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1911 78001F80001FE000070000078000292679A42F>I<00007F000003FFC0000FC1F0001F00 |
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1912 F8007E007C00FC007C01F8007E03F0003E07F0003F07E0003F0FE0003F1FC0003F1FC000 |
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1913 3F3F80003F3F80007F7F80007F7F00007F7F00007F7F0000FFFF0000FFFE0000FEFE0000 |
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1914 FEFE0001FEFE0001FCFC0001FCFC0003F8FC0003F8FC0007F0FC0007E0FC000FE07C000F |
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1915 C07E001F803E003F003E007E001F00F8000F83F00003FFC00000FE0000202677A42A>I< |
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1916 00078007C000001FE03FF000003CF0787C000038F8E03E0000787FC03E0000707F801F00 |
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1917 0070FF001F0000F0FE001F8000E0FE001F8000E0FC001F8001E1FC001F8001C1FC001F80 |
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1918 01C1F8001F8001C1F8001F800003F8003F800003F8003F800003F0003F800003F0003F80 |
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1919 0007F0007F800007F0007F800007E0007F000007E0007F00000FE000FF00000FE000FE00 |
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1920 000FC000FE00000FC001FC00001FC001FC00001FC001F800001F8003F800001F8003F000 |
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1921 003F8007E000003FC00FC000003FC00F8000003FE01F0000007FE03E0000007F70FC0000 |
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1922 007E3FF00000007E0F80000000FE0000000000FE0000000000FC0000000000FC00000000 |
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1923 01FC0000000001FC0000000001F80000000001F80000000003F80000000003F800000000 |
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1924 03F00000000007F000000000FFFFC0000000FFFFC0000000FFFFC0000000293580A42A> |
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1925 I<0000F8030007FE07001F871F003E03BF007C03FE00F801FE01F801FE03F000FE07E000 |
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1926 FC0FE000FC0FC000FC1FC001FC1FC001F83F8001F83F8001F87F8003F87F0003F07F0003 |
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1927 F07F0003F0FF0007F0FE0007E0FE0007E0FE0007E0FE000FE0FC000FC0FC000FC0FC000F |
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1928 C0FC001FC0FC001F80FC003F80FC003F807C007F807C00FF003E01FF003E03BF001F0F7F |
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1929 0007FE7E0001F07E0000007E000000FE000000FC000000FC000000FC000001FC000001F8 |
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1930 000001F8000001F8000003F8000003F0000007F00001FFFFC001FFFFC001FFFFC0203577 |
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1931 A426>I<03C003F0000FF01FFC001E783C0F001C7C700F003C3EE03F80383FC03F80387F |
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1932 803F80787F803F00707F003F00707F001C00F07E000000E0FE000000E0FC000000E0FC00 |
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1933 000000FC00000001FC00000001F800000001F800000001F800000003F800000003F00000 |
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1934 0003F000000003F000000007F000000007E000000007E000000007E00000000FE0000000 |
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1935 0FC00000000FC00000000FC00000001FC00000001F800000001F800000001F800000003F |
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1936 800000003F000000000E00000000212679A423>I<0000FE000007FF80000F83C0003E00 |
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1937 E0007C00F00078007000F800F001F803F001F003F001F003F001F003E003F001C003F800 |
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1938 0003FC000003FFC00001FFF80001FFFE0000FFFF00007FFF80003FFF80000FFFC00000FF |
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1939 C000003FC000001FC000000FC03E000FC07E000FC0FE000F80FE000F80FE000F80FC001F |
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1940 00E0001F00F0003E0070007C003800F8001E03E0000FFFC00001FE00001C267AA422>I< |
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1941 0003800007C0000FC0000FC0000FC0000FC0001FC0001F80001F80001F80003F80003F00 |
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1942 003F00003F00007F00007E007FFFFF7FFFFFFFFFFF00FC0000FC0000FC0001FC0001F800 |
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1943 01F80001F80003F80003F00003F00003F00007F00007E00007E00007E0000FE0000FC000 |
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1944 0FC0000FC0001FC0001F801C1F801C1F803C3F80383F00783F00703F00F03F00E03F01C0 |
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1945 3E03C01F07800F0F0007FC0001F000183579B31C>I<00F800000003FE0001C0078F0003 |
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1946 E00E0F8007E01E0F8007E01C0F8007E0380F800FE0381F800FE0781F800FC0701F800FC0 |
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1947 703F801FC0F03F001FC0E03F001F80E07F001F80007E003F80007E003F8000FE003F0000 |
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1948 FC003F0000FC007F0001FC007F0001F8007E0001F8007E0001F800FE0003F800FE0003F0 |
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1949 00FC0E03F000FC0E03F001FC1E03F001FC1C03F001F81C03E001F81C03E001F83C03F003 |
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1950 F83803F007F83801F007F07001F01EF8F000F83C78E0003FF03FC0000FC00F00272679A4 |
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1951 2D>I<00F0000E0003FC003F00071E007F800E1F007F801C1F007F803C1F007F80381F00 |
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1952 3F80383F001F80703F000F80703F000F80707F000F80F07E000F00E07E000700E0FE0007 |
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1953 0000FC000F0000FC000E0001FC000E0001F8000E0001F8001E0003F8001C0003F0001C00 |
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1954 03F0001C0003F0003C0007F000380007E000380007E000700007E000700007E000700007 |
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1955 E000E00007E000E00007E001C00007E003C00003E003800003F007000001F00E000000F8 |
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1956 3C0000007FF80000000FC00000212679A426>I<00F0000000070003FC0003801F80071E |
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1957 0007C03FC00E1F000FC03FC01C1F000FC03FC03C1F000FC03FC0381F001FC01FC0383F00 |
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1958 1F800FC0703F001F8007C0703F001F8007C0707F003F8007C0F07E003F800780E07E003F |
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1959 000380E0FE003F00038000FC007F00078000FC007F00070001FC007E00070001F8007E00 |
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1960 070001F800FE000F0003F800FE000E0003F000FC000E0003F000FC000E0003F000FC001E |
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1961 0007F001FC001C0007E001F8001C0007E001F8003C0007E001F800380007E001F8003800 |
