 -656 6 23 v 4596 -666 25 3 v 59 w Ff(f2)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35406
Ff(f3)q FZ(;)j Ff(f3)e FV(=)f Ff(1)p FZ(;)j Ff(f2)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35407
Ff(1)5273 -689 y FC(\033)5321 -678 y Fd(L)p 3533 -612
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35408
1623 4 v 3523 -558 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35409
FV(=)f Ff(0)p FC(^)p Ff(f2)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35410
Ff(f2)g FV(=)g Ff(fy)q FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35411
FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35412
FV(\))p 4647 -570 6 23 v 4652 -580 25 3 v 59 w Ff(f2)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35413
FV(=)f Ff(f3)p FZ(;)k Ff(f3)d FV(=)f Ff(1)p FZ(;)j Ff(f2)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35414
FV(=)f Ff(1)5155 -600 y FC(8)5190 -589 y Fd(L)p 3514
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35415
-526 1662 4 v 3503 -472 a FC(8)p Ff(x)p FZ(:)p Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35416
FZ(:)p FV(\(\()p Ff(fy)i FV(=)e Ff(0)p FC(^)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35417
FV(=)g Ff(0)p FV(\))p FC(\033)p Ff(fx)h FV(=)f Ff(fy)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35418
FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35419
Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p 4666 -484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35420
6 23 v 4672 -493 25 3 v 59 w Ff(f2)h FV(=)f Ff(f3)p FZ(;)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35421
Ff(f3)c FV(=)g Ff(1)p FZ(;)k Ff(f2)d FV(=)f Ff(1)5175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35422
-514 y FC(8)5210 -503 y Fd(L)p 3514 -439 1662 4 v 3829
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35423
-385 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35424
FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(T)p 4340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35425
-397 6 23 v 4346 -407 25 3 v 67 w Ff(f2)d FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35426
FZ(;)k Ff(f3)c FV(=)g Ff(1)p FZ(;)k Ff(f2)d FV(=)f Ff(1)5175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35427
-427 y FC(8)5210 -416 y Fd(L)p 2484 -353 2367 4 v 3001
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35428
-299 a FZ(M)3059 -289 y FP(1)3091 -299 y FZ(;)p FC(8)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35429
Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35430
FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 3688 -311 6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35431
23 v 3694 -321 25 3 v 68 w Ff(f3)e FV(=)f Ff(1)p FZ(;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35432
Ff(f2)e FV(=)f Ff(1)p FZ(;)k Ff(2)l FZ(<)l Ff(3)p FC(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35433
Ff(f2)d FV(=)f Ff(f3)4849 -345 y FC(^)4891 -333 y Fd(R)2191
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35434
-254 y Fc(|)p 2215 -254 1540 8 v 1540 w({z)p 3803 -254
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35435
V 1540 w(})3745 -193 y Fd(Y)3779 -181 y FP(9)2786 478
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35436
y F3(.)2786 511 y(.)2786 544 y(.)2786 578 y(.)2758 629
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35437
y FZ(Y)2794 639 y FP(9)p 2986 64 247 4 v 2986 115 a Ff(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35438
FC(\024)l Ff(1)p 3094 103 6 23 v 3099 93 25 3 v 60 w(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35439
FC(\024)l Ff(1)p 2894 145 431 4 v 2883 210 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35440
FZ(:)p FV(\()p Ff(x)l FC(\024)l Ff(m)3091 189 y Fe(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35441
Fd(;)p Fe(0)3148 210 y FV(\))p 3186 198 6 23 v 3191 188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35442
25 3 v 58 w Ff(1)l FC(\024)l Ff(1)3323 157 y FC(8)3358
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35443
168 y Fd(L)p 2894 242 431 4 v 2987 294 a FZ(M)3045 304
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35444
y FP(1)p 3092 282 6 23 v 3098 272 25 3 v 3137 294 a Ff(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35445
FC(\024)l Ff(1)3323 254 y FC(8)3358 265 y Fd(L)p 3452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35446
243 247 4 v 3452 294 a Ff(0)l FZ(<)l Ff(1)p 3560 282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35447
6 23 v 3566 272 25 3 v 61 w(0)l FZ(<)l Ff(1)p 2987 324
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35448
712 4 v 3089 375 a(1)l FC(\024)l Ff(1)p FC(\033)r Ff(0)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35449
FZ(<)l Ff(1)p FZ(;)12 b(M)3412 385 y FP(1)p 3459 363
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35450
6 23 v 3464 353 25 3 v 3503 375 a Ff(0)l FZ(<)l Ff(1)3698
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35451
333 y FC(\033)3746 344 y Fd(L)p 3032 405 622 4 v 3021
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35452
459 a FC(8)p Ff(y)q FZ(:)p FV(\()p Ff(1)l FC(\024)l Ff(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35453
FC(\033)r Ff(0)l FZ(<)l Ff(y)q FV(\))p FZ(;)g(M)3469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35454
469 y FP(1)p 3515 447 6 23 v 3521 437 25 3 v 3560 459
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35455
a Ff(0)l FZ(<)l Ff(1)3653 417 y FC(8)3688 428 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35456
3032 492 622 4 v 3187 543 a FZ(M)3245 553 y FP(1)3277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35457
543 y FZ(;)e(S)p 3361 531 6 23 v 3367 521 25 3 v 62 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35458
Ff(0)l FZ(<)l Ff(1)3653 503 y FC(8)3688 514 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35459
3876 -95 319 4 v 3876 -44 a Ff(f1)e FV(=)f Ff(0)p 4020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35460
-56 6 23 v 4026 -66 25 3 v 59 w(f1)h FV(=)f Ff(0)p 4245
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35461
-95 319 4 v 50 w(f1)g FV(=)g Ff(1)p 4389 -56 6 23 v 4394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35462
-66 25 3 v 59 w(f1)h FV(=)f Ff(1)p 3876 -24 687 4 v 3895
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35463
27 a(f1)h FV(=)f Ff(0)p FC(_)o Ff(f1)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35464
4210 15 6 23 v 4216 5 25 3 v 59 w(f1)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35465
FZ(;)k Ff(f1)c FV(=)g Ff(1)4562 -16 y FC(_)4604 -5 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35466
Fd(L)p 3839 57 761 4 v 3829 111 a FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35467
FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35468
FV(\))p 4266 99 6 23 v 4272 89 25 3 v 59 w Ff(f1)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35469
Ff(0)p FZ(;)j Ff(f1)e FV(=)f Ff(1)4599 69 y FC(8)4634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35470
80 y Fd(L)p 4765 -95 319 4 v 4765 -44 a Ff(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35471
Ff(0)p 4909 -56 6 23 v 4915 -66 25 3 v 59 w(f0)g FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35472
Ff(0)p 5133 -95 319 4 v 50 w(f0)h FV(=)f Ff(1)p 5277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35473
-56 6 23 v 5283 -66 25 3 v 59 w(f0)h FV(=)f Ff(1)p 4765
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35474
-24 687 4 v 4784 27 a(f0)g FV(=)g Ff(0)p FC(_)p Ff(f0)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35475
FV(=)f Ff(1)p 5099 15 6 23 v 5105 5 25 3 v 59 w(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35476
FV(=)g Ff(0)p FZ(;)k Ff(f0)d FV(=)f Ff(1)5451 -16 y FC(_)5492
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35477
-5 y Fd(L)p 4728 57 761 4 v 4717 111 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35478
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35479
Ff(1)p FV(\))p 5155 99 6 23 v 5161 89 25 3 v 59 w Ff(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35480
FV(=)g Ff(0)p FZ(;)k Ff(f0)d FV(=)f Ff(1)5488 69 y FC(8)5523
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35481
80 y Fd(L)p 3839 143 1650 4 v 4108 197 a FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35482
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35483
Ff(1)p FV(\))p 4545 185 6 23 v 4551 175 25 3 v 59 w Ff(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35484
FV(=)f Ff(0)p FC(^)o Ff(f0)h FV(=)f Ff(0)p FZ(;)k Ff(f1)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35485
FV(=)g Ff(1)p FZ(;)k Ff(f0)d FV(=)f Ff(1)5488 151 y FC(^)5529
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35486
163 y Fd(R)p 5626 146 351 4 v 5626 197 a Ff(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35487
Ff(f1)p 5786 185 6 23 v 5792 175 25 3 v 59 w(f0)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35488
Ff(f1)p 4118 230 1859 4 v 4293 284 a FV(\()p Ff(f1)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35489
FV(=)f Ff(0)p FC(^)o Ff(f0)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35490
Ff(f0)h FV(=)f Ff(f1)p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35491
Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35492
5293 272 6 23 v 5298 262 25 3 v 59 w Ff(f0)h FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35493
FZ(;)k Ff(f1)d FV(=)f Ff(1)p FZ(;)j Ff(f0)e FV(=)f Ff(1)5976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35494
239 y FC(\033)6024 250 y Fd(L)p 4236 316 1623 4 v 4226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35495
370 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)h FV(=)f Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35496
FC(^)p Ff(f0)g FV(=)g Ff(0)p FV(\))p FC(\033)q Ff(f0)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35497
FV(=)g Ff(fy)q FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35498
Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35499
5350 358 6 23 v 5355 348 25 3 v 59 w Ff(f0)h FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35500
FZ(;)k Ff(f1)d FV(=)f Ff(1)p FZ(;)j Ff(f0)e FV(=)f Ff(1)5858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35501
328 y FC(8)5893 339 y Fd(L)p 4217 402 1662 4 v 4206 456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35502
a FC(8)p Ff(x)p FZ(:)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35503
FV(=)e Ff(0)p FC(^)o Ff(fx)h FV(=)f Ff(0)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35504
Ff(fx)h FV(=)f Ff(fy)q FV(\))p FZ(;)p FC(8)p Ff(x)p FZ(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35505
FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35506
FV(\))p 5369 444 6 23 v 5375 434 25 3 v 59 w Ff(f0)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35507
FV(=)f Ff(f1)p FZ(;)j Ff(f1)e FV(=)f Ff(1)p FZ(;)k Ff(f0)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35508
FV(=)g Ff(1)5878 414 y FC(8)5913 425 y Fd(L)p 4217 489
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35509
1662 4 v 4532 543 a FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35510
FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)k(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35511
5043 531 6 23 v 5049 521 25 3 v 67 w Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35512
FZ(;)k Ff(f1)c FV(=)g Ff(1)p FZ(;)k Ff(f0)d FV(=)f Ff(1)5878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35513
500 y FC(8)5913 511 y Fd(L)p 3187 575 2367 4 v 3704 629
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35514
a FZ(M)3762 639 y FP(1)3794 629 y FZ(;)p FC(8)p Ff(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35515
FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35516
Ff(1)p FV(\))p FZ(;)k(S)o(;)f(T)p 4391 617 6 23 v 4397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35517
607 25 3 v 68 w Ff(f1)d FV(=)g Ff(1)p FZ(;)k Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35518
Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)k Ff(f0)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35519
Ff(1)5552 583 y FC(^)5593 594 y Fd(R)p 2740 661 2296
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35520
4 v 2901 715 a FZ(M)2959 725 y FP(1)2991 715 y FZ(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35521
FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)h FV(=)f Ff(0)p FC(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35522
Ff(fx)g FV(=)g Ff(1)p FV(\))p FZ(;)k(S)o(;)g(T)p 3588
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35523
703 6 23 v 3594 693 25 3 v 67 w Ff(f3)d FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35524
FC(^)o Ff(f1)h FV(=)f Ff(1)p FZ(;)k Ff(f2)c FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35525
FZ(;)k Ff(2)l FZ(<)l Ff(3)p FC(^)q Ff(f2)d FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35526
FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35527
FZ(;)k Ff(f0)c FV(=)g Ff(1)5035 669 y FC(^)5076 681 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35528
Fd(R)2740 760 y Fc(|)p 2764 760 1617 8 v 1617 w({z)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35529
4429 760 V 1617 w(})4356 821 y Fd(Y)4390 833 y FP(10)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35530
2500 935 247 4 v 2500 987 a Ff(2)l FC(\024)l Ff(3)p 2607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35531
975 6 23 v 2613 965 25 3 v 60 w(2)l FC(\024)l Ff(3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35532
2406 1017 433 4 v 2396 1081 a FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35533
Ff(2)l FC(\024)l Ff(m)2606 1061 y Fe(x)p Fd(;)p Fe(2)2662
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35534
1081 y FV(\))p 2701 1069 6 23 v 2706 1060 25 3 v 59 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35535
Ff(2)l FC(\024)l Ff(3)2838 1028 y FC(8)2873 1039 y Fd(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35536
2406 1114 433 4 v 2501 1165 a FZ(M)2559 1175 y FP(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35537
2606 1153 6 23 v 2611 1143 25 3 v 2650 1165 a Ff(2)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35538
FC(\024)l Ff(3)2838 1125 y FC(8)2873 1136 y Fd(L)p 2967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35539
1115 247 4 v 2967 1165 a Ff(1)l FZ(<)l Ff(3)p 3075 1153
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35540
6 23 v 3081 1143 25 3 v 61 w(1)l FZ(<)l Ff(3)p 2501 1195
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35541
713 4 v 2603 1247 a(2)l FC(\024)l Ff(3)p FC(\033)r Ff(1)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35542
FZ(<)l Ff(3)p FZ(;)12 b(M)2926 1257 y FP(2)p 2973 1235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35543
6 23 v 2979 1225 25 3 v 3018 1247 a Ff(1)l FZ(<)l Ff(3)3213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35544
1204 y FC(\033)3261 1216 y Fd(L)p 2546 1277 622 4 v 2536
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35545
1331 a FC(8)p Ff(y)q FZ(:)p FV(\()p Ff(2)l FC(\024)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35546
Ff(y)q FC(\033)q Ff(1)l FZ(<)l Ff(y)q FV(\))p FZ(;)g(M)2983
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35547
1341 y FP(2)p 3030 1319 6 23 v 3035 1309 25 3 v 3075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35548
1331 a Ff(1)l FZ(<)l Ff(3)3167 1289 y FC(8)3202 1300
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35549
y Fd(L)p 2546 1363 622 4 v 2701 1414 a FZ(M)2759 1424
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35550
y FP(2)2791 1414 y FZ(;)f(S)p 2875 1402 6 23 v 2881 1392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35551
25 3 v 61 w Ff(1)l FZ(<)l Ff(3)3167 1375 y FC(8)3202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35552
1386 y Fd(L)4420 921 y F3(.)4420 954 y(.)4420 987 y(.)4420
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35553
1020 y(.)4377 1072 y FZ(Y)4413 1082 y FP(10)p 4542 1020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35554
351 4 v 4542 1072 a Ff(f1)c FV(=)g Ff(f3)p 4702 1060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35555
6 23 v 4708 1050 25 3 v 60 w(f1)g FV(=)g Ff(f3)p 3431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35556
1101 2392 4 v 3431 1155 a FV(\()p Ff(f3)g FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35557
FC(^)p Ff(f1)h FV(=)f Ff(1)p FV(\))p FC(\033)p Ff(f1)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35558
FV(=)g Ff(f3)q FZ(;)j(M)4061 1165 y FP(1)4094 1155 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35559
FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)d FV(=)g Ff(0)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35560
FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35561
4691 1143 6 23 v 4696 1133 25 3 v 67 w Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35562
Ff(f3)p FZ(;)k Ff(f2)d FV(=)f Ff(1)p FZ(;)j Ff(2)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35563
Ff(3)p FC(^)r Ff(f2)d FV(=)g Ff(f3)q FZ(;)j Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35564
Ff(1)p FC(^)q Ff(f0)e FV(=)f Ff(f1)p FZ(;)k Ff(f0)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35565
Ff(1)5821 1110 y FC(\033)5869 1121 y Fd(L)p 3374 1187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35566
2505 4 v 3363 1241 a FC(8)p Ff(y)q FZ(:)p FV(\(\()p Ff(fy)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35567
FV(=)e Ff(1)p FC(^)o Ff(f1)h FV(=)f Ff(1)p FV(\))p FC(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35568
Ff(f1)h FV(=)f Ff(fy)q FV(\))p FZ(;)j(M)4118 1251 y FP(1)4150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35569
1241 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35570
Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35571
4747 1229 6 23 v 4753 1220 25 3 v 67 w Ff(f1)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35572
Ff(f3)p FZ(;)k Ff(f2)c FV(=)g Ff(1)p FZ(;)k Ff(2)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35573
Ff(3)p FC(^)q Ff(f2)d FV(=)f Ff(f3)p FZ(;)k Ff(0)l FZ(<)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35574
Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p FZ(;)k Ff(f0)c FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35575
Ff(1)5878 1199 y FC(8)5913 1210 y Fd(L)p 3354 1274 2544
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35576
4 v 3344 1328 a FC(8)p Ff(x)p FZ(:)p Ff(y)q FZ(:)p FV(\(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35577
Ff(fy)h FV(=)f Ff(1)p FC(^)p Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35578
FC(\033)p Ff(fx)g FV(=)g Ff(fy)q FV(\))p FZ(;)k(M)4138
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35579
1338 y FP(1)4170 1328 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35580
Ff(fx)d FV(=)f Ff(0)p FC(_)o Ff(fx)h FV(=)f Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35581
FZ(;)k(S)o(;)f(T)p 4767 1316 6 23 v 4773 1306 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35582
68 w Ff(f1)d FV(=)g Ff(f3)q FZ(;)j Ff(f2)e FV(=)f Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35583
FZ(;)k Ff(2)l FZ(<)l Ff(3)p FC(^)q Ff(f2)d FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35584
FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35585
FZ(;)j Ff(f0)e FV(=)f Ff(1)5897 1285 y FC(8)5932 1296
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35586
y Fd(L)p 3354 1360 2544 4 v 3717 1414 a FZ(M)3775 1424
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35587
y FP(1)3807 1414 y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35588
Ff(fx)h FV(=)f Ff(0)p FC(_)p Ff(fx)g FV(=)g Ff(1)p FV(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35589
FZ(;)k(S)o(;)g(T)p 4404 1402 6 23 v 4410 1392 25 3 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35590
67 w Ff(f1)d FV(=)f Ff(f3)p FZ(;)k Ff(f2)c FV(=)g Ff(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35591
FZ(;)k Ff(2)l FZ(<)l Ff(3)p FC(^)q Ff(f2)d FV(=)f Ff(f3)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35592
FZ(;)k Ff(0)l FZ(<)l Ff(1)p FC(^)q Ff(f0)d FV(=)f Ff(f1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35593
FZ(;)k Ff(f0)c FV(=)g Ff(1)5897 1372 y FC(8)5932 1383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35594
y Fd(L)p 2701 1446 2835 4 v 3081 1500 a FZ(M)3139 1510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35595
y FP(1)3172 1500 y FZ(;)j(M)3259 1510 y FP(2)3291 1500
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35596
y FZ(;)p FC(8)p Ff(x)p FZ(:)p FV(\()p Ff(fx)e FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35597
Ff(0)p FC(_)p Ff(fx)h FV(=)f Ff(1)p FV(\))p FZ(;)j(S)o(;)h(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35598
3888 1488 6 23 v 3894 1479 25 3 v 67 w Ff(f2)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35599
Ff(1)p FZ(;)k Ff(2)l FZ(<)l Ff(3)p FC(^)q Ff(f2)d FV(=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35600
Ff(f3)p FZ(;)j Ff(0)l FZ(<)l Ff(1)p FC(^)r Ff(f0)d FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35601
Ff(f1)q FZ(;)j Ff(1)l FZ(<)l Ff(3)p FC(^)r Ff(f1)d FV(=)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35602
Ff(f3)q FZ(;)j Ff(f0)e FV(=)f Ff(1)5534 1455 y FC(^)5576
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35603
1466 y Fd(R)2406 1545 y Fc(|)p 2430 1545 1737 8 v 1737
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35604
w({z)p 4215 1545 V 1737 w(})4144 1607 y Fd(Y)4178 1619
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35605
y FP(11)2026 1798 y Gc(M)2094 1807 y FV(1)2148 1798 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35606
F9(=)20 b FX(8)p Fu(y)q Gc(:)p Fu(x)p Gc(:)m F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35607
Fu(x)8 b FX(\024)g Fu(m)2555 1774 y Ff(x)p FZ(;)p Ff(y)2626
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35608
1798 y F9(\))101 b Gc(T)30 b F9(=)19 b FX(8)p Fu(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35609
Gc(:)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\(\()p Fu(fy)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35610
F9(=)e Fu(i)p FX(^)p Fu(fx)g F9(=)g Fu(i)p F9(\))12 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35611
FX(\033)g Fu(fx)19 b F9(=)g Fu(fy)p F9(\))2026 1924 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35612
Gc(M)2094 1933 y FV(2)2148 1924 y F9(=)h FX(8)p Fu(y)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35613
Gc(:)p Fu(x)p Gc(:)m F9(\()p Fu(y)9 b FX(\024)f Fu(m)2556
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35614
1900 y Ff(x)p FZ(;)p Ff(y)2627 1924 y F9(\))100 b Gc(S)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35615
b F9(=)d FX(8)p Fu(x)p Gc(:)p Fu(y)q Gc(:)m F9(\()p Fu(sx)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35616
b FX(\024)g Fu(y)k FX(\033)g Fu(x)c Gc(<)g Fu(y)q F9(\))6506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35617
465 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35618
 currentpoint grestore moveto
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35619
 6506 465 a 3543 5116 4 4724 v 321 5119 3226 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35620
Black 1260 5273 a Gg(Figure)24 b(9:)29 b(Subproofs)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35621
Ga(Y)2068 5287 y F9(9)2108 5273 y Gg(,)d Ga(Y)2206 5287
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35622
y F9(10)2303 5273 y Gg(and)i Ga(Y)2510 5287 y F9(11)2585
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35623
5273 y Gg(.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35624
%%Page: 141 153
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35625
TeXDict begin 141 152 bop Black Black 277 1027 a F8(A)l(ppendix)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35626
b(B)277 1459 y Gf(Details)53 b(f)-5 b(or)51 b(some)h(Pr)l(oofs)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35627
1957 y Ge(B.1)119 b(Pr)n(oofs)29 b(of)g(Chapter)i(2)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35628
2181 y Gg(In)24 b(this)h(section)h(we)d(are)h(mainly)h(concerned)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35629
(with)d(pro)o(ving)i(strong)f(normalisation)j(for)c F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35630
FY(T)t Ga(;)3305 2144 y Gc(cut)3274 2181 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35631
b(\000)f(!)p F4(\))p Gg(,)277 2294 y F4(\()p FY(T)t Ga(;)430
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35632
2257 y Gc(aux)410 2294 y F6(\000)h(\000)f(!)p F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35633
b Gg(and)h F4(\()p FY(T)888 2261 y FX($)959 2294 y Ga(;)1042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35634
2257 y Gc(l)q(oc)1000 2294 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35635
F4(\))p Gg(.)p Black 277 2450 a Gb(Pr)n(oof)24 b(of)g(Pr)n(oposition)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35636
(2.2.10.)p Black 35 w Gg(One)f(case)h(is)f(as)h(follo)n(ws.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35637
Black 277 2639 a Gb(Case)p Black 46 w Ga(M)60 b F6(\021)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35638
b FL(Cut)p F4(\()951 2627 y FX(h)979 2639 y Ga(a)1027
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35639
2627 y FX(i)1054 2639 y Ga(S)5 b(;)1155 2627 y F9(\()1183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35640
2639 y Ga(x)1235 2627 y F9(\))1262 2639 y Ga(T)13 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35641
Gg(:)56 b(Suppose)38 b F4(\000)1845 2653 y F9(1)1909
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35642
2627 y Gc(.)1964 2639 y Ga(S)2050 2627 y Gc(.)2105 2639
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35643
y F4(\001)2181 2653 y F9(1)2256 2639 y Gg(and)g Ga(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35644
b F4(:)g Ga(B)5 b(;)15 b F4(\000)2706 2653 y F9(2)2771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35645
2627 y Gc(.)2826 2639 y Ga(T)2917 2627 y Gc(.)2972 2639
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35646
y F4(\001)3048 2653 y F9(2)3123 2639 y Gg(are)37 b(typing)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35647
2751 y(judgements)27 b(for)e Ga(S)j Gg(and)d Ga(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35648
b Gg(,)23 b(respecti)n(v)o(ely)-6 b(.)35 b(Furthermore,)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35649
b(assume)f(that)g Ga(a)e Gg(is)i(not)f(free)h(in)f Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35650
b Gg(.)504 2864 y(By)23 b(the)h(inference)i(rules)e(we)f(ha)n(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35651
(for)g(the)g(term)f Ga(M)33 b Gg(the)24 b(typing)h(judgement)1353
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35652
3050 y F4(\000)1410 3064 y F9(1)1450 3050 y Ga(;)15 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35653
F4(\000)1547 3064 y F9(2)1611 3038 y Gc(.)1666 3050 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35654
FL(Cut)p F4(\()1839 3038 y FX(h)1867 3050 y Ga(a)1915
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35655
3038 y FX(i)1943 3050 y Ga(S)t(;)2043 3038 y F9(\()2071
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35656
3050 y Ga(x)2123 3038 y F9(\))2151 3050 y Ga(T)d F4(\))2277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35657
3038 y Gc(.)2332 3050 y F4(\001)2408 3064 y F9(1)2447
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35658
3050 y Ga(;)j F4(\001)2563 3064 y F9(2)2628 3050 y Ga(:)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35659
3235 y Gg(Let)25 b FL(Cut)p F4(\()823 3223 y FX(h)851
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35660
3235 y Ga(a)899 3223 y FX(i)926 3235 y Ga(S)5 b(;)1027
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35661
3223 y F9(\()1055 3235 y Ga(x)1107 3223 y F9(\))1135
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35662
3235 y Ga(T)12 b F4(\))25 b Gg(reduce)i(to)f Ga(S)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35663
F4([)p Ga(a)29 b F4(:=)1911 3223 y F9(\()1938 3235 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35664
Ga(x)1990 3223 y F9(\))2018 3235 y Ga(T)13 b F4(])p Gg(,)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35665
b(which,)h(because)h(we)e(assumed)i(that)f Ga(a)f Gg(is)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35666
3348 y(not)g(free)g(in)f Ga(S)5 b Gg(,)24 b(is)g(equi)n(v)n(alent)j(to)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35667
d Ga(S)5 b Gg(.)31 b(From)24 b(our)g(assumption)j(that)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35668
F4(\000)2722 3362 y F9(1)2787 3336 y Gc(.)2842 3348 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35669
Ga(S)2928 3336 y Gc(.)2983 3348 y F4(\001)3059 3362 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35670
F9(1)3122 3348 y Gg(is)f(a)g(typing)504 3461 y(judgement)f(for)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35671
Ga(S)5 b Gg(,)20 b(we)g(ha)n(v)o(e)h(by)g(Lemma)e(2.2.9)i(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35672
F4(\000)2275 3475 y F9(1)2314 3461 y Ga(;)15 b F4(\000)2411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35673
3475 y F9(2)2476 3449 y Gc(.)2531 3461 y Ga(S)2617 3449
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35674
y Gc(.)2672 3461 y F4(\001)2748 3475 y F9(1)2787 3461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35675
y Ga(;)g F4(\001)2903 3475 y F9(2)2963 3461 y Gg(is)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35676
b(also)i(a)e(typing)504 3574 y(judgement)26 b(for)e Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35677
b Gg(.)28 b(Thus)23 b(we)g(are)h(done.)p 3436 3574 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35678
62 v 3440 3516 55 4 v 3440 3574 V 3494 3574 4 62 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35679
277 3811 a Gb(Pr)n(oof)j(of)g(Lemma)f(2.3.1.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35680
34 w Gg(The)g(only)h(case)h(that)f(is)g(non-tri)n(vial)i(is)e(where)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35681
Ga(M)36 b Gg(is)26 b(an)h(axiom)g(\(since)277 3924 y(axioms)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35682
(introduce)j(tw)o(o)c(labels\).)p Black 277 4113 a Gb(Case)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35683
Ga(M)35 b F6(\021)25 b FL(Ax)p F4(\()p Ga(x;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35684
F4(\))p Gb(:)p Black 37 w Gg(Suppose)22 b F6(f)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35685
F6(g)f Gg(and)g F6(f)p Ga(\034)10 b F6(g)21 b Gg(are)g(of)f(the)h(form)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35686
15 b F6(f)-7 b Ga(x)27 b F4(:=)2669 4101 y FX(h)2696
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35687
4113 y Ga(a)2744 4101 y FX(i)2772 4113 y Ga(P)s F6(g)21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35688
b Gg(and)15 b F6(f)-7 b Ga(b)27 b F4(:=)3269 4101 y F9(\()3296
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35689
4113 y Ga(y)3344 4101 y F9(\))3372 4113 y Ga(Q)-10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35690
F6(g)p Gg(,)504 4226 y(respecti)n(v)o(ely)k(.)31 b(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35691
b(e)20 b(analyse)i(in)f(turn)g(the)g(cases)h FL(Ax)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35692
F4(\()p Ga(x;)15 b(b)p F4(\))p F6(f)p Ga(\033)s F6(g)-15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35693
b(f)p Ga(\034)10 b F6(g)23 b Gg(and)e FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35694
Ga(x;)15 b(b)p F4(\))p F6(f)p Ga(\034)10 b F6(g)-15 b(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35695
Ga(\033)s F6(f)p Ga(\034)10 b F6(gg)p Gg(.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35696
728 4433 a Ga(M)g F6(f)p Ga(\033)s F6(gf)p Ga(\034)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35697
F6(g)32 b(\021)e FL(Ax)o F4(\()p Ga(x;)15 b(b)p F4(\))-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35698
b F6(f)e Ga(x)27 b F4(:=)1781 4421 y FX(h)1808 4433 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35699
Ga(a)1856 4421 y FX(i)1884 4433 y Ga(P)s F6(g)-5 b(f)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35700
Ga(b)26 b F4(:=)2210 4421 y F9(\()2237 4433 y Ga(y)2285
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35701
4421 y F9(\))2312 4433 y Ga(Q)-9 b F6(g)1143 4570 y F4(=)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35702
b FL(Cut)p F4(\()1417 4558 y FX(h)1445 4570 y Ga(a)1493
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35703
4558 y FX(i)1520 4570 y Ga(P)13 b(;)1631 4558 y F9(\()1659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35704
4570 y Ga(x)1711 4558 y F9(\))1738 4570 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35705
Ga(x;)i(b)p F4(\))q(\))-5 b F6(f)e Ga(b)26 b F4(:=)2297
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35706
4558 y F9(\()2325 4570 y Ga(y)2373 4558 y F9(\))2400
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35707
4570 y Ga(Q)-9 b F6(g)1143 4706 y F4(=)30 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35708
F4(\()1417 4694 y FX(h)1445 4706 y Ga(a)1493 4694 y FX(i)1520
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35709
4706 y Ga(P)8 b F6(f)-7 b Ga(b)26 b F4(:=)1810 4694 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35710
F9(\()1838 4706 y Ga(y)1886 4694 y F9(\))1913 4706 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35711
Ga(Q)-10 b F6(g)q Ga(;)2061 4694 y F9(\()2089 4706 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35712
Ga(y)2137 4694 y F9(\))2164 4706 y Ga(Q)p F4(\))587 4937
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35713
y Ga(M)10 b F6(f)p Ga(\034)g F6(gf)p Ga(\033)s F6(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35714
Ga(\034)g F6(gg)33 b(\021)d FL(Ax)o F4(\()p Ga(x;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35715
b(b)p F4(\))-5 b F6(f)e Ga(b)27 b F4(:=)1768 4925 y F9(\()1795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35716
4937 y Ga(y)1843 4925 y F9(\))1870 4937 y Ga(Q)-9 b F6(g)k(f)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35717
Ga(x)26 b F4(:=)2210 4925 y FX(h)2238 4937 y Ga(a)2286
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35718
4925 y FX(i)2313 4937 y Ga(P)8 b F6(f)-7 b Ga(b)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35719
F4(:=)2603 4925 y F9(\()2631 4937 y Ga(y)2679 4925 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35720
F9(\))2706 4937 y Ga(Q)-10 b F6(g)i(g)1143 5074 y F4(=)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35721
b FL(Cut)p F4(\()1417 5062 y FX(h)1445 5074 y Ga(b)1484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35722
5062 y FX(i)1511 5074 y FL(Ax)p F4(\()p Ga(x;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35723
F4(\))q Ga(;)1856 5062 y F9(\()1883 5074 y Ga(y)1931
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35724
5062 y F9(\))1959 5074 y Ga(Q)p F4(\))-5 b F6(f)e Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35725
b F4(:=)2298 5062 y FX(h)2326 5074 y Ga(a)2374 5062 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35726
FX(i)2401 5074 y Ga(P)8 b F6(f)-7 b Ga(b)26 b F4(:=)2691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35727
5062 y F9(\()2719 5074 y Ga(y)2767 5062 y F9(\))2794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35728
5074 y Ga(Q)-10 b F6(g)h(g)1143 5210 y F4(=)30 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35729
F4(\()1417 5198 y FX(h)1445 5210 y Ga(a)1493 5198 y FX(i)1520
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35730
5210 y Ga(P)8 b F6(f)-7 b Ga(b)26 b F4(:=)1810 5198 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35731
F9(\()1838 5210 y Ga(y)1886 5198 y F9(\))1913 5210 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35732
Ga(Q)-10 b F6(g)q Ga(;)2061 5198 y F9(\()2089 5210 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35733
Ga(y)2137 5198 y F9(\))2164 5210 y Ga(Q)-5 b F6(f)e Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35734
b F4(:=)2468 5198 y FX(h)2495 5210 y Ga(a)2543 5198 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35735
FX(i)2571 5210 y Ga(P)8 b F6(f)-7 b Ga(b)26 b F4(:=)2861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35736
5198 y F9(\()2888 5210 y Ga(y)2936 5198 y F9(\))2964
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35737
5210 y Ga(Q)-10 b F6(g)h(g)q F4(\))1136 5291 y FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35738
FC(\003)p FV(\))1143 5374 y F6(\021)30 b FL(Cut)p F4(\()1417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35739
5362 y FX(h)1445 5374 y Ga(a)1493 5362 y FX(i)1520 5374
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35740
y Ga(P)8 b F6(f)-7 b Ga(b)26 b F4(:=)1810 5362 y F9(\()1838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35741
5374 y Ga(y)1886 5362 y F9(\))1913 5374 y Ga(Q)-10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35742
F6(g)q Ga(;)2061 5362 y F9(\()2089 5374 y Ga(y)2137 5362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35743
y F9(\))2164 5374 y Ga(Q)p F4(\))1060 5505 y F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35744
FX(\003)p F9(\))1177 5538 y Gg(because)26 b(by)e(assumption)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35745
Ga(x)h F6(62)e Ga(F)13 b(N)d F4(\()2390 5526 y F9(\()2418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35746
5538 y Ga(y)2466 5526 y F9(\))2494 5538 y Ga(Q)o F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35747
3436 5645 V 3440 5587 55 4 v 3440 5645 V 3494 5645 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35748
62 v Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35749
%%Page: 142 154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35750
TeXDict begin 142 153 bop Black -144 51 a Gb(142)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35751
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35752
4 v Black Black 321 388 a(Pr)n(oof)h(of)f(Lemma)f(2.3.11.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35753
Black 35 w Gg(W)-7 b(e)23 b(illustrate)j(the)f(proof)g(with)f(tw)o(o)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35754
(cases.)31 b(Some)n(what)24 b(surprisingly)-6 b(,)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35755
501 y(it)24 b(is)f(possible)j(that)e Ga(M)10 b F6(f)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35756
Ga(\033)s F6(g)26 b(\021)f Ga(M)1421 468 y FX(0)1445
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35757
501 y F6(f)p Ga(\033)s F6(g)p Gg(,)e(as)h(sho)n(wn)f(in)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35758
(\002rst)f(case.)p Black 321 762 a Gb(Case)h Ga(M)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35759
b F6(\021)25 b FL(Cut)p F4(\()923 750 y FX(h)951 762
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35760
y Ga(a)999 750 y FX(i)1026 762 y FL(Ax)p F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35761
b(a)p F4(\))q Ga(;)1376 750 y F9(\()1404 762 y Ga(x)1456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35762
750 y F9(\))1483 762 y Ga(S)5 b F4(\))p Gb(:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35763
46 w Gg(Let)25 b Ga(S)k Gg(freshly)e(introduce)h Ga(x)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35764
Gg(and)h(suppose)j F6(f)p Ga(\033)s F6(g)d Gg(is)g(of)g(the)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35765
875 y(form)31 b F6(f)-7 b Ga(y)53 b F4(:=)1037 863 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35766
FX(h)1064 875 y Ga(c)1103 863 y FX(i)1131 875 y Ga(T)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35767
F6(g)p Gg(.)67 b(Furthermore,)42 b(assume)37 b Ga(M)2309
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35768
838 y Gc(aux)2289 875 y F6(\000)-31 b(\000)f(!)50 b Ga(M)2607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35769
842 y FX(0)2666 875 y Gg(with)36 b Ga(M)2961 842 y FX(0)3033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35770
875 y F6(\021)49 b Ga(S)5 b F4([)p Ga(x)35 b F6(7!)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35771
Ga(y)s F4(])p Gg(.)549 988 y(Ob)o(viously)-6 b(,)27 b(we)f(ha)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35772
Ga(M)1391 955 y FX(0)1414 988 y F6(f)p Ga(\033)s F6(g)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35773
b(\021)e Ga(S)5 b F4([)p Ga(x)14 b F6(7!)g Ga(y)s F4(])-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35774
b F6(f)e Ga(y)34 b F4(:=)2258 976 y FX(h)2285 988 y Ga(c)2324
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35775
976 y FX(i)2352 988 y Ga(T)s F6(g)p Gg(.)i(In)25 b(the)h(follo)n(wing)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35776
(calculation,)549 1101 y(the)23 b(equi)n(v)n(alence)k(\(`0'-case\))f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35777
(occurs)f(if)e Ga(S)q F6(f)-7 b Ga(y)28 b F4(:=)2138
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35778
1089 y FX(h)2165 1101 y Ga(c)2204 1089 y FX(i)2232 1101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35779
y Ga(T)s F6(g)c Gg(freshly)h(introduces)h Ga(x)p Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35780
Black Black 1059 1394 a Ga(M)10 b F6(f)p Ga(\033)s F6(g)206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35781
b(\021)e FL(Cut)p F4(\()1956 1382 y FX(h)1983 1394 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35782
Ga(a)2031 1382 y FX(i)2059 1394 y FL(Ax)o F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35783
b(a)p F4(\))q Ga(;)2408 1382 y F9(\()2436 1394 y Ga(x)2488
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35784
1382 y F9(\))2516 1394 y Ga(S)5 b F4(\))-5 b F6(f)e Ga(y)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35785
b F4(:=)2840 1382 y FX(h)2867 1394 y Ga(c)2906 1382 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35786
FX(i)2934 1394 y Ga(T)s F6(g)1508 1531 y F4(=)204 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35787
F4(\()1956 1519 y FX(h)1983 1531 y Ga(c)2022 1519 y FX(i)2050
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35788
1531 y Ga(T)13 b(;)2156 1519 y F9(\()2184 1531 y Ga(x)2236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35789
1519 y F9(\))2263 1531 y Ga(S)q F6(f)-7 b Ga(y)28 b F4(:=)2552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35790
1519 y FX(h)2580 1531 y Ga(c)2619 1519 y FX(i)2646 1531
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35791
y Ga(T)t F6(g)p F4(\))1423 1634 y Gc(aux)1403 1671 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35792
F6(\000)-31 b(\000)f(!)1573 1638 y F9(0)p Gc(=)p F9(1)1783
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35793
1671 y Ga(S)q F6(f)-7 b Ga(y)27 b F4(:=)2072 1659 y FX(h)2099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35794
1671 y Ga(c)2138 1659 y FX(i)2166 1671 y Ga(T)s F6(g)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35795
b(f)e Ga(x)26 b F4(:=)2500 1659 y FX(h)2527 1671 y Ga(c)2566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35796
1659 y FX(i)2594 1671 y Ga(T)s F6(g)1501 1752 y FV(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35797
FC(\003)p FV(\))1508 1835 y F6(\021)204 b Ga(S)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35798
Ga(x)10 b F6(7!)g Ga(y)s F4(])-5 b F6(f)e Ga(y)29 b F4(:=)2333
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35799
1823 y FX(h)2360 1835 y Ga(c)2399 1823 y FX(i)2427 1835
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35800
y Ga(T)t F6(g)792 1951 y F9(\()p FX(\003)p F9(\))986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35801
1984 y Gg(because)d(by)d(Barendre)o(gt-style)28 b(naming)c(con)l(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35802
(ention)j Ga(x)e F6(62)g Ga(F)13 b(N)d F4(\()3107 1972
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35803
y FX(h)3135 1984 y Ga(c)3174 1972 y FX(i)3201 1984 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35804
Ga(T)j F4(\))p Black 321 2278 a Gb(Case)24 b Ga(M)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35805
b F6(\021)25 b FL(Cut)p F4(\()923 2266 y FX(h)951 2278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35806
y Ga(a)999 2266 y FX(i)1026 2278 y Ga(S)5 b(;)1127 2266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35807
y F9(\()1155 2278 y Ga(x)1207 2266 y F9(\))1235 2278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35808
y Ga(T)12 b F4(\))p Gb(:)p Black 46 w Gg(Suppose)35 b(the)e(term)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35809
Ga(S)38 b Gg(does)c(not)g(freshly)g(introduce)i Ga(a)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35810
Gg(and)g(as-)549 2391 y(sume)23 b Ga(M)907 2354 y Gc(aux)887
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35811
2391 y F6(\000)-32 b(\000)h(!)25 b Ga(M)1180 2358 y FX(0)1226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35812
2391 y Gg(with)f Ga(M)1509 2358 y FX(0)1557 2391 y F6(\021)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35813
Ga(S)q F6(f)-7 b Ga(a)25 b F4(:=)1942 2379 y FX(h)1970
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35814
2391 y Ga(x)2022 2379 y FX(i)2049 2391 y Ga(T)s F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35815
Gg(.)p Black Black 1111 2685 a Ga(M)10 b F6(f)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35816
F6(g)151 b(\021)e FL(Cut)p F4(\()1898 2673 y FX(h)1926
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35817
2685 y Ga(a)1974 2673 y FX(i)2001 2685 y Ga(S)5 b(;)2102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35818
2673 y F9(\()2130 2685 y Ga(x)2182 2673 y F9(\))2210
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35819
2685 y Ga(T)12 b F4(\))p F6(f)p Ga(\033)s F6(g)1505 2821
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35820
y F4(=)149 b FL(Cut)p F4(\()1898 2809 y FX(h)1926 2821
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35821
y Ga(a)1974 2809 y FX(i)2001 2821 y Ga(S)5 b F6(f)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35822
F6(g)q Ga(;)2248 2809 y F9(\()2276 2821 y Ga(x)2328 2809
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35823
y F9(\))2356 2821 y Ga(T)13 b F6(f)p Ga(\033)s F6(g)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35824
F4(\))1476 2921 y Gc(aux)1455 2958 y F6(\000)-31 b(\000)g(!)99
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35825
b Ga(S)5 b F6(f)p Ga(\033)s F6(g)-5 b(f)e Ga(a)27 b F4(:=)2160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35826
2946 y F9(\()2188 2958 y Ga(x)2240 2946 y F9(\))2267
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35827
2958 y Ga(T)13 b F6(f)p Ga(\033)s F6(g)-8 b(g)1498 3038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35828
y FV(\()p FC(\003)p FV(\))1505 3122 y F6(\021)149 b Ga(S)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35829
F6(f)-7 b Ga(a)25 b F4(:=)2014 3110 y FX(h)2042 3122
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35830
y Ga(x)2094 3110 y FX(i)2121 3122 y Ga(T)s F6(gf)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35831
F6(g)290 3274 y F9(\()p FX(\003)p F9(\))408 3307 y Gg(by)e(Barendre)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35832
(gt-style)28 b(naming)c(con)l(v)o(ention)j Ga(a)e F6(62)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35833
Ga(F)13 b(C)7 b F4(\()p Ga(dom)p F4(\()p F6(f)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35834
F6(g)p F4(\)\))25 b Gg(and)f(hence)h(by)e(Lem.)f(2.3.1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35835
3480 3460 4 62 v 3484 3402 55 4 v 3484 3460 V 3538 3460
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35836
4 62 v Black 321 3673 a Gb(Pr)n(oof)j(of)g(Lemma)e(2.3.12.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35837
Black 34 w Gg(From)h(Lemma)f(2.3.6\(i\),)i(it)g(follo)n(ws)g(that)g(if)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35838
2708 3661 y FX(h)2735 3673 y Ga(a)r F4(:)r Ga(B)2886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35839
3661 y FX(i)2913 3673 y Ga(M)37 b F6(2)27 b FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35840
Ga(B)5 b F6(i)p FB(K)o Gg(,)24 b(then)321 3774 y FX(h)349
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35841
3786 y Ga(a)r F4(:)r Ga(B)500 3774 y FX(i)527 3786 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35842
Ga(M)45 b F6(2)35 b FW(N)t(E)t(G)916 3804 y FX(h)p Gc(B)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35843
FX(i)1033 3786 y F4(\()p FB(J)p F4(\()p Ga(B)5 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35844
FB(K)p F4(\))p Gg(.)43 b(So)27 b(depending)32 b(on)c(the)h(structure)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35845
(of)f Ga(B)j Gg(we)c(need)h(to)f(analyse)i(all)321 3898
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35846
y(possible)36 b(sets)f(\(generated)h(by)e(the)g(set)g(operators\))j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35847
(where)2297 3886 y FX(h)2324 3898 y Ga(a)r F4(:)r Ga(B)2475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35848
3886 y FX(i)2502 3898 y Ga(M)44 b Gg(may)33 b(be)h(member)g(in.)59
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35849
b(Six)321 4011 y(cases)25 b(are)f(gi)n(v)o(en)g(for)f(\(i\);)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35850
(ar)n(guments)i(for)e(\(ii\))f(are)h(similar)g(and)g(omitted.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35851
Black 321 4312 a Gb(Case)p Black 49 w FW(A)t(X)t(I)t(O)t(M)t(S)866
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35852
4330 y FX(h)p Gc(B)s FX(i)983 4312 y Gg(:)1036 4300 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35853
FX(h)1064 4312 y Ga(a)r F4(:)r Ga(B)1215 4300 y FX(i)1242
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35854
4312 y Ga(M)33 b Gg(cannot)25 b(be)f(in)h FW(A)t(X)t(I)t(O)t(M)t(S)2142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35855
4330 y FX(h)p Gc(B)s FX(i)2282 4312 y Gg(because)h(axioms)e(do)g(not)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35856
(reduce.)p Black 321 4556 a Gb(Case)p Black 49 w FW(B)t(I)t(N)t(D)t(I)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35857
(N)t(G)891 4575 y FX(h)p Gc(B)s FX(i)1008 4556 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35858
FB(J)p F4(\()p Ga(B)5 b F4(\))p FB(K)o F4(\))q Gg(:)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35859
4748 y(\(1\))754 4736 y FX(h)782 4748 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35860
Ga(B)933 4736 y FX(i)960 4748 y Ga(M)35 b F6(2)27 b FW(B)t(I)t(N)t(D)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35861
(I)t(N)t(G)1506 4767 y FX(h)p Gc(B)s FX(i)1623 4748 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35862
F4(\()p FB(J)p F4(\()p Ga(B)5 b F4(\))p FB(K)p F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35863
b Gg(and)h Ga(M)2232 4711 y Gc(aux)2211 4748 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35864
b(\000)g(!)25 b Ga(M)2505 4715 y FX(0)3019 4748 y Gg(by)e(assumption)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35865
549 4867 y(\(2\))100 b Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)1080
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35866
4855 y F9(\()1108 4867 y Ga(x)r F4(:)r Ga(B)1263 4855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35867
y F9(\))1290 4867 y Ga(P)t F6(g)f(2)g Ga(S)5 b(N)1642
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35868
4881 y Gc(aux)1787 4867 y Gg(for)24 b(all)2030 4855 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35869
F9(\()2057 4867 y Ga(x)r F4(:)r Ga(B)2212 4855 y F9(\))2239
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35870
4867 y Ga(P)39 b F6(2)24 b FB(J)p F4(\()p Ga(B)5 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35871
FB(K)220 b Gg(by)24 b(De\002nition)h(2.3.3)549 4986 y(\(3\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35872
b Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)1080 4974 y F9(\()1108
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35873
4986 y Ga(x)r F4(:)r Ga(B)1263 4974 y F9(\))1290 4986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35874
y Ga(P)t F6(g)1443 4948 y Gc(aux)1422 4986 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35875
b(\000)g(!)1593 4953 y F9(0)p Gc(=)p F9(1)1728 4986 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35876
Ga(M)1826 4953 y FX(0)1845 4986 y F6(f)-7 b Ga(a)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35877
F4(:=)2077 4974 y F9(\()2105 4986 y Ga(x)r F4(:)r Ga(B)2260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35878
4974 y F9(\))2287 4986 y Ga(P)t F6(g)511 b Gg(by)24 b(Lemma)e(2.3.11)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35879
549 5104 y(\(4\))100 b Ga(M)852 5071 y FX(0)871 5104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35880
y F6(f)-7 b Ga(a)26 b F4(:=)1104 5092 y F9(\()1131 5104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35881
y Ga(x)r F4(:)r Ga(B)1286 5092 y F9(\))1313 5104 y Ga(P)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35882
F6(g)g(2)f Ga(S)5 b(N)1666 5118 y Gc(aux)1811 5104 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35883
Gg(for)23 b(all)2053 5092 y F9(\()2080 5104 y Ga(x)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35884
F4(:)r Ga(B)2235 5092 y F9(\))2263 5104 y Ga(P)38 b F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35885
b FB(J)p F4(\()p Ga(B)5 b F4(\))p FB(K)381 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35886
b(\(2\))f(and)h(\(3\))549 5223 y(\(5\))754 5211 y FX(h)782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35887
5223 y Ga(a)r F4(:)r Ga(B)933 5211 y FX(i)960 5223 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35888
Ga(M)1058 5190 y FX(0)1107 5223 y F6(2)j FW(B)t(I)t(N)t(D)t(I)t(N)t(G)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35889
1530 5241 y FX(h)p Gc(B)s FX(i)1646 5223 y F4(\()p FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35890
F4(\()p Ga(B)5 b F4(\))q FB(K)o F4(\))925 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35891
b(De\002nition)h(2.3.3)549 5341 y(\(6\))754 5329 y FX(h)782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35892
5341 y Ga(a)r F4(:)r Ga(B)933 5329 y FX(i)960 5341 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35893
Ga(M)1058 5308 y FX(0)1107 5341 y F6(2)g FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35894
Ga(B)5 b F6(i)o FB(K)1449 b Gg(by)24 b(De\002nition)h(2.3.4)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35895
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35896
%%Page: 143 155
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35897
TeXDict begin 143 154 bop Black 277 51 a Gb(B.1)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35898
(Chapter)f(2)2639 b(143)p 277 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35899
277 388 a(Case)p Black 48 w FW(N)t(O)q(T)t(R)t(I)t(G)t(H)t(T)911
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35900
407 y FX(h:)p Gc(C)5 b FX(i)1075 388 y F4(\()p FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35901
F4(\()p Ga(C)i F4(\))p FB(K)p F4(\))p Gg(,)23 b Ga(B)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35902
b F6(\021)c(:)p Ga(C)7 b Gg(:)504 544 y(\(1\))101 b Ga(M)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35903
b F6(\021)25 b FL(Not)1072 558 y Gc(R)1130 544 y F4(\()1165
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35904
532 y F9(\()1193 544 y Ga(x)1245 532 y F9(\))1272 544
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35905
y Ga(S)5 b(;)15 b(a)p F4(\))p Gg(,)48 b Ga(M)1625 511
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35906
y FX(0)1674 544 y F6(\021)25 b FL(Not)1913 558 y Gc(R)1970
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35907
544 y F4(\()2005 532 y F9(\()2033 544 y Ga(x)2085 532
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35908
y F9(\))2113 544 y Ga(S)2174 511 y FX(0)2197 544 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35909
b(a)p F4(\))p Gg(,)48 b Ga(S)2498 507 y Gc(aux)2478 544
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35910
y F6(\000)-32 b(\000)h(!)25 b Ga(S)2734 511 y FX(0)2780
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35911
544 y Gg(and)710 650 y FX(h)737 662 y Ga(a)r F4(:)r F6(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35912
Ga(C)947 650 y FX(i)974 662 y Ga(M)36 b F6(2)27 b FW(N)t(O)q(T)t(R)t(I)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35913
t(G)t(H)t(T)1586 681 y FX(h:)p Gc(C)5 b FX(i)1749 662
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35914
y F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))p FB(K)p F4(\))939
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35915
b Gg(by)24 b(assumption)504 781 y(\(2\))101 b Ga(M)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35916
b Gg(freshly)25 b(introduces)h Ga(a)d Gg(and)1732 769
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35917
y F9(\()1760 781 y Ga(x)r F4(:)r Ga(C)1913 769 y F9(\))1940
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35918
781 y Ga(S)30 b F6(2)25 b FB(J)p F4(\()p Ga(C)7 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35919
FB(K)487 b Gg(by)24 b(De\002nition)g(2.3.3)504 899 y(\(3\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35920
b Ga(M)808 866 y FX(0)854 899 y Gg(freshly)25 b(introduces)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35921
Ga(a)1327 b Gg(by)24 b(Lemma)e(2.3.8)504 1018 y(\(4\))710
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35922
1006 y F9(\()737 1018 y Ga(x)r F4(:)r Ga(C)890 1006 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35923
F9(\))918 1018 y Ga(S)979 985 y FX(0)1027 1018 y F6(2)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35924
FB(J)p F4(\()p Ga(C)7 b F4(\))p FB(K)1716 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35925
b(induction)504 1136 y(\(5\))710 1124 y FX(h)737 1136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35926
y Ga(a)r F4(:)r F6(:)p Ga(C)947 1124 y FX(i)974 1136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35927
y Ga(M)1072 1103 y FX(0)1121 1136 y F6(2)j FW(N)t(O)q(T)t(R)t(I)t(G)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35928
(H)t(T)1609 1155 y FX(h:)p Gc(C)5 b FX(i)1773 1136 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35929
F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))p FB(K)o F4(\))323
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35930
b Gg(by)24 b(\(3\),)f(\(4\))h(and)g(De\002nition)g(2.3.3)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35931
1255 y(\(6\))710 1243 y FX(h)737 1255 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35932
F6(:)p Ga(C)947 1243 y FX(i)974 1255 y Ga(M)1072 1222
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35933
y FX(0)1121 1255 y F6(2)h FB(J)p F6(h:)p Ga(C)7 b F6(i)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35934
FB(K)1332 b Gg(by)24 b(De\002nition)g(2.3.4)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35935
1425 a Gb(Case)p Black 48 w FW(A)t(N)t(D)t(R)t(I)t(G)t(H)t(T)923
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35936
1444 y FX(h)p Gc(C)5 b FX(^)s Gc(D)r FX(i)1146 1425 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35937
F4(\()p FB(J)p F6(h)p Ga(C)i F6(i)p FB(K)p Ga(;)15 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35938
FB(J)p F6(h)p Ga(D)s F6(i)p FB(K)p F4(\))p Gg(,)23 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35939
Ga(B)29 b F6(\021)c Ga(C)7 b F6(^)o Ga(D)s Gg(:)504 1581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35940
y(\(1\))101 b Ga(M)35 b F6(\021)25 b FL(And)1084 1595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35941
y Gc(R)1142 1581 y F4(\()1177 1569 y FX(h)1204 1581 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35942
Ga(d)1251 1569 y FX(i)1279 1581 y Ga(S)5 b(;)1380 1569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35943
y FX(h)1408 1581 y Ga(e)1450 1569 y FX(i)1478 1581 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35944
Ga(T)13 b(;)i(a)p F4(\))p Gg(,)48 b Ga(M)1836 1548 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35945
FX(0)1885 1581 y F6(\021)25 b FL(And)2135 1595 y Gc(R)2193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35946
1581 y F4(\()2228 1569 y FX(h)2256 1581 y Ga(d)2303 1569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35947
y FX(i)2330 1581 y Ga(S)2391 1548 y FX(0)2415 1581 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35948
Ga(;)2455 1569 y FX(h)2483 1581 y Ga(e)2525 1569 y FX(i)2552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35949
1581 y Ga(T)2618 1548 y FX(0)2641 1581 y Ga(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35950
F4(\))24 b Gg(and)710 1687 y FX(h)737 1699 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35951
Ga(C)7 b F6(^)p Ga(D)1025 1687 y FX(i)1052 1699 y Ga(M)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35952
b F6(2)26 b FW(A)t(N)t(D)t(R)t(I)t(G)t(H)t(T)1675 1718
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35953
y FX(h)p Gc(C)5 b FX(^)s Gc(D)r FX(i)1898 1699 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35954
FB(J)p F6(h)p Ga(C)i F6(i)p FB(K)p Ga(;)15 b FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35955
Ga(D)s F6(i)p FB(K)p F4(\))528 b Gg(by)24 b(assumption)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35956
1818 y(\(2\))101 b Ga(S)816 1781 y Gc(aux)796 1818 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35957
F6(\000)-31 b(\000)g(!)25 b Ga(S)1053 1785 y FX(0)1099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35958
1818 y Gg(and)f Ga(T)38 b F6(\021)25 b Ga(T)1506 1785
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35959
y FX(0)1552 1818 y Gg(\(the)f(other)g(case)g(being)h(similar\))306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35960
b(ne)n(w)23 b(assumption)504 1936 y(\(3\))101 b Ga(M)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35961
b Gg(freshly)25 b(introduces)h Ga(a)p Gg(,)1601 1924
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35962
y FX(h)1628 1936 y Ga(d)r F4(:)r Ga(C)1776 1924 y FX(i)1804
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35963
1936 y Ga(S)k F6(2)25 b FB(J)p F6(h)p Ga(C)7 b F6(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35964
FB(K)22 b Gg(and)2368 1924 y FX(h)2396 1936 y Ga(e)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35965
F4(:)r Ga(D)2545 1924 y FX(i)2573 1936 y Ga(T)38 b F6(2)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35966
b FB(J)p F6(h)p Ga(D)s F6(i)q FB(K)53 b Gg(by)24 b(Def.)e(2.3.3)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35967
2055 y(\(4\))101 b Ga(M)808 2022 y FX(0)854 2055 y Gg(freshly)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35968
b(introduces)i Ga(a)1327 b Gg(by)24 b(Lemma)e(2.3.8)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35969
2173 y(\(5\))710 2161 y FX(h)737 2173 y Ga(d)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35970
Ga(C)885 2161 y FX(i)913 2173 y Ga(S)974 2140 y FX(0)1023
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35971
2173 y F6(2)i FB(J)p F6(h)p Ga(C)7 b F6(i)p FB(K)1721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35972
b Gg(by)24 b(induction)504 2292 y(\(6\))710 2280 y FX(h)737
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35973
2292 y Ga(a)r F4(:)r Ga(C)7 b F6(^)p Ga(D)1025 2280 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35974
FX(i)1052 2292 y Ga(M)1150 2259 y FX(0)1199 2292 y F6(2)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35975
b FW(A)t(N)t(D)t(R)t(I)t(G)t(H)t(T)1699 2310 y FX(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35976
Gc(C)5 b FX(^)r Gc(D)r FX(i)1921 2292 y F4(\()p FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35977
F6(h)p Ga(C)i F6(i)p FB(K)p Ga(;)15 b FB(J)p F6(h)p Ga(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35978
F6(i)q FB(K)o F4(\))122 b Gg(by)23 b(\(4\),)h(\(5\))g(and)g(Def.)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35979
(2.3.3)504 2411 y(\(7\))710 2399 y FX(h)737 2411 y Ga(a)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35980
F4(:)r Ga(C)7 b F6(^)p Ga(D)1025 2399 y FX(i)1052 2411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35981
y Ga(M)1150 2378 y FX(0)1199 2411 y F6(2)25 b FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35982
F6(h)p Ga(C)7 b F6(^)o Ga(D)s F6(i)p FB(K)1176 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35983
b(De\002nition)g(2.3.4)p Black 277 2589 a Gb(Case)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35984
48 w FW(O)t(R)t(R)t(I)t(G)t(H)t(T)862 2556 y Gc(i)862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35985
2621 y FX(h)p Gc(C)939 2630 y FV(1)976 2621 y FX(_)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35986
Gc(C)1073 2630 y FV(2)1108 2621 y FX(i)1140 2589 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35987
FB(J)p F6(h)p Ga(C)1312 2603 y Gc(i)1340 2589 y F6(i)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35988
FB(K)o F4(\))q Gg(,)e Ga(B)30 b F6(\021)25 b Ga(C)1753
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35989
2603 y F9(1)1792 2589 y F6(_)p Ga(C)1918 2603 y F9(2)1957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35990
2589 y Gg(,)e F4(\()p Ga(i)j F4(=)f(1)p Ga(;)15 b F4(2\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35991
Gg(:)504 2758 y(\(1\))101 b Ga(M)35 b F6(\021)25 b FL(Or)1029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35992
2721 y Gc(i)1029 2781 y(R)1086 2758 y F4(\()1121 2746
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35993
y FX(h)1149 2758 y Ga(d)1196 2746 y FX(i)1224 2758 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35994
Ga(S)5 b(;)15 b(a)p F4(\))p Gg(,)48 b Ga(M)1577 2725
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35995
y FX(0)1626 2758 y F6(\021)25 b FL(Or)1821 2721 y Gc(i)1821
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35996
2781 y(R)1879 2758 y F4(\()1914 2746 y FX(h)1942 2758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35997
y Ga(d)1989 2746 y FX(i)2016 2758 y Ga(S)2077 2725 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35998
FX(0)2101 2758 y Ga(;)15 b(a)p F4(\))p Gg(,)48 b Ga(S)2402
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 35999
2721 y Gc(aux)2382 2758 y F6(\000)-32 b(\000)h(!)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36000
Ga(S)2638 2725 y FX(0)2684 2758 y Gg(and)710 2864 y FX(h)737
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36001
2876 y Ga(a)r F4(:)r Ga(C)879 2890 y F9(1)919 2876 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36002
F6(_)p Ga(C)1045 2890 y F9(2)1084 2864 y FX(i)1112 2876
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36003
y Ga(M)35 b F6(2)27 b FW(O)t(R)t(R)t(I)t(G)t(H)t(T)1674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36004
2843 y Gc(i)1674 2908 y FX(h)p Gc(C)1751 2917 y FV(1)1788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36005
2908 y FX(_)p Gc(C)1885 2917 y FV(2)1920 2908 y FX(i)1951
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36006
2876 y F4(\()p FB(J)p F6(h)p Ga(C)2123 2890 y Gc(i)2152
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36007
2876 y F6(i)p FB(K)p F4(\))715 b Gg(by)24 b(assumption)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36008
3007 y(\(2\))101 b Ga(M)33 b Gg(freshly)25 b(introduces)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36009
Ga(a)d Gg(and)1732 2995 y FX(h)1760 3007 y Ga(d)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36010
Ga(C)1901 3021 y Gc(i)1930 2995 y FX(i)1957 3007 y Ga(S)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36011
b F6(2)25 b FB(J)p F6(h)p Ga(C)2266 3021 y Gc(i)2294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36012
3007 y F6(i)q FB(K)448 b Gg(by)24 b(De\002nition)g(2.3.3)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36013
3126 y(\(3\))101 b Ga(M)808 3093 y FX(0)854 3126 y Gg(freshly)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36014
b(introduces)i Ga(a)1327 b Gg(by)24 b(Lemma)e(2.3.8)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36015
3244 y(\(4\))710 3232 y FX(h)737 3244 y Ga(d)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36016
Ga(C)878 3258 y Gc(i)907 3232 y FX(i)935 3244 y Ga(S)996
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36017
3211 y FX(0)1044 3244 y F6(2)j FB(J)p F6(h)p Ga(C)1267
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36018
3258 y Gc(i)1295 3244 y F6(i)q FB(K)1677 b Gg(by)24 b(induction)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36019
3363 y(\(5\))710 3351 y FX(h)737 3363 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36020
Ga(C)879 3377 y F9(1)919 3363 y F6(_)p Ga(C)1045 3377
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36021
y F9(2)1084 3351 y FX(i)1112 3363 y Ga(M)1210 3330 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36022
FX(0)1258 3363 y F6(2)j FW(O)t(R)t(R)t(I)t(G)t(H)t(T)1697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36023
3330 y Gc(i)1697 3394 y FX(h)p Gc(C)1774 3403 y FV(1)1811
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36024
3394 y FX(^)p Gc(C)1908 3403 y FV(2)1943 3394 y FX(i)1975
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36025
3363 y F4(\()p FB(J)p F6(h)p Ga(C)2147 3377 y Gc(i)2175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36026
3363 y F6(i)q FB(K)o F4(\))99 b Gg(by)24 b(\(3\),)f(\(4\))h(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36027
(De\002nition)g(2.3.3)504 3494 y(\(6\))710 3482 y FX(h)737
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36028
3494 y Ga(a)r F4(:)r Ga(C)879 3508 y F9(1)919 3494 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36029
F6(_)p Ga(C)1045 3508 y F9(2)1084 3482 y FX(i)1112 3494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36030
y Ga(M)1210 3461 y FX(0)1258 3494 y F6(2)h FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36031
Ga(C)1481 3508 y F9(1)1521 3494 y F6(_)o Ga(C)1646 3508
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36032
y F9(2)1686 3494 y F6(i)p FB(K)1057 b Gg(by)24 b(De\002nition)g(2.3.4)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36033
Black 277 3709 a Gb(Case)p Black 48 w FW(I)t(M)t(P)t(R)t(I)t(G)t(H)t(T)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36034
893 3727 y FX(h)p Gc(C)5 b FX(\033)t Gc(D)r FX(i)1126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36035
3709 y F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))p FB(K)o Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36036
b FB(J)p F6(h)p Ga(D)s F6(i)q FB(K)p F4(\))p Gg(,)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36037
b Ga(B)30 b F6(\021)25 b Ga(C)7 b F6(\033)o Ga(D)s Gg(:)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36038
3864 y(\(1\))101 b Ga(M)35 b F6(\021)25 b FL(Imp)1074
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36039
3886 y Gc(R)1132 3864 y F4(\()1167 3852 y F9(\()1194
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36040
3864 y Ga(x)1246 3852 y F9(\))q FX(h)1301 3864 y Ga(d)1348
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36041
3852 y FX(i)1376 3864 y Ga(S)5 b(;)15 b(a)p F4(\))p Gg(,)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36042
b Ga(M)1729 3831 y FX(0)1778 3864 y F6(\021)25 b FL(Imp)2018
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36043
3886 y Gc(R)2076 3864 y F4(\()2111 3852 y F9(\()2139
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36044
3864 y Ga(x)2191 3852 y F9(\))p FX(h)2246 3864 y Ga(d)2293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36045
3852 y FX(i)2321 3864 y Ga(S)2382 3831 y FX(0)2405 3864
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36046
y Ga(;)15 b(a)p F4(\))p Gg(,)48 b Ga(S)2706 3827 y Gc(aux)2686
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36047
3864 y F6(\000)-32 b(\000)h(!)25 b Ga(S)2942 3831 y FX(0)2988
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36048
3864 y Gg(and)710 3971 y FX(h)737 3983 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36049
Ga(C)7 b F6(\033)p Ga(D)1035 3971 y FX(i)1062 3983 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36050
Ga(M)36 b F6(2)26 b FW(I)t(M)t(P)t(R)t(I)t(G)t(H)t(T)1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36051
4001 y FX(h)p Gc(C)5 b FX(\033)t Gc(C)g FX(i)1883 3983
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36052
y F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))p FB(K)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36053
b FB(J)p F6(h)p Ga(D)s F6(i)p FB(K)p F4(\))543 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36054
b(assumption)504 4101 y(\(2\))101 b Ga(M)33 b Gg(freshly)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36055
b(introduces)h Ga(a)p Gg(,)710 4208 y FX(h)737 4220 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36056
Ga(d)r F4(:)r Ga(D)891 4208 y FX(i)919 4220 y Ga(S)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36057
F6(f)-7 b Ga(x)25 b F4(:=)1212 4208 y FX(h)1240 4220
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36058
y Ga(e)r F4(:)r Ga(C)1383 4208 y FX(i)1410 4220 y Ga(Q)-9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36059
b F6(g)26 b(2)e FB(J)p F6(h)p Ga(D)s F6(i)q FB(K)e Gg(for)i(all)2116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36060
4208 y FX(h)2144 4220 y Ga(e)r F4(:)r Ga(C)2287 4208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36061
y FX(i)2314 4220 y Ga(Q)h F6(2)g FB(J)p F6(h)p Ga(C)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36062
b F6(i)p FB(K)p Gg(,)22 b(and)710 4326 y F9(\()737 4338
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36063
y Ga(x)r F4(:)r Ga(C)890 4326 y F9(\))918 4338 y Ga(S)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36064
F6(f)-7 b Ga(d)26 b F4(:=)1206 4326 y F9(\()1234 4338
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36065
y Ga(z)6 b F4(:)r Ga(D)1387 4326 y F9(\))1414 4338 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36066
Ga(P)t F6(g)26 b(2)e FB(J)p F4(\()p Ga(C)7 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36067
FB(K)23 b Gg(for)g(all)2113 4326 y F9(\()2140 4338 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36068
Ga(z)6 b F4(:)r Ga(D)2293 4326 y F9(\))2321 4338 y Ga(P)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36069
b F6(2)24 b FB(J)p F4(\()p Ga(D)s F4(\))p FB(K)90 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36070
b(De\002nition)g(2.3.3)504 4457 y(\(3\))101 b Ga(M)808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36071
4424 y FX(0)854 4457 y Gg(freshly)25 b(introduces)i Ga(a)1327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36072
b Gg(by)24 b(Lemma)e(2.3.8)504 4576 y(\(4\))101 b Ga(S)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36073
F6(f)-7 b Ga(x)25 b F4(:=)1003 4564 y FX(h)1031 4576
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36074
y Ga(e)r F4(:)r Ga(C)1174 4564 y FX(i)1201 4576 y Ga(Q)-9
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36075
b F6(g)1354 4538 y Gc(aux)1334 4576 y F6(\000)-31 b(\000)g(!)1504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36076
4543 y F9(0)p Gc(=)p F9(1)1640 4576 y Ga(S)1701 4543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36077
y FX(0)1719 4576 y F6(f)-7 b Ga(x)26 b F4(:=)1956 4564
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36078
y FX(h)1984 4576 y Ga(e)r F4(:)r Ga(C)2127 4564 y FX(i)2154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36079
4576 y Ga(Q)-9 b F6(g)p Gg(,)710 4694 y Ga(S)q F6(f)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36080
Ga(d)25 b F4(:=)998 4682 y F9(\()1026 4694 y Ga(z)6 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36081
F4(:)r Ga(D)1179 4682 y F9(\))1207 4694 y Ga(P)s F6(g)1359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36082
4657 y Gc(aux)1339 4694 y F6(\000)-31 b(\000)f(!)1509
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36083
4661 y F9(0)p Gc(=)p F9(1)1644 4694 y Ga(S)1705 4661
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36084
y FX(0)1724 4694 y F6(f)-7 b Ga(d)26 b F4(:=)1956 4682
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36085
y F9(\()1984 4694 y Ga(z)6 b F4(:)r Ga(D)2137 4682 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36086
F9(\))2164 4694 y Ga(P)t F6(g)590 b Gg(by)23 b(Lemma)g(2.3.11)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36087
4813 y(\(5\))710 4801 y FX(h)737 4813 y Ga(d)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36088
Ga(D)891 4801 y FX(i)919 4813 y Ga(S)980 4780 y FX(0)999
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36089
4813 y F6(f)-7 b Ga(x)25 b F4(:=)1236 4801 y FX(h)1263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36090
4813 y Ga(e)r F4(:)r Ga(C)1406 4801 y FX(i)1434 4813
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36091
y Ga(Q)-10 b F6(g)26 b(2)f FB(J)p F6(h)p Ga(D)s F6(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36092
FB(K)d Gg(for)i(all)2140 4801 y FX(h)2167 4813 y Ga(e)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36093
F4(:)r Ga(C)2310 4801 y FX(i)2338 4813 y Ga(Q)h F6(2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36094
FB(J)p F6(h)p Ga(C)7 b F6(i)o FB(K)p Gg(,)710 4919 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36095
F9(\()737 4931 y Ga(x)r F4(:)r Ga(C)890 4919 y F9(\))918
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36096
4931 y Ga(S)979 4898 y FX(0)997 4931 y F6(f)-7 b Ga(d)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36097
b F4(:=)1229 4919 y F9(\()1257 4931 y Ga(z)6 b F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36098
Ga(D)1410 4919 y F9(\))1438 4931 y Ga(P)s F6(g)26 b(2)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36099
FB(J)p F4(\()p Ga(C)7 b F4(\))p FB(K)22 b Gg(for)i(all)2136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36100
4919 y F9(\()2164 4931 y Ga(z)6 b F4(:)r Ga(D)2317 4919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36101
y F9(\))2345 4931 y Ga(P)38 b F6(2)25 b FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36102
Ga(D)s F4(\))p FB(K)1598 5050 y Gg(by)f(\(2\))f(and)h(\(4\):)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36103
b(`0'-case)25 b(tri)n(vial,)g(`1'-case)g(by)f(induction)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36104
5168 y(\(6\))710 5156 y FX(h)737 5168 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36105
Ga(C)7 b F6(\033)p Ga(D)1035 5156 y FX(i)1062 5168 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36106
Ga(M)1160 5135 y FX(0)1209 5168 y F6(2)27 b FW(I)t(M)t(P)t(R)t(I)t(G)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36107
(H)t(T)1679 5187 y FX(h)p Gc(C)5 b FX(\033)t Gc(C)g FX(i)1906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36108
5168 y F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))p FB(K)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36109
b FB(J)p F6(h)p Ga(D)s F6(i)p FB(K)p F4(\))137 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36110
b(\(3\),)h(\(5\))g(and)g(Def.)e(2.3.3)504 5287 y(\(7\))710
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36111
5275 y FX(h)737 5287 y Ga(a)r F4(:)r Ga(C)7 b F6(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36112
Ga(D)1035 5275 y FX(i)1062 5287 y Ga(M)1160 5254 y FX(0)1209
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36113
5287 y F6(2)25 b FB(J)p F6(h)p Ga(C)7 b F6(\033)o Ga(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36114
F6(i)p FB(K)1156 b Gg(by)24 b(De\002nition)g(2.3.4)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36115
3436 5392 4 62 v 3440 5334 55 4 v 3440 5392 V 3494 5392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36116
4 62 v Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36117
%%Page: 144 156
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36118
TeXDict begin 144 155 bop Black -144 51 a Gb(144)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36119
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36120
4 v Black Black 321 388 a(Pr)n(oof)34 b(of)e(Lemma)g(2.3.13.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36121
Black 34 w Gg(Analogous)j(to)d(the)h(proof)h(of)f(the)g(preceding)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36122
(lemma,)f(we)e(need)i(to)321 501 y(analyse)c(all)f(possible)h(sets)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36123
(where)1479 489 y FX(h)1506 501 y Ga(a)r F4(:)r Ga(B)1657
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36124
489 y FX(i)1685 501 y Ga(M)37 b Gg(may)28 b(be)h(member)f(in.)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36125
b(Six)28 b(cases)h(are)g(gi)n(v)o(en)g(for)f(\(i\);)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36126
614 y(the)c(ar)n(guments)i(for)e(\(ii\))f(are)h(similar)g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36127
(omitted.)p Black 321 793 a Gb(Case)p Black 49 w FW(A)t(X)t(I)t(O)t(M)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36128
(S)866 812 y FX(h)p Gc(B)s FX(i)983 793 y Gg(:)k(In)c(this)g(case)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36129
Ga(M)33 b Gg(is)23 b(an)h(axiom)g(and)g(therefore)i(strongly)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36130
(normalising.)p Black 321 964 a Gb(Case)p Black 49 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36131
FW(B)t(I)t(N)t(D)t(I)t(N)t(G)891 983 y FX(h)p Gc(B)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36132
FX(i)1008 964 y F4(\()p FB(J)p F4(\()p Ga(B)5 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36133
FB(K)o F4(\))q Gg(:)549 1120 y(\(1\))754 1108 y FX(h)782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36134
1120 y Ga(a)r F4(:)r Ga(B)933 1108 y FX(i)960 1120 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36135
Ga(M)35 b F6(2)27 b FW(B)t(I)t(N)t(D)t(I)t(N)t(G)1506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36136
1138 y FX(h)p Gc(B)s FX(i)1623 1120 y F4(\()p FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36137
F4(\()p Ga(B)5 b F4(\))p FB(K)p F4(\))1108 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36138
b(assumption)549 1238 y(\(2\))100 b Ga(M)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36139
b Ga(a)26 b F4(:=)1080 1226 y F9(\()1108 1238 y Ga(x)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36140
F4(:)r Ga(B)1263 1226 y F9(\))1290 1238 y Ga(P)t F6(g)f(2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36141
Ga(S)5 b(N)1642 1252 y Gc(aux)1787 1238 y Gg(for)24 b(all)2030
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36142
1226 y F9(\()2057 1238 y Ga(x)r F4(:)r Ga(B)2212 1226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36143
y F9(\))2239 1238 y Ga(P)39 b F6(2)24 b FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36144
Ga(B)5 b F4(\))p FB(K)220 b Gg(by)24 b(De\002nition)h(2.3.3)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36145
1357 y(\(3\))754 1345 y F9(\()782 1357 y Ga(x)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36146
Ga(B)937 1345 y F9(\))964 1357 y FL(Ax)o F4(\()p Ga(x;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36147
b(a)p F4(\))27 b F6(2)e FB(J)p F4(\()p Ga(B)5 b F4(\))o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36148
FB(K)1233 b Gg(by)24 b(Lemma)e(2.3.6\(ii\))549 1475 y(\(4\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36149
b Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)1080 1463 y F9(\()1108
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36150
1475 y Ga(x)r F4(:)r Ga(B)1263 1463 y F9(\))1290 1475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36151
y FL(Ax)p F4(\()p Ga(x;)15 b(a)p F4(\))-9 b F6(g)26 b(2)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36152
Ga(S)5 b(N)1885 1489 y Gc(aux)2493 1475 y Gg(by)23 b(\(2\),)h(\(3\))f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36153
(and)h Ga(P)39 b F6(\021)25 b FL(Ax)o F4(\()p Ga(x;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36154
b(a)p F4(\))549 1594 y Gg(\(5\))100 b Ga(M)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36155
b Ga(a)26 b F4(:=)1080 1582 y F9(\()1108 1594 y Ga(x)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36156
F4(:)r Ga(B)1263 1582 y F9(\))1290 1594 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36157
Ga(x;)15 b(a)p F4(\))-9 b F6(g)1685 1557 y Gc(aux)1665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36158
1594 y F6(\000)-31 b(\000)f(!)1835 1561 y FX(\003)1900
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36159
1594 y Ga(M)962 b Gg(by)24 b(Lemma)f(2.3.9)549 1713 y(\(6\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36160
b Ga(M)36 b F6(2)24 b Ga(S)5 b(N)1097 1727 y Gc(aux)3044
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36161
1713 y Gg(by)24 b(\(4\))f(and)h(\(5\))p Black 321 1883
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36162
a Gb(Case)p Black 49 w FW(N)t(O)q(T)t(R)t(I)t(G)t(H)t(T)956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36163
1901 y FX(h:)p Gc(C)5 b FX(i)1119 1883 y F4(\()p FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36164
F4(\()p Ga(C)i F4(\))p FB(K)p F4(\))p Gg(,)23 b Ga(B)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36165
b F6(\021)25 b(:)p Ga(C)7 b Gg(:)549 2038 y(\(1\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36166
b Ga(M)36 b F6(\021)25 b FL(Not)1116 2052 y Gc(R)1174
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36167
2038 y F4(\()1209 2026 y F9(\()1237 2038 y Ga(x)1289
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36168
2026 y F9(\))1316 2038 y Ga(S)5 b(;)15 b(a)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36169
Gg(,)754 2145 y FX(h)782 2157 y Ga(a)r F4(:)r F6(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36170
Ga(C)992 2145 y FX(i)1019 2157 y Ga(M)35 b F6(2)27 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36171
FW(N)t(O)q(T)t(R)t(I)t(G)t(H)t(T)1630 2175 y FX(h:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36172
Gc(C)5 b FX(i)1793 2157 y F4(\()p FB(J)p F4(\()p Ga(C)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36173
F4(\))q FB(K)o F4(\))940 b Gg(by)23 b(assumption)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36174
2275 y(\(2\))754 2263 y F9(\()782 2275 y Ga(x)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36175
Ga(C)935 2263 y F9(\))962 2275 y Ga(S)30 b F6(2)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36176
FB(J)p F4(\()p Ga(C)7 b F4(\))p FB(K)1509 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36177
b(De\002nition)h(2.3.3)549 2394 y(\(3\))100 b Ga(S)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36178
b F6(2)25 b Ga(S)5 b(N)1060 2408 y Gc(aux)3089 2394 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36179
Gg(by)24 b(induction)549 2512 y(\(4\))100 b FL(Not)897
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36180
2526 y Gc(R)955 2512 y F4(\()990 2500 y F9(\()1017 2512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36181
y Ga(x)1069 2500 y F9(\))1097 2512 y Ga(S)5 b(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36182
F4(\))26 b F6(2)f Ga(S)5 b(N)1527 2526 y Gc(aux)3327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36183
2512 y Gg(by)23 b(\(3\))p Black 321 2683 a Gb(Case)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36184