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1962 07E001F800780007E001F800700007E003F800F00007E003F800E00003E007F801E00003 |
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1963 F00FFC03C00001F01E7C07800000FC3C3E0F0000003FF81FFE0000000FE003F000003226 |
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1964 79A437>I<0007E007C0001FF81FF800787C383C00F03E703C01E01EE0FE03C01FE0FE03 |
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1965 801FC0FE07001FC0FC0F001F80FC0E001F80700E003F80001E003F00001C003F00001C00 |
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1966 3F000000007F000000007F000000007E000000007E00000000FE00000000FE00000000FC |
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1967 00000000FC00000001FC00000001FC00000001F800700001F800700003F800F00003F800 |
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1968 E01C03F000E07E03F001E07E07F001C0FE07F00380FE0FF00780FE0EF00F00781CF81E00 |
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1969 78387C3C003FF03FF00007C00FC00027267CA427>I<00F000000003FC0001C0071E0003 |
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1970 E00E1F0007E01C1F0007E03C1F0007E0381F000FE0383F000FC0703F000FC0703F000FC0 |
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1971 707F001FC0F07E001F80E07E001F80E0FE001F8000FC003F8000FC003F0001FC003F0001 |
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1972 F8003F0001F8007F0003F8007E0003F0007E0003F0007E0003F000FE0007F000FC0007E0 |
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1973 00FC0007E000FC0007E001FC0007E001F80007E001F80007E001F80007E003F80007E003 |
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1974 F00007E007F00003E00FF00003F01FF00001F87FE000007FF7E000001FC7E00000000FE0 |
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1975 0000000FC00000000FC00000001FC0003F001F80007F003F80007F003F00007F007E0000 |
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1976 7F007C00007E00FC00007001F800007003E000003807C000003C1F8000000FFE00000003 |
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1977 F0000000233679A428>I E |
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1978 %EndDVIPSBitmapFont |
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1979 %DVIPSBitmapFont: Ft cmbx10 10 46 |
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1980 /Ft 46 122 df<000003FFC00000007FFFF0000003FFFFFC00000FFF00FE00003FF8003F |
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1981 00007FE000FF8000FFC001FF8001FFC001FF8001FF8001FF8003FF8003FFC003FF0001FF |
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1982 8003FF0001FF8003FF0001FF8003FF0000FF0003FF00007E0003FF0000000003FF000000 |
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1983 0003FF0000000003FF0000000003FF0000000003FF003FFFC0FFFFFFFFFFC0FFFFFFFFFF |
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1984 C0FFFFFFFFFFC0FFFFFFFFFFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FF |
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1985 C003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FF |
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1986 C003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FF |
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1987 C003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FF |
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1988 C003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FFC003FF0000FF |
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1989 C003FF0000FFC0FFFFFC3FFFFFFFFFFC3FFFFFFFFFFC3FFFFFFFFFFC3FFFFF303A7EB935 |
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1990 >12 D<00001C00003C0000F80001F00003E00007E0000FC0001F80001F00003F00007E00 |
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1991 00FE0000FC0001FC0001F80003F80003F80007F00007F0000FF0000FE0001FE0001FE000 |
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1992 1FC0003FC0003FC0003FC0003FC0007F80007F80007F80007F80007F8000FF8000FF8000 |
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1993 FF8000FF8000FF8000FF8000FF8000FF8000FF8000FF8000FF8000FF8000FF8000FF8000 |
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1994 FF8000FF8000FF80007F80007F80007F80007F80007F80003FC0003FC0003FC0003FC000 |
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1995 1FC0001FE0001FE0000FE0000FF00007F00007F00003F80003F80001F80001FC0000FC00 |
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1996 00FE00007E00003F00001F00001F80000FC00007E00003E00001F00000F800003C00001C |
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1997 165377BD25>40 D<E00000F000007C00003E00001F00001F80000FC00007E00003E00003 |
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1998 F00001F80001FC0000FC0000FE00007E00007F00007F00003F80003F80003FC0001FC000 |
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1999 1FE0001FE0000FE0000FF0000FF0000FF0000FF00007F80007F80007F80007F80007F800 |
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2000 07FC0007FC0007FC0007FC0007FC0007FC0007FC0007FC0007FC0007FC0007FC0007FC00 |
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2001 07FC0007FC0007FC0007FC0007FC0007F80007F80007F80007F80007F8000FF0000FF000 |
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2002 0FF0000FF0000FE0001FE0001FE0001FC0003FC0003F80003F80007F00007F00007E0000 |
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2003 FE0000FC0001FC0001F80003F00003E00007E0000FC0001F80001F00003E00007C0000F0 |
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2004 0000E0000016537BBD25>I<0F803FE07FF07FF0FFF8FFF8FFF8FFF8FFF87FF07FF03FE0 |
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2005 0F800D0D798C1B>46 D<00001E000000003E00000000FE00000007FE0000003FFE0000FF |
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2006 FFFE0000FFFFFE0000FFFFFE0000FFCFFE0000000FFE0000000FFE0000000FFE0000000F |
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2007 FE0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE |
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2008 0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE00 |
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2009 00000FFE0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE0000 |
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2010 000FFE0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE000000 |
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2011 0FFE0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE0000000FFE0000000F |
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2012 FE0000000FFE0000000FFE00007FFFFFFF807FFFFFFF807FFFFFFF807FFFFFFF80213779 |
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2013 B630>49 D<000FFC0000007FFFC00001FFFFF00007FFFFFC000FF03FFF001F800FFF803F |