Black 49 w FW(A)t(N)t(D)t(R)t(I)t(G)t(H)t(T)968 2701
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36185
y FX(h)p Gc(C)5 b FX(^)r Gc(D)r FX(i)1190 2683 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36186
FB(J)p F6(h)p Ga(C)i F6(i)p FB(K)p Ga(;)15 b FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36187
Ga(D)s F6(i)p FB(K)p F4(\))p Gg(,)23 b Ga(B)30 b F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36188
b Ga(C)7 b F6(^)o Ga(D)s Gg(:)549 2838 y(\(1\))100 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36189
Ga(M)36 b F6(\021)25 b FL(And)1128 2852 y Gc(R)1186 2838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36190
y F4(\()1221 2826 y FX(h)1249 2838 y Ga(d)1296 2826 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36191
FX(i)1323 2838 y Ga(S)5 b(;)1424 2826 y FX(h)1452 2838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36192
y Ga(e)1494 2826 y FX(i)1522 2838 y Ga(T)13 b(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36193
F4(\))p Gg(,)754 2945 y FX(h)782 2957 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36194
Ga(C)7 b F6(^)o Ga(D)1069 2945 y FX(i)1096 2957 y Ga(M)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36195
b F6(2)27 b FW(A)t(N)t(D)t(R)t(I)t(G)t(H)t(T)1720 2975
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36196
y FX(h)p Gc(C)5 b FX(^)r Gc(D)r FX(i)1942 2957 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36197
FB(J)p F6(h)p Ga(C)i F6(i)p FB(K)p Ga(;)15 b FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36198
Ga(D)s F6(i)p FB(K)p F4(\))529 b Gg(by)23 b(assumption)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36199
3075 y(\(2\))754 3063 y FX(h)782 3075 y Ga(d)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36200
Ga(C)930 3063 y FX(i)957 3075 y Ga(S)30 b F6(2)25 b FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36201
F6(h)p Ga(C)7 b F6(i)p FB(K)22 b Gg(and)1521 3063 y FX(h)1549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36202
3075 y Ga(e)r F4(:)r Ga(D)1698 3063 y FX(i)1726 3075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36203
y Ga(T)38 b F6(2)25 b FB(J)p F6(h)p Ga(D)s F6(i)p FB(K)734
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36204
b Gg(by)24 b(De\002nition)h(2.3.3)549 3194 y(\(3\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36205
b Ga(S;)15 b(T)39 b F6(2)24 b Ga(S)5 b(N)1161 3208 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36206
Gc(aux)3089 3194 y Gg(by)24 b(induction)549 3312 y(\(4\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36207
b FL(And)909 3326 y Gc(R)966 3312 y F4(\()1001 3300 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36208
FX(h)1029 3312 y Ga(d)1076 3300 y FX(i)1104 3312 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36209
b(;)1205 3300 y FX(h)1233 3312 y Ga(e)1275 3300 y FX(i)1303
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36210
3312 y Ga(T)12 b(;)j(a)p F4(\))26 b F6(2)f Ga(S)5 b(N)1737
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36211
3326 y Gc(aux)3327 3312 y Gg(by)23 b(\(3\))p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36212
3491 a Gb(Case)p Black 49 w FW(O)t(R)t(R)t(I)t(G)t(H)t(T)907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36213
3458 y Gc(i)907 3523 y FX(h)p Gc(C)984 3532 y FV(1)1021
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36214
3523 y FX(_)p Gc(C)1118 3532 y FV(2)1152 3523 y FX(i)1184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36215
3491 y F4(\()p FB(J)p F6(h)p Ga(C)1356 3505 y Gc(i)1385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36216
3491 y F6(i)p FB(K)p F4(\))p Gg(,)f Ga(B)30 b F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36217
b Ga(C)1797 3505 y F9(1)1836 3491 y F6(_)p Ga(C)1962
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36218
3505 y F9(2)2001 3491 y Gg(,)e F4(\()p Ga(i)j F4(=)f(1)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36219
Ga(;)15 b F4(2\))p Gg(:)549 3659 y(\(1\))100 b Ga(M)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36220
b F6(\021)25 b FL(Or)1073 3623 y Gc(i)1073 3683 y(R)1130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36221
3659 y F4(\()1165 3647 y FX(h)1193 3659 y Ga(d)1240 3647
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36222
y FX(i)1268 3659 y Ga(S)5 b(;)15 b(a)p F4(\))p Gg(,)754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36223
3766 y FX(h)782 3778 y Ga(a)r F4(:)r Ga(C)924 3792 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36224
F9(1)963 3778 y F6(_)p Ga(C)1089 3792 y F9(2)1128 3766
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36225
y FX(i)1156 3778 y Ga(M)35 b F6(2)27 b FW(O)t(R)t(R)t(I)t(G)t(H)t(T)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36226
1718 3745 y Gc(i)1718 3809 y FX(h)p Gc(C)1795 3818 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36227
FV(1)1832 3809 y FX(_)p Gc(C)1929 3818 y FV(2)1964 3809
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36228
y FX(i)1996 3778 y F4(\()p FB(J)p F6(h)p Ga(C)2168 3792
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36229
y Gc(i)2196 3778 y F6(i)p FB(K)p F4(\))716 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36230
b(assumption)549 3909 y(\(2\))754 3897 y FX(h)782 3909
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36231
y Ga(d)r F4(:)r Ga(C)923 3923 y Gc(i)951 3897 y FX(i)979
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36232
3909 y Ga(S)30 b F6(2)25 b FB(J)p F6(h)p Ga(C)1288 3923
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36233
y Gc(i)1316 3909 y F6(i)p FB(K)1471 b Gg(by)24 b(De\002nition)h(2.3.3)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36234
549 4028 y(\(3\))100 b Ga(S)30 b F6(2)25 b Ga(S)5 b(N)1060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36235
4042 y Gc(aux)3089 4028 y Gg(by)24 b(induction)549 4146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36236
y(\(4\))100 b FL(Or)853 4109 y Gc(i)853 4169 y(R)911
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36237
4146 y F4(\()946 4134 y FX(h)974 4146 y Ga(d)1021 4134
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36238
y FX(i)1049 4146 y Ga(S)5 b(;)15 b(a)p F4(\))26 b F6(2)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36239
Ga(S)5 b(N)1479 4160 y Gc(aux)3327 4146 y Gg(by)23 b(\(3\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36240
Black 321 4316 a Gb(Case)p Black 49 w FW(I)t(M)t(P)t(R)t(I)t(G)t(H)t(T)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36241
938 4335 y FX(h)p Gc(C)5 b FX(\033)s Gc(D)r FX(i)1170
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36242
4316 y F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))p FB(K)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36243
b FB(J)p F6(h)p Ga(D)s F6(i)p FB(K)p F4(\))p Gg(,)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36244
b Ga(B)29 b F6(\021)c Ga(C)7 b F6(\033)o Ga(D)s Gg(:)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36245
4472 y(\(1\))100 b Ga(M)36 b F6(\021)25 b FL(Imp)1118
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36246
4494 y Gc(R)1176 4472 y F4(\()1211 4460 y F9(\()1238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36247
4472 y Ga(x)1290 4460 y F9(\))q FX(h)1345 4472 y Ga(d)1392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36248
4460 y FX(i)1420 4472 y Ga(S)5 b(;)15 b(a)p F4(\))p Ga(;)754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36249
4578 y FX(h)782 4590 y Ga(a)r F4(:)r Ga(C)7 b F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36250
Ga(D)1079 4578 y FX(i)1106 4590 y Ga(M)36 b F6(2)27 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36251
FW(I)t(M)t(P)t(R)t(I)t(G)t(H)t(T)1700 4609 y FX(h)p Gc(C)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36252
b FX(\033)s Gc(C)g FX(i)1927 4590 y F4(\()p FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36253
Ga(C)i F4(\))p FB(K)p Ga(;)15 b FB(J)p F6(h)p Ga(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36254
F6(i)p FB(K)p F4(\))544 b Gg(by)23 b(assumption)549 4709
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36255
y(\(2\))754 4697 y F9(\()782 4709 y Ga(x)r F4(:)r Ga(C)935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36256
4697 y F9(\))962 4709 y Ga(S)p F6(f)-7 b Ga(d)26 b F4(:=)1250
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36257
4697 y F9(\()1278 4709 y Ga(z)6 b F4(:)r Ga(D)1431 4697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36258
y F9(\))1459 4709 y Ga(P)s F6(g)26 b(2)f FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36259
Ga(C)7 b F4(\))o FB(K)23 b Gg(for)h(all)2157 4697 y F9(\()2185
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36260
4709 y Ga(z)6 b F4(:)r Ga(D)2338 4697 y F9(\))2365 4709
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36261
y Ga(P)39 b F6(2)25 b FB(J)p F4(\()p Ga(D)s F4(\))p FB(K)89
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36262
b Gg(by)24 b(De\002nition)h(2.3.3)549 4827 y(\(3\))754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36263
4815 y F9(\()782 4827 y Ga(z)6 b F4(:)r Ga(D)935 4815
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36264
y F9(\))962 4827 y FL(Ax)p F4(\()p Ga(z)t(;)15 b(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36265
F4(\))27 b F6(2)e FB(J)p F4(\()p Ga(D)s F4(\))p FB(K)1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36266
b Gg(by)24 b(Lemma)e(2.3.6\(ii\))549 4946 y(\(4\))754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36267
4934 y F9(\()782 4946 y Ga(x)r F4(:)r Ga(C)935 4934 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36268
F9(\))962 4946 y Ga(S)q F6(f)-7 b Ga(d)25 b F4(:=)1250
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36269
4934 y F9(\()1278 4946 y Ga(z)6 b F4(:)r Ga(D)1431 4934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36270
y F9(\))1459 4946 y FL(Ax)o F4(\()p Ga(z)t(;)15 b(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36271
F4(\))-8 b F6(g)26 b(2)f FB(J)p F4(\()p Ga(C)7 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36272
FB(K)370 b Gg(by)24 b(\(2\),)f(\(3\))h(and)g Ga(P)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36273
b F6(\021)25 b FL(Ax)p F4(\()p Ga(z)t(;)15 b(d)p F4(\))549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36274
5064 y Gg(\(5\))100 b Ga(S)q F6(f)-7 b Ga(d)25 b F4(:=)1043
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36275
5052 y F9(\()1070 5064 y Ga(z)6 b F4(:)r Ga(D)1223 5052
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36276
y F9(\))1251 5064 y FL(Ax)o F4(\()p Ga(z)t(;)15 b(d)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36277
F4(\))-7 b F6(g)25 b(2)g Ga(S)5 b(N)1839 5078 y Gc(aux)3089
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36278
5064 y Gg(by)24 b(induction)549 5183 y(\(6\))100 b Ga(S)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36279
F6(f)-7 b Ga(d)25 b F4(:=)1043 5171 y F9(\()1070 5183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36280
y Ga(z)6 b F4(:)r Ga(D)1223 5171 y F9(\))1251 5183 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36281
FL(Ax)o F4(\()p Ga(z)t(;)15 b(d)p F4(\))-7 b F6(g)1640
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36282
5146 y Gc(aux)1619 5183 y F6(\000)-31 b(\000)g(!)1790
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36283
5150 y FX(\003)1854 5183 y Ga(S)1040 b Gg(by)24 b(Lemma)f(2.3.9)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36284
5302 y(\(7\))100 b Ga(S)30 b F6(2)25 b Ga(S)5 b(N)1060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36285
5316 y Gc(aux)3044 5302 y Gg(by)24 b(\(5\))f(and)h(\(6\))549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36286
5420 y(\(8\))100 b FL(Imp)899 5442 y Gc(R)956 5420 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36287
F4(\()991 5408 y F9(\()1019 5420 y Ga(x)1071 5408 y F9(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36288
FX(h)1126 5420 y Ga(d)1173 5408 y FX(i)1201 5420 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36289
b(;)15 b(a)p F4(\))26 b F6(2)f Ga(S)5 b(N)1631 5434 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36290
Gc(aux)3327 5420 y Gg(by)23 b(\(7\))p 3480 5525 4 62
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36291
v 3484 5467 55 4 v 3484 5525 V 3538 5525 4 62 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36292
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36293
%%Page: 145 157
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36294
TeXDict begin 145 156 bop Black 277 51 a Gb(B.1)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36295
(Chapter)f(2)2639 b(145)p 277 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36296
277 388 a(Pr)n(oof)28 b(of)g(Lemma)e(2.3.14.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36297
34 w Gg(The)h(induction)j(proceeds)g(o)o(v)o(er)e(the)f(le)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36298
(xicographically)33 b(ordered)c(in-)277 501 y(duction)d(v)n(alue)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36299
F4(\()p Ga(\016)n(;)15 b(\026;)g(\027)6 b F4(\))24 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36300
Gg(assigned)i(to)e(each)g(term)f(of)h(the)f(form)h FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36301
F4(\()2521 489 y FX(h)2548 501 y Ga(a)r F4(:)r Ga(B)2699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36302
489 y FX(i)2727 501 y Ga(M)10 b(;)2865 489 y F9(\()2893
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36303
501 y Ga(x)r F4(:)r Ga(B)3048 489 y F9(\))3075 501 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36304
Ga(N)g F4(\))p Gg(;)23 b Ga(\016)k Gg(is)c(the)277 614
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36305
y(de)o(gree)i(of)e(the)g(cut-formula)j Ga(B)5 b Gg(,)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36306
b(and)h Ga(\026)g Gg(and)g Ga(\027)29 b Gg(are)23 b(the)h(longest)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36307
(reduction)h(sequences)g(\(relati)n(v)o(e)e(to)297 690
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36308
y Gc(aux)277 727 y F6(\000)-31 b(\000)f(!)p Gg(\))27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36309
b(starting)j(from)d Ga(M)37 b Gg(and)28 b Ga(N)10 b Gg(,)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36310
b(respecti)n(v)o(ely)-6 b(.)43 b(By)27 b(assumption)j(both)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36311
Ga(\026)d Gg(and)i Ga(\027)33 b Gg(are)27 b(\002nite.)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36312
b(W)-7 b(e)277 840 y(shall)34 b(sho)n(w)e(that)h(e)n(v)o(ery)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36313
(immediate)h(reduct)f(to)g(which)g FL(Cut)p F4(\()2304
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36314
828 y FX(h)2332 840 y Ga(a)2380 828 y FX(i)2407 840 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36315
Ga(M)10 b(;)2545 828 y F9(\()2573 840 y Ga(x)2625 828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36316
y F9(\))2652 840 y Ga(N)g F4(\))32 b Gg(reduces)j(is)d(strongly)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36317
953 y(normalising.)277 1086 y(First,)23 b(we)g(shall)h(gi)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36318
(one)g(case)g(where)g(an)g(inner)g(reduction)i(occurs.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36319
Black 277 1333 a Gb(Inner)d(Reduction:)p Black 504 1474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36320
a Gg(\(1\))101 b FL(Cut)p F4(\()883 1462 y FX(h)911 1474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36321
y Ga(a)959 1462 y FX(i)986 1474 y Ga(M)10 b(;)1124 1462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36322
y F9(\()1152 1474 y Ga(x)1204 1462 y F9(\))1232 1474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36323
y Ga(N)g F4(\))1395 1437 y Gc(aux)1375 1474 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36324
b(\000)f(!)26 b FL(Cut)o F4(\()1743 1462 y FX(h)1771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36325
1474 y Ga(a)1819 1462 y FX(i)1847 1474 y Ga(M)1945 1441
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36326
y FX(0)1968 1474 y Ga(;)2008 1462 y F9(\()2036 1474 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36327
Ga(x)2088 1462 y F9(\))2115 1474 y Ga(N)10 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36328
Ga(;)710 1586 y FX(h)737 1598 y Ga(a)r F4(:)r Ga(B)888
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36329
1586 y FX(i)916 1598 y Ga(M)35 b F6(2)25 b FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36330
Ga(B)5 b F6(i)p FB(K)22 b Gg(and)1519 1586 y F9(\()1547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36331
1598 y Ga(x)r F4(:)r Ga(B)1702 1586 y F9(\))1729 1598
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36332
y Ga(N)35 b F6(2)25 b FB(J)p F4(\()p Ga(B)5 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36333
FB(K)833 b Gg(by)24 b(assumption)504 1723 y(\(2\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36334
b Ga(M)854 1685 y Gc(aux)833 1723 y F6(\000)-31 b(\000)g(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36335
b Ga(M)1127 1690 y FX(0)3283 1723 y Gg(by)e(\(1\))504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36336
1847 y(\(3\))710 1835 y FX(h)737 1847 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36337
Ga(B)888 1835 y FX(i)916 1847 y Ga(M)1014 1814 y FX(0)1062
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36338
1847 y F6(2)i FB(J)p F6(h)p Ga(B)5 b F6(i)p FB(K)1495
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36339
b Gg(by)23 b(Lemma)g(2.3.12)504 1971 y(\(4\))101 b Ga(M)808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36340
1938 y FX(0)857 1971 y F6(2)25 b Ga(S)5 b(N)1077 1985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36341
y Gc(aux)2861 1971 y Gg(by)23 b(Lemma)g(2.3.13)504 2095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36342
y(\(5\))101 b FL(Cut)p F4(\()883 2083 y FX(h)911 2095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36343
y Ga(a)959 2083 y FX(i)986 2095 y Ga(M)1084 2062 y FX(0)1108
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36344
2095 y Ga(;)1148 2083 y F9(\()1175 2095 y Ga(x)1227 2083
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36345
y F9(\))1255 2095 y Ga(N)10 b F4(\))25 b F6(2)g Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36346
b(N)1618 2109 y Gc(aux)3022 2095 y Gg(by)24 b(induction,)915
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36347
2219 y(the)g(de)o(gree)h(of)e(the)h(cut-formula)i(is)d(equal)i(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36348
(both)i(terms,)e(b)n(ut)h Ga(l)r F4(\()p Ga(M)3124 2186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36349
y FX(0)3148 2219 y F4(\))h Ga(<)g(l)r F4(\()p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36350
b F4(\))277 2461 y Gg(Ne)o(xt,)23 b(we)g(shall)h(gi)n(v)o(e)g(one)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36351
(case)g(where)g(a)f(commuting)i(reduction)h(occurs)f(on)e(the)h(top-le)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36352
n(v)o(el.)p Black 277 2708 a Gb(Commuting)e(Reduction:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36353
Black 504 2852 a Gg(\(1\))101 b FL(Cut)p F4(\()883 2840
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36354
y FX(h)911 2852 y Ga(a)959 2840 y FX(i)986 2852 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36355
b(;)1124 2840 y F9(\()1152 2852 y Ga(x)1204 2840 y F9(\))1232
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36356
2852 y Ga(N)g F4(\))1434 2815 y Gc(c)1465 2791 y FC(0)1375
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36357
2852 y F6(\000)-31 b(\000)f(!)26 b Ga(M)5 b F6(f)-7 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36358
Ga(a)26 b F4(:=)1897 2840 y F9(\()1924 2852 y Ga(x)1976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36359
2840 y F9(\))2004 2852 y Ga(N)p F6(g)d Gg(and)2299 2840
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36360
y FX(h)2327 2852 y Ga(a)r F4(:)r Ga(B)2478 2840 y FX(i)2505
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36361
2852 y Ga(M)36 b F6(2)24 b FB(J)p F6(h)p Ga(B)5 b F6(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36362
FB(K)42 b Gg(by)24 b(assumption)504 3007 y(W)-7 b(e)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36363
b(kno)n(w)h(that)h(only)f(the)h(commuting)g(reduction)h(is)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36364
(applicable,)k(if)28 b Ga(M)39 b Gg(does)29 b(not)h(freshly)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36365
3120 y(introduce)f Ga(a)p Gg(.)34 b(This)25 b(implies)i(that)f(there)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36366
(are)g(only)g(tw)o(o)g(possibilities)j(for)2897 3108
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36367
y FX(h)2924 3120 y Ga(a)r F4(:)r Ga(B)3075 3108 y FX(i)3102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36368
3120 y Ga(M)35 b Gg(to)26 b(be)f(in)504 3233 y FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36369
F6(h)p Ga(B)5 b F6(i)p FB(K)p Gg(:)28 b(it)c(can)f(be)h(in)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36370
FW(A)t(X)t(I)t(O)t(M)t(S)1511 3252 y FX(h)p Gc(B)s FX(i)1652
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36371
3233 y Gg(or)e(in)i FW(B)t(I)t(N)t(D)t(I)t(N)t(G)2180
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36372
3252 y FX(h)p Gc(B)s FX(i)2297 3233 y F4(\()p FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36373
F4(\()p Ga(B)5 b F4(\))q FB(K)o F4(\))q Gg(.)504 3391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36374
y(In)27 b(the)g(\002rst)g(case)g Ga(M)36 b Gg(is)27 b(an)f(axiom)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36375
(that)f(does)h(not)f(introduce)i Ga(a)p Gg(.)38 b(Thus)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36376
b Ga(M)5 b F6(f)-7 b Ga(a)33 b F4(:=)3190 3379 y F9(\()3217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36377
3391 y Ga(x)3269 3379 y F9(\))3297 3391 y Ga(N)p F6(g)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36378
b Gg(is)504 3503 y(equi)n(v)n(alent)f(to)e Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36379
Gg(,)22 b(which)i(we)f(kno)n(w)g(by)h(assumption)i(is)d(strongly)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36380
(normalising.)504 3661 y(The)d(proof)i(for)f(the)g(second)h(case)f(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36381
(as)h(follo)n(ws.)p Black Black 504 3935 a(\(2\))710
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36382
3923 y FX(h)737 3935 y Ga(a)r F4(:)r Ga(B)888 3923 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36383
FX(i)916 3935 y Ga(M)35 b F6(2)27 b FW(B)t(I)t(N)t(D)t(I)t(N)t(G)1462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36384
3953 y FX(h)p Gc(B)s FX(i)1579 3935 y F4(\()p FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36385
F4(\()p Ga(B)5 b F4(\))p FB(K)p F4(\))1049 b Gg(ne)n(w)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36386
b(assumption)504 4059 y(\(3\))101 b Ga(M)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36387
b Ga(a)26 b F4(:=)1036 4047 y F9(\()1064 4059 y Ga(y)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36388
b F4(:)r Ga(B)1215 4047 y F9(\))1242 4059 y Ga(P)s F6(g)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36389
b(2)f Ga(S)5 b(N)1594 4073 y Gc(aux)1739 4059 y Gg(for)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36390
b(all)1981 4047 y F9(\()2009 4059 y Ga(y)5 b F4(:)r Ga(B)2160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36391
4047 y F9(\))2187 4059 y Ga(P)38 b F6(2)25 b FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36392
Ga(B)5 b F4(\))p FB(K)228 b Gg(by)24 b(De\002nition)g(2.3.3)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36393
4183 y(\(4\))710 4171 y F9(\()737 4183 y Ga(x)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36394
Ga(B)892 4171 y F9(\))920 4183 y Ga(N)35 b F6(2)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36395
FB(J)p F4(\()p Ga(B)5 b F4(\))p FB(K)1642 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36396
b(assumption)504 4307 y(\(5\))101 b Ga(M)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36397
b Ga(a)26 b F4(:=)1036 4295 y F9(\()1064 4307 y Ga(x)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36398
F4(:)r Ga(B)1219 4295 y F9(\))1246 4307 y Ga(N)p F6(g)g(2)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36399
Ga(S)5 b(N)1610 4321 y Gc(aux)2470 4307 y Gg(by)23 b(\(3\),)h(\(4\))f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36400
(and)3017 4295 y F9(\()3045 4307 y Ga(y)3093 4295 y F9(\))3120
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36401
4307 y Ga(P)38 b F6(\021)3312 4295 y F9(\()3340 4307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36402
y Ga(x)3392 4295 y F9(\))3419 4307 y Ga(N)277 4573 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36403
Gg(It)d(remains)i(to)e(check)i(for)f(e)n(v)o(ery)g(logical)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36404
(cut-reduction)j(rule)c(that)g(the)g(immediate)g(reducts)h(of)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36405
4686 y FL(Cut)p F4(\()450 4674 y FX(h)478 4686 y Ga(a)526
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36406
4674 y FX(i)553 4686 y Ga(M)10 b(;)691 4674 y F9(\()719
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36407
4686 y Ga(x)771 4674 y F9(\))799 4686 y Ga(N)g F4(\))34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36408
b Gg(are)h(strongly)i(normalising.)65 b(W)-7 b(e)34 b(shall)i(gi)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36409
(e)f(three)g(cases)h(to)f(illustrate)i(the)277 4799 y(proof.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36410
b(The)20 b(dif)n(\002cult)h(case)g(is)g(the)f(logical)j(reduction)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36411
F6(\033)2078 4813 y Gc(R)2136 4799 y Ga(=)p F6(\033)2252
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36412
4813 y Gc(L)2304 4799 y Gg(,)d(because)j(both)e(immediate)h(reducts)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36413
4912 y(ha)n(v)o(e)i(tw)o(o)f(nested)i(cuts.)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36414
5159 a Gb(Logical)g(Reduction)d(with)h(Axioms:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36415
47 w FL(Cut)o F4(\()1745 5147 y FX(h)1773 5159 y Ga(a)1821
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36416
5147 y FX(i)1849 5159 y FL(Ax)o F4(\()p Ga(y)s(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36417
F4(\))q Ga(;)2198 5147 y F9(\()2226 5159 y Ga(x)2278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36418
5147 y F9(\))2306 5159 y Ga(N)9 b F4(\))24 b Gg(reduces)h(to)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36419
b Ga(N)10 b F4([)p Ga(x)g F6(7!)g Ga(y)s F4(])16 b Gg(.)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36420
b(By)23 b(as-)504 5272 y(sumption)h(we)d(kno)n(w)h(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36421
Ga(N)31 b Gg(is)21 b(strongly)j(normalising,)h(and)d(therefore)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36422
Ga(N)10 b F4([)p Ga(x)g F6(7!)g Ga(y)s F4(])22 b Gg(must)f(be)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36423
5385 y(strongly)26 b(normalising.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36424
%%Page: 146 158
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36425
TeXDict begin 146 157 bop Black -144 51 a Gb(146)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36426
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36427
4 v Black Black 321 388 a(Logical)h(Reduction)e Fb(^)1123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36428
402 y Fy(R)1189 388 y FO(=)p Fb(^)1311 402 y Fy(L)1364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36429
410 y Fa(1)1407 388 y Gb(,)f FO(B)34 b Fb(\021)29 b FO(C)6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36430
b Fb(^)p FO(D)s Gb(:)p Black 549 546 a Gg(\(1\))100 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36431
FL(Cut)p F4(\()927 534 y FX(h)955 546 y Ga(c)994 534
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36432
y FX(i)1022 546 y Ga(M)10 b(;)1160 534 y F9(\()1188 546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36433
y Ga(y)1236 534 y F9(\))1263 546 y Ga(N)g F4(\))1481
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36434
509 y Gc(l)1406 546 y F6(\000)-31 b(\000)g(!)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36435
F4(\()1775 534 y FX(h)1803 546 y Ga(a)1851 534 y FX(i)1878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36436
546 y Ga(S)5 b(;)1979 534 y F9(\()2007 546 y Ga(x)2059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36437
534 y F9(\))2086 546 y Ga(U)10 b F4(\))p Gg(,)754 670
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36438
y Ga(M)36 b F6(\021)25 b FL(And)1128 684 y Gc(R)1186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36439
670 y F4(\()1221 658 y FX(h)1249 670 y Ga(a)1297 658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36440
y FX(i)1324 670 y Ga(S)5 b(;)1425 658 y FX(h)1453 670
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36441
y Ga(b)1492 658 y FX(i)1519 670 y Ga(T)13 b(;)i(c)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36442
Gg(,)49 b Ga(N)35 b F6(\021)25 b FL(And)2130 633 y F9(1)2130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36443
693 y Gc(L)2182 670 y F4(\()2217 658 y F9(\()2245 670
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36444
y Ga(x)2297 658 y F9(\))2324 670 y Ga(U)10 b(;)15 b(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36445
F4(\))p Gg(,)754 794 y Ga(M)33 b Gg(and)24 b Ga(N)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36446
b Gg(freshly)25 b(introduce)h Ga(c)d Gg(and)h Ga(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36447
Gg(,)e(respecti)n(v)o(ely)-6 b(,)754 906 y FX(h)782 918
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36448
y Ga(c)r F4(:)r Ga(C)7 b F6(^)o Ga(D)1060 906 y FX(i)1088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36449
918 y Ga(M)35 b F6(2)25 b FB(J)p F6(h)p Ga(C)7 b F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36450
Ga(D)s F6(i)p FB(K)22 b Gg(and)1827 906 y F9(\()1855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36451
918 y Ga(y)5 b F4(:)r Ga(C)i F6(^)o Ga(D)2142 906 y F9(\))2169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36452
918 y Ga(N)36 b F6(2)24 b FB(J)p F4(\()p Ga(C)7 b F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36453
Ga(D)s F4(\))q FB(K)301 b Gg(by)23 b(assumption)549 1066
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36454
y(Since)g Ga(M)33 b Gg(and)24 b Ga(N)33 b Gg(are)24 b(not)f(axioms,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36455
(it)e(follo)n(ws)h(from)f(Lemma)g(2.3.6\(i\))h(that)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36456
1213 y FX(h)576 1225 y Ga(c)r F4(:)r Ga(C)7 b F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36457
Ga(D)855 1213 y FX(i)882 1225 y Ga(M)35 b F6(2)27 b FW(B)t(I)t(N)t(D)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36458
(I)t(N)t(G)1428 1243 y FX(h)p Gc(C)5 b FX(^)r Gc(D)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36459
FX(i)1651 1225 y F4(\()p FB(J)p F4(\()p Ga(C)i F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36460
Ga(D)s F4(\))p FB(K)p F4(\))46 b F6([)h FW(A)t(N)t(D)t(R)t(I)t(G)t(H)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36461
(T)2641 1243 y FX(h)p Gc(C)5 b FX(^)r Gc(D)r FX(i)2863
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36462
1225 y F4(\()p FB(J)p F6(h)p Ga(C)i F6(i)p FB(K)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36463
b FB(J)p F6(h)p Ga(D)s F6(i)p FB(K)p F4(\))549 1348 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36464
F9(\()576 1360 y Ga(y)5 b F4(:)r Ga(C)i F6(^)o Ga(D)863
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36465
1348 y F9(\))891 1360 y Ga(N)35 b F6(2)27 b FW(B)t(I)t(N)t(D)t(I)t(N)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36466
(G)1422 1379 y F9(\()p Gc(C)5 b FX(^)q Gc(D)r F9(\))1644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36467
1360 y F4(\()p FB(J)p F6(h)p Ga(C)i F6(^)o Ga(D)s F6(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36468
FB(K)p F4(\))46 b F6([)h FW(A)t(N)t(D)t(L)t(E)t(F)t(T)2579
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36469
1327 y F9(1)2579 1391 y(\()p Gc(C)5 b FX(^)t Gc(D)r F9(\))2804
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36470
1360 y F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))p FB(K)o F4(\))549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36471
1518 y Gg(No)n(w)27 b(our)h(ar)n(gument)j(splits)e(into)h(tw)o(o)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36472
(cases)h(depending)i(on)e(whether)g(at)g(least)g(one)g(of)f(the)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36473
1619 y FX(h)576 1631 y Ga(c)r F4(:)r Ga(C)7 b F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36474
Ga(D)855 1619 y FX(i)882 1631 y Ga(M)35 b Gg(and)1161
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36475
1619 y F9(\()1189 1631 y Ga(y)5 b F4(:)r Ga(C)i F6(^)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36476
Ga(D)1476 1619 y F9(\))1503 1631 y Ga(N)35 b Gg(belong)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36477
b(to)h FW(B)t(I)t(N)t(D)t(I)t(N)t(G)r Gg(.)35 b(Let)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36478
b(us)h(assume)2919 1619 y FX(h)2946 1631 y Ga(c)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36479
Ga(C)7 b F6(^)p Ga(D)3225 1619 y FX(i)3252 1631 y Ga(M)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36480
b Gg(is)26 b(an)549 1744 y(element)e(in)i FW(B)t(I)t(N)t(D)t(I)t(N)t(G)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36481
1290 1762 y FX(h)p Gc(C)5 b FX(^)r Gc(D)r FX(i)1512 1744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36482
y F4(\()p FB(J)p F4(\()p Ga(C)i F6(^)o Ga(D)s F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36483
FB(K)o F4(\))p Gg(.)549 1914 y(\(2.1\))822 1902 y FX(h)850
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36484
1914 y Ga(c)r F4(:)r Ga(C)g F6(^)p Ga(D)1129 1902 y FX(i)1156
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36485
1914 y Ga(M)35 b F6(2)27 b FW(B)t(I)t(N)t(D)t(I)t(N)t(G)1702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36486
1932 y FX(h)p Gc(C)5 b FX(^)r Gc(D)r FX(i)1924 1914 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36487
F4(\()p FB(J)p F4(\()p Ga(C)i F6(^)p Ga(D)s F4(\))p FB(K)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36488
F4(\))611 b Gg(ne)n(w)23 b(assumption)549 2038 y(\(2.2\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36489
b Ga(M)5 b F6(f)-7 b Ga(c)27 b F4(:=)1140 2026 y F9(\()1167
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36490
2038 y Ga(z)1213 2026 y F9(\))1241 2038 y Ga(P)t F6(g)e(2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36491
Ga(S)5 b(N)1593 2052 y Gc(aux)1738 2038 y Gg(for)24 b(all)1981
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36492
2026 y F9(\()2008 2038 y Ga(z)6 b F4(:)r Ga(C)h F6(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36493
Ga(D)2294 2026 y F9(\))2321 2038 y Ga(P)38 b F6(2)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36494
b FB(J)p F4(\()p Ga(C)7 b F6(^)o Ga(D)s F4(\))p FB(K)212
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36495
b Gg(by)24 b(Def.)f(2.3.3)549 2162 y(\(2.3\))100 b Ga(M)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36496
b F6(f)-7 b Ga(c)27 b F4(:=)1140 2150 y F9(\()1167 2162
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36497
y Ga(y)1215 2150 y F9(\))1243 2162 y Ga(N)p F6(g)f(2)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36498
Ga(S)5 b(N)1607 2176 y Gc(aux)2451 2162 y Gg(by)24 b(\(1\),)f(\(2.2\))h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36499
(and)3067 2150 y F9(\()3095 2162 y Ga(z)3141 2150 y F9(\))3168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36500
2162 y Ga(P)39 b F6(\021)3361 2150 y F9(\()3388 2162
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36501
y Ga(y)3436 2150 y F9(\))3463 2162 y Ga(N)549 2286 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36502
Gg(\(2.4\))100 b(The)23 b(follo)n(wing)i(calculation)i(sho)n(ws)d(that)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36503
g FL(Cut)p F4(\()2362 2274 y FX(h)2390 2286 y Ga(c)2429
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36504
2274 y FX(i)2456 2286 y Ga(M)10 b(;)2594 2274 y F9(\()2622
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36505
2286 y Ga(y)2670 2274 y F9(\))2698 2286 y Ga(N)g F4(\))25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36506
b F6(2)g Ga(S)5 b(N)3061 2300 y Gc(aux)3183 2286 y Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36507
Black Black 643 2458 a Ga(M)g F6(f)-7 b Ga(c)27 b F4(:=)961
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36508
2446 y F9(\()988 2458 y Ga(y)1036 2446 y F9(\))1064 2458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36509
y Ga(N)p F6(g)k(\021)f FL(And)1468 2472 y Gc(R)1526 2458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36510
y F4(\()1561 2446 y FX(h)1589 2458 y Ga(a)1637 2446 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36511
FX(i)1664 2458 y Ga(S)5 b(;)1765 2446 y FX(h)1793 2458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36512
y Ga(b)1832 2446 y FX(i)1860 2458 y Ga(T)12 b(;)j(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36513
F4(\))-5 b F6(f)e Ga(c)28 b F4(:=)2260 2446 y F9(\()2287
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36514
2458 y Ga(y)2335 2446 y F9(\))2363 2458 y Ga(N)p F6(g)1213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36515
2594 y F4(=)i FL(Cut)p F4(\()1487 2582 y FX(h)1514 2594
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36516
y Ga(c)1553 2582 y FX(i)1581 2594 y FL(And)1736 2608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36517
y Gc(R)1793 2594 y F4(\()1828 2582 y FX(h)1856 2594 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36518
Ga(a)1904 2582 y FX(i)1932 2594 y Ga(S)q F6(f)-7 b Ga(c)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36519
b F4(:=)2212 2582 y F9(\()2240 2594 y Ga(y)2288 2582
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36520
y F9(\))2315 2594 y Ga(N)p F6(g)q Ga(;)2474 2582 y FX(h)2502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36521
2594 y Ga(b)2541 2582 y FX(i)2568 2594 y Ga(T)8 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36522
b Ga(c)26 b F4(:=)2853 2582 y F9(\()2881 2594 y Ga(y)2929
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36523
2582 y F9(\))2956 2594 y Ga(N)q F6(g)p Ga(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36524
F4(\))q Ga(;)3230 2582 y F9(\()3258 2594 y Ga(y)3306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36525
2582 y F9(\))3333 2594 y Ga(N)10 b F4(\))1206 2675 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36526
FV(\()p FC(\003)p FV(\))1213 2758 y F6(\021)30 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36527
F4(\()1487 2746 y FX(h)1514 2758 y Ga(c)1553 2746 y FX(i)1581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36528
2758 y FL(And)1736 2772 y Gc(R)1793 2758 y F4(\()1828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36529
2746 y FX(h)1856 2758 y Ga(a)1904 2746 y FX(i)1932 2758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36530
y Ga(S)5 b(;)2033 2746 y FX(h)2061 2758 y Ga(b)2100 2746
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36531
y FX(i)2127 2758 y Ga(T)13 b(;)i(c)p F4(\))q Ga(;)2348
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36532
2746 y F9(\()2376 2758 y Ga(y)2424 2746 y F9(\))2451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36533
2758 y Ga(N)10 b F4(\))1213 2895 y F6(\021)30 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36534
F4(\()1487 2883 y FX(h)1514 2895 y Ga(c)1553 2883 y FX(i)1581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36535
2895 y Ga(M)10 b(;)1719 2883 y F9(\()1747 2895 y Ga(y)1795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36536
2883 y F9(\))1822 2895 y Ga(N)g F4(\))1106 3002 y F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36537
FX(\003)p F9(\))1314 3035 y Gg(because)25 b Ga(M)33 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36538
Gg(freshly)25 b(introduces)i Ga(c)p Gg(.)549 3192 y(If)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36539
FL(Cut)p F4(\()810 3180 y FX(h)838 3192 y Ga(c)877 3180
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36540
y FX(i)905 3192 y Ga(M)10 b(;)1043 3180 y F9(\()1071
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36541
3192 y Ga(y)1119 3180 y F9(\))1146 3192 y Ga(N)g F4(\))29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36542
b Gg(is)g(strongly)i(normalising,)i(then)d(its)f(reduct)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36543
FL(Cut)p F4(\()2922 3180 y FX(h)2950 3192 y Ga(a)2998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36544
3180 y FX(i)3026 3192 y Ga(S)5 b(;)3127 3180 y F9(\()3154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36545
3192 y Ga(x)3206 3180 y F9(\))3234 3192 y Ga(U)10 b F4(\))29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36546
b Gg(must)549 3305 y(be)d(strongly)j(normalising)h(too.)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36547
b(In)27 b(case)1906 3293 y F9(\()1933 3305 y Ga(y)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36548
F4(:)r Ga(C)i F6(^)o Ga(D)2220 3293 y F9(\))2247 3305
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36549
y Ga(N)36 b Gg(is)27 b(in)i FW(B)t(I)t(N)t(D)t(I)t(N)t(G)2877
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36550
3324 y F9(\()p Gc(C)5 b FX(^)r Gc(D)r F9(\))3099 3305
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36551
y F4(\()p FB(J)p F6(h)p Ga(C)i F6(^)o Ga(D)s F6(i)q FB(K)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36552
F4(\))q Gg(,)549 3418 y(we)22 b(reason)j(analogous.)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36553
3571 y(If)c(neither)902 3559 y FX(h)930 3571 y Ga(c)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36554
F4(:)r Ga(C)7 b F6(^)p Ga(D)1209 3559 y FX(i)1236 3571
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36555
y Ga(M)30 b Gg(nor)1496 3559 y F9(\()1523 3571 y Ga(y)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36556
b F4(:)r Ga(C)i F6(^)p Ga(D)1811 3559 y F9(\))1838 3571
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36557
y Ga(N)30 b Gg(are)21 b(in)j FW(B)t(I)t(N)t(D)t(I)t(N)t(G)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36558
Gg(,)e(then)g(we)e(proceed)j(as)e(follo)n(ws.)549 3725
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36559
y(\(3.1\))822 3713 y FX(h)850 3725 y Ga(c)r F4(:)r Ga(C)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36560
b F6(^)p Ga(D)1129 3713 y FX(i)1156 3725 y Ga(M)35 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36561
F6(2)27 b FW(A)t(N)t(D)t(R)t(I)t(G)t(H)t(T)1779 3744
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36562
y FX(h)p Gc(C)5 b FX(^)r Gc(D)r FX(i)2002 3725 y F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36563
FB(J)p F6(h)p Ga(C)i F6(i)p FB(K)o Ga(;)15 b FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36564
Ga(D)s F6(i)q FB(K)o F4(\))q Gg(,)822 3838 y F9(\()850
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36565
3850 y Ga(y)5 b F4(:)r Ga(C)i F6(^)o Ga(D)1137 3838 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36566
F9(\))1164 3850 y Ga(N)36 b F6(2)26 b FW(A)t(N)t(D)t(L)t(E)t(F)t(T)1717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36567
3817 y F9(1)1717 3881 y(\()p Gc(C)5 b FX(^)t Gc(D)r F9(\))1942
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36568
3850 y F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))p FB(K)p F4(\))732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36569
b Gg(ne)n(w)23 b(assumption)549 3984 y(\(3.2\))822 3972
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36570
y FX(h)850 3984 y Ga(a)17 b F4(:)h Ga(C)1030 3972 y FX(i)1056
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36571
3984 y Ga(S)31 b F6(2)25 b FB(J)p F6(h)p Ga(C)7 b F6(i)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36572
FB(K)23 b Gg(and)1621 3972 y F9(\()1648 3984 y Ga(x)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36573
b F4(:)h Ga(C)1832 3972 y F9(\))1859 3984 y Ga(U)35 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36574
F6(2)25 b FB(J)p F4(\()p Ga(C)7 b F4(\))p FB(K)601 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36575
Gg(by)24 b(De\002nition)h(2.3.3)549 4109 y(\(3.3\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36576
b Ga(S;)15 b(U)36 b F6(2)25 b Ga(S)5 b(N)1236 4123 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36577
Gc(aux)2905 4109 y Gg(by)24 b(Lemma)e(2.3.13)549 4233
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36578
y(\(3.4\))100 b FL(Cut)p F4(\()995 4221 y FX(h)1023 4233
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36579
y Ga(a)r F4(:)r Ga(C)1172 4221 y FX(i)1199 4233 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36580
b(;)1300 4221 y F9(\()1328 4233 y Ga(x)r F4(:)r Ga(C)1481
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36581
4221 y F9(\))1508 4233 y Ga(U)10 b F4(\))26 b F6(2)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36582
Ga(S)5 b(N)1861 4247 y Gc(aux)2250 4233 y Gg(by)24 b(induction)i(\(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36583
e(de)o(gree)h(decreased\))p Black 321 4426 a Gb(Logical)g(Reduction)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36584
Fb(\033)1135 4440 y Fy(R)1200 4426 y FO(=)p Fb(\033)1334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36585
4440 y Fy(L)1391 4426 y Gb(,)g FO(B)33 b Fb(\021)d FO(C)6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36586
b Fb(\033)p FO(D)s Gb(:)p Black 549 4583 a Gg(\(1\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36587
b Ga(M)36 b F6(\021)25 b FL(Imp)1118 4605 y Gc(R)1176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36588
4583 y F4(\()1211 4571 y F9(\()1238 4583 y Ga(x)1290
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36589
4571 y F9(\))q FX(h)1345 4583 y Ga(a)1393 4571 y FX(i)1421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36590
4583 y Ga(S)5 b(;)15 b(b)p F4(\))p Gg(,)23 b Ga(N)60
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36591
b F6(\021)25 b FL(Imp)2016 4605 y Gc(L)2068 4583 y F4(\()2103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36592
4571 y FX(h)2131 4583 y Ga(c)2170 4571 y FX(i)2198 4583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36593
y Ga(T)12 b(;)2303 4571 y F9(\()2331 4583 y Ga(y)2379
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36594
4571 y F9(\))2407 4583 y Ga(U)d(;)15 b(z)t F4(\))p Gg(,)754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36595
4708 y Ga(M)33 b Gg(and)24 b Ga(N)33 b Gg(freshly)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36596
b(introduce)h Ga(b)d Gg(and)h Ga(z)t Gg(,)e(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36597
b(,)754 4820 y FX(h)782 4832 y Ga(c)r F4(:)r Ga(C)7 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36598
F6(\033)p Ga(D)1071 4820 y FX(i)1098 4832 y Ga(M)35 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36599
F6(2)25 b FB(J)p F6(h)p Ga(C)7 b F6(\033)o Ga(D)s F6(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36600
FB(K)23 b Gg(and)1848 4820 y F9(\()1875 4832 y Ga(y)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36601
b F4(:)r Ga(C)i F6(\033)o Ga(D)2172 4820 y F9(\))2200
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36602
4832 y Ga(N)35 b F6(2)25 b FB(J)p F4(\()p Ga(C)7 b F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36603
Ga(D)s F4(\))p FB(K)261 b Gg(by)23 b(assumption)549 4985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36604
y(The)g(term)g FL(Cut)p F4(\()1075 4973 y FX(h)1103 4985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36605
y Ga(b)1142 4973 y FX(i)1169 4985 y Ga(M)10 b(;)1307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36606
4973 y F9(\()1335 4985 y Ga(z)1381 4973 y F9(\))1409
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36607
4985 y Ga(N)g F4(\))23 b Gg(reduces)i(to)f(either:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36608
Black Black 757 5216 a FL(Cut)p F4(\()930 5204 y FX(h)958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36609
5216 y Ga(a)1006 5204 y FX(i)1059 5216 y FL(Cut)p F4(\()1232
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36610
5204 y FX(h)1259 5216 y Ga(c)1298 5204 y FX(i)1326 5216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36611
y Ga(T)13 b(;)1432 5204 y F9(\()1460 5216 y Ga(x)1512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36612
5204 y F9(\))1539 5216 y Ga(S)5 b F4(\))q Ga(;)1676 5204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36613
y F9(\()1703 5216 y Ga(y)1751 5204 y F9(\))1779 5216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36614
y Ga(U)10 b F4(\))100 b Gg(or)g FL(Cut)p F4(\()2334 5204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36615
y FX(h)2362 5216 y Ga(c)2401 5204 y FX(i)2428 5216 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36616
Ga(T)13 b(;)2534 5204 y F9(\()2562 5216 y Ga(x)2614 5204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36617
y F9(\))2667 5216 y FL(Cut)p F4(\()2840 5204 y FX(h)2867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36618
5216 y Ga(a)2915 5204 y FX(i)2943 5216 y Ga(S)5 b(;)3044
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36619
5204 y F9(\()3072 5216 y Ga(y)3120 5204 y F9(\))3147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36620
5216 y Ga(U)10 b F4(\)\))26 b Gg(.)p Black Black eop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36621
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36622
%%Page: 147 159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36623
TeXDict begin 147 158 bop Black 277 51 a Gb(B.1)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36624
(Chapter)f(2)2639 b(147)p 277 88 3691 4 v Black 504 388
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36625
a Gg(W)-7 b(e)34 b(ha)n(v)o(e)i(to)e(sho)n(w)h(that)g(both)h(reducts)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36626
(are)f(strongly)i(normalising.)65 b(W)-7 b(e)34 b(shall)h(gi)n(v)o(e)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36627
(the)504 501 y(details)25 b(for)f(the)g(\002rst)f(case.)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36628
659 y(Since)h Ga(M)33 b Gg(and)24 b Ga(N)33 b Gg(are)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36629
b(not)h(axioms,)g(it)g(follo)n(ws)g(from)f(Lemma)g(2.3.6\(i\))h(that)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36630
504 812 y FX(h)532 824 y Ga(b)r F4(:)r Ga(C)7 b F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36631
Ga(D)820 812 y FX(i)848 824 y Ga(M)35 b F6(2)27 b FW(B)t(I)t(N)t(D)t(I)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36632
t(N)t(G)1394 842 y FX(h)p Gc(C)5 b FX(\033)r Gc(D)r FX(i)1624
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36633
824 y F4(\()p FB(J)p F4(\()p Ga(C)i F6(\033)p Ga(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36634
F4(\))p FB(K)o F4(\))46 b F6([)h FW(I)t(M)t(P)t(R)t(I)t(G)t(H)t(T)2594
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36635
842 y FX(h)p Gc(C)5 b FX(\033)t Gc(D)r FX(i)2826 824
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36636
y F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))q FB(K)o Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36637
b FB(J)p F6(h)p Ga(D)s F6(i)q FB(K)o F4(\))504 947 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36638
F9(\()532 959 y Ga(z)6 b F4(:)r Ga(C)h F6(\033)p Ga(D)828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36639
947 y F9(\))855 959 y Ga(N)35 b F6(2)27 b FW(B)t(I)t(N)t(D)t(I)t(N)t(G)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36640
1386 978 y F9(\()p Gc(C)5 b FX(\033)r Gc(D)r F9(\))1616
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36641
959 y F4(\()p FB(J)p F6(h)p Ga(C)i F6(\033)p Ga(D)s F6(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36642
FB(K)o F4(\))46 b F6([)h FW(I)t(M)t(P)t(L)t(E)t(F)t(T)2531
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36643
978 y F9(\()p Gc(C)5 b FX(\033)h Gc(D)r F9(\))2766 959
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36644
y F4(\()p FB(J)p F6(h)p Ga(C)h F6(i)p FB(K)o Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36645
b FB(J)p F4(\()p Ga(D)s F4(\))q FB(K)o F4(\))504 1115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36646
y Gg(Again)35 b(the)g(proof)h(splits)g(into)g(tw)o(o)e(cases)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36647
(depending)i(on)d(whether)h(the)f(co-named)i(term)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36648
1216 y FX(h)532 1228 y Ga(b)17 b F4(:)g Ga(C)7 b F6(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36649
Ga(D)851 1216 y FX(i)878 1228 y Ga(M)38 b Gg(or)28 b(the)g(named)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36650
(term)1709 1216 y F9(\()1737 1228 y Ga(z)21 b F4(:)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36651
Ga(C)7 b F6(\033)o Ga(D)2063 1216 y F9(\))2090 1228 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36652
Ga(N)38 b Gg(belong)30 b(to)g FW(B)t(I)t(N)t(D)t(I)t(N)t(G)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36653
Gg(.)44 b(Let)27 b(us)i(assume)504 1329 y FX(h)532 1341
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36654
y Ga(b)r F4(:)r Ga(C)7 b F6(\033)o Ga(D)820 1329 y FX(i)848
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36655
1341 y Ga(M)32 b Gg(is)24 b(an)f(element)i(in)g FW(B)t(I)t(N)t(D)t(I)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36656
(N)t(G)1901 1360 y FX(h)p Gc(C)5 b FX(\033)r Gc(D)r FX(i)2132
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36657
1341 y F4(\()p FB(J)p F4(\()p Ga(C)i F6(\033)o Ga(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36658
F4(\))p FB(K)p F4(\))p Gg(.)504 1517 y(\(2.1\))778 1505
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36659
y FX(h)806 1517 y Ga(b)r F4(:)r Ga(C)g F6(\033)o Ga(D)1094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36660
1505 y FX(i)1121 1517 y Ga(M)36 b F6(2)27 b FW(B)t(I)t(N)t(D)t(I)t(N)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36661
(G)1668 1535 y FX(h)p Gc(C)5 b FX(\033)q Gc(D)r FX(i)1898
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36662
1517 y F4(\()p FB(J)p F4(\()p Ga(C)i F6(\033)o Ga(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36663
F4(\))q FB(K)o F4(\))584 b Gg(ne)n(w)23 b(assumption)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36664
1641 y(\(2.2\))101 b Ga(M)5 b F6(f)-7 b Ga(b)26 b F4(:=)1095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36665
1629 y F9(\()1123 1641 y Ga(v)1170 1629 y F9(\))1198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36666
1641 y Ga(P)s F6(g)g(2)f Ga(S)5 b(N)1550 1655 y Gc(aux)1695
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36667
1641 y Gg(for)23 b(all)1937 1629 y F9(\()1965 1641 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36668
Ga(v)5 b F4(:)r Ga(C)i F6(\033)o Ga(D)2261 1629 y F9(\))2289
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36669
1641 y Ga(P)38 b F6(2)25 b FB(J)p F4(\()p Ga(C)7 b F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36670
Ga(D)s F4(\))p FB(K)190 b Gg(by)24 b(Def.)e(2.3.3)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36671
1765 y(\(2.3\))101 b Ga(M)5 b F6(f)-7 b Ga(b)26 b F4(:=)1095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36672
1753 y F9(\()1123 1765 y Ga(z)1169 1753 y F9(\))1197
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36673
1765 y Ga(N)p F6(g)g(2)f Ga(S)5 b(N)1561 1779 y Gc(aux)2408
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36674
1765 y Gg(by)23 b(\(1\),)h(\(2.2\))g(and)3024 1753 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36675
F9(\()3051 1765 y Ga(v)3098 1753 y F9(\))3126 1765 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36676
Ga(P)38 b F6(\021)3318 1753 y F9(\()3345 1765 y Ga(z)3391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36677
1753 y F9(\))3419 1765 y Ga(N)504 1889 y Gg(\(2.4\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36678
b(The)23 b(follo)n(wing)i(calculation)i(sho)n(ws)d(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36679
FL(Cut)p F4(\()2318 1877 y FX(h)2345 1889 y Ga(c)2384
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36680
1877 y FX(i)2412 1889 y Ga(M)10 b(;)2550 1877 y F9(\()2578
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36681
1889 y Ga(z)2624 1877 y F9(\))2652 1889 y Ga(N)g F4(\))25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36682
b F6(2)g Ga(S)5 b(N)3015 1903 y Gc(aux)3138 1889 y Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36683
Black Black 874 2104 a Ga(M)g F6(f)-7 b Ga(b)26 b F4(:=)1192
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36684
2092 y F9(\()1219 2104 y Ga(z)1265 2092 y F9(\))1293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36685
2104 y Ga(N)p F6(g)31 b(\021)f FL(Imp)1687 2125 y Gc(R)1745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36686
2104 y F4(\()1780 2092 y F9(\()1808 2104 y Ga(x)1860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36687
2092 y F9(\))p FX(h)1915 2104 y Ga(a)1963 2092 y FX(i)1990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36688
2104 y Ga(S)5 b(;)15 b(b)p F4(\))-5 b F6(f)e Ga(b)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36689
b F4(:=)2385 2092 y F9(\()2413 2104 y Ga(z)2459 2092
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36690
y F9(\))2486 2104 y Ga(N)q F6(g)1442 2240 y F4(=)j FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36691
F4(\()1716 2228 y FX(h)1744 2240 y Ga(b)1783 2228 y FX(i)1810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36692
2240 y FL(Imp)1954 2262 y Gc(R)2012 2240 y F4(\()2047
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36693
2228 y F9(\()2075 2240 y Ga(x)2127 2228 y F9(\))p FX(h)2182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36694
2240 y Ga(a)2230 2228 y FX(i)2257 2240 y Ga(S)q F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36695
b Ga(b)25 b F4(:=)2537 2228 y F9(\()2565 2240 y Ga(z)2611
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36696
2228 y F9(\))2639 2240 y Ga(N)p F6(g)q Ga(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36697
F4(\))p Ga(;)2912 2228 y F9(\()2940 2240 y Ga(z)2986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36698
2228 y F9(\))3014 2240 y Ga(N)10 b F4(\))1435 2320 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36699
FV(\()p FC(\003)p FV(\))1442 2404 y F6(\021)30 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36700
F4(\()1716 2392 y FX(h)1744 2404 y Ga(b)1783 2392 y FX(i)1810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36701
2404 y FL(Imp)1954 2426 y Gc(R)2012 2404 y F4(\()2047
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36702
2392 y F9(\()2075 2404 y Ga(x)2127 2392 y F9(\))p FX(h)2182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36703
2404 y Ga(a)2230 2392 y FX(i)2257 2404 y Ga(S)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36704
b(b)p F4(\))q Ga(;)2473 2392 y F9(\()2501 2404 y Ga(z)2547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36705
2392 y F9(\))2575 2404 y Ga(N)10 b F4(\))1442 2541 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36706
F6(\021)30 b FL(Cut)p F4(\()1716 2529 y FX(h)1744 2541
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36707
y Ga(b)1783 2529 y FX(i)1810 2541 y Ga(M)10 b(;)1948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36708
2529 y F9(\()1976 2541 y Ga(z)2022 2529 y F9(\))2050
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36709
2541 y Ga(N)g F4(\))1336 2672 y F9(\()p FX(\003)p F9(\))1543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36710
2705 y Gg(because)25 b Ga(M)33 b Gg(freshly)25 b(introduces)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36711
Ga(b)p Gg(.)504 2905 y(Therefore)37 b(we)e(kno)n(w)g(that)h(the)f(term)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36712
g FL(Cut)p F4(\()1962 2893 y FX(h)1990 2905 y Ga(b)2029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36713
2893 y FX(i)2056 2905 y Ga(M)11 b(;)2195 2893 y F9(\()2222
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36714
2905 y Ga(z)2268 2893 y F9(\))2296 2905 y Ga(N)f F4(\))35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36715
b Gg(is)g(strongly)i(normalising,)k(and)504 3018 y(hence)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36716
b(its)e(reduct)i FL(Cut)p F4(\()1273 3006 y FX(h)1300
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36717
3018 y Ga(a)1348 3006 y FX(i)1401 3018 y FL(Cut)p F4(\()1574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36718
3006 y FX(h)1602 3018 y Ga(c)1641 3006 y FX(i)1669 3018
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36719
y Ga(T)12 b(;)1774 3006 y F9(\()1802 3018 y Ga(x)1854
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36720
3006 y F9(\))1882 3018 y Ga(S)5 b F4(\))p Ga(;)2018 3006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36721
y F9(\()2046 3018 y Ga(y)2094 3006 y F9(\))2121 3018
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36722
y Ga(U)10 b F4(\))24 b Gg(must)g(be)g(strongly)j(normalising,)f(too.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36723
504 3130 y(In)34 b(f)o(act)h(both)g(reducts)g(must)f(be)g(strongly)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36724
(normalising.)62 b(The)34 b(case)g(where)3069 3118 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36725
F9(\()3096 3130 y Ga(z)6 b F4(:)r Ga(C)h F6(\033)p Ga(D)3392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36726
3118 y F9(\))3419 3130 y Ga(N)504 3243 y Gg(belongs)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36727
b(to)f FW(B)t(I)t(N)t(D)t(I)t(N)t(G)1240 3262 y F9(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36728
Gc(C)5 b FX(\033)r Gc(D)r F9(\))1471 3243 y F4(\()p FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36729
F6(h)p Ga(C)i F6(\033)o Ga(D)s F6(i)p FB(K)p F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36730
b Gg(is)g(similar)-5 b(.)504 3401 y(W)e(e)34 b(no)n(w)f(ha)n(v)o(e)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36731
(to)f(sho)n(w)g(that)h(the)g(reduct)g(is)f(strongly)j(normalising)g(in)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36732
d(the)g(case)h(where)504 3502 y FX(h)532 3514 y Ga(b)17
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36733
b F4(:)g Ga(C)7 b F6(\033)p Ga(D)851 3502 y FX(i)878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36734
3514 y Ga(M)45 b Gg(and)1178 3502 y F9(\()1205 3514 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36735
Ga(z)22 b F4(:)17 b Ga(C)7 b F6(\033)o Ga(D)1531 3502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36736
y F9(\))1558 3514 y Ga(N)46 b Gg(belong)37 b(to)h FW(I)t(M)t(P)t(R)t(I)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36737
t(G)t(H)t(T)2449 3533 y FX(h)p Gc(C)5 b FX(\033)t Gc(D)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36738
FX(i)2681 3514 y F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36739
FB(K)o Ga(;)15 b FB(J)p F6(h)p Ga(D)s F6(i)q FB(K)o F4(\))36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36740
b Gg(and)g(to)506 3627 y FW(I)t(M)t(P)t(L)t(E)t(F)t(T)833
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36741
3646 y F9(\()p Gc(C)6 b FX(\033)f Gc(D)r F9(\))1068 3627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36742
y F4(\()p FB(J)p F6(h)p Ga(C)i F6(i)p FB(K)o Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36743
b FB(J)p F4(\()p Ga(D)s F4(\))q FB(K)p F4(\))p Gg(,)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36744
b(respecti)n(v)o(ely)-6 b(.)504 3785 y(W)f(e)23 b(\002rst)g(sho)n(w)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36745
(that)g(the)f(inner)i(cut)f(of)f(the)h(reduct)h(is)e(strongly)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36746
(normalising.)504 3945 y(\(3.1\))778 3933 y FX(h)806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36747
3945 y Ga(b)17 b F4(:)g Ga(C)7 b F6(\033)o Ga(D)1124
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36748
3933 y FX(i)1152 3945 y Ga(M)35 b F6(2)27 b FW(I)t(M)t(P)t(R)t(I)t(G)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36749
(H)t(T)1745 3964 y FX(h)p Gc(C)5 b FX(\033)t Gc(D)r FX(i)1977
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36750
3945 y F4(\()p FB(J)p F4(\()p Ga(C)i F4(\))p FB(K)p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36751
b FB(J)p F6(h)p Ga(D)s F6(i)p FB(K)p F4(\))p Gg(,)778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36752
4058 y F9(\()806 4070 y Ga(z)21 b F4(:)d Ga(C)7 b F6(\033)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36753
Ga(D)1132 4058 y F9(\))1159 4070 y Ga(N)35 b F6(2)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36754
b FW(I)t(M)t(P)t(L)t(E)t(F)t(T)1682 4088 y F9(\()p Gc(C)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36755
b FX(\033)h Gc(D)r F9(\))1916 4070 y F4(\()p FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36756
Ga(C)h F6(i)p FB(K)p Ga(;)15 b FB(J)p F4(\()p Ga(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36757
F4(\))q FB(K)o F4(\))452 b Gg(ne)n(w)23 b(assumption)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36758
4194 y(\(3.2\))778 4182 y F9(\()806 4194 y Ga(x)17 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36759
F4(:)g Ga(C)989 4182 y F9(\))1016 4194 y Ga(S)q F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36760
b Ga(a)25 b F4(:=)1305 4182 y F9(\()1333 4194 y Ga(v)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36761
b F4(:)r Ga(D)1487 4182 y F9(\))1515 4194 y Ga(P)s F6(g)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36762
b(2)f FB(J)p F4(\()p Ga(C)7 b F4(\))p FB(K)22 b Gg(for)i(all)2213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36763
4182 y F9(\()2241 4194 y Ga(v)5 b F4(:)r Ga(D)2395 4182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36764
y F9(\))2423 4194 y Ga(P)38 b F6(2)25 b FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36765
Ga(D)s F4(\))p FB(K)o Gg(,)778 4306 y FX(h)806 4318 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36766
Ga(c)r F4(:)r Ga(C)946 4306 y FX(i)973 4318 y Ga(T)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36767
b F6(2)25 b FB(J)p F6(h)p Ga(C)7 b F6(i)p FB(K)1449 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36768
Gg(by)24 b(De\002nition)g(2.3.3)504 4442 y(\(3.3\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36769
b Ga(S)q F6(f)-7 b Ga(a)25 b F4(:=)1067 4430 y F9(\()1095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36770
4442 y Ga(v)5 b F4(:)r Ga(D)1249 4430 y F9(\))1277 4442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36771
y Ga(P)s F6(g)p Ga(;)15 b(T)39 b F6(2)25 b Ga(S)5 b(N)1735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36772
4456 y Gc(aux)2861 4442 y Gg(by)23 b(Lemma)g(2.3.13)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36773
4567 y(\(3.4\))101 b FL(Cut)p F4(\()951 4555 y FX(h)979
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36774
4567 y Ga(c)1018 4555 y FX(i)1046 4567 y Ga(T)12 b(;)1151
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36775
4555 y F9(\()1179 4567 y Ga(x)1231 4555 y F9(\))1259
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36776
4567 y Ga(S)q F6(f)-7 b Ga(a)25 b F4(:=)1548 4555 y F9(\()1575
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36777
4567 y Ga(v)5 b F4(:)r Ga(D)1729 4555 y F9(\))1757 4567
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36778
y Ga(P)t F6(g)p F4(\))26 b F6(2)f Ga(S)5 b(N)2145 4581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36779
y Gc(aux)2411 4567 y Gg(by)23 b(ind.)h(\(the)g(de)o(gree)h(decreased\))
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36780
504 4719 y(As)j(stated)h(in)f(Remark)g(2.2.7,)h(we)e(require)i(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36781
Ga(a)e Gg(is)h(not)g(free)h(in)2671 4707 y FX(h)2699
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36782
4719 y Ga(c)2738 4707 y FX(i)2766 4719 y Ga(T)12 b Gg(,)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36783
b(and)h(therefore)h(we)504 4832 y(may)24 b(mo)o(v)o(e)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36784
(substitution)28 b(on)23 b(the)h(top-le)n(v)o(el.)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36785
b(Thus)24 b(we)f(ha)n(v)o(e)h(that)796 5048 y FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36786
F4(\()969 5036 y FX(h)997 5048 y Ga(c)1036 5036 y FX(i)1064
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36787
5048 y Ga(T)12 b(;)1169 5036 y F9(\()1197 5048 y Ga(x)1249
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36788
5036 y F9(\))1277 5048 y Ga(S)q F6(f)-7 b Ga(a)25 b F4(:=)1566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36789
5036 y F9(\()1593 5048 y Ga(v)5 b F4(:)r Ga(D)1747 5036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36790
y F9(\))1775 5048 y Ga(P)t F6(g)p F4(\))26 b F6(\021)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36791
FL(Cut)p F4(\()2212 5036 y FX(h)2239 5048 y Ga(c)2278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36792
5036 y FX(i)2306 5048 y Ga(T)13 b(;)2412 5036 y F9(\()2440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36793
5048 y Ga(x)2492 5036 y F9(\))2519 5048 y Ga(S)5 b F4(\))-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36794
b F6(f)e Ga(a)26 b F4(:=)2844 5036 y F9(\()2871 5048
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36795
y Ga(v)5 b F4(:)r Ga(D)3025 5036 y F9(\))3053 5048 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36796
Ga(P)s F6(g)26 b Ga(:)522 5269 y Gg(\(3.5\))100 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36797
F4(\()968 5257 y FX(h)996 5269 y Ga(c)1035 5257 y FX(i)1063
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36798
5269 y Ga(T)13 b(;)1169 5257 y F9(\()1197 5269 y Ga(x)1249
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36799
5257 y F9(\))1276 5269 y Ga(S)5 b F4(\))-5 b F6(f)e Ga(a)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36800
b F4(:=)1600 5257 y F9(\()1628 5269 y Ga(v)5 b F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36801
Ga(D)1782 5257 y F9(\))1810 5269 y Ga(P)s F6(g)26 b(2)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36802
Ga(S)5 b(N)2162 5283 y Gc(aux)2307 5269 y Gg(for)24 b(all)2549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36803
5257 y F9(\()2577 5269 y Ga(v)5 b F4(:)r Ga(D)2731 5257
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36804
y F9(\))2759 5269 y Ga(P)38 b F6(2)25 b FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36805
Ga(D)s F4(\))p FB(K)69 b Gg(by)23 b(\(3.4\))522 5394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36806
y(\(3.6\))795 5382 y FX(h)823 5394 y Ga(a)r F4(:)r Ga(D)978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36807
5382 y FX(i)1006 5394 y FL(Cut)o F4(\()1178 5382 y FX(h)1206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36808
5394 y Ga(c)1245 5382 y FX(i)1273 5394 y Ga(T)13 b(;)1379
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36809
5382 y F9(\()1407 5394 y Ga(x)1459 5382 y F9(\))1486
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36810
5394 y Ga(S)5 b F4(\))26 b F6(2)e FB(J)p F6(h)p Ga(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36811
F6(i)q FB(K)917 b Gg(by)23 b(De\002nition)i(2.3.3)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36812
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36813
%%Page: 148 160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36814
TeXDict begin 148 159 bop Black -144 51 a Gb(148)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36815
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36816
4 v Black 549 388 a Gg(No)n(w)e(we)g(sho)n(w)i(that)g(the)g(outer)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36817
(cut)g(is)f(strongly)j(normalising.)549 592 y(\(3.7\))822
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36818
580 y F9(\()850 592 y Ga(y)5 b F4(:)r Ga(D)1005 580 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36819
F9(\))1032 592 y Ga(U)35 b F6(2)25 b FB(J)p F4(\()p Ga(D)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36820
F4(\))p FB(K)1072 b Gg(by)23 b(\(3.1\))h(and)g(De\002nition)h(2.3.3)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36821
716 y(\(3.8\))100 b FL(Cut)p F4(\()995 704 y FX(h)1023
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36822
716 y Ga(c)1062 704 y FX(i)1090 716 y Ga(T)13 b(;)1196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36823
704 y F9(\()1223 716 y Ga(x)1275 704 y F9(\))1303 716
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36824
y Ga(S)5 b F4(\))p Ga(;)15 b(U)36 b F6(2)25 b Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36825
b(N)1757 730 y Gc(aux)2905 716 y Gg(by)24 b(Lemma)e(2.3.13)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36826
840 y(\(3.9\))100 b FL(Cut)p F4(\()995 828 y FX(h)1023
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36827
840 y Ga(a)1071 828 y FX(i)1099 840 y FL(Cut)o F4(\()1271
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36828
828 y FX(h)1299 840 y Ga(c)1338 828 y FX(i)1366 840 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36829
Ga(T)13 b(;)1472 828 y F9(\()1500 840 y Ga(x)1552 828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36830
y F9(\))1579 840 y Ga(S)5 b F4(\))p Ga(;)1715 828 y F9(\()1743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36831
840 y Ga(y)1791 828 y F9(\))1818 840 y Ga(U)10 b F4(\))26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36832
b F6(2)f Ga(S)5 b(N)2171 854 y Gc(aux)2455 840 y Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36833
b(ind.)h(\(the)g(de)o(gree)h(decreased\))549 986 y(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36834
b(e)22 b(reason)j(analogous)i(in)c(the)h(case)g(where)1035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36835
1249 y FL(Cut)p F4(\()1208 1237 y FX(h)1236 1249 y Ga(b)1275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36836
1237 y FX(i)1302 1249 y Ga(M)10 b(;)1440 1237 y F9(\()1468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36837
1249 y Ga(z)1514 1237 y F9(\))1542 1249 y Ga(N)g F4(\))1759
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36838
1212 y Gc(l)1685 1249 y F6(\000)-31 b(\000)g(!)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36839
F4(\()2054 1237 y FX(h)2081 1249 y Ga(c)2120 1237 y FX(i)2148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36840
1249 y Ga(T)13 b(;)2254 1237 y F9(\()2282 1249 y Ga(x)2334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36841
1237 y F9(\))2387 1249 y FL(Cut)o F4(\()2559 1237 y FX(h)2587
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36842
1249 y Ga(a)2635 1237 y FX(i)2663 1249 y Ga(S)5 b(;)2764
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36843
1237 y F9(\()2792 1249 y Ga(y)2840 1237 y F9(\))2867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36844
1249 y Ga(U)10 b F4(\)\))26 b Ga(:)321 1492 y Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36845
b(e)23 b(ha)n(v)o(e)h(sho)n(wn)f(that)h(all)f(immediate)i(reducts)g(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36846
e FL(Cut)p F4(\()2146 1480 y FX(h)2174 1492 y Ga(a)2222
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36847
1480 y FX(i)2249 1492 y Ga(M)10 b(;)2387 1480 y F9(\()2415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36848
1492 y Ga(x)2467 1480 y F9(\))2494 1492 y Ga(N)g F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36849
b Gg(are)h(strongly)h(normalising.)321 1605 y(Consequently)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36850
b(,)27 b FL(Cut)p F4(\()1034 1593 y FX(h)1062 1605 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36851
Ga(a)1110 1593 y FX(i)1137 1605 y Ga(M)10 b(;)1275 1593
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36852
y F9(\()1303 1605 y Ga(x)1355 1593 y F9(\))1382 1605
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36853
y Ga(N)g F4(\))23 b Gg(must)h(be)g(strongly)h(normalising.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36854
b(Thus)24 b(we)f(are)g(done.)p 3480 1605 4 62 v 3484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36855
1547 55 4 v 3484 1605 V 3538 1605 4 62 v Black 321 1818
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36856
a Gb(Pr)n(oof)h(of)f(Lemma)e(2.3.18.)p Black 35 w Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36857
b(e)22 b(shall)h(gi)n(v)o(e)g(\002)n(v)o(e)f(representati)n(v)o(e)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36858
(cases,)e(in)f(which)g(the)g(e)o(xpression)326 1931 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36859
F4(^)-50 b Ga(\033)t(;)15 b F6(f)p Ga(\033)s F6(g)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36860
b Gg(will)f(stand)i(for)e(the)h(set)29 b F4(^)-50 b Ga(\033)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36861
b F6([)d(f)p Ga(\033)s F6(g)k Gg(with)f F6(f)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36862
F6(g)k(62)i F4(^)-50 b Ga(\033)t Gg(.)p Black 321 2162
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36863
a Gb(Case)24 b FL(Ax)o F4(\()p Ga(x;)15 b(a)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36864
Gb(:)p Black 47 w Gg(W)-7 b(e)27 b(ha)n(v)o(e)h(to)f(pro)o(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36865
(that)g FL(Ax)o F4(\()p Ga(x;)15 b(a)p F4(\))39 b(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36866
b Ga(\033)s(;)10 b F6(f)-7 b Ga(x)34 b F4(:=)2440 2150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36867
y FX(h)2468 2162 y Ga(b)2507 2150 y FX(i)2534 2162 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36868
Ga(P)t F6(g)p Ga(;)10 b F6(f)-7 b Ga(a)34 b F4(:=)2924
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36869
2150 y F9(\()2951 2162 y Ga(y)2999 2150 y F9(\))3027
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36870
2162 y Ga(Q)-10 b F6(g)27 b Gg(is)g(strongly)549 2275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36871
y(normalising)f(for)d(arbitrary)j(\(co-\)named)g(terms)2137
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36872
2263 y FX(h)2164 2275 y Ga(b)r F4(:)r Ga(B)2306 2263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36873
y FX(i)2333 2275 y Ga(P)39 b F6(2)24 b FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36874
Ga(B)5 b F6(i)p FB(K)23 b Gg(and)2910 2263 y F9(\()2937
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36875
2275 y Ga(y)5 b F4(:)r Ga(B)3088 2263 y F9(\))3115 2275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36876
y Ga(Q)25 b F6(2)g FB(J)p F4(\()p Ga(B)5 b F4(\))p FB(K)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36877
Gg(.)549 2428 y(\(1\))48 b FL(Ax)o F4(\()p Ga(x;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36878
F4(\))32 b(^)-50 b Ga(\033)s(;)10 b F6(f)-7 b Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36879
b F4(:=)1368 2416 y FX(h)1396 2428 y Ga(b)1435 2416 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36880
FX(i)1462 2428 y Ga(P)t F6(g)p Ga(;)10 b F6(f)-7 b Ga(a)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36881
b F4(:=)1838 2416 y F9(\()1865 2428 y Ga(y)1913 2416
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36882
y F9(\))1941 2428 y Ga(Q)-10 b F6(g)26 b F4(=)f FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36883
F4(\()2343 2416 y FX(h)2370 2428 y Ga(b)2409 2416 y FX(i)2437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36884
2428 y Ga(P)13 b(;)2548 2416 y F9(\()2576 2428 y Ga(y)2624
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36885
2416 y F9(\))2651 2428 y Ga(Q)p F4(\))239 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36886
b(Def.)e(of)i F6(f)p 3438 2428 28 4 v 3456 2428 V 3474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36887
2428 V 65 w(g)549 2549 y Gg(\(2\))702 2537 y FX(h)729
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36888
2549 y Ga(b)r F4(:)r Ga(B)871 2537 y FX(i)899 2549 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36889
Ga(P)38 b F6(2)25 b FB(J)p F6(h)p Ga(B)5 b F6(i)p FB(K)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36890
b Gg(and)1475 2537 y F9(\()1502 2549 y Ga(y)5 b F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36891
Ga(B)1653 2537 y F9(\))1680 2549 y Ga(Q)26 b F6(2)e FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36892
F4(\()p Ga(B)5 b F4(\))p FB(K)938 b Gg(by)23 b(assumption)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36893
2670 y(\(3\))48 b Ga(P)38 b F6(2)25 b Ga(S)5 b(N)1018
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36894
2684 y Gc(aux)1163 2670 y Gg(and)24 b Ga(Q)h F6(2)g Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36895
b(N)1634 2684 y Gc(aux)2905 2670 y Gg(by)24 b(Lemma)e(2.3.13)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36896
2791 y(\(4\))48 b FL(Cut)p F4(\()875 2779 y FX(h)903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36897
2791 y Ga(b)942 2779 y FX(i)969 2791 y Ga(P)13 b(;)1080
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36898
2779 y F9(\()1108 2791 y Ga(y)1156 2779 y F9(\))1183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36899
2791 y Ga(Q)p F4(\))26 b F6(2)e Ga(S)5 b(N)1535 2805
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36900
y Gc(aux)2905 2791 y Gg(by)24 b(Lemma)e(2.3.14)549 2912
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36901
y(\(5\))48 b FL(Ax)o F4(\()p Ga(x;)15 b(a)p F4(\))32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36902
b(^)-50 b Ga(\033)s(;)10 b F6(f)-7 b Ga(x)26 b F4(:=)1368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36903
2900 y FX(h)1396 2912 y Ga(b)1435 2900 y FX(i)1462 2912
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36904
y Ga(P)t F6(g)p Ga(;)10 b F6(f)-7 b Ga(a)27 b F4(:=)1838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36905
2900 y F9(\()1865 2912 y Ga(y)1913 2900 y F9(\))1941
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36906
2912 y Ga(Q)-10 b F6(g)26 b(2)f Ga(S)5 b(N)2294 2926
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36907
y Gc(aux)3044 2912 y Gg(by)24 b(\(1\))f(and)h(\(4\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36908
Black 321 3148 a Gb(Case)g FL(And)685 3162 y Gc(R)743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36909
3148 y F4(\()778 3136 y FX(h)806 3148 y Ga(a)854 3136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36910
y FX(i)881 3148 y Ga(M)11 b(;)1020 3136 y FX(h)1047 3148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36911
y Ga(b)1086 3136 y FX(i)1114 3148 y Ga(N)f(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36912
F4(\))p Gb(:)p Black 47 w Gg(W)-7 b(e)24 b(ha)n(v)o(e)i(to)f(pro)o(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36913
(e)h(that)g FL(And)2363 3162 y Gc(R)2421 3148 y F4(\()2456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36914
3136 y FX(h)2484 3148 y Ga(a)2532 3136 y FX(i)2559 3148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36915
y Ga(M)10 b(;)2697 3136 y FX(h)2725 3148 y Ga(b)2764
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36916
3136 y FX(i)2792 3148 y Ga(N)f(;)15 b(c)p F4(\))35 b(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36917
b Ga(\033)s(;)10 b F6(f)-7 b Ga(c)30 b F4(:=)3339 3136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36918
y F9(\()3367 3148 y Ga(z)3413 3136 y F9(\))3441 3148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36919
y Ga(R)-9 b F6(g)549 3261 y Gg(is)23 b(strongly)j(normalising)g(for)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36920
(an)h(arbitrary)i(named)e(term)2437 3249 y F9(\()2465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36921
3261 y Ga(z)6 b F4(:)r Ga(B)f F6(^)o Ga(C)2746 3249 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36922
F9(\))2773 3261 y Ga(R)26 b F6(2)f FB(J)p F4(\()p Ga(B)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36923
b F6(^)o Ga(C)i F4(\))p FB(K)p Gg(.)549 3414 y(\(1\))48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36924
b FL(And)856 3428 y Gc(R)914 3414 y F4(\()949 3402 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36925
FX(h)977 3414 y Ga(a)1025 3402 y FX(i)1052 3414 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36926
b(;)1190 3402 y FX(h)1218 3414 y Ga(b)1257 3402 y FX(i)1285
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36927
3414 y Ga(N)g(;)15 b(c)p F4(\))21 b(^)-51 b Ga(\033)t(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36928
b F6(f)-7 b Ga(c)10 b F4(:=)1781 3402 y F9(\()1810 3414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36929
y Ga(z)1856 3402 y F9(\))1884 3414 y Ga(R)-9 b F6(g)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36930
b F4(=)702 3535 y FL(Cut)p F4(\()875 3523 y FX(h)903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36931
3535 y Ga(c)942 3523 y FX(i)969 3535 y FL(And)1124 3549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36932
y Gc(R)1182 3535 y F4(\()1217 3523 y FX(h)1244 3535 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36933
Ga(a)1292 3523 y FX(i)1345 3535 y Ga(M)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36934
b Ga(\033)s(;)10 b F6(f)-7 b Ga(c)27 b F4(:=)1758 3523
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36935
y F9(\()1786 3535 y Ga(z)1832 3523 y F9(\))1859 3535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36936
y Ga(R)-8 b F6(g)p Ga(;)2005 3523 y FX(h)2033 3535 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36937
Ga(b)2072 3523 y FX(i)2125 3535 y Ga(N)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36938
b Ga(\033)s(;)10 b F6(f)-7 b Ga(c)27 b F4(:=)2523 3523
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36939
y F9(\()2550 3535 y Ga(z)2596 3523 y F9(\))2624 3535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36940
y Ga(R)-9 b F6(g)q Ga(;)15 b(c)p F4(\))q Ga(;)2885 3523
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36941
y F9(\()2913 3535 y Ga(z)2959 3523 y F9(\))2986 3535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36942
y Ga(R)q F4(\))35 b Gg(Def.)23 b(of)8 b F6(f)p 3438 3535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36943
V 3456 3535 V 3474 3535 V 65 w(g)549 3656 y Gg(\(2\))48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36944
b Ga(M)40 b F4(^)-50 b Ga(\033)s(;)10 b F6(f)-7 b Ga(c)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36945
b F4(:=)1140 3644 y F9(\()1168 3656 y Ga(z)1214 3644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36946
y F9(\))1241 3656 y Ga(R)-8 b F6(g)p Ga(;)10 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36947