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2014 0003FFC07F0003FFC07FC001FFE0FFE000FFF0FFF000FFF0FFF000FFF0FFF0007FF8FFF0 |
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2015 007FF8FFF0007FF8FFF0007FF87FE0007FF83FC0007FF80F00007FF80000007FF8000000 |
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2016 FFF0000000FFF0000000FFE0000001FFE0000001FFC0000003FF80000003FF00000007FE |
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2017 00000007FC0000000FF80000001FF00000003FE00000007F800000007F00000000FE0000 |
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2018 0001F800000003F000780007E00078000FC00078001F800078003E0000F8007C0000F000 |
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2019 F80000F001F00001F003FFFFFFF003FFFFFFF007FFFFFFF00FFFFFFFF01FFFFFFFF03FFF |
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2020 FFFFF07FFFFFFFE0FFFFFFFFE0FFFFFFFFE0FFFFFFFFE0FFFFFFFFE025377BB630>I<00 |
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2021 03FF0000001FFFF000007FFFFC0001FC07FF0003F003FF8007E001FFC007C000FFE00FF0 |
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2022 00FFE00FFC00FFF01FFC00FFF01FFE00FFF01FFE00FFF01FFE00FFF01FFE00FFF01FFE00 |
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2023 FFF00FFC00FFE00FFC00FFE003F001FFE0000001FFC0000003FF80000003FF00000007FE |
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2024 0000000FFC0000003FF000000FFFC000000FFF0000000FFFF000000007FE00000001FF80 |
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2025 000000FFC0000000FFE00000007FF00000007FF80000007FFC0000003FFC0000003FFC00 |
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2026 00003FFE1FC0003FFE3FE0003FFE7FF0003FFEFFF8003FFEFFF8003FFEFFF8003FFEFFF8 |
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2027 003FFEFFF8003FFCFFF8003FFCFFF0007FF87FF0007FF87FC000FFF03F8000FFE01FC001 |
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2028 FFC00FFC07FF8007FFFFFF0001FFFFFC00007FFFF0000007FF000027387CB630>I<0000 |
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2029 0007C0000000000FC0000000000FC0000000001FC0000000003FC0000000007FC0000000 |
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2030 00FFC000000000FFC000000001FFC000000003FFC000000007FFC00000000FFFC0000000 |
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2031 0FFFC00000001EFFC00000003CFFC00000007CFFC0000000F8FFC0000000F0FFC0000001 |
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2032 E0FFC0000003C0FFC0000007C0FFC000000F80FFC000000F00FFC000001E00FFC000003C |
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2033 00FFC000007C00FFC00000F800FFC00000F000FFC00001E000FFC00003C000FFC00007C0 |
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2034 00FFC0000F8000FFC0000F0000FFC0001E0000FFC0003C0000FFC0007C0000FFC000F800 |
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2035 00FFC000FFFFFFFFFF80FFFFFFFFFF80FFFFFFFFFF80FFFFFFFFFF80000001FFC0000000 |
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2036 01FFC000000001FFC000000001FFC000000001FFC000000001FFC000000001FFC0000000 |
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2037 01FFC000000001FFC000000001FFC0000003FFFFFF800003FFFFFF800003FFFFFF800003 |
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2038 FFFFFF8029377DB630>I<1C000000C01F800007C01FF8007FC01FFFFFFF801FFFFFFF80 |
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2039 1FFFFFFF001FFFFFFE001FFFFFFC001FFFFFF0001FFFFFE0001FFFFF80001FFFFE00001F |
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2040 FFF000001F000000001F000000001F000000001F000000001F000000001F000000001F00 |
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2041 0000001F000000001F07FE00001F3FFFE0001FFFFFF8001FFC07FC001FE003FF001F8001 |
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2042 FF801F0000FFC00E0000FFC0000000FFE00000007FE00000007FF00000007FF00000007F |
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2043 F80000007FF80600007FF83F80007FF87FE0007FF8FFE0007FF8FFF0007FF8FFF0007FF8 |
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2044 FFF0007FF8FFF0007FF8FFE0007FF0FFE0007FF07FC000FFF07F0000FFE07C0000FFC03E |
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2045 0001FFC01F0003FF801FC007FF000FF01FFE0007FFFFF80001FFFFE000007FFF8000000F |
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2046 F8000025387BB630>I<0F803FE07FF07FF0FFF8FFF8FFF8FFF8FFF87FF07FF03FE00F80 |
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2047 000000000000000000000000000000000000000000000F803FE07FF07FF0FFF8FFF8FFF8 |
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2048 FFF8FFF87FF07FF03FE00F800D2579A41B>58 D<00000003E00000000000000007F00000 |
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2049 000000000007F0000000000000000FF8000000000000000FF8000000000000000FF80000 |
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2050 00000000001FFC000000000000001FFC000000000000003FFE000000000000003FFE0000 |
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2051 00000000003FFE000000000000007FFF000000000000007FFF00000000000000FFFF8000 |
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2052 0000000000FFFF80000000000000FFFF80000000000001FFFFC0000000000001F3FFC000 |
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2053 0000000003F3FFE0000000000003E1FFE0000000000003E1FFE0000000000007E1FFF000 |
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2054 0000000007C0FFF000000000000FC0FFF800000000000F807FF800000000000F807FF800 |
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2055 000000001F807FFC00000000001F003FFC00000000003F003FFE00000000003E001FFE00 |
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2056 000000003E001FFE00000000007E001FFF00000000007C000FFF0000000000FC000FFF80 |
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2057 00000000F80007FF8000000000F80007FF8000000001F80007FFC000000001F00003FFC0 |
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2058 00000003FFFFFFFFE000000003FFFFFFFFE000000003FFFFFFFFE000000007FFFFFFFFF0 |
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2059 00000007C00000FFF00000000FC00000FFF80000000F8000007FF80000000F8000007FF8 |
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2060 0000001F8000007FFC0000001F0000003FFC0000003F0000003FFE0000003E0000001FFE |
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2061 0000003E0000001FFE0000007E0000001FFF0000007C0000000FFF000000FE0000000FFF |
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2062 8000FFFFF80007FFFFFF80FFFFF80007FFFFFF80FFFFF80007FFFFFF80FFFFF80007FFFF |
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2063 FF80413A7DB948>65 D<0000003FFC0001C0000003FFFFC003C000001FFFFFF007C00000 |
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2064 FFFFFFFC0FC00003FFFC00FF3FC00007FFC0001FFFC0001FFE00000FFFC0003FFC000003 |