b Ga(a)26 b F4(:=)1616 3644 y F9(\()1643 3656 y Ga(y)1691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36948
3644 y F9(\))1718 3656 y Ga(P)t F6(g)g(2)f Ga(S)5 b(N)2071
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36949
3670 y Gc(aux)2216 3656 y Gg(for)23 b(arbitrary)2680
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36950
3644 y F9(\()2707 3656 y Ga(y)5 b F4(:)r Ga(B)2858 3644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36951
y F9(\))2886 3656 y Ga(P)38 b F6(2)25 b FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36952
Ga(B)5 b F4(\))o FB(K)p Gg(,)702 3777 y Ga(N)40 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36953
b Ga(\033)s(;)10 b F6(f)-7 b Ga(c)27 b F4(:=)1125 3765
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36954
y F9(\()1153 3777 y Ga(z)1199 3765 y F9(\))1226 3777
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36955
y Ga(R)-8 b F6(g)p Ga(;)10 b F6(f)-7 b Ga(b)26 b F4(:=)1592
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36956
3765 y F9(\()1619 3777 y Ga(x)1671 3765 y F9(\))1698
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36957
3777 y Ga(Q)-9 b F6(g)25 b(2)g Ga(S)5 b(N)2051 3791 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36958
Gc(aux)2196 3777 y Gg(for)24 b(arbitrary)2661 3765 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36959
F9(\()2688 3777 y Ga(x)r F4(:)r Ga(C)2841 3765 y F9(\))2868
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36960
3777 y Ga(Q)i F6(2)e FB(J)p F4(\()p Ga(C)7 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36961
FB(K)3089 3898 y Gg(by)24 b(induction)549 4018 y(\(3\))48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36962
b F4(\()p Ga(M)15 b F4(^)-50 b Ga(\033)s(;)10 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36963
b Ga(c)27 b F4(:=)1150 4006 y F9(\()1178 4018 y Ga(z)1224
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36964
4006 y F9(\))1252 4018 y Ga(R)-9 b F6(g)p F4(\))k F6(f)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36965
Ga(a)27 b F4(:=)1621 4006 y F9(\()1648 4018 y Ga(y)1696
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36966
4006 y F9(\))1723 4018 y Ga(P)t F6(g)f(2)f Ga(S)5 b(N)2076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36967
4032 y Gc(aux)2198 4018 y Gg(,)702 4139 y F4(\()p Ga(N)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36968
b F4(^)-50 b Ga(\033)s(;)10 b F6(f)-7 b Ga(c)27 b F4(:=)1135
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36969
4127 y F9(\()1163 4139 y Ga(z)1209 4127 y F9(\))1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36970
4139 y Ga(R)-8 b F6(g)p F4(\))j F6(f)e Ga(b)26 b F4(:=)1597
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36971
4127 y F9(\()1624 4139 y Ga(x)1676 4127 y F9(\))1703
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36972
4139 y Ga(Q)-9 b F6(g)26 b(2)e Ga(S)5 b(N)2056 4153 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36973
Gc(aux)3044 4139 y Gg(by)24 b(\(2\))f(and)h(sss)549 4260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36974
y(\(4\))702 4248 y FX(h)729 4260 y Ga(a)r F4(:)r Ga(B)880
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36975
4248 y FX(i)908 4260 y F4(\()p Ga(M)15 b F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36976
b F6(f)-7 b Ga(c)27 b F4(:=)1356 4248 y F9(\()1384 4260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36977
y Ga(z)1430 4248 y F9(\))1457 4260 y Ga(R)-8 b F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36978
F4(\))26 b F6(2)e FB(J)p F6(h)p Ga(B)5 b F6(i)p FB(K)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36979
Gg(,)702 4369 y FX(h)729 4381 y Ga(b)r F4(:)r Ga(C)869
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36980
4369 y FX(i)897 4381 y F4(\()p Ga(N)15 b F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36981
b F6(f)-7 b Ga(c)27 b F4(:=)1330 4369 y F9(\()1357 4381
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36982
y Ga(z)1403 4369 y F9(\))1431 4381 y Ga(R)-9 b F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36983
F4(\))26 b F6(2)f FB(J)p F6(h)p Ga(C)7 b F6(i)p FB(K)960
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36984
b Gg(by)24 b(De\002nition)h(2.3.3)549 4502 y(\(5\))48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36985
b FL(And)856 4516 y Gc(R)914 4502 y F4(\()949 4490 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36986
FX(h)977 4502 y Ga(a)1025 4490 y FX(i)1052 4502 y Ga(M)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36987
b F4(^)-50 b Ga(\033)t(;)10 b F6(f)-7 b Ga(c)26 b F4(:=)1466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36988
4490 y F9(\()1493 4502 y Ga(z)1539 4490 y F9(\))1567
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36989
4502 y Ga(R)-9 b F6(g)p Ga(;)1712 4490 y FX(h)1740 4502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36990
y Ga(b)1779 4490 y FX(i)1807 4502 y Ga(N)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36991
b Ga(\033)s(;)10 b F6(f)-7 b Ga(c)27 b F4(:=)2205 4490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36992
y F9(\()2232 4502 y Ga(z)2278 4490 y F9(\))2306 4502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36993
y Ga(R)-9 b F6(g)q Ga(;)15 b(c)p F4(\))23 b Gg(freshly)j(introduces)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36994
Ga(c)62 b Gg(by)23 b(\(1\))549 4622 y(\(6\))702 4610
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36995
y FX(h)729 4622 y Ga(c)r F4(:)r Ga(B)5 b F6(^)p Ga(C)1004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36996
4610 y FX(i)1031 4622 y FL(And)1186 4636 y Gc(R)1243
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36997
4622 y F4(\()1278 4610 y FX(h)1306 4622 y Ga(a)1354 4610
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36998
y FX(i)1382 4622 y Ga(M)15 b F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 36999
b F6(f)-7 b Ga(c)26 b F4(:=)1795 4610 y F9(\()1822 4622
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37000
y Ga(z)1868 4610 y F9(\))1896 4622 y Ga(R)-9 b F6(g)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37001
Ga(;)2042 4610 y FX(h)2069 4622 y Ga(b)2108 4610 y FX(i)2136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37002
4622 y Ga(N)15 b F4(^)-50 b Ga(\033)s(;)10 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37003
b Ga(c)27 b F4(:=)2534 4610 y F9(\()2561 4622 y Ga(z)2607
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37004
4610 y F9(\))2635 4622 y Ga(R)-9 b F6(g)q Ga(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37005
F4(\))26 b F6(2)f FB(J)p F6(h)p Ga(B)5 b F6(^)o Ga(C)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37006
F6(i)p FB(K)2425 4743 y Gg(by)24 b(\(4\),)f(\(5\))h(and)g(De\002nition)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37007
h(2.3.3)549 4864 y(\(7\))48 b FL(And)856 4878 y Gc(R)914
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37008
4864 y F4(\()949 4852 y FX(h)977 4864 y Ga(a)1025 4852
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37009
y FX(i)1052 4864 y Ga(M)15 b F4(^)-50 b Ga(\033)t(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37010
b F6(f)-7 b Ga(c)26 b F4(:=)1466 4852 y F9(\()1493 4864
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37011
y Ga(z)1539 4852 y F9(\))1567 4864 y Ga(R)-9 b F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37012
Ga(;)1712 4852 y FX(h)1740 4864 y Ga(b)1779 4852 y FX(i)1807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37013
4864 y Ga(N)15 b F4(^)-50 b Ga(\033)s(;)10 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37014
b Ga(c)27 b F4(:=)2205 4852 y F9(\()2232 4864 y Ga(z)2278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37015
4852 y F9(\))2306 4864 y Ga(R)-9 b F6(g)q Ga(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37016
F4(\))26 b F6(2)f Ga(S)5 b(N)2772 4878 y Gc(aux)2894
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37017
4864 y Gg(,)702 4985 y Ga(R)26 b F6(2)f Ga(S)5 b(N)1017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37018
4999 y Gc(aux)2905 4985 y Gg(by)24 b(Lemma)e(2.3.13)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37019
5106 y(\(8\))48 b FL(Cut)p F4(\()875 5094 y FX(h)903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37020
5106 y Ga(c)942 5094 y FX(i)969 5106 y FL(And)1124 5120
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37021
y Gc(R)1182 5106 y F4(\()1217 5094 y FX(h)1244 5106 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37022
Ga(a)1292 5094 y FX(i)1320 5106 y Ga(M)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37023
b Ga(\033)s(;)10 b F6(f)-7 b Ga(c)27 b F4(:=)1733 5094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37024
y F9(\()1760 5106 y Ga(z)1806 5094 y F9(\))1834 5106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37025
y Ga(R)-9 b F6(g)q Ga(;)1980 5094 y FX(h)2008 5106 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37026
Ga(b)2047 5094 y FX(i)2074 5106 y Ga(N)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37027
b Ga(\033)s(;)10 b F6(f)-7 b Ga(c)27 b F4(:=)2472 5094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37028
y F9(\()2500 5106 y Ga(z)2546 5094 y F9(\))2573 5106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37029
y Ga(R)-8 b F6(g)p Ga(;)15 b(c)p F4(\))q Ga(;)2834 5094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37030
y F9(\()2862 5106 y Ga(z)2908 5094 y F9(\))2936 5106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37031
y Ga(R)q F4(\))25 b F6(2)g Ga(S)5 b(N)3286 5120 y Gc(aux)2905
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37032
5226 y Gg(by)24 b(Lemma)e(2.3.14)549 5347 y(\(9\))48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37033
b FL(And)856 5361 y Gc(R)914 5347 y F4(\()949 5335 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37034
FX(h)977 5347 y Ga(a)1025 5335 y FX(i)1052 5347 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37035
b(;)1190 5335 y FX(h)1218 5347 y Ga(b)1257 5335 y FX(i)1285
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37036
5347 y Ga(N)g(;)15 b(c)p F4(\))31 b(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37037
b F6(f)-7 b Ga(c)27 b F4(:=)1823 5335 y F9(\()1851 5347
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37038
y Ga(z)1897 5335 y F9(\))1924 5347 y Ga(R)-8 b F6(g)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37039
b(2)g Ga(S)5 b(N)2275 5361 y Gc(aux)3044 5347 y Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37040
b(\(1\))f(and)h(\(8\))p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37041
%%Page: 149 161
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37042
TeXDict begin 149 160 bop Black 277 51 a Gb(B.1)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37043
(Chapter)f(2)2639 b(149)p 277 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37044
277 388 a(Case)24 b FL(And)641 351 y Gc(i)641 411 y(L)693
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37045
388 y F4(\()728 376 y F9(\()756 388 y Ga(x)808 376 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37046
F9(\))836 388 y Ga(M)10 b(;)15 b(y)s F4(\))23 b(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37047
Ga(i)j F4(=)f(1)p Ga(;)15 b F4(2\))p Gb(:)p Black 47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37048
w Gg(W)-7 b(e)28 b(ha)n(v)o(e)h(to)f(pro)o(v)o(e)h(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37049
FL(And)2502 351 y Gc(i)2502 411 y(L)2554 388 y F4(\()2589
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37050
376 y F9(\()2616 388 y Ga(x)2668 376 y F9(\))2696 388
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37051
y Ga(M)10 b(;)15 b(y)s F4(\))40 b(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37052
b F6(f)-7 b Ga(y)38 b F4(:=)3293 376 y FX(h)3321 388
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37053
y Ga(a)3369 376 y FX(i)3396 388 y Ga(R)-8 b F6(g)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37054
501 y Gg(is)21 b(strongly)h(normalising)h(for)e(an)f(arbitrary)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37055
(co-named)g(term)2484 489 y FX(h)2511 501 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37056
Ga(C)2653 515 y F9(1)2693 501 y F6(^)o Ga(C)2818 515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37057
y F9(2)2858 489 y FX(i)2885 501 y Ga(R)k F6(2)f FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37058
F6(h)p Ga(C)3203 515 y F9(1)3242 501 y F6(^)p Ga(C)3368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37059
515 y F9(2)3407 501 y F6(i)q FB(K)o Gg(.)504 733 y(\(1\))49
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37060
b FL(And)812 696 y Gc(i)812 756 y(L)864 733 y F4(\()899
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37061
721 y F9(\()927 733 y Ga(x)979 721 y F9(\))1007 733 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37062
Ga(M)10 b(;)15 b(y)s F4(\))20 b(^)-50 b Ga(\033)t(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37063
b F6(f)-7 b Ga(y)29 b F4(:=)1567 721 y FX(h)1594 733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37064
y Ga(a)1642 721 y FX(i)1670 733 y Ga(R)-9 b F6(g)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37065
F4(=)658 850 y FL(Cut)p F4(\()831 838 y FX(h)858 850
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37066
y Ga(a)906 838 y FX(i)934 850 y Ga(R)q(;)1044 838 y F9(\()1071
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37067
850 y Ga(y)1119 838 y F9(\))1147 850 y FL(And)1301 813
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37068
y Gc(i)1301 873 y(L)1353 850 y F4(\()1388 838 y F9(\()1416
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37069
850 y Ga(x)1468 838 y F9(\))1496 850 y Ga(M)k F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37070
b Ga(\033)s(;)10 b F6(f)-7 b Ga(y)29 b F4(:=)1932 838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37071
y FX(h)1960 850 y Ga(a)2008 838 y FX(i)2035 850 y Ga(R)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37072
b F6(g)p Ga(;)15 b(y)s F4(\))q(\))443 b Gg(by)24 b(De\002nition)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37073
F6(f)p 3394 850 28 4 v 3412 850 V 3429 850 V 65 w(g)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37074
968 y Gg(\(2\))49 b Ga(M)40 b F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37075
b F6(f)-7 b Ga(y)29 b F4(:=)1105 956 y FX(h)1132 968
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37076
y Ga(a)1180 956 y FX(i)1208 968 y Ga(R)-9 b F6(g)p Ga(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37077
b F6(f)-7 b Ga(x)26 b F4(:=)1586 956 y FX(h)1613 968
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37078
y Ga(b)1652 956 y FX(i)1680 968 y Ga(P)s F6(g)g(2)f Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37079
b(N)2032 982 y Gc(aux)2177 968 y Gg(for)23 b(arbitrary)2641
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37080
956 y FX(h)2669 968 y Ga(b)r F4(:)r Ga(C)2802 982 y Gc(i)2830
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37081
956 y FX(i)2858 968 y Ga(P)38 b F6(2)25 b FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37082
Ga(C)3177 982 y Gc(i)3205 968 y F6(i)p FB(K)3045 1085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37083
y Gg(by)f(induction)504 1202 y(\(3\))49 b F4(\()p Ga(M)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37084
b F4(^)-50 b Ga(\033)t(;)10 b F6(f)-7 b Ga(y)29 b F4(:=)1140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37085
1190 y FX(h)1167 1202 y Ga(a)1215 1190 y FX(i)1243 1202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37086
y Ga(R)-9 b F6(g)p F4(\))k F6(f)e Ga(x)27 b F4(:=)1616
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37087
1190 y FX(h)1643 1202 y Ga(b)1682 1190 y FX(i)1710 1202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37088
y Ga(P)s F6(g)f(2)f Ga(S)5 b(N)2062 1216 y Gc(aux)3000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37089
1202 y Gg(by)23 b(\(2\))h(and)g(sss)504 1320 y(\(4\))658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37090
1308 y F9(\()685 1320 y Ga(x)r F4(:)r Ga(C)831 1334 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37091
Gc(i)860 1308 y F9(\))887 1320 y F4(\()p Ga(M)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37092
b Ga(\033)t(;)10 b F6(f)-7 b Ga(y)29 b F4(:=)1344 1308
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37093
y FX(h)1371 1320 y Ga(a)1419 1308 y FX(i)1447 1320 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37094
Ga(R)-9 b F6(g)p F4(\))26 b F6(2)f FB(J)p F4(\()p Ga(C)1836
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37095
1334 y Gc(i)1864 1320 y F4(\))q FB(K)878 b Gg(by)24 b(De\002nition)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37096
(2.3.3)504 1437 y(\(5\))49 b FL(And)812 1400 y Gc(i)812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37097
1460 y(L)864 1437 y F4(\()899 1425 y F9(\()927 1437 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37098
Ga(x)979 1425 y F9(\))1007 1437 y Ga(M)30 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37099
b Ga(\033)s(;)10 b F6(f)-7 b Ga(y)13 b F4(:=)1412 1425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37100
y FX(h)1440 1437 y Ga(a)1488 1425 y FX(i)1516 1437 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37101
Ga(R)-9 b F6(g)q Ga(;)15 b(y)s F4(\))23 b Gg(freshly)i(introduces)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37102
Ga(y)793 b Gg(by)23 b(\(1\))504 1555 y(\(6\))658 1543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37103
y F9(\()685 1555 y Ga(y)5 b F4(:)r Ga(C)827 1569 y F9(1)867
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37104
1555 y F6(^)o Ga(C)992 1569 y F9(2)1032 1543 y(\))1059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37105
1555 y FL(And)1214 1518 y Gc(i)1214 1578 y(L)1266 1555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37106
y F4(\()1301 1543 y F9(\()1329 1555 y Ga(x)1381 1543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37107
y F9(\))1408 1555 y Ga(M)30 b F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37108
b F6(f)-7 b Ga(y)13 b F4(:=)1813 1543 y FX(h)1842 1555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37109
y Ga(a)1890 1543 y FX(i)1917 1555 y Ga(R)-8 b F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37110
Ga(;)15 b(y)s F4(\))26 b F6(2)f FB(J)p F4(\()p Ga(C)2395
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37111
1569 y F9(1)2434 1555 y F6(^)p Ga(C)2560 1569 y F9(2)2599
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37112
1555 y F4(\))q FB(K)2591 1672 y Gg(by)e(\(4\),)h(\(5\))g(and)g(Def.)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37113
(2.3.3)504 1790 y(\(7\))49 b FL(And)812 1753 y Gc(i)812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37114
1813 y(L)864 1790 y F4(\()899 1778 y F9(\()927 1790 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37115
Ga(x)979 1778 y F9(\))1007 1790 y Ga(M)30 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37116
b Ga(\033)s(;)10 b F6(f)-7 b Ga(y)13 b F4(:=)1412 1778
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37117
y FX(h)1440 1790 y Ga(a)1488 1778 y FX(i)1516 1790 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37118
Ga(R)-9 b F6(g)q Ga(;)15 b(y)s F4(\))p Ga(;)41 b(R)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37119
b F6(2)f Ga(S)5 b(N)2126 1804 y Gc(aux)2861 1790 y Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37120
b(Lemma)g(2.3.13)504 1907 y(\(8\))49 b FL(Cut)p F4(\()831
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37121
1895 y FX(h)858 1907 y Ga(a)906 1895 y FX(i)934 1907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37122
y Ga(R)q(;)1044 1895 y F9(\()1071 1907 y Ga(y)1119 1895
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37123
y F9(\))1147 1907 y FL(And)1301 1870 y Gc(i)1301 1930
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37124
y(L)1353 1907 y F4(\()1388 1895 y F9(\()1416 1907 y Ga(x)1468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37125
1895 y F9(\))1496 1907 y Ga(M)30 b F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37126
b F6(f)-7 b Ga(y)13 b F4(:=)1901 1895 y FX(h)1930 1907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37127
y Ga(a)1978 1895 y FX(i)2005 1907 y Ga(R)-9 b F6(g)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37128
Ga(;)15 b(y)s F4(\)\))26 b F6(2)f Ga(S)5 b(N)2515 1921
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37129
y Gc(aux)2861 1907 y Gg(by)23 b(Lemma)g(2.3.14)504 2024
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37130
y(\(9\))49 b FL(And)812 1988 y Gc(i)812 2047 y(L)864
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37131
2024 y F4(\()899 2012 y F9(\()927 2024 y Ga(x)979 2012
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37132
y F9(\))1007 2024 y Ga(M)10 b(;)15 b(y)s F4(\))20 b(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37133
b Ga(\033)t(;)10 b F6(f)-7 b Ga(y)13 b F4(:=)1536 2012
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37134
y FX(h)1564 2024 y Ga(a)1612 2012 y FX(i)1640 2024 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37135
Ga(R)-9 b F6(g)26 b(2)e Ga(S)5 b(N)1990 2038 y Gc(aux)3000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37136
2024 y Gg(by)23 b(\(1\))h(and)g(\(8\))p Black 277 2420
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37137
a Gb(Case)g FL(Imp)631 2442 y Gc(R)689 2420 y F4(\()724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37138
2408 y F9(\()752 2420 y Ga(x)804 2408 y F9(\))p FX(h)858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37139
2420 y Ga(a)906 2408 y FX(i)934 2420 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37140
b(b)p F4(\))p Gb(:)p Black 47 w Gg(W)-7 b(e)36 b(pro)o(v)o(e)h(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37141
FL(Imp)1935 2442 y Gc(R)1992 2420 y F4(\()2027 2408 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37142
F9(\()2055 2420 y Ga(x)2107 2408 y F9(\))q FX(h)2162
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37143
2420 y Ga(a)2210 2408 y FX(i)2238 2420 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37144
b(b)p F4(\))55 b(^)-50 b Ga(\033)t(;)10 b F6(f)-7 b Ga(b)51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37145
b F4(:=)2865 2408 y F9(\()2892 2420 y Ga(z)2938 2408
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37146
y F9(\))2966 2420 y Ga(R)-9 b F6(g)36 b Gg(is)h(strongly)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37147
2533 y(normalising)26 b(for)e(an)g(arbitrary)h(named)g(term)1989
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37148
2521 y F9(\()2016 2533 y Ga(z)6 b F4(:)r Ga(B)f F6(\033)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37149
Ga(C)2308 2521 y F9(\))2335 2533 y Ga(R)24 b Gg(that)g(belongs)h(to)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37150
FB(J)p F4(\()p Ga(B)5 b F6(\033)o Ga(C)i F4(\))p FB(K)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37151
Gg(.)504 2765 y(\(1\))49 b FL(Imp)802 2787 y Gc(R)860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37152
2765 y F4(\()895 2753 y F9(\()923 2765 y Ga(x)975 2753
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37153
y F9(\))p FX(h)1029 2765 y Ga(a)1077 2753 y FX(i)1105
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37154
2765 y Ga(M)10 b(;)15 b(b)p F4(\))31 b(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37155
b F6(f)-7 b Ga(b)27 b F4(:=)1658 2753 y F9(\()1685 2765
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37156
y Ga(z)1731 2753 y F9(\))1759 2765 y Ga(R)-9 b F6(g)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37157
b F4(=)658 2884 y FL(Cut)p F4(\()831 2872 y FX(h)858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37158
2884 y Ga(c)897 2872 y FX(i)925 2884 y FL(Imp)1070 2906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37159
y Gc(R)1127 2884 y F4(\()1162 2872 y F9(\()1190 2884
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37160
y Ga(x)1242 2872 y F9(\))p FX(h)1297 2884 y Ga(a)1345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37161
2872 y FX(i)1372 2884 y Ga(M)15 b F4(^)-50 b Ga(\033)t(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37162
b F6(f)-7 b Ga(b)26 b F4(:=)1785 2872 y F9(\()1813 2884
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37163
y Ga(z)1859 2872 y F9(\))1886 2884 y Ga(R)-8 b F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37164
Ga(;)15 b(b)p F4(\))q Ga(;)2147 2872 y F9(\()2175 2884
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37165
y Ga(z)2221 2872 y F9(\))2248 2884 y Ga(R)q F4(\))390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37166
b Gg(by)24 b(De\002nition)g(of)g F6(f)p 3394 2884 V 3412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37167
2884 V 3429 2884 V 65 w(g)504 3002 y Gg(\(2\))49 b Ga(M)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37168
b F4(^)-50 b Ga(\033)s(;)10 b F6(f)-7 b Ga(b)27 b F4(:=)1096
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37169
2990 y F9(\()1123 3002 y Ga(z)1169 2990 y F9(\))1197
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37170
3002 y Ga(R)-9 b F6(g)p Ga(;)10 b F6(f)-7 b Ga(a)27 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37171
F4(:=)1571 2990 y F9(\()1599 3002 y Ga(y)1647 2990 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37172
F9(\))1674 3002 y Ga(P)s F6(g)p Ga(;)10 b F6(f)-7 b Ga(x)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37173
b F4(:=)2053 2990 y FX(h)2081 3002 y Ga(c)2120 2990 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37174
FX(i)2148 3002 y Ga(Q)-10 b F6(g)26 b(2)f Ga(S)5 b(N)2501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37175
3016 y Gc(aux)658 3121 y Gg(for)23 b(arbitrary)1122 3109
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37176
y F9(\()1150 3121 y Ga(y)5 b F4(:)r Ga(B)1301 3109 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37177
F9(\))1328 3121 y Ga(P)38 b F6(2)25 b FB(J)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37178
Ga(B)5 b F4(\))p FB(K)22 b Gg(and)1904 3109 y FX(h)1931
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37179
3121 y Ga(c)r F4(:)r Ga(C)2071 3109 y FX(i)2099 3121
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37180
y Ga(Q)j F6(2)g FB(J)p F6(h)p Ga(C)7 b F6(i)p FB(K)547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37181
b Gg(by)24 b(induction)504 3239 y(\(3\))49 b F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37182
Ga(M)15 b F4(^)-50 b Ga(\033)s(;)10 b F6(f)-7 b Ga(b)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37183
b F4(:=)1106 3227 y F9(\()1133 3239 y Ga(z)1179 3227
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37184
y F9(\))1207 3239 y Ga(R)-9 b F6(g)p Ga(;)10 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37185
b Ga(x)27 b F4(:=)1585 3227 y FX(h)1613 3239 y Ga(c)1652
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37186
3227 y FX(i)1679 3239 y Ga(Q)-9 b F6(g)p F4(\))k F6(f)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37187
Ga(a)27 b F4(:=)2051 3227 y F9(\()2078 3239 y Ga(y)2126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37188
3227 y F9(\))2154 3239 y Ga(P)s F6(g)f(2)f Ga(S)5 b(N)2506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37189
3253 y Gc(aux)2628 3239 y Gg(,)658 3358 y F4(\()p Ga(M)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37190
b F4(^)-50 b Ga(\033)s(;)10 b F6(f)-7 b Ga(b)27 b F4(:=)1106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37191
3346 y F9(\()1133 3358 y Ga(z)1179 3346 y F9(\))1207
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37192
3358 y Ga(R)-9 b F6(g)p Ga(;)10 b F6(f)-7 b Ga(a)27 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37193
F4(:=)1581 3346 y F9(\()1609 3358 y Ga(y)1657 3346 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37194
F9(\))1684 3358 y Ga(P)t F6(g)p F4(\))-5 b F6(f)e Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37195
b F4(:=)2058 3346 y FX(h)2086 3358 y Ga(c)2125 3346 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37196
FX(i)2153 3358 y Ga(Q)-10 b F6(g)26 b(2)f Ga(S)5 b(N)2506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37197
3372 y Gc(aux)3000 3358 y Gg(by)23 b(\(2\))h(and)g(sss)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37198
3476 y(\(4\))658 3464 y FX(h)685 3476 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37199
Ga(B)836 3464 y FX(i)889 3476 y Ga(M)15 b F4(^)-50 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37200
Ga(\033)s(;)10 b F6(f)-7 b Ga(b)26 b F4(:=)1302 3464
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37201
y F9(\()1329 3476 y Ga(z)1375 3464 y F9(\))1403 3476
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37202
y Ga(R)-9 b F6(g)p Ga(;)10 b F6(f)-7 b Ga(x)27 b F4(:=)1781
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37203
3464 y FX(h)1808 3476 y Ga(c)1847 3464 y FX(i)1875 3476
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37204
y Ga(Q)-9 b F6(g)25 b(2)g FB(J)p F6(h)p Ga(B)5 b F6(i)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37205
FB(K)658 3583 y F9(\()685 3595 y Ga(x)r F4(:)r Ga(C)838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37206
3583 y F9(\))891 3595 y Ga(M)15 b F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37207
b F6(f)-7 b Ga(b)26 b F4(:=)1303 3583 y F9(\()1331 3595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37208
y Ga(z)1377 3583 y F9(\))1405 3595 y Ga(R)-9 b F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37209
Ga(;)10 b F6(f)-7 b Ga(a)27 b F4(:=)1779 3583 y F9(\()1806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37210
3595 y Ga(y)1854 3583 y F9(\))1882 3595 y Ga(P)s F6(g)f(2)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37211
FB(J)p F4(\()p Ga(C)7 b F4(\))o FB(K)500 b Gg(by)24 b(De\002nition)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37212
(2.3.3)504 3714 y(\(5\))49 b FL(Imp)802 3735 y Gc(R)860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37213
3714 y F4(\()895 3702 y F9(\()923 3714 y Ga(x)975 3702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37214
y F9(\))p FX(h)1029 3714 y Ga(a)1077 3702 y FX(i)1105
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37215
3714 y Ga(M)15 b F4(^)-50 b Ga(\033)s(;)10 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37216
b Ga(b)27 b F4(:=)1518 3702 y F9(\()1545 3714 y Ga(z)1591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37217
3702 y F9(\))1619 3714 y Ga(R)-9 b F6(g)q Ga(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37218
F4(\))23 b Gg(freshly)i(introduces)i Ga(b)705 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37219
b(\(1\))504 3832 y(\(6\))658 3820 y FX(h)685 3832 y Ga(b)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37220
F4(:)r Ga(B)5 b F6(\033)p Ga(C)970 3820 y FX(i)997 3832
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37221
y FL(Imp)1141 3854 y Gc(R)1199 3832 y F4(\()1234 3820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37222
y F9(\()1262 3832 y Ga(x)1314 3820 y F9(\))p FX(h)1368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37223
3832 y Ga(a)1416 3820 y FX(i)1444 3832 y Ga(M)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37224
b Ga(\033)s(;)10 b F6(f)-7 b Ga(b)27 b F4(:=)1857 3820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37225
y F9(\()1884 3832 y Ga(z)1930 3820 y F9(\))1958 3832
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37226
y Ga(R)-9 b F6(g)q Ga(;)15 b(b)p F4(\))26 b F6(2)e FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37227
F6(h)p Ga(B)5 b F6(\033)p Ga(C)i F6(i)p FB(K)2250 3951
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37228
y Gg(by)23 b(\(4\))h(and)g(\(5\))g(and)g(De\002nition)g(2.3.3)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37229
4069 y(\(7\))49 b FL(Imp)802 4091 y Gc(R)860 4069 y F4(\()895
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37230
4057 y F9(\()923 4069 y Ga(x)975 4057 y F9(\))p FX(h)1029
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37231
4069 y Ga(a)1077 4057 y FX(i)1105 4069 y Ga(M)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37232
b Ga(\033)s(;)10 b F6(f)-7 b Ga(b)27 b F4(:=)1518 4057
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37233
y F9(\()1545 4069 y Ga(z)1591 4057 y F9(\))1619 4069
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37234
y Ga(R)-9 b F6(g)q Ga(;)15 b(b)p F4(\))p Ga(;)41 b(R)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37235
b F6(2)f Ga(S)5 b(N)2220 4083 y Gc(aux)2861 4069 y Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37236
b(Lemma)g(2.3.13)504 4188 y(\(8\))49 b FL(Cut)p F4(\()831
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37237
4176 y FX(h)858 4188 y Ga(b)897 4176 y FX(i)925 4188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37238
y FL(Imp)1069 4210 y Gc(R)1127 4188 y F4(\()1162 4176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37239
y F9(\()1190 4188 y Ga(x)1242 4176 y F9(\))p FX(h)1297
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37240
4188 y Ga(a)1345 4176 y FX(i)1372 4188 y Ga(M)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37241
b Ga(\033)s(;)10 b F6(f)-7 b Ga(b)27 b F4(:=)1785 4176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37242
y F9(\()1812 4188 y Ga(z)1858 4176 y F9(\))1886 4188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37243
y Ga(R)-9 b F6(g)q Ga(;)15 b(b)p F4(\))q Ga(;)2147 4176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37244
y F9(\()2174 4188 y Ga(z)2220 4176 y F9(\))2248 4188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37245
y Ga(R)q F4(\))25 b F6(2)g Ga(S)5 b(N)2598 4202 y Gc(aux)2861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37246
4188 y Gg(by)23 b(Lemma)g(2.3.14)504 4306 y(\(9\))49
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37247
b FL(Imp)802 4328 y Gc(R)860 4306 y F4(\()895 4294 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37248
F9(\()923 4306 y Ga(x)975 4294 y F9(\))p FX(h)1029 4306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37249
y Ga(a)1077 4294 y FX(i)1105 4306 y Ga(M)10 b(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37250
F4(\))31 b(^)-50 b Ga(\033)s(;)10 b F6(f)-7 b Ga(b)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37251
b F4(:=)1658 4294 y F9(\()1685 4306 y Ga(z)1731 4294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37252
y F9(\))1759 4306 y Ga(R)-9 b F6(g)26 b(2)f Ga(S)5 b(N)2110
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37253
4320 y Gc(aux)3000 4306 y Gg(by)23 b(\(1\))h(and)g(\(8\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37254
Black 277 4702 a Gb(Case)g FL(Cut)p F4(\()660 4690 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37255
FX(h)687 4702 y Ga(a)735 4690 y FX(i)763 4702 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37256
b(;)901 4690 y F9(\()929 4702 y Ga(x)981 4690 y F9(\))1008
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37257
4702 y Ga(N)g F4(\))p Gb(:)p Black 46 w Gg(Since)20 b(we)f(introduced)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37258
(in)d(the)g(de\002nition)i(of)e F6(f)p 2686 4702 V 2704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37259
4702 V 2722 4702 V 65 w(g)f Gg(tw)o(o)h(special)i(clauses)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37260
4815 y(for)i(cuts)g(with)g(an)f(axiom)h(as)g(immediate)g(subterm,)h(we)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37261
d(ha)n(v)o(e)i(to)g(distinguish)j(tw)o(o)c(cases.)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37262
5046 y Gb(Subcase)h(I:)h Ga(M)34 b Gg(is)25 b(an)f(axiom)h(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37263
(freshly)g(introduces)i(the)d(label)g(of)g(the)g(cut-formula)i(\(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37264
504 5159 y(case)35 b(where)g Ga(N)43 b Gg(is)34 b(such)h(an)f(axiom)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37265
(is)f(similar\).)61 b(W)-7 b(e)33 b(ha)n(v)o(e)i(to)f(sho)n(w)g(that)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37266
(for)f(arbitrary)504 5260 y FX(h)532 5272 y Ga(b)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37267
Ga(B)674 5260 y FX(i)701 5272 y Ga(R)h F6(2)g FB(J)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37268
F6(h)p Ga(B)5 b F6(i)o FB(K)28 b Gg(the)h(term)f FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37269
F4(\()1652 5260 y FX(h)1680 5272 y Ga(a)1728 5260 y FX(i)1755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37270
5272 y FL(Ax)p F4(\()p Ga(x;)15 b(a)p F4(\))q Ga(;)2109
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37271
5260 y F9(\()2137 5272 y Ga(y)2185 5260 y F9(\))2212
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37272
5272 y Ga(N)10 b F4(\))40 b(^)-50 b Ga(\033)s(;)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37273
F6(f)-7 b Ga(x)36 b F4(:=)2711 5260 y FX(h)2738 5272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37274
y Ga(b)2777 5260 y FX(i)2805 5272 y Ga(R)-9 b F6(g)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37275
b Gg(is)h(strongly)i(nor)n(-)504 5385 y(malising.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37276
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37277
%%Page: 150 162
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37278
TeXDict begin 150 161 bop Black -144 51 a Gb(150)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37279
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37280
4 v Black 549 380 a Gg(\(1\))48 b FL(Cut)p F4(\()875
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37281
368 y FX(h)903 380 y Ga(a)951 368 y FX(i)978 380 y FL(Ax)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37282
F4(\()p Ga(x;)15 b(a)p F4(\))q Ga(;)1332 368 y F9(\()1359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37283
380 y Ga(y)1407 368 y F9(\))1435 380 y Ga(N)10 b F4(\))30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37284
b(^)-50 b Ga(\033)t(;)10 b F6(f)-7 b Ga(x)26 b F4(:=)1906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37285
368 y FX(h)1933 380 y Ga(b)1972 368 y FX(i)2000 380 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37286
Ga(R)-9 b F6(g)26 b F4(=)702 498 y FL(Cut)p F4(\()875
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37287
486 y FX(h)903 498 y Ga(b)942 486 y FX(i)969 498 y Ga(R)q(;)1079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37288
486 y F9(\()1107 498 y Ga(y)1155 486 y F9(\))1207 498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37289
y Ga(N)15 b F4(^)-50 b Ga(\033)s(;)10 b F6(f)-7 b Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37290
b F4(:=)1618 486 y FX(h)1645 498 y Ga(b)1684 486 y FX(i)1712
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37291
498 y Ga(R)-9 b F6(g)p F4(\))935 b Gg(by)24 b(De\002nition)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37292
F6(f)p 3438 498 28 4 v 3456 498 V 3474 498 V 65 w(g)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37293
617 y Gg(\(2\))48 b Ga(N)40 b F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37294
b F6(f)-7 b Ga(x)26 b F4(:=)1138 605 y FX(h)1165 617
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37295
y Ga(b)1204 605 y FX(i)1232 617 y Ga(R)-9 b F6(g)p Ga(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37296
b F6(f)-7 b Ga(y)29 b F4(:=)1606 605 y FX(h)1633 617
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37297
y Ga(c)1672 605 y FX(i)1700 617 y Ga(P)s F6(g)d(2)f Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37298
b(N)2052 631 y Gc(aux)2197 617 y Gg(for)24 b(arbitrary)2661
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37299
605 y FX(h)2689 617 y Ga(c)r F4(:)r Ga(B)2831 605 y FX(i)2859
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37300
617 y Ga(P)38 b F6(2)25 b FB(J)p F6(h)p Ga(B)5 b F6(i)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37301
FB(K)3089 736 y Gg(by)24 b(induction)549 854 y(\(3\))48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37302
b F4(\()p Ga(N)40 b F4(^)-50 b Ga(\033)t(;)10 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37303
b Ga(x)26 b F4(:=)1173 842 y FX(h)1201 854 y Ga(b)1240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37304
842 y FX(i)1267 854 y Ga(R)-9 b F6(g)p F4(\))k F6(f)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37305
Ga(y)30 b F4(:=)1636 842 y FX(h)1663 854 y Ga(c)1702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37306
842 y FX(i)1730 854 y Ga(P)t F6(g)25 b(2)g Ga(S)5 b(N)2082
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37307
868 y Gc(aux)3044 854 y Gg(by)24 b(\(2\))f(and)h(sss)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37308
973 y(\(4\))702 961 y F9(\()729 973 y Ga(y)5 b F4(:)18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37309
b Ga(B)896 961 y F9(\))948 973 y Ga(N)d F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37310
b F6(f)-7 b Ga(x)26 b F4(:=)1358 961 y FX(h)1386 973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37311
y Ga(b)1425 961 y FX(i)1452 973 y Ga(R)-8 b F6(g)25 b(2)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37312
FB(J)p F4(\()p Ga(B)5 b F4(\))p FB(K)972 b Gg(by)24 b(De\002nition)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37313
(2.3.3)549 1091 y(\(5\))48 b Ga(N)40 b F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37314
b F6(f)-7 b Ga(x)26 b F4(:=)1138 1079 y FX(h)1165 1091
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37315
y Ga(b)1204 1079 y FX(i)1232 1091 y Ga(R)-9 b F6(g)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37316
b(2)e Ga(S)5 b(N)1582 1105 y Gc(aux)1727 1091 y Gg(and)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37317
b Ga(R)i F6(2)f Ga(S)5 b(N)2196 1105 y Gc(aux)2905 1091
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37318
y Gg(by)24 b(Lemma)e(2.3.13)549 1210 y(\(6\))48 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37319
F4(\()875 1198 y FX(h)903 1210 y Ga(b)942 1198 y FX(i)969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37320
1210 y Ga(R)q(;)1079 1198 y F9(\()1107 1210 y Ga(y)1155
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37321
1198 y F9(\))1207 1210 y Ga(N)15 b F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37322
b F6(f)-7 b Ga(x)26 b F4(:=)1618 1198 y FX(h)1645 1210
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37323
y Ga(b)1684 1198 y FX(i)1712 1210 y Ga(R)-9 b F6(g)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37324
F4(\))26 b F6(2)f Ga(S)5 b(N)2098 1224 y Gc(aux)2905
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37325
1210 y Gg(by)24 b(Lemma)e(2.3.14)549 1328 y(\(7\))48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37326
b FL(Cut)p F4(\()875 1316 y FX(h)903 1328 y Ga(a)951
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37327
1316 y FX(i)978 1328 y FL(Ax)p F4(\()p Ga(x;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37328
F4(\))q Ga(;)1332 1316 y F9(\()1359 1328 y Ga(y)1407
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37329
1316 y F9(\))1435 1328 y Ga(N)10 b F4(\))30 b(^)-50 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37330
Ga(\033)t(;)10 b F6(f)-7 b Ga(x)26 b F4(:=)1906 1316
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37331
y FX(h)1933 1328 y Ga(b)1972 1316 y FX(i)2000 1328 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37332
Ga(R)-9 b F6(g)26 b(2)f Ga(S)5 b(N)2351 1342 y Gc(aux)3044
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37333
1328 y Gg(by)24 b(\(1\))f(and)h(\(6\))549 1511 y Gb(Subcase)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37334
b(II:)h Ga(M)39 b Gg(and)30 b Ga(N)39 b Gg(are)29 b(not)h(axioms)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37335
(that)f(freshly)i(introduce)g(the)e(label)g(of)g(the)g(cut-)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37336
1624 y(formula.)f(W)-7 b(e)23 b(ha)n(v)o(e)h(to)g(pro)o(v)o(e)g(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37337
FL(Cut)p F4(\()1873 1612 y FX(h)1900 1624 y Ga(a)1948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37338
1612 y FX(i)1976 1624 y Ga(M)10 b(;)2114 1612 y F9(\()2142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37339
1624 y Ga(x)2194 1612 y F9(\))2221 1624 y Ga(N)g F4(\))21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37340
b(^)-51 b Ga(\033)27 b Gg(is)c(strongly)j(normalising.)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37341
1767 y(\(1\))48 b FL(Cut)p F4(\()875 1755 y FX(h)903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37342
1767 y Ga(a)951 1755 y FX(i)978 1767 y Ga(M)10 b(;)1116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37343
1755 y F9(\()1144 1767 y Ga(x)1196 1755 y F9(\))1223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37344
1767 y Ga(N)g F4(\))31 b(^)-50 b Ga(\033)28 b F4(=)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37345
FL(Cut)p F4(\()1716 1755 y FX(h)1744 1767 y Ga(a)1792
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37346
1755 y FX(i)1820 1767 y Ga(M)15 b F4(^)-50 b Ga(\033)s(;)2013
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37347
1755 y F9(\()2041 1767 y Ga(x)2093 1755 y F9(\))2120
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37348
1767 y Ga(N)15 b F4(^)-50 b Ga(\033)s F4(\))494 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37349
b(De\002nition)g(of)g F6(f)p 3438 1767 V 3456 1767 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37350
3474 1767 V 65 w(g)549 1886 y Gg(\(2\))48 b Ga(M)40 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37351
F4(^)-50 b Ga(\033)s(;)10 b F6(f)-7 b Ga(a)27 b F4(:=)1149
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37352
1874 y F9(\()1176 1886 y Ga(y)1224 1874 y F9(\))1252
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37353
1886 y Ga(S)-5 b F6(g)26 b(2)f Ga(S)5 b(N)1594 1900 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37354
Gc(aux)1739 1886 y Gg(for)23 b(arbitrary)2203 1874 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37355
F9(\()2231 1886 y Ga(y)5 b F4(:)r Ga(B)2382 1874 y F9(\))2409
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37356
1886 y Ga(S)30 b F6(2)25 b FB(J)p F4(\()p Ga(B)5 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37357
FB(K)o Gg(,)702 2004 y Ga(N)40 b F4(^)-50 b Ga(\033)s(;)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37358
b F6(f)-7 b Ga(x)26 b F4(:=)1138 1992 y FX(h)1165 2004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37359
y Ga(b)1204 1992 y FX(i)1232 2004 y Ga(T)s F6(g)g(2)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37360
Ga(S)5 b(N)1578 2018 y Gc(aux)1723 2004 y Gg(for)24 b(arbitrary)2188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37361
1992 y FX(h)2215 2004 y Ga(b)r F4(:)r Ga(B)2357 1992
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37362
y FX(i)2385 2004 y Ga(T)38 b F6(2)25 b FB(J)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37363
Ga(B)5 b F6(i)o FB(K)310 b Gg(by)24 b(induction)549 2123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37364
y(\(3\))48 b F4(\()p Ga(M)15 b F4(^)-50 b Ga(\033)s F4(\))-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37365
b F6(f)e Ga(a)27 b F4(:=)1154 2111 y F9(\()1181 2123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37366
y Ga(y)1229 2111 y F9(\))1257 2123 y Ga(S)-5 b F6(g)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37367
b(2)f Ga(S)5 b(N)1599 2137 y Gc(aux)1744 2123 y Gg(and)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37368
b F4(\()p Ga(N)15 b F4(^)-50 b Ga(\033)s F4(\))-5 b F6(f)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37369
Ga(x)27 b F4(:=)2339 2111 y FX(h)2366 2123 y Ga(b)2405
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37370
2111 y FX(i)2433 2123 y Ga(T)s F6(g)f(2)f Ga(S)5 b(N)2780
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37371
2137 y Gc(aux)3044 2123 y Gg(by)24 b(\(2\))f(and)h(sss)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37372
2241 y(\(4\))702 2229 y FX(h)729 2241 y Ga(a)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37373
Ga(B)880 2229 y FX(i)933 2241 y Ga(M)15 b F4(^)-50 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37374
Ga(\033)28 b F6(2)d FB(J)p F6(h)p Ga(B)5 b F6(i)p FB(K)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37375
b Gg(and)1591 2229 y F9(\()1619 2241 y Ga(x)r F4(:)r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37376
Ga(B)1774 2229 y F9(\))1826 2241 y Ga(N)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37377
b Ga(\033)29 b F6(2)c FB(J)p F4(\()p Ga(B)5 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37378
FB(K)565 b Gg(by)24 b(De\002nition)h(2.3.3)549 2360 y(\(5\))48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37379
b Ga(M)15 b F4(^)-50 b Ga(\033)28 b F6(2)d Ga(S)5 b(N)1100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37380
2374 y Gc(aux)1245 2360 y Gg(and)24 b Ga(N)15 b F4(^)-50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37381
b Ga(\033)29 b F6(2)24 b Ga(S)5 b(N)1782 2374 y Gc(aux)2905
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37382
2360 y Gg(by)24 b(Lemma)e(2.3.13)549 2478 y(\(6\))48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37383
b FL(Cut)p F4(\()875 2466 y FX(h)903 2478 y Ga(a)951
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37384
2466 y FX(i)1003 2478 y Ga(M)15 b F4(^)-50 b Ga(\033)t(;)1197
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37385
2466 y F9(\()1224 2478 y Ga(x)1276 2466 y F9(\))1329
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37386
2478 y Ga(N)15 b F4(^)-50 b Ga(\033)s F4(\))26 b F6(2)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37387
Ga(S)5 b(N)1748 2492 y Gc(aux)2905 2478 y Gg(by)24 b(Lemma)e(2.3.14)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37388
2597 y(\(7\))48 b FL(Cut)p F4(\()875 2585 y FX(h)903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37389
2597 y Ga(a)951 2585 y FX(i)978 2597 y Ga(M)10 b(;)1116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37390
2585 y F9(\()1144 2597 y Ga(x)1196 2585 y F9(\))1223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37391
2597 y Ga(N)g F4(\))31 b(^)-50 b Ga(\033)28 b F6(2)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37392
Ga(S)5 b(N)1667 2611 y Gc(aux)3044 2597 y Gg(by)24 b(\(1\))f(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37393
(\(6\))p 3480 2684 4 62 v 3484 2626 55 4 v 3484 2684
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37394
V 3538 2684 4 62 v Black 321 2922 a Gb(Pr)n(oof)j(of)g(Lemma)e(2.3.20.)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37395
p Black 34 w Gg(The)h(non-tri)n(vial)j(cases)f(are)e(where)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37396
(de\002nitions)h(of)e F6(f)p 3140 2922 28 4 v 3159 2922
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37397
V 3176 2922 V 65 w(g)g Gg(and)h F4([)p 3458 2922 V 3476
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37398
2922 V 3494 2922 V 65 w(])321 3034 y Gg(dif)n(fer;)e(tw)o(o)e(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37399
(them)f(are)h(listed)h(belo)n(w)-6 b(.)p Black 321 3197
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37400
a Gb(Case)24 b Ga(M)35 b F6(\021)25 b FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37401
Ga(y)s(;)15 b(a)p F4(\))p Gb(:)p Black 46 w Gg(Consider)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37402
b(the)f(substitutions:)p Black Black 792 3369 a Ga(M)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37403
b F6(f)-7 b Ga(a)26 b F4(:=)1119 3357 y F9(\()1146 3369
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37404
y Ga(x)1198 3357 y F9(\))1225 3369 y Ga(N)q F6(g)51 b(\021)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37405
FL(Ax)p F4(\()p Ga(y)s(;)15 b(a)p F4(\))-5 b F6(f)e Ga(a)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37406
b F4(:=)2054 3357 y F9(\()2081 3369 y Ga(x)2133 3357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37407
y F9(\))2161 3369 y Ga(N)p F6(g)51 b F4(=)f FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37408
F4(\()2624 3357 y FX(h)2652 3369 y Ga(a)2700 3357 y FX(i)2727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37409
3369 y FL(Ax)p F4(\()p Ga(y)s(;)15 b(a)p F4(\))q Ga(;)3077
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37410
3357 y F9(\()3105 3369 y Ga(x)3157 3357 y F9(\))3184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37411
3369 y Ga(N)10 b F4(\))792 3494 y Ga(M)g F4([)p Ga(a)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37412
b F4(:=)1110 3482 y F9(\()1138 3494 y Ga(x)1190 3482
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37413
y F9(\))1217 3494 y Ga(N)10 b F4(])66 b F6(\021)50 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37414
FL(Ax)p F4(\()p Ga(y)s(;)15 b(a)p F4(\)[)p Ga(a)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37415
F4(:=)2041 3482 y F9(\()2069 3494 y Ga(x)2121 3482 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37416
F9(\))2148 3494 y Ga(N)10 b F4(])81 b(=)50 b Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37417
b F4([)p Ga(x)g F6(7!)g Ga(y)s F4(])549 3661 y Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37418
b(e)31 b(ha)n(v)o(e)i(tw)o(o)g(cases)g(depending)j(on)c(whether)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37419
Ga(N)41 b Gg(freshly)35 b(introduces)g Ga(x)p Gg(.)55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37420
b(If)32 b Ga(N)42 b Gg(freshly)549 3774 y(introduces)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37421
b Ga(x)p Gg(,)e(then)h FL(Cut)p F4(\()1414 3762 y FX(h)1442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37422
3774 y Ga(a)1490 3762 y FX(i)1517 3774 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37423
Ga(y)s(;)15 b(a)p F4(\))q Ga(;)1867 3762 y F9(\()1895
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37424
3774 y Ga(x)1947 3762 y F9(\))1974 3774 y Ga(N)10 b F4(\))2146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37425
3737 y Gc(aux)2126 3774 y F6(\000)-31 b(\000)f(!)34 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37426
Ga(N)10 b F4([)p Ga(x)18 b F6(7!)g Ga(y)s F4(])27 b Gg(by)i(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37427
(logical)j(reduction;)549 3887 y(in)23 b(the)h(other)g(case)773
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37428
4041 y FL(Cut)p F4(\()946 4029 y FX(h)974 4041 y Ga(a)1022
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37429
4029 y FX(i)1050 4041 y FL(Ax)o F4(\()p Ga(y)s(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37430
F4(\))q Ga(;)1399 4029 y F9(\()1427 4041 y Ga(x)1479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37431
4029 y F9(\))1506 4041 y Ga(N)10 b F4(\))1670 4004 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37432
Gc(aux)1650 4041 y F6(\000)-31 b(\000)f(!)25 b Ga(N)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37433
b F6(f)-7 b Ga(x)26 b F4(:=)2160 4029 y FX(h)2188 4041
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37434
y Ga(a)2236 4029 y FX(i)2263 4041 y FL(Ax)p F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37435
b(a)p F4(\))-8 b F6(g)2689 4004 y Gc(aux)2659 4041 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37436
F6(\000)-27 b(\000)f(!)2618 4101 y F5(\(by)17 b(2.3.9\))2837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37437
4003 y FX(\003)2927 4041 y Ga(N)10 b F4([)p Ga(x)g F6(7!)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37438
Ga(y)s F4(])25 b Ga(:)549 4243 y Gg(Thus)e(we)g(ha)n(v)o(e)h(that)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37439
FL(Ax)p F4(\()p Ga(y)s(;)15 b(a)p F4(\)[)p Ga(a)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37440
F4(:=)1761 4231 y F9(\()1789 4243 y Ga(x)1841 4231 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37441
F9(\))1868 4243 y Ga(N)10 b F4(])2022 4206 y Gc(aux)2002
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37442
4243 y F6(\000)-32 b(\000)h(!)2172 4210 y FX(\003)2237
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37443
4243 y FL(Ax)o F4(\()p Ga(y)s(;)15 b(a)p F4(\))-5 b F6(f)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37444
Ga(a)27 b F4(:=)2774 4231 y F9(\()2802 4243 y Ga(x)2854
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37445
4231 y F9(\))2881 4243 y Ga(N)q F6(g)p Gg(.)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37446
4414 a Gb(Case)d Ga(M)35 b F6(\021)25 b FL(Cut)p F4(\()923
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37447
4402 y FX(h)951 4414 y Ga(b)990 4402 y FX(i)1017 4414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37448
y Ga(P)13 b(;)1128 4402 y F9(\()1156 4414 y Ga(y)1204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37449
4402 y F9(\))1231 4414 y FL(Ax)p F4(\()p Ga(y)s(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37450
F4(\))q(\))p Gb(:)p Black 46 w Gg(Consider)25 b(the)f(substitutions:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37451
Black Black 674 4586 a Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)1001
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37452
4574 y F9(\()1028 4586 y Ga(x)1080 4574 y F9(\))1108
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37453
4586 y Ga(N)p F6(g)189 b(\021)f FL(Cut)p F4(\()1847 4574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37454
y FX(h)1875 4586 y Ga(b)1914 4574 y FX(i)1941 4586 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37455
Ga(P)13 b(;)2052 4574 y F9(\()2080 4586 y Ga(y)2128 4574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37456
y F9(\))2155 4586 y FL(Ax)p F4(\()p Ga(y)s(;)i(a)p F4(\))q(\))-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37457
b F6(f)e Ga(a)26 b F4(:=)2729 4574 y F9(\()2756 4586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37458
y Ga(x)2808 4574 y F9(\))2835 4586 y Ga(N)q F6(g)1415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37459
4723 y F4(=)188 b FL(Cut)p F4(\()1847 4711 y FX(h)1875
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37460
4723 y Ga(b)1914 4711 y FX(i)1967 4723 y Ga(P)8 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37461
b Ga(a)26 b F4(:=)2266 4711 y F9(\()2293 4723 y Ga(x)2345
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37462
4711 y F9(\))2373 4723 y Ga(N)p F6(g)q Ga(;)2532 4711
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37463
y F9(\()2559 4723 y Ga(x)2611 4711 y F9(\))2639 4723
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37464
y Ga(N)10 b F4(\))1372 4822 y Gc(aux)1326 4859 y F6(\000)-37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37465
b(\000)-20 b(\000)-38 b(!)1332 4920 y F5(\(by)17 b(IH\))1535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37466
4826 y FX(\003)1674 4859 y FL(Cut)p F4(\()1847 4847 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37467
FX(h)1875 4859 y Ga(b)1914 4847 y FX(i)1967 4859 y Ga(P)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37468
F4([)p Ga(a)25 b F4(:=)2257 4847 y F9(\()2285 4859 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37469
Ga(x)2337 4847 y F9(\))2364 4859 y Ga(N)10 b F4(])p Ga(;)2512
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37470
4847 y F9(\()2540 4859 y Ga(x)2592 4847 y F9(\))2620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37471
4859 y Ga(N)g F4(\))694 5022 y Ga(M)g F4([)p Ga(a)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37472
b F4(:=)1011 5010 y F9(\()1039 5022 y Ga(x)1091 5010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37473
y F9(\))1118 5022 y Ga(N)10 b F4(])189 b F6(\021)f FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37474
F4(\()1847 5010 y FX(h)1875 5022 y Ga(b)1914 5010 y FX(i)1941
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37475
5022 y Ga(P)13 b(;)2052 5010 y F9(\()2080 5022 y Ga(y)2128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37476
5010 y F9(\))2155 5022 y FL(Ax)p F4(\()p Ga(y)s(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37477
F4(\))q(\)[)p Ga(a)26 b F4(:=)2720 5010 y F9(\()2748
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37478
5022 y Ga(x)2800 5010 y F9(\))2827 5022 y Ga(N)10 b F4(])1415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37479
5159 y(=)188 b FL(Cut)p F4(\()1847 5147 y FX(h)1875 5159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37480
y Ga(b)1914 5147 y FX(i)1967 5159 y Ga(P)13 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37481
Ga(a)25 b F4(:=)2257 5147 y F9(\()2285 5159 y Ga(x)2337
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37482
5147 y F9(\))2364 5159 y Ga(N)10 b F4(])p Ga(;)2512 5147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37483
y F9(\()2540 5159 y Ga(y)2588 5147 y F9(\))2641 5159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37484
y FL(Ax)o F4(\()p Ga(y)s(;)15 b(a)p F4(\)[)p Ga(a)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37485
b F4(:=)3170 5147 y F9(\()3198 5159 y Ga(x)3250 5147
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37486
y F9(\))3277 5159 y Ga(N)10 b F4(]\))1415 5295 y(=)188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37487
b FL(Cut)p F4(\()1847 5283 y FX(h)1875 5295 y Ga(b)1914
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37488
5283 y FX(i)1967 5295 y Ga(P)13 b F4([)p Ga(a)25 b F4(:=)2257
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37489
5283 y F9(\()2285 5295 y Ga(x)2337 5283 y F9(\))2364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37490
5295 y Ga(N)10 b F4(])p Ga(;)2512 5283 y F9(\()2540 5295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37491
y Ga(y)2588 5283 y F9(\))2641 5295 y Ga(N)g F4([)p Ga(x)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37492
F6(7!)g Ga(y)s F4(]\))1408 5352 y FV(\()p FC(\003)p FV(\))1415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37493
5436 y F6(\021)188 b FL(Cut)p F4(\()1847 5424 y FX(h)1875
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37494
5436 y Ga(b)1914 5424 y FX(i)1967 5436 y Ga(P)13 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37495
Ga(a)25 b F4(:=)2257 5424 y F9(\()2285 5436 y Ga(x)2337
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37496
5424 y F9(\))2364 5436 y Ga(N)10 b F4(])p Ga(;)2512 5424
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37497
y F9(\()2540 5436 y Ga(x)2592 5424 y F9(\))2620 5436
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37498
y Ga(N)g F4(\))510 5528 y F9(\()p FX(\003)p F9(\))674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37499
5561 y Gg(because)25 b(by)f(the)g(Barendre)o(gt-style)j(naming)d(con)l
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37500
(v)o(ention)j Ga(y)f Gg(cannot)f(be)e(free)h(in)3278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37501
5549 y F9(\()3305 5561 y Ga(x)3357 5549 y F9(\))3385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37502
5561 y Ga(N)10 b Gg(.)p 3480 5699 4 62 v 3484 5641 55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37503
4 v 3484 5699 V 3538 5699 4 62 v Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37504
%%Page: 151 163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37505
TeXDict begin 151 162 bop Black 277 51 a Gb(B.1)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37506
(Chapter)f(2)2639 b(151)p 277 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37507
277 388 a(Pr)n(oof)24 b(of)g(Lemma)e(2.3.21.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37508
35 w Gg(W)-7 b(e)22 b(shall)j(analyse)g(all)f(possible)h(cases)g(of)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37509
2627 351 y Gc(cut)2596 388 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37510
Gg(-reductions.)p Black 277 589 a Gb(Inner)23 b(Reduction:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37511
Black 46 w Gg(Gi)n(v)o(en)i(that)g Ga(M)1553 552 y Gc(cut)1522
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37512
589 y F6(\000)-32 b(\000)h(!)28 b Ga(N)10 b Gg(,)25 b(there)h(is)f(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37513
(proper)j(subterm)f(in)f Ga(M)10 b Gg(,)25 b(say)h Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37514
b Gg(,)24 b(which)504 702 y(reduces)g(to)e Ga(S)956 669
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37515
y FX(0)979 702 y Gg(.)27 b(This)22 b(term)g Ga(S)1461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37516
669 y FX(0)1505 702 y Gg(is)g(a)f(subterm)i(of)f Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37517
b Gg(.)27 b(W)-7 b(e)21 b(kno)n(w)h(by)g(induction)i(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37518
Ga(S)3293 665 y Gc(aux)3273 702 y F6(\000)-31 b(\000)f(!)3443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37519
669 y F9(+)504 815 y Ga(S)565 782 y FX(0)611 815 y Gg(and)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37520
b(by)g(conte)o(xt)h(closure)g(that)f Ga(M)1756 778 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37521
Gc(aux)1736 815 y F6(\000)-31 b(\000)f(!)1906 782 y F9(+)1990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37522
815 y Ga(N)10 b Gg(.)p Black 277 1002 a Gb(Logical)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37523
b(Reduction:)p Black 45 w Gg(This)e(case)g(is)g(ob)o(vious,)h(because)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37524
(both)2362 965 y Gc(cut)2331 1002 y F6(\000)-31 b(\000)f(!)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37525
b Gg(and)2696 965 y Gc(aux)2676 1002 y F6(\000)-31 b(\000)f(!)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37526
b Gg(perform)h(the)g(same)504 1115 y(logical)j(reductions.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37527
Black 277 1301 a Gb(Commuting)c(Reduction:)p Black 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37528
w Gg(Suppose)36 b Ga(M)1796 1264 y Gc(c)1726 1301 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37529
b(\000)f(!)47 b Ga(N)d Gg(with)34 b Ga(M)57 b F6(\021)46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37530
b FL(Cut)p F4(\()2690 1289 y FX(h)2718 1301 y Ga(a)2766
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37531
1289 y FX(i)2793 1301 y Ga(S)5 b(;)2894 1289 y F9(\()2922
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37532
1301 y Ga(x)2974 1289 y F9(\))3001 1301 y Ga(T)13 b F4(\))35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37533
b Gg(and)g Ga(N)56 b F6(\021)504 1414 y Ga(S)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37534
Ga(a)29 b F4(:=)792 1402 y F9(\()819 1414 y Ga(x)871
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37535
1402 y F9(\))899 1414 y Ga(T)12 b F4(])p Gg(,)25 b(then)i(we)d(kno)n(w)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37536
h(that)h Ga(M)1920 1377 y Gc(c)1951 1353 y FC(0)1861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37537
1414 y F6(\000)-31 b(\000)f(!)29 b Ga(S)q F6(f)-7 b Ga(a)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37538
b F4(:=)2356 1402 y F9(\()2383 1414 y Ga(x)2435 1402
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37539
y F9(\))2462 1414 y Ga(T)t F6(g)p Gg(.)34 b(From)24 b(Lemma)g(2.3.20)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37540
(we)504 1527 y(ha)n(v)o(e)g(that)g Ga(S)q F6(f)-7 b Ga(a)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37541
b F4(:=)1155 1515 y F9(\()1182 1527 y Ga(x)1234 1515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37542
y F9(\))1262 1527 y Ga(T)s F6(g)1412 1490 y Gc(aux)1392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37543
1527 y F6(\000)-31 b(\000)f(!)1562 1494 y FX(\003)1631
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37544
1527 y Ga(N)34 b Gg(and)26 b(therefore)i Ga(M)2394 1490
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37545
y Gc(aux)2374 1527 y F6(\000)-32 b(\000)h(!)2544 1494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37546
y F9(+)2632 1527 y Ga(N)10 b Gg(.)33 b(The)25 b(symmetric)i(case)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37547
1640 y(is)d(analogous.)p 3436 1640 4 62 v 3440 1582 55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37548
4 v 3440 1640 V 3494 1640 4 62 v Black 277 1852 a Gb(Pr)n(oof)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37549
(Lemma)e(2.6.4.)p Black 34 w Gg(The)h(follo)n(wing)i(measure)g(reduces)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37550
g(in)f(e)n(v)o(ery)2716 1815 y Gc(x)2651 1852 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37551
b(\000)h(!)p Gg(-reduction.)p Black Black 749 2020 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37552
Fs([)p FL(Ax)p F4(\()p Ga(x;)15 b(a)p F4(\))p Fs(])1161
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37553
1969 y F5(def)1168 2020 y F4(=)39 b(1)498 2164 y Fs([)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37554
FL(Not)673 2178 y Gc(R)731 2164 y F4(\()766 2152 y F9(\()794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37555
2164 y Ga(x)846 2152 y F9(\))873 2164 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37556
b(a)p F4(\))p Fs(])1161 2112 y F5(def)1168 2164 y F4(=)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37557
b Fs([)q Ga(M)10 b Fs(])21 b F4(+)f(1)282 2307 y Fs([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37558
FL(And)469 2321 y Gc(R)526 2307 y F4(\()561 2295 y FX(h)589
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37559
2307 y Ga(a)637 2295 y FX(i)665 2307 y Ga(M)10 b(;)803
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37560
2295 y FX(h)831 2307 y Ga(b)870 2295 y FX(i)897 2307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37561
y Ga(N)g(;)15 b(c)p F4(\))p Fs(])1161 2255 y F5(def)1168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37562
2307 y F4(=)39 b Fs([)q Ga(M)10 b Fs(])21 b F4(+)f Fs([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37563
Ga(N)10 b Fs(])21 b F4(+)f(1)554 2450 y Fs([)q FL(Or)686
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37564
2413 y Gc(i)686 2473 y(R)744 2450 y F4(\()779 2438 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37565
FX(h)807 2450 y Ga(a)855 2438 y FX(i)882 2450 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37566
b(;)15 b(b)p F4(\))p Fs(])1161 2398 y F5(def)1168 2450
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37567
y F4(=)39 b Fs([)q Ga(M)10 b Fs(])21 b F4(+)f(1)402 2593
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37568
y Fs([)q FL(Imp)579 2615 y Gc(R)637 2593 y F4(\()672
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37569
2581 y F9(\()700 2593 y Ga(x)752 2581 y F9(\))p FX(h)807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37570
2593 y Ga(a)855 2581 y FX(i)882 2593 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37571
b(b)p F4(\))p Fs(])1161 2542 y F5(def)1168 2593 y F4(=)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37572
b Fs([)q Ga(M)10 b Fs(])21 b F4(+)f(1)421 2760 y Fs([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37573
FL(Cut)532 2713 y FC(!)593 2760 y F4(\()628 2748 y FX(h)656
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37574
2760 y Ga(a)704 2748 y FX(i)731 2760 y Ga(M)11 b(;)870
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37575
2748 y F9(\()897 2760 y Ga(x)949 2748 y F9(\))977 2760
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37576
y Ga(N)f F4(\))p Fs(])1161 2708 y F5(def)1168 2760 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37577
F4(=)39 b(\()p Fs([)q Ga(M)10 b Fs(])21 b F4(+)f(1\))h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37578
F6(\001)f F4(\(4)p Fs([)r Ga(N)10 b Fs(])20 b F4(+)g(1\))2231
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37579
2020 y Fs([)q FL(Cut)o F4(\()2436 2008 y FX(h)2464 2020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37580
y Ga(a)2512 2008 y FX(i)2540 2020 y Ga(M)10 b(;)2678
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37581
2008 y F9(\()2706 2020 y Ga(x)2758 2008 y F9(\))2785
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37582
2020 y Ga(N)g F4(\))p Fs(])2969 1969 y F5(def)2976 2020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37583
y F4(=)40 b Fs([)p Ga(M)10 b Fs(])21 b F4(+)f Fs([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37584
Ga(N)10 b Fs(])21 b F4(+)f(1)2312 2164 y Fs([)p FL(Not)2487
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37585
2178 y Gc(L)2539 2164 y F4(\()2574 2152 y FX(h)2602 2164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37586
y Ga(a)2650 2152 y FX(i)2678 2164 y Ga(M)10 b(;)15 b(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37587
F4(\))p Fs(])2969 2112 y F5(def)2976 2164 y F4(=)40 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37588
Fs([)p Ga(M)10 b Fs(])21 b F4(+)f(1)2300 2307 y Fs([)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37589
FL(And)2487 2270 y Gc(i)2487 2330 y(L)2540 2307 y F4(\()2575
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37590
2295 y F9(\()2602 2307 y Ga(x)2654 2295 y F9(\))2682
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37591
2307 y Ga(M)10 b(;)15 b(y)s F4(\))p Fs(])2969 2255 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37592
F5(def)2976 2307 y F4(=)40 b Fs([)p Ga(M)10 b Fs(])21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37593
b F4(+)f(1)2131 2450 y Fs([)q FL(Or)2263 2464 y Gc(L)2315
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37594
2450 y F4(\()2350 2438 y F9(\()2378 2450 y Ga(x)2430
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37595
2438 y F9(\))2457 2450 y Ga(M)10 b(;)2595 2438 y F9(\()2623
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37596
2450 y Ga(y)2671 2438 y F9(\))2698 2450 y Ga(N)g(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37597
b(z)t F4(\))p Fs(])2969 2398 y F5(def)2976 2450 y F4(=)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37598
b Fs([)p Ga(M)10 b Fs(])21 b F4(+)f Fs([)p Ga(N)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37599
Fs(])21 b F4(+)f(1)2084 2593 y Fs([)q FL(Imp)2261 2615
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37600
y Gc(L)2313 2593 y F4(\()2348 2581 y FX(h)2376 2593 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37601
Ga(a)2424 2581 y FX(i)2452 2593 y Ga(M)10 b(;)2590 2581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37602
y F9(\()2617 2593 y Ga(x)2669 2581 y F9(\))2697 2593
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37603
y Ga(N)g(;)15 b(y)s F4(\))p Fs(])2969 2542 y F5(def)2976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37604
2593 y F4(=)40 b Fs([)p Ga(M)10 b Fs(])21 b F4(+)f Fs([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37605
Ga(N)10 b Fs(])21 b F4(+)f(1)2229 2760 y Fs([)q FL(Cut)2341
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37606
2713 y FC( )2401 2760 y F4(\()2436 2748 y FX(h)2464 2760
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37607
y Ga(a)2512 2748 y FX(i)2540 2760 y Ga(M)10 b(;)2678
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37608
2748 y F9(\()2706 2760 y Ga(x)2758 2748 y F9(\))2785
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37609
2760 y Ga(N)g F4(\))p Fs(])2969 2708 y F5(def)2976 2760
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37610
y F4(=)40 b(\(4)p Fs([)q Ga(M)10 b Fs(])21 b F4(+)f(1\))h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37611
F6(\001)f F4(\()p Fs([)q Ga(N)10 b Fs(])21 b F4(+)f(1\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37612
3436 2912 V 3440 2854 55 4 v 3440 2912 V 3494 2912 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37613
62 v Black 277 3124 a Gb(Pr)n(oof)k(of)e(Lemma)g(2.6.10.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37614
Black 34 w Gg(The)h(calculation)i(for)e(\(i\))g(is)g(as)f(follo)n(ws,)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37615
(where)f F4(\()p F6(\003)p F4(\))g Gg(is)f(by)h(con\003uence)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37616
3237 y(of)441 3200 y Gc(x)376 3237 y F6(\000)-32 b(\000)h(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37617
Gg(.)483 3465 y F6(j)p FL(Cut)587 3417 y FC( )648 3465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37618
y F4(\()683 3453 y FX(h)711 3465 y Ga(a)759 3453 y FX(i)786
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37619
3465 y Ga(M)10 b(;)924 3453 y F9(\()952 3465 y Ga(x)1004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37620
3453 y F9(\))1032 3465 y Ga(N)g F4(\))p F6(j)1175 3479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37621
y Gc(x)1244 3381 y FV(\()p FC(\003)p FV(\))1251 3465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37622
y F6(\021)31 b(j)p FL(Cut)1458 3417 y FC( )1518 3465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37623
y F4(\()1553 3453 y FX(h)1581 3465 y Ga(a)1629 3453 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37624
FX(i)1672 3465 y F6(j)p Ga(M)10 b F6(j)1820 3479 y Gc(x)1864
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37625
3465 y Ga(;)1904 3453 y F9(\()1932 3465 y Ga(x)1984 3453
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37626
y F9(\))2027 3465 y F6(j)p Ga(N)g F6(j)2160 3479 y Gc(x)2204
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37627
3465 y F4(\))p F6(j)2264 3479 y Gc(x)2334 3381 y FV(\(2)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37628
FZ(:)p FV(6)p FZ(:)p FV(9\))2389 3465 y F6(\021)80 b(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37629
Ga(M)10 b F6(j)2688 3479 y Gc(x)2728 3465 y F6(f)-7 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37630
Ga(a)26 b F4(:=)2961 3453 y F9(\()2988 3465 y Ga(x)3040
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37631
3453 y F9(\))3083 3465 y F6(j)p Ga(N)10 b F6(j)3216 3479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37632
y Gc(x)3251 3465 y F6(g)p 3436 3665 V 3440 3607 55 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37633
v 3440 3665 V 3494 3665 4 62 v Black 277 3878 a Gb(Pr)n(oof)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37634
b(of)g(Lemma)e(2.6.11.)p Black 35 w Gg(W)-7 b(e)22 b(shall)j(analyse)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37635
(same)f(cases)g(of)2413 3841 y Gc(l)q(oc)2372 3878 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37636
F6(\000)-32 b(\000)h(!)p Gg(-reductions.)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37637
4079 a Gb(Inner)23 b(Reduction:)p Black 46 w Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37638
b(e)22 b(gi)n(v)o(e)i(tw)o(o)f(cases.)504 4231 y Ga(M)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37639
b F6(\021)25 b FL(And)878 4245 y Gc(R)936 4231 y F4(\()971
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37640
4219 y FX(h)999 4231 y Ga(a)1047 4219 y FX(i)1074 4231
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37641
y Ga(S)5 b(;)1175 4219 y FX(h)1203 4231 y Ga(x)1255 4219
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37642
y FX(i)1283 4231 y Ga(T)12 b(;)j(b)p F4(\))24 b Gg(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37643
Ga(N)35 b F6(\021)25 b FL(And)1998 4245 y Gc(R)2056 4231
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37644
y F4(\()2091 4219 y FX(h)2119 4231 y Ga(a)2167 4219 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37645
FX(i)2194 4231 y Ga(S)2255 4198 y FX(0)2279 4231 y Ga(;)2319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37646
4219 y FX(h)2347 4231 y Ga(x)2399 4219 y FX(i)2426 4231
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37647
y Ga(T)13 b(;)i(b)p F4(\))504 4355 y Gg(\(1\))101 b Ga(S)838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37648
4318 y Gc(l)q(oc)796 4355 y F6(\000)-31 b(\000)g(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37649
b Ga(S)1053 4322 y FX(0)2974 4355 y Gg(by)f(assumption)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37650
4479 y(\(2\))101 b F6(j)p Ga(M)10 b F6(j)858 4493 y Gc(x)928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37651
4479 y F6(\021)25 b FL(And)1178 4493 y Gc(R)1236 4479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37652
y F4(\()1271 4467 y FX(h)1299 4479 y Ga(a)1347 4467 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37653
FX(i)1374 4479 y F6(j)p Ga(S)5 b F6(j)1485 4493 y Gc(x)1530
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37654
4479 y Ga(;)1570 4467 y FX(h)1597 4479 y Ga(x)1649 4467
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37655
y FX(i)1677 4479 y F6(j)p Ga(T)13 b F6(j)1793 4493 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37656
Gc(x)1837 4479 y Ga(;)i(b)p F4(\))955 b Gg(by)24 b(Lemma)e(2.6.7)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37657
4603 y(\(3\))101 b FL(And)865 4617 y Gc(R)922 4603 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37658
F4(\()957 4591 y FX(h)985 4603 y Ga(a)1033 4591 y FX(i)1061
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37659
4603 y F6(j)p Ga(S)5 b F6(j)1172 4617 y Gc(x)1216 4603
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37660
y Ga(;)1256 4591 y FX(h)1284 4603 y Ga(x)1336 4591 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37661
FX(i)1363 4603 y F6(j)p Ga(T)13 b F6(j)1479 4617 y Gc(x)1523
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37662
4603 y Ga(;)i(b)p F4(\))1684 4566 y Gc(aux)1663 4603
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37663
y F6(\000)-31 b(\000)g(!)1834 4570 y FX(\003)1898 4603
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37664
y FL(And)2053 4617 y Gc(R)2111 4603 y F4(\()2146 4591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37665
y FX(h)2173 4603 y Ga(a)2221 4591 y FX(i)2249 4603 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37666
F6(j)p Ga(S)2335 4570 y FX(0)2358 4603 y F6(j)2383 4617
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37667
y Gc(x)2428 4603 y Ga(;)2468 4591 y FX(h)2495 4603 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37668
Ga(x)2547 4591 y FX(i)2575 4603 y F6(j)p Ga(T)13 b F6(j)2691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37669
4617 y Gc(x)2735 4603 y Ga(;)i(b)p F4(\))196 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37670
b(induction)504 4727 y(\(4\))101 b FL(And)865 4741 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37671
Gc(R)922 4727 y F4(\()957 4715 y FX(h)985 4727 y Ga(a)1033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37672
4715 y FX(i)1061 4727 y F6(j)p Ga(S)1147 4694 y FX(0)1170
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37673
4727 y F6(j)1195 4741 y Gc(x)1239 4727 y Ga(;)1279 4715
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37674
y FX(h)1307 4727 y Ga(x)1359 4715 y FX(i)1386 4727 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37675
F6(j)p Ga(T)13 b F6(j)1502 4741 y Gc(x)1547 4727 y Ga(;)i(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37676
F4(\))26 b F6(\021)f(j)p Ga(N)10 b F6(j)1916 4741 y Gc(x)2906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37677
4727 y Gg(by)24 b(Lemma)e(2.6.7)504 4897 y Ga(M)36 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37678
F6(\021)25 b FL(Cut)803 4849 y FC( )863 4897 y F4(\()898
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37679
4885 y FX(h)926 4897 y Ga(a)974 4885 y FX(i)1002 4897
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37680
y Ga(S)5 b(;)1103 4885 y F9(\()1131 4897 y Ga(x)1183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37681
4885 y F9(\))1210 4897 y Ga(T)13 b F4(\))23 b Gg(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37682
Ga(N)35 b F6(\021)25 b FL(Cut)1771 4849 y FC( )1832 4897
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37683
y F4(\()1867 4885 y FX(h)1894 4897 y Ga(a)1942 4885 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37684
FX(i)1970 4897 y Ga(S)2031 4864 y FX(0)2054 4897 y Ga(;)2094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37685
4885 y F9(\()2122 4897 y Ga(x)2174 4885 y F9(\))2201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37686
4897 y Ga(T)13 b F4(\))504 5021 y Gg(\(1\))101 b Ga(S)838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37687
4984 y Gc(l)q(oc)796 5021 y F6(\000)-31 b(\000)g(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37688
b Ga(S)1053 4988 y FX(0)2974 5021 y Gg(by)f(assumption)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37689
5145 y(\(2\))101 b F6(j)p Ga(M)10 b F6(j)858 5159 y Gc(x)928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37690
5145 y F6(\021)25 b(j)p Ga(S)5 b F6(j)1135 5159 y Gc(x)1174
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37691
5145 y F6(f)-7 b Ga(a)26 b F4(:=)1407 5133 y F9(\()1435
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37692
5145 y Ga(x)1487 5133 y F9(\))1529 5145 y F6(j)p Ga(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37693
b F6(j)1645 5159 y Gc(x)1680 5145 y F6(g)1136 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37694
b(Lemma)g(2.6.10)504 5269 y(\(3\))101 b F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37695
b F6(j)821 5283 y Gc(x)861 5269 y F6(f)-7 b Ga(a)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37696
F4(:=)1094 5257 y F9(\()1121 5269 y Ga(x)1173 5257 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37697
F9(\))1216 5269 y F6(j)p Ga(T)13 b F6(j)1332 5283 y Gc(x)1366
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37698
5269 y F6(g)1457 5232 y Gc(aux)1437 5269 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37699
b(\000)f(!)1607 5236 y FX(\003)1672 5269 y F6(j)p Ga(S)1758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37700
5236 y FX(0)1781 5269 y F6(j)1806 5283 y Gc(x)1846 5269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37701
y F6(f)-7 b Ga(a)25 b F4(:=)2079 5257 y F9(\()2106 5269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37702
y Ga(x)2158 5257 y F9(\))2201 5269 y F6(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37703
F6(j)2317 5283 y Gc(x)2351 5269 y F6(g)56 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37704
b(induction)k(and)d(Lem)e(2.3.11)504 5394 y(\(4\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37705
b F6(j)p Ga(S)796 5361 y FX(0)820 5394 y F6(j)845 5408
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37706
y Gc(x)884 5394 y F6(f)-7 b Ga(a)26 b F4(:=)1117 5382
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37707
y F9(\()1144 5394 y Ga(x)1196 5382 y F9(\))1239 5394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37708
y F6(j)p Ga(T)13 b F6(j)1355 5408 y Gc(x)1390 5394 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37709
F6(g)25 b(\021)g(j)p Ga(N)10 b F6(j)1689 5408 y Gc(x)2861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37710
5394 y Gg(by)23 b(Lemma)g(2.6.10)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37711
%%Page: 152 164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37712
TeXDict begin 152 163 bop Black -144 51 a Gb(152)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37713
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37714
4 v Black Black 321 388 a(Commuting)f(Reduction:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37715
45 w Gg(W)-7 b(e)23 b(gi)n(v)o(e)h(one)g(case.)549 562
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37716
y Ga(M)35 b F6(\021)25 b FL(Cut)p F4(\()941 550 y FX(h)969