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2065 FFC0007FF0000001FFC000FFE0000000FFC001FFC0000000FFC003FFC00000007FC007FF |
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2066 800000003FC00FFF000000003FC00FFF000000001FC01FFE000000001FC01FFE00000000 |
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2067 0FC03FFE000000000FC03FFC000000000FC03FFC0000000007C07FFC0000000007C07FFC |
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2068 0000000007C07FF8000000000000FFF8000000000000FFF8000000000000FFF800000000 |
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2069 0000FFF8000000000000FFF8000000000000FFF8000000000000FFF8000000000000FFF8 |
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2070 000000000000FFF8000000000000FFF8000000000000FFF8000000000000FFF800000000 |
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2071 0000FFF80000000000007FF80000000000007FFC0000000000007FFC0000000003C03FFC |
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2072 0000000003C03FFC0000000003C03FFE0000000003C01FFE0000000007C01FFE00000000 |
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2073 07C00FFF0000000007800FFF000000000F8007FF800000000F0003FFC00000001F0001FF |
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2074 C00000003E0000FFE00000007E00007FF8000000FC00003FFC000001F800001FFF000003 |
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2075 F0000007FFC0000FE0000003FFFC00FF80000000FFFFFFFE000000001FFFFFF800000000 |
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2076 03FFFFE000000000003FFE0000003A3B7BB945>67 D<FFFFFFFFFF800000FFFFFFFFFFF8 |
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2077 0000FFFFFFFFFFFF0000FFFFFFFFFFFFC000007FF80007FFF000007FF800007FF800007F |
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2078 F800001FFE00007FF800000FFF00007FF8000003FF80007FF8000001FFC0007FF8000001 |
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2079 FFC0007FF8000000FFE0007FF80000007FF0007FF80000007FF0007FF80000003FF8007F |
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2080 F80000003FF8007FF80000003FFC007FF80000001FFC007FF80000001FFC007FF8000000 |
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2081 1FFE007FF80000001FFE007FF80000001FFE007FF80000001FFE007FF80000001FFF007F |
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2082 F80000001FFF007FF80000001FFF007FF80000001FFF007FF80000001FFF007FF8000000 |
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2083 1FFF007FF80000001FFF007FF80000001FFF007FF80000001FFF007FF80000001FFF007F |
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2084 F80000001FFF007FF80000001FFF007FF80000001FFE007FF80000001FFE007FF8000000 |
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2085 1FFE007FF80000001FFE007FF80000001FFC007FF80000003FFC007FF80000003FFC007F |
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2086 F80000003FF8007FF80000007FF8007FF80000007FF0007FF8000000FFE0007FF8000000 |
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2087 FFE0007FF8000001FFC0007FF8000003FF80007FF8000007FF00007FF800001FFE00007F |
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2088 F800007FFC00007FF80007FFF000FFFFFFFFFFFFC000FFFFFFFFFFFF0000FFFFFFFFFFFC |
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2089 0000FFFFFFFFFF80000040397DB849>I<FFFFFFFFFFFFF0FFFFFFFFFFFFF0FFFFFFFFFF |
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2090 FFF0FFFFFFFFFFFFF0007FF80000FFF0007FF800001FF8007FF8000007F8007FF8000003 |
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2091 F8007FF8000001F8007FF8000001F8007FF8000000F8007FF8000000F8007FF800000078 |
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2092 007FF800000078007FF800000078007FF8003C007C007FF8003C003C007FF8003C003C00 |
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2093 7FF8003C003C007FF8003C003C007FF8003C0000007FF8007C0000007FF8007C0000007F |
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2094 F800FC0000007FF803FC0000007FFFFFFC0000007FFFFFFC0000007FFFFFFC0000007FFF |
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2095 FFFC0000007FF803FC0000007FF800FC0000007FF8007C0000007FF8007C0000007FF800 |
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2096 3C0000007FF8003C000F007FF8003C000F007FF8003C000F007FF8003C001E007FF8003C |
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2097 001E007FF80000001E007FF80000001E007FF80000003E007FF80000003E007FF8000000 |
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2098 3C007FF80000007C007FF80000007C007FF8000000FC007FF8000000FC007FF8000001F8 |
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2099 007FF8000003F8007FF800000FF8007FF800003FF8007FF80001FFF8FFFFFFFFFFFFF8FF |
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2100 FFFFFFFFFFF0FFFFFFFFFFFFF0FFFFFFFFFFFFF038397DB83F>I<FFFFFFFFFFFFC0FFFF |
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2101 FFFFFFFFC0FFFFFFFFFFFFC0FFFFFFFFFFFFC0007FF80003FFC0007FF800007FE0007FF8 |
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2102 00001FE0007FF800000FE0007FF8000007E0007FF8000007E0007FF8000003E0007FF800 |
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2103 0003E0007FF8000001E0007FF8000001E0007FF8000001E0007FF8000001F0007FF80078 |
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2104 00F0007FF8007800F0007FF8007800F0007FF8007800F0007FF800780000007FF8007800 |
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2105 00007FF800F80000007FF800F80000007FF801F80000007FF807F80000007FFFFFF80000 |
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2106 007FFFFFF80000007FFFFFF80000007FFFFFF80000007FF807F80000007FF801F8000000 |
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2107 7FF800F80000007FF800F80000007FF800780000007FF800780000007FF800780000007F |
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2108 F800780000007FF800780000007FF800780000007FF800000000007FF800000000007FF8 |
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2109 00000000007FF800000000007FF800000000007FF800000000007FF800000000007FF800 |
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2110 000000007FF800000000007FF800000000007FF800000000007FF800000000007FF80000 |
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2111 0000FFFFFFFF000000FFFFFFFF000000FFFFFFFF000000FFFFFFFF00000034397DB83C> |
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2112 I<FFFFFFFF000000FFFFFFFF000000FFFFFFFF000000FFFFFFFF000000007FF800000000 |
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2113 007FF800000000007FF800000000007FF800000000007FF800000000007FF80000000000 |
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2114 7FF800000000007FF800000000007FF800000000007FF800000000007FF800000000007F |
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2115 F800000000007FF800000000007FF800000000007FF800000000007FF800000000007FF8 |
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2116 00000000007FF800000000007FF800000000007FF800000000007FF800000000007FF800 |
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2117 000000007FF800000000007FF800000000007FF800000000007FF800000000007FF80000 |
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2118 0000007FF800000000007FF800000000007FF800000000007FF800000780007FF8000007 |
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2119 80007FF800000780007FF800000780007FF800000780007FF800000F80007FF800000F00 |
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2120 007FF800000F00007FF800000F00007FF800001F00007FF800001F00007FF800003F0000 |
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2121 7FF800003F00007FF800007F00007FF80000FF00007FF80001FF00007FF80003FF00007F |
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2122 F8000FFE00007FF8007FFE00FFFFFFFFFFFE00FFFFFFFFFFFE00FFFFFFFFFFFE00FFFFFF |
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2123 FFFFFE0031397DB839>76 D<000000FFF800000000000FFFFF80000000007FFFFFF00000 |
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2124 0001FFC01FFC00000007FF0007FF0000001FFC0001FFC000003FF000007FE000007FE000 |
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2125 003FF00000FFC000001FF80001FF8000000FFC0003FF8000000FFE0007FF00000007FF00 |
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2126 07FF00000007FF000FFE00000003FF800FFE00000003FF801FFC00000001FFC01FFC0000 |
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2127 0001FFC03FFC00000001FFE03FFC00000001FFE03FFC00000001FFE07FF800000000FFF0 |
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2128 7FF800000000FFF07FF800000000FFF07FF800000000FFF0FFF800000000FFF8FFF80000 |
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2129 0000FFF8FFF800000000FFF8FFF800000000FFF8FFF800000000FFF8FFF800000000FFF8 |
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2130 FFF800000000FFF8FFF800000000FFF8FFF800000000FFF8FFF800000000FFF8FFF80000 |
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2131 0000FFF8FFF800000000FFF87FF800000000FFF07FFC00000001FFF07FFC00000001FFF0 |
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2132 7FFC00000001FFF03FFC00000001FFE03FFC00000001FFE03FFE00000003FFE01FFE0000 |
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2133 0003FFC01FFE00000003FFC00FFF00000007FF8007FF00000007FF0007FF8000000FFF00 |
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2134 03FFC000001FFE0001FFC000001FFC0000FFE000003FF800007FF000007FF000003FFC00 |
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4542 end |
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4543 %%EndProlog |
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4544 %%BeginSetup |
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4545 %%Feature: *Resolution 600dpi |
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4546 TeXDict begin |
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4547 %%PaperSize: A4 |
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4548 end |
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4549 %%EndSetup |
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4550 %%Page: 1 1 |
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4551 TeXDict begin 1 0 bop Black Black Black Black 728 448 |
|
4552 a FA(Barendregt's)46 b(V)-11 b(ariable)46 b(Con)l(v)l(en)l(tion)g(in)f |
|
4553 (Rule)1655 598 y(Inductions)886 886 y Fz(Christian)27 |
|
4554 b(Urban)1480 856 y Fy(1)1517 886 y Fz(,)g(Stefan)h(Berghofer)2177 |
|
4555 856 y Fy(1)2213 886 y Fz(,)g(and)f(Mic)n(hael)g(Norrish)3006 |
|
4556 856 y Fy(2)1555 1029 y Fx(1)1628 1060 y Fw(TU)f(Munic)n(h,)f(German)n |
|
4557 (y)1625 1120 y Fx(2)1698 1152 y Fw(NICT)-6 b(A,)26 b(Australia)p |
|
4558 Black 759 1477 a Fv(Abstract.)p Black 43 w Fw(Inductiv)n(e)c |
|
4559 (de\014nitions)j(and)e(rule)i(inductions)f(are)h(t)n(w)n(o)f(fundamen-) |
|
4560 759 1569 y(tal)f(reasoning)g(to)r(ols)g(in)f(logic)i(and)d(computer)g |
|
4561 (science.)i(When)f(inductiv)n(e)f(de\014ni-)759 1660 |
|
4562 y(tions)j(in)n(v)n(olv)n(e)f(binders,)g(then)f(Barendregt's)j(v)l |
|
4563 (ariable)f(con)n(v)n(en)n(tion)e(is)i(nearly)f(al-)759 |
|
4564 1751 y(w)n(a)n(ys)g(emplo)n(y)n(ed)d(\(explicitly)j(or)f(implicitly\))g |
|
4565 (in)g(order)g(to)h(obtain)f(simple)g(pro)r(ofs.)759 1843 |
|
4566 y(Using)c(this)f(con)n(v)n(en)n(tion,)g(one)g(do)r(es)h(not)f(consider) |
|
4567 g(truly)g(arbitrary)g(b)r(ound)f(names,)759 1934 y(as)39 |
|
4568 b(required)f(b)n(y)f(the)h(rule)g(induction)g(principle,)g(but)g |
|
4569 (rather)g(b)r(ound)f(names)759 2025 y(ab)r(out)21 b(whic)n(h)f(v)l |
|
4570 (arious)g(freshness)i(assumptions)e(are)g(made.)g(Unfortunately)-6 |
|
4571 b(,)20 b(nei-)759 2117 y(ther)34 b(Barendregt)h(nor)f(others)g(giv)n(e) |
|
4572 g(a)g(formal)h(justi\014cation)g(for)g(the)e(v)l(ariable)759 |
|
4573 2208 y(con)n(v)n(en)n(tion,)i(whic)n(h)f(mak)n(es)g(it)h(hard)f(to)h |
|
4574 (formalise)h(suc)n(h)e(pro)r(ofs.)i(In)e(this)h(pa-)759 |
|
4575 2299 y(p)r(er)28 b(w)n(e)h(iden)n(tify)f(conditions)h(an)f(inductiv)n |
|
4576 (e)g(de\014nition)g(has)g(to)h(satisfy)g(so)g(that)759 |
|
4577 2391 y(a)k(form)g(of)h(the)e(v)l(ariable)h(con)n(v)n(en)n(tion)g(can)f |
|
4578 (b)r(e)h(built)g(in)n(to)g(the)f(rule)h(induction)759 |
|
4579 2482 y(principle.)23 b(In)e(practice)h(this)g(means)g(w)n(e)g(come)f |
|
4580 (quite)h(close)h(to)f(the)f(informal)i(rea-)759 2573 |
|
4581 y(soning)h(of)g(\\p)r(encil-and-pap)r(er")f(pro)r(ofs,)i(while)f |
|
4582 (remaining)f(completely)g(formal.)759 2665 y(Our)h(conditions)i(also)g |
|
4583 (rev)n(eal)f(circumstances)g(in)f(whic)n(h)h(Barendregt's)h(v)l |
|
4584 (ariable)759 2756 y(con)n(v)n(en)n(tion)f(is)h(not)g(applicable,)h(and) |
|
4585 e(can)h(ev)n(en)f(lead)h(to)g(fault)n(y)g(reasoning.)523 |
|
4586 3031 y Fu(1)112 b(In)m(tro)s(duction)523 3225 y Fz(In)31 |
|
4587 b(informal)f(pro)r(ofs)g(ab)r(out)h(languages)e(that)i(feature)g(b)r |
|
4588 (ound)g(v)-5 b(ariables,)29 b(one)i(often)g(as-)523 3325 |
|
4589 y(sumes)19 b(\(explicitly)g(or)f(implicitly\))h(a)g(rather)f(con)n(v)n |
|
4590 (enien)n(t)f(con)n(v)n(en)n(tion)g(ab)r(out)i(those)g(b)r(ound)523 |
|
4591 3425 y(v)-5 b(ariables.)26 b(Barendregt's)g(statemen)n(t)i(of)f(the)h |
|
4592 (con)n(v)n(en)n(tion)e(is:)p Black Black 664 3601 a Ft(V)-8 |
|
4593 b(ariable)32 b(Con)m(v)m(en)m(tion)p Fs(:)e(If)f Fr(M)1737 |
|
4594 3613 y Fy(1)1774 3601 y Fr(;)14 b(:)g(:)g(:)g(;)g(M)2040 |
|
4595 3613 y Fq(n)2114 3601 y Fs(o)l(c)l(cur)29 b(in)g(a)h(c)l(ertain)f |
|
4596 (mathematic)l(al)664 3700 y(c)l(ontext)34 b(\(e.g.)h(de\014nition,)h |
|
4597 (pr)l(o)l(of)9 b(\),)36 b(then)e(in)h(these)f(terms)h(al)t(l)g(b)l |
|
4598 (ound)g(variables)664 3800 y(ar)l(e)30 b(chosen)h(to)f(b)l(e)f |
|
4599 (di\013er)l(ent)h(fr)l(om)h(the)e(fr)l(e)l(e)i(variables.)402 |