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37717
562 y Ga(a)1017 550 y FX(i)1044 562 y Ga(S)5 b(;)1145
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37718
550 y F9(\()1173 562 y Ga(x)1225 550 y F9(\))1252 562
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37719
y Ga(T)13 b F4(\))23 b Gg(and)h Ga(N)35 b F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37720
b FL(Cut)1813 515 y FC( )1874 562 y F4(\()1909 550 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37721
FX(h)1937 562 y Ga(a)1985 550 y FX(i)2012 562 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37722
b(;)2113 550 y F9(\()2141 562 y Ga(x)2193 550 y F9(\))2221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37723
562 y Ga(T)12 b F4(\))549 686 y Gg(\(1\))100 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37724
Ga(M)10 b F6(j)902 700 y Gc(x)972 686 y F6(\021)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37725
FL(Cut)p F4(\()1241 674 y FX(h)1269 686 y Ga(a)1317 674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37726
y FX(i)1344 686 y F6(j)p Ga(S)5 b F6(j)1455 700 y Gc(x)1499
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37727
686 y Ga(;)1539 674 y F9(\()1567 686 y Ga(x)1619 674
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37728
y F9(\))1647 686 y F6(j)p Ga(T)13 b F6(j)1763 700 y Gc(x)1807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37729
686 y F4(\))1108 b Gg(by)24 b(Lemma)f(2.6.7)549 810 y(\(2\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37730
b FL(Cut)p F4(\()927 798 y FX(h)955 810 y Ga(a)1003 798
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37731
y FX(i)1030 810 y F6(j)p Ga(S)5 b F6(j)1141 824 y Gc(x)1186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37732
810 y Ga(;)1226 798 y F9(\()1254 810 y Ga(x)1306 798
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37733
y F9(\))1333 810 y F6(j)p Ga(T)13 b F6(j)1449 824 y Gc(x)1493
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37734
810 y F4(\))1574 773 y Gc(aux)1554 810 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37735
b(\000)f(!)1724 777 y F9(0)p Gc(=)p F9(1)1859 810 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37736
Ga(S)5 b F6(j)1970 824 y Gc(x)2010 810 y F6(f)-7 b Ga(a)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37737
b F4(:=)2243 798 y F9(\()2270 810 y Ga(x)2322 798 y F9(\))2365
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37738
810 y F6(j)p Ga(T)13 b F6(j)2481 824 y Gc(x)2515 810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37739
y F6(g)2219 935 y Gg(`0'-case:)31 b(if)24 b F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37740
b F6(j)2755 949 y Gc(x)2822 935 y Gg(freshly)25 b(introduces)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37741
Ga(a)549 1059 y Gg(\(3\))100 b F6(j)p Ga(S)5 b F6(j)865
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37742
1073 y Gc(x)905 1059 y F6(f)-7 b Ga(a)26 b F4(:=)1138
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37743
1047 y F9(\()1165 1059 y Ga(x)1217 1047 y F9(\))1260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37744
1059 y F6(j)p Ga(T)13 b F6(j)1376 1073 y Gc(x)1410 1059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37745
y F6(g)26 b(\021)f(j)p Ga(N)10 b F6(j)1710 1073 y Gc(x)2905
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37746
1059 y Gg(by)24 b(Lemma)e(2.6.10)p Black 321 1290 a Gb(Labelled)h(Cut)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37747
(Reduction:)p Black 45 w Gg(T)m(ri)n(vial,)h(because)h(if)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37748
Ga(M)2161 1253 y Gc(x)2096 1290 y F6(\000)-32 b(\000)h(!)-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37749
b Ga(N)10 b Gg(,)22 b(we)h(ha)n(v)o(e)h F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37750
b F6(j)2857 1304 y Gc(x)2927 1290 y F6(\021)25 b(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37751
Ga(N)10 b F6(j)3156 1304 y Gc(x)3200 1290 y Gg(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37752
321 1521 a Gb(Logical)25 b(Reduction:)p Black 46 w Gg(Routine)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37753
(calculation)j(using)e(Lemma)d(2.6.7.)p Black 321 1751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37754
a Gb(Garbage)i(Reduction:)p Black 46 w Gg(T)-7 b(ak)o(e)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37755
b(for)h(e)o(xample)h Ga(M)35 b F6(\021)25 b FL(Cut)2107
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37756
1704 y FC( )2168 1751 y F4(\()2203 1739 y FX(h)2231 1751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37757
y Ga(a)2279 1739 y FX(i)2306 1751 y Ga(S)5 b(;)2407 1739
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37758
y F9(\()2435 1751 y Ga(x)2487 1739 y F9(\))2514 1751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37759
y Ga(T)13 b F4(\))22 b Gg(with)h Ga(a)e Gg(not)i(free)h(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37760
Ga(S)5 b Gg(,)22 b(then)549 1864 y(by)h(Lemma)g(2.6.10)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37761
F6(j)p Ga(M)10 b F6(j)1361 1878 y Gc(x)1430 1864 y F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37762
b(j)p Ga(N)10 b F6(j)1659 1878 y Gc(x)1704 1864 y Gg(.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37763
3480 1864 4 62 v 3484 1806 55 4 v 3484 1864 V 3538 1864
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37764
4 62 v Black 321 2077 a Gb(Pr)n(oof)33 b(of)f(Lemma)g(2.6.16.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37765
Black 34 w Gg(W)-7 b(e)31 b(shall)i(analyse)h(in)f(detail)g(one)g(case)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37766
g(where)f(an)3068 2040 y Gc(x)3002 2077 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37767
b(\000)g(!)p Gg(-reduction)321 2190 y(occurs)25 b(on)f(the)g(top-le)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37768
(v)o(el.)30 b(Suppose)p Black Black 1051 2431 a Ga(M)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37769
b F6(\021)10 b FL(Cut)1318 2383 y FC( )1379 2431 y F4(\()1414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37770
2419 y FX(h)1442 2431 y Ga(a)1490 2419 y FX(i)1518 2431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37771
y FL(Not)1660 2445 y Gc(R)1718 2431 y F4(\()1753 2419
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37772
y F9(\()1781 2431 y Ga(x)1833 2419 y F9(\))1860 2431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37773
y Ga(S)5 b(;)15 b(a)p F4(\))q Ga(;)2085 2419 y F9(\()2113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37774
2431 y Ga(y)2161 2419 y F9(\))2188 2431 y Ga(T)e F4(\))48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37775
b Gg(and)1051 2567 y Ga(N)20 b F6(\021)10 b FL(Cut)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37776
F4(\()1397 2555 y FX(h)1425 2567 y Ga(a)1473 2555 y FX(i)1501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37777
2567 y FL(Not)1643 2581 y Gc(R)1701 2567 y F4(\()1736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37778
2555 y F9(\()1764 2567 y Ga(x)1816 2555 y F9(\))1843
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37779
2567 y FL(Cut)1922 2520 y FC( )1983 2567 y F4(\()2018
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37780
2555 y FX(h)2046 2567 y Ga(a)2094 2555 y FX(i)2121 2567
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37781
y Ga(S)5 b(;)2222 2555 y F9(\()2250 2567 y Ga(y)2298
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37782
2555 y F9(\))2325 2567 y Ga(T)13 b F4(\))p Ga(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37783
F4(\))q Ga(;)2590 2555 y F9(\()2618 2567 y Ga(y)2666
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37784
2555 y F9(\))2693 2567 y Ga(T)e F4(\))p Gg(.)321 2806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37785
y(Hence)24 b(we)f(ha)n(v)o(e)h Ga(M)1085 2769 y Gc(x)1019
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37786
2806 y F6(\000)-31 b(\000)g(!)25 b Ga(N)10 b Gg(.)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37787
b(The)23 b(subterms)i(of)e Ga(N)33 b Gg(are:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37788
Black 944 3064 a(\(1\))101 b(all)23 b(subterms)i(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37789
Ga(S)j Gg(and)d Ga(T)13 b Gg(,)944 3177 y(\(2\))101 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37790
FL(Cut)1228 3130 y FC( )1289 3177 y F4(\()1324 3165 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37791
FX(h)1352 3177 y Ga(a)1400 3165 y FX(i)1428 3177 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37792
b(;)1529 3165 y F9(\()1556 3177 y Ga(y)1604 3165 y F9(\))1632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37793
3177 y Ga(T)12 b F4(\))p Gg(,)944 3290 y(\(3\))101 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37794
FL(Not)1292 3304 y Gc(R)1350 3290 y F4(\()1385 3278 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37795
F9(\()1413 3290 y Ga(x)1465 3278 y F9(\))1492 3290 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37796
FL(Cut)1571 3243 y FC( )1632 3290 y F4(\()1667 3278 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37797
FX(h)1695 3290 y Ga(a)1743 3278 y FX(i)1770 3290 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37798
b(;)1871 3278 y F9(\()1899 3290 y Ga(y)1947 3278 y F9(\))1974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37799
3290 y Ga(T)13 b F4(\))p Ga(;)i(a)p F4(\))24 b Gg(and)944
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37800
3403 y(\(4\))101 b Ga(N)35 b F6(\021)25 b FL(Cut)p F4(\()1527
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37801
3391 y FX(h)1554 3403 y Ga(a)1602 3391 y FX(i)1630 3403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37802
y FL(Not)1773 3417 y Gc(R)1830 3403 y F4(\()1865 3391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37803
y F9(\()1893 3403 y Ga(x)1945 3391 y F9(\))1972 3403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37804
y FL(Cut)2051 3355 y FC( )2112 3403 y F4(\()2147 3391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37805
y FX(h)2175 3403 y Ga(a)2223 3391 y FX(i)2250 3403 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37806
Ga(S)5 b(;)2351 3391 y F9(\()2379 3403 y Ga(y)2427 3391
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37807
y F9(\))2455 3403 y Ga(T)12 b F4(\))q Ga(;)j(a)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37808
Ga(;)2719 3391 y F9(\()2747 3403 y Ga(y)2795 3391 y F9(\))2823
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37809
3403 y Ga(T)d F4(\))321 3677 y Gg(W)-7 b(e)33 b(ha)n(v)o(e)i(to)e(sho)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37810
(w)h(that)g(their)h Ga(x)p Gg(-normal)g(form)e(is)h(an)g(element)h(in)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37811
Ga(S)5 b(N)2723 3691 y Gc(aux)2846 3677 y Gg(.)58 b(Case)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37812
b(\(1\))g(follo)n(ws)321 3790 y(by)28 b(assumption:)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37813
b(the)27 b Ga(x)p Gg(-normal)i(form)e(of)h(e)n(v)o(ery)g(subterm)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37814
Ga(M)37 b Gg(is)27 b(an)g(element)i(in)e Ga(S)5 b(N)3242
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37815
3804 y Gc(aux)3365 3790 y Gg(,)27 b(and)321 3903 y(therefore)j(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37816
Ga(x)p Gg(-normal)g(form)g(of)f(e)n(v)o(ery)h(subterm)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37817
Ga(S)j Gg(and)d Ga(T)40 b Gg(must)27 b(be)h(in)f Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37818
b(N)3018 3917 y Gc(aux)3140 3903 y Gg(,)28 b(too.)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37819
b(Case)321 4016 y(\(4\))20 b(follo)n(ws)h(by)f(Lemma)f(2.6.11,)i(which)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37820
f(says)h(that)f F6(j)p Ga(M)10 b F6(j)2133 4030 y Gc(x)2223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37821
3978 y(aux)2203 4016 y F6(\000)-31 b(\000)f(!)2373 3983
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37822
y FX(\003)2438 4016 y F6(j)p Ga(N)10 b F6(j)2571 4030
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37823
y Gc(x)2615 4016 y Gg(,)20 b(and)g(thus)h F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37824
b F6(j)3112 4030 y Gc(x)3181 4016 y F6(2)25 b Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37825
b(N)3401 4030 y Gc(aux)3524 4016 y Gg(.)321 4128 y(By)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37826
b(Lemma)g(2.6.7)g(we)g(ha)n(v)o(e)h(the)g(identities)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37827
Black Black 1140 4370 a F6(j)p FL(Cut)p F4(\()1338 4358
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37828
y FX(h)1366 4370 y Ga(a)1414 4358 y FX(i)1441 4370 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37829
FL(Not)1584 4384 y Gc(R)1641 4370 y F4(\()1676 4358 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37830
F9(\()1704 4370 y Ga(x)1756 4358 y F9(\))1784 4370 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37831
FL(Cut)1862 4322 y FC( )1923 4370 y F4(\()1958 4358 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37832
FX(h)1986 4370 y Ga(a)2034 4358 y FX(i)2062 4370 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37833
b(;)2163 4358 y F9(\()2190 4370 y Ga(y)2238 4358 y F9(\))2266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37834
4370 y Ga(T)12 b F4(\))q Ga(;)j(a)p F4(\))q Ga(;)2531
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37835
4358 y F9(\()2558 4370 y Ga(y)2606 4358 y F9(\))2634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37836
4370 y Ga(T)e F4(\))p F6(j)2760 4384 y Gc(x)969 4483
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37837
y F6(\021)100 b FL(Cut)o F4(\()1312 4471 y FX(h)1340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37838
4483 y Ga(a)1388 4471 y FX(i)1416 4483 y F6(j)p FL(Not)1584
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37839
4497 y Gc(R)1641 4483 y F4(\()1676 4471 y F9(\()1704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37840
4483 y Ga(x)1756 4471 y F9(\))1784 4483 y FL(Cut)1862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37841
4435 y FC( )1923 4483 y F4(\()1958 4471 y FX(h)1986 4483
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37842
y Ga(a)2034 4471 y FX(i)2062 4483 y Ga(S)5 b(;)2163 4471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37843
y F9(\()2190 4483 y Ga(y)2238 4471 y F9(\))2266 4483
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37844
y Ga(T)12 b F4(\))q Ga(;)j(a)p F4(\))p F6(j)2515 4497
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37845
y Gc(x)2560 4483 y Ga(;)2600 4471 y F9(\()2628 4483 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37846
Ga(y)2676 4471 y F9(\))2703 4483 y F6(j)p Ga(T)e F6(j)2819
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37847
4497 y Gc(x)2863 4483 y F4(\))969 4595 y F6(\021)100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37848
b FL(Cut)o F4(\()1312 4583 y FX(h)1340 4595 y Ga(a)1388
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37849
4583 y FX(i)1416 4595 y FL(Not)1558 4609 y Gc(R)1616
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37850
4595 y F4(\()1651 4583 y F9(\()1679 4595 y Ga(x)1731
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37851
4583 y F9(\))1758 4595 y F6(j)p FL(Cut)1862 4548 y FC( )1923
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37852
4595 y F4(\()1958 4583 y FX(h)1986 4595 y Ga(a)2034 4583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37853
y FX(i)2062 4595 y Ga(S)5 b(;)2163 4583 y F9(\()2190
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37854
4595 y Ga(y)2238 4583 y F9(\))2266 4595 y Ga(T)12 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37855
F6(j)2391 4609 y Gc(x)2436 4595 y Ga(;)j(a)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37856
Ga(;)2600 4583 y F9(\()2628 4595 y Ga(y)2676 4583 y F9(\))2703
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37857
4595 y F6(j)p Ga(T)e F6(j)2819 4609 y Gc(x)2863 4595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37858
y F4(\))321 4834 y Gg(and)24 b(therefore)i(the)e Ga(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37859
Gg(-normal)g(form)g(of)f(\(2\))h(and)g(\(3\))g(are)g(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37860
Ga(S)5 b(N)2393 4848 y Gc(aux)2515 4834 y Gg(;)23 b(so)h(we)e(are)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37861
(done.)p 3480 4834 V 3484 4776 55 4 v 3484 4834 V 3538
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37862
4834 4 62 v Black 321 5046 a Gb(Pr)n(oof)34 b(of)f(Lemma)f(2.6.20.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37863
Black 34 w Gg(F)o(or)g(e)n(v)o(ery)h(reduction)j(we)c(ha)n(v)o(e)h(to)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37864
(do)g(check)h(whether)g(the)f(corre-)321 5159 y(sponding)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37865
b(terms)e(are)g(ordered)h(decreasingly)i(according)g(to)c(de\002nition)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37866
i(of)f Ga(>)2843 5126 y Gc(r)r(po)2950 5159 y Gg(.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37867
b(Since)25 b(there)g(are)321 5272 y(man)o(y)19 b(cases,)i(we)e(shall)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37868
(present)h(only)f(a)f(fe)n(w)f(representati)n(v)o(e)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37869
b(of)c(them.)27 b(W)-7 b(e)19 b(write)g(rpo)h(as)f(shorthand)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37870
5385 y(for)24 b(De\002nition)g(2.6.12.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37871
eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37872
%%Page: 153 165
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37873
TeXDict begin 153 164 bop Black 277 51 a Gb(B.1)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37874
(Chapter)f(2)2639 b(153)p 277 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37875
277 388 a(Inner)23 b(Reduction:)p Black 46 w Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37876
b(e)22 b(gi)n(v)o(e)i(one)g(case.)504 533 y Ga(M)36 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37877
F6(\021)25 b FL(Cut)p F4(\()897 521 y FX(h)924 533 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37878
Ga(a)972 521 y FX(i)1000 533 y Ga(S)5 b(;)1101 521 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37879
F9(\()1129 533 y Ga(x)1181 521 y F9(\))1208 533 y Ga(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37880
b F4(\))1376 496 y Gc(l)q(oc)1334 533 y F6(\000)-31 b(\000)g(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37881
b FL(Cut)p F4(\()1703 521 y FX(h)1731 533 y Ga(a)1779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37882
521 y FX(i)1806 533 y Ga(S)1867 500 y FX(0)1890 533 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37883
Ga(;)1930 521 y F9(\()1958 533 y Ga(x)2010 521 y F9(\))2038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37884
533 y Ga(T)12 b F4(\))26 b F6(\021)f Ga(N)504 646 y Gg(\(1\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37885
b Ga(S)838 608 y Gc(l)q(oc)796 646 y F6(\000)-31 b(\000)g(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37886
b Ga(S)1053 613 y FX(0)1099 646 y Gg(and)f Ga(S)p 1253
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37887
661 61 4 v 30 w(>)1410 613 y Gc(r)r(po)1543 646 y Ga(S)1604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37888
613 y FX(0)p 1543 661 85 4 v 2454 646 a Gg(by)g(assumption)i(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37889
(induction)504 758 y(\(2\))101 b Ga(M)p 710 773 99 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37890
v 35 w F4(=)25 b Ga(S)p 929 773 61 4 v 26 w F6(\001)1036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37891
772 y Gc(m)1123 758 y Ga(T)p 1123 773 66 4 v 60 w Gg(and)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37892
b Ga(N)p 1416 773 83 4 v 35 w F4(=)25 b Ga(S)1681 725
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37893
y FX(0)p 1620 773 85 4 v 1724 758 a F6(\001)1749 772
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37894
y Gc(n)1817 758 y Ga(T)p 1817 773 66 4 v 900 w Gg(by)e(De\002nition)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37895
(2.6.17)504 871 y(\(3\))101 b Ga(m)25 b F6(\025)g Ga(n)1809
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37896
b Gg(by)24 b(Lemma)e(2.6.19\(i\))504 984 y(\(4\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37897
b Ga(S)p 710 999 61 4 v 25 w F6(\001)816 998 y Gc(m)903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37898
984 y Ga(T)p 903 999 66 4 v 38 w(>)1065 951 y Gc(r)r(po)1198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37899
984 y Ga(S)1259 951 y FX(0)p 1198 999 85 4 v 1330 984
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37900
a Gg(and)49 b Ga(S)p 1509 999 61 4 v 25 w F6(\001)1615
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37901
998 y Gc(m)1703 984 y Ga(T)p 1703 999 66 4 v 38 w(>)1865
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37902
951 y Gc(r)r(po)1997 984 y Ga(T)p 1997 999 V 849 w Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37903
b(\(1\))h(and)g(rpo\(i\))504 1097 y(\(5\))101 b F6(f)-24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37904
b(j)p Ga(S)p 756 1112 61 4 v 6 w(;)15 b(T)p 858 1112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37905
66 4 v 13 w F6(j)-24 b(g)27 b Ga(>)1068 1053 y Gc(r)r(po)1068
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37906
1127 y(mul)q(t)1247 1097 y F6(f)-24 b(j)p Ga(S)1354 1064
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37907
y FX(0)p 1293 1112 85 4 v 1379 1097 a Ga(;)15 b(T)p 1419
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37908
1112 66 4 v 13 w F6(j)-24 b(g)1368 b Gg(by)23 b(\(1\))h(and)g(rpo\(i\))
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37909
504 1210 y(\(6\))101 b Ga(M)p 710 1225 99 4 v 35 w(>)904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37910
1177 y Gc(r)r(po)1037 1210 y Ga(N)p 1037 1225 83 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37911
1180 w Gg(if)23 b Ga(m)i F4(=)g Ga(n)p Gg(,)d(then)i(by)g(\(5\))g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37912
(rpo\(iii\))2315 1323 y(if)f Ga(m)i(>)g(n)p Gg(,)d(then)j(by)e(\(4\))h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37913
(and)g(rpo\(ii\))p Black 277 1503 a Gb(Labelled)f(Cut)f(Reduction:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37914
Black 46 w Gg(W)-7 b(e)23 b(gi)n(v)o(e)g(\002)n(v)o(e)g(typical)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37915
(cases.)504 1655 y Ga(M)36 b F6(\021)22 b FL(Cut)800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37916
1608 y FC( )861 1655 y F4(\()896 1643 y FX(h)924 1655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37917
y Ga(c)963 1643 y FX(i)990 1655 y FL(And)1145 1669 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37918
Gc(R)1203 1655 y F4(\()1238 1643 y FX(h)1266 1655 y Ga(a)1314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37919
1643 y FX(i)1341 1655 y Ga(S)5 b(;)1442 1643 y FX(h)1470
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37920
1655 y Ga(b)1509 1643 y FX(i)1536 1655 y Ga(T)13 b(;)i(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37921
F4(\))q Ga(;)1757 1643 y F9(\()1785 1655 y Ga(x)1837
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37922
1643 y F9(\))1864 1655 y Ga(U)10 b F4(\))782 1742 y Gc(x)717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37923
1779 y F6(\000)-31 b(\000)f(!)26 b FL(Cut)o F4(\()1085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37924
1767 y FX(h)1113 1779 y Ga(c)1152 1767 y FX(i)1180 1779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37925
y FL(And)1335 1793 y Gc(R)1392 1779 y F4(\()1427 1767
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37926
y FX(h)1455 1779 y Ga(a)1503 1767 y FX(i)1531 1779 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37927
FL(Cut)1609 1732 y FC( )1670 1779 y F4(\()1705 1767 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37928
FX(h)1733 1779 y Ga(c)1772 1767 y FX(i)1800 1779 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37929
b(;)1901 1767 y F9(\()1929 1779 y Ga(x)1981 1767 y F9(\))2008
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37930
1779 y Ga(U)10 b F4(\))p Ga(;)2155 1767 y FX(h)2183 1779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37931
y Ga(b)2222 1767 y FX(i)2250 1779 y FL(Cut)2329 1732
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37932
y FC( )2389 1779 y F4(\()2424 1767 y FX(h)2452 1779 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37933
Ga(c)2491 1767 y FX(i)2519 1779 y Ga(T)j(;)2625 1767
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37934
y F9(\()2653 1779 y Ga(x)2705 1767 y F9(\))2732 1779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37935
y Ga(U)d F4(\))p Ga(;)15 b(c)p F4(\))q Ga(;)2994 1767
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37936
y F9(\()3022 1779 y Ga(x)3074 1767 y F9(\))3102 1779
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37937
y Ga(U)10 b F4(\))25 b F6(\021)g Ga(N)504 1903 y Gg(\(1\))146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37938
b Ga(M)p 755 1918 99 4 v 36 w F4(=)25 b FB(L)p Ga(S)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37939
1010 1918 61 4 v 5 w(;)15 b(T)p 1111 1918 66 4 v 13 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37940
FB(M)g F6(h)p Ga(U)p 1262 1918 72 4 v 11 w F6(i)1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37941
1917 y Gc(m)1485 1903 y Gg(and)49 b Ga(N)p 1664 1918
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37942
83 4 v 36 w F4(=)24 b FB(L)p Ga(S)p 1903 1918 61 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37943
6 w F6(h)p Ga(U)p 2000 1918 72 4 v 10 w F6(i)2107 1917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37944
y Gc(r)2145 1903 y Ga(;)15 b(T)p 2185 1918 66 4 v 14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37945
w F6(h)p Ga(U)p 2287 1918 72 4 v 10 w F6(i)2394 1917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37946
y Gc(s)2431 1903 y FB(M)21 b F6(\001)2512 1917 y Gc(t)2562
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37947
1903 y Ga(U)p 2562 1918 V 146 w Gg(by)i(De\002nition)i(2.6.17)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37948
2027 y(\(2\))146 b Ga(m)26 b F6(\025)f Ga(t;)15 b(r)m(;)g(s)1551
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37949
b Gg(by)23 b(Lemma)g(2.6.19\(i,ii\))504 2152 y(\(3\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37950
758 2152 28 4 v 775 2152 V 793 2152 V 211 w F6(h)p 857
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37951
2152 V 875 2152 V 893 2152 V 65 w(i)955 2166 y Gc(m)1047
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37952
2152 y F6(\035)p 1165 2152 V 1183 2152 V 1201 2152 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37953
110 w(\001)1273 2166 y Gc(t)p 1325 2152 V 1343 2152 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37954
1360 2152 V 2487 2152 a Gg(by)h(\(2\))f(and)h(De\002nition)h(2.6.14)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37955
2276 y(\(4\))146 b FB(L)p Ga(S)p 790 2291 61 4 v 6 w(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37956
b(T)p 892 2291 66 4 v 13 w FB(M)p F6(h)p Ga(U)p 1028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37957
2291 72 4 v 11 w F6(i)1136 2290 y Gc(m)1228 2276 y Ga(>)1299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37958
2243 y Gc(r)r(po)1431 2276 y Ga(S)p 1431 2291 61 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37959
53 w Gg(and)50 b FB(L)p Ga(S)p 1755 2291 V 5 w(;)15 b(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37960
1856 2291 66 4 v 13 w FB(M)p F6(h)p Ga(U)p 1992 2291
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37961
72 4 v 11 w F6(i)2100 2290 y Gc(m)2192 2276 y Ga(>)2263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37962
2243 y Gc(r)r(po)2396 2276 y Ga(U)p 2396 2291 V 724 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37963
Gg(by)23 b(rpo\(i\))504 2400 y(\(5\))146 b F6(f)-24 b(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37964
FB(L)p Ga(S)p 836 2415 61 4 v 7 w(;)15 b(T)p 939 2415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37965
66 4 v 13 w FB(M)p Ga(;)g(U)p 1080 2415 72 4 v 11 w F6(j)-24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37966
b(g)26 b Ga(>)1296 2356 y Gc(r)r(po)1296 2429 y(mul)q(t)1476
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37967
2400 y F6(f)-24 b(j)p Ga(S)p 1522 2415 61 4 v 6 w(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37968
b(U)p 1624 2415 72 4 v 10 w F6(j)-24 b(g)1440 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37969
b(rpo\(i\))504 2524 y(\(6\))146 b FB(L)p Ga(S)p 790 2539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37970
61 4 v 6 w(;)15 b(T)p 892 2539 66 4 v 13 w FB(M)p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37971
Ga(U)p 1028 2539 72 4 v 11 w F6(i)1136 2538 y Gc(m)1228
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37972
2524 y Ga(>)1299 2491 y Gc(r)r(po)1431 2524 y Ga(S)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37973
1431 2539 61 4 v 5 w F6(h)p Ga(U)p 1527 2539 72 4 v 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37974
w F6(i)1635 2538 y Gc(r)2301 2524 y Gg(if)23 b Ga(m)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37975
F4(=)g Ga(r)s Gg(,)d(then)i(by)g(\(5\))g(and)g(rpo\(iii\))2326
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37976
2648 y(if)f Ga(m)i(>)g(r)s Gg(,)d(then)j(by)e(\(4\))h(and)g(rpo\(ii\))
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37977
504 2773 y(\(7\))146 b FB(L)p Ga(S)p 790 2788 61 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37978
6 w(;)15 b(T)p 892 2788 66 4 v 13 w FB(M)p F6(h)p Ga(U)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37979
1028 2788 72 4 v 11 w F6(i)1136 2787 y Gc(m)1228 2773
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37980
y Ga(>)1299 2740 y Gc(r)r(po)1431 2773 y Ga(T)p 1431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37981
2788 66 4 v 13 w F6(h)p Ga(U)p 1532 2788 72 4 v 10 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37982
F6(i)1639 2787 y Gc(s)2775 2773 y Gg(analogous)26 b(to)e(\(4,5,6\))504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37983
2897 y(\(8\))146 b FB(L)p Ga(S)p 790 2912 61 4 v 6 w(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37984
b(T)p 892 2912 66 4 v 13 w FB(M)p F6(h)p Ga(U)p 1028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37985
2912 72 4 v 11 w F6(i)1136 2911 y Gc(m)1228 2897 y Ga(>)1299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37986
2864 y Gc(r)r(po)1431 2897 y FB(L)p Ga(S)p 1466 2912
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37987
61 4 v 6 w F6(h)p Ga(U)p 1563 2912 72 4 v 10 w F6(i)1670
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37988
2911 y Gc(r)1708 2897 y Ga(;)g(T)p 1748 2912 66 4 v 14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37989
w F6(h)p Ga(U)p 1850 2912 72 4 v 10 w F6(i)1957 2911
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37990
y Gc(s)1994 2897 y FB(M)776 b Gg(by)24 b(\(6,7\))g(and)g(rpo\(ii\))504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37991
3021 y(\(9\))146 b FB(L)p Ga(S)p 790 3036 61 4 v 6 w(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37992
b(T)p 892 3036 66 4 v 13 w FB(M)p F6(h)p Ga(U)p 1028
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37993
3036 72 4 v 11 w F6(i)1136 3035 y Gc(m)1228 3021 y Ga(>)1299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37994
2988 y Gc(r)r(po)1431 3021 y Ga(U)p 1431 3036 V 1689
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37995
w Gg(by)23 b(rpo\(i\))504 3145 y(\(10\))101 b Ga(M)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37996
755 3160 99 4 v 36 w(>)950 3112 y Gc(r)r(po)1082 3145
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37997
y Ga(N)p 1082 3160 83 4 v 1582 w Gg(by)24 b(\(3,8,9\))g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37998
(rpo\(ii\))504 3386 y Ga(M)36 b F6(\021)25 b FL(Cut)803
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 37999
3339 y FC( )863 3386 y F4(\()898 3374 y FX(h)926 3386
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38000
y Ga(d)973 3374 y FX(i)1001 3386 y FL(And)1156 3400 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38001
Gc(R)1213 3386 y F4(\()1248 3374 y FX(h)1276 3386 y Ga(a)1324
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38002
3374 y FX(i)1352 3386 y Ga(S)5 b(;)1453 3374 y FX(h)1480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38003
3386 y Ga(b)1519 3374 y FX(i)1547 3386 y Ga(T)13 b(;)i(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38004
F4(\))q Ga(;)1768 3374 y F9(\()1796 3386 y Ga(x)1848
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38005
3374 y F9(\))1875 3386 y Ga(U)10 b F4(\))782 3473 y Gc(x)717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38006
3510 y F6(\000)-31 b(\000)f(!)26 b FL(And)1067 3524 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38007
Gc(R)1125 3510 y F4(\()1160 3498 y FX(h)1188 3510 y Ga(a)1236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38008
3498 y FX(i)1263 3510 y FL(Cut)1342 3463 y FC( )1403
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38009
3510 y F4(\()1438 3498 y FX(h)1466 3510 y Ga(d)1513 3498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38010
y FX(i)1540 3510 y Ga(S)5 b(;)1641 3498 y F9(\()1669
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38011
3510 y Ga(x)1721 3498 y F9(\))1749 3510 y Ga(U)10 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38012
Ga(;)1896 3498 y FX(h)1924 3510 y Ga(b)1963 3498 y FX(i)1990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38013
3510 y FL(Cut)2069 3463 y FC( )2130 3510 y F4(\()2165
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38014
3498 y FX(h)2193 3510 y Ga(d)2240 3498 y FX(i)2267 3510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38015
y Ga(T)j(;)2373 3498 y F9(\()2401 3510 y Ga(x)2453 3498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38016
y F9(\))2480 3510 y Ga(U)d F4(\))q Ga(;)15 b(c)p F4(\))26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38017
b F6(\021)f Ga(N)504 3634 y Gg(\(1\))101 b Ga(M)p 710
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38018
3649 99 4 v 35 w F4(=)25 b FB(L)p Ga(S)p 964 3649 61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38019
4 v 6 w(;)15 b(T)p 1066 3649 66 4 v 13 w FB(M)g F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38020
Ga(U)p 1217 3649 72 4 v 11 w F6(i)1325 3648 y Gc(m)1440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38021
3634 y Gg(and)49 b Ga(N)p 1619 3649 83 4 v 35 w F4(=)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38022
b FB(L)p Ga(S)p 1858 3649 61 4 v 5 w F6(h)p Ga(U)p 1954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38023
3649 72 4 v 11 w F6(i)2062 3648 y Gc(r)2100 3634 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38024
b(T)p 2140 3649 66 4 v 13 w F6(h)p Ga(U)p 2241 3649 72
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38025
4 v 10 w F6(i)2348 3648 y Gc(s)2386 3634 y FB(M)349 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38026
Gg(by)23 b(De\002nition)i(2.6.17)504 3759 y(\(2\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38027
b Ga(m)25 b F6(\025)g Ga(r)m(;)15 b(s)1670 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38028
b(Lemma)g(2.6.19\(i,ii\))504 3883 y(\(3\))101 b FB(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38029
Ga(S)p 745 3898 61 4 v 5 w(;)15 b(T)p 846 3898 66 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38030
14 w FB(M)p F6(h)p Ga(U)p 983 3898 72 4 v 10 w F6(i)1090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38031
3897 y Gc(m)1182 3883 y Ga(>)1253 3850 y Gc(r)r(po)1386
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38032
3883 y Ga(S)p 1386 3898 61 4 v 53 w Gg(and)49 b FB(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38033
Ga(S)p 1709 3898 V 5 w(;)15 b(T)p 1810 3898 66 4 v 14
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38034
w FB(M)p F6(h)p Ga(U)p 1947 3898 72 4 v 10 w F6(i)2054
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38035
3897 y Gc(m)2147 3883 y Ga(>)2218 3850 y Gc(r)r(po)2350
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38036
3883 y Ga(U)p 2350 3898 V 770 w Gg(by)23 b(rpo\(i\))504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38037
4007 y(\(4\))101 b F6(f)-24 b(j)p FB(L)p Ga(S)p 791 4022
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38038
61 4 v 6 w(;)15 b(T)p 893 4022 66 4 v 14 w FB(M)p Ga(;)g(U)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38039
1035 4022 72 4 v 10 w F6(j)-24 b(g)27 b Ga(>)1251 3963
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38040
y Gc(r)r(po)1251 4036 y(mul)q(t)1430 4007 y F6(f)-24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38041
b(j)p Ga(S)p 1476 4022 61 4 v 6 w(;)15 b(U)p 1578 4022
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38042
72 4 v 11 w F6(j)-24 b(g)1485 b Gg(by)23 b(rpo\(i\))504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38043
4131 y(\(5\))101 b FB(L)p Ga(S)p 745 4146 61 4 v 5 w(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38044
b(T)p 846 4146 66 4 v 14 w FB(M)p F6(h)p Ga(U)p 983 4146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38045
72 4 v 10 w F6(i)1090 4145 y Gc(m)1182 4131 y Ga(>)1253
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38046
4098 y Gc(r)r(po)1386 4131 y Ga(S)p 1386 4146 61 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38047
5 w F6(h)p Ga(U)p 1482 4146 72 4 v 10 w F6(i)1589 4145
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38048
y Gc(r)2301 4131 y Gg(if)23 b Ga(m)i F4(=)g Ga(r)s Gg(,)d(then)i(by)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38049
(\(4\))g(and)g(rpo\(iii\))2326 4255 y(if)f Ga(m)i(>)g(r)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38050
Gg(,)d(then)j(by)e(\(3\))h(and)g(rpo\(ii\))504 4380 y(\(6\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38051
b FB(L)p Ga(S)p 745 4395 61 4 v 5 w(;)15 b(T)p 846 4395
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38052
66 4 v 14 w FB(M)p F6(h)p Ga(U)p 983 4395 72 4 v 10 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38053
F6(i)1090 4394 y Gc(m)1182 4380 y Ga(>)1253 4347 y Gc(r)r(po)1386
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38054
4380 y Ga(T)p 1386 4395 66 4 v 13 w F6(h)p Ga(U)p 1487
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38055
4395 72 4 v 10 w F6(i)1594 4394 y Gc(s)2775 4380 y Gg(analogous)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38056
b(to)e(\(3,4,5\))504 4504 y(\(7\))101 b Ga(M)p 710 4519
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38057
99 4 v 35 w(>)904 4471 y Gc(r)r(po)1037 4504 y Ga(N)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38058
1037 4519 83 4 v 1695 w Gg(by)24 b(\(5,6\))g(and)g(rpo\(ii\))504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38059
4745 y Ga(M)36 b F6(\021)25 b FL(Cut)803 4697 y FC( )863
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38060
4745 y F4(\()898 4733 y FX(h)926 4745 y Ga(a)974 4733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38061
y FX(i)1002 4745 y FL(Cut)p F4(\()1175 4733 y FX(h)1202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38062
4745 y Ga(b)1241 4733 y FX(i)1269 4745 y Ga(S)5 b(;)1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38063
4733 y F9(\()1398 4745 y Ga(x)1450 4733 y F9(\))1477
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38064
4745 y Ga(T)13 b F4(\))p Ga(;)1618 4733 y F9(\()1646
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38065
4745 y Ga(y)1694 4733 y F9(\))1721 4745 y Ga(U)d F4(\))782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38066
4832 y Gc(x)717 4869 y F6(\000)-31 b(\000)f(!)26 b FL(Cut)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38067
F4(\()1085 4857 y FX(h)1113 4869 y Ga(b)1152 4857 y FX(i)1180
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38068
4869 y FL(Cut)1258 4821 y FC( )1319 4869 y F4(\()1354
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38069
4857 y FX(h)1382 4869 y Ga(a)1430 4857 y FX(i)1458 4869
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38070
y Ga(S)5 b(;)1559 4857 y F9(\()1586 4869 y Ga(y)1634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38071
4857 y F9(\))1662 4869 y Ga(U)10 b F4(\))p Ga(;)1809
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38072
4857 y F9(\()1837 4869 y Ga(x)1889 4857 y F9(\))1916
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38073
4869 y FL(Cut)1995 4821 y FC( )2056 4869 y F4(\()2091
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38074
4857 y FX(h)2119 4869 y Ga(a)2167 4857 y FX(i)2194 4869
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38075
y Ga(T)j(;)2300 4857 y F9(\()2328 4869 y Ga(y)2376 4857
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38076
y F9(\))2403 4869 y Ga(U)d F4(\))q(\))25 b F6(\021)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38077
Ga(N)504 4993 y Gg(\(1\))101 b Ga(M)p 710 5008 99 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38078
35 w F4(=)25 b(\()p Ga(S)p 964 5008 61 4 v 26 w F6(\001)1071
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38079
5007 y Gc(m)1158 4993 y Ga(T)p 1158 5008 66 4 v 13 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38080
F4(\))p F6(h)p Ga(U)p 1294 5008 72 4 v 10 w F6(i)1401
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38081
5007 y Gc(n)1497 4993 y Gg(and)49 b Ga(N)p 1676 5008
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38082
83 4 v 35 w F4(=)25 b Ga(S)p 1880 5008 61 4 v 5 w F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38083
Ga(U)p 1976 5008 72 4 v 10 w F6(i)2083 5007 y Gc(r)2142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38084
4993 y F6(\001)2167 5007 y Gc(s)2224 4993 y Ga(T)p 2224
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38085
5008 66 4 v 13 w F6(h)p Ga(U)p 2325 5008 72 4 v 10 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38086
F6(i)2432 5007 y Gc(t)2770 4993 y Gg(by)e(De\002nition)i(2.6.17)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38087
5117 y(\(2\))101 b Ga(n)25 b F6(\025)g Ga(s;)15 b(r)m(;)g(t)1622
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38088
b Gg(by)23 b(Lemma)g(2.6.19\(i,ii\))504 5241 y(\(3\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38089
712 5241 28 4 v 730 5241 V 747 5241 V 166 w F6(h)p 812
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38090
5241 V 830 5241 V 847 5241 V 64 w(i)909 5255 y Gc(n)982
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38091
5241 y F6(\035)p 1100 5241 V 1118 5241 V 1135 5241 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38092
110 w(\001)1208 5255 y Gc(s)p 1267 5241 V 1285 5241 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38093
1302 5241 V 2487 5241 a Gg(by)h(\(2\))f(and)h(De\002nition)h(2.6.14)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38094
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38095
%%Page: 154 166
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38096
TeXDict begin 154 165 bop Black -144 51 a Gb(154)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38097
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38098
4 v Black 549 384 a Gg(\(4\))100 b F4(\()p Ga(S)p 789
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38099
399 61 4 v 26 w F6(\001)896 398 y Gc(m)983 384 y Ga(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38100
983 399 66 4 v 13 w F4(\))p F6(h)p Ga(U)p 1119 399 72
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38101
4 v 10 w F6(i)1226 398 y Gc(n)1299 384 y Ga(>)1370 351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38102
y Gc(r)r(po)1502 384 y Ga(S)p 1502 399 61 4 v 53 w Gg(and)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38103
b F4(\()p Ga(S)p 1826 399 V 25 w F6(\001)1932 398 y Gc(m)2019
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38104
384 y Ga(T)p 2019 399 66 4 v 13 w F4(\))p F6(h)p Ga(U)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38105
2155 399 72 4 v 11 w F6(i)2263 398 y Gc(n)2335 384 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38106
Ga(>)2406 351 y Gc(r)r(po)2539 384 y Ga(U)p 2539 399
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38107
V 625 w Gg(by)23 b(rpo\(i\))549 508 y(\(5\))100 b F6(f)-24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38108
b(j)p F4(\()p Ga(S)p 835 523 61 4 v 27 w F6(\001)943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38109
522 y Gc(m)1030 508 y Ga(T)p 1030 523 66 4 v 13 w F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38110
Ga(;)15 b(U)p 1171 523 72 4 v 10 w F6(j)-24 b(g)27 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38111
Ga(>)1387 464 y Gc(r)r(po)1387 537 y(mul)q(t)1566 508
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38112
y F6(f)-24 b(j)p Ga(S)p 1612 523 61 4 v 6 w(;)15 b(U)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38113
1714 523 72 4 v 11 w F6(j)-24 b(g)1393 b Gg(by)23 b(rpo\(i\))549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38114
632 y(\(6\))100 b F4(\()p Ga(S)p 789 647 61 4 v 26 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38115
F6(\001)896 646 y Gc(m)983 632 y Ga(T)p 983 647 66 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38116
v 13 w F4(\))p F6(h)p Ga(U)p 1119 647 72 4 v 10 w F6(i)1226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38117
646 y Gc(n)1299 632 y Ga(>)1370 599 y Gc(r)r(po)1502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38118
632 y Ga(S)p 1502 647 61 4 v 5 w F6(h)p Ga(U)p 1598 647
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38119
72 4 v 11 w F6(i)1706 646 y Gc(r)2370 632 y Gg(if)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38120
b Ga(n)i F4(=)g Ga(r)s Gg(,)d(then)i(by)g(\(5\))g(and)g(rpo\(iii\))2395
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38121
756 y(if)g Ga(n)g(>)h(r)s Gg(,)d(then)j(by)e(\(4\))h(and)g(rpo\(ii\))
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38122
549 881 y(\(7\))100 b F4(\()p Ga(S)p 789 896 61 4 v 26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38123
w F6(\001)896 895 y Gc(m)983 881 y Ga(T)p 983 896 66
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38124
4 v 13 w F4(\))p F6(h)p Ga(U)p 1119 896 72 4 v 10 w F6(i)1226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38125
895 y Gc(n)1299 881 y Ga(>)1370 848 y Gc(r)r(po)1502
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38126
881 y Ga(T)p 1502 896 66 4 v 13 w F6(h)p Ga(U)p 1603
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38127
896 72 4 v 10 w F6(i)1710 895 y Gc(t)2819 881 y Gg(analogous)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38128
b(to)c(\(4,5,6\))549 1005 y(\(8\))100 b Ga(M)p 754 1020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38129
99 4 v 36 w(>)949 972 y Gc(r)r(po)1081 1005 y Ga(N)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38130
1081 1020 83 4 v 1627 w Gg(by)24 b(\(3,6,7\))g(and)g(rpo\(ii\))549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38131
1294 y Ga(M)35 b F6(\021)25 b FL(Cut)847 1247 y FC( )908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38132
1294 y F4(\()943 1282 y FX(h)970 1294 y Ga(a)1018 1282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38133
y FX(i)1046 1294 y FL(Cut)p F4(\()1219 1282 y FX(h)1247
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38134
1294 y Ga(b)1286 1282 y FX(i)1313 1294 y Ga(S)5 b(;)1414
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38135
1282 y F9(\()1442 1294 y Ga(x)1494 1282 y F9(\))1521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38136
1294 y FL(Ax)p F4(\()p Ga(x;)15 b(a)p F4(\))q(\))p Ga(;)1910
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38137
1282 y F9(\()1938 1294 y Ga(y)1986 1282 y F9(\))2013
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38138
1294 y Ga(U)10 b F4(\))826 1381 y Gc(x)761 1418 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38139
b(\000)g(!)25 b FL(Cut)p F4(\()1130 1406 y FX(h)1157
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38140
1418 y Ga(b)1196 1406 y FX(i)1224 1418 y FL(Cut)1303
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38141
1371 y FC( )1364 1418 y F4(\()1399 1406 y FX(h)1426 1418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38142
y Ga(a)1474 1406 y FX(i)1502 1418 y Ga(S)5 b(;)1603 1406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38143
y F9(\()1631 1418 y Ga(y)1679 1406 y F9(\))1706 1418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38144
y Ga(U)10 b F4(\))p Ga(;)1853 1406 y F9(\()1881 1418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38145
y Ga(y)1929 1406 y F9(\))1956 1418 y Ga(U)g F4(\))26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38146
b F6(\021)f Ga(N)549 1543 y Gg(\(1\))100 b Ga(M)p 754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38147
1558 99 4 v 36 w F4(=)25 b(\()p Ga(S)p 1009 1558 61 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38148
v 25 w F6(\001)1115 1557 y Gc(m)1202 1543 y Ga(?)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38149
F6(h)p Ga(U)p 1317 1558 72 4 v 11 w F6(i)1425 1557 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38150
Gc(n)1521 1543 y Gg(and)49 b Ga(N)p 1700 1558 83 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38151
35 w F4(=)25 b Ga(S)p 1904 1558 61 4 v 5 w F6(h)p Ga(U)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38152
2000 1558 72 4 v 10 w F6(i)2107 1557 y Gc(r)2166 1543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38153
y F6(\001)2191 1557 y Gc(s)2248 1543 y Ga(U)p 2248 1558
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38154
V 504 w Gg(by)f(De\002nition)g(2.6.17)549 1667 y(\(2\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38155
b Ga(n)25 b F6(\025)g Ga(s;)15 b(r)1692 b Gg(by)24 b(Lemma)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38156
(2.6.19\(i,ii\))549 1791 y(\(3\))p 756 1791 28 4 v 774
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38157
1791 V 791 1791 V 165 w F6(h)p 856 1791 V 874 1791 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38158
891 1791 V 65 w(i)954 1805 y Gc(n)1026 1791 y F6(\035)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38159
1144 1791 V 1162 1791 V 1180 1791 V 110 w(\001)1252 1805
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38160
y Gc(s)p 1311 1791 V 1329 1791 V 1347 1791 V 2531 1791
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38161
a Gg(by)i(\(2\))g(and)g(De\002nition)g(2.6.14)549 1915
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38162
y(\(4\))100 b F4(\()p Ga(S)p 789 1930 61 4 v 26 w F6(\001)896
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38163
1929 y Gc(m)983 1915 y Ga(?)p F4(\))p F6(h)p Ga(U)p 1098
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38164
1930 72 4 v 11 w F6(i)1206 1929 y Gc(n)1278 1915 y Ga(>)1349
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38165
1882 y Gc(r)r(po)1482 1915 y Ga(S)p 1482 1930 61 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38166
53 w Gg(and)49 b F4(\()p Ga(S)p 1805 1930 V 26 w F6(\001)1912
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38167
1929 y Gc(m)1999 1915 y Ga(?)p F4(\))p F6(h)p Ga(U)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38168
2114 1930 72 4 v 11 w F6(i)2222 1929 y Gc(n)2295 1915
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38169
y Ga(>)2366 1882 y Gc(r)r(po)2498 1915 y Ga(U)p 2498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38170
1930 V 666 w Gg(by)23 b(rpo\(i\))549 2040 y(\(5\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38171
b F6(f)-24 b(j)p F4(\()p Ga(S)p 835 2055 61 4 v 27 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38172
F6(\001)943 2054 y Gc(m)1030 2040 y Ga(?)p F4(\))p Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38173
b(U)p 1150 2055 72 4 v 11 w F6(j)-24 b(g)26 b Ga(>)1366
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38174
1995 y Gc(r)r(po)1366 2069 y(mul)q(t)1546 2040 y F6(f)-24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38175
b(j)p Ga(S)p 1592 2055 61 4 v 6 w(;)15 b(U)p 1694 2055
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38176
72 4 v 11 w F6(j)-24 b(g)1413 b Gg(by)23 b(rpo\(i\))549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38177
2164 y(\(6\))100 b F4(\()p Ga(S)p 789 2179 61 4 v 26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38178
w F6(\001)896 2178 y Gc(m)983 2164 y Ga(?)p F4(\))p F6(h)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38179
Ga(U)p 1098 2179 72 4 v 11 w F6(i)1206 2178 y Gc(n)1278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38180
2164 y Ga(>)1349 2131 y Gc(r)r(po)1482 2164 y Ga(S)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38181
1482 2179 61 4 v 5 w F6(h)p Ga(U)p 1578 2179 72 4 v 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38182
w F6(i)1685 2178 y Gc(r)2370 2164 y Gg(if)23 b Ga(n)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38183
F4(=)g Ga(r)s Gg(,)d(then)i(by)g(\(5\))g(and)g(rpo\(iii\))2395
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38184
2288 y(if)g Ga(n)g(>)h(r)s Gg(,)d(then)j(by)e(\(4\))h(and)g(rpo\(ii\))
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38185
549 2412 y(\(7\))100 b Ga(M)p 754 2427 99 4 v 36 w(>)949
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38186
2379 y Gc(r)r(po)1081 2412 y Ga(N)p 1081 2427 83 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38187
1627 w Gg(by)24 b(\(3,4,6\))g(and)g(rpo\(ii\))549 2702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38188
y Ga(M)35 b F6(\021)25 b FL(Cut)847 2654 y FC( )908 2702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38189
y F4(\()943 2690 y FX(h)970 2702 y Ga(a)1018 2690 y FX(i)1046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38190
2702 y FL(Ax)o F4(\()p Ga(x;)15 b(a)p F4(\))r Ga(;)1400
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38191
2690 y F9(\()1427 2702 y Ga(y)1475 2690 y F9(\))1503
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38192
2702 y Ga(S)5 b F4(\))1690 2664 y Gc(x)1624 2702 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38193
b(\000)g(!)25 b FL(Cut)p F4(\()1993 2690 y FX(h)2020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38194
2702 y Ga(a)2068 2690 y FX(i)2096 2702 y FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38195
Ga(x;)15 b(a)p F4(\))r Ga(;)2450 2690 y F9(\()2477 2702
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38196
y Ga(y)2525 2690 y F9(\))2553 2702 y Ga(S)5 b F4(\))25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38197
b F6(\021)g Ga(N)549 2826 y Gg(\(1\))100 b Ga(M)p 754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38198
2841 99 4 v 36 w F4(=)25 b Ga(?)15 b F6(h)p Ga(S)p 1069
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38199
2841 61 4 v 5 w F6(i)1165 2840 y Gc(m)1280 2826 y Gg(and)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38200
b Ga(N)p 1460 2841 83 4 v 35 w F4(=)25 b Ga(?)20 b F6(\001)1754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38201
2840 y Gc(n)1822 2826 y Ga(S)p 1822 2841 61 4 v 936 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38202
Gg(by)k(De\002nition)g(2.6.17)549 2950 y(\(2\))100 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38203
Ga(m)25 b F6(\025)g Ga(n)1809 b Gg(by)24 b(Lemma)e(2.6.19\(i\))549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38204
3074 y(\(3\))p 756 3074 28 4 v 774 3074 V 791 3074 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38205
165 w F6(h)p 856 3074 V 874 3074 V 891 3074 V 65 w(i)954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38206
3088 y Gc(m)1046 3074 y F6(\035)p 1164 3074 V 1182 3074
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38207
V 1199 3074 V 110 w(\001)1272 3088 y Gc(n)p 1341 3074
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38208
V 1359 3074 V 1376 3074 V 2531 3074 a Gg(by)i(\(2\))g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38209
(De\002nition)g(2.6.14)549 3198 y(\(4\))100 b Ga(?)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38210
b F6(h)p Ga(S)p 849 3213 61 4 v 6 w F6(i)946 3212 y Gc(m)1038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38211
3198 y Ga(>)1109 3165 y Gc(r)r(po)1242 3198 y Ga(?)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38212
b Gg(and)i Ga(?)15 b F6(h)p Ga(S)p 1610 3213 V 6 w F6(i)1707
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38213
3212 y Gc(m)1799 3198 y Ga(>)1870 3165 y Gc(r)r(po)2002
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38214
3198 y Ga(S)p 2002 3213 V 1168 w Gg(by)23 b(rpo\(i\))549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38215
3323 y(\(5\))100 b Ga(M)p 754 3338 99 4 v 36 w(>)949
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38216
3290 y Gc(r)r(po)1081 3323 y Ga(N)p 1081 3338 83 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38217
1696 w Gg(by)23 b(\(3,4\))h(and)g(rpo\(ii\))p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38218
3603 a Gb(Commuting)f(Reduction:)p Black 45 w Gg(One)h(case)g(is)f(as)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38219
(follo)n(ws.)549 3803 y Ga(M)35 b F6(\021)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38220
F4(\()941 3791 y FX(h)969 3803 y Ga(a)1017 3791 y FX(i)1044
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38221
3803 y Ga(S)5 b(;)1145 3791 y F9(\()1173 3803 y Ga(x)1225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38222
3791 y F9(\))1252 3803 y Ga(T)13 b F4(\))1428 3766 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38223
Gc(c)1459 3742 y FC(00)1379 3803 y F6(\000)-32 b(\000)h(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38224
b FL(Cut)1653 3755 y FC( )1714 3803 y F4(\()1749 3791
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38225
y FX(h)1777 3803 y Ga(a)1825 3791 y FX(i)1852 3803 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38226
Ga(S)5 b(;)1953 3791 y F9(\()1981 3803 y Ga(x)2033 3791
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38227
y F9(\))2060 3803 y Ga(T)13 b F4(\))26 b F6(\021)f Ga(N)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38228
3927 y Gg(\(1\))100 b Ga(M)p 754 3942 99 4 v 36 w F4(=)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38229
b Ga(S)p 974 3942 61 4 v 25 w F6(\001)1080 3941 y Gc(m)1167
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38230
3927 y Ga(T)p 1167 3942 66 4 v 60 w Gg(and)50 b Ga(N)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38231
1460 3942 83 4 v 35 w F4(=)25 b Ga(S)p 1664 3942 61 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38232
v 5 w F6(h)p Ga(T)p 1760 3942 66 4 v 13 w F6(i)1861 3941
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38233
y Gc(n)2814 3927 y Gg(by)f(De\002nition)g(2.6.17)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38234
4051 y(\(2\))100 b Ga(m)25 b(>)g(n)1759 b Gg(by)23 b(Lemma)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38235
(2.6.19\(iii\))549 4175 y(\(3\))p 756 4175 28 4 v 774
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38236
4175 V 791 4175 V 185 w F6(\001)864 4189 y Gc(m)p 953
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38237
4175 V 971 4175 V 988 4175 V 1041 4175 a F6(\035)p 1159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38238
4175 V 1177 4175 V 1194 4175 V 89 w(h)p 1258 4175 V 1277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38239
4175 V 1294 4175 V 65 w(i)1356 4189 y Gc(n)2531 4175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38240
y Gg(by)h(\(2\))g(and)g(De\002nition)g(2.6.14)549 4299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38241
y(\(4\))100 b Ga(S)p 754 4314 61 4 v 25 w F6(\001)860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38242
4313 y Gc(m)947 4299 y Ga(T)p 947 4314 66 4 v 38 w(>)1109
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38243
4266 y Gc(r)r(po)1242 4299 y Ga(S)p 1242 4314 61 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38244
53 w Gg(and)49 b Ga(S)p 1530 4314 V 25 w F6(\001)1636
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38245
4313 y Gc(m)1723 4299 y Ga(T)p 1723 4314 66 4 v 38 w(>)1885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38246
4266 y Gc(r)r(po)2018 4299 y Ga(T)p 2018 4314 V 1155
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38247
w Gg(by)23 b(rpo\(i\))549 4424 y(\(5\))100 b Ga(M)p 754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38248
4439 99 4 v 36 w(>)949 4391 y Gc(r)r(po)1081 4424 y Ga(N)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38249
1081 4439 83 4 v 1696 w Gg(by)23 b(\(3,4\))h(and)g(rpo\(ii\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38250
Black 321 4704 a Gb(Logical)h(Reduction:)p Black 46 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38251
Gg(W)-7 b(e)22 b(tackle)j(three)g(representati)n(v)o(e)i(cases.)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38252
4905 y Ga(M)35 b F6(\021)25 b FL(Cut)p F4(\()941 4893
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38253
y FX(h)969 4905 y Ga(a)1017 4893 y FX(i)1044 4905 y FL(Ax)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38254
F4(\()p Ga(x;)15 b(a)p F4(\))q Ga(;)1398 4893 y F9(\()1426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38255
4905 y Ga(y)1474 4893 y F9(\))1501 4905 y Ga(S)5 b F4(\))1697
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38256
4868 y Gc(l)1622 4905 y F6(\000)-31 b(\000)g(!)25 b Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38257
b F4([)p Ga(y)13 b F6(7!)d Ga(x)p F4(])26 b F6(\021)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38258
Ga(N)549 5029 y Gg(\(1\))100 b Ga(M)p 754 5044 99 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38259
36 w F4(=)25 b Ga(?)20 b F6(\001)1064 5043 y Gc(n)1132
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38260
5029 y Ga(S)p 1132 5044 61 4 v 53 w Gg(and)49 b Ga(N)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38261
1420 5044 83 4 v 35 w F4(=)25 b Ga(S)p 1624 5044 61 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38262
v 1134 w Gg(by)f(De\002nition)g(2.6.17)549 5153 y(\(2\))100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38263
b Ga(M)p 754 5168 99 4 v 36 w(>)949 5120 y Gc(r)r(po)1081
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38264
5153 y Ga(N)p 1081 5168 83 4 v 1764 w Gg(by)23 b(\(1\))h(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38265
(rpo\(ii\))p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38266
%%Page: 155 167
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38267
TeXDict begin 155 166 bop Black 277 51 a Gb(B.1)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38268
(Chapter)f(2)2639 b(155)p 277 88 3691 4 v Black 504 384
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38269
a Ga(M)36 b F6(\021)25 b FL(Cut)p F4(\()897 372 y FX(h)924
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38270
384 y Ga(c)963 372 y FX(i)991 384 y FL(And)1146 398 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38271
Gc(R)1203 384 y F4(\()1238 372 y FX(h)1266 384 y Ga(a)1314
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38272
372 y FX(i)1342 384 y Ga(S)5 b(;)1443 372 y FX(h)1471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38273
384 y Ga(b)1510 372 y FX(i)1537 384 y Ga(T)13 b(;)i(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38274
F4(\))q Ga(;)1758 372 y F9(\()1786 384 y Ga(y)1834 372
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38275
y F9(\))1861 384 y FL(And)2016 347 y F9(1)2016 407 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38276
Gc(L)2068 384 y F4(\()2103 372 y F9(\()2130 384 y Ga(x)2182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38277
372 y F9(\))2210 384 y Ga(U)10 b(;)15 b(y)s F4(\)\))2540
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38278
347 y Gc(l)2466 384 y F6(\000)-31 b(\000)f(!)26 b FL(Cut)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38279
F4(\()2834 372 y FX(h)2862 384 y Ga(a)2910 372 y FX(i)2938
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38280
384 y Ga(S)5 b(;)3039 372 y F9(\()3067 384 y Ga(x)3119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38281
372 y F9(\))3146 384 y Ga(U)10 b F4(\))26 b F6(\021)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38282
Ga(N)504 508 y Gg(\(1\))101 b Ga(M)p 710 523 99 4 v 35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38283
w F4(=)25 b FB(L)p Ga(S)p 964 523 61 4 v 6 w(;)15 b(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38284
1066 523 66 4 v 13 w FB(M)20 b F6(\001)1212 522 y Gc(m)1299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38285
508 y FB(L)p Ga(U)p 1334 523 72 4 v 11 w FB(M)j Gg(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38286
Ga(N)p 1619 523 83 4 v 35 w F4(=)h Ga(S)p 1823 523 61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38287
4 v 25 w F6(\001)1929 522 y Gc(n)1997 508 y Ga(U)p 1997
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38288
523 72 4 v 711 w Gg(by)e(De\002nition)i(2.6.17)504 632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38289
y(\(2\))101 b Ga(m)25 b(>)g(n)1758 b Gg(by)24 b(Lemma)f(2.6.19\(iii\))
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38290
504 756 y(\(3\))101 b FB(L)p Ga(S)p 745 771 61 4 v 5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38291
w(;)15 b(T)p 846 771 66 4 v 14 w FB(M)20 b F6(\001)993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38292
770 y Gc(m)1080 756 y FB(L)p Ga(U)p 1115 771 72 4 v 10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38293
w FB(M)26 b Ga(>)1319 723 y Gc(r)r(po)1452 756 y Ga(S)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38294
1452 771 61 4 v 53 w Gg(,)47 b FB(L)p Ga(S)p 1666 771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38295
V 6 w(;)15 b(T)p 1768 771 66 4 v 13 w FB(M)20 b F6(\001)1914
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38296
770 y Gc(m)2001 756 y FB(L)p Ga(U)p 2036 771 72 4 v 11
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38297
w FB(M)25 b Ga(>)2240 723 y Gc(r)r(po)2373 756 y Ga(U)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38298
2373 771 V 747 w Gg(by)e(rpo\(i\))504 881 y(\(4\))101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38299
b Ga(M)p 710 896 99 4 v 35 w(>)904 848 y Gc(r)r(po)1037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38300
881 y Ga(N)p 1037 896 83 4 v 1695 w Gg(by)24 b(\(2,3\))g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38301
(rpo\(ii\))504 1116 y Ga(M)36 b F6(\021)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38302
F4(\()897 1104 y FX(h)924 1116 y Ga(b)963 1104 y FX(i)991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38303
1116 y FL(Imp)1135 1138 y Gc(R)1193 1116 y F4(\()1228
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38304
1104 y F9(\()1256 1116 y Ga(x)1308 1104 y F9(\))p FX(h)1363
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38305
1116 y Ga(a)1411 1104 y FX(i)1438 1116 y Ga(S)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38306
b(b)p F4(\))q Ga(;)1654 1104 y F9(\()1682 1116 y Ga(z)1728
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38307
1104 y F9(\))1755 1116 y FL(Imp)1900 1138 y Gc(L)1952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38308
1116 y F4(\()1987 1104 y FX(h)2015 1116 y Ga(c)2054 1104
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38309
y FX(i)2082 1116 y Ga(T)d(;)2187 1104 y F9(\()2215 1116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38310
y Ga(y)2263 1104 y F9(\))2291 1116 y Ga(U)e(;)15 b(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38311
F4(\))q(\))791 1203 y Gc(l)717 1240 y F6(\000)-31 b(\000)f(!)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38312
b FL(Cut)o F4(\()1085 1228 y FX(h)1113 1240 y Ga(a)1161
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38313
1228 y FX(i)1189 1240 y FL(Cut)p F4(\()1362 1228 y FX(h)1389
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38314
1240 y Ga(c)1428 1228 y FX(i)1456 1240 y Ga(T)13 b(;)1562
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38315
1228 y F9(\()1590 1240 y Ga(x)1642 1228 y F9(\))1669
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38316
1240 y Ga(S)5 b F4(\))p Ga(;)1805 1228 y F9(\()1833 1240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38317
y Ga(y)1881 1228 y F9(\))1909 1240 y Ga(U)10 b F4(\))25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38318
b F6(\021)g Ga(N)504 1364 y Gg(\(1\))101 b Ga(M)p 710
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38319
1379 99 4 v 35 w F4(=)25 b FB(L)p Ga(S)p 964 1379 61
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38320
4 v 6 w FB(M)20 b F6(\001)1106 1378 y Gc(m)1193 1364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38321
y FB(L)p Ga(T)p 1228 1379 66 4 v 13 w(;)15 b(U)p 1334
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38322
1379 72 4 v 11 w FB(M)23 b Gg(and)h Ga(N)p 1619 1379
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38323
83 4 v 35 w F4(=)h(\()p Ga(T)p 1858 1379 66 4 v 33 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38324
F6(\001)1969 1378 y Gc(s)2027 1364 y Ga(S)p 2027 1379
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38325
61 4 v 5 w F4(\))20 b F6(\001)2168 1378 y Gc(t)2218 1364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38326
y Ga(U)p 2218 1379 72 4 v 490 w Gg(by)j(De\002nition)i(2.6.17)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38327
1488 y(\(2\))101 b Ga(m)25 b(>)g(s;)15 b(t)1624 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38328
b(Lemma)e(2.6.19\(iii,ii\))504 1613 y(\(3\))101 b FB(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38329
Ga(S)p 745 1628 61 4 v 5 w FB(M)21 b F6(\001)887 1627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38330
y Gc(m)974 1613 y FB(L)p Ga(T)p 1009 1628 66 4 v 13 w(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38331
b(U)p 1115 1628 72 4 v 10 w FB(M)26 b Ga(>)1319 1580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38332
y Gc(r)r(po)1452 1613 y Ga(S)p 1452 1628 61 4 v 53 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38333
Gg(,)47 b FB(L)p Ga(S)p 1666 1628 V 6 w FB(M)20 b F6(\001)1808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38334
1627 y Gc(m)1895 1613 y FB(L)p Ga(T)p 1930 1628 66 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38335
v 13 w(;)15 b(U)p 2036 1628 72 4 v 11 w FB(M)25 b Ga(>)2240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38336
1580 y Gc(r)r(po)2373 1613 y Ga(T)p 2373 1628 66 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38337
756 w Gg(by)e(rpo\(i\))504 1737 y(\(4\))101 b FB(L)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38338
Ga(S)p 745 1752 61 4 v 5 w FB(M)21 b F6(\001)887 1751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38339
y Gc(m)974 1737 y FB(L)p Ga(T)p 1009 1752 66 4 v 13 w(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38340
b(U)p 1115 1752 72 4 v 10 w FB(M)26 b Ga(>)1319 1704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38341
y Gc(r)r(po)1452 1737 y Ga(T)p 1452 1752 66 4 v 33 w
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38342
F6(\001)1563 1751 y Gc(s)1620 1737 y Ga(S)p 1620 1752
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38343
61 4 v 1129 w Gg(by)e(\(2,3\))g(and)g(rpo\(ii\))504 1861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38344
y(\(5\))101 b FB(L)p Ga(S)p 745 1876 V 5 w FB(M)21 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38345
F6(\001)887 1875 y Gc(m)974 1861 y FB(L)p Ga(T)p 1009
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38346
1876 66 4 v 13 w(;)15 b(U)p 1115 1876 72 4 v 10 w FB(M)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38347
b Ga(>)1319 1828 y Gc(r)r(po)1452 1861 y Ga(U)p 1452
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38348
1876 V 1170 w Gg(by)e(De\002nition)g(2.6.12\(i\))504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38349
1985 y(\(6\))101 b Ga(M)p 710 2000 99 4 v 35 w(>)904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38350
1952 y Gc(r)r(po)1037 1985 y Ga(N)p 1037 2000 83 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38351
1627 w Gg(by)24 b(\(2,4,5\))g(and)g(rpo\(ii\))p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38352
277 2157 a Gb(Garbage)g(Reduction:)p Black 46 w Gg(Finally)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38353
b(,)23 b(we)e(gi)n(v)o(e)h(one)h(case)g(where)f(a)g(garbage)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38354
(reduction)i(is)d(performed.)504 2303 y Ga(M)36 b F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38355
b FL(Cut)803 2256 y FC( )863 2303 y F4(\()898 2291 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38356
FX(h)926 2303 y Ga(a)974 2291 y FX(i)1002 2303 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38357
b(;)1103 2291 y F9(\()1131 2303 y Ga(x)1183 2291 y F9(\))1210
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38358
2303 y Ga(T)13 b F4(\))1388 2266 y Gc(g)r(c)1336 2303
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38359
y F6(\000)-31 b(\000)g(!)25 b Ga(S)30 b F6(\021)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38360
Ga(N)504 2427 y Gg(\(1\))101 b Ga(M)p 710 2442 99 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38361
35 w F4(=)25 b Ga(S)p 929 2442 61 4 v 5 w F6(h)p Ga(T)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38362
1025 2442 66 4 v 13 w F6(i)1126 2441 y Gc(m)1241 2427
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38363
y Gg(and)50 b Ga(N)p 1421 2442 83 4 v 35 w F4(=)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38364
Ga(S)p 1625 2442 61 4 v 1089 w Gg(by)e(De\002nition)i(2.6.17)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38365
2552 y(\(2\))101 b Ga(M)p 710 2567 99 4 v 35 w(>)904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38366
2519 y Gc(r)r(po)1037 2552 y Ga(N)p 1037 2567 83 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38367
1789 w Gg(by)23 b(\(1\))h(and)g(rpo\(i\))p 3436 2659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38368
4 62 v 3440 2600 55 4 v 3440 2659 V 3494 2659 4 62 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38369
Black 277 2871 a Gb(Pr)n(oof)k(of)g(Lemma)f(2.6.22.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38370
Black 34 w Gg(W)-7 b(e)27 b(gi)n(v)o(e)g(the)h(details)h(of)f(ho)n(w)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38371
(to)g(replace)j FL(Cut)2707 2824 y FC( )2767 2871 y Gg(-instances.)44
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38372
b(Suppose)277 2984 y FL(Cut)356 2937 y FC( )417 2984
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38373
y F4(\()452 2972 y FX(h)480 2984 y Ga(a)528 2972 y FX(i)555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38374
2984 y Ga(S)5 b(;)656 2972 y F9(\()684 2984 y Ga(x)736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38375
2972 y F9(\))763 2984 y Ga(T)13 b F4(\))23 b Gg(is)h(a)f(subterm)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38376
Ga(M)33 b Gg(with)23 b(the)h(typing)h(judgement)1271
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38377
3151 y F4(\000)1328 3165 y F9(1)1393 3139 y Gc(.)1448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38378
3151 y Ga(S)1534 3139 y Gc(.)1589 3151 y F4(\001)1665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38379
3165 y F9(1)1845 3151 y F4(\000)1902 3165 y F9(2)1967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38380
3139 y Gc(.)2022 3151 y Ga(T)2113 3139 y Gc(.)2168 3151
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38381
y F4(\001)2244 3165 y F9(2)p 1151 3184 1252 4 v 1151
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38382
3269 a F4(\000)1208 3283 y F9(1)1248 3269 y Ga(;)15 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38383
F4(\000)1345 3283 y F9(2)1410 3257 y Gc(.)1464 3269 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38384
FL(Cut)1543 3222 y FC( )1604 3269 y F4(\()1639 3257 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38385
FX(h)1667 3269 y Ga(a)1715 3257 y FX(i)1742 3269 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38386
b(;)1843 3257 y F9(\()1871 3269 y Ga(x)1923 3257 y F9(\))1951
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38387
3269 y Ga(T)12 b F4(\))2077 3257 y Gc(.)2132 3269 y F4(\001)2208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38388
3283 y F9(1)2247 3269 y Ga(;)j F4(\001)2363 3283 y F9(2)2444
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38389
3217 y Gg(Cut)2516 3170 y FC( )2603 3217 y Ga(:)277 3456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38390
y Gg(Our)23 b(ar)n(gument)j(splits)e(into)h(tw)o(o)e(cases:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38391
Black 414 3650 a F6(\017)p Black 45 w Gg(If)35 b(the)g(subterm)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38392
Ga(S)j Gg(does)c(not)h(freshly)g(introduce)h Ga(a)p Gg(,)g(then)e(we)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38393
(replace)j FL(Cut)2994 3603 y FC( )3055 3650 y F4(\()3090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38394
3638 y FX(h)3117 3650 y Ga(a)3165 3638 y FX(i)3193 3650
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38395
y Ga(S)5 b(;)3294 3638 y F9(\()3322 3650 y Ga(x)3374
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38396
3638 y F9(\))3401 3650 y Ga(T)13 b F4(\))504 3763 y Gg(simply)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38397
b(by)f FL(Cut)p F4(\()1074 3751 y FX(h)1102 3763 y Ga(a)1150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38398
3751 y FX(i)1177 3763 y Ga(S)5 b(;)1278 3751 y F9(\()1306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38399
3763 y Ga(x)1358 3751 y F9(\))1385 3763 y Ga(T)13 b F4(\))29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38400
b Gg(\(both)i(terms)f(ha)n(v)o(e)h(the)f(same)g(typing)h(judgement\).)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38401
50 b(In)30 b(this)504 3876 y(case)25 b(we)d(ha)n(v)o(e)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38402
FL(Cut)p F4(\()1176 3864 y FX(h)1204 3876 y Ga(a)1252
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38403
3864 y FX(i)1280 3876 y Ga(S)5 b(;)1381 3864 y F9(\()1409
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38404
3876 y Ga(x)1461 3864 y F9(\))1488 3876 y Ga(T)13 b F4(\))1664
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38405
3839 y Gc(c)1695 3816 y FC(00)1614 3876 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38406
b(\000)g(!)25 b FL(Cut)1889 3829 y FC( )1949 3876 y F4(\()1984
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38407
3864 y FX(h)2012 3876 y Ga(a)2060 3864 y FX(i)2088 3876
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38408
y Ga(S)5 b(;)2189 3864 y F9(\()2217 3876 y Ga(x)2269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38409
3864 y F9(\))2296 3876 y Ga(T)13 b F4(\))p Gg(.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38410
414 4052 a F6(\017)p Black 45 w Gg(The)24 b(more)g(interesting)k(case)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38411
(is)h(where)f Ga(S)k Gg(freshly)e(introduces)h Ga(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38412
Gg(.)j(Here)24 b(we)g(cannot)i(simply)504 4165 y(replace)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38413
FL(Cut)867 4118 y FC( )948 4165 y Gg(with)e FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38414
Gg(,)f(because)j(there)e(is)g(no)g(reduction)i(that)e(gi)n(v)o(es)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38415
Ga(N)2899 4128 y Gc(l)q(oc)2857 4165 y F6(\000)-31 b(\000)f(!)3027
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38416
4132 y F9(+)3111 4165 y Ga(M)10 b Gg(.)28 b(There-)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38417
4278 y(fore)k(we)e(replace)j FL(Cut)1189 4231 y FC( )1250
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38418
4278 y F4(\()1285 4266 y FX(h)1312 4278 y Ga(a)1360 4266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38419
y FX(i)1388 4278 y Ga(S)5 b(;)1489 4266 y F9(\()1517
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38420
4278 y Ga(x)1569 4266 y F9(\))1596 4278 y Ga(T)13 b F4(\))30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38421
b Gg(by)i FL(Cut)p F4(\()2022 4266 y FX(h)2049 4278 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38422
Ga(a)2097 4266 y FX(i)2125 4278 y FL(Cut)p F4(\()2298
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38423
4266 y FX(h)2325 4278 y Ga(b)2364 4266 y FX(i)2392 4278
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38424
y Ga(S)5 b(;)2493 4266 y F9(\()2521 4278 y Ga(y)2569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38425
4266 y F9(\))2596 4278 y FL(Ax)p F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38426
b(c)p F4(\))q(\))p Ga(;)2972 4266 y F9(\()3000 4278 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38427
Ga(x)3052 4266 y F9(\))3079 4278 y Ga(T)e F4(\))31 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38428
Gg(where)g Ga(b)504 4391 y Gg(and)i Ga(c)f Gg(are)h(fresh)g(co-names)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38429
(that)f(do)g(not)f(occur)i(an)o(ywhere)f(else.)56 b(On)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38430
b(the)h(le)n(v)o(el)f(of)h(type)504 4504 y(deri)n(v)n(ations)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38431
b(this)d(means)g(that)g(an)f(instance)j(of)d(the)h(form)1410
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38432
4652 y F4(\000)1467 4666 y F9(1)1532 4640 y Gc(.)1587
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38433
4652 y Ga(S)1673 4640 y Gc(.)1728 4652 y F4(\001)1804
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38434
4666 y F9(1)1984 4652 y F4(\000)2041 4666 y F9(2)2106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38435
4640 y Gc(.)2161 4652 y Ga(T)2252 4640 y Gc(.)2307 4652
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38436
y F4(\001)2383 4666 y F9(2)p 1290 4686 V 1290 4770 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38437
F4(\000)1347 4784 y F9(1)1387 4770 y Ga(;)15 b F4(\000)1484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38438
4784 y F9(2)1548 4758 y Gc(.)1603 4770 y FL(Cut)1682
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38439
4723 y FC( )1743 4770 y F4(\()1778 4758 y FX(h)1806 4770
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38440
y Ga(a)1854 4758 y FX(i)1881 4770 y Ga(S)5 b(;)1982 4758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38441
y F9(\()2010 4770 y Ga(x)2062 4758 y F9(\))2090 4770
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38442
y Ga(T)12 b F4(\))2216 4758 y Gc(.)2271 4770 y F4(\001)2347
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38443
4784 y F9(1)2386 4770 y Ga(;)j F4(\001)2502 4784 y F9(2)2583
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38444
4718 y Gg(Cut)2655 4671 y FC( )504 4940 y Gg(is)24 b(replaced)h(by)782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38445
5130 y F4(\000)839 5144 y F9(1)903 5118 y Gc(.)958 5130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38446
y Ga(S)1044 5118 y Gc(.)1099 5130 y F4(\001)1175 5144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38447
y F9(1)1215 5130 y Ga(;)15 b(b)i F4(:)h Ga(B)p 1518 5045
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38448
815 4 v 95 w(y)i F4(:)d Ga(B)1724 5118 y Gc(.)1779 5130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38449
y FL(Ax)p F4(\()p Ga(y)s(;)e(c)p F4(\))2105 5118 y Gc(.)2160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38450
5130 y Ga(c)j F4(:)f Ga(B)p 782 5173 1551 4 v 861 5258
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38451
a F4(\000)918 5272 y F9(1)983 5246 y Gc(.)1038 5258 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38452
FL(Cut)o F4(\()1210 5246 y FX(h)1238 5258 y Ga(b)1277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38453
5246 y FX(i)1305 5258 y Ga(S)5 b(;)1406 5246 y F9(\()1434
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38454
5258 y Ga(y)1482 5246 y F9(\))1509 5258 y FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38455
Ga(y)s(;)15 b(c)p F4(\))r(\))1870 5246 y Gc(.)1925 5258
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38456
y F4(\001)2001 5272 y F9(1)2040 5258 y Ga(;)g(c)j F4(:)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38457
Ga(B)2374 5203 y Gg(Cut)2596 5258 y F4(\000)2653 5272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38458
y F9(2)2718 5246 y Gc(.)2773 5258 y Ga(T)2864 5246 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38459
Gc(.)2919 5258 y F4(\001)2995 5272 y F9(2)p 843 5300
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38460
2209 4 v 843 5385 a F4(\000)900 5399 y F9(1)939 5385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38461
y Ga(;)d F4(\000)1036 5399 y F9(2)1101 5373 y Gc(.)1156
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38462
5385 y FL(Cut)p F4(\()1329 5373 y FX(h)1357 5385 y Ga(a)1405
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38463
5373 y FX(i)1432 5385 y FL(Cut)p F4(\()1605 5373 y FX(h)1633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38464
5385 y Ga(b)1672 5373 y FX(i)1699 5385 y Ga(S)5 b(;)1800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38465
5373 y F9(\()1828 5385 y Ga(y)1876 5373 y F9(\))1904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38466
5385 y FL(Ax)o F4(\()p Ga(y)s(;)15 b(c)p F4(\))q(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38467
Ga(;)2280 5373 y F9(\()2308 5385 y Ga(x)2360 5373 y F9(\))2387
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38468
5385 y Ga(T)e F4(\))2513 5373 y Gc(.)2568 5385 y F4(\001)2644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38469
5399 y F9(1)2683 5385 y Ga(;)i F4(\001)2799 5399 y F9(2)2839
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38470
5385 y Ga(;)g(c)j F4(:)f Ga(B)3093 5331 y Gg(Cut)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38471
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38472
%%Page: 156 168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38473
TeXDict begin 156 167 bop Black -144 51 a Gb(156)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38474
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38475
4 v Black 549 388 a Gg(It)j(is)g(routine)i(to)e(v)o(erify)h(that)g(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38476
f(ne)n(w)g(cut-instances)k(are)c(all)h(well-typed.)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38477
b(No)n(w)26 b(we)h(sho)n(w)549 501 y(ho)n(w)35 b(the)h(ne)n(w)f(term)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38478
(can)g(reduce.)67 b(Because)37 b FL(Cut)p F4(\()2272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38479
489 y FX(h)2299 501 y Ga(b)2338 489 y FX(i)2366 501 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38480
Ga(S)5 b(;)2467 489 y F9(\()2495 501 y Ga(y)2543 489
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38481
y F9(\))2570 501 y FL(Ax)o F4(\()p Ga(y)s(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38482
F4(\))r(\))35 b Gg(does)i(not)f(freshly)549 614 y(introduce)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38483
b Ga(a)p Gg(,)c(we)h(can)h(\002rst)f(perform)i(a)e(commuting)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38484
(reduction)604 768 y FL(Cut)p F4(\()777 756 y FX(h)805
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38485
768 y Ga(a)853 756 y FX(i)880 768 y FL(Cut)p F4(\()1053
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38486
756 y FX(h)1081 768 y Ga(b)1120 756 y FX(i)1148 768 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38487
Ga(S)5 b(;)1249 756 y F9(\()1276 768 y Ga(y)1324 756
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38488
y F9(\))1352 768 y FL(Ax)o F4(\()p Ga(y)s(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38489
F4(\))q(\))q Ga(;)1728 756 y F9(\()1756 768 y Ga(x)1808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38490
756 y F9(\))1835 768 y Ga(T)e F4(\))2011 731 y Gc(c)2042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38491
708 y FC(00)1961 768 y F6(\000)-31 b(\000)g(!)25 b FL(Cut)2236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38492
721 y FC( )2297 768 y F4(\()2332 756 y FX(h)2359 768
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38493
y Ga(a)2407 756 y FX(i)2435 768 y FL(Cut)p F4(\()2608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38494
756 y FX(h)2636 768 y Ga(b)2675 756 y FX(i)2702 768 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38495
Ga(S)5 b(;)2803 756 y F9(\()2831 768 y Ga(y)2879 756
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38496
y F9(\))2906 768 y FL(Ax)p F4(\()p Ga(y)s(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38497
F4(\))q(\))p Ga(;)3282 756 y F9(\()3310 768 y Ga(x)3362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38498
756 y F9(\))3390 768 y Ga(T)d F4(\))549 923 y Gg(and)19
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38499
b(subsequently)24 b(we)18 b(can)i(remo)o(v)o(e)g(the)g(cut)f(on)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38500
(axiom)h(by)g(a)2712 886 y Gc(c)2743 862 y FC(00)2663
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38501
923 y F6(\000)-31 b(\000)f(!)p Gg(-reduction)22 b(follo)n(wed)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38502
1036 y(by)h(a)777 999 y Gc(g)r(c)725 1036 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38503
b(\000)g(!)p Gg(-reduction.)p Black Black 1515 1205 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38504
FL(Cut)1593 1158 y FC( )1654 1205 y F4(\()1689 1193 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38505
FX(h)1717 1205 y Ga(a)1765 1193 y FX(i)1793 1205 y FL(Cut)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38506
F4(\()1965 1193 y FX(h)1993 1205 y Ga(b)2032 1193 y FX(i)2060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38507
1205 y Ga(S)5 b(;)2161 1193 y F9(\()2189 1205 y Ga(y)2237
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38508
1193 y F9(\))2264 1205 y FL(Ax)o F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38509
b(c)p F4(\))r(\))p Ga(;)2640 1193 y F9(\()2668 1205 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38510
Ga(x)2720 1193 y F9(\))2747 1205 y Ga(T)e F4(\))1294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38511
1293 y Gc(c)1325 1269 y FC(00)1245 1330 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38512
b(\000)h(!)100 b FL(Cut)1593 1283 y FC( )1654 1330 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38513
F4(\()1689 1318 y FX(h)1717 1330 y Ga(a)1765 1318 y FX(i)1793
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38514
1330 y FL(Cut)1871 1283 y FC( )1932 1330 y F4(\()1967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38515
1318 y FX(h)1995 1330 y Ga(b)2034 1318 y FX(i)2062 1330
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38516
y Ga(S)5 b(;)2163 1318 y F9(\()2190 1330 y Ga(y)2238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38517
1318 y F9(\))2266 1330 y FL(Ax)o F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38518
b(c)p F4(\))q(\))q Ga(;)2642 1318 y F9(\()2670 1330 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38519
Ga(x)2722 1318 y F9(\))2749 1330 y Ga(T)e F4(\))1297
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38520
1417 y Gc(g)r(c)1245 1455 y F6(\000)-32 b(\000)h(!)100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38521
b FL(Cut)1593 1407 y FC( )1654 1455 y F4(\()1689 1443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38522
y FX(h)1717 1455 y Ga(a)1765 1443 y FX(i)1793 1455 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38523
Ga(S)5 b(;)1894 1443 y F9(\()1921 1455 y Ga(x)1973 1443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38524
y F9(\))2001 1455 y Ga(T)13 b F4(\))549 1621 y Gg(Thus)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38525
b(we)g(are)h(done.)p 3480 1621 4 62 v 3484 1563 55 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38526
v 3484 1621 V 3538 1621 4 62 v 321 1922 a Ge(B.2)119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38527
b(Pr)n(oofs)29 b(of)g(Chapter)i(3)321 2146 y Gg(In)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38528
b(this)g(section)i(we)c(are)i(mainly)h(concerned)h(with)e(translations)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38529
j(between)d(natural)i(deduction)g(cal-)321 2258 y(culi)d(and)g(sequent)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38530
i(calculi.)k(First)24 b(let)f(us)h(gi)n(v)o(e)f(tw)o(o)h(f)o(acts,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38531
(which)g(will)f(be)g(useful)i(later)-5 b(.)p Black 321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38532
2479 a Gb(F)n(act)23 b(B.2.1:)p Black Black 417 2655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38533
a Gg(\(i\))p Black 47 w FL(Cut)o F4(\()721 2643 y FX(h)749
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38534
2655 y Ga(a)797 2643 y FX(i)825 2655 y Ga(T)13 b(;)931
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38535
2643 y F9(\()958 2655 y Ga(x)1010 2643 y F9(\))1038 2655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38536
y Ga(S)5 b F4(\))1198 2618 y Gc(int)1159 2655 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38537
b(\000)g(!)1330 2622 y FX(\003)1394 2655 y Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38538
F4([)p Ga(a)26 b F4(:=)1680 2643 y F9(\()1707 2655 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38539
Ga(x)1759 2643 y F9(\))1787 2655 y Ga(S)5 b F4(])p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38540
392 2797 a Gg(\(ii\))p Black 47 w FL(Cut)o F4(\()721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38541
2785 y FX(h)749 2797 y Ga(a)797 2785 y FX(i)825 2797
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38542
y Ga(T)13 b(;)931 2785 y F9(\()958 2797 y Ga(x)1010 2785
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38543
y F9(\))1038 2797 y Ga(S)5 b F4(\))1198 2760 y Gc(int)1159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38544
2797 y F6(\000)-31 b(\000)g(!)1330 2764 y FX(\003)1394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38545
2797 y Ga(S)5 b F4([)p Ga(x)26 b F4(:=)1679 2785 y FX(h)1706
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38546
2797 y Ga(a)1754 2785 y FX(i)1782 2797 y Ga(T)13 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38547
Black 321 3010 a F7(Pr)l(oof)o(.)p Black 34 w Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38548
b(e)22 b(gi)n(v)o(e)g(the)h(details)h(for)f(\(i\).)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38549
b(If)22 b Ga(T)34 b Gg(does)24 b(not)f(freshly)h(introduce)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38550
Ga(a)p Gg(,)d(then)h(by)g(a)f(commuting)321 3123 y(reduction)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38551
b FL(Cut)p F4(\()863 3111 y FX(h)891 3123 y Ga(a)939
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38552
3111 y FX(i)967 3123 y Ga(T)12 b(;)1072 3111 y F9(\()1100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38553
3123 y Ga(x)1152 3111 y F9(\))1180 3123 y Ga(S)5 b F4(\))1346
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38554
3085 y Gc(int)1307 3123 y F6(\000)-32 b(\000)h(!)31 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38555
Ga(T)13 b F4([)p Ga(a)30 b F4(:=)1804 3111 y F9(\()1831
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38556
3123 y Ga(x)1883 3111 y F9(\))1911 3123 y Ga(S)5 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38557
Gg(.)37 b(Otherwise,)28 b(if)e Ga(T)38 b Gg(freshly)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38558
b(introduces)i Ga(a)p Gg(,)c(b)n(ut)321 3235 y(is)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38559
b(not)g(an)f(axiom,)i(then)g FL(Cut)p F4(\()1314 3223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38560
y FX(h)1341 3235 y Ga(a)1389 3223 y FX(i)1417 3235 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38561
Ga(T)13 b(;)1523 3223 y F9(\()1550 3235 y Ga(x)1602 3223
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38562
y F9(\))1630 3235 y Ga(S)5 b F4(\))38 b F6(\021)g Ga(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38563
b F4([)p Ga(a)38 b F4(:=)2185 3223 y F9(\()2212 3235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38564
y Ga(x)2264 3223 y F9(\))2291 3235 y Ga(S)5 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38565
Gg(;)34 b(if)c Ga(T)43 b Gg(is)30 b(an)h(axiom,)h(for)f(e)o(xample)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38566
3348 y FL(Ax)p F4(\()p Ga(z)t(;)15 b(a)p F4(\))p Gg(,)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38567
b(then)920 3503 y FL(Cut)p F4(\()1093 3491 y FX(h)1120
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38568
3503 y Ga(a)1168 3491 y FX(i)1196 3503 y Ga(T)13 b(;)1302
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38569
3491 y F9(\()1330 3503 y Ga(x)1382 3491 y F9(\))1409
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38570
3503 y Ga(S)5 b F4(\))1569 3466 y Gc(int)1531 3503 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38571
F6(\000)-32 b(\000)h(!)1701 3465 y F9(+)1785 3503 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38572
b F4([)p Ga(x)10 b F6(7!)g Ga(z)t F4(])26 b(=)f FL(Ax)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38573
F4(\()p Ga(z)t(;)15 b(a)p F4(\)[)p Ga(a)27 b F4(:=)2755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38574
3491 y F9(\()2782 3503 y Ga(x)2834 3491 y F9(\))2862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38575
3503 y Ga(S)5 b F4(])321 3657 y Gg(by)34 b(a)f(logical)i(reduction)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38576
(or)d(by)g(a)g(commuting)i(reduction)h(and)e(Lemma)e(2.2.11.)58
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38577
b(Thus)34 b(we)e(are)321 3770 y(done.)p 3480 3770 V 3484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38578
3712 55 4 v 3484 3770 V 3538 3770 4 62 v Black 321 3958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38579
a Gb(Pr)n(oof)24 b(of)g(Lemma)f(3.2.1.)p Black 33 w Gg(The)g(dif)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38580
(\002cult)i(case)f(is)f(as)h(follo)n(ws.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38581
1196 4096 a FL(Ax)o F4(\()p Ga(x;)15 b(a)p F4(\)[)p Ga(a)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38582
b F4(:=)1729 4084 y F9(\()1756 4096 y Ga(y)1804 4084
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38583
y F9(\))1832 4096 y Ga(S)5 b F4(][)p Ga(x)25 b F4(:=)2142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38584
4084 y FX(h)2169 4096 y Ga(b)2208 4084 y FX(i)2235 4096
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38585
y Ga(T)13 b F4(])1078 4209 y(=)47 b Ga(S)5 b F4([)p Ga(y)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38586
b F6(7!)d Ga(x)p F4(][)p Ga(x)25 b F4(:=)1742 4197 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38587
FX(h)1769 4209 y Ga(b)1808 4197 y FX(i)1836 4209 y Ga(T)12
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38588
b F4(])1078 4322 y F6(\021)47 b Ga(S)5 b F4([)p Ga(y)28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38589
b F4(:=)1476 4310 y FX(h)1504 4322 y Ga(b)1543 4310 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38590
FX(i)1570 4322 y Ga(T)13 b F4(][)p Ga(x)25 b F4(:=)1885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38591
4310 y FX(h)1912 4322 y Ga(b)1951 4310 y FX(i)1979 4322
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38592
y Ga(T)12 b F4(])1078 4439 y(=)47 b FL(Cut)o F4(\()1368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38593
4427 y FX(h)1396 4439 y Ga(b)1435 4427 y FX(i)1463 4439
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38594
y Ga(T)12 b(;)1568 4427 y F9(\()1596 4439 y Ga(y)1644
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38595
4427 y F9(\))1672 4439 y Ga(S)5 b F4(\)[)p Ga(x)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38596
F4(:=)1992 4427 y FX(h)2019 4439 y Ga(b)2058 4427 y FX(i)2085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38597
4439 y Ga(T)13 b F4(])2835 4406 y F9(\(1\))1078 4552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38598
y F4(=)47 b FL(Cut)o F4(\()1368 4540 y FX(h)1396 4552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38599
y Ga(b)1435 4540 y FX(i)1463 4552 y Ga(T)12 b(;)1568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38600
4540 y F9(\()1596 4552 y Ga(y)1644 4540 y F9(\))1672
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38601
4552 y Ga(S)5 b F4([)p Ga(x)25 b F4(:=)1956 4540 y FX(h)1984
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38602
4552 y Ga(b)2023 4540 y FX(i)2050 4552 y Ga(T)13 b F4(]\))978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38603
4631 y Gc(int)939 4668 y F6(\000)-32 b(\000)h(!)1109
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38604
4635 y FX(\003)1196 4668 y Ga(T)13 b F4([)p Ga(b)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38605
F4(:=)1472 4656 y F9(\()1499 4668 y Ga(y)1547 4656 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38606
F9(\))1575 4668 y Ga(S)5 b F4([)p Ga(x)25 b F4(:=)1859
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38607
4656 y FX(h)1887 4668 y Ga(b)1926 4656 y FX(i)1953 4668
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38608
y Ga(T)13 b F4(]])2835 4635 y F9(\(2\))1078 4785 y F6(\021)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38609
b Ga(T)13 b F4([)p Ga(b)d F6(7!)g Ga(a)p F4(][)p Ga(a)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38610
b F4(:=)1730 4773 y F9(\()1757 4785 y Ga(y)1805 4773
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38611
y F9(\))1832 4785 y Ga(S)5 b F4([)p Ga(x)26 b F4(:=)2117
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38612
4773 y FX(h)2145 4785 y Ga(b)2184 4773 y FX(i)2211 4785
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38613
y Ga(T)13 b F4(]])2835 4752 y F9(\(3\))1078 4898 y F4(=)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38614
b FL(Ax)o F4(\()p Ga(x;)15 b(a)p F4(\)[)p Ga(x)27 b F4(:=)1733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38615
4886 y FX(h)1760 4898 y Ga(b)1799 4886 y FX(i)1827 4898
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38616
y Ga(T)13 b F4(][)p Ga(a)25 b F4(:=)2138 4886 y F9(\()2165
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38617
4898 y Ga(y)2213 4886 y F9(\))2240 4898 y Ga(S)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38618
Ga(x)26 b F4(:=)2525 4886 y FX(h)2552 4898 y Ga(b)2591
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38619
4886 y FX(i)2619 4898 y Ga(T)13 b F4(]])p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38620
790 5056 a F9(\(1\))984 5089 y Gg(by)24 b(assumption)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38621
Ga(S)h Gg(freshly)e(introduces)i Ga(y)e Gg(and)f(is)g(not)g(an)f(axiom)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38622
790 5173 y F9(\(2\))984 5206 y Gg(by)h(F)o(act)f(B.2.1)790
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38623
5289 y F9(\(3\))984 5322 y Gg(by)h(assumption)i Ga(a)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38624
F6(62)g Ga(F)13 b(C)7 b F4(\()1872 5310 y FX(h)1899 5322
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38625
y Ga(b)1938 5310 y FX(i)1966 5322 y Ga(T)12 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38626
3480 5492 V 3484 5434 55 4 v 3484 5492 V 3538 5492 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38627
62 v Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38628
%%Page: 157 169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38629
TeXDict begin 157 168 bop Black 277 51 a Gb(B.2)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38630
(Chapter)f(3)2639 b(157)p 277 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38631
277 388 a(Pr)n(oof)28 b(of)e(Lemma)g(3.2.3.)p Black 34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38632
w Gg(If)h Ga(M)36 b Gg(is)27 b(a)f(v)n(ariable,)j(then)f(the)f(lemma)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38633
(is)h(by)g(routine)i(calculation.)41 b(In)277 501 y(case)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38634
b Ga(M)36 b Gg(corresponds)30 b(to)d(a)f(term)g(gi)n(v)o(en)h(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38635
(Lemma)g(3.2.2,)h(then)g F6(j)p Ga(M)10 b F6(j)2588 468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38636
y Fu(s)2588 524 y Gc(a)2656 501 y Gg(cannot)28 b(introduce)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38637
Ga(x)d Gg(and)277 614 y(hence)f(the)e(substitution)k(is)c(mo)o(v)o(ed)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38638
(inside)h(the)e(subterm\(s\).)31 b(In)23 b(consequence,)k(one)d(can)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38639
(apply)i(the)277 727 y(induction)30 b(hypothesis.)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38640
b(Otherwise,)28 b(the)f(top-le)n(v)o(el)h(term)f(constructor)j(of)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38641
Ga(M)36 b Gg(corresponds)31 b(to)26 b(an)277 840 y(elimination)g(rule,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38642
e(for)g(which)f(we)g(illustrate)j(the)e(lemma)f(with)g(one)h(case.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38643
Black 277 1044 a Gb(Case)p Black 46 w Ga(M)36 b F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38644
b Ga(S)20 b(T)13 b Gg(:)582 1174 y F6(j)p Ga(S)20 b(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38645
b F6(j)774 1141 y Fu(s)774 1196 y Gc(a)816 1174 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38646
Ga(\033)s F4(])1659 b Gg(with)24 b F4([)p Ga(\033)s F4(])i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38647
F6(\021)f F4([)p Ga(x)g F4(:=)3215 1162 y FX(h)3243 1174
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38648
y Ga(b)3282 1162 y FX(i)3309 1174 y F6(j)p Ga(N)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38649
F6(j)3442 1141 y Fu(s)3442 1201 y Gc(b)3477 1174 y F4(])439
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38650
1298 y(=)72 b F6(j)p Ga(S)5 b F6(j)693 1265 y Fu(s)693
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38651
1320 y Gc(c)728 1298 y F4([)p Ga(c)26 b F4(:=)939 1286
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38652
y F9(\()967 1298 y Ga(y)1015 1286 y F9(\))1042 1298 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38653
FL(Imp)1186 1320 y Gc(L)1238 1298 y F4(\()1273 1286 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38654
FX(h)1301 1298 y Ga(d)1348 1286 y FX(i)1376 1298 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38655
Ga(T)13 b F6(j)1492 1265 y Fu(s)1492 1326 y Gc(d)1533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38656
1298 y Ga(;)1573 1286 y F9(\()1601 1298 y Ga(z)1647 1286
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38657
y F9(\))1674 1298 y FL(Ax)p F4(\()p Ga(z)t(;)i(a)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38658
Ga(;)g(y)s F4(\))q(][)p Ga(\033)s F4(])779 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38659
b(\(6\))f(of)h F6(j)p 3383 1298 28 4 v 3401 1298 V 3418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38660
1298 V 65 w(j)3471 1265 y Fu(s)378 1385 y Gc(int)339
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38661
1422 y F6(\000)-31 b(\000)f(!)519 1389 y FX(\003)582
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38662
1422 y F6(j)p Ga(S)5 b F6(j)693 1389 y Fu(s)693 1445
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38663
y Gc(c)728 1422 y F4([)p Ga(\033)s F4(][)p Ga(c)27 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38664
F4(:=)1045 1410 y F9(\()1072 1422 y Ga(y)1120 1410 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38665
F9(\))1148 1422 y FL(Imp)1292 1444 y Gc(L)1344 1422 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38666
F4(\()1379 1410 y FX(h)1407 1422 y Ga(d)1454 1410 y FX(i)1482
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38667
1422 y F6(j)p Ga(T)13 b F6(j)1598 1389 y Fu(s)1598 1450
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38668
y Gc(d)1638 1422 y Ga(;)1678 1410 y F9(\()1706 1422 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38669
Ga(z)1752 1410 y F9(\))1780 1422 y FL(Ax)o F4(\()p Ga(z)t(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38670
F4(\))r Ga(;)g(y)s F4(\)[)p Ga(\033)s F4(]])565 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38671
b(Lemma)e(3.2.1)439 1546 y F4(=)72 b F6(j)p Ga(S)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38672
F6(j)693 1513 y Fu(s)693 1569 y Gc(c)728 1546 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38673
Ga(\033)s F4(][)p Ga(c)27 b F4(:=)1045 1534 y F9(\()1072
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38674
1546 y Ga(y)1120 1534 y F9(\))1148 1546 y FL(Imp)1292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38675
1568 y Gc(L)1344 1546 y F4(\()1379 1534 y FX(h)1407 1546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38676
y Ga(d)1454 1534 y FX(i)1482 1546 y F6(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38677
F6(j)1598 1513 y Fu(s)1598 1574 y Gc(d)1638 1546 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38678
Ga(\033)s F4(])q Ga(;)1784 1534 y F9(\()1812 1546 y Ga(z)1858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38679
1534 y F9(\))1886 1546 y FL(Ax)o F4(\()p Ga(z)t(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38680
F4(\))r Ga(;)g(y)s F4(\)])378 1633 y Gc(int)339 1671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38681
y F6(\000)-31 b(\000)f(!)519 1638 y FX(\003)582 1671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38682
y F6(j)p Ga(S)5 b F4([)p Ga(x)26 b F4(:=)f Ga(N)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38683
F4(])p F6(j)1025 1638 y Fu(s)1025 1693 y Gc(c)1060 1671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38684
y F4([)p Ga(c)26 b F4(:=)1271 1659 y F9(\()1298 1671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38685
y Ga(y)1346 1659 y F9(\))1374 1671 y FL(Imp)1518 1692
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38686
y Gc(L)1570 1671 y F4(\()1605 1659 y FX(h)1633 1671 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38687
Ga(d)1680 1659 y FX(i)1708 1671 y F6(j)p Ga(T)13 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38688
Ga(x)25 b F4(:=)g Ga(N)10 b F4(])p F6(j)2155 1638 y Fu(s)2155
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38689
1698 y Gc(d)2196 1671 y Ga(;)2236 1659 y F9(\()2264 1671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38690
y Ga(z)2310 1659 y F9(\))2338 1671 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38691
Ga(z)t(;)15 b(a)p F4(\))q Ga(;)g(y)s F4(\))q(])498 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38692
Gg(by)23 b(IH)439 1795 y F4(=)72 b F6(j)p Ga(S)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38693
Ga(x)26 b F4(:=)f Ga(N)10 b F4(])25 b Ga(T)13 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38694
Ga(x)25 b F4(:=)h Ga(N)10 b F4(])p F6(j)1448 1762 y Fu(s)1448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38695
1817 y Gc(a)3015 1795 y Gg(by)24 b(\(6\))f(of)h F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38696
3383 1795 V 3401 1795 V 3418 1795 V 65 w(j)3471 1762
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38697
y Fu(s)439 1919 y F6(\021)72 b(j)p F4(\()p Ga(S)21 b(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38698
b F4(\)[)p Ga(x)25 b F4(:=)g Ga(N)10 b F4(])p F6(j)1176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38699
1886 y Fu(s)1176 1941 y Gc(a)p 3436 2026 4 62 v 3440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38700
1968 55 4 v 3440 2026 V 3494 2026 4 62 v Black 277 2238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38701
a Gb(Pr)n(oof)32 b(of)f(Pr)n(oposition)i(3.2.4.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38702
33 w Gg(W)-7 b(e)31 b(shall)h(be)o(gin)g(by)f(gi)n(ving)i(in)e(turn)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38703
(the)f(details)i(where)f(a)e(beta-)277 2351 y(reduction)c(occurs)f(on)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38704
(the)g(top-le)n(v)o(el.)p Black 277 2555 a Gb(Case)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38705
Black 46 w Ga(M)36 b F6(\021)25 b F4(\()p Ga(\025x:S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38706
b F4(\))15 b Ga(T)1160 2518 y Gc(\014)1097 2555 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38707
b(\000)f(!)25 b Ga(S)5 b F4([)p Ga(x)26 b F4(:=)f Ga(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38708
b F4(])25 b F6(\021)g Ga(M)1887 2522 y FX(0)1911 2555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38709
y Gg(:)315 2706 y(\(i\))201 b F6(j)p F4(\()p Ga(\025)q(x:S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38710
b F4(\))15 b Ga(T)e F6(j)994 2673 y Fu(s)994 2729 y Gc(c)439
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38711
2830 y F4(=)91 b FL(Imp)746 2852 y Gc(R)804 2830 y F4(\()839
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38712
2818 y F9(\()866 2830 y Ga(x)918 2818 y F9(\))q FX(h)973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38713
2830 y Ga(a)1021 2818 y FX(i)1049 2830 y F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38714
b F6(j)1160 2797 y Fu(s)1160 2853 y Gc(a)1202 2830 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38715
Ga(;)15 b(b)p F4(\)[)p Ga(b)26 b F4(:=)1527 2818 y F9(\()1555
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38716
2830 y Ga(y)1603 2818 y F9(\))1630 2830 y FL(Imp)1774
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38717
2852 y Gc(L)1827 2830 y F4(\()1862 2818 y FX(h)1889 2830
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38718
y Ga(d)1936 2818 y FX(i)1964 2830 y F6(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38719
F6(j)2080 2797 y Fu(s)2080 2858 y Gc(d)2121 2830 y Ga(;)2161
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38720
2818 y F9(\()2189 2830 y Ga(z)2235 2818 y F9(\))2262
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38721
2830 y FL(Ax)p F4(\()p Ga(z)t(;)i(c)p F4(\))r Ga(;)g(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38722
F4(\)])237 b Gg(by)24 b(\(2,6\))f(of)h F6(j)p 3383 2830
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38723
28 4 v 3401 2830 V 3418 2830 V 65 w(j)3471 2797 y Fu(s)439
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38724
2955 y F4(=)91 b FL(Cut)p F4(\()774 2943 y FX(h)802 2955
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38725
y Ga(b)841 2943 y FX(i)869 2955 y FL(Imp)1013 2976 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38726
Gc(R)1071 2955 y F4(\()1106 2943 y F9(\()1133 2955 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38727
Ga(x)1185 2943 y F9(\))q FX(h)1240 2955 y Ga(a)1288 2943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38728
y FX(i)1316 2955 y F6(j)p Ga(S)5 b F6(j)1427 2922 y Fu(s)1427
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38729
2977 y Gc(a)1469 2955 y Ga(;)15 b(b)p F4(\))q Ga(;)1624
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38730
2943 y F9(\()1652 2955 y Ga(y)1700 2943 y F9(\))1727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38731
2955 y FL(Imp)1871 2976 y Gc(L)1923 2955 y F4(\()1958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38732
2943 y FX(h)1986 2955 y Ga(d)2033 2943 y FX(i)2061 2955
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38733
y F6(j)p Ga(T)e F6(j)2177 2922 y Fu(s)2177 2982 y Gc(d)2218
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38734
2955 y Ga(;)2258 2943 y F9(\()2286 2955 y Ga(z)2332 2943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38735
y F9(\))2359 2955 y FL(Ax)p F4(\()p Ga(z)t(;)i(c)p F4(\))r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38736
Ga(;)g(y)s F4(\)\))178 b Gg(by)23 b(Lem.)f(3.2.2)378
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38737
3042 y Gc(int)339 3079 y F6(\000)-31 b(\000)f(!)92 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38738
FL(Cut)p F4(\()774 3067 y FX(h)802 3079 y Ga(a)850 3067
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38739
y FX(i)878 3079 y FL(Cut)o F4(\()1050 3067 y FX(h)1078
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38740
3079 y Ga(d)1125 3067 y FX(i)1153 3079 y F6(j)p Ga(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38741
b F6(j)1269 3046 y Fu(s)1269 3107 y Gc(d)1310 3079 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38742
Ga(;)1350 3067 y F9(\()1378 3079 y Ga(x)1430 3067 y F9(\))1457
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38743
3079 y F6(j)p Ga(S)5 b F6(j)1568 3046 y Fu(s)1568 3101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38744
y Gc(a)1610 3079 y F4(\))p Ga(;)1685 3067 y F9(\()1713
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38745
3079 y Ga(z)1759 3067 y F9(\))1787 3079 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38746
Ga(z)t(;)15 b(c)p F4(\))q(\))53 b Ga(y)26 b Gg(is)d(freshly)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38747
(introduced)g(by)e(\(6\))f(of)h F6(j)p 3383 3079 V 3401
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38748
3079 V 3418 3079 V 65 w(j)3471 3046 y Fu(s)378 3166 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38749
Gc(int)339 3203 y F6(\000)-31 b(\000)f(!)519 3170 y F9(+)601
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38750
3203 y FL(Cut)p F4(\()774 3191 y FX(h)802 3203 y Ga(d)849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38751
3191 y FX(i)877 3203 y F6(j)p Ga(T)13 b F6(j)993 3170
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38752
y Fu(s)993 3231 y Gc(d)1034 3203 y Ga(;)1074 3191 y F9(\()1101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38753
3203 y Ga(x)1153 3191 y F9(\))1181 3203 y F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38754
b F6(j)1292 3170 y Fu(s)1292 3226 y Gc(a)1334 3203 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38755
F4(\)[)p Ga(a)10 b F6(7!)g Ga(c)p F4(])476 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38756
b(a)f(comm.)g(reduction)j(and)e(Lem.)e(2.2.11)439 3327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38757
y F6(\021)91 b FL(Cut)p F4(\()774 3315 y FX(h)802 3327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38758
y Ga(d)849 3315 y FX(i)877 3327 y F6(j)p Ga(T)13 b F6(j)993
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38759
3294 y Fu(s)993 3355 y Gc(d)1034 3327 y Ga(;)1074 3315
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38760
y F9(\()1101 3327 y Ga(x)1153 3315 y F9(\))1181 3327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38761
y F6(j)p Ga(S)5 b F6(j)1292 3294 y Fu(s)1292 3350 y Gc(c)1327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38762
3327 y F4(\))880 b Ga(a)25 b F6(62)g Ga(F)13 b(C)7 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38763
F4(\()2579 3315 y FX(h)2607 3327 y Ga(d)2654 3315 y FX(i)2681
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38764
3327 y F6(j)p Ga(T)13 b F6(j)2797 3294 y Fu(s)2797 3355
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38765
y Gc(d)2838 3327 y F4(\))23 b Gg(by)h(Remark)f(2.2.7)378
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38766
3414 y Gc(int)339 3451 y F6(\000)-31 b(\000)f(!)519 3418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38767
y FX(\003)601 3451 y F6(j)p Ga(S)5 b F6(j)712 3418 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38768
Fu(s)712 3474 y Gc(c)748 3451 y F4([)p Ga(x)25 b F4(:=)971
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38769
3439 y FX(h)999 3451 y Ga(d)1046 3439 y FX(i)1073 3451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38770
y F6(j)p Ga(T)13 b F6(j)1189 3418 y Fu(s)1189 3479 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38771
Gc(d)1230 3451 y F4(])1759 b Gg(by)23 b(F)o(act)g(B.2.1)378
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38772
3539 y Gc(int)339 3576 y F6(\000)-31 b(\000)f(!)519 3543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38773
y FX(\003)601 3576 y F6(j)p Ga(S)5 b F4([)p Ga(x)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38774
F4(:=)f Ga(T)13 b F4(])p F6(j)1027 3543 y Fu(s)1027 3598
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38775
y Gc(c)2906 3576 y Gg(by)24 b(Lemma)e(3.2.3)315 3741
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38776
y(\(ii\))157 b F6(j)p F4(\()p Ga(\025x:S)5 b F4(\))15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38777
b Ga(T)e F6(j)974 3708 y Fu(s)974 3764 y Gc(c)1010 3741
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38778
y F4([)p Ga(\033)s F4(])1554 b Gg(with)23 b F4([)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38779
F4(])j F6(\021)f F4([)p Ga(c)h F4(:=)3291 3729 y F9(\()3319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38780
3741 y Ga(y)3367 3729 y F9(\))3394 3741 y Ga(N)10 b F4(])439
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38781
3866 y(=)72 b FL(Imp)726 3887 y Gc(R)784 3866 y F4(\()819
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38782
3854 y F9(\()847 3866 y Ga(x)899 3854 y F9(\))p FX(h)954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38783
3866 y Ga(c)993 3854 y FX(i)1020 3866 y F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38784
b F6(j)1131 3833 y Fu(s)1131 3888 y Gc(c)1167 3866 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38785
Ga(;)15 b(a)p F4(\)[)p Ga(a)26 b F4(:=)1510 3854 y F9(\()1538
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38786
3866 y Ga(z)1584 3854 y F9(\))1611 3866 y FL(Imp)1756
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38787
3887 y Gc(L)1808 3866 y F4(\()1843 3854 y FX(h)1871 3866
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38788
y Ga(b)1910 3854 y FX(i)1937 3866 y F6(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38789
F6(j)2053 3833 y Fu(s)2053 3893 y Gc(b)2088 3866 y Ga(;)2128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38790
3854 y F9(\()2156 3866 y Ga(x)2208 3854 y F9(\))2235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38791
3866 y FL(Ax)p F4(\()p Ga(x;)i(c)p F4(\))q Ga(;)g(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38792
F4(\))q(][)p Ga(\033)s F4(])155 b Gg(by)24 b(\(2,6\))f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38793
F6(j)p 3383 3866 V 3401 3866 V 3418 3866 V 65 w(j)3471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38794
3833 y Fu(s)439 3990 y F4(=)72 b FL(Cut)p F4(\()755 3978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38795
y FX(h)782 3990 y Ga(a)830 3978 y FX(i)858 3990 y FL(Imp)1002
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38796
4012 y Gc(R)1060 3990 y F4(\()1095 3978 y F9(\()1123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38797
3990 y Ga(x)1175 3978 y F9(\))p FX(h)1230 3990 y Ga(c)1269
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38798
3978 y FX(i)1297 3990 y F6(j)p Ga(S)5 b F6(j)1408 3957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38799
y Fu(s)1408 4012 y Gc(c)1443 3990 y Ga(;)15 b(a)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38800
Ga(;)1607 3978 y F9(\()1634 3990 y Ga(z)1680 3978 y F9(\))1708
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38801
3990 y FL(Imp)1853 4012 y Gc(L)1905 3990 y F4(\()1940
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38802
3978 y FX(h)1968 3990 y Ga(b)2007 3978 y FX(i)2034 3990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38803
y F6(j)p Ga(T)e F6(j)2150 3957 y Fu(s)2150 4017 y Gc(b)2185
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38804
3990 y Ga(;)2225 3978 y F9(\()2252 3990 y Ga(x)2304 3978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38805
y F9(\))2332 3990 y FL(Ax)o F4(\()p Ga(x;)i(c)p F4(\))r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38806
Ga(;)g(z)t F4(\))q(\)[)p Ga(\033)s F4(])96 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38807
b(Lem.)f(3.2.2)439 4114 y F4(=)72 b FL(Cut)p F4(\()755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38808
4102 y FX(h)782 4114 y Ga(a)830 4102 y FX(i)858 4114
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38809
y FL(Imp)1002 4136 y Gc(R)1060 4114 y F4(\()1095 4102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38810
y F9(\()1123 4114 y Ga(x)1175 4102 y F9(\))p FX(h)1230
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38811
4114 y Ga(c)1269 4102 y FX(i)1297 4114 y F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38812
b F6(j)1408 4081 y Fu(s)1408 4136 y Gc(c)1443 4114 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38813
Ga(;)15 b(a)p F4(\))q Ga(;)1607 4102 y F9(\()1634 4114
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38814
y Ga(z)1680 4102 y F9(\))1708 4114 y FL(Imp)1853 4136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38815
y Gc(L)1905 4114 y F4(\()1940 4102 y FX(h)1968 4114 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38816
Ga(b)2007 4102 y FX(i)2034 4114 y F6(j)p Ga(T)e F6(j)2150
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38817
4081 y Fu(s)2150 4142 y Gc(b)2185 4114 y Ga(;)2225 4102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38818
y F9(\()2252 4114 y Ga(y)2300 4102 y F9(\))2328 4114
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38819
y Ga(N)d(;)15 b(z)t F4(\))q(\))45 b Ga(c)26 b F6(62)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38820
Ga(F)13 b(C)7 b F4(\()2942 4102 y FX(h)2969 4114 y Ga(b)3008
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38821
4102 y FX(i)3035 4114 y F6(j)p Ga(T)13 b F6(j)3151 4081
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38822
y Fu(s)3151 4142 y Gc(b)3186 4114 y Ga(;)3226 4102 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38823
FX(h)3254 4114 y Ga(c)3293 4102 y FX(i)3321 4114 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38824
Ga(S)5 b F6(j)3432 4081 y Fu(s)3432 4136 y Gc(c)3467
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38825
4114 y F4(\))378 4201 y Gc(int)339 4238 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38826
b(\000)f(!)73 b FL(Cut)p F4(\()755 4226 y FX(h)782 4238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38827
y Ga(b)821 4226 y FX(i)849 4238 y F6(j)p Ga(T)13 b F6(j)965
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38828
4205 y Fu(s)965 4266 y Gc(b)1000 4238 y Ga(;)1040 4226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38829
y F9(\()1067 4238 y Ga(x)1119 4226 y F9(\))1147 4238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38830
y FL(Cut)p F4(\()1320 4226 y FX(h)1347 4238 y Ga(c)1386
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38831
4226 y FX(i)1414 4238 y F6(j)p Ga(S)5 b F6(j)1525 4205
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38832
y Fu(s)1525 4261 y Gc(c)1560 4238 y Ga(;)1600 4226 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38833
F9(\()1628 4238 y Ga(y)1676 4226 y F9(\))1704 4238 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38834
Ga(N)10 b F4(\)\))319 b Ga(z)27 b Gg(is)c(freshly)j(introduced)g(by)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38835
(\(6\))f(of)h F6(j)p 3383 4238 V 3401 4238 V 3418 4238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38836
V 65 w(j)3471 4205 y Fu(s)378 4325 y Gc(int)339 4362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38837
y F6(\000)-31 b(\000)f(!)73 b FL(Cut)p F4(\()755 4350
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38838
y FX(h)782 4362 y Ga(c)821 4350 y FX(i)849 4362 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38839
Ga(S)5 b F6(j)960 4329 y Fu(s)960 4385 y Gc(c)995 4362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38840
y Ga(;)1035 4350 y F9(\()1063 4362 y Ga(y)1111 4350 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38841
F9(\))1139 4362 y Ga(N)10 b F4(\)[)p Ga(x)25 b F4(:=)1481
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38842
4350 y FX(h)1508 4362 y Ga(b)1547 4350 y FX(i)1574 4362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38843
y F6(j)p Ga(T)13 b F6(j)1690 4329 y Fu(s)1690 4390 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38844
Gc(b)1725 4362 y F4(])439 4487 y(=)72 b FL(Cut)p F4(\()755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38845
4475 y FX(h)782 4487 y Ga(c)821 4475 y FX(i)849 4487
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38846
y F6(j)p Ga(S)5 b F6(j)960 4454 y Fu(s)960 4509 y Gc(c)995
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38847
4487 y F4([)p Ga(x)26 b F4(:=)1219 4475 y FX(h)1247 4487
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38848
y Ga(b)1286 4475 y FX(i)1313 4487 y F6(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38849
F6(j)1429 4454 y Fu(s)1429 4514 y Gc(b)1464 4487 y F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38850
Ga(;)1529 4475 y F9(\()1557 4487 y Ga(y)1605 4475 y F9(\))1632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38851
4487 y Ga(N)d F4(\))550 b Ga(x)25 b F6(62)g Ga(F)13 b(N)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38852
F4(\()2652 4475 y F9(\()2680 4487 y Ga(y)2728 4475 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38853
F9(\))2755 4487 y Ga(N)g F4(\))23 b Gg(by)h(Remark)f(2.2.7)378
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38854
4574 y Gc(int)339 4611 y F6(\000)-31 b(\000)f(!)519 4578
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38855
y FX(\003)582 4611 y FL(Cut)p F4(\()755 4599 y FX(h)782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38856
4611 y Ga(c)821 4599 y FX(i)849 4611 y F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38857
b F4([)p Ga(x)26 b F4(:=)f Ga(T)13 b F4(])p F6(j)1275
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38858
4578 y Fu(s)1275 4633 y Gc(c)1310 4611 y Ga(;)1350 4599
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38859
y F9(\()1378 4611 y Ga(y)1426 4599 y F9(\))1453 4611
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38860
y Ga(N)d F4(\))1335 b Gg(by)24 b(Lemma)e(3.2.3)378 4698
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38861
y Gc(int)339 4735 y F6(\000)-31 b(\000)f(!)519 4702 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38862
FX(\003)582 4735 y F6(j)p Ga(S)5 b F4([)p Ga(x)26 b F4(:=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38863
Ga(T)13 b F4(])p F6(j)1008 4702 y Fu(s)1008 4757 y Gc(c)1043
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38864
4735 y F4([)p Ga(\033)s F4(])1866 b Gg(by)23 b(F)o(act)g(B.2.1)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38865
4933 y(W)-7 b(e)21 b(omit)g(the)g(other)i(base)f(cases)g(and)g(proceed)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38866
h(with)e(the)g(details)i(for)f(inner)g(reductions.)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38867
b(In)21 b(case)h Ga(M)277 5046 y Gg(corresponds)29 b(to)d(a)f(term)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38868
(gi)n(v)o(en)i(in)e(Lemma)g(3.2.2,)h(\(i\))f(and)i(\(ii\))e(are)h(by)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38869
(routine)h(calculation)i(using)277 5159 y(the)f(induction)j(hypothesis)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38870
g(\(note)d(in)g(\(ii\))h F6(j)p Ga(M)10 b F6(j)1800 5126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38871
y Fu(s)1800 5182 y Gc(c)1835 5159 y F4([)p Ga(c)34 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38872
F4(:=)2062 5147 y F9(\()2090 5159 y Ga(x)2142 5147 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38873
F9(\))2169 5159 y Ga(N)10 b F4(])27 b Gg(e)o(xpands)j(to)e(a)f(cut\).)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38874
43 b(W)-7 b(e)27 b(are)h(left)g(to)277 5272 y(check)d(the)f(cases)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38875
(where)f(the)g(top-le)n(v)o(el)h(term)f(constructor)j(of)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38876
Ga(M)33 b Gg(corresponds)28 b(to)23 b(an)h(elimination)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38877
5385 y(rule.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38878
%%Page: 158 170
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38879
TeXDict begin 158 169 bop Black -144 51 a Gb(158)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38880
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38881
4 v Black Black 321 388 a(Case)p Black 47 w Ga(M)35 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38882
F6(\021)25 b Ga(S)20 b(T)1004 351 y Gc(\014)940 388 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38883
F6(\000)-31 b(\000)f(!)26 b Ga(S)1197 355 y FX(0)1235
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38884
388 y Ga(T)38 b F6(\021)25 b Ga(M)1520 355 y FX(0)1566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38885
388 y Gg(with)e Ga(S)1900 351 y Gc(\014)1837 388 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38886
b(\000)h(!)25 b Ga(S)2093 355 y FX(0)2116 388 y Gg(:)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38887
539 y(By)k(IH)g(we)g(ha)n(v)o(e)h F6(j)p Ga(S)5 b F6(j)1252
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38888
506 y Fu(s)1252 562 y Gc(c)1363 502 y(int)1324 539 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38889
F6(\000)-31 b(\000)g(!)1494 506 y F9(+)1590 539 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38890
Ga(S)5 b F6(j)1701 506 y Fu(s)1701 562 y Gc(c)1766 539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38891
y Gg(and)30 b F6(j)p Ga(S)5 b F6(j)2037 506 y Fu(s)2037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38892
562 y Gc(c)2072 539 y F4([)p Ga(c)38 b F4(:=)2306 527
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38893
y F9(\()2334 539 y Ga(y)2382 527 y F9(\))2409 539 y Ga(P)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38894
b F4(])2581 502 y Gc(int)2542 539 y F6(\000)-31 b(\000)g(!)2713
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38895
506 y F9(+)2808 539 y F6(j)p Ga(S)2894 506 y FX(0)2918
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38896
539 y F6(j)2943 506 y Fu(s)2943 562 y Gc(c)2978 539 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38897
F4([)p Ga(c)37 b F4(:=)3212 527 y F9(\()3240 539 y Ga(y)3288
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38898
527 y F9(\))3315 539 y Ga(P)13 b F4(])29 b Gg(for)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38899
652 y(an)o(y)22 b(substitution)j(where)d Ga(P)35 b Gg(freshly)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38900
b(introduces)i Ga(y)f Gg(and)e(is)g(not)g(an)g(axiom.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38901
b(T)-7 b(ak)o(e)3174 640 y F9(\()3201 652 y Ga(y)3249
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38902
640 y F9(\))3276 652 y Ga(P)35 b Gg(to)22 b(be)549 753
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38903
y F9(\()576 765 y Ga(y)624 753 y F9(\))651 765 y FL(Imp)796
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38904
787 y Gc(L)848 765 y F4(\()883 753 y FX(h)911 765 y Ga(b)950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38905
753 y FX(i)977 765 y F6(j)p Ga(T)13 b F6(j)1093 732 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38906
Fu(s)1093 793 y Gc(b)1128 765 y Ga(;)1168 753 y F9(\()1196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38907
765 y Ga(z)1242 753 y F9(\))1269 765 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38908
Ga(z)t(;)i(c)p F4(\))r Ga(;)g(y)s F4(\))p Gg(,)23 b(then)h(we)f(ha)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38909
o(e)h(the)g(follo)n(wing)h(reduction.)360 926 y(\(i\))201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38910
b F6(j)p Ga(S)20 b(T)13 b F6(j)838 893 y Fu(s)838 949
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38911
y Gc(c)483 1051 y F4(=)92 b F6(j)p Ga(S)5 b F6(j)757
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38912
1018 y Fu(s)757 1073 y Gc(a)799 1051 y F4([)p Ga(a)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38913
b F4(:=)1018 1039 y F9(\()1046 1051 y Ga(y)1094 1039
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38914
y F9(\))1121 1051 y FL(Imp)1266 1072 y Gc(L)1318 1051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38915
y F4(\()1353 1039 y FX(h)1381 1051 y Ga(b)1420 1039 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38916
FX(i)1447 1051 y F6(j)p Ga(T)13 b F6(j)1563 1018 y Fu(s)1563
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38917
1078 y Gc(b)1598 1051 y Ga(;)1638 1039 y F9(\()1665 1051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38918
y Ga(z)1711 1039 y F9(\))1739 1051 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38919
Ga(z)t(;)i(c)p F4(\))q Ga(;)g(y)s F4(\))q(])872 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38920
b(\(6\))g(of)f F6(j)p 3427 1051 28 4 v 3445 1051 V 3463
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38921
1051 V 65 w(j)3515 1018 y Fu(s)422 1138 y Gc(int)383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38922
1175 y F6(\000)-31 b(\000)g(!)563 1142 y F9(+)646 1175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38923
y F6(j)p Ga(S)732 1142 y FX(0)755 1175 y F6(j)780 1142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38924
y Fu(s)780 1197 y Gc(a)822 1175 y F4([)p Ga(a)26 b F4(:=)1042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38925
1163 y F9(\()1069 1175 y Ga(y)1117 1163 y F9(\))1144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38926
1175 y FL(Imp)1289 1197 y Gc(L)1341 1175 y F4(\()1376
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38927
1163 y FX(h)1404 1175 y Ga(b)1443 1163 y FX(i)1470 1175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38928
y F6(j)p Ga(T)13 b F6(j)1586 1142 y Fu(s)1586 1203 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38929
Gc(b)1621 1175 y Ga(;)1661 1163 y F9(\()1689 1175 y Ga(z)1735
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38930
1163 y F9(\))1762 1175 y FL(Ax)p F4(\()p Ga(z)t(;)i(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38931
F4(\))r Ga(;)g(y)s F4(\)])1127 b Gg(by)23 b(IH)483 1299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38932
y F4(=)92 b F6(j)p Ga(S)732 1266 y FX(0)770 1299 y Ga(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38933
b F6(j)861 1266 y Fu(s)861 1322 y Gc(c)3059 1299 y Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38934
b(\(6\))g(of)f F6(j)p 3427 1299 V 3445 1299 V 3463 1299
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38935
V 65 w(j)3515 1266 y Fu(s)549 1451 y Gg(In)g(\(ii\))h(we)f(tak)o(e)1083
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38936
1439 y F9(\()1110 1451 y Ga(y)1158 1439 y F9(\))1186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38937
1451 y Ga(P)35 b Gg(to)24 b(be)1481 1439 y F9(\()1509
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38938
1451 y Ga(z)1555 1439 y F9(\))1582 1451 y FL(Imp)1727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38939
1473 y Gc(L)1779 1451 y F4(\()1814 1439 y FX(h)1842 1451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38940
y Ga(b)1881 1439 y FX(i)1908 1451 y F6(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38941
F6(j)2024 1418 y Fu(s)2024 1479 y Gc(b)2059 1451 y Ga(;)2099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38942
1439 y F9(\()2127 1451 y Ga(u)2179 1439 y F9(\))2206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38943
1451 y FL(Ax)p F4(\()p Ga(u;)i(c)p F4(\))q Ga(;)g(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38944
F4(\))q([)p Ga(c)26 b F4(:=)2844 1439 y F9(\()2872 1451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38945
y Ga(y)2920 1439 y F9(\))2947 1451 y Ga(N)10 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38946
Gg(.)360 1612 y(\(ii\))176 b F6(j)p Ga(S)20 b(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38947
F6(j)838 1579 y Fu(s)838 1635 y Gc(c)873 1612 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38948
Ga(c)26 b F4(:=)1084 1600 y F9(\()1111 1612 y Ga(y)1159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38949
1600 y F9(\))1186 1612 y Ga(N)10 b F4(])483 1736 y(=)92
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38950
b F6(j)p Ga(S)5 b F6(j)757 1704 y Fu(s)757 1759 y Gc(a)799
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38951
1736 y F4([)p Ga(a)25 b F4(:=)1018 1724 y F9(\()1046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38952
1736 y Ga(z)1092 1724 y F9(\))1120 1736 y FL(Imp)1264
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38953
1758 y Gc(L)1316 1736 y F4(\()1351 1724 y FX(h)1379 1736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38954
y Ga(b)1418 1724 y FX(i)1445 1736 y F6(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38955
F6(j)1561 1704 y Fu(s)1561 1764 y Gc(b)1596 1736 y Ga(;)1636
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38956
1724 y F9(\()1664 1736 y Ga(u)1716 1724 y F9(\))1743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38957
1736 y FL(Ax)p F4(\()p Ga(u;)i(c)p F4(\))q Ga(;)g(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38958
F4(\))q(][)p Ga(c)27 b F4(:=)2407 1724 y F9(\()2434 1736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38959
y Ga(y)2482 1724 y F9(\))2509 1736 y Ga(N)10 b F4(])442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38960
b Gg(by)24 b(\(6\))g(of)f F6(j)p 3427 1736 V 3445 1736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38961
V 3463 1736 V 65 w(j)3515 1704 y Fu(s)483 1861 y F6(\021)92
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38962
b(j)p Ga(S)5 b F6(j)757 1828 y Fu(s)757 1883 y Gc(a)799
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38963
1861 y F4([)p Ga(a)25 b F4(:=)1018 1849 y F9(\()1046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38964
1861 y Ga(z)1092 1849 y F9(\))1120 1861 y FL(Imp)1264
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38965
1883 y Gc(L)1316 1861 y F4(\()1351 1849 y FX(h)1379 1861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38966
y Ga(b)1418 1849 y FX(i)1445 1861 y F6(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38967
F6(j)1561 1828 y Fu(s)1561 1888 y Gc(b)1596 1861 y Ga(;)1636
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38968
1849 y F9(\()1664 1861 y Ga(u)1716 1849 y F9(\))1743
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38969
1861 y FL(Ax)p F4(\()p Ga(u;)i(c)p F4(\))q Ga(;)g(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38970
F4(\))q([)p Ga(c)26 b F4(:=)2382 1849 y F9(\()2409 1861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38971
y Ga(y)2457 1849 y F9(\))2484 1861 y Ga(N)10 b F4(]])422
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38972
1948 y Gc(int)383 1985 y F6(\000)-31 b(\000)g(!)563 1952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38973
y F9(+)646 1985 y F6(j)p Ga(S)732 1952 y FX(0)755 1985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38974
y F6(j)780 1952 y Fu(s)780 2007 y Gc(a)822 1985 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38975
Ga(a)26 b F4(:=)1042 1973 y F9(\()1069 1985 y Ga(z)1115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38976
1973 y F9(\))1143 1985 y FL(Imp)1287 2007 y Gc(L)1340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38977
1985 y F4(\()1375 1973 y FX(h)1402 1985 y Ga(b)1441 1973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38978
y FX(i)1469 1985 y F6(j)p Ga(T)13 b F6(j)1585 1952 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38979
Fu(s)1585 2013 y Gc(b)1619 1985 y Ga(;)1659 1973 y F9(\()1687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38980
1985 y Ga(u)1739 1973 y F9(\))1767 1985 y FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38981
Ga(u;)i(c)p F4(\))r Ga(;)g(z)t F4(\))q([)p Ga(c)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38982
F4(:=)2405 1973 y F9(\()2432 1985 y Ga(y)2480 1973 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38983
F9(\))2508 1985 y Ga(N)9 b F4(]])697 b Gg(by)23 b(IH)483
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38984
2109 y F6(\021)92 b(j)p Ga(S)732 2076 y FX(0)755 2109
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38985
y F6(j)780 2076 y Fu(s)780 2132 y Gc(a)822 2109 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38986
Ga(a)26 b F4(:=)1042 2097 y F9(\()1069 2109 y Ga(z)1115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38987
2097 y F9(\))1143 2109 y FL(Imp)1287 2131 y Gc(L)1340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38988
2109 y F4(\()1375 2097 y FX(h)1402 2109 y Ga(b)1441 2097
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38989
y FX(i)1469 2109 y F6(j)p Ga(T)13 b F6(j)1585 2076 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38990
Fu(s)1585 2137 y Gc(b)1619 2109 y Ga(;)1659 2097 y F9(\()1687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38991
2109 y Ga(u)1739 2097 y F9(\))1767 2109 y FL(Ax)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38992
Ga(u;)i(c)p F4(\))r Ga(;)g(z)t F4(\))q(][)p Ga(c)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38993
F4(:=)2430 2097 y F9(\()2458 2109 y Ga(y)2506 2097 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38994
F9(\))2533 2109 y Ga(N)10 b F4(])483 2233 y(=)92 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38995
Ga(S)732 2200 y FX(0)770 2233 y Ga(T)13 b F6(j)861 2200
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38996
y Fu(s)861 2256 y Gc(c)896 2233 y F4([)p Ga(c)26 b F4(:=)1107
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38997
2221 y F9(\()1134 2233 y Ga(y)1182 2221 y F9(\))1210
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38998
2233 y Ga(N)10 b F4(])1741 b Gg(by)24 b(\(6\))g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 38999
F6(j)p 3427 2233 V 3445 2233 V 3463 2233 V 65 w(j)3515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39000
2200 y Fu(s)p Black 321 2425 a Gb(Case)p Black 47 w Ga(M)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39001
b F6(\021)25 b Ga(S)20 b(T)1004 2388 y Gc(\014)940 2425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39002
y F6(\000)-31 b(\000)f(!)26 b Ga(S)20 b(T)1278 2392 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39003
FX(0)1326 2425 y F6(\021)25 b Ga(M)1520 2392 y FX(0)1566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39004
2425 y Gg(with)e Ga(T)1905 2388 y Gc(\014)1841 2425 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39005
F6(\000)-31 b(\000)g(!)25 b Ga(T)2103 2392 y FX(0)2126
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39006
2425 y Gg(:)549 2577 y(By)j(IH)g(we)g(ha)n(v)o(e)i F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39007
Ga(T)13 b F6(j)1254 2544 y Fu(s)1254 2599 y Gc(c)1363
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39008
2539 y(int)1324 2577 y F6(\000)-31 b(\000)f(!)1494 2544
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39009
y F9(+)1588 2577 y F6(j)p Ga(T)13 b F6(j)1704 2544 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39010
Fu(s)1704 2599 y Gc(c)1768 2577 y Gg(and)29 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39011
Ga(T)13 b F6(j)2043 2544 y Fu(s)2043 2599 y Gc(c)2078
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39012
2577 y F4([)p Ga(c)36 b F4(:=)2309 2565 y F9(\()2336
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39013
2577 y Ga(y)2384 2565 y F9(\))2412 2577 y Ga(P)13 b F4(])2582
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39014
2539 y Gc(int)2543 2577 y F6(\000)-31 b(\000)g(!)2714
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39015
2544 y F9(+)2808 2577 y F6(j)p Ga(T)2899 2544 y FX(0)2922
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39016
2577 y F6(j)2947 2544 y Fu(s)2947 2599 y Gc(c)2982 2577
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39017
y F4([)p Ga(c)36 b F4(:=)3213 2565 y F9(\()3241 2577
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39018
y Ga(y)3289 2565 y F9(\))3316 2577 y Ga(P)13 b F4(])28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39019
b Gg(for)549 2689 y(an)o(y)23 b(substitution)k(where)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39020
Ga(P)36 b Gg(freshly)25 b(introduces)h Ga(y)g Gg(and)e(is)f(not)h(an)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39021
(axiom.)360 2843 y(\(i\))201 b F6(j)p Ga(S)20 b(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39022
b F6(j)838 2810 y Fu(s)838 2865 y Gc(c)483 2967 y F4(=)92
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39023
b F6(j)p Ga(S)5 b F6(j)757 2934 y Fu(s)757 2989 y Gc(a)799
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39024
2967 y F4([)p Ga(a)25 b F4(:=)1018 2955 y F9(\()1046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39025
2967 y Ga(y)1094 2955 y F9(\))1121 2967 y FL(Imp)1266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39026
2989 y Gc(L)1318 2967 y F4(\()1353 2955 y FX(h)1381 2967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39027
y Ga(b)1420 2955 y FX(i)1447 2967 y F6(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39028
F6(j)1563 2934 y Fu(s)1563 2994 y Gc(b)1598 2967 y Ga(;)1638
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39029
2955 y F9(\()1665 2967 y Ga(z)1711 2955 y F9(\))1739
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39030
2967 y FL(Ax)p F4(\()p Ga(z)t(;)i(c)p F4(\))q Ga(;)g(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39031
F4(\))q(])872 b Gg(by)24 b(\(6\))g(of)f F6(j)p 3427 2967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39032
V 3445 2967 V 3463 2967 V 65 w(j)3515 2934 y Fu(s)422
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39033
3054 y Gc(int)383 3091 y F6(\000)-31 b(\000)g(!)563 3058
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39034
y F9(+)646 3091 y F6(j)p Ga(S)5 b F6(j)757 3058 y Fu(s)757
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39035
3113 y Gc(a)799 3091 y F4([)p Ga(a)25 b F4(:=)1018 3079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39036
y F9(\()1046 3091 y Ga(y)1094 3079 y F9(\))1121 3091
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39037
y FL(Imp)1266 3113 y Gc(L)1318 3091 y F4(\()1353 3079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39038
y FX(h)1381 3091 y Ga(b)1420 3079 y FX(i)1447 3091 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39039
F6(j)p Ga(T)1538 3058 y FX(0)1561 3091 y F6(j)1586 3058
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39040
y Fu(s)1586 3119 y Gc(b)1621 3091 y Ga(;)1661 3079 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39041
F9(\()1689 3091 y Ga(z)1735 3079 y F9(\))1762 3091 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39042
FL(Ax)p F4(\()p Ga(z)t(;)15 b(c)p F4(\))r Ga(;)g(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39043
F4(\)])1127 b Gg(by)23 b(IH)483 3215 y F4(=)92 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39044
Ga(S)20 b(T)813 3182 y FX(0)836 3215 y F6(j)861 3182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39045
y Fu(s)861 3238 y Gc(c)3059 3215 y Gg(by)k(\(6\))g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39046
F6(j)p 3427 3215 V 3445 3215 V 3463 3215 V 65 w(j)3515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39047
3182 y Fu(s)360 3386 y Gg(\(ii\))176 b F6(j)p Ga(S)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39048
b(T)13 b F6(j)838 3353 y Fu(s)838 3408 y Gc(c)873 3386
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39049
y F4([)p Ga(c)26 b F4(:=)1084 3374 y F9(\()1111 3386
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39050
y Ga(y)1159 3374 y F9(\))1186 3386 y Ga(N)10 b F4(])483
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39051
3510 y(=)92 b F6(j)p Ga(S)5 b F6(j)757 3477 y Fu(s)757
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39052
3533 y Gc(a)799 3510 y F4([)p Ga(a)25 b F4(:=)1018 3498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39053
y F9(\()1046 3510 y Ga(y)1094 3498 y F9(\))1121 3510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39054
y FL(Imp)1266 3532 y Gc(L)1318 3510 y F4(\()1353 3498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39055
y FX(h)1381 3510 y Ga(b)1420 3498 y FX(i)1447 3510 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39056
F6(j)p Ga(T)13 b F6(j)1563 3477 y Fu(s)1563 3538 y Gc(b)1598
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39057
3510 y Ga(;)1638 3498 y F9(\()1665 3510 y Ga(z)1711 3498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39058
y F9(\))1739 3510 y FL(Ax)p F4(\()p Ga(z)t(;)i(c)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39059
Ga(;)g(y)s F4(\))q(][)p Ga(c)26 b F4(:=)2398 3498 y F9(\()2426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39060
3510 y Ga(y)2474 3498 y F9(\))2501 3510 y Ga(N)10 b F4(])450
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39061
b Gg(by)24 b(\(6\))g(of)f F6(j)p 3427 3510 V 3445 3510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39062
V 3463 3510 V 65 w(j)3515 3477 y Fu(s)483 3634 y F6(\021)92
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39063
b(j)p Ga(S)5 b F6(j)757 3601 y Fu(s)757 3657 y Gc(a)799
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39064
3634 y F4([)p Ga(a)25 b F4(:=)1018 3622 y F9(\()1046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39065
3634 y Ga(y)1094 3622 y F9(\))1121 3634 y FL(Imp)1266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39066
3656 y Gc(L)1318 3634 y F4(\()1353 3622 y FX(h)1381 3634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39067
y Ga(b)1420 3622 y FX(i)1447 3634 y F6(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39068
F6(j)1563 3601 y Fu(s)1563 3662 y Gc(b)1598 3634 y Ga(;)1638
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39069
3622 y F9(\()1665 3634 y Ga(z)1711 3622 y F9(\))1739
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39070
3634 y FL(Ax)p F4(\()p Ga(z)t(;)i(c)p F4(\))q Ga(;)g(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39071
F4(\))q([)p Ga(c)26 b F4(:=)2373 3622 y F9(\()2401 3634
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39072
y Ga(y)2449 3622 y F9(\))2476 3634 y Ga(N)10 b F4(]])422
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39073
3722 y Gc(int)383 3759 y F6(\000)-31 b(\000)g(!)563 3726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39074
y F9(+)646 3759 y F6(j)p Ga(S)5 b F6(j)757 3726 y Fu(s)757
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39075
3781 y Gc(a)799 3759 y F4([)p Ga(a)25 b F4(:=)1018 3747
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39076
y F9(\()1046 3759 y Ga(y)1094 3747 y F9(\))1121 3759
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39077
y FL(Imp)1266 3780 y Gc(L)1318 3759 y F4(\()1353 3747
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39078
y FX(h)1381 3759 y Ga(b)1420 3747 y FX(i)1447 3759 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39079
F6(j)p Ga(T)1538 3726 y FX(0)1561 3759 y F6(j)1586 3726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39080
y Fu(s)1586 3786 y Gc(b)1621 3759 y Ga(;)1661 3747 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39081
F9(\()1689 3759 y Ga(z)1735 3747 y F9(\))1762 3759 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39082
FL(Ax)p F4(\()p Ga(z)t(;)15 b(c)p F4(\))r Ga(;)g(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39083
F4(\)[)p Ga(c)26 b F4(:=)2396 3747 y F9(\()2424 3759
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39084
y Ga(y)2472 3747 y F9(\))2499 3759 y Ga(N)10 b F4(]])705
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39085
b Gg(by)23 b(IH)483 3883 y F6(\021)92 b(j)p Ga(S)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39086
F6(j)757 3850 y Fu(s)757 3905 y Gc(a)799 3883 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39087
Ga(a)25 b F4(:=)1018 3871 y F9(\()1046 3883 y Ga(y)1094
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39088
3871 y F9(\))1121 3883 y FL(Imp)1266 3905 y Gc(L)1318
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39089
3883 y F4(\()1353 3871 y FX(h)1381 3883 y Ga(b)1420 3871
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39090
y FX(i)1447 3883 y F6(j)p Ga(T)1538 3850 y FX(0)1561
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39091
3883 y F6(j)1586 3850 y Fu(s)1586 3910 y Gc(b)1621 3883
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39092
y Ga(;)1661 3871 y F9(\()1689 3883 y Ga(z)1735 3871 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39093
F9(\))1762 3883 y FL(Ax)p F4(\()p Ga(z)t(;)15 b(c)p F4(\))r
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39094
Ga(;)g(y)s F4(\)][)p Ga(c)26 b F4(:=)2422 3871 y F9(\()2449
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39095
3883 y Ga(y)2497 3871 y F9(\))2524 3883 y Ga(N)10 b F4(])483
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39096
4007 y(=)92 b F6(j)p Ga(S)20 b(T)813 3974 y FX(0)836
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39097
4007 y F6(j)861 3974 y Fu(s)861 4029 y Gc(c)896 4007
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39098
y F4([)p Ga(c)26 b F4(:=)1107 3995 y F9(\()1134 4007
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39099
y Ga(y)1182 3995 y F9(\))1210 4007 y Ga(N)10 b F4(])1741
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39100
b Gg(by)24 b(\(6\))g(of)f F6(j)p 3427 4007 V 3445 4007
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39101
V 3463 4007 V 65 w(j)3515 3974 y Fu(s)p 3480 4114 4 62
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39102
v 3484 4056 55 4 v 3484 4114 V 3538 4114 4 62 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39103
321 4351 a Gb(Pr)n(oof)29 b(of)g(Pr)n(oposition)h(3.2.5.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39104
Black 33 w Gg(By)e(induction)j(on)e(the)g(de\002nition)h(of)2689
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39105
4314 y Gc(\015)2624 4351 y F6(\000)-32 b(\000)h(!)p Gg(.)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39106
b(Here)28 b(the)h(induction)321 4464 y(steps)c(are)f(tri)n(vial.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39107
b(W)-7 b(e)23 b(illustrate)j(the)e(proof)g(with)g(one)g(base)g(case.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39108
Black 321 4670 a Gb(Case)p Black 47 w Fp(case)n F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39109
Ga(P)s(;)15 b(\025)q(x:M)5 b(;)15 b(\025)q(y)s(:N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39110
b F4(\))31 b Ga(Q)1582 4633 y Gc(\015)1517 4670 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39111
b(\000)h(!)25 b Fp(case)o F4(\()p Ga(P)s(;)15 b(\025)q(x:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39112
F4(\()p Ga(M)26 b(Q)p F4(\))p Ga(;)15 b(\025y)s(:)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39113
Ga(N)26 b(Q)p F4(\)\))p Gg(:)487 4827 y F6(j)p Fp(case)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39114
F4(\()p Ga(P)s(;)15 b(\025)q(x:M)5 b(;)15 b(\025)q(y)s(:N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39115
b F4(\))15 b Ga(Q)p F6(j)1460 4794 y Fu(s)1460 4849 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39116
Gc(c)3059 4827 y Gg(by)24 b(\(6\))g(of)f F6(j)p 3427
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39117
4827 28 4 v 3445 4827 V 3463 4827 V 65 w(j)3515 4794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39118
y Fu(s)383 4951 y F4(=)33 b F6(j)p Fp(case)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39119
Ga(P)s(;)15 b(\025)q(x:M)5 b(;)15 b(\025)q(y)s(:N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39120
b F4(\))p F6(j)1373 4918 y Fu(s)1373 4973 y Gc(a)1430
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39121
4951 y F4([)p Ga(\033)s F4(])205 b Gg(with)24 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39122
Ga(\033)s F4(])i F6(\021)f F4([)p Ga(a)g F4(:=)2371 4939
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39123
y F9(\()2399 4951 y Ga(y)2447 4939 y F9(\))2474 4951
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39124
y FL(Imp)2619 4973 y Gc(L)2671 4951 y F4(\()2706 4939
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39125
y FX(h)2733 4951 y Ga(b)2772 4939 y FX(i)2800 4951 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39126
F6(j)p Ga(Q)p F6(j)2922 4918 y Fu(s)2922 4978 y Gc(b)2957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39127
4951 y Ga(;)2997 4939 y F9(\()3025 4951 y Ga(z)3071 4939
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39128
y F9(\))3098 4951 y FL(Ax)p F4(\()p Ga(z)t(;)15 b(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39129
F4(\))q Ga(;)g(y)s F4(\))q(])383 5075 y(=)33 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39130
Ga(P)13 b F6(j)608 5042 y Fu(s)608 5103 y Gc(d)664 5075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39131
y F4([)p Ga(d)26 b F4(:=)883 5063 y F9(\()911 5075 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39132
Ga(u)963 5063 y F9(\))990 5075 y FL(Or)1089 5089 y Gc(L)1141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39133
5075 y F4(\()1176 5063 y F9(\()1204 5075 y Ga(x)1256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39134
5063 y F9(\))1284 5075 y F6(j)p Ga(M)10 b F6(j)1432 5042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39135
y Fu(s)1432 5097 y Gc(a)1474 5075 y Ga(;)1514 5063 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39136
F9(\()1542 5075 y Ga(y)1590 5063 y F9(\))1617 5075 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39137
F6(j)p Ga(N)g F6(j)1750 5042 y Fu(s)1750 5097 y Gc(a)1792
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39138
5075 y Ga(;)15 b(u)p F4(\))q(])g([)p Ga(\033)s F4(])994
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39139
b Gg(by)24 b(\(9\))g(of)f F6(j)p 3427 5075 V 3445 5075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39140
V 3463 5075 V 65 w(j)3515 5042 y Fu(s)383 5199 y F6(\021)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39141
b(j)p Ga(P)13 b F6(j)608 5166 y Fu(s)608 5227 y Gc(d)664
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39142
5199 y F4([)p Ga(d)26 b F4(:=)883 5187 y F9(\()911 5199
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39143
y Ga(u)963 5187 y F9(\))990 5199 y FL(Or)1089 5213 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39144
Gc(L)1141 5199 y F4(\()1176 5187 y F9(\()1204 5199 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39145
Ga(x)1256 5187 y F9(\))1299 5199 y F6(j)p Ga(M)10 b F6(j)1447
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39146
5166 y Fu(s)1447 5222 y Gc(a)1489 5199 y F4([)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39147
F4(])q Ga(;)1635 5187 y F9(\()1663 5199 y Ga(y)1711 5187
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39148
y F9(\))1753 5199 y F6(j)p Ga(N)g F6(j)1886 5166 y Fu(s)1886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39149
5222 y Gc(a)1928 5199 y F4([)p Ga(\033)s F4(])q Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39150
b(u)p F4(\))q(])383 5323 y(=)33 b F6(j)p Ga(P)13 b F6(j)608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39151
5290 y Fu(s)608 5351 y Gc(d)664 5323 y F4([)p Ga(d)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39152
b F4(:=)883 5311 y F9(\()911 5323 y Ga(u)963 5311 y F9(\))990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39153
5323 y FL(Or)1089 5337 y Gc(L)1141 5323 y F4(\()1176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39154
5311 y F9(\()1204 5323 y Ga(x)1256 5311 y F9(\))1284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39155
5323 y F6(j)p Ga(M)f(Q)p F6(j)1519 5290 y Fu(s)1519 5346
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39156
y Gc(c)1554 5323 y Ga(;)1594 5311 y F9(\()1622 5323 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39157
Ga(y)1670 5311 y F9(\))1697 5323 y F6(j)p Ga(N)g(Q)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39158
F6(j)1917 5290 y Fu(s)1917 5346 y Gc(c)1952 5323 y Ga(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39159
b(u)p F4(\))q(])954 b Gg(by)24 b(\(6\))g(of)f F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39160
3427 5323 V 3445 5323 V 3463 5323 V 65 w(j)3515 5290
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39161
y Fu(s)383 5448 y F4(=)33 b F6(j)p Fp(case)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39162
Ga(P)s(;)15 b(\025)q(x:)p F4(\()p Ga(M)26 b(Q)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39163
Ga(;)15 b(\025y)s(:)p F4(\()p Ga(N)26 b(Q)p F4(\)\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39164
F6(j)1693 5415 y Fu(s)1693 5470 y Gc(c)3059 5448 y Gg(by)e(\(9\))g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39165
F6(j)p 3427 5448 V 3445 5448 V 3463 5448 V 65 w(j)3515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39166
5415 y Fu(s)p 3480 5554 4 62 v 3484 5496 55 4 v 3484
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39167
5554 V 3538 5554 4 62 v Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39168
%%Page: 159 171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39169
TeXDict begin 159 170 bop Black 277 51 a Gb(B.2)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39170
(Chapter)f(3)2639 b(159)p 277 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39171
277 388 a(Pr)n(oof)24 b(of)g(Theor)n(em)f(3.2.6.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39172
34 w Gg(The)g(follo)n(wing)i(measure)g(reduces)g(in)f(e)n(v)o(ery)2770
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39173
351 y Gc(\015)2705 388 y F6(\000)-32 b(\000)h(!)p Gg(-reduction.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39174
Black Black 613 541 a Fs([)p Ga(x)p Fs(])763 489 y F5(def)770
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39175
541 y F4(=)40 b(1)436 680 y Fs([)q Ga(\025x:M)10 b Fs(])763
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39176
629 y F5(def)770 680 y F4(=)40 b Fs([)p Ga(M)10 b Fs(])21
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39177
b F4(+)f(1)378 820 y Fs([)p F6(h)p Ga(M)5 b(;)15 b(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39178
b F6(i)p Fs(])763 769 y F5(def)770 820 y F4(=)40 b Fs([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39179
Ga(M)10 b Fs(])21 b F4(+)f Fs([)p Ga(N)10 b Fs(])21 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39180
F4(+)f(1)353 960 y Fs([)p Fp(inl)o F4(\()p Ga(M)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39181
F4(\))p Fs(])763 908 y F5(def)770 960 y F4(=)40 b Fs([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39182
Ga(M)10 b Fs(])21 b F4(+)f(1)353 1100 y Fs([)p Fp(inr)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39183
F4(\()p Ga(M)10 b F4(\))p Fs(])763 1048 y F5(def)770
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39184
1100 y F4(=)40 b Fs([)p Ga(M)10 b Fs(])21 b F4(+)f(1)2193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39185
646 y Fs([)q Ga(M)25 b(N)10 b Fs(])2487 594 y F5(def)2494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39186
646 y F4(=)40 b Fs([)q Ga(M)10 b Fs(])21 b F6(\003)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39187
F4(\()p Fs([)q Ga(N)10 b Fs(])21 b F4(+)f(1\))2077 786
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39188
y Fs([)q Fp(fst)o F4(\()p Ga(M)10 b F4(\))p Fs(])2487
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39189
734 y F5(def)2494 786 y F4(=)40 b(3)p Fs([)q Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39190
b Fs(])21 b F4(+)f(1)2077 925 y Fs([)q Fp(snd)o F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39191
Ga(M)10 b F4(\))p Fs(])2487 874 y F5(def)2494 925 y F4(=)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39192
b(3)p Fs([)q Ga(M)10 b Fs(])21 b F4(+)f(1)1554 1065 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39193
Fs([)p Fp(case)o F4(\()p Ga(P)s(;)15 b(\025)q(x:M)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39194
b(\025y)s(:N)10 b F4(\))p Fs(])2487 1014 y F5(def)2494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39195
1065 y F4(=)40 b Fs([)q Ga(P)13 b Fs(])20 b F6(\003)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39196
F4(\()p Fs([)q Ga(M)10 b Fs(])21 b F4(+)f Fs([)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39197
b Fs(])21 b F4(+)f(1\))p 3436 1191 4 62 v 3440 1133 55
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39198
4 v 3440 1191 V 3494 1191 4 62 v Black 277 1404 a Gb(Pr)n(oof)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39199
b(of)f(Lemma)f(3.2.7.)p Black 34 w Gg(Both)h(by)g(induction)i(on)e(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39200
h(structure)h(of)e Ga(M)10 b Gg(.)44 b(W)-7 b(e)28 b(\002rst)h(tackle)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39201
(three)277 1517 y(cases)25 b(of)e(\(i\).)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39202
1701 a Gb(Case)p Black 46 w Ga(M)36 b F6(\021)25 b FL(Ax)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39203
F4(\()p Ga(x;)15 b(b)p F4(\))p Gg(:)658 1848 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39204
FL(Ax)p F4(\()p Ga(x;)g(b)p F4(\)[)p Ga(x)26 b F4(:=)1212
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39205
1836 y FX(h)1239 1848 y Ga(a)1287 1836 y FX(i)1314 1848
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39206
y Ga(N)10 b F4(])p F6(j)1447 1815 y Fu(n)554 1966 y F4(=)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39207
b F6(j)p Ga(N)10 b F4([)p Ga(a)g F6(7!)g Ga(b)p F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39208
F6(j)1039 1933 y Fu(n)554 2085 y F6(\021)33 b(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39209
b F6(j)791 2052 y Fu(n)554 2204 y F6(\021)33 b Ga(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39210
F4([)p Ga(x)25 b F4(:=)h F6(j)p Ga(N)10 b F6(j)1067 2171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39211
y Fu(n)1108 2204 y F4(])554 2322 y(=)33 b F6(j)p FL(Ax)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39212
F4(\()p Ga(x;)15 b(b)p F4(\))p F6(j)1012 2289 y Fu(n)1054
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39213
2322 y F4([)p Ga(x)25 b F4(:=)g F6(j)p Ga(N)10 b F6(j)1410
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39214
2289 y Fu(n)1451 2322 y F4(])1530 b Gg(by)23 b(\(1\))h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39215
F6(j)p 3374 2322 28 4 v 3391 2322 V 3409 2322 V 64 w(j)3461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39216
2289 y Fu(n)p Black 277 2501 a Gb(Case)p Black 46 w Ga(M)58
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39217
b F6(\021)47 b FL(Imp)918 2523 y Gc(L)970 2501 y F4(\()1005
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39218
2489 y FX(h)1033 2501 y Ga(b)1072 2489 y FX(i)1099 2501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39219
y Ga(S)5 b(;)1200 2489 y F9(\()1228 2501 y Ga(y)1276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39220
2489 y F9(\))1303 2501 y Ga(T)13 b(;)i(z)t F4(\))p Gg(:)53
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39221
b(Let)35 b F4([)p Ga(\033)s F4(])h Gg(and)g F4([)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39222
F4(])2136 2468 y Fu(n)2212 2501 y Gg(be)f F4([)p Ga(x)48
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39223
b F4(:=)2600 2489 y FX(h)2628 2501 y Ga(a)2676 2489 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39224
FX(i)2703 2501 y Ga(N)10 b F4(])35 b Gg(and)h F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39225
Ga(x)48 b F4(:=)f F6(j)p Ga(N)10 b F6(j)3413 2468 y Fu(n)3454
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39226
2501 y F4(])p Gg(,)504 2614 y(respecti)n(v)o(ely)-6 b(.)658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39227
2772 y F6(j)p FL(Imp)828 2794 y Gc(L)880 2772 y F4(\()915
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39228
2760 y FX(h)943 2772 y Ga(b)982 2760 y FX(i)1009 2772
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39229
y Ga(S)5 b(;)1110 2760 y F9(\()1138 2772 y Ga(y)1186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39230
2760 y F9(\))1213 2772 y Ga(T)13 b(;)i(z)t F4(\)[)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39231
F4(])p F6(j)1530 2739 y Fu(n)554 2890 y F4(=)33 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39232
FL(Imp)828 2912 y Gc(L)880 2890 y F4(\()915 2878 y FX(h)943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39233
2890 y Ga(b)982 2878 y FX(i)1024 2890 y Ga(S)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39234
Ga(\033)s F4(])q Ga(;)1231 2878 y F9(\()1259 2890 y Ga(y)1307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39235
2878 y F9(\))1349 2890 y Ga(T)13 b F4([)p Ga(\033)s F4(])q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39236
Ga(;)i(z)t F4(\))p F6(j)1667 2857 y Fu(n)554 3009 y F4(=)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39237
b F6(j)p Ga(T)13 b F4([)p Ga(\033)s F4(])p F6(j)879 2976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39238
y Fu(n)921 3009 y F4([)p Ga(y)28 b F4(:=)d Ga(z)20 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39239
F6(j)p Ga(S)5 b F4([)p Ga(\033)s F4(])p F6(j)1418 2976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39240
y Fu(n)1459 3009 y F4(])1476 b Gg(by)24 b(\(10\))g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39241
F6(j)p 3374 3009 V 3391 3009 V 3409 3009 V 64 w(j)3461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39242
2976 y Fu(n)554 3128 y F6(\021)33 b(j)p Ga(T)13 b F6(j)774
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39243
3095 y Fu(n)815 3128 y F4([)p Ga(\033)s F4(])920 3095
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39244
y Fu(n)961 3128 y F4([)p Ga(y)28 b F4(:=)e Ga(z)19 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39245
F6(j)p Ga(S)5 b F6(j)1353 3095 y Fu(n)1394 3128 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39246