|
4600 b Fz([2,)27 b(P)n(age)f(26])523 3976 y(The)f(reason)e(for)i(this)g(con) |
|
4601 n(v)n(en)n(tion)e(is)i(that)g(it)h(leads)e(to)h(v)n(ery)f(slic)n(k)g |
|
4602 (informal)g(pro)r(ofs|one)523 4076 y(can)j(a)n(v)n(oid)f(ha)n(ving)h |
|
4603 (to)g(rename)g(b)r(ound)h(v)-5 b(ariables.)648 4167 y(One)29 |
|
4604 b(example)g(of)h(suc)n(h)g(a)f(slic)n(k)g(informal)g(pro)r(of)g(is)h |
|
4605 (giv)n(en)f(in)h([2,)g(P)n(age)e(60],)h(pro)n(ving)523 |
|
4606 4258 y(the)g(substitutivit)n(y)f(prop)r(ert)n(y)f(of)h(the)h |
|
4607 Fp(\000)-14 b(!)1835 4276 y Fo(1)1789 4258 y Fp(\000)g(!)28 |
|
4608 b Fz(\(or)g(\\parallel)e(reduction"\))i(relation,)f(whic)n(h)523 |
|
4609 4350 y(is)h(de\014ned)f(b)n(y)h(the)g(rules:)p 892 4496 |
|
4610 360 4 v 892 4564 a Fn(M)h Fm(\000)-13 b(!)1064 4580 y |
|
4611 Fo(1)1023 4564 y Fm(\000)g(!)22 b Fn(M)1292 4518 y Fw(One)1429 |
|
4612 4526 y Fx(1)2233 4462 y Fn(M)30 b Fm(\000)-13 b(!)2406 |
|
4613 4478 y Fo(1)2365 4462 y Fm(\000)g(!)21 b Fn(M)2592 4430 |
|
4614 y Fl(0)p 1984 4488 880 4 v 1984 4564 a Fk(lam)6 b Fw(\()p |
|
4615 Fn(y)s(:M)i Fw(\))21 b Fm(\000)-13 b(!)2406 4580 y Fo(1)2365 |
|
4616 4564 y Fm(\000)g(!)21 b Fk(lam)6 b Fw(\()p Fn(y)s(:M)2812 |
|
4617 4532 y Fl(0)2834 4564 y Fw(\))2905 4509 y(One)3042 4518 |
|
4618 y Fx(2)664 4698 y Fn(M)30 b Fm(\000)-13 b(!)837 4714 |
|
4619 y Fo(1)796 4698 y Fm(\000)g(!)21 b Fn(M)1023 4666 y Fl(0)1122 |
|
4620 4698 y Fn(N)30 b Fm(\000)-13 b(!)1282 4714 y Fo(1)1241 |
|
4621 4698 y Fm(\000)g(!)21 b Fn(N)1455 4666 y Fl(0)p 586 4724 |
|
4622 971 4 v 586 4800 a Fk(app)5 b Fw(\()p Fn(M)t(;)13 b(N)8 |
|
4623 b Fw(\))22 b Fm(\000)-13 b(!)1040 4816 y Fo(1)999 4800 |
|
4624 y Fm(\000)g(!)21 b Fk(app)5 b Fw(\()p Fn(M)1378 4769 |
|
4625 y Fl(0)1401 4800 y Fn(;)13 b(N)1504 4769 y Fl(0)1527 |
|
4626 4800 y Fw(\))1598 4746 y(One)1735 4754 y Fx(3)2017 4698 |
|
4627 y Fn(M)29 b Fm(\000)-13 b(!)2189 4714 y Fo(1)2149 4698 |
|
4628 y Fm(\000)f(!)22 b Fn(M)2376 4666 y Fl(0)2475 4698 y |
|
4629 Fn(N)30 b Fm(\000)-13 b(!)2635 4714 y Fo(1)2594 4698 |
|
4630 y Fm(\000)g(!)21 b Fn(N)2808 4666 y Fl(0)p 1817 4724 |
|
4631 1215 4 v 1817 4800 a Fk(app)5 b Fw(\()p Fk(lam)g Fw(\()p |
|
4632 Fn(y)s(:M)j Fw(\))p Fn(;)13 b(N)8 b Fw(\))22 b Fm(\000)-13 |
|
4633 b(!)2524 4816 y Fo(1)2483 4800 y Fm(\000)g(!)21 b Fn(M)2710 |
|
4634 4769 y Fl(0)2733 4800 y Fw([)p Fn(y)j Fw(:=)d Fn(N)2987 |
|
4635 4769 y Fl(0)3010 4800 y Fw(])3073 4746 y(One)3210 4754 |
|
4636 y Fx(4)3307 4634 y Fw(\(1\))523 4924 y Fz(The)28 b(substitutivit)n(y)g |
|
4637 (prop)r(ert)n(y)e(states:)p Black Black eop end |
|
4638 %%Page: 2 2 |
|
4639 TeXDict begin 2 1 bop Black 523 232 a Fw(2)237 b(Christian)27 |
|
4640 b(Urban,)e(Stefan)h(Berghofer,)i(and)d(Mic)n(hael)i(Norrish)p |
|
4641 Black 523 448 a Ft(Lemma.)e Fz(If)j Fr(M)k Fp(\000)-15 |
|
4642 b(!)1153 466 y Fo(1)1108 448 y Fp(\000)g(!)24 b Fr(M)1355 |
|
4643 418 y Fj(0)1405 448 y Fz(and)k Fr(N)k Fp(\000)-15 b(!)1739 |
|
4644 466 y Fo(1)1694 448 y Fp(\000)g(!)24 b Fr(N)1927 418 |
|
4645 y Fj(0)1950 448 y Fz(,)j(then)h Fr(M)9 b Fz([)p Fr(x)24 |
|
4646 b Fz(:=)e Fr(N)9 b Fz(])23 b Fp(\000)-14 b(!)2679 466 |
|
4647 y Fo(1)2633 448 y Fp(\000)g(!)23 b Fr(M)2880 418 y Fj(0)2903 |
|
4648 448 y Fz([)p Fr(x)h Fz(:=)e Fr(N)3183 418 y Fj(0)3207 |
|
4649 448 y Fz(].)523 570 y(In)32 b([2],)g(the)g(pro)r(of)f(of)h(this)g |
|
4650 (lemma)g(pro)r(ceeds)f(b)n(y)h(an)f(induction)h(o)n(v)n(er)f(the)h |
|
4651 (de\014nition)g(of)523 670 y Fr(M)g Fp(\000)-15 b(!)709 |
|
4652 687 y Fo(1)664 670 y Fp(\000)g(!)24 b Fr(M)911 639 y |
|
4653 Fj(0)934 670 y Fz(.)j(Though)g(Barendregt)f(do)r(es)h(not)h(ac)n(kno)n |
|
4654 (wledge)d(the)j(fact)f(explicitly)-7 b(,)28 b(there)523 |
|
4655 769 y(are)37 b(t)n(w)n(o)g(places)h(in)g(his)g(pro)r(of)f(where)h(the)g |
|
4656 (v)-5 b(ariable)37 b(con)n(v)n(en)n(tion)g(is)h(used.)g(In)g(case)f(of) |
|
4657 523 869 y(rule)23 b(One)833 881 y Fy(2)870 869 y Fz(,)h(for)f(example,) |
|
4658 g(Barendregt)f(writes)h(\(sligh)n(tly)h(c)n(hanged)e(to)i(conform)e |
|
4659 (with)j(the)523 968 y(syn)n(tax)i(w)n(e)g(shall)g(emplo)n(y)g(for)g |
|
4660 Fr(\025)p Fz(-terms\):)523 1112 y Ft(Case)20 b Fz(One)888 |
|
4661 1124 y Fy(2)925 1112 y Fz(.)43 b Fr(M)32 b Fp(\000)-15 |
|
4662 b(!)1177 1130 y Fo(1)1132 1112 y Fp(\000)g(!)24 b Fr(M)1379 |
|
4663 1082 y Fj(0)1422 1112 y Fz(is)33 b Fs(lam)7 b Fz(\()p |
|
4664 Fr(y)s(:P)12 b Fz(\))23 b Fp(\000)-14 b(!)1942 1130 y |
|
4665 Fo(1)1897 1112 y Fp(\000)f(!)24 b Fs(lam)6 b Fz(\()p |
|
4666 Fr(y)s(:P)2355 1082 y Fj(0)2379 1112 y Fz(\))34 b(and)20 |
|
4667 b(is)g(a)f(direct)h(consequence)523 1212 y(of)28 b Fr(P)36 |
|
4668 b Fp(\000)-14 b(!)781 1230 y Fo(1)735 1212 y Fp(\000)g(!)24 |
|
4669 b Fr(P)958 1182 y Fj(0)981 1212 y Fz(.)k(By)g(induction)h(h)n(yp)r |
|
4670 (othesis)e(one)h(has)g Fr(P)12 b Fz([)p Fr(x)24 b Fz(:=)g |
|
4671 Fr(N)9 b Fz(])23 b Fp(\000)-14 b(!)2711 1230 y Fo(1)2666 |
|
4672 1212 y Fp(\000)f(!)24 b Fr(P)12 b Fz([)p Fr(x)25 b Fz(:=)e |
|
4673 Fr(N)3170 1182 y Fj(0)3193 1212 y Fz(].)29 b(But)523 |
|
4674 1311 y(then)j Fs(lam)7 b Fz(\()p Fr(y)s(:P)12 b Fz([)p |
|
4675 Fr(x)30 b Fz(:=)f Fr(N)9 b Fz(]\))30 b Fp(\000)-14 b(!)1470 |
|
4676 1329 y Fo(1)1424 1311 y Fp(\000)g(!)30 b Fs(lam)7 b Fz(\()p |
|
4677 Fr(y)s(:P)1890 1281 y Fj(0)1913 1311 y Fz([)p Fr(x)30 |
|
4678 b Fz(:=)f Fr(N)2206 1281 y Fj(0)2229 1311 y Fz(]\),)j(i.e.)g |
|
4679 Fr(M)9 b Fz([)p Fr(x)30 b Fz(:=)f Fr(N)9 b Fz(])30 b |
|
4680 Fp(\000)-15 b(!)2986 1329 y Fo(1)2941 1311 y Fp(\000)g(!)30 |
|
4681 b Fr(M)3194 1281 y Fj(0)3217 1311 y Fz([)p Fr(x)g Fz(:=)523 |
|
4682 1411 y Fr(N)599 1381 y Fj(0)622 1411 y Fz(].)2682 b Fp(u)-55 |
|
4683 b(t)523 1546 y Fz(Ho)n(w)n(ev)n(er,)26 b(the)i(last)f(step)h(in)g(this) |
|
4684 f(case)g(only)g(w)n(orks)f(if)i(one)g(kno)n(ws)e(that)1249 |
|
4685 1694 y Fk(lam)6 b Fw(\()p Fn(y)s(:P)11 b Fw([)p Fn(x)20 |
|
4686 b Fw(:=)i Fn(N)8 b Fw(]\))24 b(=)f Fk(lam)5 b Fw(\()p |
|
4687 Fn(y)s(:P)11 b Fw(\)[)p Fn(x)21 b Fw(:=)g Fn(N)8 b Fw(])69 |
|
4688 b(and)1205 1809 y Fk(lam)5 b Fw(\()p Fn(y)s(:P)1484 1777 |
|
4689 y Fl(0)1506 1809 y Fw([)p Fn(x)21 b Fw(:=)g Fn(N)1763 |
|
4690 1777 y Fl(0)1786 1809 y Fw(]\))j(=)f Fk(lam)5 b Fw(\()p |
|
4691 Fn(y)s(:P)2223 1777 y Fl(0)2245 1809 y Fw(\)[)p Fn(x)21 |
|
4692 b Fw(:=)g Fn(N)2532 1777 y Fl(0)2555 1809 y Fw(])523 |
|
4693 1961 y Fz(whic)n(h)31 b(only)g(holds)g(when)g(the)h(b)r(ound)g(v)-5 |
|
4694 b(ariable)30 b Fr(y)k Fz(is)d(not)h(equal)e(to)h Fr(x)p |
|
4695 Fz(,)h(and)f(not)h(free)f(in)523 2060 y Fr(N)40 b Fz(and)30 |
|
4696 b Fr(N)870 2030 y Fj(0)894 2060 y Fz(.)h(These)f(assumptions)g(migh)n |
|
4697 (t)h(b)r(e)g(inferred)g(from)f(the)i(v)-5 b(ariable)29 |
|
4698 b(con)n(v)n(en)n(tion,)523 2160 y(pro)n(vided)36 b(one)g(has)h(a)f |
|
4699 (formal)h(justi\014cation)g(for)f(this)h(con)n(v)n(en)n(tion.)f(Since)h |