Ga(\033)s F4(])1499 3095 y Fu(n)1541 3128 y F4(])1727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39247
b Gg(by)23 b(IH)554 3246 y F6(\021)33 b(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39248
F6(j)774 3213 y Fu(n)815 3246 y F4([)p Ga(y)28 b F4(:=)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39249
Ga(z)20 b F6(j)p Ga(S)5 b F6(j)1207 3213 y Fu(n)1248
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39250
3246 y F4(][)p Ga(\033)s F4(])1378 3213 y Fu(n)554 3365
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39251
y F4(=)33 b F6(j)p FL(Imp)828 3386 y Gc(L)880 3365 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39252
F4(\()915 3353 y FX(h)943 3365 y Ga(b)982 3353 y FX(i)1009
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39253
3365 y Ga(S)5 b(;)1110 3353 y F9(\()1138 3365 y Ga(y)1186
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39254
3353 y F9(\))1213 3365 y Ga(T)13 b(;)i(z)t F4(\))p F6(j)1425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39255
3332 y Fu(n)1467 3365 y F4([)p Ga(\033)s F4(])1572 3332
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39256
y Fu(n)2960 3365 y Gg(by)24 b(\(10\))g(of)g F6(j)p 3374
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39257
3365 V 3391 3365 V 3409 3365 V 64 w(j)3461 3332 y Fu(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39258
Black 277 3543 a Gb(Case)p Black 46 w Ga(M)36 b F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39259
b FL(Imp)873 3565 y Gc(L)925 3543 y F4(\()960 3531 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39260
FX(h)988 3543 y Ga(b)1027 3531 y FX(i)1055 3543 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39261
b(;)1156 3531 y F9(\()1183 3543 y Ga(y)1231 3531 y F9(\))1259
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39262
3543 y Ga(T)12 b(;)j(x)p F4(\))p Gg(:)30 b(Again,)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39263
b(let)g F4([)p Ga(\033)s F4(])g Gg(and)h F4([)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39264
F4(])2273 3510 y Fu(n)2337 3543 y Gg(be)f F4([)p Ga(x)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39265
F4(:=)2669 3531 y FX(h)2696 3543 y Ga(a)2744 3531 y FX(i)2772
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39266
3543 y Ga(N)10 b F4(])23 b Gg(and)g F4([)p Ga(x)j F4(:=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39267
F6(j)p Ga(N)10 b F6(j)3413 3510 y Fu(n)3454 3543 y F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39268
Gg(,)504 3656 y(respecti)n(v)o(ely)-6 b(.)658 3814 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39269
F6(j)p FL(Imp)828 3836 y Gc(L)880 3814 y F4(\()915 3802
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39270
y FX(h)943 3814 y Ga(b)982 3802 y FX(i)1009 3814 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39271
b(;)1110 3802 y F9(\()1138 3814 y Ga(y)1186 3802 y F9(\))1213
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39272
3814 y Ga(T)13 b(;)i(x)p F4(\)[)p Ga(\033)s F4(])p F6(j)1536
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39273
3781 y Fu(n)554 3933 y F4(=)33 b F6(j)p FL(Cut)p F4(\()856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39274
3921 y FX(h)884 3933 y Ga(a)932 3921 y FX(i)959 3933
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39275
y Ga(N)10 b(;)1082 3921 y F9(\()1110 3933 y Ga(x)1162
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39276
3921 y F9(\))1190 3933 y FL(Imp)1334 3955 y Gc(L)1386
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39277
3933 y F4(\()1421 3921 y FX(h)1449 3933 y Ga(b)1488 3921
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39278
y FX(i)1531 3933 y Ga(S)5 b F4([)p Ga(\033)s F4(])p Ga(;)1737
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39279
3921 y F9(\()1765 3933 y Ga(y)1813 3921 y F9(\))1856
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39280
3933 y Ga(T)13 b F4([)p Ga(\033)s F4(])p Ga(;)i(x)p F4(\))q(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39281
F6(j)2215 3900 y Fu(n)554 4051 y F4(=)33 b F6(j)p Ga(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39282
b F4([)p Ga(\033)s F4(])p F6(j)879 4018 y Fu(n)921 4051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39283
y F4([)p Ga(y)28 b F4(:=)d Ga(x)15 b F6(j)p Ga(S)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39284
F4([)p Ga(\033)s F4(])p F6(j)1423 4018 y Fu(n)1465 4051
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39285
y F4(][)p Ga(x)26 b F4(:=)f F6(j)p Ga(N)10 b F6(j)1847
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39286
4018 y Fu(n)1888 4051 y F4(])979 b Gg(by)24 b(\(2,10\))g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39287
F6(j)p 3374 4051 V 3391 4051 V 3409 4051 V 64 w(j)3461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39288
4018 y Fu(n)554 4170 y F4(=)33 b F6(j)p Ga(T)13 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39289
Ga(\033)s F4(])p F6(j)879 4137 y Fu(n)921 4170 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39290
Ga(y)28 b F4(:=)d F6(j)p Ga(N)10 b F6(j)1273 4137 y Fu(n)1344
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39291
4170 y F6(j)p Ga(S)5 b F4([)p Ga(\033)s F4(])p F6(j)1560
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39292
4137 y Fu(n)1602 4170 y F4(])936 b Ga(x)26 b F6(62)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39293
Ga(F)13 b(V)21 b F4(\()p F6(j)p Ga(T)13 b F4([)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39294
F4(])p F6(j)3127 4137 y Fu(n)3169 4170 y Ga(;)i F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39295
Ga(S)5 b F4([)p Ga(\033)s F4(])p F6(j)3425 4137 y Fu(n)3467
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39296
4170 y F4(\))554 4289 y F6(\021)33 b(j)p Ga(T)13 b F6(j)774
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39297
4256 y Fu(n)815 4289 y F4([)p Ga(\033)s F4(])920 4256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39298
y Fu(n)961 4289 y F4([)p Ga(y)28 b F4(:=)e F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39299
b F6(j)1314 4256 y Fu(n)1385 4289 y F6(j)p Ga(S)5 b F6(j)1496
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39300
4256 y Fu(n)1537 4289 y F4([)p Ga(\033)s F4(])1642 4256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39301
y Fu(n)1683 4289 y F4(])1585 b Gg(by)23 b(IH)554 4407
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39302
y F6(\021)33 b(j)p Ga(T)13 b F6(j)774 4374 y Fu(n)815
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39303
4407 y F4([)p Ga(y)28 b F4(:=)d Ga(x)15 b F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39304
b F6(j)1212 4374 y Fu(n)1254 4407 y F4(][)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39305
F4(])1384 4374 y Fu(n)554 4526 y F4(=)33 b F6(j)p FL(Imp)828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39306
4547 y Gc(L)880 4526 y F4(\()915 4514 y FX(h)943 4526
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39307
y Ga(b)982 4514 y FX(i)1009 4526 y Ga(S)5 b(;)1110 4514
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39308
y F9(\()1138 4526 y Ga(y)1186 4514 y F9(\))1213 4526
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39309
y Ga(T)13 b(;)i(x)p F4(\))p F6(j)1431 4493 y Fu(n)1473
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39310
4526 y F4([)p Ga(\033)s F4(])1578 4493 y Fu(n)2960 4526
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39311
y Gg(by)24 b(\(10\))g(of)g F6(j)p 3374 4526 V 3391 4526
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39312
V 3409 4526 V 64 w(j)3461 4493 y Fu(n)277 4709 y Gg(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39313
b(e)23 b(proceed)i(with)f(three)g(cases)h(of)e(\(ii\).)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39314
b(Notice)24 b(that)g Ga(a)i F6(2)e Ga(F)13 b(C)7 b F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39315
Ga(M)j F4(\))p Gg(.)p Black 277 4892 a Gb(Case)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39316
46 w Ga(M)36 b F6(\021)25 b FL(Ax)o F4(\()p Ga(x;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39317
b(a)p F4(\))p Gg(:)658 5040 y F6(j)p FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39318
Ga(y)s(;)g(a)p F4(\)[)p Ga(a)26 b F4(:=)1213 5028 y F9(\()1240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39319
5040 y Ga(x)1292 5028 y F9(\))1319 5040 y Ga(N)10 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39320
F6(j)1452 5007 y Fu(n)554 5158 y F4(=)33 b F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39321
b F4([)p Ga(x)g F6(7!)g Ga(y)s F4(])p F6(j)1052 5125
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39322
y Fu(n)554 5277 y F6(\021)33 b(j)p Ga(N)10 b F6(j)791
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39323
5244 y Fu(n)832 5277 y F4([)p Ga(x)26 b F4(:=)f Ga(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39324
F4(])554 5395 y(=)33 b F6(j)p Ga(N)10 b F6(j)791 5362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39325
y Fu(n)832 5395 y F4([)p Ga(x)26 b F4(:=)f F6(j)p FL(Ax)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39326
F4(\()p Ga(y)s(;)15 b(a)p F4(\))p F6(j)1415 5362 y Fu(n)1456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39327
5395 y F4(])1525 b Gg(by)23 b(\(1\))h(of)g F6(j)p 3374
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39328
5395 V 3391 5395 V 3409 5395 V 64 w(j)3461 5362 y Fu(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39329
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39330
%%Page: 160 172
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39331
TeXDict begin 160 171 bop Black -144 51 a Gb(160)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39332
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39333
4 v Black Black 321 388 a(Case)p Black 47 w Ga(M)35 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39334
F6(\021)25 b FL(And)927 402 y Gc(R)985 388 y F4(\()1020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39335
376 y FX(h)1048 388 y Ga(c)1087 376 y FX(i)1115 388 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39336
Ga(S)5 b(;)1216 376 y FX(h)1244 388 y Ga(d)1291 376 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39337
FX(i)1318 388 y Ga(T)13 b(;)i(a)p F4(\))p Gg(:)29 b(Notice)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39338
b(that)h Ga(M)32 b Gg(has)24 b(to)g(freshly)h(introduce)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39339
Ga(a)p Gg(.)702 550 y F6(j)p FL(And)882 564 y Gc(R)940
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39340
550 y F4(\()975 538 y FX(h)1002 550 y Ga(c)1041 538 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39341
FX(i)1069 550 y Ga(S)5 b(;)1170 538 y FX(h)1198 550 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39342
Ga(d)1245 538 y FX(i)1273 550 y Ga(T)13 b(;)i(a)p F4(\)[)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39343
Ga(a)26 b F4(:=)1682 538 y F9(\()1710 550 y Ga(x)1762
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39344
538 y F9(\))1789 550 y Ga(N)10 b F4(])p F6(j)1922 517
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39345
y Fu(n)598 668 y F4(=)33 b F6(j)p FL(Cut)p F4(\()900
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39346
656 y FX(h)928 668 y Ga(a)976 656 y FX(i)1004 668 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39347
b(;)1142 656 y F9(\()1170 668 y Ga(x)1222 656 y F9(\))1249
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39348
668 y Ga(N)g F4(\))p F6(j)1392 635 y Fu(n)598 787 y F4(=)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39349
b F6(j)p Ga(N)10 b F6(j)835 754 y Fu(n)876 787 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39350
Ga(x)26 b F4(:=)f F6(j)p Ga(M)10 b F6(j)1248 754 y Fu(n)1289
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39351
787 y F4(])1736 b Gg(by)24 b(\(2\))f(of)h F6(j)p 3418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39352
787 28 4 v 3436 787 V 3453 787 V 64 w(j)3505 754 y Fu(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39353
Black 321 957 a Gb(Case)p Black 47 w Ga(M)35 b F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39354
b FL(Cut)p F4(\()946 945 y FX(h)974 957 y Ga(c)1013 945
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39355
y FX(i)1040 957 y Ga(S)5 b(;)1141 945 y F9(\()1169 957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39356
y Ga(y)1217 945 y F9(\))1244 957 y Ga(T)13 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39357
Gg(:)702 1118 y F6(j)p FL(Cut)p F4(\()900 1106 y FX(h)928
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39358
1118 y Ga(c)967 1106 y FX(i)995 1118 y Ga(S)5 b(;)1096
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39359
1106 y F9(\()1124 1118 y Ga(y)1172 1106 y F9(\))1199
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39360
1118 y Ga(T)13 b F4(\)[)p Ga(a)26 b F4(:=)1520 1106 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39361
F9(\()1547 1118 y Ga(x)1599 1106 y F9(\))1627 1118 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39362
Ga(N)10 b F4(])p F6(j)1760 1085 y Fu(n)598 1237 y F4(=)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39363
b F6(j)p FL(Cut)p F4(\()900 1225 y FX(h)928 1237 y Ga(c)967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39364
1225 y FX(i)995 1237 y Ga(S)5 b(;)1096 1225 y F9(\()1124
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39365
1237 y Ga(y)1172 1225 y F9(\))1214 1237 y Ga(T)13 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39366
Ga(a)25 b F4(:=)1500 1225 y F9(\()1527 1237 y Ga(x)1579
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39367
1225 y F9(\))1607 1237 y Ga(N)9 b F4(])q(\))p F6(j)1775
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39368
1204 y Fu(n)2250 1237 y Ga(a)25 b F6(62)g Ga(F)13 b(C)7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39369
b F4(\()2587 1225 y FX(h)2614 1237 y Ga(c)2653 1225 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39370
FX(i)2681 1237 y Ga(S)e F4(\))23 b Gg(because)j Ga(M)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39371
b F6(2)25 b FY(J)3366 1204 y F9(\()p FX(\033)p Gc(;)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39372
FX(^)p F9(\))598 1355 y F6(\021)33 b(j)p Ga(T)13 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39373
Ga(a)26 b F4(:=)1013 1343 y F9(\()1040 1355 y Ga(x)1092
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39374
1343 y F9(\))1120 1355 y Ga(N)10 b F4(])p F6(j)1253 1322
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39375
y Fu(n)1294 1355 y F4([)p Ga(y)28 b F4(:=)d F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39376
b F6(j)1624 1322 y Fu(n)1665 1355 y F4(])1360 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39377
b(\(2\))f(of)h F6(j)p 3418 1355 V 3436 1355 V 3453 1355
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39378
V 64 w(j)3505 1322 y Fu(n)598 1474 y F4(=)33 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39379
Ga(N)10 b F6(j)835 1441 y Fu(n)876 1474 y F4([)p Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39380
b F4(:=)f F6(j)p Ga(T)13 b F6(j)1216 1441 y Fu(n)1257
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39381
1474 y F4(][)p Ga(y)28 b F4(:=)e F6(j)p Ga(S)5 b F6(j)1613
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39382
1441 y Fu(n)1654 1474 y F4(])1658 b Gg(by)23 b(IH)598
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39383
1592 y F6(\021)33 b(j)p Ga(N)10 b F6(j)835 1559 y Fu(n)876
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39384
1592 y F4([)p Ga(x)26 b F4(:=)f F6(j)p Ga(T)13 b F6(j)1216
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39385
1559 y Fu(n)1257 1592 y F4([)p Ga(y)28 b F4(:=)d F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39386
Ga(S)5 b F6(j)1587 1559 y Fu(n)1628 1592 y F4(]])1043
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39387
b Ga(y)29 b F6(62)24 b Ga(F)13 b(N)d F4(\()3069 1580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39388
y F9(\()3097 1592 y Ga(x)3149 1580 y F9(\))3177 1592
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39389
y Ga(N)g F4(\))23 b Gg(by)g F4([)p 3458 1592 V 3476 1592
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39390
V 3494 1592 V 65 w(])598 1711 y(=)33 b F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39391
b F6(j)835 1678 y Fu(n)876 1711 y F4([)p Ga(x)26 b F4(:=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39392
F6(j)p FL(Cut)p F4(\()1298 1699 y FX(h)1326 1711 y Ga(c)1365
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39393
1699 y FX(i)1393 1711 y Ga(S)5 b(;)1494 1699 y F9(\()1521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39394
1711 y Ga(y)1569 1699 y F9(\))1597 1711 y Ga(T)12 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39395
F6(j)1722 1678 y Fu(n)1764 1711 y F4(])1261 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39396
b(\(2\))f(of)h F6(j)p 3418 1711 V 3436 1711 V 3453 1711
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39397
V 64 w(j)3505 1678 y Fu(n)p 3480 1816 4 62 v 3484 1758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39398
55 4 v 3484 1816 V 3538 1816 4 62 v Black 321 2029 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39399
Gb(Pr)n(oof)k(of)f(Pr)n(oposition)h(3.2.8.)p Black 34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39400
w Gg(The)e(inducti)n(v)o(e)j(steps)f(are)g(quite)g(routine;)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39401
(therefore)e(we)d(gi)n(v)o(e)h(the)321 2142 y(calculations)g(for)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39402
(four)g(base)g(cases)h(only)-6 b(.)p Black 321 2304 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39403
Gb(Logical)25 b(Cut)d(with)h(Axiom:)p Black 549 2417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39404
a FL(Cut)o F4(\()721 2405 y FX(h)749 2417 y Ga(a)797
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39405
2405 y FX(i)825 2417 y Ga(M)10 b(;)963 2405 y F9(\()991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39406
2417 y Ga(x)1043 2405 y F9(\))1070 2417 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39407
Ga(x;)15 b(b)p F4(\))q(\))1474 2380 y Gc(int)1435 2417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39408
y F6(\000)-31 b(\000)g(!)25 b Ga(M)10 b F4([)p Ga(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39409
F6(7!)g Ga(b)p F4(])808 2564 y F6(j)p FL(Cut)p F4(\()1006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39410
2552 y FX(h)1034 2564 y Ga(a)1082 2552 y FX(i)1110 2564
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39411
y Ga(M)g(;)1248 2552 y F9(\()1276 2564 y Ga(x)1328 2552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39412
y F9(\))1355 2564 y FL(Ax)p F4(\()p Ga(x;)15 b(b)p F4(\))q(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39413
F6(j)1720 2531 y Fu(n)686 2688 y F4(=)51 b Ga(x)p F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39414
Ga(x)26 b F4(:=)f F6(j)p Ga(M)10 b F6(j)1232 2655 y Fu(n)1273
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39415
2688 y F4(])1684 b Gg(by)23 b(\(1,2\))h(of)g F6(j)p 3418
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39416
2688 28 4 v 3436 2688 V 3453 2688 V 64 w(j)3505 2655
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39417
y Fu(n)686 2813 y F4(=)51 b F6(j)p Ga(M)10 b F6(j)956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39418
2780 y Fu(n)686 2937 y F6(\021)51 b(j)p Ga(M)10 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39419
Ga(a)g F6(7!)g Ga(b)p F4(])p F6(j)1204 2904 y Fu(n)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39420
Black 321 3109 a Gb(Logical)25 b(Cut)d Fb(^)871 3123
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39421
y Fy(R)936 3109 y FO(=)p Fb(^)1058 3123 y Fy(L)1111 3131
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39422
y Fa(1)1154 3109 y Gb(:)p Black 549 3238 a FL(Cut)o F4(\()721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39423
3226 y FX(h)749 3238 y Ga(c)788 3226 y FX(i)816 3238
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39424
y FL(And)971 3252 y Gc(R)1028 3238 y F4(\()1063 3226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39425
y FX(h)1091 3238 y Ga(a)1139 3226 y FX(i)1167 3238 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39426
Ga(M)10 b(;)1305 3226 y FX(h)1333 3238 y Ga(b)1372 3226
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39427
y FX(i)1399 3238 y Ga(N)g(;)15 b(c)p F4(\))q Ga(;)1637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39428
3226 y F9(\()1665 3238 y Ga(y)1713 3226 y F9(\))1740
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39429
3238 y FL(And)1895 3201 y F9(1)1895 3261 y Gc(L)1947
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39430
3238 y F4(\()1982 3226 y F9(\()2010 3238 y Ga(x)2062
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39431
3226 y F9(\))2089 3238 y Ga(P)e(;)i(y)s F4(\))q(\))2383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39432
3201 y Gc(int)2344 3238 y F6(\000)-31 b(\000)g(!)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39433
FL(Cut)p F4(\()2713 3226 y FX(h)2740 3238 y Ga(a)2788
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39434
3226 y FX(i)2816 3238 y Ga(M)10 b(;)2954 3226 y F9(\()2982
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39435
3238 y Ga(x)3034 3226 y F9(\))3061 3238 y Ga(P)j F4(\))810
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39436
3385 y F6(j)p FL(Cut)p F4(\()1008 3373 y FX(h)1036 3385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39437
y Ga(c)1075 3373 y FX(i)1103 3385 y FL(And)1257 3399
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39438
y Gc(R)1315 3385 y F4(\()1350 3373 y FX(h)1378 3385 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39439
Ga(a)1426 3373 y FX(i)1454 3385 y Ga(M)d(;)1592 3373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39440
y FX(h)1619 3385 y Ga(b)1658 3373 y FX(i)1686 3385 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39441
Ga(N)g(;)15 b(c)p F4(\))q Ga(;)1924 3373 y F9(\()1952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39442
3385 y Ga(y)2000 3373 y F9(\))2027 3385 y FL(And)2182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39443
3349 y F9(1)2182 3409 y Gc(L)2234 3385 y F4(\()2269 3373
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39444
y F9(\()2297 3385 y Ga(x)2349 3373 y F9(\))2376 3385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39445
y Ga(P)e(;)i(y)s F4(\))q(\))p F6(j)2631 3352 y Fu(n)648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39446
3510 y F4(=)91 b F6(j)p Ga(P)13 b F6(j)931 3477 y Fu(n)972
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39447
3510 y F4([)p Ga(x)26 b F4(:=)f Fp(fst)o F4(\()p Ga(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39448
F4(\)][)p Ga(y)k F4(:=)d F6(hj)p Ga(M)10 b F6(j)1886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39449
3477 y Fu(n)1927 3510 y Ga(;)15 b F6(j)p Ga(N)10 b F6(j)2100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39450
3477 y Fu(n)2142 3510 y F6(i)p F4(])712 b Gg(by)23 b(\(2,3,7\))h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39451
F6(j)p 3418 3510 V 3436 3510 V 3453 3510 V 64 w(j)3505
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39452
3477 y Fu(n)648 3634 y F6(\021)91 b(j)p Ga(P)13 b F6(j)931
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39453
3601 y Fu(n)972 3634 y F4([)p Ga(x)26 b F4(:=)f Fp(fst)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39454
F4(\()p F6(hj)p Ga(M)10 b F6(j)1557 3601 y Fu(n)1599
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39455
3634 y Ga(;)15 b F6(j)p Ga(N)10 b F6(j)1772 3601 y Fu(n)1814
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39456
3634 y F6(i)p F4(\)])250 b Ga(y)28 b F6(62)d Ga(F)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39457
b(V)20 b F4(\()p F6(j)p Ga(P)13 b F6(j)2618 3601 y Fu(n)2659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39458
3634 y F4(\))24 b Gg(by)f(logical)i(cut)f F6(^)3296 3648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39459
y Gc(R)3354 3634 y Ga(=)p F6(^)3460 3648 y Gc(L)3508
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39460
3657 y FV(1)621 3721 y Gc(\023)549 3758 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39461
b(\000)h(!)728 3725 y FX(\003)810 3758 y F6(j)p Ga(P)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39462
b F6(j)931 3725 y Fu(n)972 3758 y F4([)p Ga(x)26 b F4(:=)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39463
F6(j)p Ga(M)10 b F6(j)1344 3725 y Fu(n)1385 3758 y F4(])648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39464
3882 y(=)91 b F6(j)p FL(Cut)p F4(\()1008 3870 y FX(h)1036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39465
3882 y Ga(a)1084 3870 y FX(i)1112 3882 y Ga(M)10 b(;)1250
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39466
3870 y F9(\()1278 3882 y Ga(x)1330 3870 y F9(\))1357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39467
3882 y Ga(P)j F4(\))p F6(j)1488 3849 y Fu(n)3050 3882
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39468
y Gg(by)24 b(\(2\))f(of)h F6(j)p 3418 3882 V 3436 3882
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39469
V 3453 3882 V 64 w(j)3505 3849 y Fu(n)p Black 321 4054
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39470
a Gb(Logical)h(Cut)d Fb(\033)882 4068 y Fy(R)948 4054
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39471
y FO(=)p Fb(\033)1081 4068 y Fy(L)1139 4054 y Gb(:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39472
Black 598 4181 a FL(Cut)p F4(\()771 4169 y FX(h)799 4181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39473
y Ga(b)838 4169 y FX(i)866 4181 y FL(Imp)1010 4203 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39474
Gc(R)1068 4181 y F4(\()1103 4169 y F9(\()1130 4181 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39475
Ga(x)1182 4169 y F9(\))q FX(h)1237 4181 y Ga(a)1285 4169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39476
y FX(i)1313 4181 y Ga(M)10 b(;)15 b(b)p F4(\))q Ga(;)1566
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39477
4169 y F9(\()1594 4181 y Ga(z)1640 4169 y F9(\))1667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39478
4181 y FL(Imp)1812 4203 y Gc(L)1864 4181 y F4(\()1899
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39479
4169 y FX(h)1927 4181 y Ga(c)1966 4169 y FX(i)1993 4181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39480
y Ga(P)e(;)2104 4169 y F9(\()2132 4181 y Ga(y)2180 4169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39481
y F9(\))2208 4181 y Ga(Q)o(;)i(z)t F4(\))q(\))728 4257
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39482
y Gc(int)689 4294 y F6(\000)-31 b(\000)g(!)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39483
F4(\()1058 4282 y FX(h)1085 4294 y Ga(a)1133 4282 y FX(i)1161
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39484
4294 y FL(Cut)p F4(\()1334 4282 y FX(h)1362 4294 y Ga(c)1401
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39485
4282 y FX(i)1428 4294 y Ga(P)13 b(;)1539 4282 y F9(\()1567
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39486
4294 y Ga(x)1619 4282 y F9(\))1647 4294 y Ga(M)d F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39487
Ga(;)1820 4282 y F9(\()1848 4294 y Ga(y)1896 4282 y F9(\))1923
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39488
4294 y Ga(Q)p F4(\))74 b Gg(or)728 4370 y Gc(int)689
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39489
4407 y F6(\000)-31 b(\000)g(!)25 b FL(Cut)p F4(\()1058
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39490
4395 y FX(h)1085 4407 y Ga(c)1124 4395 y FX(i)1152 4407
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39491
y Ga(P)13 b(;)1263 4395 y F9(\()1291 4407 y Ga(x)1343
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39492
4395 y F9(\))1371 4407 y FL(Cut)o F4(\()1543 4395 y FX(h)1571
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39493
4407 y Ga(a)1619 4395 y FX(i)1647 4407 y Ga(M)d(;)1785
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39494
4395 y F9(\()1813 4407 y Ga(y)1861 4395 y F9(\))1888
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39495
4407 y Ga(Q)p F4(\)\))598 4561 y Gg(F)o(or)23 b(the)h(\002rst)f(reduct)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39496
i(we)e(ha)n(v)o(e:)810 4679 y F6(j)p FL(Cut)p F4(\()1008
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39497
4667 y FX(h)1036 4679 y Ga(b)1075 4667 y FX(i)1103 4679
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39498
y FL(Imp)1247 4701 y Gc(R)1305 4679 y F4(\()1340 4667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39499
y F9(\()1368 4679 y Ga(x)1420 4667 y F9(\))p FX(h)1474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39500
4679 y Ga(a)1522 4667 y FX(i)1550 4679 y Ga(M)10 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39501
b(b)p F4(\))q Ga(;)1803 4667 y F9(\()1831 4679 y Ga(z)1877
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39502
4667 y F9(\))1904 4679 y FL(Imp)2049 4701 y Gc(L)2101
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39503
4679 y F4(\()2136 4667 y FX(h)2164 4679 y Ga(c)2203 4667
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39504
y FX(i)2231 4679 y Ga(P)e(;)2342 4667 y F9(\()2369 4679
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39505
y Ga(y)2417 4667 y F9(\))2445 4679 y Ga(Q;)i(z)t F4(\))q(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39506
F6(j)2699 4646 y Fu(n)648 4798 y F4(=)91 b F6(j)p Ga(Q)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39507
F6(j)932 4765 y Fu(n)973 4798 y F4([)p Ga(y)29 b F4(:=)c(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39508
Ga(z)45 b F6(j)p Ga(P)13 b F6(j)1436 4765 y Fu(n)1477
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39509
4798 y F4(\)][)p Ga(z)30 b F4(:=)25 b Ga(\025x:)p F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39510
Ga(M)10 b F6(j)2033 4765 y Fu(n)2075 4798 y F4(])768
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39511
b Gg(by)24 b(\(2,6,10\))g(of)g F6(j)p 3418 4798 V 3436
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39512
4798 V 3453 4798 V 64 w(j)3505 4765 y Fu(n)648 4917 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39513
F6(\021)91 b(j)p Ga(Q)p F6(j)932 4884 y Fu(n)973 4917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39514
y F4([)p Ga(y)29 b F4(:=)c(\()p Ga(\025x:)p F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39515
b F6(j)1506 4884 y Fu(n)1548 4917 y F4(\))40 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39516
Ga(P)13 b F6(j)1744 4884 y Fu(n)1785 4917 y F4(])481
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39517
b Gg(by)23 b(logical)j(cut)d Ga(z)30 b F6(62)25 b Ga(F)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39518
b(V)20 b F4(\()p F6(j)p Ga(Q)p F6(j)3267 4884 y Fu(n)3308
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39519
4917 y Ga(;)15 b F6(j)p Ga(P)e F6(j)3469 4884 y Fu(n)3511
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39520
4917 y F4(\))621 4998 y Gc(\023)549 5035 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39521
b(\000)h(!)728 5002 y FX(\003)810 5035 y F6(j)p Ga(Q)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39522
F6(j)932 5002 y Fu(n)973 5035 y F4([)p Ga(y)29 b F4(:=)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39523
F6(j)p Ga(M)10 b F6(j)1341 5002 y Fu(n)1382 5035 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39524
Ga(x)25 b F4(:=)h F6(j)p Ga(P)13 b F6(j)1727 5002 y Fu(n)1768
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39525
5035 y F4(]])648 5154 y(=)91 b F6(j)p FL(Cut)p F4(\()1008
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39526
5142 y FX(h)1036 5154 y Ga(a)1084 5142 y FX(i)1112 5154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39527
y FL(Cut)p F4(\()1285 5142 y FX(h)1312 5154 y Ga(c)1351
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39528
5142 y FX(i)1379 5154 y Ga(P)13 b(;)1490 5142 y F9(\()1518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39529
5154 y Ga(x)1570 5142 y F9(\))1597 5154 y Ga(M)d F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39530
Ga(;)1771 5142 y F9(\()1799 5154 y Ga(y)1847 5142 y F9(\))1874
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39531
5154 y Ga(Q)p F4(\))p F6(j)2006 5121 y Fu(n)3050 5154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39532
y Gg(by)24 b(\(2\))f(of)h F6(j)p 3418 5154 V 3436 5154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39533
V 3453 5154 V 64 w(j)3505 5121 y Fu(n)598 5319 y Gg(F)o(or)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39534
(other)g(reduct)h(we)e(ha)n(v)o(e:)810 5438 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39535
FL(Cut)p F4(\()1008 5426 y FX(h)1036 5438 y Ga(c)1075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39536
5426 y FX(i)1103 5438 y Ga(P)13 b(;)1214 5426 y F9(\()1242
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39537
5438 y Ga(x)1294 5426 y F9(\))1321 5438 y FL(Cut)p F4(\()1494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39538
5426 y FX(h)1522 5438 y Ga(a)1570 5426 y FX(i)1597 5438
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39539
y Ga(M)d(;)1735 5426 y F9(\()1763 5438 y Ga(y)1811 5426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39540
y F9(\))1839 5438 y Ga(Q)o F4(\))q(\))p F6(j)2006 5405
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39541
y Fu(n)648 5557 y F4(=)91 b F6(j)p Ga(Q)p F6(j)932 5524
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39542
y Fu(n)973 5557 y F4([)p Ga(y)29 b F4(:=)c F6(j)p Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39543
b F6(j)1341 5524 y Fu(n)1382 5557 y F4(][)p Ga(x)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39544
F4(:=)f F6(j)p Ga(P)13 b F6(j)1752 5524 y Fu(n)1793 5557
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39545
y F4(])1232 b Gg(by)24 b(\(2\))f(of)h F6(j)p 3418 5557
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39546
V 3436 5557 V 3453 5557 V 64 w(j)3505 5524 y Fu(n)648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39547
5675 y F6(\021)91 b(j)p Ga(Q)p F6(j)932 5642 y Fu(n)973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39548
5675 y F4([)p Ga(y)29 b F4(:=)c F6(j)p Ga(M)10 b F6(j)1341
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39549
5642 y Fu(n)1382 5675 y F4([)p Ga(x)25 b F4(:=)h F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39550
Ga(P)13 b F6(j)1727 5642 y Fu(n)1768 5675 y F4(]])559
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39551
b Ga(x)25 b F6(62)g Ga(F)13 b(V)20 b F4(\()p F6(j)p Ga(Q)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39552
F6(j)2841 5642 y Fu(n)2882 5675 y F4(\))j Gg(by)h(Remark)g(2.2.7)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39553
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39554
%%Page: 161 173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39555
TeXDict begin 161 172 bop Black 277 51 a Gb(B.2)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39556
(Chapter)f(3)2639 b(161)p 277 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39557
277 388 a(Commuting)22 b(Cut:)p Black 554 525 a FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39558
F4(\()727 513 y FX(h)755 525 y Ga(a)803 513 y FX(i)830
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39559
525 y Ga(M)11 b(;)969 513 y F9(\()996 525 y Ga(x)1048
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39560
513 y F9(\))1076 525 y Ga(N)f F4(\))1333 488 y Gc(int)1294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39561
525 y F6(\000)-32 b(\000)h(!)100 b Ga(M)10 b F4([)p Ga(a)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39562
b F4(:=)1881 513 y F9(\()1909 525 y Ga(x)1961 513 y F9(\))1988
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39563
525 y Ga(N)10 b F4(])74 b Gg(or)1333 601 y Gc(int)1294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39564
638 y F6(\000)-32 b(\000)h(!)100 b Ga(N)10 b F4([)p Ga(x)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39565
b F4(:=)1870 626 y FX(h)1898 638 y Ga(a)1946 626 y FX(i)1973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39566
638 y Ga(M)10 b F4(])556 872 y F6(j)p FL(Cut)q F4(\()755
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39567
860 y FX(h)782 872 y Ga(a)830 860 y FX(i)858 872 y Ga(M)g(;)996
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39568
860 y F9(\()1024 872 y Ga(x)1076 860 y F9(\))1103 872
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39569
y Ga(N)g F4(\))p F6(j)1246 839 y Fu(n)1313 872 y F4(=)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39570
b F6(j)p Ga(N)10 b F6(j)1542 839 y Fu(n)1583 872 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39571
Ga(x)25 b F4(:=)g F6(j)p Ga(M)10 b F6(j)1954 839 y Fu(n)1995
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39572
872 y F4(])49 b F6(\021)2190 743 y FK(\032)2258 813 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39573
F6(j)p Ga(M)10 b F4([)p Ga(a)26 b F4(:=)2601 801 y F9(\()2629
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39574
813 y Ga(x)2681 801 y F9(\))2708 813 y Ga(N)10 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39575
F6(j)2841 780 y Fu(n)2258 932 y F6(j)p Ga(N)g F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39576
Ga(x)26 b F4(:=)2590 920 y FX(h)2617 932 y Ga(a)2665
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39577
920 y FX(i)2693 932 y Ga(M)10 b F4(])p F6(j)2841 899
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39578
y Fu(n)2995 872 y Gg(by)23 b(Lem.)f(3.2.7)p 3436 1048
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39579
4 62 v 3440 990 55 4 v 3440 1048 V 3494 1048 4 62 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39580
277 1260 a Gb(Pr)n(oof)30 b(of)f(Lemma)g(3.4.1.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39581
33 w Gg(First)h(we)e(notice)j(that)f(whene)n(v)o(er)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39582
F6(j)p Ga(M)10 b F6(j)2461 1227 y Fu(S)2533 1260 y Gg(does)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39583
b(not)g(introduce)i Ga(a)p Gg(,)e(then)277 1373 y(the)d(substitution)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39584
F4([)p 891 1373 28 4 v 909 1373 V 926 1373 V 64 w(])c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39585
Gg(is)g(pushed)i(inside)g(the)f(subterms)h(and)e(the)h(lemma)f(is)g(by)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39586
h(routine)h(induction)277 1486 y(and)h(calculation.)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39587
b(So)28 b(we)g(are)h(left)g(with)f(the)h(cases)h(where)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39588
F6(j)p Ga(M)10 b F6(j)2392 1453 y Fu(S)2464 1486 y Gg(introduces)32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39589
b Ga(a)p Gg(.)43 b(If)28 b Ga(M)38 b Gg(is)29 b(of)f(the)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39590
1599 y(form)c FL(Id)p F4(\()p Ga(y)s(;)15 b(a)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39591
Gg(,)23 b(we)g(reason)i(as)e(follo)n(ws.)p Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39592
1860 a Gb(Case)p Black 46 w Ga(M)36 b F6(\021)25 b FL(Id)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39593
F4(\()p Ga(y)s(;)15 b(a)p F4(\))675 2024 y F6(j)p FL(Id)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39594
F4(\()p Ga(y)s(;)g(a)p F4(\))p F6(j)1003 1991 y Fu(S)1048
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39595
2024 y F4([)p Ga(a)25 b F4(:=)1268 2012 y F9(\()1295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39596
2024 y Ga(x)1347 2012 y F9(\))1374 2024 y F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39597
b F6(j)1507 1991 y Fu(S)1551 2024 y F4(])504 2142 y(=)100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39598
b FL(Ax)o F4(\()p Ga(y)s(;)15 b(a)p F4(\)[)p Ga(a)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39599
b F4(:=)1204 2130 y F9(\()1231 2142 y Ga(x)1283 2130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39600
y F9(\))1311 2142 y F6(j)p Ga(N)10 b F6(j)1444 2109 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39601
Fu(S)1488 2142 y F4(])1490 b Gg(by)24 b(\(1\))f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39602
F6(j)p 3371 2142 V 3389 2142 V 3406 2142 V 65 w(j)3459
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39603
2109 y Fu(S)504 2261 y F4(=)100 b F6(j)p Ga(N)10 b F6(j)808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39604
2228 y Fu(S)852 2261 y F4([)p Ga(x)g F6(7!)g Ga(y)s F4(])504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39605
2379 y F6(\021)100 b(j)p Ga(N)10 b F4([)p Ga(x)g F6(7!)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39606
Ga(y)s F4(])p F6(j)1069 2346 y Fu(S)504 2498 y F4(=)100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39607
b F6(j)p FL(Id)p F4(\()p Ga(y)s(;)15 b(a)p F4(\))p Fs(\()-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39608
b Ga(a)25 b F4(:=)1199 2486 y F9(\()1227 2498 y Ga(x)1279
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39609
2486 y F9(\))1306 2498 y Ga(N)s Fs(\))p F6(j)1439 2465
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39610
y Fu(S)3032 2498 y Gg(by)f(\(2\))f(of)h Fs(\()p 3407
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39611
2498 V 3425 2498 V 3442 2498 V 65 w(\))277 2769 y Gg(F)o(or)j(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39612
(other)i(cases)f(we)e(illustrate)k(the)e(calculations)j(with)c(one)h(e)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39613
o(xample.)41 b(Let)26 b F4([)p Ga(\033)s F4(])3041 2736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39614
y Fu(S)3112 2769 y Gg(and)i Fs(\()p Ga(\033)s Fs(\))f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39615
Gg(be)277 2882 y F4([)p Ga(a)f F4(:=)497 2870 y F9(\()524
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39616
2882 y Ga(x)576 2870 y F9(\))604 2882 y F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39617
b F6(j)737 2849 y Fu(S)781 2882 y F4(])23 b Gg(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39618
Fs(\()-7 b Ga(a)25 b F4(:=)1202 2870 y F9(\()1230 2882
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39619
y Ga(x)1282 2870 y F9(\))1309 2882 y Ga(N)s Fs(\))q Gg(,)d(respecti)n
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39620
(v)o(ely)-6 b(.)p Black 277 3142 a Gb(Case)p Black 46
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39621
w Ga(M)36 b F6(\021)25 b FL(And)883 3156 y Gc(I)923 3142
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39622
y F4(\()958 3130 y FX(h)986 3142 y Ga(c)1025 3130 y FX(i)1053
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39623
3142 y Ga(S)5 b(;)1154 3130 y FX(h)1182 3142 y Ga(b)1221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39624
3130 y FX(i)1248 3142 y Ga(T)13 b(;)i(a)p F4(\))675 3306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39625
y F6(j)p FL(And)855 3320 y Gc(I)894 3306 y F4(\()929
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39626
3294 y FX(h)957 3306 y Ga(c)996 3294 y FX(i)1024 3306
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39627
y Ga(S)5 b(;)1125 3294 y FX(h)1153 3306 y Ga(b)1192 3294
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39628
y FX(i)1219 3306 y Ga(T)13 b(;)i(a)p F4(\))p F6(j)1433
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39629
3273 y Fu(S)1478 3306 y F4([)p Ga(\033)s F4(])1583 3273
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39630
y Fu(S)504 3425 y F4(=)100 b FL(And)829 3439 y Gc(R)887
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39631
3425 y F4(\()922 3413 y FX(h)950 3425 y Ga(c)989 3413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39632
y FX(i)1017 3425 y F6(j)p Ga(S)5 b F6(j)1128 3392 y Fu(S)1171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39633
3425 y Ga(;)1211 3413 y FX(h)1239 3425 y Ga(b)1278 3413
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39634
y FX(i)1306 3425 y F6(j)p Ga(T)13 b F6(j)1422 3392 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39635
Fu(S)1465 3425 y Ga(;)i(a)p F4(\)[)p Ga(\033)s F4(])1693
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39636
3392 y Fu(S)3003 3425 y Gg(by)24 b(\(3\))f(of)h F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39637
3371 3425 V 3389 3425 V 3406 3425 V 65 w(j)3459 3392
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39638
y Fu(S)504 3543 y F4(=)100 b FL(Cut)p F4(\()848 3531
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39639
y FX(h)875 3543 y Ga(a)923 3531 y FX(i)951 3543 y FL(And)1105
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39640
3557 y Gc(R)1163 3543 y F4(\()1198 3531 y FX(h)1226 3543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39641
y Ga(c)1265 3531 y FX(i)1293 3543 y F6(j)p Ga(S)5 b F6(j)1404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39642
3510 y Fu(S)1448 3543 y F4([)p Ga(\033)s F4(])1553 3510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39643
y Fu(S)1597 3543 y Ga(;)1637 3531 y FX(h)1665 3543 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39644
Ga(b)1704 3531 y FX(i)1731 3543 y F6(j)p Ga(T)13 b F6(j)1847
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39645
3510 y Fu(S)1891 3543 y F4([)p Ga(\033)s F4(])1996 3510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39646
y Fu(S)2040 3543 y Ga(;)i(a)p F4(\))p Ga(;)2203 3531
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39647
y F9(\()2231 3543 y Ga(x)2283 3531 y F9(\))2311 3543
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39648
y F6(j)p Ga(N)10 b F6(j)2444 3510 y Fu(S)2488 3543 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39649
F4(\))504 3662 y F6(\021)100 b FL(Cut)p F4(\()848 3650
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39650
y FX(h)875 3662 y Ga(a)923 3650 y FX(i)951 3662 y FL(And)1105
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39651
3676 y Gc(R)1163 3662 y F4(\()1198 3650 y FX(h)1226 3662
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39652
y Ga(c)1265 3650 y FX(i)1293 3662 y F6(j)p Ga(S)5 b Fs(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39653
Ga(\033)s Fs(\))q F6(j)1524 3629 y Fu(S)1568 3662 y Ga(;)1608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39654
3650 y FX(h)1636 3662 y Ga(b)1675 3650 y FX(i)1702 3662
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39655
y F6(j)p Ga(T)13 b Fs(\()q Ga(\033)s Fs(\))p F6(j)1938
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39656
3629 y Fu(S)1982 3662 y Ga(;)i(a)p F4(\))q Ga(;)2146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39657
3650 y F9(\()2173 3662 y Ga(x)2225 3650 y F9(\))2253
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39658
3662 y F6(j)p Ga(N)10 b F6(j)2386 3629 y Fu(S)2430 3662
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39659
y F4(\))828 b Gg(by)23 b(IH)504 3780 y F4(=)100 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39660
FL(Subst)o F4(\()947 3768 y FX(h)975 3780 y Ga(a)1023
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39661
3768 y FX(i)1050 3780 y FL(And)1205 3794 y Gc(I)1245
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39662
3780 y F4(\()1280 3768 y FX(h)1308 3780 y Ga(c)1347 3768
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39663
y FX(i)1374 3780 y Ga(S)5 b Fs(\()q Ga(\033)s Fs(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39664
Ga(;)1596 3768 y FX(h)1623 3780 y Ga(b)1662 3768 y FX(i)1690
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39665
3780 y Ga(T)13 b Fs(\()p Ga(\033)s Fs(\))q Ga(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39666
F4(\))q Ga(;)2040 3768 y F9(\()2067 3780 y Ga(x)2119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39667
3768 y F9(\))2147 3780 y Ga(N)10 b F4(\))p F6(j)2290
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39668
3747 y Fu(S)2935 3780 y Gg(by)23 b(\(2,3\))h(of)g F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39669
3371 3780 V 3389 3780 V 3406 3780 V 65 w(j)3459 3747
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39670
y Fu(S)504 3899 y F4(=)100 b F6(j)p FL(And)855 3913 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39671
Gc(I)894 3899 y F4(\()929 3887 y FX(h)957 3899 y Ga(c)996
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39672
3887 y FX(i)1024 3899 y Ga(S)5 b(;)1125 3887 y FX(h)1153
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39673
3899 y Ga(b)1192 3887 y FX(i)1219 3899 y Ga(T)13 b(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39674
F4(\))p Fs(\()q Ga(\033)s Fs(\))q F6(j)1554 3866 y Fu(S)3032
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39675
3899 y Gg(by)24 b(\(3\))f(of)h Fs(\()p 3407 3899 V 3425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39676
3899 V 3442 3899 V 65 w(\))p 3436 4004 4 62 v 3440 3946
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39677
55 4 v 3440 4004 V 3494 4004 4 62 v Black 277 4217 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39678
Gb(Pr)n(oof)c(of)f(Lemma)f(3.4.2.)p Black 33 w Gg(Again,)i(whene)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39679
(er)g F6(j)p Ga(M)10 b F6(j)1938 4184 y Fu(S)2000 4217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39680
y Gg(does)19 b(not)h(introduce)h Ga(x)p Gg(,)e(then)h(the)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39681
(substitution)277 4330 y F4([)p 304 4330 28 4 v 322 4330
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39682
V 340 4330 V 65 w(])24 b Gg(is)h(pushed)i(inside)g(the)e(subterms)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39683
(and)e(the)h(lemma)e(is)h(by)g(routine)i(induction)h(and)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39684
(calculation.)277 4443 y(So)g(we)f(are)i(left)g(with)f(the)h(cases)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39685
(where)g F6(j)p Ga(M)10 b F6(j)1751 4410 y Fu(S)1820
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39686
4443 y Gg(introduces)30 b Ga(x)p Gg(.)36 b(If)26 b Ga(M)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39687
b Gg(is)26 b(of)h(the)f(form)h FL(Id)p F4(\()p Ga(x;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39688
b(b)p F4(\))p Gg(,)27 b(we)277 4555 y(reason)e(as)f(follo)n(ws.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39689
Black 277 4785 a Gb(Case)p Black 46 w Ga(M)36 b F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39690
b FL(Id)p F4(\()p Ga(x;)15 b(b)p F4(\))675 4945 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39691
FL(Id)p F4(\()p Ga(x;)g(b)p F4(\))p F6(j)998 4912 y Fu(S)1043
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39692
4945 y F4([)p Ga(x)25 b F4(:=)1267 4933 y FX(h)1294 4945
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39693
y Ga(a)1342 4933 y FX(i)1370 4945 y F6(j)p Ga(N)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39694
F6(j)1503 4912 y Fu(S)1546 4945 y F4(])504 5058 y(=)100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39695
b FL(Ax)o F4(\()p Ga(x;)15 b(b)p F4(\)[)p Ga(x)27 b F4(:=)1203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39696
5046 y FX(h)1230 5058 y Ga(a)1278 5046 y FX(i)1306 5058
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39697
y F6(j)p Ga(N)10 b F6(j)1439 5025 y Fu(S)1483 5058 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39698
F4(])1495 b Gg(by)24 b(\(1\))f(of)h F6(j)p 3371 5058
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39699
V 3389 5058 V 3406 5058 V 65 w(j)3459 5025 y Fu(S)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39700
5171 y F4(=)100 b F6(j)p Ga(N)10 b F6(j)808 5138 y Fu(S)852
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39701
5171 y F4([)p Ga(a)g F6(7!)g Ga(b)p F4(])504 5284 y F6(\021)100
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39702
b(j)p Ga(N)10 b F4([)p Ga(a)g F6(7!)g Ga(b)p F4(])p F6(j)1056
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39703
5251 y Fu(S)504 5397 y F4(=)100 b F6(j)p FL(Id)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39704
Ga(x;)15 b(b)p F4(\))p Fs(\()-5 b Ga(x)25 b F4(:=)1198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39705
5385 y FX(h)1226 5397 y Ga(a)1274 5385 y FX(i)1301 5397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39706
y Ga(N)s Fs(\))p F6(j)1434 5364 y Fu(S)3032 5397 y Gg(by)f(\(1\))f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39707
Fs(\()p 3407 5397 V 3425 5397 V 3442 5397 V 65 w(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39708
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39709
%%Page: 162 174
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39710
TeXDict begin 162 173 bop Black -144 51 a Gb(162)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39711
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39712
4 v Black 321 388 a Gg(The)e(only)g(other)h(cases)g(which)f(need)h(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39713
(be)h(considered)j(are)d(where)g F6(j)p Ga(M)10 b F6(j)2659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39714
355 y Fu(S)2724 388 y Gg(introduces)24 b Ga(x)d Gg(and)i(is)e(not)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39715
501 y(an)30 b(axiom.)49 b(W)-7 b(e)29 b(illustrate)j(the)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39716
(calculations)k(with)c Ga(M)47 b F6(\021)37 b FL(And)2421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39717
464 y Gc(i)2421 524 y(E)2481 501 y F4(\()2516 489 y FX(h)2544
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39718
501 y Ga(b)2583 489 y FX(i)2610 501 y FL(Id)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39719
Ga(x;)15 b(b)p F4(\))q Ga(;)2924 489 y F9(\()2952 501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39720
y Ga(y)3000 489 y F9(\))3028 501 y Ga(S)5 b F4(\))p Gg(.)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39721
b(There)30 b(are)321 614 y(tw)o(o)23 b(subcases)i(depending)h(on)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39722
(whether)h Ga(N)32 b Gg(is)22 b(an)h(axiom)h(that)f(introduces)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39723
Ga(a)p Gg(.)i(Let)22 b F4([)p Ga(\033)s F4(])3099 581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39724
y Fu(S)3165 614 y Gg(and)h Fs(\()q Ga(\033)s Fs(\))g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39725
Gg(be)321 727 y F4([)p Ga(x)j F4(:=)545 715 y FX(h)572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39726
727 y Ga(a)620 715 y FX(i)648 727 y F6(j)p Ga(N)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39727
F6(j)781 694 y Fu(S)825 727 y F4(])23 b Gg(and)h Fs(\()-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39728
b Ga(x)25 b F4(:=)1251 715 y FX(h)1278 727 y Ga(a)1326
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39729
715 y FX(i)1353 727 y Ga(N)s Fs(\))q Gg(,)d(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39730
b(.)p Black 321 931 a Gb(Subcase)p Black 46 w Ga(M)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39731
b F6(\021)25 b FL(And)1054 894 y Gc(i)1054 954 y(E)1113
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39732
931 y F4(\()1148 919 y FX(h)1176 931 y Ga(b)1215 919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39733
y FX(i)1243 931 y FL(Id)p F4(\()p Ga(x;)15 b(b)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39734
Ga(;)1557 919 y F9(\()1585 931 y Ga(y)1633 919 y F9(\))1660
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39735
931 y Ga(S)5 b F4(\))23 b Gg(and)h Ga(N)35 b F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39736
b FL(Id)q F4(\()p Ga(z)t(;)15 b(a)p F4(\))794 1076 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39737
F6(j)p FL(And)973 1039 y Gc(i)973 1099 y(E)1033 1076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39738
y F4(\()1068 1064 y FX(h)1096 1076 y Ga(b)1135 1064 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39739
FX(i)1162 1076 y FL(Id)q F4(\()p Ga(x;)g(b)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39740
Ga(;)1477 1064 y F9(\()1504 1076 y Ga(y)1552 1064 y F9(\))1580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39741
1076 y Ga(S)5 b F4(\))p F6(j)1701 1043 y Fu(S)1745 1076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39742
y F4([)p Ga(\033)s F4(])1850 1043 y Fu(S)648 1190 y F4(=)75
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39743
b FL(And)948 1154 y Gc(i)948 1214 y(L)1000 1190 y F4(\()1035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39744
1178 y F9(\()1063 1190 y Ga(y)1111 1178 y F9(\))1138
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39745
1190 y F6(j)p Ga(S)5 b F6(j)1249 1157 y Fu(S)1293 1190
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39746
y Ga(;)15 b(x)p F4(\)[)p Ga(\033)s F4(])1525 1157 y Fu(S)3047
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39747
1190 y Gg(by)24 b(\(7\))f(of)h F6(j)p 3415 1190 28 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39748
v 3433 1190 V 3450 1190 V 65 w(j)3503 1157 y Fu(S)648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39749
1305 y F4(=)75 b FL(Cut)p F4(\()967 1293 y FX(h)994 1305
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39750
y Ga(a)1042 1293 y FX(i)1070 1305 y F6(j)p FL(Id)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39751
Ga(z)t(;)15 b(a)p F4(\))p F6(j)1396 1272 y Fu(S)1441
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39752
1305 y Ga(;)1481 1293 y F9(\()1509 1305 y Ga(x)1561 1293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39753
y F9(\))1589 1305 y FL(And)1743 1268 y Gc(i)1743 1328
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39754
y(L)1795 1305 y F4(\()1830 1293 y F9(\()1858 1305 y Ga(y)1906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39755
1293 y F9(\))1933 1305 y F6(j)p Ga(S)5 b F6(j)2044 1272
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39756
y Fu(S)2088 1305 y F4([)p Ga(\033)s F4(])2193 1272 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39757
Fu(S)2237 1305 y Ga(;)15 b(x)p F4(\))q(\))619 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39758
b(assumption)579 1383 y Gc(cut)549 1420 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39759
b(\000)h(!)731 1387 y FX(\003)794 1420 y FL(Cut)p F4(\()967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39760
1408 y FX(h)994 1420 y Ga(a)1042 1408 y FX(i)1070 1420
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39761
y F6(j)p FL(Id)p F4(\()p Ga(z)t(;)15 b(a)p F4(\))p F6(j)1396
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39762
1387 y Fu(S)1441 1420 y Ga(;)1481 1408 y F9(\()1509 1420
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39763
y Ga(x)1561 1408 y F9(\))1589 1420 y FL(And)1743 1383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39764
y Gc(i)1743 1443 y(L)1795 1420 y F4(\()1830 1408 y F9(\()1858
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39765
1420 y Ga(y)1906 1408 y F9(\))1933 1420 y F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39766
b Fs(\()q Ga(\033)s Fs(\))q F6(j)2165 1387 y Fu(S)2208
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39767
1420 y Ga(;)15 b(x)p F4(\))q(\))966 b Gg(by)23 b(IH)579
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39768
1497 y Gc(cut)549 1534 y F6(\000)-32 b(\000)h(!)75 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39769
FL(And)948 1497 y Gc(i)948 1557 y(L)1000 1534 y F4(\()1035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39770
1522 y F9(\()1063 1534 y Ga(y)1111 1522 y F9(\))1138
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39771
1534 y F6(j)p Ga(S)5 b Fs(\()q Ga(\033)s Fs(\))q F6(j)1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39772
1501 y Fu(S)1413 1534 y Ga(;)15 b(x)p F4(\)[)p Ga(x)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39773
b F6(7!)g Ga(z)t F4(])648 1649 y F6(\021)75 b FL(And)948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39774
1612 y Gc(i)948 1672 y(L)1000 1649 y F4(\()1035 1637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39775
y F9(\()1063 1649 y Ga(y)1111 1637 y F9(\))1138 1649
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39776
y F6(j)p Ga(S)5 b Fs(\()q Ga(\033)s Fs(\))q F6(j)1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39777
1616 y Fu(S)1413 1649 y Ga(;)15 b(z)t F4(\))1350 b Ga(x)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39778
b F6(62)g Ga(F)13 b(N)d F4(\()p F6(j)p Ga(S)5 b Fs(\()q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39779
Ga(\033)s Fs(\))p F6(j)3467 1616 y Fu(S)3511 1649 y F4(\))648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39780
1764 y(=)75 b F6(j)p FL(And)973 1727 y Gc(i)973 1787
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39781
y(E)1033 1764 y F4(\()1068 1752 y FX(h)1096 1764 y Ga(a)1144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39782
1752 y FX(i)1171 1764 y FL(Id)q F4(\()p Ga(z)t(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39783
F4(\))q Ga(;)1489 1752 y F9(\()1517 1764 y Ga(y)1565
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39784
1752 y F9(\))1592 1764 y Ga(S)5 b Fs(\()q Ga(\033)s Fs(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39785
F4(\))p F6(j)1833 1731 y Fu(S)3047 1764 y Gg(by)24 b(\(7\))f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39786
F6(j)p 3415 1764 V 3433 1764 V 3450 1764 V 65 w(j)3503
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39787
1731 y Fu(S)648 1878 y F4(=)75 b F6(j)p FL(And)973 1841
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39788
y Gc(i)973 1901 y(E)1033 1878 y F4(\()1068 1866 y FX(h)1096
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39789
1878 y Ga(b)1135 1866 y FX(i)1162 1878 y FL(Id)q F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39790
Ga(x;)15 b(b)p F4(\))q Ga(;)1477 1866 y F9(\()1504 1878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39791
y Ga(y)1552 1866 y F9(\))1580 1878 y Ga(S)5 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39792
Fs(\()q Ga(\033)s Fs(\))p F6(j)1821 1845 y Fu(S)321 2073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39793
y Gg(Otherwise)25 b(the)e(lemma)h(follo)n(ws)g(by)f(the)h(calculation)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39794
(gi)n(v)o(en)d(belo)n(w)-6 b(.)p Black 321 2277 a Gb(Subcase)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39795
Black 46 w Ga(M)35 b F6(\021)25 b FL(And)1054 2240 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39796
Gc(i)1054 2300 y(E)1113 2277 y F4(\()1148 2265 y FX(h)1176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39797
2277 y Ga(b)1215 2265 y FX(i)1243 2277 y FL(Id)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39798
Ga(x;)15 b(b)p F4(\))q Ga(;)1557 2265 y F9(\()1585 2277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39799
y Ga(y)1633 2265 y F9(\))1660 2277 y Ga(S)5 b F4(\))23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39800
b Gg(and)h Ga(N)35 b F6(6\021)25 b FL(Id)q F4(\()p Ga(z)t(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39801
b(a)p F4(\))794 2422 y F6(j)p FL(And)973 2385 y Gc(i)973
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39802
2445 y(E)1033 2422 y F4(\()1068 2410 y FX(h)1096 2422
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39803
y Ga(b)1135 2410 y FX(i)1162 2422 y FL(Id)q F4(\()p Ga(x;)g(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39804
F4(\))q Ga(;)1477 2410 y F9(\()1504 2422 y Ga(y)1552
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39805
2410 y F9(\))1580 2422 y Ga(S)5 b F4(\))p F6(j)1701 2389
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39806
y Fu(S)1745 2422 y F4([)p Ga(\033)s F4(])1850 2389 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39807
Fu(S)648 2536 y F4(=)75 b FL(And)948 2500 y Gc(i)948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39808
2560 y(L)1000 2536 y F4(\()1035 2524 y F9(\()1063 2536
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39809
y Ga(y)1111 2524 y F9(\))1138 2536 y F6(j)p Ga(S)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39810
F6(j)1249 2503 y Fu(S)1293 2536 y Ga(;)15 b(x)p F4(\)[)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39811
Ga(\033)s F4(])1525 2503 y Fu(S)3047 2536 y Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39812
b(\(7\))f(of)h F6(j)p 3415 2536 V 3433 2536 V 3450 2536
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39813
V 65 w(j)3503 2503 y Fu(S)648 2651 y F4(=)75 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39814
F4(\()967 2639 y FX(h)994 2651 y Ga(a)1042 2639 y FX(i)1070
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39815
2651 y F6(j)p Ga(N)10 b F6(j)1203 2618 y Fu(S)1247 2651
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39816
y Ga(;)1287 2639 y F9(\()1315 2651 y Ga(x)1367 2639 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39817
F9(\))1394 2651 y FL(And)1548 2614 y Gc(i)1548 2674 y(L)1601
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39818
2651 y F4(\()1636 2639 y F9(\()1663 2651 y Ga(y)1711
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39819
2639 y F9(\))1739 2651 y F6(j)p Ga(S)5 b F6(j)1850 2618
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39820
y Fu(S)1894 2651 y F4([)p Ga(\033)s F4(])1999 2618 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39821
Fu(S)2043 2651 y Ga(;)15 b(x)p F4(\)\))814 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39822
b(assumption)579 2729 y Gc(cut)549 2766 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39823
b(\000)h(!)731 2733 y FX(\003)794 2766 y FL(Cut)p F4(\()967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39824
2754 y FX(h)994 2766 y Ga(a)1042 2754 y FX(i)1070 2766
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39825
y F6(j)p Ga(N)10 b F6(j)1203 2733 y Fu(S)1247 2766 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39826
Ga(;)1287 2754 y F9(\()1315 2766 y Ga(x)1367 2754 y F9(\))1394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39827
2766 y FL(And)1548 2729 y Gc(i)1548 2789 y(L)1601 2766
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39828
y F4(\()1636 2754 y F9(\()1663 2766 y Ga(y)1711 2754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39829
y F9(\))1739 2766 y F6(j)p Ga(S)5 b Fs(\()p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39830
Fs(\))q F6(j)1970 2733 y Fu(S)2014 2766 y Ga(;)15 b(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39831
F4(\)\))1161 b Gg(by)23 b(IH)648 2880 y F4(=)75 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39832
FL(And)973 2843 y Gc(i)973 2903 y(E)1033 2880 y F4(\()1068
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39833
2868 y FX(h)1096 2880 y Ga(a)1144 2868 y FX(i)1171 2880
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39834
y Ga(N)10 b(;)1294 2868 y F9(\()1322 2880 y Ga(y)1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39835
2868 y F9(\))1397 2880 y Ga(S)5 b Fs(\()q Ga(\033)s Fs(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39836
F4(\))p F6(j)1639 2847 y Fu(S)3002 2880 y Gg(by)23 b(\(11\))h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39837
F6(j)p 3415 2880 V 3433 2880 V 3450 2880 V 65 w(j)3503
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39838
2847 y Fu(S)648 2995 y F4(=)75 b F6(j)p FL(And)973 2958
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39839
y Gc(i)973 3018 y(E)1033 2995 y F4(\()1068 2983 y FX(h)1096
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39840
2995 y Ga(b)1135 2983 y FX(i)1162 2995 y FL(Id)q F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39841
Ga(x;)15 b(b)p F4(\))q Ga(;)1477 2983 y F9(\()1504 2995
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39842
y Ga(y)1552 2983 y F9(\))1580 2995 y Ga(S)5 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39843
Fs(\()q Ga(\033)s Fs(\))p F6(j)1821 2962 y Fu(S)p 3480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39844
3098 4 62 v 3484 3040 55 4 v 3484 3098 V 3538 3098 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39845
62 v Black 321 3311 a Gb(Pr)n(oof)21 b(of)f(Lemma)f(3.4.3.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39846
Black 34 w Gg(As)g(in)h(Lemma)g(3.4.1,)g(the)g(dif)n(\002cult)h(cases)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39847
(are)e(where)g F6(j)p Ga(M)10 b F6(j)3104 3278 y Fu(S)3168
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39848
3311 y Gg(introduces)321 3424 y Ga(a)p Gg(.)58 b(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39849
b(e)32 b(illustrate)k(the)e(calculations)j(with)c(tw)o(o)g(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39850
(xamples.)59 b(Let)33 b Fs(^)p Ga(\033)s Fs(_)h Gg(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39851
F4([)p Ga(\033)s F4(])2923 3391 y Fu(S)2999 3424 y Gg(be)g(of)f(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39852
(form)321 3537 y Fs(^)q FL(And)508 3500 y Gc(i)508 3560
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39853
y(E)568 3537 y F4(\()603 3525 y FX(h)631 3537 y Ga(a)679
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39854
3525 y FX(i)p 708 3537 28 4 v 726 3537 V 743 3537 V 771
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39855
3537 a Ga(;)811 3525 y F9(\()839 3537 y Ga(x)891 3525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39856
y F9(\))918 3537 y Ga(N)10 b F4(\))p Fs(_)23 b Gg(and)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39857
F4([)p Ga(a)i F4(:=)1465 3525 y F9(\()1493 3537 y Ga(z)1539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39858
3525 y F9(\))1566 3537 y FL(And)1721 3500 y Gc(i)1721
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39859
3560 y(L)1773 3537 y F4(\()1808 3525 y F9(\()1836 3537
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39860
y Ga(x)1888 3525 y F9(\))1915 3537 y F6(j)p Ga(N)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39861
F6(j)2048 3504 y Fu(S)2092 3537 y Ga(;)15 b(z)t F4(\))q(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39862
Gg(,)23 b(respecti)n(v)o(ely)-6 b(.)p Black 321 3740
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39863
a Gb(Case)p Black 47 w Ga(M)35 b F6(\021)25 b FL(Id)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39864
F4(\()p Ga(y)s(;)15 b(a)p F4(\))662 3887 y F6(j)p FL(Id)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39865
F4(\()p Ga(y)s(;)g(a)p F4(\))p F6(j)990 3854 y Fu(S)1035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39866
3887 y F4([)p Ga(\033)s F4(])1140 3854 y Fu(S)549 4006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39867
y F4(=)42 b FL(Ax)o F4(\()p Ga(y)s(;)15 b(a)p F4(\)[)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39868
Ga(a)27 b F4(:=)1191 3994 y F9(\()1219 4006 y Ga(z)1265
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39869
3994 y F9(\))1292 4006 y FL(And)1447 3969 y Gc(i)1447
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39870
4029 y(L)1499 4006 y F4(\()1534 3994 y F9(\()1562 4006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39871
y Ga(x)1614 3994 y F9(\))1641 4006 y F6(j)p Ga(N)10 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39872
F6(j)1774 3973 y Fu(S)1818 4006 y Ga(;)15 b(z)t F4(\))q(])1082
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39873
b Gg(by)24 b(\(1\))f(of)h F6(j)p 3415 4006 V 3433 4006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39874
V 3450 4006 V 65 w(j)3503 3973 y Fu(S)549 4125 y F4(=)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39875
b FL(And)816 4088 y Gc(i)816 4148 y(L)869 4125 y F4(\()904
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39876
4113 y F9(\()931 4125 y Ga(x)983 4113 y F9(\))1011 4125
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39877
y F6(j)p Ga(N)10 b F6(j)1144 4092 y Fu(S)1188 4125 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39878
Ga(;)15 b(y)s F4(\))1019 b Gg(by)24 b(assumption)i Ga(z)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39879
F6(62)c Ga(F)13 b(N)d F4(\()3227 4113 y F9(\()3255 4125
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39880
y Ga(x)3307 4113 y F9(\))3334 4125 y F6(j)p Ga(N)g F6(j)3467
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39881
4092 y Fu(S)3511 4125 y F4(\))549 4243 y(=)42 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39882
FL(And)842 4206 y Gc(i)842 4266 y(E)901 4243 y F4(\()936
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39883
4231 y FX(h)964 4243 y Ga(a)1012 4231 y FX(i)1039 4243
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39884
y FL(Id)q F4(\()p Ga(y)s(;)15 b(a)p F4(\))q Ga(;)1359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39885
4231 y F9(\()1387 4243 y Ga(x)1439 4231 y F9(\))1466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39886
4243 y Ga(N)10 b F4(\))p F6(j)1609 4210 y Fu(S)549 4362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39887
y F4(=)42 b F6(j)p FL(Id)p F4(\()p Ga(y)s(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39888
F4(\))p Fs(^)r Ga(\033)s Fs(_)p F6(j)1111 4329 y Fu(S)3076
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39889
4362 y Gg(by)24 b(\(1\))f(of)h Fs(^)p 3451 4362 V 3469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39890
4362 V 3487 4362 V 65 w(_)p Black 321 4540 a Gb(Case)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39891
Black 47 w Ga(M)35 b F6(\021)25 b FL(And)927 4554 y Gc(I)967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39892
4540 y F4(\()1002 4528 y FX(h)1030 4540 y Ga(c)1069 4528
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39893
y FX(i)1097 4540 y Ga(S)5 b(;)1198 4528 y FX(h)1226 4540
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39894
y Ga(d)1273 4528 y FX(i)1301 4540 y Ga(T)12 b(;)j(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39895
F4(\))662 4687 y F6(j)p FL(And)842 4701 y Gc(I)882 4687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39896
y F4(\()917 4675 y FX(h)944 4687 y Ga(c)983 4675 y FX(i)1011
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39897
4687 y F6(j)p Ga(S)5 b F6(j)1122 4654 y Fu(S)1166 4687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39898
y Ga(;)1206 4675 y FX(h)1234 4687 y Ga(d)1281 4675 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39899
FX(i)1309 4687 y F6(j)p Ga(T)13 b F6(j)1425 4654 y Fu(S)1468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39900
4687 y Ga(;)i(a)p F4(\))p F6(j)1616 4654 y Fu(S)1661
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39901
4687 y F4([)p Ga(\033)s F4(])1766 4654 y Fu(S)549 4806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39902
y F4(=)42 b FL(And)816 4820 y Gc(R)874 4806 y F4(\()909
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39903
4794 y FX(h)937 4806 y Ga(c)976 4794 y FX(i)1004 4806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39904
y F6(j)p Ga(S)5 b F6(j)1115 4773 y Fu(S)1158 4806 y Ga(;)1198
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39905
4794 y FX(h)1226 4806 y Ga(d)1273 4794 y FX(i)1301 4806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39906
y F6(j)p Ga(T)13 b F6(j)1417 4773 y Fu(S)1461 4806 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39907
Ga(;)i(a)p F4(\)[)p Ga(\033)s F4(])1689 4773 y Fu(S)3047
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39908
4806 y Gg(by)24 b(\(3\))f(of)h F6(j)p 3415 4806 V 3433
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39909
4806 V 3450 4806 V 65 w(j)3503 4773 y Fu(S)549 4924 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39910
F4(=)42 b FL(Cut)p F4(\()835 4912 y FX(h)862 4924 y Ga(a)910
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39911
4912 y FX(i)938 4924 y FL(And)1093 4938 y Gc(R)1150 4924
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39912
y F4(\()1185 4912 y FX(h)1213 4924 y Ga(c)1252 4912 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39913
FX(i)1280 4924 y F6(j)p Ga(S)5 b F6(j)1391 4891 y Fu(S)1435
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39914
4924 y F4([)p Ga(\033)s F4(])1540 4891 y Fu(S)1584 4924
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39915
y Ga(;)1624 4912 y FX(h)1652 4924 y Ga(d)1699 4912 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39916
FX(i)1726 4924 y F6(j)p Ga(T)13 b F6(j)1842 4891 y Fu(S)1886
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39917
4924 y F4([)p Ga(\033)s F4(])1991 4891 y Fu(S)2035 4924
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39918
y Ga(;)i(a)p F4(\))q Ga(;)2199 4912 y F9(\()2227 4924
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39919
y Ga(z)2273 4912 y F9(\))2300 4924 y FL(And)2455 4887
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39920
y Gc(i)2455 4947 y(L)2507 4924 y F4(\()2542 4912 y F9(\()2570
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39921
4924 y Ga(x)2622 4912 y F9(\))2649 4924 y F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39922
b F6(j)2782 4891 y Fu(S)2826 4924 y Ga(;)15 b(z)t F4(\))q(\))549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39923
5043 y F6(\021)42 b FL(Cut)p F4(\()835 5031 y FX(h)862
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39924
5043 y Ga(a)910 5031 y FX(i)938 5043 y FL(And)1093 5057
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39925
y Gc(R)1150 5043 y F4(\()1185 5031 y FX(h)1213 5043 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39926
Ga(c)1252 5031 y FX(i)1280 5043 y F6(j)p Ga(S)5 b Fs(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39927
Ga(\033)s Fs(_)p F6(j)1511 5010 y Fu(S)1555 5043 y Ga(;)1595
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39928
5031 y FX(h)1623 5043 y Ga(d)1670 5031 y FX(i)1697 5043
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39929
y F6(j)p Ga(T)13 b Fs(^)q Ga(\033)s Fs(_)q F6(j)1934
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39930
5010 y Fu(S)1977 5043 y Ga(;)i(a)p F4(\))q Ga(;)2141
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39931
5031 y F9(\()2169 5043 y Ga(z)2215 5031 y F9(\))2243
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39932
5043 y FL(And)2397 5006 y Gc(i)2397 5066 y(L)2449 5043
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39933
y F4(\()2484 5031 y F9(\()2512 5043 y Ga(x)2564 5031
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39934
y F9(\))2591 5043 y F6(j)p Ga(N)10 b F6(j)2724 5010 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39935
Fu(S)2768 5043 y Ga(;)15 b(z)t F4(\))q(\))412 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39936
b(IH)549 5161 y F4(=)42 b F6(j)p FL(And)842 5124 y Gc(i)842
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39937
5184 y(E)901 5161 y F4(\()936 5149 y FX(h)964 5161 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39938
Ga(a)1012 5149 y FX(i)1039 5161 y FL(And)1194 5175 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39939
Gc(I)1234 5161 y F4(\()1269 5149 y FX(h)1297 5161 y Ga(c)1336
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39940
5149 y FX(i)1364 5161 y Ga(S)5 b Fs(^)p Ga(\033)s Fs(_)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39941
Ga(;)1585 5149 y FX(h)1613 5161 y Ga(d)1660 5149 y FX(i)1687
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39942
5161 y Ga(T)13 b Fs(^)p Ga(\033)s Fs(_)q Ga(;)i(a)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39943
Ga(;)2037 5149 y F9(\()2065 5161 y Ga(x)2117 5149 y F9(\))2144
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39944
5161 y Ga(N)10 b F4(\))p F6(j)2287 5128 y Fu(S)2933 5161
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39945
y Gg(by)24 b(\(2,11\))g(of)g F6(j)p 3415 5161 V 3433
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39946
5161 V 3450 5161 V 65 w(j)3503 5128 y Fu(S)549 5280 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39947
F4(=)42 b F6(j)p FL(And)842 5294 y Gc(I)882 5280 y F4(\()917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39948
5268 y FX(h)944 5280 y Ga(c)983 5268 y FX(i)1011 5280
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39949
y Ga(S)5 b(;)1112 5268 y FX(h)1140 5280 y Ga(d)1187 5268
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39950
y FX(i)1215 5280 y Ga(T)12 b(;)j(a)p F4(\))p Fs(^)r Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39951
Fs(_)p F6(j)1549 5247 y Fu(S)3076 5280 y Gg(by)24 b(\(2\))f(of)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39952
Fs(^)p 3451 5280 V 3469 5280 V 3487 5280 V 65 w(_)p 3480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39953
5385 4 62 v 3484 5327 55 4 v 3484 5385 V 3538 5385 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39954
62 v Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39955
%%Page: 163 175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39956
TeXDict begin 163 174 bop Black 277 51 a Gb(B.2)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39957
(Chapter)f(3)2639 b(163)p 277 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39958
277 388 a(Pr)n(oof)24 b(of)g(Theor)n(em)f(3.4.4.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39959
34 w Gg(W)-7 b(e)23 b(gi)n(v)o(e)g(some)h(illustrati)n(v)o(e)i(e)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39960
(xamples.)p Black 277 612 a Gb(Beta-Reduction:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39961
410 748 a FL(Imp)554 770 y Gc(E)614 748 y F4(\()649 736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39962
y FX(h)677 748 y Ga(b)716 736 y FX(i)743 748 y FL(Imp)888
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39963
770 y Gc(I)928 748 y F4(\()963 736 y F9(\()990 748 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39964
Ga(x)1042 736 y F9(\))q FX(h)1097 748 y Ga(a)1145 736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39965
y FX(i)1173 748 y Ga(M)10 b(;)15 b(b)p F4(\))q Ga(;)1426
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39966
736 y FX(h)1453 748 y Ga(c)1492 736 y FX(i)1520 748 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39967
Ga(N)10 b(;)1643 736 y F9(\()1671 748 y Ga(y)1719 736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39968
y F9(\))1746 748 y Ga(P)j F4(\))1940 711 y Gc(\014)1876
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39969
748 y F6(\000)-31 b(\000)g(!)23 b FL(Subst)o F4(\()2317
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39970
736 y FX(h)2345 748 y Ga(c)2384 736 y FX(i)2412 748 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39971
Ga(N)10 b(;)2535 736 y F9(\()2563 748 y Ga(x)2615 736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39972
y F9(\))2642 748 y FL(Subst)o F4(\()2889 736 y FX(h)2917
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39973
748 y Ga(a)2965 736 y FX(i)2992 748 y Ga(M)h(;)3131 736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39974
y F9(\()3158 748 y Ga(y)3206 736 y F9(\))3234 748 y Ga(P)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39975
F4(\)\))698 920 y F6(j)p FL(Imp)868 942 y Gc(E)928 920
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39976
y F4(\()963 908 y FX(h)990 920 y Ga(b)1029 908 y FX(i)1057
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39977
920 y FL(Imp)1201 942 y Gc(I)1241 920 y F4(\()1276 908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39978
y F9(\()1304 920 y Ga(x)1356 908 y F9(\))p FX(h)1411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39979
920 y Ga(a)1459 908 y FX(i)1486 920 y Ga(M)d(;)15 b(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39980
F4(\))q Ga(;)1739 908 y FX(h)1767 920 y Ga(c)1806 908
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39981
y FX(i)1834 920 y Ga(N)10 b(;)1957 908 y F9(\()1985 920
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39982
y Ga(y)2033 908 y F9(\))2060 920 y Ga(P)j F4(\))p F6(j)2191
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39983
887 y Fu(S)604 1039 y F4(=)23 b FL(Cut)p F4(\()871 1027
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39984
y FX(h)899 1039 y Ga(b)938 1027 y FX(i)965 1039 y FL(Imp)1110
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39985
1061 y Gc(R)1168 1039 y F4(\()1203 1027 y F9(\()1230
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39986
1039 y Ga(x)1282 1027 y F9(\))q FX(h)1337 1039 y Ga(a)1385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39987
1027 y FX(i)1413 1039 y F6(j)p Ga(M)10 b F6(j)1561 1006
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39988
y Fu(S)1605 1039 y Ga(;)15 b(b)p F4(\))p Ga(;)1759 1027
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39989
y F9(\()1787 1039 y Ga(z)1833 1027 y F9(\))1861 1039
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39990
y FL(Imp)2005 1061 y Gc(L)2058 1039 y F4(\()2093 1027
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39991
y FX(h)2120 1039 y Ga(c)2159 1027 y FX(i)2187 1039 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39992
F6(j)p Ga(N)10 b F6(j)2320 1006 y Fu(S)2364 1039 y Ga(;)2404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39993
1027 y F9(\()2432 1039 y Ga(y)2480 1027 y F9(\))2507
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39994
1039 y F6(j)p Ga(P)j F6(j)2628 1006 y Fu(S)2672 1039
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39995
y Ga(;)i(z)t F4(\))q(\))60 b Gg(by)24 b(\(5,13\))g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39996
F6(j)p 3371 1039 28 4 v 3389 1039 V 3406 1039 V 65 w(j)3459
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39997
1006 y Fu(S)535 1120 y Gc(cut)504 1157 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39998
b(\000)g(!)23 b FL(Cut)p F4(\()871 1145 y FX(h)899 1157
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 39999
y Ga(c)938 1145 y FX(i)966 1157 y F6(j)p Ga(N)10 b F6(j)1099
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40000
1124 y Fu(S)1143 1157 y Ga(;)1183 1145 y F9(\()1210 1157
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40001
y Ga(x)1262 1145 y F9(\))1290 1157 y FL(Cut)p F4(\()1463
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40002
1145 y FX(h)1490 1157 y Ga(a)1538 1145 y FX(i)1566 1157
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40003
y F6(j)p Ga(M)g F6(j)1714 1124 y Fu(S)1758 1157 y Ga(;)1798
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40004
1145 y F9(\()1826 1157 y Ga(y)1874 1145 y F9(\))1901
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40005
1157 y F6(j)p Ga(P)j F6(j)2022 1124 y Fu(S)2066 1157
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40006
y F4(\)\))1575 1276 y Ga(z)27 b Gg(is)d(freshly)h(introduced)i(by)c
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40007
(side-condition)28 b(\(13\))c(of)g F6(j)p 3371 1276 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40008
3389 1276 V 3406 1276 V 65 w(j)3459 1243 y Fu(S)2232
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40009
1394 y Gg(and)g Ga(b)f Gg(is)h(freshly)h(introduced)h(by)3396
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40010
1357 y Gc(\014)3332 1394 y F6(\000)-31 b(\000)f(!)604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40011
1513 y F4(=)23 b F6(j)p FL(Subst)p F4(\()971 1501 y FX(h)998