|
4700 (one)f(usually)523 2260 y(assumes)29 b(that)i Fr(\025)p |
|
4701 Fz(-terms)f(are)f Fr(\013)p Fz(-equated,)i(one)f(migh)n(t)g(think)h(a)f |
|
4702 (simple)g(justi\014cation)h(for)523 2359 y(the)19 b(v)-5 |
|
4703 b(ariable)17 b(con)n(v)n(en)n(tion)g(is)i(along)e(the)i(lines)f(that)h |
|
4704 (one)f(can)g(alw)n(a)n(ys)f(rename)g(binders)h(with)523 |
|
4705 2459 y(fresh)j(names.)g(This)g(is)g(ho)n(w)n(ev)n(er)f |
|
4706 Fs(not)29 b Fz(su\016cien)n(t)21 b(in)h(the)f(con)n(text)g(of)g |
|
4707 (inductiv)n(e)h(de\014nitions,)523 2559 y(b)r(ecause)33 |
|
4708 b(there)f(rules)h(can)f(ha)n(v)n(e)g(the)h(same)f(v)-5 |
|
4709 b(ariable)32 b(o)r(ccurring)g(b)r(oth)h(in)g(binding)g(and)523 |
|
4710 2658 y(non-binding)24 b(p)r(ositions.)f(In)i(rule)e(One)1760 |
|
4711 2670 y Fy(4)1797 2658 y Fz(,)h(for)g(example,)f Fr(y)k |
|
4712 Fz(o)r(ccurs)c(in)i(binding)f(p)r(osition)g(in)523 2758 |
|
4713 y(the)31 b(subterm)f Fs(lam)7 b Fz(\()p Fr(y)s(:M)i Fz(\),)31 |
|
4714 b(and)f(in)h(the)f(subterm)h Fr(M)2234 2728 y Fj(0)2257 |
|
4715 2758 y Fz([)p Fr(y)f Fz(:=)e Fr(N)2543 2728 y Fj(0)2566 |
|
4716 2758 y Fz(])i(it)h(is)f(in)h(a)f Fs(non)6 b Fz(-binding)523 |
|
4717 2857 y(p)r(osition.)28 b(Both)f(o)r(ccurrences)f(m)n(ust)i(refer)f(to)g |
|
4718 (the)h(same)f(v)-5 b(ariable)27 b(as)f(the)i(rule)1413 |
|
4719 3013 y Fr(M)j Fp(\000)-14 b(!)1599 3031 y Fo(1)1553 3013 |
|
4720 y Fp(\000)g(!)23 b Fr(M)1800 2983 y Fj(0)1906 3013 y |
|
4721 Fr(N)32 b Fp(\000)-14 b(!)2079 3031 y Fo(1)2033 3013 |
|
4722 y Fp(\000)g(!)23 b Fr(N)2266 2983 y Fj(0)p 1195 3040 |
|
4723 1311 4 v 1195 3119 a Fs(app)7 b Fz(\()p Fs(lam)g Fz(\()p |
|
4724 Fr(z)t(:M)i Fz(\))p Fr(;)14 b(N)9 b Fz(\))23 b Fp(\000)-15 |
|
4725 b(!)1959 3137 y Fo(1)1914 3119 y Fp(\000)g(!)24 b Fr(M)2161 |
|
4726 3089 y Fj(0)2184 3119 y Fz([)p Fr(y)h Fz(:=)e Fr(N)2460 |
|
4727 3089 y Fj(0)2483 3119 y Fz(])2548 3064 y(One)2695 3028 |
|
4728 y Fj(0)2695 3084 y Fy(4)523 3287 y Fz(leads)k(to)h(a)f(nonsensical)f |
|
4729 (reduction)h(relation.)648 3378 y(In)e(the)g(absence,)f(ho)n(w)n(ev)n |
|
4730 (er,)f(of)i(a)f(formal)g(justi\014cation)h(for)f(the)i(v)-5 |
|
4731 b(ariable)23 b(con)n(v)n(en)n(tion,)523 3470 y(Barendregt's)h(argumen)n |
|
4732 (t)g(considering)h(only)g(a)g(w)n(ell-c)n(hosen)f Fr(y)29 |
|
4733 b Fz(seems)c(dubious,)g(b)r(ecause)523 3561 y(the)j(induction)g |
|
4734 (principle)g(that)f(comes)g(with)h(the)g(inductiv)n(e)g(de\014nition)g |
|
4735 (of)g Fp(\000)-15 b(!)3030 3579 y Fo(1)2985 3561 y Fp(\000)g(!)28 |
|
4736 b Fz(is:)674 3708 y Fm(8)p Fn(M)t(:)42 b(P)24 b(M)d(M)674 |
|
4737 3823 y Fm(8)p Fn(y)15 b(M)21 b(M)947 3791 y Fl(0)969 |
|
4738 3823 y Fn(:)43 b(P)24 b(M)d(M)1283 3791 y Fl(0)1348 3823 |
|
4739 y Fm(\))43 b Fn(P)23 b Fw(\()p Fk(lam)6 b Fw(\()p Fn(y)s(:M)i |
|
4740 Fw(\)\))k(\()p Fk(lam)5 b Fw(\()p Fn(y)s(:M)2275 3791 |
|
4741 y Fl(0)2298 3823 y Fw(\)\))674 3938 y Fm(8)p Fn(M)20 |
|
4742 b(M)893 3906 y Fl(0)929 3938 y Fn(N)h(N)1080 3906 y Fl(0)1103 |
|
4743 3938 y Fn(:)43 b(P)23 b(M)f(M)1417 3906 y Fl(0)1478 3938 |
|
4744 y Fm(^)38 b Fn(P)24 b(N)d(N)1791 3906 y Fl(0)1856 3938 |
|
4745 y Fm(\))43 b Fn(P)23 b Fw(\()p Fk(app)6 b Fw(\()p Fn(M)t(;)13 |
|
4746 b(N)8 b Fw(\)\))13 b(\()p Fk(app)5 b Fw(\()p Fn(M)2749 |
|
4747 3906 y Fl(0)2771 3938 y Fn(;)13 b(N)2874 3906 y Fl(0)2897 |
|
4748 3938 y Fw(\)\))674 4053 y Fm(8)p Fn(y)i(M)21 b(M)947 |
|
4749 4021 y Fl(0)982 4053 y Fn(N)g(N)1133 4021 y Fl(0)1156 |
|
4750 4053 y Fn(:)43 b(P)24 b(M)d(M)1470 4021 y Fl(0)1531 4053 |
|
4751 y Fm(^)38 b Fn(P)24 b(N)d(N)1844 4021 y Fl(0)1910 4053 |
|
4752 y Fm(\))42 b Fn(P)24 b Fw(\()p Fk(app)5 b Fw(\()p Fk(lam)g |
|
4753 Fw(\()p Fn(y)s(:M)j Fw(\))p Fn(;)13 b(N)8 b Fw(\)\))13 |
|
4754 b(\()p Fn(M)2903 4021 y Fl(0)2926 4053 y Fw([)p Fn(y)24 |
|
4755 b Fw(:=)d Fn(N)3180 4021 y Fl(0)3203 4053 y Fw(]\))p |
|
4756 674 4100 2581 4 v 1591 4168 a Fn(M)30 b Fm(\000)-13 b(!)1764 |
|
4757 4184 y Fo(1)1723 4168 y Fm(\000)g(!)21 b Fn(N)51 b Fm(\))43 |
|
4758 b Fn(P)23 b(M)e(N)523 4327 y Fz(where)38 b(b)r(oth)h(cases)e(One)1348 |
|
4759 4339 y Fy(2)1424 4327 y Fz(and)h(One)1744 4339 y Fy(4)1819 |
|
4760 4327 y Fz(require)g(that)h(the)f(corresp)r(onding)f(implication)523 |
|
4761 4426 y(holds)29 b(for)f Ft(all)g Fr(y)s Fz(,)h(not)g(just)g(the)g(ones) |
|
4762 f(with)i Fr(y)e Fp(6)p Fz(=)c Fr(x)30 b Fz(and)e Fr(y)g |
|
4763 Fp(62)e Fr(F)12 b(V)18 b Fz(\()p Fr(N)t(;)c(N)2823 4396 |
|
4764 y Fj(0)2847 4426 y Fz(\).)29 b(Nev)n(ertheless,)523 4526 |
|
4765 y(w)n(e)e(will)g(sho)n(w)f(that)h(Barendregt's)e(apparen)n(tly)g |
|
4766 (dubious)i(step)g(can)f(b)r(e)i(giv)n(en)e(a)g(faithful,)523 |
|
4767 4625 y(and)32 b(sound,)g(mec)n(hanisation.)f(Being)g(able)h(to)g |
|
4768 (restrict)f(the)i(argumen)n(t)e(in)h(general)e(to)i(a)523 |
|
4769 4725 y(suitably)25 b(c)n(hosen)e(b)r(ound)i(v)-5 b(ariable)24 |
|
4770 b(will,)h(ho)n(w)n(ev)n(er,)e(dep)r(end)i(on)f(the)i(form)e(of)h(the)g |
|
4771 (rules)f(in)523 4825 y(an)h(inductiv)n(e)g(de\014nition.)h(In)f(this)g |
|
4772 (pap)r(er)g(w)n(e)f(will)i(mak)n(e)e(precise)g(what)h(this)h(form)e(is) |
|
4773 h(and)523 4924 y(will)j(sho)n(w)e(ho)n(w)h(the)g(v)-5 |
|
4774 b(ariable)26 b(con)n(v)n(en)n(tion)g(can)h(b)r(e)h(built)g(in)n(to)f |
|
4775 (the)g(induction)h(principle.)p Black Black eop end |
|
4776 %%Page: 3 3 |
|
4777 TeXDict begin 3 2 bop Black 1347 232 a Fw(Barendregt's)27 |
|
4778 b(V)-6 b(ariable)26 b(Con)n(v)n(en)n(tion)f(in)h(Rule)f(Inductions)237 |
|
4779 b(3)p Black 648 448 a Fz(The)27 b(in)n(teractions)f(b)r(et)n(w)n(een)i |
|
4780 (b)r(ound)g(and)f(free)g(o)r(ccurrences)f(of)h(v)-5 b(ariables,)27 |
|
4781 b(and)g(their)523 548 y(consequences)19 b(for)g(obtaining)h(a)f(formal) |
|
4782 h(argumen)n(t,)f(seem)h(to)f(often)i(b)r(e)f(o)n(v)n(erlo)r(ok)n(ed)d |
|
4783 (in)k(the)523 648 y(literature)i(when)h(claiming)f(that)h(pro)r(ofs)f |
|
4784 (b)n(y)h(rule)f(inductions)h(are)f(straigh)n(tforw)n(ard.)d(One)523 |
|
4785 747 y(example)27 b(of)h(this)g(comes)f(with)h(a)f(w)n(eak)n(ening)f |
|
4786 (result)i(for)f(con)n(texts)g(in)h(the)g(simply-t)n(yp)r(ed)523 |
|
4787 847 y Fr(\025)p Fz(-calculus.)648 939 y(W)-7 b(e)39 b(assume)e(t)n(yp)r |
|
4788 (es)i(are)f(of)g(the)h(form)f Fr(T)94 b Fz(::=)82 b Fr(X)48 |
|
4789 b Fp(j)41 b Fr(T)53 b Fp(!)41 b Fr(T)12 b Fz(,)38 b(and)g(that)h(t)n |
|
4790 (yping)523 1030 y(con)n(texts)27 b(\(\014nite)h(lists)f(of)g(v)-5 |
|
4791 b(ariable-t)n(yp)r(e)26 b(pairs\))g(are)h Fs(valid)37 |
|
4792 b Fz(if)28 b(no)f(v)-5 b(ariable)26 b(o)r(ccurs)g(t)n(wice.)523 |
|
4793 1121 y(The)i(t)n(yping)f(relation)g(can)g(then)h(b)r(e)g(de\014ned)g(b) |
|
4794 n(y)f(the)h(rules)770 1274 y Fk(valid)7 b Fw(\()p Fn(\000)k |
|
4795 Fw(\))98 b(\()p Fn(x)9 b Fw(:)g Fn(T)i Fw(\))19 b Fm(2)j |
|
4796 Fn(\000)p 770 1312 729 4 v 892 1385 a(\000)32 b Fm(`)21 |
|