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40012
1513 y Ga(c)1037 1501 y FX(i)1065 1513 y Ga(N)10 b(;)1188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40013
1501 y F9(\()1216 1513 y Ga(x)1268 1501 y F9(\))1295
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40014
1513 y FL(Subst)p F4(\()1543 1501 y FX(h)1570 1513 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40015
Ga(a)1618 1501 y FX(i)1646 1513 y Ga(M)g(;)1784 1501
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40016
y F9(\()1812 1513 y Ga(y)1860 1501 y F9(\))1887 1513
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40017
y Ga(P)j F4(\)\))p F6(j)2053 1480 y Fu(S)3003 1513 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40018
Gg(by)24 b(\(2\))f(of)h F6(j)p 3371 1513 V 3389 1513
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40019
V 3406 1513 V 65 w(j)3459 1480 y Fu(S)p Black 277 1713
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40020
a Gb(Commuting)e(Reduction:)p Black 504 1850 a FL(And)659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40021
1813 y Gc(i)659 1873 y(E)719 1850 y F4(\()754 1838 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40022
FX(h)781 1850 y Ga(a)829 1838 y FX(i)857 1850 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40023
b(;)995 1838 y F9(\()1023 1850 y Ga(x)1075 1838 y F9(\))1102
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40024
1850 y Ga(N)g F4(\))1311 1813 y Gc(\015)1246 1850 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40025
b(\000)f(!)25 b Ga(M)10 b Fs(^)q FL(And)1726 1813 y Gc(i)1726
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40026
1873 y(E)1786 1850 y F4(\()1821 1838 y FX(h)1849 1850
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40027
y Ga(a)1897 1838 y FX(i)p 1926 1850 V 1944 1850 V 1962
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40028
1850 V 1989 1850 a Ga(;)2029 1838 y F9(\()2057 1850 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40029
Ga(x)2109 1838 y F9(\))2136 1850 y Ga(N)g F4(\))p Fs(_)717
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40030
2011 y F6(j)p FL(And)897 1974 y Gc(i)897 2034 y(E)957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40031
2011 y F4(\()992 1999 y FX(h)1019 2011 y Ga(a)1067 1999
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40032
y FX(i)1095 2011 y Ga(M)g(;)1233 1999 y F9(\()1261 2011
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40033
y Ga(x)1313 1999 y F9(\))1340 2011 y Ga(N)g F4(\))p F6(j)1483
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40034
1978 y Fu(S)604 2130 y F4(=)42 b FL(Cut)p F4(\()890 2118
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40035
y FX(h)918 2130 y Ga(a)966 2118 y FX(i)993 2130 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40036
Ga(M)10 b F6(j)1141 2097 y Fu(S)1185 2130 y Ga(;)1225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40037
2118 y F9(\()1253 2130 y Ga(z)1299 2118 y F9(\))1327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40038
2130 y FL(And)1481 2093 y Gc(i)1481 2153 y(L)1534 2130
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40039
y F4(\()1569 2118 y F9(\()1596 2130 y Ga(x)1648 2118
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40040
y F9(\))1676 2130 y F6(j)p Ga(N)g F6(j)1809 2097 y Fu(S)1853
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40041
2130 y Ga(;)15 b(z)t F4(\))q(\))947 b Gg(by)24 b(\(11\))g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40042
F6(j)p 3371 2130 V 3389 2130 V 3406 2130 V 65 w(j)3459
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40043
2097 y Fu(S)535 2211 y Gc(cut)504 2248 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40044
b(\000)g(!)42 b(j)p Ga(M)10 b F6(j)865 2215 y Fu(S)909
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40045
2248 y F4([)p Ga(a)26 b F4(:=)1129 2236 y F9(\()1156
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40046
2248 y Ga(z)1202 2236 y F9(\))1230 2248 y FL(And)1385
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40047
2211 y Gc(i)1385 2271 y(L)1437 2248 y F4(\()1472 2236
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40048
y F9(\()1500 2248 y Ga(x)1552 2236 y F9(\))1579 2248
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40049
y F6(j)p Ga(N)10 b F6(j)1712 2215 y Fu(S)1756 2248 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40050
Ga(;)15 b(z)t F4(\))q(])335 b Ga(a)23 b Gg(is)h(not)g(freshly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40051
(introduced)h(by)3397 2211 y Gc(\015)3332 2248 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40052
b(\000)f(!)604 2367 y(\021)42 b(j)p Ga(M)10 b Fs(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40053
FL(And)1028 2330 y Gc(i)1028 2390 y(E)1087 2367 y F4(\()1122
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40054
2355 y FX(h)1150 2367 y Ga(a)1198 2355 y FX(i)p 1228
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40055
2367 V 1245 2367 V 1263 2367 V 1290 2367 a Ga(;)1330
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40056
2355 y F9(\()1358 2367 y Ga(x)1410 2355 y F9(\))1437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40057
2367 y Ga(N)g F4(\))p Fs(_)q F6(j)1613 2334 y Fu(S)2906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40058
2367 y Gg(by)24 b(Lemma)e(3.4.3)p Black 277 2567 a Gb(Substitution)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40059
(Elimination:)p Black 504 2704 a FL(Subst)p F4(\()752
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40060
2692 y FX(h)779 2704 y Ga(a)827 2692 y FX(i)855 2704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40061
y Ga(M)10 b(;)993 2692 y F9(\()1021 2704 y Ga(x)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40062
2692 y F9(\))1100 2704 y Ga(N)g F4(\))1382 2667 y Gc(\033)1318
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40063
2704 y F6(\000)-31 b(\000)f(!)100 b Ga(M)10 b Fs(\()-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40064
b Ga(a)26 b F4(:=)1906 2692 y F9(\()1933 2704 y Ga(x)1985
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40065
2692 y F9(\))2013 2704 y Ga(N)s Fs(\))717 2880 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40066
FL(Subst)p F4(\()990 2868 y FX(h)1017 2880 y Ga(a)1065
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40067
2868 y FX(i)1093 2880 y Ga(M)10 b(;)1231 2868 y F9(\()1259
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40068
2880 y Ga(x)1311 2868 y F9(\))1338 2880 y Ga(N)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40069
F6(j)1481 2847 y Fu(S)604 3004 y F4(=)42 b FL(Cut)p F4(\()890
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40070
2992 y FX(h)918 3004 y Ga(a)966 2992 y FX(i)993 3004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40071
y F6(j)p Ga(M)10 b F6(j)1141 2971 y Fu(S)1185 3004 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40072
Ga(;)1225 2992 y F9(\()1253 3004 y Ga(x)1305 2992 y F9(\))1333
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40073
3004 y F6(j)p Ga(N)g F6(j)1466 2971 y Fu(S)1509 3004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40074
y F4(\))1459 b Gg(by)24 b(\(2\))f(of)h F6(j)p 3371 3004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40075
V 3389 3004 V 3406 3004 V 65 w(j)3459 2971 y Fu(S)535
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40076
3091 y Gc(cut)504 3128 y F6(\000)-31 b(\000)g(!)42 b(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40077
Ga(M)10 b F6(j)865 3095 y Fu(S)909 3128 y F4([)p Ga(a)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40078
b F4(:=)1129 3116 y F9(\()1156 3128 y Ga(x)1208 3116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40079
y F9(\))1236 3128 y F6(j)p Ga(N)10 b F6(j)1369 3095 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40080
Fu(S)1413 3128 y F4(])800 b Ga(a)23 b Gg(is)h(not)g(freshly)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40081
(introduced)h(by)3396 3091 y Gc(\033)3332 3128 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40082
b(\000)f(!)604 3252 y(\021)42 b(j)p Ga(M)10 b Fs(\()-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40083
b Ga(a)25 b F4(:=)1060 3240 y F9(\()1088 3252 y Ga(x)1140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40084
3240 y F9(\))1167 3252 y Ga(N)s Fs(\))q F6(j)1301 3219
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40085
y Fu(S)2906 3252 y Gg(by)f(Lemma)e(3.4.1)504 3424 y FL(Subst)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40086
F4(\()752 3412 y FX(h)779 3424 y Ga(a)827 3412 y FX(i)855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40087
3424 y Ga(M)10 b(;)993 3412 y F9(\()1021 3424 y Ga(x)1073
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40088
3412 y F9(\))1100 3424 y Ga(N)g F4(\))1382 3387 y Gc(\033)1318
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40089
3424 y F6(\000)-31 b(\000)f(!)100 b Ga(N)10 b Fs(\()-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40090
b Ga(x)26 b F4(:=)1895 3412 y FX(h)1922 3424 y Ga(a)1970
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40091
3412 y FX(i)1998 3424 y Ga(M)s Fs(\))741 3600 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40092
FL(Subst)p F4(\()1014 3588 y FX(h)1042 3600 y Ga(a)1090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40093
3588 y FX(i)1117 3600 y Ga(M)10 b(;)1255 3588 y F9(\()1283
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40094
3600 y Ga(x)1335 3588 y F9(\))1362 3600 y Ga(N)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40095
F6(j)1505 3567 y Fu(S)628 3724 y F4(=)42 b FL(Cut)p F4(\()914
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40096
3712 y FX(h)942 3724 y Ga(a)990 3712 y FX(i)1018 3724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40097
y F6(j)p Ga(M)10 b F6(j)1166 3691 y Fu(S)1210 3724 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40098
Ga(;)1250 3712 y F9(\()1277 3724 y Ga(x)1329 3712 y F9(\))1357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40099
3724 y F6(j)p Ga(N)g F6(j)1490 3691 y Fu(S)1534 3724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40100
y F4(\))1434 b Gg(by)24 b(\(2\))f(of)h F6(j)p 3371 3724
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40101
V 3389 3724 V 3406 3724 V 65 w(j)3459 3691 y Fu(S)559
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40102
3811 y Gc(cut)529 3848 y F6(\000)-32 b(\000)h(!)42 b(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40103
Ga(N)10 b F6(j)874 3815 y Fu(S)918 3848 y F4([)p Ga(x)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40104
b F4(:=)1142 3836 y FX(h)1169 3848 y Ga(a)1217 3836 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40105
FX(i)1245 3848 y F6(j)p Ga(M)10 b F6(j)1393 3815 y Fu(S)1437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40106
3848 y F4(])773 b Ga(x)22 b Gg(is)i(not)g(freshly)h(introduced)h(by)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40107
3396 3811 y Gc(\033)3332 3848 y F6(\000)-31 b(\000)f(!)550
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40108
3936 y Gc(cut)520 3973 y F6(\000)g(\000)h(!)690 3940
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40109
y FX(\003)741 3973 y F6(j)p Ga(N)10 b Fs(\()-6 b Ga(x)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40110
b F4(:=)1073 3961 y FX(h)1101 3973 y Ga(a)1149 3961 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40111
FX(i)1176 3973 y Ga(M)s Fs(\))q F6(j)1325 3940 y Fu(S)2906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40112
3973 y Gg(by)f(Lemma)e(3.4.2)p Black 277 4175 a Gb(Inner)h(Reduction:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40113
Black 46 w Gg(The)38 b(non-tri)n(vial)k(cases)d(are)g(where)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40114
Ga(M)49 b Gg(is)38 b(translated)k(using)e(the)f(clauses)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40115
4288 y(\(11\)\226\(14\))26 b(of)e F6(j)p 1001 4288 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40116
1019 4288 V 1036 4288 V 65 w(j)1089 4255 y Fu(S)1132
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40117
4288 y Gg(,)f(while)h Ga(N)32 b Gg(using)25 b(\(7\)\226\(10\).)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40118
b(W)-7 b(e)22 b(gi)n(v)o(e)i(one)g(case.)504 4447 y FL(And)659
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40119
4411 y Gc(i)659 4471 y(E)719 4447 y F4(\()754 4435 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40120
FX(h)781 4447 y Ga(a)829 4435 y FX(i)857 4447 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40121
b(;)958 4435 y F9(\()986 4447 y Ga(x)1038 4435 y F9(\))1065
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40122
4447 y Ga(T)13 b F4(\))1256 4410 y Gc(\024)1191 4447
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40123
y F6(\000)-31 b(\000)g(!)25 b FL(And)1541 4411 y Gc(i)1541
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40124
4471 y(E)1601 4447 y F4(\()1636 4435 y FX(h)1664 4447
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40125
y Ga(a)1712 4435 y FX(i)1739 4447 y Ga(S)1800 4414 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40126
FX(0)1824 4447 y Ga(;)1864 4435 y F9(\()1891 4447 y Ga(x)1943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40127
4435 y F9(\))1971 4447 y Ga(T)13 b F4(\))23 b Gg(with)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40128
Ga(S)2340 4414 y FX(0)2389 4447 y F6(\021)i FL(Id)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40129
Ga(y)s(;)15 b(a)p F4(\))741 4609 y F6(j)p FL(And)921
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40130
4572 y Gc(i)921 4632 y(E)981 4609 y F4(\()1016 4597 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40131
FX(h)1044 4609 y Ga(a)1092 4597 y FX(i)1119 4609 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40132
b(;)1220 4597 y F9(\()1248 4609 y Ga(x)1300 4597 y F9(\))1327
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40133
4609 y Ga(T)13 b F4(\))p F6(j)1453 4576 y Fu(S)628 4727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40134
y F4(=)42 b FL(Cut)p F4(\()914 4715 y FX(h)942 4727 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40135
Ga(a)990 4715 y FX(i)1018 4727 y F6(j)p Ga(S)5 b F6(j)1129
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40136
4694 y Fu(S)1172 4727 y Ga(;)1212 4715 y F9(\()1240 4727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40137
y Ga(z)1286 4715 y F9(\))1314 4727 y FL(And)1469 4691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40138
y Gc(i)1469 4750 y(L)1521 4727 y F4(\()1556 4715 y F9(\()1584
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40139
4727 y Ga(x)1636 4715 y F9(\))1663 4727 y F6(j)p Ga(T)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40140
b F6(j)1779 4694 y Fu(S)1823 4727 y Ga(;)i(z)t F4(\))q(\))977
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40141
b Gg(by)24 b(\(11\))g(of)g F6(j)p 3371 4727 V 3389 4727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40142
V 3406 4727 V 65 w(j)3459 4694 y Fu(S)550 4809 y Gc(cut)520
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40143
4846 y F6(\000)-32 b(\000)h(!)690 4813 y FX(\003)741
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40144
4846 y FL(Cut)p F4(\()914 4834 y FX(h)942 4846 y Ga(a)990
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40145
4834 y FX(i)1018 4846 y F6(j)p Ga(S)1104 4813 y FX(0)1127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40146
4846 y F6(j)1152 4813 y Fu(S)1196 4846 y Ga(;)1236 4834
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40147
y F9(\()1264 4846 y Ga(z)1310 4834 y F9(\))1337 4846
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40148
y FL(And)1492 4809 y Gc(i)1492 4869 y(L)1544 4846 y F4(\()1579
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40149
4834 y F9(\()1607 4846 y Ga(x)1659 4834 y F9(\))1686
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40150
4846 y F6(j)p Ga(T)13 b F6(j)1802 4813 y Fu(S)1846 4846
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40151
y Ga(;)i(z)t F4(\))q(\))1290 b Gg(by)23 b(IH)559 4927
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40152
y Gc(cut)529 4964 y F6(\000)-32 b(\000)h(!)42 b FL(And)896
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40153
4928 y Gc(i)896 4988 y(L)948 4964 y F4(\()983 4952 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40154
F9(\()1011 4964 y Ga(x)1063 4952 y F9(\))1090 4964 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40155
F6(j)p Ga(T)13 b F6(j)1206 4931 y Fu(S)1250 4964 y Ga(;)i(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40156
F4(\)[)p Ga(z)f F6(7!)c Ga(y)s F4(])628 5083 y F6(\021)42
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40157
b FL(And)896 5046 y Gc(i)896 5106 y(L)948 5083 y F4(\()983
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40158
5071 y F9(\()1011 5083 y Ga(x)1063 5071 y F9(\))1090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40159
5083 y F6(j)p Ga(T)13 b F6(j)1206 5050 y Fu(S)1250 5083
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40160
y Ga(;)i(y)s F4(\))1020 b Ga(z)30 b F6(62)24 b Ga(F)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40161
b(N)d F4(\()p F6(j)p Ga(T)j F6(j)2855 5050 y Fu(S)2899
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40162
5083 y F4(\))23 b Gg(by)h(\(11\))g(of)g F6(j)p 3371 5083
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40163
V 3389 5083 V 3406 5083 V 65 w(j)3459 5050 y Fu(S)628
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40164
5202 y F4(=)42 b F6(j)p FL(And)921 5165 y Gc(i)921 5225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40165
y(E)981 5202 y F4(\()1016 5190 y FX(h)1044 5202 y Ga(a)1092
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40166
5190 y FX(i)1119 5202 y FL(Id)p F4(\()p Ga(y)s(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40167
F4(\))q Ga(;)1438 5190 y F9(\()1466 5202 y Ga(x)1518
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40168
5190 y F9(\))1546 5202 y Ga(T)d F4(\))p F6(j)1671 5169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40169
y Fu(S)1741 5202 y F6(\021)25 b(j)p Ga(N)10 b F6(j)1970
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40170
5169 y Fu(S)3003 5202 y Gg(by)24 b(\(7\))f(of)h F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40171
3371 5202 V 3389 5202 V 3406 5202 V 65 w(j)3459 5169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40172
y Fu(S)p 3436 5307 4 62 v 3440 5249 55 4 v 3440 5307
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40173
V 3494 5307 4 62 v Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40174
%%Page: 164 176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40175
TeXDict begin 164 175 bop Black -144 51 a Gb(164)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40176
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40177
4 v Black Black 321 412 a(Pr)n(oof)i(of)f(Lemma)f(3.4.6.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40178
Black 34 w Gg(The)h(cases)h(corresponding)j(to)c(the)g(clauses)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40179
(\(1\)\226\(5\))g(and)f(\(14\))f(of)g F6(j)p 3404 412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40180
28 4 v 3422 412 V 3440 412 V 65 w(j)3492 379 y Fu(N)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40181
525 y Gg(are)j(routine.)43 b(Belo)n(w)27 b(we)f(gi)n(v)o(e)i(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40182
(details)h(for)f(the)g(clauses)h(\(6\))f(and)g(\(10\).)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40183
b(Let)27 b F4([)p Ga(\033)s F4(])h Gg(and)g Fs(\()p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40184
Fs(\))3380 483 y Fu(N)3461 525 y Gg(be)321 638 y F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40185
Ga(x)e F4(:=)545 626 y FX(h)572 638 y Ga(a)620 626 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40186
FX(i)648 638 y Ga(N)10 b F4(])23 b Gg(and)h Fs(\()-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40187
b Ga(x)25 b F4(:=)1157 626 y FX(h)1184 638 y Ga(a)1232
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40188
626 y FX(i)1260 638 y F6(j)p Ga(N)10 b F6(j)1393 605
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40189
y Fu(N)1440 638 y Fs(\))p Gg(,)23 b(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40190
b(.)p Black 321 907 a Gb(Case)p Black 47 w Ga(M)35 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40191
F6(\021)25 b FL(And)927 870 y Gc(i)927 930 y(L)980 907
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40192
y F4(\()1015 895 y F9(\()1042 907 y Ga(y)1090 895 y F9(\))1118
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40193
907 y Ga(S)5 b(;)15 b(x)p F4(\))674 1074 y F6(j)p FL(And)853
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40194
1038 y Gc(i)853 1098 y(L)906 1074 y F4(\()941 1062 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40195
F9(\()968 1074 y Ga(y)1016 1062 y F9(\))1044 1074 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40196
b(;)15 b(x)p F4(\)[)p Ga(\033)s F4(])p F6(j)1362 1042
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40197
y Fu(N)549 1193 y F4(=)54 b F6(j)p FL(Cut)p F4(\()872
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40198
1181 y FX(h)900 1193 y Ga(a)948 1181 y FX(i)975 1193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40199
y Ga(N)10 b(;)1098 1181 y F9(\()1126 1193 y Ga(x)1178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40200
1181 y F9(\))1205 1193 y FL(And)1360 1156 y Gc(i)1360
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40201
1216 y(L)1412 1193 y F4(\()1447 1181 y F9(\()1475 1193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40202
y Ga(y)1523 1181 y F9(\))1550 1193 y Ga(S)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40203
Ga(\033)s F4(])q Ga(;)15 b(x)p F4(\)\))p F6(j)1904 1160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40204
y Fu(N)549 1312 y F4(=)54 b FL(And)828 1275 y Gc(i)828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40205
1335 y(E)888 1312 y F4(\()923 1300 y FX(h)951 1312 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40206
Ga(a)999 1300 y FX(i)1026 1312 y F6(j)p Ga(N)10 b F6(j)1159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40207
1279 y Fu(N)1213 1312 y Ga(;)1253 1300 y F9(\()1281 1312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40208
y Ga(y)1329 1300 y F9(\))1357 1312 y F6(j)p Ga(S)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40209
F4([)p Ga(\033)s F4(])p F6(j)1573 1279 y Fu(N)1628 1312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40210
y F4(\))1328 b Gg(by)24 b(\(10\))g(of)f F6(j)p 3404 1312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40211
V 3422 1312 V 3440 1312 V 65 w(j)3492 1279 y Fu(N)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40212
1435 y F6(\021)54 b FL(And)828 1398 y Gc(i)828 1458 y(E)888
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40213
1435 y F4(\()923 1423 y FX(h)951 1435 y Ga(a)999 1423
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40214
y FX(i)1026 1435 y F6(j)p Ga(N)10 b F6(j)1159 1402 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40215
Fu(N)1213 1435 y Ga(;)1253 1423 y F9(\()1281 1435 y Ga(y)1329
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40216
1423 y F9(\))1357 1435 y F6(j)p Ga(S)5 b F6(j)1468 1402
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40217
y Fu(N)1522 1435 y Fs(\()q Ga(\033)s Fs(\))1642 1393
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40218
y Fu(N)1696 1435 y F4(\))1606 b Gg(by)23 b(IH)549 1559
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40219
y F6(\021)54 b FL(And)828 1522 y Gc(i)828 1582 y(E)888
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40220
1559 y F4(\()923 1547 y FX(h)951 1559 y Ga(a)999 1547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40221
y FX(i)1026 1559 y FL(Id)p F4(\()p Ga(x;)15 b(a)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40222
Fs(\()r Ga(\033)s Fs(\))1430 1517 y Fu(N)1483 1559 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40223
Ga(;)1523 1547 y F9(\()1551 1559 y Ga(y)1599 1547 y F9(\))1627
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40224
1559 y F6(j)p Ga(S)5 b F6(j)1738 1526 y Fu(N)1792 1559
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40225
y Fs(\()q Ga(\033)s Fs(\))1912 1517 y Fu(N)1966 1559
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40226
y F4(\))549 1682 y(=)54 b FL(And)828 1646 y Gc(i)828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40227
1705 y(E)888 1682 y F4(\()923 1670 y FX(h)951 1682 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40228
Ga(a)999 1670 y FX(i)1026 1682 y FL(Id)p F4(\()p Ga(x;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40229
b(a)p F4(\))q Ga(;)1349 1670 y F9(\()1377 1682 y Ga(y)1425
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40230
1670 y F9(\))1453 1682 y F6(j)p Ga(S)5 b F6(j)1564 1649
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40231
y Fu(N)1618 1682 y F4(\))p Fs(\()q Ga(\033)s Fs(\))1773
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40232
1640 y Fu(N)549 1806 y F4(=)54 b F6(j)p FL(And)853 1769
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40233
y Gc(i)853 1829 y(L)906 1806 y F4(\()941 1794 y F9(\()968
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40234
1806 y Ga(y)1016 1794 y F9(\))1044 1806 y Ga(S)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40235
b(x)p F4(\))p F6(j)1257 1773 y Fu(N)1312 1806 y Fs(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40236
Ga(\033)s Fs(\))1432 1764 y Fu(N)3036 1806 y Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40237
b(\(6\))g(of)f F6(j)p 3404 1806 V 3422 1806 V 3440 1806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40238
V 65 w(j)3492 1773 y Fu(N)p Black 321 2037 a Gb(Case)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40239
Black 47 w Ga(M)35 b F6(\021)25 b FL(Cut)p F4(\()946
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40240
2025 y FX(h)974 2037 y Ga(b)1013 2025 y FX(i)1040 2037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40241
y Ga(S)5 b(;)1141 2025 y F9(\()1169 2037 y Ga(y)1217
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40242
2025 y F9(\))1244 2037 y FL(And)1399 2000 y Gc(i)1399
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40243
2060 y(L)1451 2037 y F4(\()1486 2025 y F9(\()1514 2037
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40244
y Ga(z)1560 2025 y F9(\))1587 2037 y Ga(T)13 b(;)i(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40245
F4(\))q(\))674 2205 y F6(j)p FL(Cut)p F4(\()872 2193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40246
y FX(h)900 2205 y Ga(b)939 2193 y FX(i)966 2205 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40247
b(;)1067 2193 y F9(\()1095 2205 y Ga(y)1143 2193 y F9(\))1170
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40248
2205 y FL(And)1325 2168 y Gc(i)1325 2228 y(L)1377 2205
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40249
y F4(\()1412 2193 y F9(\()1440 2205 y Ga(z)1486 2193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40250
y F9(\))1513 2205 y Ga(T)13 b(;)i(y)s F4(\))q(\)[)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40251
F4(])p F6(j)1868 2172 y Fu(N)549 2324 y F4(=)54 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40252
FL(Cut)p F4(\()872 2312 y FX(h)900 2324 y Ga(b)939 2312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40253
y FX(i)966 2324 y Ga(S)5 b F4([)p Ga(\033)s F4(])q Ga(;)1173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40254
2312 y F9(\()1201 2324 y Ga(y)1249 2312 y F9(\))1276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40255
2324 y FL(And)1430 2287 y Gc(i)1430 2347 y(L)1483 2324
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40256
y F4(\()1518 2312 y F9(\()1545 2324 y Ga(z)1591 2312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40257
y F9(\))1619 2324 y Ga(T)13 b F4([)p Ga(\033)s F4(])q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40258
Ga(;)i(y)s F4(\)\))p F6(j)1974 2291 y Fu(N)549 2442 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40259
F4(=)54 b FL(And)828 2405 y Gc(i)828 2465 y(E)888 2442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40260
y F4(\()923 2430 y FX(h)951 2442 y Ga(b)990 2430 y FX(i)1017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40261
2442 y F6(j)p Ga(S)5 b F4([)p Ga(\033)s F4(])p F6(j)1233
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40262
2409 y Fu(N)1288 2442 y Ga(;)1328 2430 y F9(\()1356 2442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40263
y Ga(z)1402 2430 y F9(\))1430 2442 y F6(j)p Ga(T)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40264
F4([)p Ga(\033)s F4(])p F6(j)1651 2409 y Fu(N)1706 2442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40265
y F4(\))1250 b Gg(by)24 b(\(10\))g(of)f F6(j)p 3404 2442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40266
V 3422 2442 V 3440 2442 V 65 w(j)3492 2409 y Fu(N)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40267
2566 y F6(\021)54 b FL(And)828 2529 y Gc(i)828 2589 y(E)888
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40268
2566 y F4(\()923 2554 y FX(h)951 2566 y Ga(b)990 2554
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40269
y FX(i)1017 2566 y F6(j)p Ga(S)5 b F6(j)1128 2533 y Fu(N)1182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40270
2566 y Fs(\()q Ga(\033)s Fs(\))1303 2524 y Fu(N)1357
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40271
2566 y Ga(;)1397 2554 y F9(\()1424 2566 y Ga(z)1470 2554
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40272
y F9(\))1498 2566 y F6(j)p Ga(T)13 b F6(j)1614 2533 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40273
Fu(N)1668 2566 y Fs(\()q Ga(\033)s Fs(\))1788 2524 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40274
Fu(N)1842 2566 y F4(\))1460 b Gg(by)23 b(IH)549 2689
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40275
y F4(=)54 b FL(And)828 2652 y Gc(i)828 2712 y(E)888 2689
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40276
y F4(\()923 2677 y FX(h)951 2689 y Ga(b)990 2677 y FX(i)1017
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40277
2689 y F6(j)p Ga(S)5 b F6(j)1128 2656 y Fu(N)1182 2689
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40278
y Ga(;)1222 2677 y F9(\()1250 2689 y Ga(z)1296 2677 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40279
F9(\))1324 2689 y F6(j)p Ga(T)13 b F6(j)1440 2656 y Fu(N)1494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40280
2689 y F4(\))p Fs(\()q Ga(\033)s Fs(\))1650 2647 y Fu(N)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40281
2813 y F4(=)54 b F6(j)p FL(Cut)p F4(\()872 2801 y FX(h)900
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40282
2813 y Ga(b)939 2801 y FX(i)966 2813 y Ga(S)5 b(;)1067
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40283
2801 y F9(\()1095 2813 y Ga(y)1143 2801 y F9(\))1170
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40284
2813 y FL(And)1325 2776 y Gc(i)1325 2836 y(L)1377 2813
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40285
y F4(\()1412 2801 y F9(\()1440 2813 y Ga(z)1486 2801
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40286
y F9(\))1513 2813 y Ga(T)13 b(;)i(y)s F4(\))q(\))p F6(j)1763
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40287
2780 y Fu(N)1817 2813 y Fs(\()q Ga(\033)s Fs(\))1937
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40288
2771 y Fu(N)2991 2813 y Gg(by)24 b(\(10\))g(of)f F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40289
3404 2813 V 3422 2813 V 3440 2813 V 65 w(j)3492 2780
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40290
y Fu(N)p 3480 2918 4 62 v 3484 2860 55 4 v 3484 2918
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40291
V 3538 2918 4 62 v Black 321 3131 a Gb(Pr)n(oof)32 b(of)g(Lemma)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40292
(3.4.7.)p Black 34 w Gg(There)i(are)f(23)h(cases)g(to)f(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40293
(considered,)k(three)c(of)g(which)f(are)h(gi)n(v)o(en)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40294
3244 y(belo)n(w)-6 b(.)76 b(W)-7 b(e)38 b(assume)i(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40295
F4([)p Ga(\033)s F4(])g Gg(and)h Fs(^)p Ga(\033)s Fs(_)1704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40296
3202 y Fu(N)1796 3244 y Gg(are)g(of)f(the)g(form)g F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40297
Ga(a)54 b F4(:=)2701 3232 y F9(\()2729 3244 y Ga(x)2781
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40298
3232 y F9(\))2808 3244 y FL(And)2963 3207 y Gc(i)2963
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40299
3267 y(L)3015 3244 y F4(\()3050 3232 y F9(\()3078 3244
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40300
y Ga(y)3126 3232 y F9(\))3153 3244 y Ga(P)13 b(;)i(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40301
F4(\))q(])38 b Gg(and)321 3356 y Fs(^)q FL(And)508 3320
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40302
y Gc(i)508 3380 y(E)568 3356 y F4(\()603 3344 y FX(h)631
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40303
3356 y Ga(a)679 3344 y FX(i)p 708 3356 28 4 v 726 3356
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40304
V 743 3356 V 771 3356 a Ga(;)811 3344 y F9(\()839 3356
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40305
y Ga(y)887 3344 y F9(\))914 3356 y F6(j)p Ga(P)13 b F6(j)1035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40306
3323 y Fu(N)1089 3356 y F4(\))p Fs(_)q Gg(,)23 b(respecti)n(v)o(ely)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40307
b(.)p Black 321 3593 a Gb(Case)p Black 47 w Ga(M)35 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40308
F6(\021)25 b FL(And)927 3607 y Gc(R)985 3593 y F4(\()1020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40309
3581 y FX(h)1048 3593 y Ga(d)1095 3581 y FX(i)1123 3593
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40310
y Ga(S)5 b(;)1224 3581 y FX(h)1252 3593 y Ga(e)1294 3581
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40311
y FX(i)1321 3593 y Ga(T)13 b(;)i(a)p F4(\))813 3765 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40312
F6(j)p FL(And)992 3779 y Gc(R)1050 3765 y F4(\()1085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40313
3753 y FX(h)1113 3765 y Ga(d)1160 3753 y FX(i)1188 3765
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40314
y Ga(S)5 b(;)1289 3753 y FX(h)1316 3765 y Ga(e)1358 3753
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40315
y FX(i)1386 3765 y Ga(T)13 b(;)i(a)p F4(\))p F6(j)1600
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40316
3732 y Fu(N)1655 3765 y Fs(^)q Ga(\033)s Fs(_)1775 3723
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40317
y Fu(N)648 3890 y F4(=)94 b FL(And)967 3904 y Gc(I)1007
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40318
3890 y F4(\()1042 3878 y FX(h)1070 3890 y Ga(d)1117 3878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40319
y FX(i)1145 3890 y F6(j)p Ga(S)5 b F6(j)1256 3857 y Fu(N)1310
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40320
3890 y Ga(;)1350 3878 y FX(h)1378 3890 y Ga(e)1420 3878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40321
y FX(i)1448 3890 y F6(j)p Ga(T)13 b F6(j)1564 3857 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40322
Fu(N)1618 3890 y Ga(;)i(a)p F4(\))p Fs(^)q Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40323
Fs(_)1862 3848 y Fu(N)3036 3890 y Gg(by)24 b(\(2\))g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40324
F6(j)p 3404 3890 V 3422 3890 V 3440 3890 V 65 w(j)3492
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40325
3857 y Fu(N)648 4016 y F4(=)94 b FL(And)967 3979 y Gc(i)967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40326
4039 y(E)1027 4016 y F4(\()1062 4004 y FX(h)1090 4016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40327
y Ga(a)1138 4004 y FX(i)1165 4016 y FL(And)1320 4030
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40328
y Gc(I)1360 4016 y F4(\()1395 4004 y FX(h)1422 4016 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40329
Ga(d)1469 4004 y FX(i)1497 4016 y F6(j)p Ga(S)5 b F6(j)1608
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40330
3983 y Fu(N)1663 4016 y Fs(^)p Ga(\033)s Fs(_)1783 3974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40331
y Fu(N)1837 4016 y Ga(;)1877 4004 y FX(h)1904 4016 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40332
Ga(e)1946 4004 y FX(i)1974 4016 y F6(j)p Ga(T)13 b F6(j)2090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40333
3983 y Fu(N)2144 4016 y Fs(^)q Ga(\033)s Fs(_)2265 3974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40334
y Fu(N)2319 4016 y Ga(;)i(a)p F4(\))p Ga(;)2482 4004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40335
y F9(\()2510 4016 y Ga(y)2558 4004 y F9(\))2585 4016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40336
y F6(j)p Ga(P)e F6(j)2706 3983 y Fu(N)2761 4016 y F4(\))280
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40337
b Gg(by)24 b(\(2\))f(of)h Fs(^)p 3451 4016 V 3469 4016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40338
V 3487 4016 V 65 w(_)613 4103 y Gc(\024)549 4140 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40339
b(\000)h(!)731 4107 y FX(\003)813 4140 y FL(And)967 4103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40340
y Gc(i)967 4163 y(E)1027 4140 y F4(\()1062 4128 y FX(h)1090
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40341
4140 y Ga(a)1138 4128 y FX(i)1165 4140 y FL(And)1320
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40342
4154 y Gc(I)1360 4140 y F4(\()1395 4128 y FX(h)1422 4140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40343
y Ga(d)1469 4128 y FX(i)1497 4140 y F6(j)p Ga(S)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40344
Ga(\033)s F4(])p F6(j)1713 4107 y Fu(N)1768 4140 y Ga(;)1808
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40345
4128 y FX(h)1836 4140 y Ga(e)1878 4128 y FX(i)1906 4140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40346
y F6(j)p Ga(T)13 b F4([)p Ga(\033)s F4(])p F6(j)2127
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40347
4107 y Fu(N)2182 4140 y Ga(;)i(a)p F4(\))q Ga(;)2346
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40348
4128 y F9(\()2373 4140 y Ga(y)2421 4128 y F9(\))2449
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40349
4140 y F6(j)p Ga(P)e F6(j)2570 4107 y Fu(N)2624 4140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40350
y F4(\))678 b Gg(by)23 b(IH)648 4264 y F4(=)94 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40351
FL(Cut)p F4(\()1011 4252 y FX(h)1039 4264 y Ga(a)1087
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40352
4252 y FX(i)1114 4264 y FL(And)1269 4278 y Gc(R)1326
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40353
4264 y F4(\()1361 4252 y FX(h)1389 4264 y Ga(d)1436 4252
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40354
y FX(i)1464 4264 y Ga(S)5 b F4([)p Ga(\033)s F4(])q Ga(;)1671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40355
4252 y FX(h)1698 4264 y Ga(e)1740 4252 y FX(i)1768 4264
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40356
y Ga(T)13 b F4([)p Ga(\033)s F4(])q Ga(;)i(a)p F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40357
Ga(;)2103 4252 y F9(\()2131 4264 y Ga(x)2183 4252 y F9(\))2211
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40358
4264 y FL(And)2365 4227 y Gc(i)2365 4287 y(L)2417 4264
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40359
y F4(\()2452 4252 y F9(\()2480 4264 y Ga(y)2528 4252
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40360
y F9(\))2555 4264 y Ga(P)e(;)i(x)p F4(\))q(\))p F6(j)2814
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40361
4231 y Fu(N)2923 4264 y Gg(by)23 b(\(2,10\))i(of)e F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40362
3404 4264 V 3422 4264 V 3440 4264 V 65 w(j)3492 4231
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40363
y Fu(N)648 4388 y F4(=)94 b F6(j)p FL(And)992 4402 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40364
Gc(R)1050 4388 y F4(\()1085 4376 y FX(h)1113 4388 y Ga(d)1160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40365
4376 y FX(i)1188 4388 y Ga(S)5 b(;)1289 4376 y FX(h)1316
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40366
4388 y Ga(e)1358 4376 y FX(i)1386 4388 y Ga(T)13 b(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40367
F4(\)[)p Ga(\033)s F4(])p F6(j)1705 4355 y Fu(N)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40368
321 4621 a Gb(Case)p Black 47 w Ga(M)35 b F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40369
b FL(Cut)p F4(\()946 4609 y FX(h)974 4621 y Ga(b)1013
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40370
4609 y FX(i)1040 4621 y Ga(S)5 b(;)1141 4609 y F9(\()1169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40371
4621 y Ga(z)1215 4609 y F9(\))1243 4621 y FL(And)1397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40372
4584 y Gc(i)1397 4644 y(L)1449 4621 y F4(\()1484 4609
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40373
y F9(\()1512 4621 y Ga(u)1564 4609 y F9(\))1592 4621
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40374
y Ga(T)12 b(;)j(z)t F4(\))q(\))813 4794 y F6(j)p FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40375
F4(\()1011 4782 y FX(h)1039 4794 y Ga(b)1078 4782 y FX(i)1105
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40376
4794 y Ga(S)5 b(;)1206 4782 y F9(\()1234 4794 y Ga(z)1280
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40377
4782 y F9(\))1308 4794 y FL(And)1462 4757 y Gc(i)1462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40378
4817 y(L)1514 4794 y F4(\()1549 4782 y F9(\()1577 4794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40379
y Ga(u)1629 4782 y F9(\))1657 4794 y Ga(T)12 b(;)j(z)t
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40380
F4(\))q(\))p F6(j)1904 4761 y Fu(N)1959 4794 y Fs(^)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40381
Ga(\033)s Fs(_)2079 4752 y Fu(N)648 4919 y F4(=)94 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40382
FL(And)967 4882 y Gc(i)967 4942 y(E)1027 4919 y F4(\()1062
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40383
4907 y FX(h)1090 4919 y Ga(b)1129 4907 y FX(i)1156 4919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40384
y F6(j)p Ga(S)5 b F6(j)1267 4886 y Fu(N)1321 4919 y Ga(;)1361
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40385
4907 y F9(\()1389 4919 y Ga(u)1441 4907 y F9(\))1469
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40386
4919 y F6(j)p Ga(T)13 b F6(j)1585 4886 y Fu(N)1639 4919
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40387
y F4(\))p Fs(^)q Ga(\033)s Fs(_)1794 4877 y Fu(N)2991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40388
4919 y Gg(by)24 b(\(10\))g(of)f F6(j)p 3404 4919 V 3422
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40389
4919 V 3440 4919 V 65 w(j)3492 4886 y Fu(N)648 5045 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40390
F4(=)94 b FL(And)967 5008 y Gc(i)967 5068 y(E)1027 5045
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40391
y F4(\()1062 5033 y FX(h)1090 5045 y Ga(b)1129 5033 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40392
FX(i)1156 5045 y F6(j)p Ga(S)5 b F6(j)1267 5012 y Fu(N)1321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40393
5045 y Fs(^)q Ga(\033)s Fs(_)1442 5003 y Fu(N)1496 5045
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40394
y Ga(;)1536 5033 y F9(\()1563 5045 y Ga(u)1615 5033 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40395
F9(\))1643 5045 y F6(j)p Ga(T)13 b F6(j)1759 5012 y Fu(N)1813
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40396
5045 y Fs(^)q Ga(\033)s Fs(_)1933 5003 y Fu(N)1987 5045
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40397
y F4(\))613 5132 y Gc(\024)549 5169 y F6(\000)-32 b(\000)h(!)731
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40398
5136 y FX(\003)813 5169 y FL(And)967 5132 y Gc(i)967
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40399
5192 y(E)1027 5169 y F4(\()1062 5157 y FX(h)1090 5169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40400
y Ga(b)1129 5157 y FX(i)1156 5169 y F6(j)p Ga(S)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40401
Ga(\033)s F4(])p F6(j)1372 5136 y Fu(N)1427 5169 y Ga(;)1467
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40402
5157 y F9(\()1495 5169 y Ga(u)1547 5157 y F9(\))1574
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40403
5169 y F6(j)p Ga(T)13 b F4([)p Ga(\033)s F4(])p F6(j)1795
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40404
5136 y Fu(N)1850 5169 y F4(\))1452 b Gg(by)23 b(IH)648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40405
5293 y F4(=)94 b F6(j)p FL(Cut)p F4(\()1011 5281 y FX(h)1039
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40406
5293 y Ga(b)1078 5281 y FX(i)1105 5293 y Ga(S)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40407
Ga(\033)s F4(])q Ga(;)1312 5281 y F9(\()1340 5293 y Ga(z)1386
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40408
5281 y F9(\))1413 5293 y FL(And)1568 5256 y Gc(i)1568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40409
5316 y(L)1620 5293 y F4(\()1655 5281 y F9(\()1683 5293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40410
y Ga(u)1735 5281 y F9(\))1762 5293 y Ga(T)13 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40411
Ga(\033)s F4(])q Ga(;)i(z)t F4(\))q(\))p F6(j)2116 5260
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40412
y Fu(N)2991 5293 y Gg(by)24 b(\(10\))g(of)f F6(j)p 3404
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40413
5293 V 3422 5293 V 3440 5293 V 65 w(j)3492 5260 y Fu(N)648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40414
5417 y F4(=)94 b F6(j)p FL(Cut)p F4(\()1011 5405 y FX(h)1039
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40415
5417 y Ga(b)1078 5405 y FX(i)1105 5417 y Ga(S)5 b(;)1206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40416
5405 y F9(\()1234 5417 y Ga(z)1280 5405 y F9(\))1308
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40417
5417 y FL(And)1462 5380 y Gc(i)1462 5440 y(L)1514 5417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40418
y F4(\()1549 5405 y F9(\()1577 5417 y Ga(u)1629 5405
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40419
y F9(\))1657 5417 y Ga(T)12 b(;)j(z)t F4(\))q(\)[)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40420
F4(])p F6(j)2009 5384 y Fu(N)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40421
%%Page: 165 177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40422
TeXDict begin 165 176 bop Black 277 51 a Gb(B.2)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40423
(Chapter)f(3)2639 b(165)p 277 88 3691 4 v Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40424
277 412 a(Case)p Black 46 w Ga(M)36 b F6(\021)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40425
F4(\()902 400 y FX(h)929 412 y Ga(b)968 400 y FX(i)996
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40426
412 y Ga(S)5 b(;)1097 400 y F9(\()1125 412 y Ga(z)1171
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40427
400 y F9(\))1198 412 y FL(Ax)p F4(\()p Ga(z)t(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40428
F4(\))q(\))768 596 y F6(j)p FL(Cut)q F4(\()967 584 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40429
FX(h)994 596 y Ga(b)1033 584 y FX(i)1061 596 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40430
b(;)1162 584 y F9(\()1190 596 y Ga(z)1236 584 y F9(\))1263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40431
596 y FL(Ax)p F4(\()p Ga(z)t(;)15 b(a)p F4(\))q(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40432
F6(j)1631 563 y Fu(N)1686 596 y Fs(^)p Ga(\033)s Fs(_)1806
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40433
554 y Fu(N)604 722 y F4(=)93 b FL(Subst)p F4(\()1016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40434
710 y FX(h)1043 722 y Ga(b)1082 710 y FX(i)1110 722 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40435
F6(j)p Ga(S)5 b F6(j)1221 689 y Fu(N)1275 722 y Ga(;)1315
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40436
710 y F9(\()1343 722 y Ga(z)1389 710 y F9(\))1417 722
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40437
y FL(Id)p F4(\()p Ga(z)t(;)15 b(a)p F4(\))q(\))p Fs(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40438
Ga(\033)s Fs(_)1850 680 y Fu(N)2879 722 y Gg(by)23 b(\(1,14\))i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40439
F6(j)p 3360 722 28 4 v 3378 722 V 3396 722 V 65 w(j)3448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40440
689 y Fu(N)604 847 y F4(=)93 b FL(Subst)p F4(\()1016
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40441
835 y FX(h)1043 847 y Ga(b)1082 835 y FX(i)1110 847 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40442
F6(j)p Ga(S)5 b F6(j)1221 814 y Fu(N)1275 847 y Fs(^)q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40443
Ga(\033)s Fs(_)1395 805 y Fu(N)1449 847 y Ga(;)1489 835
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40444
y F9(\()1517 847 y Ga(z)1563 835 y F9(\))1591 847 y FL(And)1745
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40445
810 y Gc(i)1745 870 y(E)1805 847 y F4(\()1840 835 y FX(h)1868
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40446
847 y Ga(a)1916 835 y FX(i)1943 847 y FL(Id)q F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40447
Ga(z)t(;)15 b(a)p F4(\))q Ga(;)2261 835 y F9(\()2289
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40448
847 y Ga(y)2337 835 y F9(\))2364 847 y F6(j)p Ga(P)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40449
F6(j)2485 814 y Fu(N)2540 847 y F4(\)\))422 b Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40450
b(\(1\))f(of)h Fs(^)p 3407 847 V 3425 847 V 3442 847
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40451
V 65 w(_)569 934 y Gc(\024)504 971 y F6(\000)-31 b(\000)g(!)686
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40452
938 y FX(\003)768 971 y FL(Subst)p F4(\()1016 959 y FX(h)1043
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40453
971 y Ga(b)1082 959 y FX(i)1110 971 y F6(j)p Ga(S)5 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40454
F4([)p Ga(\033)s F4(])p F6(j)1326 938 y Fu(N)1381 971
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40455
y Ga(;)1421 959 y F9(\()1449 971 y Ga(z)1495 959 y F9(\))1523
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40456
971 y FL(And)1677 934 y Gc(i)1677 994 y(E)1737 971 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40457
F4(\()1772 959 y FX(h)1799 971 y Ga(a)1847 959 y FX(i)1875
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40458
971 y FL(Id)p F4(\()p Ga(z)t(;)15 b(a)p F4(\))r Ga(;)2193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40459
959 y F9(\()2220 971 y Ga(y)2268 959 y F9(\))2296 971
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40460
y F6(j)p Ga(P)e F6(j)2417 938 y Fu(N)2471 971 y F4(\))q(\))751
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40461
b Gg(by)23 b(IH)569 1058 y Gc(\024)504 1095 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40462
b(\000)g(!)93 b FL(And)923 1058 y Gc(i)923 1118 y(E)983
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40463
1095 y F4(\()1018 1083 y FX(h)1045 1095 y Ga(b)1084 1083
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40464
y FX(i)1112 1095 y F6(j)p Ga(S)5 b F4([)p Ga(\033)s F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40465
F6(j)1328 1062 y Fu(N)1383 1095 y Ga(;)1423 1083 y F9(\()1451
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40466
1095 y Ga(y)1499 1083 y F9(\))1526 1095 y F6(j)p Ga(P)13
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40467
b F6(j)1647 1062 y Fu(N)1702 1095 y F4(\))839 b Ga(z)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40468
b F6(62)24 b Ga(F)13 b(N)d F4(\()2922 1083 y F9(\()2950
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40469
1095 y Ga(y)2998 1083 y F9(\))3026 1095 y F6(j)p Ga(P)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40470
F6(j)3147 1062 y Fu(N)3201 1095 y F4(\))23 b Gg(by)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40471
Fs(^)p 3407 1095 V 3425 1095 V 3442 1095 V 65 w(_)604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40472
1220 y F4(=)93 b F6(j)p FL(Cut)q F4(\()967 1208 y FX(h)994
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40473
1220 y Ga(b)1033 1208 y FX(i)1061 1220 y Ga(S)5 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40474
Ga(\033)s F4(])q Ga(;)1268 1208 y F9(\()1295 1220 y Ga(x)1347
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40475
1208 y F9(\))1375 1220 y FL(And)1529 1183 y Gc(i)1529
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40476
1243 y(L)1581 1220 y F4(\()1616 1208 y F9(\()1644 1220
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40477
y Ga(y)1692 1208 y F9(\))1720 1220 y Ga(P)13 b(;)i(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40478
F4(\)\))p F6(j)1978 1187 y Fu(N)2947 1220 y Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40479
b(\(10\))g(of)f F6(j)p 3360 1220 V 3378 1220 V 3396 1220
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40480
V 65 w(j)3448 1187 y Fu(N)604 1344 y F4(=)93 b F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40481
FL(Cut)q F4(\()967 1332 y FX(h)994 1344 y Ga(b)1033 1332
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40482
y FX(i)1061 1344 y Ga(S)5 b(;)1162 1332 y F9(\()1190
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40483
1344 y Ga(z)1236 1332 y F9(\))1263 1344 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40484
Ga(z)t(;)15 b(a)p F4(\))q(\)[)p Ga(\033)s F4(])p F6(j)1736
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40485
1311 y Fu(N)p 3436 1451 4 62 v 3440 1393 55 4 v 3440
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40486
1451 V 3494 1451 4 62 v Black 277 1663 a Gb(Pr)n(oof)20
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40487
b(of)e(Lemma)g(3.4.8.)p Black 34 w Gg(Let)g F4([)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40488
F4(])h Gg(and)g Fs(\()q Ga(\033)s Fs(\))1690 1621 y Fu(N)1762
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40489
1663 y Gg(be)g F4([)p Ga(a)26 b F4(:=)2086 1651 y F9(\()2114
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40490
1663 y Ga(x)2166 1651 y F9(\))2193 1663 y Ga(N)10 b F4(])18
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40491
b Gg(and)h Fs(\()-6 b Ga(a)25 b F4(:=)2688 1651 y F9(\()2716
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40492
1663 y Ga(x)2768 1651 y F9(\))2795 1663 y F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40493
b F6(j)2928 1630 y Fu(N)2976 1663 y Fs(\))p Gg(,)19 b(respecti)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40494
(ely)-6 b(.)277 1776 y(There)24 b(are)g(20)f(cases)i(to)e(be)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40495
(considered,)i(tw)o(o)d(of)h(which)g(are)f(gi)n(v)o(en)h(belo)n(w)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40496
b(.)p Black 277 2075 a Gb(Case)p Black 46 w Ga(M)36 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40497
F6(\021)25 b FL(And)883 2089 y Gc(R)941 2075 y F4(\()976
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40498
2063 y FX(h)1004 2075 y Ga(c)1043 2063 y FX(i)1071 2075
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40499
y Ga(S)5 b(;)1172 2063 y FX(h)1199 2075 y Ga(d)1246 2063
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40500
y FX(i)1274 2075 y Ga(T)13 b(;)i(a)p F4(\))535 2259 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40501
F6(j)p FL(And)715 2273 y Gc(R)772 2259 y F4(\()807 2247
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40502
y FX(h)835 2259 y Ga(d)882 2247 y FX(i)910 2259 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40503
b(;)1011 2247 y FX(h)1039 2259 y Ga(e)1081 2247 y FX(i)1109
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40504
2259 y Ga(T)12 b(;)j(a)p F4(\)[)p Ga(\033)s F4(])p F6(j)1427
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40505
2226 y Fu(N)410 2383 y F4(=)54 b F6(j)p FL(Cut)p F4(\()733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40506
2371 y FX(h)761 2383 y Ga(a)809 2371 y FX(i)836 2383
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40507
y FL(And)991 2397 y Gc(R)1049 2383 y F4(\()1084 2371
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40508
y FX(h)1111 2383 y Ga(d)1158 2371 y FX(i)1186 2383 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40509
Ga(S)5 b F4([)p Ga(\033)s F4(])q Ga(;)1393 2371 y FX(h)1421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40510
2383 y Ga(e)1463 2371 y FX(i)1491 2383 y Ga(T)13 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40511
Ga(\033)s F4(])p Ga(;)i(a)p F4(\))q Ga(;)1826 2371 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40512
F9(\()1854 2383 y Ga(x)1906 2371 y F9(\))1933 2383 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40513
Ga(N)10 b F4(\))p F6(j)2076 2350 y Fu(N)2177 2383 y Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40514
b(ass.)h(\(10\22613\))h(of)f F6(j)p 2883 2383 28 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40515
2901 2383 V 2918 2383 V 65 w(j)2971 2350 y Fu(N)3048
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40516
2383 y Gg(do)f(not)h(apply)410 2507 y F4(=)54 b FL(Subst)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40517
F4(\()782 2495 y FX(h)810 2507 y Ga(a)858 2495 y FX(i)885
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40518
2507 y FL(And)1040 2521 y Gc(I)1080 2507 y F4(\()1115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40519
2495 y FX(h)1143 2507 y Ga(d)1190 2495 y FX(i)1217 2507
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40520
y F6(j)p Ga(S)5 b F4([)p Ga(\033)s F4(])p F6(j)1433 2474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40521
y Fu(N)1489 2507 y Ga(;)1529 2495 y FX(h)1556 2507 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40522
Ga(e)1598 2495 y FX(i)1626 2507 y F6(j)p Ga(T)13 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40523
Ga(\033)s F4(])p F6(j)1847 2474 y Fu(N)1902 2507 y Ga(;)i(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40524
F4(\))q Ga(;)2066 2495 y F9(\()2094 2507 y Ga(x)2146
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40525
2495 y F9(\))2173 2507 y F6(j)p Ga(N)10 b F6(j)2306 2474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40526
y Fu(N)2361 2507 y F4(\))483 b Gg(by)23 b(\(2,14\))i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40527
F6(j)p 3360 2507 V 3378 2507 V 3396 2507 V 65 w(j)3448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40528
2474 y Fu(N)410 2632 y F6(\021)54 b FL(Subst)o F4(\()782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40529
2620 y FX(h)810 2632 y Ga(a)858 2620 y FX(i)885 2632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40530
y FL(And)1040 2646 y Gc(I)1080 2632 y F4(\()1115 2620
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40531
y FX(h)1143 2632 y Ga(d)1190 2620 y FX(i)1217 2632 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40532
F6(j)p Ga(S)5 b F6(j)1328 2599 y Fu(N)1383 2632 y Fs(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40533
Ga(\033)s Fs(\))1503 2591 y Fu(N)1557 2632 y Ga(;)1597
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40534
2620 y FX(h)1625 2632 y Ga(e)1667 2620 y FX(i)1695 2632
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40535
y F6(j)p Ga(T)13 b F6(j)1811 2599 y Fu(N)1865 2632 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40536
Fs(\()p Ga(\033)s Fs(\))1985 2591 y Fu(N)2039 2632 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40537
Ga(;)i(a)p F4(\))q Ga(;)2203 2620 y F9(\()2231 2632 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40538
Ga(x)2283 2620 y F9(\))2310 2632 y F6(j)p Ga(N)10 b F6(j)2443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40539
2599 y Fu(N)2497 2632 y F4(\))761 b Gg(by)23 b(IH)410
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40540
2758 y F4(=)54 b FL(And)690 2772 y Gc(I)729 2758 y F4(\()764
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40541
2746 y FX(h)792 2758 y Ga(d)839 2746 y FX(i)867 2758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40542
y F6(j)p Ga(S)5 b F6(j)978 2725 y Fu(N)1032 2758 y Ga(;)1072
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40543
2746 y FX(h)1100 2758 y Ga(e)1142 2746 y FX(i)1170 2758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40544
y F6(j)p Ga(T)13 b F6(j)1286 2725 y Fu(N)1340 2758 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40545
Ga(;)i(a)p F4(\))p Fs(\()r Ga(\033)s Fs(\))1584 2716
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40546
y Fu(N)3032 2758 y Gg(by)24 b(\(3\))f(of)h Fs(\()p 3407
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40547
2758 V 3425 2758 V 3442 2758 V 65 w(\))410 2883 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40548
b F6(j)p FL(And)715 2897 y Gc(R)772 2883 y F4(\()807
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40549
2871 y FX(h)835 2883 y Ga(d)882 2871 y FX(i)910 2883
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40550
y Ga(S)5 b(;)1011 2871 y FX(h)1039 2883 y Ga(e)1081 2871
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40551
y FX(i)1109 2883 y Ga(T)12 b(;)j(a)p F4(\))p F6(j)1322
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40552
2850 y Fu(N)1377 2883 y Fs(\()q Ga(\033)s Fs(\))1498
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40553
2841 y Fu(N)2992 2883 y Gg(by)24 b(\(2\))g(of)f F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40554
3360 2883 V 3378 2883 V 3396 2883 V 65 w(j)3448 2850
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40555
y Fu(N)p Black 277 3128 a Gb(Case)p Black 46 w Ga(M)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40556
b F6(\021)25 b FL(Cut)p F4(\()902 3116 y FX(h)929 3128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40557
y Ga(b)968 3116 y FX(i)996 3128 y Ga(S)5 b(;)1097 3116
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40558
y F9(\()1125 3128 y Ga(y)1173 3116 y F9(\))1200 3128
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40559
y FL(Ax)o F4(\()p Ga(y)s(;)15 b(a)p F4(\))q(\))535 3311
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40560
y F6(j)p FL(Cut)p F4(\()733 3299 y FX(h)761 3311 y Ga(b)800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40561
3299 y FX(i)827 3311 y Ga(S)5 b(;)928 3299 y F9(\()956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40562
3311 y Ga(y)1004 3299 y F9(\))1031 3311 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40563
Ga(y)s(;)15 b(a)p F4(\))q(\)[)p Ga(\033)s F4(])p F6(j)1506
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40564
3278 y Fu(N)410 3435 y F4(=)54 b F6(j)p FL(Cut)p F4(\()733
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40565
3423 y FX(h)761 3435 y Ga(b)800 3423 y FX(i)827 3435
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40566
y Ga(S)5 b F4([)p Ga(\033)s F4(])q Ga(;)1034 3423 y F9(\()1062
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40567
3435 y Ga(y)1110 3423 y F9(\))1137 3435 y Ga(N)10 b F4([)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40568
Ga(x)g F6(7!)g Ga(y)s F4(])q(\))p F6(j)1542 3402 y Fu(N)1942
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40569
3435 y Gg(by)23 b(assumtion)j(\(10\22613\))f(of)f F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40570
2883 3435 V 2901 3435 V 2918 3435 V 65 w(j)2971 3402
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40571
y Fu(N)3048 3435 y Gg(do)f(not)h(apply)410 3559 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40572
b FL(Subst)o F4(\()782 3547 y FX(h)810 3559 y Ga(b)849
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40573
3547 y FX(i)876 3559 y F6(j)p Ga(S)5 b F4([)p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40574
F4(])p F6(j)1092 3526 y Fu(N)1148 3559 y Ga(;)1188 3547
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40575
y F9(\()1215 3559 y Ga(y)1263 3547 y F9(\))1291 3559
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40576
y F6(j)p Ga(N)10 b F6(j)1424 3526 y Fu(N)1478 3559 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40577
F4([)p Ga(x)g F6(7!)g Ga(y)s F4(]\))1173 b Gg(by)24 b(\(14\))g(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40578
F6(j)p 3360 3559 V 3378 3559 V 3396 3559 V 65 w(j)3448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40579
3526 y Fu(N)410 3685 y F6(\021)54 b FL(Subst)o F4(\()782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40580
3673 y FX(h)810 3685 y Ga(b)849 3673 y FX(i)876 3685
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40581
y F6(j)p Ga(S)5 b F6(j)987 3652 y Fu(N)1042 3685 y Fs(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40582
Ga(\033)s Fs(\))1162 3643 y Fu(N)1216 3685 y Ga(;)1256
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40583
3673 y F9(\()1284 3685 y Ga(y)1332 3673 y F9(\))1359
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40584
3685 y F6(j)p Ga(N)10 b F6(j)1492 3652 y Fu(N)1546 3685
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40585
y F4([)p Ga(x)g F6(7!)g Ga(y)s F4(])q(\))1450 b Gg(by)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40586
b(IH)410 3810 y F4(=)54 b FL(Subst)o F4(\()782 3798 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40587
FX(h)810 3810 y Ga(b)849 3798 y FX(i)876 3810 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40588
Ga(S)5 b F6(j)987 3777 y Fu(N)1042 3810 y Fs(\()p Ga(\033)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40589
Fs(\))1162 3768 y Fu(N)1216 3810 y Ga(;)1256 3798 y F9(\()1284
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40590
3810 y Ga(y)1332 3798 y F9(\))1359 3810 y FL(Id)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40591
Ga(y)s(;)15 b(a)p F4(\))p Fs(\()r Ga(\033)s Fs(\))1758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40592
3768 y Fu(N)1812 3810 y F4(\))410 3935 y(=)54 b FL(Subst)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40593
F4(\()782 3923 y FX(h)810 3935 y Ga(b)849 3923 y FX(i)876
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40594
3935 y F6(j)p Ga(S)5 b F6(j)987 3902 y Fu(N)1042 3935
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40595
y Ga(;)1082 3923 y F9(\()1110 3935 y Ga(y)1158 3923 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40596
F9(\))1185 3935 y FL(Id)p F4(\()p Ga(y)s(;)15 b(a)p F4(\))q(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40597
Fs(\()q Ga(\033)s Fs(\))1620 3893 y Fu(N)410 4061 y F4(=)54
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40598
b F6(j)p FL(Cut)p F4(\()733 4049 y FX(h)761 4061 y Ga(b)800
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40599
4049 y FX(i)827 4061 y Ga(S)5 b(;)928 4049 y F9(\()956
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40600
4061 y Ga(y)1004 4049 y F9(\))1031 4061 y FL(Ax)p F4(\()p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40601
Ga(y)s(;)15 b(a)p F4(\))q(\))p F6(j)1401 4028 y Fu(N)1456
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40602
4061 y Fs(\()p Ga(\033)s Fs(\))1576 4019 y Fu(N)2879
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40603
4061 y Gg(by)23 b(\(1,14\))i(of)e F6(j)p 3360 4061 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40604
3378 4061 V 3396 4061 V 65 w(j)3448 4028 y Fu(N)p 3436
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40605
4233 4 62 v 3440 4175 55 4 v 3440 4233 V 3494 4233 4
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40606
62 v Black 277 4445 a Gb(Pr)n(oof)h(of)g(Theor)n(em)f(3.4.9.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40607
Black 34 w Gg(W)-7 b(e)23 b(\002rst)g(gi)n(v)o(e)h(the)f(calculations)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40608
(for)d(some)g(base)g(cases.)p Black 277 4706 a Gb(Logical)h(Cut)d(with)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40609
g(Axiom:)p Black 504 4861 a Ga(M)36 b F6(\021)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40610
F4(\()897 4849 y FX(h)924 4861 y Ga(a)972 4849 y FX(i)1000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40611
4861 y Ga(S)5 b(;)1101 4849 y F9(\()1129 4861 y Ga(y)1177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40612
4849 y F9(\))1204 4861 y FL(Ax)p F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40613
b(b)p F4(\)\))1596 4824 y Gc(cut)1565 4861 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40614
b(\000)f(!)26 b Ga(S)5 b F4([)p Ga(a)10 b F6(7!)g Ga(b)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40615
F4(])26 b F6(\021)e Ga(N)727 5045 y F6(j)p FL(Cut)p F4(\()925
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40616
5033 y FX(h)953 5045 y Ga(a)1001 5033 y FX(i)1028 5045
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40617
y Ga(M)10 b(;)1166 5033 y F9(\()1194 5045 y Ga(y)1242
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40618
5033 y F9(\))1269 5045 y FL(Ax)p F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40619
b(b)p F4(\))q(\))p F6(j)1630 5012 y Fu(N)604 5169 y F4(=)52
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40620
b FL(Subst)o F4(\()974 5157 y FX(h)1002 5169 y Ga(a)1050
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40621
5157 y FX(i)1077 5169 y F6(j)p Ga(S)5 b F6(j)1188 5136
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40622
y Fu(N)1242 5169 y Ga(;)1282 5157 y F9(\()1310 5169 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40623
Ga(y)1358 5157 y F9(\))1386 5169 y FL(Id)p F4(\()p Ga(y)s(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40624
b(b)p F4(\))q(\))1188 b Gg(by)23 b(\(1,14\))i(of)e F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40625
3360 5169 28 4 v 3378 5169 V 3396 5169 V 65 w(j)3448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40626
5136 y Fu(N)569 5256 y Gc(\024)504 5293 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40627
b(\000)g(!)52 b(j)p Ga(S)5 b F6(j)838 5260 y Fu(N)892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40628
5293 y F4([)p Ga(a)10 b F6(7!)g Ga(b)p F4(])604 5417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40629
y F6(\021)52 b(j)p Ga(S)5 b F4([)p Ga(a)10 b F6(7!)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40630
Ga(b)p F4(])p F6(j)1086 5384 y Fu(N)p Black Black eop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40631
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40632
%%Page: 166 178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40633
TeXDict begin 166 177 bop Black -144 51 a Gb(166)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40634
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40635
4 v Black 549 412 a Ga(M)41 b F6(\021)31 b FL(Cut)o F4(\()952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40636
400 y FX(h)980 412 y Ga(a)1028 400 y FX(i)1056 412 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40637
FL(Ax)o F4(\()p Ga(y)s(;)15 b(a)p F4(\))q Ga(;)1405 400
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40638
y F9(\()1433 412 y Ga(x)1485 400 y F9(\))1513 412 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40639
b F4(\))1671 375 y Gc(cut)1640 412 y F6(\000)-31 b(\000)f(!)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40640
b Ga(S)5 b F4([)p Ga(x)16 b F6(7!)g Ga(y)s F4(])32 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40641
F6(\021)e Ga(N)36 b Gg(and)27 b(one)g(of)g(the)g(clauses)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40642
(\(10\22613\))549 525 y(of)23 b F6(j)p 674 525 28 4 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40643
692 525 V 710 525 V 65 w(j)762 492 y Fu(N)839 525 y Gg(applies)i(to)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40644
Ga(M)10 b Gg(,)23 b(for)h(e)o(xample)g(\(10\):)772 678
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40645
y F6(j)p FL(Cut)p F4(\()970 666 y FX(h)998 678 y Ga(a)1046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40646
666 y FX(i)1074 678 y FL(Ax)o F4(\()p Ga(y)s(;)15 b(a)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40647
F4(\))q Ga(;)1423 666 y F9(\()1451 678 y Ga(x)1503 666
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40648
y F9(\))1530 678 y Ga(S)5 b F4(\))p F6(j)1651 645 y Fu(N)650
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40649
803 y F4(=)51 b FL(And)927 766 y Gc(i)927 826 y(E)986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40650
803 y F4(\()1021 791 y FX(h)1049 803 y Ga(a)1097 791
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40651
y FX(i)1125 803 y FL(Id)p F4(\()p Ga(y)s(;)15 b(a)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40652
Ga(;)1444 791 y F9(\()1472 803 y Ga(x)1524 791 y F9(\))1551
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40653
803 y F6(j)p Ga(S)5 b F6(j)1662 770 y Fu(N)1717 803 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40654
F4(\))1171 b Gg(by)23 b(\(1,10\))i(of)e F6(j)p 3404 803
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40655
V 3422 803 V 3440 803 V 65 w(j)3492 770 y Fu(N)650 927
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40656
y F6(\021)51 b(j)p Ga(N)10 b F6(j)905 894 y Fu(N)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40657
321 1112 a Gb(Logical)25 b(Cut)d Fb(^)871 1126 y Fy(R)936
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40658
1112 y FO(=)p Fb(^)1058 1126 y Fy(L)1111 1134 y Fa(1)1154
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40659
1112 y Gb(:)p Black 549 1267 a Ga(M)35 b F6(\021)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40660
FL(Cut)p F4(\()941 1255 y FX(h)969 1267 y Ga(c)1008 1255
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40661
y FX(i)1035 1267 y FL(And)1190 1281 y Gc(R)1248 1267
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40662
y F4(\()1283 1255 y FX(h)1310 1267 y Ga(a)1358 1255 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40663
FX(i)1386 1267 y Ga(S)5 b(;)1487 1255 y FX(h)1515 1267
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40664
y Ga(b)1554 1255 y FX(i)1581 1267 y Ga(T)13 b(;)i(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40665
F4(\))q Ga(;)1802 1255 y F9(\()1830 1267 y Ga(y)1878
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40666
1255 y F9(\))1905 1267 y FL(And)2060 1231 y F9(1)2060
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40667
1291 y Gc(L)2112 1267 y F4(\()2147 1255 y F9(\()2175
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40668
1267 y Ga(x)2227 1255 y F9(\))2254 1267 y Ga(P)e(;)i(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40669
F4(\))q(\))2540 1230 y Gc(cut)2509 1267 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40670
b(\000)g(!)25 b FL(Cut)p F4(\()2878 1255 y FX(h)2905
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40671
1267 y Ga(a)2953 1255 y FX(i)2981 1267 y Ga(S)5 b(;)3082
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40672
1255 y F9(\()3110 1267 y Ga(x)3162 1255 y F9(\))3189
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40673
1267 y Ga(P)13 b F4(\))26 b F6(\021)f Ga(N)771 1421 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40674
F6(j)p FL(Cut)p F4(\()969 1409 y FX(h)997 1421 y Ga(c)1036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40675
1409 y FX(i)1064 1421 y FL(And)1218 1435 y Gc(R)1276
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40676
1421 y F4(\()1311 1409 y FX(h)1339 1421 y Ga(a)1387 1409
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40677
y FX(i)1414 1421 y Ga(S)5 b(;)1515 1409 y FX(h)1543 1421
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40678
y Ga(b)1582 1409 y FX(i)1609 1421 y Ga(T)13 b(;)i(c)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40679
F4(\))q Ga(;)1830 1409 y F9(\()1858 1421 y Ga(y)1906
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40680
1409 y F9(\))1933 1421 y FL(And)2088 1385 y F9(1)2088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40681
1445 y Gc(L)2140 1421 y F4(\()2175 1409 y F9(\()2203
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40682
1421 y Ga(x)2255 1409 y F9(\))2282 1421 y Ga(P)e(;)i(y)s
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40683
F4(\))q(\))p F6(j)2537 1388 y Fu(N)648 1546 y F4(=)52
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40684
b FL(And)925 1509 y Gc(i)925 1569 y(E)985 1546 y F4(\()1020
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40685
1534 y FX(h)1048 1546 y Ga(c)1087 1534 y FX(i)1115 1546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40686
y FL(And)1269 1560 y Gc(I)1309 1546 y F4(\()1344 1534
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40687
y FX(h)1372 1546 y Ga(a)1420 1534 y FX(i)1447 1546 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40688
F6(j)p Ga(S)5 b F6(j)1558 1513 y Fu(N)1613 1546 y Ga(;)1653
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40689
1534 y FX(h)1681 1546 y Ga(b)1720 1534 y FX(i)1747 1546
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40690
y F6(j)p Ga(T)13 b F6(j)1863 1513 y Fu(N)1917 1546 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40691
Ga(;)i(c)p F4(\))q Ga(;)2072 1534 y F9(\()2100 1546 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40692
Ga(x)2152 1534 y F9(\))2180 1546 y F6(j)p Ga(P)e F6(j)2301
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40693
1513 y Fu(N)2355 1546 y F4(\))533 b Gg(by)23 b(\(2,10\))i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40694
F6(j)p 3404 1546 V 3422 1546 V 3440 1546 V 65 w(j)3492
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40695
1513 y Fu(N)613 1633 y Gc(\024)549 1670 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40696
b(\000)h(!)52 b FL(Subst)o F4(\()1018 1658 y FX(h)1046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40697
1670 y Ga(a)1094 1658 y FX(i)1121 1670 y F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40698
b F6(j)1232 1637 y Fu(N)1287 1670 y Ga(;)1327 1658 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40699
F9(\()1355 1670 y Ga(x)1407 1658 y F9(\))1434 1670 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40700
F6(j)p Ga(P)13 b F6(j)1555 1637 y Fu(N)1609 1670 y F4(\))786
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40701
b Ga(c)23 b Gg(is)h(freshly)h(introduced)h(by)3407 1633
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40702
y Gc(cut)3376 1670 y F6(\000)-31 b(\000)f(!)549 1813
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40703
y Gg(No)n(w)28 b(we)h(ha)n(v)o(e)i(tw)o(o)f(subcases)i(depending)h(on)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40704
(whether)h Ga(P)42 b Gg(freshly)32 b(introduces)h Ga(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40705
Gg(,)d(b)n(ut)h(is)549 1926 y(not)23 b(an)h(axiom.)29
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40706
b(In)24 b(this)g(case)g(we)f(reason)i(as)e(follo)n(ws.)580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40707
2181 y FL(Subst)o F4(\()827 2169 y FX(h)855 2181 y Ga(a)903
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40708
2169 y FX(i)930 2181 y F6(j)p Ga(S)5 b F6(j)1041 2148
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40709
y Fu(N)1096 2181 y Ga(;)1136 2169 y F9(\()1164 2181 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40710
Ga(x)1216 2169 y F9(\))1243 2181 y F6(j)p Ga(P)13 b F6(j)1364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40711
2148 y Fu(N)1418 2181 y F4(\))1544 2144 y Gc(\024)1479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40712
2181 y F6(\000)-31 b(\000)f(!)26 b(j)p Ga(P)13 b F6(j)1796
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40713
2143 y Fu(N)1850 2181 y Fs(\()-7 b Ga(x)26 b F4(:=)2074
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40714
2169 y FX(h)2101 2181 y Ga(a)2149 2169 y FX(i)2177 2181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40715
y F6(j)p Ga(S)5 b F6(j)2288 2148 y Fu(N)2335 2181 y Fs(\))2393
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40716
2112 y F9(\(3)p Gc(:)p F9(4)p Gc(:)p F9(6\))2457 2181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40717
y F6(\021)90 b(j)p Ga(P)13 b F4([)p Ga(x)26 b F4(:=)2938
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40718
2169 y FX(h)2965 2181 y Ga(a)3013 2169 y FX(i)3041 2181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40719
y Ga(S)5 b F4(])p F6(j)3152 2143 y Fu(N)3232 2181 y F6(\021)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40720
b(j)p Ga(N)10 b F6(j)3461 2143 y Fu(N)549 2411 y Gg(Otherwise)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40721
b(by)g(\(14\))g(of)f F6(j)p 1358 2411 V 1376 2411 V 1394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40722
2411 V 65 w(j)1446 2378 y Fu(N)1500 2411 y Gg(:)1048
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40723
2663 y FL(Subst)o F4(\()1295 2651 y FX(h)1323 2663 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40724
Ga(a)1371 2651 y FX(i)1398 2663 y F6(j)p Ga(S)5 b F6(j)1509
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40725
2630 y Fu(N)1564 2663 y Ga(;)1604 2651 y F9(\()1632 2663
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40726
y Ga(x)1684 2651 y F9(\))1711 2663 y F6(j)p Ga(P)13 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40727
F6(j)1832 2630 y Fu(N)1887 2663 y F4(\))25 b(=)g F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40728
FL(Cut)p F4(\()2241 2651 y FX(h)2269 2663 y Ga(a)2317
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40729
2651 y FX(i)2345 2663 y Ga(S)5 b(;)2446 2651 y F9(\()2473
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40730
2663 y Ga(x)2525 2651 y F9(\))2553 2663 y Ga(P)13 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40731
F6(j)2684 2626 y Fu(N)2764 2663 y F6(\021)25 b(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40732
b F6(j)2993 2626 y Fu(N)p Black 321 2893 a Gb(Logical)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40733
b(Cut)d Fb(\033)882 2907 y Fy(R)948 2893 y FO(=)p Fb(\033)1081
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40734
2907 y Fy(L)1139 2893 y Gb(:)p Black 598 3046 a Ga(M)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40735
b F6(\021)f FL(Cut)p F4(\()1001 3034 y FX(h)1029 3046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40736
y Ga(b)1068 3034 y FX(i)1095 3046 y FL(Imp)1239 3068
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40737
y Gc(R)1297 3046 y F4(\()1332 3034 y F9(\()1360 3046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40738
y Ga(x)1412 3034 y F9(\))p FX(h)1467 3046 y Ga(a)1515
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40739
3034 y FX(i)1542 3046 y Ga(S)5 b(;)15 b(b)p F4(\))q Ga(;)1758
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40740
3034 y F9(\()1786 3046 y Ga(z)1832 3034 y F9(\))1860
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40741
3046 y FL(Imp)2004 3068 y Gc(L)2056 3046 y F4(\()2091
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40742
3034 y FX(h)2119 3046 y Ga(c)2158 3034 y FX(i)2186 3046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40743
y Ga(T)e(;)2292 3034 y F9(\()2319 3046 y Ga(y)2367 3034
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40744
y F9(\))2395 3046 y Ga(P)g(;)i(z)t F4(\))q(\))859 3122
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40745
y Gc(cut)828 3159 y F6(\000)-31 b(\000)f(!)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40746
F4(\()1196 3147 y FX(h)1224 3159 y Ga(c)1263 3147 y FX(i)1291
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40747
3159 y Ga(T)13 b(;)1397 3147 y F9(\()1424 3159 y Ga(x)1476
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40748
3147 y F9(\))1504 3159 y FL(Cut)p F4(\()1677 3147 y FX(h)1705
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40749
3159 y Ga(a)1753 3147 y FX(i)1780 3159 y Ga(S)5 b(;)1881
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40750
3147 y F9(\()1909 3159 y Ga(y)1957 3147 y F9(\))1984
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40751
3159 y Ga(P)13 b F4(\)\))26 b F6(\021)f Ga(N)771 3324
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40752
y F6(j)p FL(Cut)p F4(\()969 3312 y FX(h)997 3324 y Ga(b)1036
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40753
3312 y FX(i)1063 3324 y FL(Imp)1208 3346 y Gc(R)1265
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40754
3324 y F4(\()1300 3312 y F9(\()1328 3324 y Ga(x)1380
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40755
3312 y F9(\))q FX(h)1435 3324 y Ga(a)1483 3312 y FX(i)1510
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40756
3324 y Ga(S)5 b(;)15 b(b)p F4(\))q Ga(;)1726 3312 y F9(\()1754
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40757
3324 y Ga(z)1800 3312 y F9(\))1828 3324 y FL(Imp)1972
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40758
3346 y Gc(L)2024 3324 y F4(\()2059 3312 y FX(h)2087 3324
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40759
y Ga(c)2126 3312 y FX(i)2154 3324 y Ga(T)e(;)2260 3312
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40760
y F9(\()2288 3324 y Ga(y)2336 3312 y F9(\))2363 3324
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40761
y Ga(P)g(;)i(z)t F4(\))q(\))p F6(j)2616 3291 y Fu(N)648
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40762
3448 y F4(=)52 b FL(Imp)915 3470 y Gc(E)975 3448 y F4(\()1010
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40763
3436 y FX(h)1038 3448 y Ga(b)1077 3436 y FX(i)1104 3448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40764
y FL(Imp)1249 3470 y Gc(I)1288 3448 y F4(\()1323 3436
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40765
y F9(\()1351 3448 y Ga(x)1403 3436 y F9(\))q FX(h)1458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40766
3448 y Ga(a)1506 3436 y FX(i)1534 3448 y F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40767
b F6(j)1645 3415 y Fu(N)1699 3448 y Ga(;)15 b(b)p F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40768
Ga(;)1854 3436 y FX(h)1882 3448 y Ga(c)1921 3436 y FX(i)1948
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40769
3448 y F6(j)p Ga(T)e F6(j)2064 3415 y Fu(N)2119 3448
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40770
y Ga(;)2159 3436 y F9(\()2187 3448 y Ga(y)2235 3436 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40771
F9(\))2262 3448 y F6(j)p Ga(P)g F6(j)2383 3415 y Fu(N)2437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40772
3448 y F4(\))451 b Gg(by)23 b(\(4,12\))i(of)e F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40773
3404 3448 V 3422 3448 V 3440 3448 V 65 w(j)3492 3415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40774
y Fu(N)613 3535 y Gc(\024)549 3572 y F6(\000)-32 b(\000)h(!)52
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40775
b FL(Subst)o F4(\()1018 3560 y FX(h)1046 3572 y Ga(c)1085
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40776
3560 y FX(i)1113 3572 y F6(j)p Ga(N)10 b F6(j)1246 3539
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40777
y Fu(N)1300 3572 y Ga(;)1340 3560 y F9(\()1368 3572 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40778
Ga(x)1420 3560 y F9(\))1447 3572 y FL(Subst)o F4(\()1694
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40779
3560 y FX(h)1722 3572 y Ga(a)1770 3560 y FX(i)1798 3572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40780
y F6(j)p Ga(M)g F6(j)1946 3539 y Fu(N)2000 3572 y Ga(;)2040
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40781
3560 y F9(\()2068 3572 y Ga(y)2116 3560 y F9(\))2143
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40782
3572 y F6(j)p Ga(P)j F6(j)2264 3539 y Fu(N)2319 3572
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40783
y F4(\)\))42 b Ga(b)22 b Gg(is)i(freshly)h(introduced)h(by)3407
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40784
3535 y Gc(cut)3376 3572 y F6(\000)-31 b(\000)f(!)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40785
3715 y Gg(Again)33 b(we)g(ha)n(v)o(e)h(tw)o(o)f(subcases)j(depending)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40786
(on)e(whether)g Ga(P)46 b Gg(freshly)35 b(introduces)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40787
Ga(x)p Gg(,)f(b)n(ut)549 3828 y(is)f(not)h(an)g(axiom.)62
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40788
b(In)35 b(this)g(case,)j(we)33 b(reason)j(analogous)i(as)c(in)h(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40789
(case)g(gi)n(v)o(en)g(abo)o(v)o(e.)549 3941 y(Otherwise)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40790
b(by)g(\(14\))g(of)f F6(j)p 1358 3941 V 1376 3941 V 1394
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40791
3941 V 65 w(j)1446 3908 y Fu(N)1500 3941 y Gg(:)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40792
Black 1323 4154 a FL(Subst)o F4(\()1570 4142 y FX(h)1598
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40793
4154 y Ga(c)1637 4142 y FX(i)1665 4154 y F6(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40794
b F6(j)1798 4121 y Fu(N)1852 4154 y Ga(;)1892 4142 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40795
F9(\()1920 4154 y Ga(x)1972 4142 y F9(\))2000 4154 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40796
FL(Subst)o F4(\()2247 4142 y FX(h)2275 4154 y Ga(a)2323
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40797
4142 y FX(i)2350 4154 y F6(j)p Ga(M)g F6(j)2498 4121
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40798
y Fu(N)2553 4154 y Ga(;)2593 4142 y F9(\()2621 4154 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40799
Ga(y)2669 4142 y F9(\))2696 4154 y F6(j)p Ga(P)j F6(j)2817
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40800
4121 y Fu(N)2871 4154 y F4(\))q(\))1153 4267 y(=)99 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40801
F6(j)p FL(Cut)p F4(\()1521 4255 y FX(h)1549 4267 y Ga(c)1588
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40802
4255 y FX(i)1616 4267 y Ga(N)10 b(;)1739 4255 y F9(\()1767
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40803
4267 y Ga(x)1819 4255 y F9(\))1846 4267 y FL(Cut)p F4(\()2019
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40804
4255 y FX(h)2047 4267 y Ga(a)2095 4255 y FX(i)2122 4267
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40805
y Ga(M)g(;)2260 4255 y F9(\()2288 4267 y Ga(y)2336 4255
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40806
y F9(\))2364 4267 y Ga(P)j F4(\)\))p F6(j)2530 4234 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40807
Fu(N)2610 4267 y F6(\021)25 b(j)p Ga(N)10 b F6(j)2839
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40808
4234 y Fu(N)p Black 321 4474 a Gb(Commuting)23 b(Cut:)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40809
Black 549 4629 a Ga(M)35 b F6(\021)25 b FL(Cut)p F4(\()941
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40810
4617 y FX(h)969 4629 y Ga(a)1017 4617 y FX(i)1044 4629
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40811
y Ga(S)5 b(;)1145 4617 y F9(\()1173 4629 y Ga(x)1225
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40812
4617 y F9(\))1252 4629 y Ga(T)13 b F4(\))1409 4592 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40813
Gc(cut)1379 4629 y F6(\000)-32 b(\000)h(!)25 b Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40814
b F4([)p Ga(a)26 b F4(:=)1855 4617 y F9(\()1882 4629
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40815
y Ga(x)1934 4617 y F9(\))1962 4629 y Ga(T)13 b F4(])25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40816
b F6(\021)g Ga(N)549 4778 y Gg(If)33 b(one)g(of)h(the)f(clauses)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40817
(\(10\22613\))h(applies)f(to)e Ga(M)10 b Gg(,)35 b(then)f(we)e(ha)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40818
(e)i(for)g(e)o(xample)g(for)f Ga(T)57 b F6(\021)549 4890
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40819
y FL(And)703 4854 y Gc(i)703 4914 y(L)755 4890 y F4(\()790
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40820
4878 y F9(\()818 4890 y Ga(y)866 4878 y F9(\))893 4890
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40821
y Ga(P)13 b(;)i(x)p F4(\))810 5045 y F6(j)p FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40822
F4(\()1008 5033 y FX(h)1036 5045 y Ga(a)1084 5033 y FX(i)1112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40823
5045 y Ga(S)5 b(;)1213 5033 y F9(\()1240 5045 y Ga(x)1292
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40824
5033 y F9(\))1320 5045 y FL(And)1474 5008 y Gc(i)1474
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40825
5068 y(L)1527 5045 y F4(\()1562 5033 y F9(\()1589 5045
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40826
y Ga(y)1637 5033 y F9(\))1665 5045 y Ga(P)13 b(;)i(x)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40827
F4(\)\))p F6(j)1923 5012 y Fu(N)648 5169 y F4(=)91 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40828
FL(And)965 5132 y Gc(i)965 5192 y(E)1024 5169 y F4(\()1059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40829
5157 y FX(h)1087 5169 y Ga(a)1135 5157 y FX(i)1163 5169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40830
y F6(j)p Ga(S)5 b F6(j)1274 5136 y Fu(N)1328 5169 y Ga(;)1368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40831
5157 y F9(\()1396 5169 y Ga(y)1444 5157 y F9(\))1471
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40832
5169 y F6(j)p Ga(P)13 b F6(j)1592 5136 y Fu(N)1647 5169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40833
y F4(\))1309 b Gg(by)24 b(\(10\))g(of)f F6(j)p 3404 5169
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40834
V 3422 5169 V 3440 5169 V 65 w(j)3492 5136 y Fu(N)613
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40835
5256 y Gc(\024)549 5293 y F6(\000)-32 b(\000)h(!)91 b(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40836
Ga(S)5 b F6(j)921 5260 y Fu(N)976 5293 y Fs(^)p FL(And)1163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40837
5256 y Gc(i)1163 5316 y(E)1222 5293 y F4(\()1257 5281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40838
y FX(h)1285 5293 y Ga(a)1333 5281 y FX(i)p 1363 5293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40839
V 1380 5293 V 1398 5293 V 1425 5293 a Ga(;)1465 5281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40840
y F9(\()1493 5293 y Ga(y)1541 5281 y F9(\))1568 5293
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40841
y F6(j)p Ga(P)13 b F6(j)1689 5260 y Fu(N)1744 5293 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40842
F4(\))p Fs(_)613 5380 y Gc(\024)549 5417 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40843
b(\000)h(!)728 5384 y FX(\003)810 5417 y F6(j)p Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40844
b F4([)p Ga(a)26 b F4(:=)1116 5405 y F9(\()1144 5417
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40845
y Ga(x)1196 5405 y F9(\))1223 5417 y Ga(T)13 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40846
F6(j)1339 5384 y Fu(N)1419 5417 y F6(\021)25 b(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40847
b F6(j)1648 5384 y Fu(N)2950 5417 y Gg(by)24 b(Lemma)f(3.4.7)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40848
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40849
%%Page: 167 179
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40850
TeXDict begin 167 178 bop Black 277 51 a Gb(B.3)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40851
(Chapter)f(4)2639 b(167)p 277 88 3691 4 v Black 504 412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40852
a Gg(Otherwise)25 b(we)d(ha)n(v)o(e)727 545 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40853
FL(Cut)p F4(\()925 533 y FX(h)953 545 y Ga(a)1001 533
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40854
y FX(i)1028 545 y Ga(S)5 b(;)1129 533 y F9(\()1157 545
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40855
y Ga(x)1209 533 y F9(\))1236 545 y Ga(T)13 b F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40856
F6(j)1362 512 y Fu(N)604 670 y F4(=)52 b FL(Subst)o F4(\()974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40857
658 y FX(h)1002 670 y Ga(a)1050 658 y FX(i)1077 670 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40858
F6(j)p Ga(S)5 b F6(j)1188 637 y Fu(N)1242 670 y Ga(;)1282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40859
658 y F9(\()1310 670 y Ga(x)1362 658 y F9(\))1390 670
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40860
y F6(j)p Ga(T)13 b F6(j)1506 637 y Fu(N)1560 670 y F4(\))1352
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40861
b Gg(by)24 b(\(14\))g(of)f F6(j)p 3360 670 28 4 v 3378
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40862
670 V 3396 670 V 65 w(j)3448 637 y Fu(N)569 757 y Gc(\024)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40863
794 y F6(\000)-31 b(\000)g(!)52 b(j)p Ga(S)5 b F6(j)838
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40864
761 y Fu(N)892 794 y Fs(\()-7 b Ga(a)26 b F4(:=)1112
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40865
782 y F9(\()1139 794 y Ga(x)1191 782 y F9(\))1219 794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40866
y F6(j)p Ga(T)13 b F6(j)1335 761 y Fu(N)1382 794 y Fs(\))604
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40867
918 y F6(\021)52 b(j)p Ga(S)5 b F4([)p Ga(a)25 b F4(:=)1033
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40868
906 y F9(\()1060 918 y Ga(x)1112 906 y F9(\))1139 918
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40869
y Ga(T)13 b F4(])p F6(j)1255 885 y Fu(N)2906 918 y Gg(by)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40870
b(Lemma)e(3.4.8)504 1080 y Ga(M)36 b F6(\021)25 b FL(Cut)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40871
F4(\()897 1068 y FX(h)924 1080 y Ga(a)972 1068 y FX(i)1000
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40872
1080 y Ga(S)5 b(;)1101 1068 y F9(\()1129 1080 y Ga(x)1181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40873
1068 y F9(\))1208 1080 y Ga(T)13 b F4(\))1365 1042 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40874
Gc(cut)1334 1079 y F6(\000)-31 b(\000)g(!)1530 1080 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40875
Ga(T)13 b F4([)p Ga(x)25 b F4(:=)1819 1068 y FX(h)1847
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40876
1080 y Ga(a)1895 1068 y FX(i)1922 1080 y Ga(S)5 b F4(])26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40877
b F6(\021)f Ga(N)727 1246 y F6(j)p FL(Cut)p F4(\()925
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40878
1234 y FX(h)953 1246 y Ga(a)1001 1234 y FX(i)1028 1246
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40879
y Ga(S)5 b(;)1129 1234 y F9(\()1157 1246 y Ga(x)1209
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40880
1234 y F9(\))1236 1246 y Ga(T)13 b F4(\))p F6(j)1362
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40881
1213 y Fu(N)604 1370 y F4(=)52 b FL(Subst)o F4(\()974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40882
1358 y FX(h)1002 1370 y Ga(a)1050 1358 y FX(i)1077 1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40883
y F6(j)p Ga(S)5 b F6(j)1188 1337 y Fu(N)1242 1370 y Ga(;)1282
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40884
1358 y F9(\()1310 1370 y Ga(x)1362 1358 y F9(\))1390
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40885
1370 y F6(j)p Ga(T)13 b F6(j)1506 1337 y Fu(N)1560 1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40886
y F4(\))1352 b Gg(by)24 b(\(14\))g(of)f F6(j)p 3360 1370
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40887
V 3378 1370 V 3396 1370 V 65 w(j)3448 1337 y Fu(N)569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40888
1457 y Gc(\024)504 1494 y F6(\000)-31 b(\000)g(!)52 b(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40889
Ga(T)13 b F6(j)843 1461 y Fu(N)897 1494 y Fs(\()-7 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40890
Ga(x)25 b F4(:=)1121 1482 y FX(h)1148 1494 y Ga(a)1196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40891
1482 y FX(i)1224 1494 y F6(j)p Ga(S)5 b F6(j)1335 1461
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40892
y Fu(N)1382 1494 y Fs(\))604 1618 y F6(\021)52 b(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40893
Ga(T)13 b F4([)p Ga(x)25 b F4(:=)1041 1606 y FX(h)1069
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40894
1618 y Ga(a)1117 1606 y FX(i)1144 1618 y Ga(S)5 b F4(])p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40895
F6(j)1255 1585 y Fu(N)1335 1618 y F6(\021)25 b(j)p Ga(N)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40896
b F6(j)1564 1585 y Fu(N)2906 1618 y Gg(by)24 b(Lemma)e(3.4.6)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40897
Black 277 1800 a Gb(Inner)h(Reduction:)p Black 46 w Gg(The)d(inner)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40898
(reductions)i(are)d(by)g(simple)g(induction)j(ar)n(guments)f(e)o(xcept)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40899
f(where)504 1913 y Ga(M)38 b Gg(is)28 b(translated)j(by)e(clause)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40900
(\(14\))g(of)g F6(j)p 1782 1913 V 1800 1913 V 1817 1913
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40901
V 65 w(j)1870 1880 y Fu(N)1951 1913 y Gg(and)g Ga(N)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40902
b Gg(by)28 b(one)h(of)f(the)h(clauses)h(\(10\22613\).)45
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40903
b(W)-7 b(e)504 2026 y(gi)n(v)o(e)24 b(the)g(details)h(for)f(one)g(such)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40904
g(case.)29 b(Suppose)c Ga(y)h Gg(is)d(free)h(in)2496
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40905
2014 y F9(\()2523 2026 y Ga(x)2575 2014 y F9(\))2602
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40906
2026 y Ga(T)13 b Gg(,)23 b(b)n(ut)h(not)g(in)3083 2014
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40907
y F9(\()3111 2026 y Ga(x)3163 2014 y F9(\))3190 2026
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40908
y Ga(T)3256 1993 y FX(0)3279 2026 y Gg(.)504 2176 y Ga(M)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40909
b F6(\021)25 b FL(Cut)p F4(\()897 2164 y FX(h)924 2176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40910
y Ga(c)963 2164 y FX(i)991 2176 y Ga(S)5 b(;)1092 2164
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40911
y F9(\()1120 2176 y Ga(y)1168 2164 y F9(\))1195 2176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40912
y FL(And)1350 2139 y Gc(i)1350 2199 y(L)1402 2176 y F4(\()1437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40913
2164 y F9(\()1465 2176 y Ga(x)1517 2164 y F9(\))1544
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40914
2176 y Ga(T)13 b(;)i(y)s F4(\))q(\))1825 2139 y Gc(cut)1794
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40915
2176 y F6(\000)-31 b(\000)f(!)26 b FL(Cut)p F4(\()2163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40916
2164 y FX(h)2190 2176 y Ga(c)2229 2164 y FX(i)2257 2176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40917
y Ga(S)5 b(;)2358 2164 y F9(\()2386 2176 y Ga(y)2434
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40918
2164 y F9(\))2461 2176 y FL(And)2616 2139 y Gc(i)2616
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40919
2199 y(L)2668 2176 y F4(\()2703 2164 y F9(\()2731 2176
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40920
y Ga(x)2783 2164 y F9(\))2810 2176 y Ga(T)2876 2143 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40921
FX(0)2899 2176 y Ga(;)15 b(y)s F4(\))q(\))25 b F6(\021)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40922
Ga(N)766 2331 y F6(j)p FL(Cut)p F4(\()964 2319 y FX(h)992
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40923
2331 y Ga(c)1031 2319 y FX(i)1059 2331 y Ga(S)5 b(;)1160
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40924
2319 y F9(\()1188 2331 y Ga(y)1236 2319 y F9(\))1263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40925
2331 y FL(And)1417 2294 y Gc(i)1417 2354 y(L)1470 2331
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40926
y F4(\()1505 2319 y F9(\()1532 2331 y Ga(x)1584 2319
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40927
y F9(\))1612 2331 y Ga(T)13 b(;)i(y)s F4(\)\))p F6(j)1861
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40928
2298 y Fu(N)604 2455 y F4(=)91 b FL(Subst)o F4(\()1013
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40929
2443 y FX(h)1041 2455 y Ga(c)1080 2443 y FX(i)1108 2455
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40930
y F6(j)p Ga(S)5 b F6(j)1219 2422 y Fu(N)1273 2455 y Ga(;)1313
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40931
2443 y F9(\()1341 2455 y Ga(y)1389 2443 y F9(\))1416
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40932
2455 y FL(And)1571 2418 y Gc(i)1571 2478 y(E)1630 2455
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40933
y F4(\()1665 2443 y FX(h)1693 2455 y Ga(a)1741 2443 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40934
FX(i)1769 2455 y FL(Id)p F4(\()p Ga(a;)15 b(y)s F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40935
Ga(;)2088 2443 y F9(\()2116 2455 y Ga(x)2168 2443 y F9(\))2195
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40936
2455 y F6(j)p Ga(T)e F6(j)2311 2422 y Fu(N)2365 2455
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40937
y F4(\))q(\))443 b Gg(by)23 b(\(6,14\))i(of)e F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40938
3360 2455 V 3378 2455 V 3396 2455 V 65 w(j)3448 2422
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40939
y Fu(N)569 2542 y Gc(\024)504 2580 y F6(\000)-31 b(\000)g(!)684
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40940
2547 y FX(\003)766 2580 y FL(Subst)o F4(\()1013 2568
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40941
y FX(h)1041 2580 y Ga(c)1080 2568 y FX(i)1108 2580 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40942
F6(j)p Ga(S)5 b F6(j)1219 2547 y Fu(N)1273 2580 y Ga(;)1313
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40943
2568 y F9(\()1341 2580 y Ga(y)1389 2568 y F9(\))1416
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40944
2580 y FL(And)1571 2543 y Gc(i)1571 2603 y(E)1630 2580
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40945
y F4(\()1665 2568 y FX(h)1693 2580 y Ga(a)1741 2568 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40946
FX(i)1769 2580 y FL(Id)p F4(\()p Ga(a;)15 b(y)s F4(\))q
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40947
Ga(;)2088 2568 y F9(\()2116 2580 y Ga(x)2168 2568 y F9(\))2195
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40948
2580 y F6(j)p Ga(T)2286 2547 y FX(0)2310 2580 y F6(j)2335
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40949
2547 y Fu(N)2389 2580 y F4(\)\))834 b Gg(by)23 b(IH)569
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40950
2667 y Gc(\024)504 2704 y F6(\000)-31 b(\000)g(!)91 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40951
FL(And)921 2667 y Gc(i)921 2727 y(E)980 2704 y F4(\()1015
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40952
2692 y FX(h)1043 2704 y Ga(a)1091 2692 y FX(i)1118 2704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40953
y FL(Id)q F4(\()p Ga(a;)15 b(y)s F4(\))q Ga(;)1438 2692
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40954
y F9(\()1466 2704 y Ga(x)1518 2692 y F9(\))1545 2704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40955
y F6(j)p Ga(T)1636 2671 y FX(0)1659 2704 y F6(j)1684
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40956
2671 y Fu(N)1738 2704 y F4(\))p Fs(\()-6 b Ga(y)28 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40957
F4(:=)1993 2692 y FX(h)2021 2704 y Ga(c)2060 2692 y FX(i)2088
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40958
2704 y F6(j)p Ga(S)5 b F6(j)2199 2671 y Fu(N)2246 2704
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40959
y Fs(\))604 2828 y F4(=)91 b FL(And)921 2791 y Gc(i)921
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40960
2851 y(E)980 2828 y F4(\()1015 2816 y FX(h)1043 2828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40961
y Ga(c)1082 2816 y FX(i)1110 2828 y F6(j)p Ga(S)5 b F6(j)1221
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40962
2795 y Fu(N)1275 2828 y Ga(;)1315 2816 y F9(\()1343 2828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40963
y Ga(x)1395 2816 y F9(\))1422 2828 y F6(j)p Ga(T)1513
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40964
2795 y FX(0)1537 2828 y F6(j)1562 2795 y Fu(N)1616 2828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40965
y F4(\))721 b Gg(by)24 b(assumption)i Ga(y)i F6(62)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40966
Ga(F)13 b(N)d F4(\()3271 2816 y F9(\()3298 2828 y Ga(x)3350
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40967
2816 y F9(\))3378 2828 y Ga(T)3444 2795 y FX(0)3467 2828
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40968
y F4(\))604 2952 y(=)91 b F6(j)p FL(Cut)p F4(\()964 2940
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40969
y FX(h)992 2952 y Ga(c)1031 2940 y FX(i)1059 2952 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40970
b(;)1160 2940 y F9(\()1188 2952 y Ga(y)1236 2940 y F9(\))1263
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40971
2952 y FL(And)1417 2915 y Gc(i)1417 2975 y(L)1470 2952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40972
y F4(\()1505 2940 y F9(\()1532 2952 y Ga(x)1584 2940
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40973
y F9(\))1612 2952 y Ga(T)1678 2919 y FX(0)1701 2952 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40974
Ga(;)15 b(y)s F4(\)\))p F6(j)1884 2919 y Fu(N)1964 2952
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40975
y F6(\021)25 b(j)p Ga(N)10 b F6(j)2193 2919 y Fu(N)2947
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40976
2952 y Gg(by)24 b(\(10\))g(of)f F6(j)p 3360 2952 V 3378
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40977
2952 V 3396 2952 V 65 w(j)3448 2919 y Fu(N)p 3436 3059
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40978
4 62 v 3440 3001 55 4 v 3440 3059 V 3494 3059 4 62 v
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40979
277 3368 a Ge(B.3)119 b(Pr)n(oofs)29 b(of)g(Chapter)i(4)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40980
3591 y Gg(In)i(this)h(section)h(we)e(pro)o(v)o(e)h(that)g(a)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40981
(leftmost-outermost)38 b(reduction)e(strate)o(gy)f(of)3004
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40982
3554 y Gc(aux)2984 3591 y F6(\000)-31 b(\000)f(!)33 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40983
Gg(does)h(not)277 3704 y(restrict)25 b(the)f(collection)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40984
(cut-free)h(proofs)g(reachable)h(from)e(a)f(gi)n(v)o(en)h(proof.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40985
Black 277 3842 a Gb(Pr)n(oof)i(of)e(Lemma)g(4.1.2.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40986
Black 34 w Gg(By)g(induction)k(on)d(the)g(structure)i(of)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40987
Ga(M)2454 3856 y F9(1)2494 3842 y Gg(.)32 b(W)-7 b(e)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40988
b(analyse)i(four)g(represen-)277 3955 y(tati)n(v)o(e)e(cases.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40989
Black 414 4157 a F6(\017)p Black 45 w Gg(Suppose)h(we)e(ha)n(v)o(e)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40990
(the)g(term)f(with)h(the)f(reduction)k(sequence)649 4415
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40991
y FL(And)804 4429 y Gc(R)861 4415 y F4(\()896 4403 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40992
FX(h)924 4415 y Ga(a)972 4403 y FX(i)1000 4415 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40993
b(;)1138 4403 y FX(h)1166 4415 y Ga(b)1205 4403 y FX(i)1232
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40994
4415 y Ga(N)g(;)15 b(c)p F4(\))1483 4378 y Gc(bad)1455
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40995
4415 y F6(\000)-31 b(\000)g(!)25 b FL(And)1805 4429 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40996
Gc(R)1863 4415 y F4(\()1898 4403 y FX(h)1926 4415 y Ga(a)1974
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40997
4403 y FX(i)2001 4415 y Ga(M)11 b(;)2140 4403 y FX(h)2167
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40998
4415 y Ga(b)2206 4403 y FX(i)2234 4415 y Ga(S)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 40999
b(c)p F4(\))p 2455 4340 119 3 v 2455 4378 a Gc(aux)2435
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41000
4415 y F6(\000)-31 b(\000)g(!)25 b FL(And)2785 4429 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41001
Gc(R)2843 4415 y F4(\()2878 4403 y FX(h)2906 4415 y Ga(a)2954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41002
4403 y FX(i)2981 4415 y Ga(T)13 b(;)3087 4403 y FX(h)3115
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41003
4415 y Ga(b)3154 4403 y FX(i)3181 4415 y Ga(S)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41004
b(c)p F4(\))504 4654 y Gg(where)31 b(a)e(reduction)j(is)e(performed)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41005
(in)e(the)g(right)h(subterm,)h(although)g(the)e(left)h(subterm)g(is)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41006
4766 y(not)24 b(yet)g(normal.)30 b(In)23 b(this)h(case)g(we)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41007
(transform)i(the)f(reduction)i(sequence)g(into)629 5025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41008
y FL(And)784 5039 y Gc(R)841 5025 y F4(\()876 5013 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41009
FX(h)904 5025 y Ga(a)952 5013 y FX(i)980 5025 y Ga(M)10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41010
b(;)1118 5013 y FX(h)1146 5025 y Ga(b)1185 5013 y FX(i)1212
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41011
5025 y Ga(N)g(;)15 b(c)p F4(\))p 1456 4950 V 1456 4988
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41012
a Gc(aux)1435 5025 y F6(\000)-31 b(\000)g(!)25 b FL(And)1785
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41013
5039 y Gc(R)1843 5025 y F4(\()1878 5013 y FX(h)1906 5025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41014
y Ga(a)1954 5013 y FX(i)1981 5025 y Ga(T)13 b(;)2087
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41015
5013 y FX(h)2115 5025 y Ga(b)2154 5013 y FX(i)2181 5025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41016
y Ga(N)d(;)15 b(c)p F4(\))2425 4988 y Gc(aux)2405 5025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41017
y F6(\000)-31 b(\000)f(!)25 b FL(And)2755 5039 y Gc(R)2813
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41018
5025 y F4(\()2848 5013 y FX(h)2875 5025 y Ga(a)2923 5013
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41019
y FX(i)2951 5025 y Ga(T)13 b(;)3057 5013 y FX(h)3084
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41020
5025 y Ga(b)3123 5013 y FX(i)3151 5025 y Ga(S)5 b(;)15
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41021
b(c)p F4(\))26 b Ga(:)p Black 414 5263 a F6(\017)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41022
45 w Gg(Suppose)f(we)e(ha)n(v)o(e)h(a)f(term)h(with)f(the)h(reduction)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41023
(sequence)646 5522 y FL(And)800 5536 y Gc(R)858 5522
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41024
y F4(\()893 5510 y FX(h)921 5522 y Ga(a)969 5510 y FX(i)996
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41025
5522 y Ga(M)10 b(;)1134 5510 y FX(h)1162 5522 y Ga(b)1201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41026
5510 y FX(i)1229 5522 y Ga(N)g(;)15 b(c)p F4(\))1479
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41027
5485 y Gc(bad)1452 5522 y F6(\000)-31 b(\000)f(!)26 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41028
FL(And)1802 5536 y Gc(R)1860 5522 y F4(\()1895 5510 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41029
FX(h)1923 5522 y Ga(a)1971 5510 y FX(i)1998 5522 y Ga(S)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41030
b(;)2099 5510 y FX(h)2127 5522 y Ga(b)2166 5510 y FX(i)2193
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41031
5522 y Ga(N)10 b(;)15 b(c)p F4(\))p 2437 5447 V 2437
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41032
5485 a Gc(aux)2417 5522 y F6(\000)-32 b(\000)h(!)25 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41033
FL(And)2767 5536 y Gc(R)2824 5522 y F4(\()2859 5510 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41034
FX(h)2887 5522 y Ga(a)2935 5510 y FX(i)2963 5522 y Ga(T)12
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41035
b(;)3068 5510 y FX(h)3096 5522 y Ga(b)3135 5510 y FX(i)3163
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41036
5522 y Ga(N)e(;)15 b(c)p F4(\))p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41037
%%Page: 168 180
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41038
TeXDict begin 168 179 bop Black -144 51 a Gb(168)2658
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41039
b(Details)24 b(f)n(or)g(some)g(Pr)n(oofs)p -144 88 3691
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41040
4 v Black 549 412 a Gg(where)30 b(\002rst)g(a)g(bad)h(reduction)h(is)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41041
(performed)h(in)e(the)g(left)h(subterm)g(and)g(then)g(a)f(reduction)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41042
525 y(in)d(the)h(same)f(term)g(which)h(respects)h(the)f(constraints)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41043
(of)e(the)f(strate)o(gy)-6 b(.)42 b(Thus)28 b(we)e(ha)n(v)o(e)i(the)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41044
638 y(reduction)21 b(sequence)g Ga(M)1411 601 y Gc(bad)1384
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41045
638 y F6(\000)-31 b(\000)f(!)25 b Ga(S)p 1686 563 119
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41046
3 v 1686 601 a Gc(aux)1666 638 y F6(\000)-32 b(\000)h(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41047
b Ga(T)13 b Gg(.)26 b(By)18 b(induction)j(hypothesis)g(there)f(is)e(a)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41048
(reduction)549 751 y Ga(M)p 692 676 V 692 714 a Gc(aux)672
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41049
751 y F6(\000)-31 b(\000)f(!)25 b Ga(R)983 714 y Gc(aux)962
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41050
751 y F6(\000)-31 b(\000)g(!)1133 718 y FX(\003)1197
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41051
751 y Ga(T)13 b Gg(,)23 b(from)g(which)h(we)f(obtain)i(the)e(reduction)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41052
k(sequence)641 954 y FL(And)795 968 y Gc(R)853 954 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41053
F4(\()888 942 y FX(h)916 954 y Ga(a)964 942 y FX(i)991
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41054
954 y Ga(M)10 b(;)1129 942 y FX(h)1157 954 y Ga(b)1196
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41055
942 y FX(i)1224 954 y Ga(N)f(;)15 b(c)p F4(\))p 1467
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41056
879 V 1467 917 a Gc(aux)1447 954 y F6(\000)-31 b(\000)f(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41057
b FL(And)1797 968 y Gc(R)1855 954 y F4(\()1890 942 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41058
FX(h)1917 954 y Ga(a)1965 942 y FX(i)1993 954 y Ga(R)q(;)2103
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41059
942 y FX(h)2130 954 y Ga(b)2169 942 y FX(i)2197 954 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41060
Ga(N)10 b(;)15 b(c)p F4(\))2440 917 y Gc(aux)2420 954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41061
y F6(\000)-31 b(\000)g(!)2590 916 y FX(\003)2655 954
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41062
y FL(And)2810 968 y Gc(R)2867 954 y F4(\()2902 942 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41063
FX(h)2930 954 y Ga(a)2978 942 y FX(i)3006 954 y Ga(T)12
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41064
b(;)3111 942 y FX(h)3139 954 y Ga(b)3178 942 y FX(i)3206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41065
954 y Ga(N)e(;)15 b(c)p F4(\))26 b Ga(:)p Black 458 1137
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41066
a F6(\017)p Black 46 w Gg(Suppose)e(we)f(ha)n(v)o(e)h(a)f(term)h(with)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41067
(the)h(reduction)i(sequence)933 1340 y FL(Cut)p F4(\()1106
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41068
1328 y FX(h)1133 1340 y Ga(a)1181 1328 y FX(i)1209 1340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41069
y Ga(M)10 b(;)1347 1328 y F9(\()1375 1340 y Ga(x)1427
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41070
1328 y F9(\))1454 1340 y Ga(N)g F4(\))1625 1303 y Gc(bad)1598
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41071
1340 y F6(\000)-31 b(\000)f(!)25 b FL(Cut)p F4(\()1966
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41072
1328 y FX(h)1994 1340 y Ga(a)2042 1328 y FX(i)2070 1340
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41073
y Ga(M)10 b(;)2208 1328 y F9(\()2236 1340 y Ga(x)2288
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41074
1328 y F9(\))2315 1340 y Ga(S)5 b F4(\))p 2457 1266 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41075
2457 1303 a Gc(aux)2437 1340 y F6(\000)-32 b(\000)h(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41076
b Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)2958 1328 y F9(\()2986
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41077
1340 y Ga(x)3038 1328 y F9(\))3065 1340 y Ga(S)l F6(g)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41078
1524 y Gg(which)d(we)g(may)g(transform)j(into)946 1727
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41079
y FL(Cut)p F4(\()1119 1715 y FX(h)1147 1727 y Ga(a)1195
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41080
1715 y FX(i)1222 1727 y Ga(M)10 b(;)1360 1715 y F9(\()1388
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41081
1727 y Ga(x)1440 1715 y F9(\))1468 1727 y Ga(N)f F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41082
1631 1652 V 1631 1690 a Gc(aux)1611 1727 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41083
b(\000)f(!)26 b Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)2133
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41084
1715 y F9(\()2160 1727 y Ga(x)2212 1715 y F9(\))2240
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41085
1727 y Ga(N)p F6(g)2404 1690 y Gc(aux)2384 1727 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41086
b(\000)f(!)2554 1690 y FX(\003)2619 1727 y Ga(M)5 b F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41087
b Ga(a)26 b F4(:=)2945 1715 y F9(\()2973 1727 y Ga(x)3025
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41088
1715 y F9(\))3052 1727 y Ga(S)-5 b F6(g)549 1911 y Gg(using)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41089
b(the)g(f)o(act)g(that)g(if)g Ga(N)1426 1873 y Gc(aux)1406
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41090
1911 y F6(\000)-31 b(\000)f(!)25 b Ga(S)j Gg(then)c Ga(M)5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41091
b F6(f)-7 b Ga(a)26 b F4(:=)2191 1899 y F9(\()2218 1911
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41092
y Ga(x)2270 1899 y F9(\))2297 1911 y Ga(N)q F6(g)2462
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41093
1873 y Gc(aux)2442 1911 y F6(\000)-32 b(\000)h(!)2612
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41094
1878 y FX(\003)2677 1911 y Ga(M)5 b F6(f)-7 b Ga(a)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41095
b F4(:=)3003 1899 y F9(\()3030 1911 y Ga(x)3082 1899
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41096
y F9(\))3110 1911 y Ga(S)-5 b F6(g)p Gg(.)p Black 458
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41097
2082 a F6(\017)p Black 46 w Gg(Suppose)24 b(we)f(ha)n(v)o(e)h(a)f(term)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41098
h(with)f(reduction)j(sequence)948 2285 y FL(Cut)p F4(\()1121
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41099
2273 y FX(h)1149 2285 y Ga(a)1197 2273 y FX(i)1224 2285
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41100
y Ga(M)10 b(;)1362 2273 y F9(\()1390 2285 y Ga(x)1442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41101
2273 y F9(\))1469 2285 y Ga(N)g F4(\))1640 2248 y Gc(bad)1613
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41102
2285 y F6(\000)-31 b(\000)f(!)26 b FL(Cut)o F4(\()1981
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41103
2273 y FX(h)2009 2285 y Ga(a)2057 2273 y FX(i)2085 2285
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41104
y Ga(S)5 b(;)2186 2273 y F9(\()2214 2285 y Ga(x)2266
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41105
2273 y F9(\))2293 2285 y Ga(N)10 b F4(\))p 2457 2210
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41106
V 2457 2248 a Gc(aux)2437 2285 y F6(\000)-32 b(\000)h(!)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41107
b Ga(S)q F6(f)-7 b Ga(a)25 b F4(:=)2921 2273 y F9(\()2949
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41108
2285 y Ga(x)3001 2273 y F9(\))3028 2285 y Ga(N)p F6(g)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41109
2468 y Gg(which)e(we)g(may)g(transform)j(into)954 2671
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41110
y FL(Cut)o F4(\()1126 2659 y FX(h)1154 2671 y Ga(a)1202
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41111
2659 y FX(i)1230 2671 y Ga(M)10 b(;)1368 2659 y F9(\()1396
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41112
2671 y Ga(x)1448 2659 y F9(\))1475 2671 y Ga(N)g F4(\))p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41113
1639 2597 V 1639 2634 a Gc(aux)1619 2671 y F6(\000)-32
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41114
b(\000)h(!)25 b Ga(M)5 b F6(f)-7 b Ga(a)26 b F4(:=)2140
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41115
2659 y F9(\()2168 2671 y Ga(x)2220 2659 y F9(\))2247
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41116
2671 y Ga(N)q F6(g)2412 2634 y Gc(aux)2391 2671 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41117
b(\000)g(!)2562 2634 y FX(\003)2626 2671 y Ga(S)q F6(f)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41118
b Ga(a)25 b F4(:=)2916 2659 y F9(\()2943 2671 y Ga(x)2995
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41119
2659 y F9(\))3022 2671 y Ga(N)q F6(g)549 2855 y Gg(using)f(Lemma)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41120
(2.3.11.)p 3480 2855 4 62 v 3484 2797 55 4 v 3484 2855
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41121
V 3538 2855 4 62 v Black 321 3092 a Gb(Pr)n(oof)31 b(of)f(Theor)n(em)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41122
(4.1.3.)p Black 34 w Gg(W)l(ithout)i(loss)g(of)f(generality)i(we)d(may)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41123
h(assume)g(that)h(the)f(reduction)321 3205 y(sequence)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41124
Ga(M)828 3168 y Gc(aux)808 3205 y F6(\000)-31 b(\000)f(!)978
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41125
3172 y FX(\003)1043 3205 y Ga(N)40 b Gg(is)31 b(of)g(the)g(form)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41126
Ga(M)p 1858 3131 119 3 v 1858 3168 a Gc(aux)1838 3205
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41127
y F6(\000)-32 b(\000)h(!)2008 3172 y FX(\003)2086 3205
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41128
y Ga(M)2174 3219 y Gc(i)2262 3168 y(aux)2242 3205 y F6(\000)f(\000)h(!)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41129
2412 3172 y FX(\003)2490 3205 y Ga(N)40 b Gg(where)31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41130
b(either)i Ga(M)3181 3219 y Gc(i)3248 3205 y F6(\021)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41131
b Ga(N)h Gg(or)321 3318 y Ga(M)409 3332 y Gc(i)508 3281
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41132
y(bad)481 3318 y F6(\000)-31 b(\000)f(!)43 b Ga(M)782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41133
3332 y Gc(i)p F9(+1)901 3318 y Gg(.)57 b(The)33 b(proof)h(then)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41134
(proceeds)h(by)f(induction)i(on)d(the)h(number)g(of)f(steps)h(of)f(the)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41135
321 3431 y(longest)28 b(reduction)g(sequence)g(starting)g(from)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41136
Ga(M)1923 3445 y Gc(i)1951 3431 y Gg(,)g(i.e.,)f(on)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41137
FW(M)t(A)t(X)t(R)t(E)t(D)2617 3445 y Gc(aux)2739 3431
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41138
y F4(\()p Ga(M)2862 3445 y Gc(i)2891 3431 y F4(\))p Gg(.)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41139
b(Since)26 b(we)f(pro)o(v)o(e)321 3544 y(the)30 b(theorem)g(for)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41140
(terms)h(belonging)i(to)d FY(T)t Gg(,)c(we)k(kno)n(w)g(their)h(longest)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41141
h(reduction)h(sequences)g(must)321 3657 y(be)24 b(\002nite.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41142
Black 458 3836 a F6(\017)p Black 46 w Gg(If)g FW(M)t(A)t(X)t(R)t(E)t(D)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41143
974 3850 y Gc(aux)1096 3836 y F4(\()p Ga(M)1219 3850
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41144
y Gc(i)1247 3836 y F4(\))i(=)f(0)p Gg(,)e(then)g Ga(M)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41145
1817 3761 V 1817 3799 a Gc(aux)1797 3836 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41146
b(\000)f(!)1967 3803 y FX(\003)2032 3836 y Ga(M)2120
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41147
3850 y Gc(i)2173 3836 y F6(\021)25 b Ga(N)33 b Gg(is)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41148
b(a)h(sequence)j(which)d(respects)i(the)549 3949 y(restrictions)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41149
(imposed)f(on)p 1440 3874 V 1440 3912 a Gc(aux)1420 3949
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41150
y F6(\000)-31 b(\000)f(!)p Gg(.)p Black 458 4120 a F6(\017)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41151
Black 46 w Gg(If)28 b FW(M)t(A)t(X)t(R)t(E)t(D)977 4134
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41152
y Gc(aux)1099 4120 y F4(\()p Ga(M)1222 4134 y Gc(i)1251
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41153
4120 y F4(\))j Ga(>)g F4(0)p Gg(,)c(we)f(\002rst)g(observ)o(e)i(that)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41154
(the)g(last)g(reduction)i(leading)g(to)d(a)h(nor)n(-)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41155
4233 y(mal)e(form)h(must)g(be)g(a)f(reduction)j(of)e(the)g(form)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41156
2117 4158 V 2117 4196 a Gc(aux)2097 4233 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41157
b(\000)f(!)p Gg(.)35 b(Therefore)28 b(the)e(reduction)i(sequence)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41158
4346 y Ga(M)637 4360 y Gc(i)724 4309 y(aux)703 4346 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41159
F6(\000)-31 b(\000)g(!)874 4313 y FX(\003)951 4346 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41160
Ga(N)40 b Gg(contains)33 b(at)d(least)i(one)p 1870 4271
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41161
V 1870 4309 a Gc(aux)1850 4346 y F6(\000)-31 b(\000)f(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41162
Gg(.)49 b(Let)30 b Ga(M)2331 4360 y Gc(j)2398 4346 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41163
Gg(be)g(the)h(leftmost)h(term)f(in)f(this)h(se-)549 4459
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41164
y(quence)e(from)e(which)h(a)f(reduction)p 1742 4384 V
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41165
1742 4422 a Gc(aux)1722 4459 y F6(\000)-32 b(\000)h(!)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41166
b Gg(is)i(performed.)42 b(Hence)28 b(the)f(reduction)j(sequence)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41167
4572 y Ga(M)637 4586 y Gc(i)723 4535 y(aux)702 4572 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41168
F6(\000)-31 b(\000)g(!)873 4539 y FX(\003)949 4572 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41169
Ga(N)40 b Gg(is)30 b(of)g(the)g(form)g Ga(M)1691 4586
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41170
y Gc(i)1784 4535 y(bad)1757 4572 y F6(\000)-32 b(\000)h(!)1927
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41171
4539 y F9(+)2023 4572 y Ga(M)2111 4586 y Gc(j)p 2206
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41172
4497 V 2206 4535 a(aux)2185 4572 y F6(\000)g(\000)g(!)37
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41173
b Ga(M)2481 4586 y Gc(j)t F9(+1)2666 4535 y Gc(aux)2645
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41174
4572 y F6(\000)-31 b(\000)g(!)2816 4539 y FX(\003)2892
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41175
4572 y Ga(N)10 b Gg(.)48 b(By)29 b(iterati)n(v)o(ely)549
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41176
4685 y(applying)i(Lemma)d(4.1.2,)j(we)d(can)i(mo)o(v)o(e)f(the)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41177
(reduction)p 2484 4610 V 2484 4647 a Gc(aux)2464 4685
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41178
y F6(\000)-31 b(\000)f(!)29 b Gg(ne)o(xt)g(to)h Ga(M)3034
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41179
4699 y Gc(i)3090 4685 y Gg(and)g(obtain)h(a)549 4797
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41180
y(reduction)26 b(sequence)g(of)d(the)h(form)1384 5001
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41181
y Ga(M)1472 5015 y Gc(i)p 1546 4926 V 1546 4964 a(aux)1525
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41182
5001 y F6(\000)-31 b(\000)g(!)25 b Ga(M)1819 4963 y FX(0)1888
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41183
4964 y Gc(aux)1868 5001 y F6(\000)-32 b(\000)h(!)2038
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41184
4963 y FX(\003)2103 5001 y Ga(M)2191 5015 y Gc(j)t F9(+1)2363
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41185
4964 y Gc(aux)2343 5001 y F6(\000)f(\000)h(!)2513 4963
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41186
y FX(\003)2578 5001 y Ga(N)35 b(:)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41187
%%Page: 169 181
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41188
TeXDict begin 169 180 bop Black 277 51 a Gb(B.3)23 b(Pr)n(oofs)h(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41189
(Chapter)f(4)2639 b(169)p 277 88 3691 4 v Black 504 412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41190
a Gg(W)l(ithout)26 b(loss)e(of)g(generality)i(we)d(may)h(assume)g(the)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41191
(sequence)h Ga(M)2659 379 y FX(0)2728 375 y Gc(aux)2708
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41192
412 y F6(\000)-31 b(\000)f(!)2878 379 y FX(\003)2943
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41193
412 y Ga(M)3031 426 y Gc(j)t F9(+1)3204 375 y Gc(aux)3184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41194
412 y F6(\000)h(\000)f(!)3354 379 y FX(\003)3419 412
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41195
y Ga(N)504 525 y Gg(is)24 b(of)f(the)h(form)1575 767
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41196
y Ga(M)1673 730 y FX(0)p 1742 693 119 3 v 50 w Gc(aux)1722
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41197
767 y F6(\000)-31 b(\000)f(!)1892 730 y FX(\003)1957
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41198
767 y Ga(M)2045 782 y Gc(k)2133 730 y(aux)2113 767 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41199
F6(\000)h(\000)g(!)2283 730 y FX(\003)2348 767 y Ga(N)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41200
1009 y Gg(where)34 b(either)g Ga(M)1086 1024 y Gc(k)1173
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41201
1009 y F6(\021)43 b Ga(N)f Gg(or)33 b Ga(M)1598 1024
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41202
y Gc(k)1712 972 y(bad)1685 1009 y F6(\000)-32 b(\000)h(!)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41203
b Ga(M)1986 1024 y Gc(k)r F9(+1)2119 1009 y Gg(.)57 b(This)34
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41204
b(means)f Ga(M)2746 1024 y Gc(k)2822 1009 y Gg(is)g(the)g(\002rst)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41205
(term)g(in)504 1122 y(this)22 b(reduction)i(sequence)g(from)e(which)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41206
(a)g(bad)h(reduction)i(is)e(performed.)30 b(Clearly)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41207
b(,)22 b(we)f(ha)n(v)o(e)506 1235 y FW(M)t(A)t(X)t(R)t(E)t(D)846
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41208
1249 y Gc(aux)969 1235 y F4(\()p Ga(M)1092 1249 y Gc(i)1120
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41209
1235 y F4(\))50 b Ga(>)i FW(M)t(A)t(X)t(R)t(E)t(D)1668
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41210
1249 y Gc(aux)1790 1235 y F4(\()p Ga(M)1923 1202 y FX(0)1947
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41211
1235 y F4(\))e F6(\025)h FW(M)t(A)t(X)t(R)t(E)t(D)2494
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41212
1249 y Gc(aux)2616 1235 y F4(\()p Ga(M)2739 1250 y Gc(k)2782
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41213
1235 y F4(\))p Gg(,)40 b(and)d(thus)g(we)f(can)504 1348
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41214
y(apply)27 b(the)f(induction)j(hypothesis)f(which)e(gi)n(v)o(es)g(us)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41215
(a)f(reduction)k(sequence)f Ga(M)3105 1315 y FX(0)p 3178
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41216
1273 V 3178 1311 a Gc(aux)3157 1348 y F6(\000)-31 b(\000)g(!)3328
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41217
1315 y FX(\003)3396 1348 y Ga(N)10 b Gg(.)504 1461 y(W)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41218
b(e)23 b(conclude)j(this)e(case)g(with)g(the)f(reduction)j(sequence)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41219
1398 1722 y Ga(M)p 1542 1647 V 1542 1685 a Gc(aux)1521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41220
1722 y F6(\000)-31 b(\000)g(!)1692 1684 y FX(\003)1756
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41221
1722 y Ga(M)1844 1736 y Gc(i)p 1918 1647 V 1918 1685
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41222
a(aux)1898 1722 y F6(\000)g(\000)f(!)25 b Ga(M)2191 1684
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41223
y FX(0)p 2260 1647 V 2260 1685 a Gc(aux)2240 1722 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41224
b(\000)f(!)2410 1684 y FX(\003)2475 1722 y Ga(N)35 b(:)504
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41225
1964 y Gg(Thus)24 b(we)f(are)g(done.)p 3436 1964 4 62
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41226
v 3440 1905 55 4 v 3440 1964 V 3494 1964 4 62 v Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41227
Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41228
%%Page: 170 182
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41229
TeXDict begin 170 181 bop Black Black Black Black eop
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41230
end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41231
%%Page: 171 183
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41232
TeXDict begin 171 182 bop Black Black 277 1093 a Gf(Bibliograph)m(y)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41233
Black Black 277 1388 a Gg(Abramsk)o(y)-6 b(,)27 b(S.)d([1993].)43
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41234
b(Computational)28 b(Interpretations)i(of)c(Linear)g(Logic.)41
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41235
b F7(Theor)m(etical)28 b(Com-)368 1501 y(puter)d(Science)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41236
Gg(,)g(111\(1\2262\):3\22657.)p Black Black 277 1637
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41237
a(Baader)l(,)j(F)-7 b(.)25 b(and)i(Nipk)o(o)n(w)-6 b(,)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41238
b(T)-7 b(.)25 b([1998].)44 b F7(Term)26 b(Re)o(writing)h(and)g(All)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41239
(That)p Gg(.)42 b(Cambridge)28 b(Uni)n(v)o(ersity)368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41240
1750 y(Press.)p Black Black 277 1885 a(Barbanera,)38
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41241
b(F)-7 b(.)32 b(and)j(Berardi,)i(S.)32 b([1994].)68 b(A)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41242
b(Symmetric)h(Lambda)g(Calculus)h(for)f(\223Classical\224)368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41243
1998 y(Program)i(Extraction.)75 b(In)35 b F7(Theor)m(etical)j(Aspects)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41244
(of)f(Computer)g(Softwar)m(e)p Gg(,)k(v)n(olume)d(789)f(of)368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41245
2111 y F7(LNCS)p Gg(,)22 b(pages)i(495\226515.)i(Springer)f(V)-10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41246
b(erlag.)p Black Black 277 2246 a(Barbanera,)27 b(F)-7
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41247
b(.,)23 b(Berardi,)i(S.,)f(and)h(Schi)n(v)n(alocchi,)i(M.)c([1997].)39
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41248
b(\223Classical\224)27 b(Programming-with-)368 2359 y(Proofs)e(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41249
Ga(\025)776 2326 y Gc(sy)r(m)913 2359 y Gg(:)30 b(An)24
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41250
b(Analysis)i(of)e(Non-Con\003uence.)39 b(In)24 b F7(Theor)m(etical)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41251
(Aspects)e(of)g(Computer)368 2472 y(Softwar)m(e)p Gg(,)f(v)n(olume)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41252
(1281)g(of)e F7(LNCS)p Gg(,)f(pages)i(365\226390.)i(Springer)f(V)-10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41253
b(erlag.)p Black Black 277 2608 a(Barendre)o(gt,)27 b(H.)d([1981].)40
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41254
b F7(The)25 b(Lambda)g(Calculus:)34 b(Its)26 b(Syntax)g(and)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41255
(Semantics)p Gg(,)h(v)n(olume)g(103)e(of)368 2721 y F7(Studies)g(in)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41256
(Lo)o(gic)g(and)g(the)g(F)-10 b(oundations)26 b(of)e(Mathematics)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41257
Gg(.)35 b(North-Holland.)p Black Black 277 2856 a(Barendre)o(gt,)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41258
b(H.)c(and)h(Ghilezan,)j(S.)25 b([2000].)47 b(Theoretical)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41259
b(Pearls:)37 b(Lambda)27 b(Terms)g(for)h(Natural)368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41260
2969 y(Deduction,)35 b(Sequent)e(Calculus)f(and)g(Cut)f(Elimination.)60
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41261
b F7(J)n(ournal)33 b(of)e(Functional)j(Pr)l(o)o(gr)o(am-)368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41262
3082 y(ming)p Gg(,)23 b(10\(1\):121\226134.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41263
277 3218 a(Benton,)34 b(P)-10 b(.)29 b(N.)g([1995].)59
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41264
b(Strong)32 b(Normalisation)h(for)e(the)h(Linear)f(Term)f(Calculus.)59
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41265
b F7(J)n(ournal)32 b(of)368 3331 y(Functional)26 b(Pr)l(o)o(gr)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41266
(amming)p Gg(,)e(5\(1\):65\22680.)p Black Black 277 3466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41267
a(Benton,)j(P)-10 b(.)24 b(N.,)g(Bierman,)i(G.)e(M.,)g(de)i(P)o(ai)n(v)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41268
n(a,)f(V)-12 b(.)24 b(C.)g(V)-12 b(.,)25 b(and)g(Hyland,)i(J.)e(M.)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41269
(E.)g([1993].)41 b(A)24 b(Term)368 3579 y(Calculus)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41270
b(for)f(Intuitionistic)j(Linear)d(Logic.)49 b(In)29 b
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41271
F7(T)-7 b(yped)29 b(Lambda)f(Calculi)i(and)f(Applications)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41272
Gg(,)368 3692 y(v)n(olume)c(664)f(of)f F7(LNCS)p Gg(,)f(pages)j
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41273
(75\22690.)f(Springer)n(-V)-10 b(erlag.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41274
277 3828 a(Ber)n(ger)l(,)29 b(U.,)e(Buchholz,)j(W)-8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41275
b(.,)26 b(and)i(Schwichtenber)n(g,)j(H.)26 b([2000].)47
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41276
b(Re\002ned)28 b(Program)f(Extraction)368 3941 y(from)c(Classical)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41277
(Proofs.)34 b F7(Annals)25 b(of)e(Pur)m(e)g(and)h(Applied)h(Lo)o(gic)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41278
Gg(.)33 b(T)-7 b(o)23 b(appear)-5 b(.)p Black Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41279
4076 a(Bierman,)33 b(G.)d(M.)g([1998].)58 b(A)30 b(Computational)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41280
(Interpretation)h(of)c(the)g Ga(\025\026)p Gg(-calculus.)60
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41281
b(In)31 b F7(Math-)368 4189 y(ematical)j(F)-10 b(oundations)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41282
b(of)c(Computer)i(Science)p Gg(,)i(v)n(olume)e(1450)f(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41283
F7(LNCS)p Gg(,)d(pages)k(336\226345.)368 4302 y(Springer)n(-V)-10
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41284
b(erlag.)p Black Black 277 4437 a(Bierman,)30 b(G.)d(M.)g([1999].)50
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41285
b(A)27 b(Classical)j(Linear)e Ga(\025)p Gg(-calculus.)51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41286
b F7(Theor)m(etical)31 b(Computer)e(Science)p Gg(,)368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41287
4550 y(227\(1\2262\):43\22678.)p Black Black 277 4686
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41288
a(Bittar)l(,)i(E.)c(T)-7 b(.)28 b([1999].)51 b(Strong)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41289
b(Normalisation)h(Proofs)f(for)f(Cut-Elimination)i(in)d(Gentzen')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41290
b(s)31 b(Se-)368 4799 y(quent)25 b(Calculi.)34 b(In)23
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41291
b F7(Lo)o(gic,)h(Alg)o(ebr)o(a)g(and)g(Computer)g(Science)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41292
Gg(,)h(v)n(olume)g(46)f(of)f F7(Banac)o(h-Center)368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41293
4912 y(Publications)p Gg(,)j(pages)f(179\226225.)p Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41294
Black 277 5047 a(Bloo,)e(R.)e([1997].)33 b F7(Pr)m(eservation)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41295
b(of)e(T)-8 b(ermination)24 b(for)f(Explicit)h(Substitution)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41296
Gg(.)35 b(PhD)21 b(thesis,)j(Eind-)368 5160 y(ho)o(v)o(en)g(Uni)n(v)o
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41297
(ersity)h(of)f(T)-6 b(echnology)g(.)p Black Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41298
5296 a(Bloo,)24 b(R.)f(and)h(Geuv)o(ers,)h(H.)e([1999].)36
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41299
b(Explicit)25 b(Substitution:)33 b(On)24 b(the)g(Edge)g(of)g(Strong)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41300
(Normali-)368 5409 y(sation.)35 b F7(Theor)m(etical)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41301
b(Computer)g(Science)p Gg(,)g(211\(1\2262\):375\226395.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41302
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41303
%%Page: 172 184
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41304
TeXDict begin 172 183 bop Black -144 51 a Gb(172)3046
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41305
b(Bibliograph)o(y)p -144 88 3691 4 v Black Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41306
321 412 a Gg(Bori)5 b(\020)-35 b(ci)5 b(\264)-35 b(c,)26
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41307
b(B.)e(R.)g([1985].)40 b(On)24 b(Sequence-Conclusion)30
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41308
b(Natural)c(Deduction)h(Systems.)39 b F7(J)n(ournal)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41309
b(of)412 525 y(Philosophical)g(Lo)o(gic)p Gg(,)d(14\(4\):359\226377.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41310
Black Black 321 671 a(Buss,)30 b(S.)d(R.,)h(editor)i([1998].)51
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41311
b F7(Handbook)30 b(of)f(Pr)l(oof)g(Theory)p Gg(,)h(v)n(olume)g(137)f
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b(oundations)27 b(of)c(Mathematics)p Gg(.)35 b(North-Holland.)p
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Christian Urban <urbanc@in.tum.de>
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b F7(Ne)o(w)23 b(Gener)n(-)412 1042 y(ation)i(Computing)p
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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b(Semantics)i(of)f(Evidence)h(of)e(Classical)j(Arithmetic.)41
Christian Urban <urbanc@in.tum.de>
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b F7(J)n(ournal)28 b(of)e(Sym-)412 1819 y(bolic)f(Lo)o(gic)p
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Gg(,)e(60\(1\):325\226337.)p Black Black 321 1965 a(Curry)-6
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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b(oundations)27 b(of)c(Mathematics)p Gg(.)35 b(North-Holland.)p
Christian Urban <urbanc@in.tum.de>
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(and)i(Schellinx,)h(H.)c([1997].)43 b(A)25 b(Ne)n(w)g(Deconstructi)n(v)
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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2595 y(Uni)n(v)o(ersity)25 b(Press.)p Black Black 321
Christian Urban <urbanc@in.tum.de>
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2741 a(de)i(Bruijn,)h(N.)d(G.)g([1980].)45 b(A)26 b(Surv)o(e)o(y)h(of)f
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Christian Urban <urbanc@in.tum.de>
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2967 y(Press.)p Black Black 321 3113 a(Dersho)n(witz,)i(N.)d([1982].)52
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b F7(Theor)m(etical)31 b(Computer)412 3226 y(Science)p
Christian Urban <urbanc@in.tum.de>
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Gg(,)25 b(17:279\226301.)p Black Black 321 3371 a(Dragalin,)31
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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b(Mathematical)h(Soci-)412 3597 y(ety)-6 b(.)p Black
Christian Urban <urbanc@in.tum.de>
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Black 321 3743 a(Dyckhof)n(f,)40 b(R.)34 b(and)i(Pinto,)j(L.)34
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b([1998].)74 b(Cut-Elimination)38 b(and)e(a)f(Permutation-Free)j
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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b F7(Studia)25 b(Lo)o(gica)p Gg(,)f(60\(1\):107\226118.)p
Christian Urban <urbanc@in.tum.de>
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(Functions.)37 b(In)412 4115 y F7(Higher)24 b(Set)g(Theory)p
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(Springer)n(-V)-10 b(erlag.)p Black Black 321 4261 a(Gabbay)k(,)31
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b F7(Theor)m(etical)26 b(Computer)e(Science)p Gg(,)h
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(110\(2\):249\226239.)p Black Black 321 5037 a(Gentzen,)c(G.)c([1935].)
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25 b(Untersuchungen)30 b(\250)-38 b(uber)20 b(das)f(logische)i
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b F7(Mathematisc)o(he)412 5409 y(Annalen)p Gg(,)25 b(112:493\226565.)p
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525 y(F)o(oundations)29 b(of)d(Mathematics.)i(North-Holland.)45
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891 y(\264)1370 890 y(erieur)m(e)p Gg(.)36 b(PhD)22 b(thesis,)j(Uni)n
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Gg(,)h(50\(1\):1\226102.)p Black Black 277 1169 a(Girard,)31
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b(J.-Y)-12 b(.)29 b([1987b].)53 b F7(Pr)l(oof)30 b(Theory)g(and)g(Lo)o
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b(.)368 1282 y(Bibliopolis.)p Black Black 277 1422 a(Girard,)29
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b(J.-Y)-12 b(.)26 b([1991].)47 b(A)26 b(Ne)n(w)g(Constructi)n(v)o(e)j
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Gg(,)h(1\(3\):255\226296.)p Black Black 277 1674 a(Girard,)44
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Christian Urban <urbanc@in.tum.de>
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Gg(,)368 1787 y(11:301\226506.)p Black Black 277 1927
Christian Urban <urbanc@in.tum.de>
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a(Girard,)22 b(J.-Y)-12 b(.,)20 b(Lafont,)h(Y)-12 b(.,)20
Christian Urban <urbanc@in.tum.de>
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b(and)i(T)-7 b(aylor)l(,)21 b(P)-10 b(.)19 b([1989].)29
Christian Urban <urbanc@in.tum.de>
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F7(Cambridg)o(e)368 2040 y(T)-5 b(r)o(acts)24 b(in)f(Theor)m(etical)j
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(Computer)e(Science)p Gg(.)35 b(Cambridge)25 b(Uni)n(v)o(ersity)g
Christian Urban <urbanc@in.tum.de>
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Christian Urban <urbanc@in.tum.de>
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2292 y(ming)c(Langua)o(g)o(es)p Gg(,)h(pages)g(47\22658.)g(A)l(CM)d
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(Press.)p Black Black 277 2432 a(Hennessy)-6 b(,)31 b(M.)c(C.)g(B.)g
Christian Urban <urbanc@in.tum.de>
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(and)i(Ashcroft,)h(E.)d(A.)g([1980].)50 b(A)27 b(Mathematical)k
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Ga(\025)p Gg(-calculus.)35 b F7(Theor)m(etical)26 b(Computer)e(Science)
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b(H.)c([1994].)50 b(A)28 b Ga(\025)p Gg(-calculus)j(Structure)f
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b(erlag.)p Black Black 277 2937 a(Herbelin,)22 b(H.)e([1995].)28
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b(du)21 b(Calcul)g(des)g(S)3229 2938 y(\264)3224 2937
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y(equents)368 3050 y(comme)34 b(Calcul)h(de)f Ga(\025)p
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3051 y(\264)2373 3050 y(egies)h(Ga)o(gnantes)p Gg(.)70
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b(PhD)33 b(thesis,)368 3163 y(Uni)n(v)o(ersit)5 b(\264)-35
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b(e)25 b(P)o(aris)e(VII.)p Black Black 277 3302 a(Hindle)o(y)-6
Christian Urban <urbanc@in.tum.de>
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b(,)30 b(J.)d(R.)g([1969].)50 b(The)27 b(Principal)j(Type)e(Scheme)g
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(of)g(an)g(Object)h(in)f(Combinatory)i(Logic.)368 3415
Christian Urban <urbanc@in.tum.de>
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y F7(T)-5 b(r)o(ansactions)26 b(of)e(the)f(American)i(Mathematical)g
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(Society)p Gg(,)g(146:29\22640.)p Black Black 277 3555
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b(In)49 b F7(To)368 3668 y(H.)25 b(B.)h(Curry:)36 b(Essays)27
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diff changeset
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b F7(Normalisation,)39 b(Cut-Elimination)d(and)e(the)g(Theory)g(of)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41580
(Pr)l(oofs)p Gg(,)j(v)n(ol-)368 1791 y(ume)23 b(28)h(of)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41581
F7(CLSI)g(Lectur)m(e)h(Notes)p Gg(.)34 b(Center)24 b(for)f(the)h(Study)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41582
g(of)g(Language)h(and)f(Information.)p Black Black 277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41583
1932 a(Urban,)i(C.)e([1998].)40 b(Implementation)28 b(of)d(Proof)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41584
(Search)h(in)f(the)h(Imperati)n(v)o(e)g(Programming)h(Lan-)368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41585
2045 y(guage)k(Pizza.)51 b(In)29 b F7(A)n(utomated)i(Reasoning)g(with)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41586
(Analytic)h(T)-8 b(ableaux)30 b(and)g(Related)g(Methods)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41587
Gg(,)368 2157 y(v)n(olume)25 b(1397)f(of)g F7(LN)n(AI)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41588
Gg(,)d(pages)j(313\226319.)i(Springer)n(-V)-10 b(erlag.)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41589
Black Black 277 2298 a(W)j(adler)l(,)37 b(P)-10 b(.)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41590
b([1990].)68 b(Linear)34 b(Types)h(can)f(Change)h(the)f(World!)94
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41591
b(In)34 b F7(W)-8 b(orking)35 b(Confer)m(ence)h(on)368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41592
2411 y(Pr)l(o)o(gr)o(amming)24 b(Concepts)i(and)e(Methods)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41593
Gg(,)g(pages)h(561\226581.)h(North-Holland.)p Black Black
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41594
277 2551 a(Zuck)o(er)l(,)39 b(J.)34 b([1974].)71 b(The)35
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41595
b(Correspondence)k(Between)c(Cut-Elimination)i(and)e(Normalisation.)368
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41596
2664 y F7(Annals)24 b(of)g(Mathematical)h(Lo)o(gic)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41597
Gg(,)f(7:1\226112.)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41598
%%Page: 176 188
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41599
TeXDict begin 176 187 bop Black Black Black 321 875 a
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41600
Gf(Index)p Black 321 1251 a Gb(Reductions)342 1327 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41601
Gc(aux)321 1365 y F6(\000)-31 b(\000)g(!)p Gg(,)22 b(32)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41602
342 1403 119 3 v 342 1441 a Gc(aux)321 1478 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41603
b(\000)g(!)p Gg(,)22 b(103)p 336 1517 V 336 1554 a Gc(aux)466
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41604
1531 y FC(0)321 1592 y F6(\000)-31 b(\000)g(!)p Gg(,)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41605
b(106)385 1668 y Gc(\014)321 1705 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41606
Gg(,)22 b(72,)h(89)391 1781 y Gc(c)321 1819 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41607
b(\000)g(!)p Gg(,)22 b(29)380 1895 y Gc(c)411 1871 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41608
FC(0)321 1932 y F6(\000)-31 b(\000)g(!)p Gg(,)22 b(32)371
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41609
2008 y Gc(c)402 1985 y FC(00)321 2046 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41610
Gg(,)22 b(49)362 2122 y Gc(c)393 2098 y FC(000)321 2159
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41611
y F6(\000)-31 b(\000)g(!)p Gg(,)22 b(104)352 2235 y Gc(cut)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41612
2272 y F6(\000)-31 b(\000)g(!)p Gg(,)22 b(28,)h(43)386
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41613
2349 y Gc(\015)321 2386 y F6(\000)-31 b(\000)g(!)p Gg(,)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41614
b(72,)h(90)373 2462 y Gc(g)r(c)321 2499 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41615
b(\000)g(!)p Gg(,)22 b(49)360 2576 y Gc(int)321 2613
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41616
y F6(\000)-31 b(\000)g(!)p Gg(,)22 b(47)394 2689 y Gc(\023)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41617
2726 y F6(\000)-31 b(\000)g(!)p Gg(,)22 b(73,)h(93)386
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41618
2803 y Gc(\024)321 2840 y F6(\000)-31 b(\000)g(!)p Gg(,)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41619
b(93)396 2916 y Gc(l)321 2953 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41620
Gg(,)22 b(29)384 3030 y Gc(l)406 3006 y FC(0)321 3067
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41621
y F6(\000)-31 b(\000)g(!)p Gg(,)22 b(49)375 3143 y Gc(l)397
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41622
3120 y FC(00)321 3180 y F6(\000)-31 b(\000)g(!)p Gg(,)22
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41623
b(104)363 3257 y Gc(l)q(oc)321 3294 y F6(\000)-31 b(\000)g(!)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41624
Gg(,)22 b(51)385 3370 y Gc(\033)321 3407 y F6(\000)-31
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41625
b(\000)g(!)p Gg(,)22 b(93)334 3484 y Gc(sy)r(m)321 3521
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41626
y F6(\000)-31 b(\000)g(!)p Gg(,)22 b(62)387 3597 y Gc(x)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41627
3634 y F6(\000)-31 b(\000)g(!)p Gg(,)22 b(50)321 3841
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41628
y Gb(T)-7 b(ranslations)321 3954 y F6(j)p 348 3954 28
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41629
4 v 366 3954 V 384 3954 V 65 w(j)436 3921 y Gc(\024)481
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41630
3954 y Gg(,)23 b(99)321 4068 y F6(j)p 348 4068 V 366
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41631
4068 V 384 4068 V 65 w(j)436 4035 y Gc(\026)483 4068
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41632
y Gg(,)f(99)321 4181 y F6(j)p 348 4181 V 366 4181 V 384
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41633
4181 V 65 w(j)436 4148 y Fu(n)477 4181 y Gg(,)g(71)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41634
4295 y F6(j)p 348 4295 V 366 4295 V 384 4295 V 65 w(j)436
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41635
4262 y Fu(N)490 4295 y Gg(,)h(87)321 4408 y F6(j)p 348
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41636
4408 V 366 4408 V 384 4408 V 65 w(j)436 4375 y Fu(s)468
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41637
4408 y Gg(,)f(71)321 4522 y F6(j)p 348 4522 V 366 4522
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41638
V 384 4522 V 65 w(j)436 4489 y Fu(s)p FV(+)514 4522 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41639
Gg(,)h(120)321 4635 y F6(j)p 348 4635 V 366 4635 V 384
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41640
4635 V 65 w(j)436 4602 y Fu(S)480 4635 y Gg(,)f(86)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41641
4749 y F6(j)p 348 4749 V 366 4749 V 384 4749 V 65 w(j)436
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41642
4716 y Fu(Sym)568 4749 y Gg(,)h(63)321 4862 y F6(j)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41643
348 4862 V 366 4862 V 384 4862 V 65 w(j)436 4876 y Gc(x)480
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41644
4862 y Gg(,)g(53)321 5068 y Gb(T)-8 b(erm)23 b(Calculi)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41645
5182 y FY(H)q Gg(,)e(55)321 5295 y FY(I)p Gg(,)i(47)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41646
5409 y F4(K)p Gg(,)f(84)p Black Black 2079 1251 a F4(\003)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41647
Gg(,)h(69)2079 1364 y F4(\003)2142 1378 y F9(+)2201 1364
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41648
y Gg(,)g(120)2079 1477 y FY(R)p Gg(,)f(21)2079 1590 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41649
F4(R)p Gg(,)g(84)2079 1703 y FY(T)t Gg(,)d(22)2079 1816
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41650
y FY(T)2136 1834 y F9(\()p Gc(B)s F9(\))2248 1816 y Gg(,)j(23)2079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41651
1929 y FY(T)2136 1947 y FX(h)p Gc(B)s FX(i)2248 1929
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41652
y Gg(,)g(23)2079 2041 y FY(T)2140 2008 y FX($)2211 2041
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41653
y Gg(,)g(49)2079 2154 y FY(T)2140 2121 y FX($)2136 2177
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41654
y Gc(<)p FX(1)2262 2154 y Gg(,)h(55)2079 2347 y Gb(Symbols)2079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41655
2460 y Ga(>)2150 2427 y Gc(r)r(po)2257 2460 y Gg(,)g(54)2081
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41656
2573 y FW(M)t(A)t(X)t(R)t(E)t(D)r Gg(,)h(12)2079 2765
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41657
y Gb(A)2079 2878 y Gg(Abramsk)o(y)-6 b(,)24 b(5,)f(21)2079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41658
2991 y(antecedent,)j(10)2079 3184 y Gb(B)2079 3297 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41659
Gg(Barbanera,)i(6,)e(8,)g(9,)h(19,)f(31,)h(61,)f(114,)h(123,)2411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41660
3410 y(125,)d(126)2079 3522 y(Barendre)o(gt,)h(21,)e(24,)h(78)2079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41661
3635 y(Barendre)o(gt-style)f(naming)c(con)l(v)o(ention,)k(24,)2411
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41662
3748 y(30)2079 3861 y(Benton,)h(2,)f(126)2079 3974 y(Bierman,)h(5,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41663
(98,)g(99,)g(122)2079 4087 y(Bloo,)g(51,)h(54,)f(125)2079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41664
4200 y(Bori)5 b(\020)-35 b(ci)5 b(\264)-35 b(c,)24 b(65,)g(81,)f(82,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41665
(97)2079 4393 y Gb(C)2079 4505 y Gg(co-name,)i(19,)e(22,)g(84)2079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41666
4618 y(conte)o(xt,)i(19)2245 4731 y(-con)l(v)o(ention,)i(19)2079
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41667
4844 y(Coquand,)e(114,)f(126,)f(127)2079 4957 y(Curry-Ho)n(w)o(ard)i
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41668
(correspondence,)j(1)2079 5070 y(cut)2245 5183 y(commuting,)d(12)2245
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41669
5296 y(labelled,)g(48)2245 5409 y(logical,)g(12,)e(26,)h(43,)f(45)p
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41670
Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41671
%%Page: 177 189
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41672
TeXDict begin 177 188 bop Black 277 51 a Gb(Index)3332
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41673
b(177)p 277 88 3691 4 v Black 277 412 a Gg(cut-elimination,)27
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41674
b(12,)c(28,)h(32,)f(43,)h(47,)f(51)277 612 y Gb(D)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41675
725 y Gg(Danos,)h(4,)f(5,)g(9,)g(15,)g(16,)g(121\226123)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41676
838 y(Dragalin,)h(6,)f(9,)g(15,)g(59,)h(121)277 952 y(Dyckhof)n(f,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41677
(16)277 1152 y Gb(E)277 1265 y Gg(e)o(xpression,)h(41)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41678
1465 y Gb(F)277 1579 y Gg(formula)443 1692 y(coloured,)g(16)443
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41679
1805 y(cut-,)e(11)443 1918 y(main,)f(10,)h(22)277 2118
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41680
y Gb(G)277 2231 y Gg(G3a,)f(19)277 2345 y(G3c,)g(19)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41681
2458 y(Gallier)l(,)h(15,)g(24,)f(41,)g(126)277 2571 y(Gentzen,)h(3,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41682
(10,)h(24,)f(65,)g(66,)h(74,)f(81,)h(121)277 2684 y(Gentzen-Sch)8
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41683
b(\250)-38 b(utte)27 b(system,)d(46)277 2797 y(Girard,)33
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41684
b(2,)e(4\2266,)i(14,)f(17,)h(24,)f(31,)g(72,)g(126,)609
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41685
2910 y(128)277 3023 y(Grif)n(\002n,)23 b(2,)g(127)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41686
3224 y Gb(H)277 3337 y Gg(Hauptsatz,)i(9)277 3450 y(Herbelin,)g(15,)e
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41687
(16,)g(78,)h(114,)g(126,)f(127)277 3563 y(Hyland,)h(126,)g(128)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41688
3763 y Gb(I)277 3877 y Gg(implicit)h(contractions,)i(19)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41689
3990 y(inconsistenc)o(y)-6 b(,)27 b F7(see)d Gg(Lafont')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41690
b(s)25 b(e)o(xample)277 4103 y(inference)h(rule,)e F7(see)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41691
Gg(cut)443 4216 y(logical,)h(3,)e(10,)g(44)443 4329 y(structural,)j(3,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41692
d(11)277 4529 y Gb(K)277 4643 y Gg(Kleene,)h(19,)f(43,)h(111,)g(128)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41693
4756 y(Kreisel,)g(4,)f(73,)g(122)277 4956 y Gb(L)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41694
5069 y Ga(\025\026)p Gg(,)f(5,)h(97\226100)277 5182 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41695
Ga(\025)330 5146 y Gc(S)t(y)r(m)481 5182 y Gg(,)f(61)277
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41696
5296 y Ga(\025)p F1(x)p Gg(,)g(48,)i(78)277 5409 y(Lafont')-5
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41697
b(s)25 b(e)o(xample,)f(4)p Black Black 2035 412 a(LJ,)e(11)2035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41698
525 y(LK,)g(9)2035 638 y(LK)2157 605 y Gc(tq)2220 638
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41699
y Gg(,)g(16,)h(122)2035 831 y Gb(M)2035 944 y Gg(maximal)h(se)o(gment,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41700
g(88)2035 1057 y(Melli)5 b(\036)-35 b(es,)24 b(48,)f(52)2035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41701
1249 y Gb(N)2035 1362 y Gg(name,)g(19,)h(22)2035 1475
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41702
y(NJ,)e(67)2035 1588 y(NK,)g(81)2035 1701 y(normal)i(form,)f(12)2201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41703
1814 y Ga(x)p Gg(-,)g(53)2201 1927 y(mix-,)g(106)2035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41704
2040 y(normalisation,)j(73,)e(93)2035 2233 y Gb(P)2035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41705
2346 y Gg(P)o(arigot,)g(5,)f(7,)f(65,)i(82,)f(95,)h(97,)f(99)2035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41706
2459 y(Pitts,)g(24,)g(127)2035 2572 y(Pottinger)l(,)i(3,)e(21,)g(73,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41707
(74)2035 2684 y(Pra)o(witz,)f(3,)g(38,)g(65,)g(66,)h(73,)f(77,)g(81)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41708
2035 2797 y(proof)i(substitution,)h(25,)e(27,)f(57,)g(91,)h(92)2201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41709
2910 y(auxiliary)-6 b(,)26 b(33,)d(58)2035 3023 y(pure-v)n(ariable)k
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41710
(con)l(v)o(ention,)f(43)2035 3216 y Gb(R)2035 3329 y
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41711
Gg(recursi)n(v)o(e)f(path)f(ordering,)i(54)2035 3442
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41712
y(rede)o(x,)e(12)2035 3555 y(reducibility)j(candidate,)f(36)2035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41713
3668 y(reduction)g(system,)e(12)2035 3861 y Gb(S)2035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41714
3974 y Gg(Schellinx,)h(5,)e(17)2035 4087 y(Schwichtenber)n(g,)j(6,)d
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41715
(19,)h(43,)f(128)2035 4200 y(sequent,)i(10,)e(19)2035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41716
4312 y(strong)k(normalisation,)h(12,)e(31,)f(40,)h(41,)g(44,)2367
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41717
4425 y(47,)d(56,)h(57,)f(77,)g(94)2035 4538 y(subformula)j(property)-6
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41718
b(,)25 b(11)2035 4651 y(subject)g(reduction,)h(30,)d(51,)h(73,)f(93)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41719
2035 4764 y(substitution,)k F7(see)d Gg(proof)g(substitution)2201
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41720
4877 y(safe,)g(38)2201 4990 y(simultaneous,)i(39)2035
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41721
5103 y(succedent,)g(10)2035 5296 y Gb(T)2035 5409 y Gg(term,)d(21,)g
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41722
(42,)h(69)p Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41723
%%Page: 178 190
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41724
TeXDict begin 178 189 bop Black -144 51 a Gb(178)3333
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41725
b(Index)p -144 88 3691 4 v Black 487 412 a Gg(co-named,)25
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41726
b(23)487 525 y(labelled,)h(49)487 638 y(named,)e(23)487
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41727
751 y(well-typed,)i(22)321 864 y(typing)f(judgement,)h(22,)d(48,)g(69,)
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41728
h(84)321 1059 y Gb(U)321 1172 y Gg(Ungar)l(,)g(7,)f(23,)g(75,)g(95,)h
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41729
(97)321 1368 y Gb(V)321 1481 y Gg(v)n(ariable,)h(41,)f(66)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41730
1677 y Gb(W)321 1790 y Gg(weak)g(normalisation,)i(14)321
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41731
1986 y Gb(Z)321 2099 y Gg(Zuck)o(er)l(,)f(3,)e(60,)g(73,)g(74,)h(95,)f
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41732
(126)p Black Black Black Black eop end
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41733
%%Trailer
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41734
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41735
userdict /end-hook known{end-hook}if
Christian Urban <urbanc@in.tum.de>
parents:
diff changeset
 41736
%%EOF