4797 b Fk(var)9 b Fw(\()p Fn(x)p Fw(\))20 b(:)i Fn(T)1540 |
|
4798 1329 y Fw(T)n(yp)r(e)1713 1347 y Fx(1)1901 1284 y Fn(\000)32 |
|
4799 b Fm(`)21 b Fn(M)30 b Fw(:)21 b Fn(T)2239 1292 y Fx(1)2295 |
|
4800 1284 y Fm(!)g Fn(T)2438 1292 y Fx(2)2549 1284 y Fn(\000)32 |
|
4801 b Fm(`)21 b Fn(N)30 b Fw(:)22 b Fn(T)2875 1292 y Fx(1)p |
|
4802 1901 1312 1009 4 v 2078 1385 a Fn(\000)32 b Fm(`)21 b |
|
4803 Fk(app)5 b Fw(\()p Fn(M)t(;)13 b(N)8 b Fw(\))22 b(:)g |
|
4804 Fn(T)2698 1393 y Fx(2)2951 1329 y Fw(T)n(yp)r(e)3124 |
|
4805 1347 y Fx(2)1360 1539 y Fn(x)f Fw(#)g Fn(\000)109 b Fw(\()p |
|
4806 Fn(x)9 b Fw(:)g Fn(T)1824 1547 y Fx(1)1857 1539 y Fw(\))g(::)g |
|
4807 Fn(\000)31 b Fm(`)22 b Fn(M)29 b Fw(:)22 b Fn(T)2285 |
|
4808 1547 y Fx(2)p 1360 1578 960 4 v 1428 1651 a Fn(\000)32 |
|
4809 b Fm(`)21 b Fk(lam)5 b Fw(\()p Fn(x:M)j Fw(\))21 b(:)h |
|
4810 Fn(T)2018 1659 y Fx(1)2074 1651 y Fm(!)f Fn(T)2217 1659 |
|
4811 y Fx(2)2361 1594 y Fw(T)n(yp)r(e)2534 1613 y Fx(3)3307 |
|
4812 1467 y Fw(\(2\))523 1821 y Fz(where)30 b(\()p Fr(x)e |
|
4813 Fz(:)f Fr(T)12 b Fz(\))27 b Fp(2)g Fr(\000)42 b Fz(stands)30 |
|
4814 b(for)f(list-mem)n(b)r(ership,)h(and)g Fr(x)e Fz(#)f |
|
4815 Fr(\000)42 b Fz(for)30 b Fr(x)g Fz(b)r(eing)h(fresh)f(for)523 |
|
4816 1921 y Fr(\000)12 b Fz(,)31 b(or)e(equiv)-5 b(alen)n(tly)31 |
|
4817 b Fr(x)g Fz(not)g(o)r(ccuring)f(in)h Fr(\000)12 b Fz(.)30 |
|
4818 b(De\014ne)i(a)e(con)n(text)g Fr(\000)2686 1891 y Fj(0)2740 |
|
4819 1921 y Fz(to)h(b)r(e)g(w)n(eak)n(er)e(than)523 2020 y |
|
4820 Fr(\000)42 b Fz(\(written)31 b Fr(\000)39 b Fp(\022)27 |
|
4821 b Fr(\000)1187 1990 y Fj(0)1210 2020 y Fz(\),)j(if)h(ev)n(ery)e(name-t) |
|
4822 n(yp)r(e)h(pair)g(in)g Fr(\000)42 b Fz(also)29 b(app)r(ears)g(in)i |
|
4823 Fr(\000)3015 1990 y Fj(0)3038 2020 y Fz(.)f(Then)g(w)n(e)523 |
|
4824 2120 y(ha)n(v)n(e)523 2271 y Ft(Lemma)f(\(W)-8 b(eak)m(ening\).)30 |
|
4825 b Fs(If)k Fr(\000)40 b Fp(`)29 b Fr(M)37 b Fz(:)29 b |
|
4826 Fr(T)44 b Fs(is)33 b(derivable,)j(and)d Fr(\000)40 b |
|
4827 Fp(\022)29 b Fr(\000)2852 2241 y Fj(0)2907 2271 y Fs(with)34 |
|
4828 b Fr(\000)3154 2241 y Fj(0)3210 2271 y Fs(valid,)523 |
|
4829 2371 y(then)c Fr(\000)771 2340 y Fj(0)817 2371 y Fp(`)22 |
|
4830 b Fr(M)32 b Fz(:)23 b Fr(T)41 b Fs(is)30 b(also)h(derivable.)523 |
|
4831 2521 y Fz(The)k(informal)f(pro)r(of)g(of)g(this)h(lemma)f(is)h(straigh) |
|
4832 n(tforw)n(ard,)c(pro)n(vided\(!\))j(one)h(uses)f(the)523 |
|
4833 2621 y(v)-5 b(ariable)27 b(con)n(v)n(en)n(tion.)523 2747 |
|
4834 y Fs(Informal)37 b(Pr)l(o)l(of)p Fz(.)f Fs(By)h(rule)f(induction)h |
|
4835 (over)g Fr(\000)46 b Fp(`)35 b Fr(M)43 b Fz(:)35 b Fr(T)47 |
|
4836 b Fs(showing)37 b(that)g Fr(\000)3020 2716 y Fj(0)3077 |
|
4837 2747 y Fp(`)e Fr(M)43 b Fz(:)35 b Fr(T)523 2846 y Fs(holds)c(for)g(al)t |
|
4838 (l)g Fr(\000)1050 2816 y Fj(0)1102 2846 y Fs(with)g Fr(\000)j |
|
4839 Fp(\022)23 b Fr(\000)1519 2816 y Fj(0)1571 2846 y Fs(and)31 |
|
4840 b Fr(\000)1796 2816 y Fj(0)1848 2846 y Fs(b)l(eing)g(valid.)523 |
|
4841 2972 y Ft(Case)e Fz(T)n(yp)r(e)936 2984 y Fy(1)974 2972 |
|
4842 y Fz(:)g Fr(\000)38 b Fp(`)26 b Fr(M)35 b Fz(:)27 b Fr(T)42 |
|
4843 b Fs(is)32 b Fr(\000)38 b Fp(`)26 b Fs(var)10 b Fz(\()p |
|
4844 Fr(x)p Fz(\))27 b(:)g Fr(T)12 b Fs(.)31 b(By)h(assumption)g(we)g(know)g |
|
4845 Fr(v)s(al)r(id)p Fz(\()p Fr(\000)3325 2942 y Fj(0)3347 |
|
4846 2972 y Fz(\))p Fs(,)523 3071 y Fz(\()p Fr(x)9 b Fz(:)g |
|
4847 Fr(T)j Fz(\))24 b Fp(2)f Fr(\000)41 b Fs(and)31 b Fr(\000)j |
|
4848 Fp(\022)23 b Fr(\000)1328 3041 y Fj(0)1351 3071 y Fs(.)30 |
|
4849 b(Ther)l(efor)l(e)i(we)e(c)l(an)f(use)h Fz(T)n(yp)r(e)2391 |
|
4850 3083 y Fy(1)2458 3071 y Fs(to)f(derive)i Fr(\000)2867 |
|
4851 3041 y Fj(0)2913 3071 y Fp(`)23 b Fs(var)10 b Fz(\()p |
|
4852 Fr(x)p Fz(\))24 b(:)f Fr(T)12 b Fs(.)523 3197 y Ft(Case)41 |
|
4853 b Fz(T)n(yp)r(e)948 3209 y Fy(2)985 3197 y Fz(:)h Fr(\000)58 |
|
4854 b Fp(`)45 b Fr(M)55 b Fz(:)46 b Fr(T)54 b Fs(is)43 b |
|
4855 Fr(\000)57 b Fp(`)46 b Fs(app)7 b Fz(\()p Fr(M)2116 3209 |
|
4856 y Fy(1)2153 3197 y Fr(;)14 b(M)2271 3209 y Fy(2)2307 |
|
4857 3197 y Fz(\))47 b(:)f Fr(T)12 b Fs(.)42 b(Case)h(fol)t(lows)i(fr)l(om)e |
|
4858 (the)523 3297 y(induction)30 b(hyp)l(otheses)i(and)e(rule)f |
|
4859 Fz(T)n(yp)r(e)1816 3309 y Fy(2)1854 3297 y Fs(.)523 3422 |
|
4860 y Ft(Case)38 b Fz(T)n(yp)r(e)945 3434 y Fy(3)983 3422 |
|
4861 y Fz(:)g Fr(\000)53 b Fp(`)41 b Fr(M)50 b Fz(:)42 b Fr(T)51 |
|
4862 b Fs(is)40 b Fr(\000)53 b Fp(`)41 b Fs(lam)7 b Fz(\()p |
|
4863 Fr(x:M)2152 3434 y Fy(1)2189 3422 y Fz(\))42 b(:)f Fr(T)2376 |
|
4864 3434 y Fy(1)2455 3422 y Fp(!)g Fr(T)2628 3434 y Fy(2)2665 |
|
4865 3422 y Fs(.)f(Using)g(the)g(variable)523 3522 y(c)l(onvention)26 |
|
4866 b(we)g(assume)g(that)f Fr(x)f Fz(#)f Fr(\000)1734 3492 |
|
4867 y Fj(0)1757 3522 y Fs(.)j(Then)g(we)g(know)g(that)g Fz(\(\()p |
|
4868 Fr(x)9 b Fz(:)g Fr(T)2714 3534 y Fy(1)2752 3522 y Fz(\))g(::)g |
|
4869 Fr(\000)2911 3492 y Fj(0)2935 3522 y Fz(\))26 b Fs(is)g(valid)h(and)523 |
|
4870 3621 y(henc)l(e)f(that)g Fz(\(\()p Fr(x)9 b Fz(:)g Fr(T)1115 |
|
4871 3633 y Fy(1)1154 3621 y Fz(\))g(::)g Fr(\000)1313 3591 |
|
4872 y Fj(0)1336 3621 y Fz(\))24 b Fp(`)f Fr(M)1547 3633 y |
|
4873 Fy(1)1606 3621 y Fz(:)h Fr(T)1702 3633 y Fy(2)1764 3621 |
|
4874 y Fs(holds.)k(By)f(app)l(e)l(aling)h(to)e(the)g(variable)i(c)l |
|
4875 (onvention)523 3721 y(again,)j(we)f(have)h(that)f Fr(\000)1320 |
|
4876 3691 y Fj(0)1366 3721 y Fp(`)23 b Fs(lam)7 b Fz(\()p |
|
4877 Fr(x:M)1761 3733 y Fy(1)1798 3721 y Fz(\))24 b(:)f Fr(T)1949 |
|
4878 3733 y Fy(1)2009 3721 y Fp(!)g Fr(T)2164 3733 y Fy(2)2230 |
|
4879 3721 y Fs(holds)31 b(using)f(rule)g Fz(T)n(yp)r(e)3017 |
|
4880 3733 y Fy(3)3350 3721 y Fp(u)-55 b(t)523 3897 y Fz(Ho)n(w)n(ev)n(er,)23 |
|
4881 b(in)i(order)e(to)i(mak)n(e)e(this)i(informal)f(pro)r(of)h(w)n(ork)e |
|
4882 (with)i(the)g(induction)g(principle)523 3997 y(that)j(comes)f(with)h |
|
4883 (the)g(rules)f(in)h(\(2\),)g(namely)605 4180 y Fp(8)p |
|
4884 Fr(\000)d(x)14 b(T)7 b(:)45 b Fs(valid)11 b Fz(\()p Fr(\000)h |
|
4885 Fz(\))18 b Fp(^)h Fz(\()p Fr(x)9 b Fz(:)g Fr(T)j Fz(\))24 |
|
4886 b Fp(2)f Fr(\000)35 b Fp(\))23 b Fr(P)35 b(\000)g Fz(\()p |
|
4887 Fs(var)9 b Fz(\()p Fr(x)p Fz(\)\))25 b Fr(T)605 4303 |
|
4888 y Fp(8)p Fr(\000)g(M)d(N)h(T)970 4315 y Fy(1)1020 4303 |
|
4889 y Fr(T)1069 4315 y Fy(2)1106 4303 y Fr(:)g(P)35 b(\000)g(M)c |
|
4890 Fz(\()p Fr(T)1519 4315 y Fy(1)1579 4303 y Fp(!)23 b Fr(T)1734 |
|
4891 4315 y Fy(2)1771 4303 y Fz(\))42 b Fp(^)g Fr(P)35 b(\000)f(N)e(T)2263 |
|
4892 4315 y Fy(1)2346 4303 y Fp(\))23 b Fr(P)35 b(\000)g Fz(\()p |
|
4893 Fs(app)7 b Fz(\()p Fr(M)t(;)14 b(N)9 b Fz(\)\))23 b Fr(T)3157 |
|
4894 4315 y Fy(2)605 4426 y Fp(8)p Fr(x)14 b(\000)25 b(M)d(T)941 |
|
4895 4438 y Fy(1)992 4426 y Fr(T)1041 4438 y Fy(2)1078 4426 |
|
4896 y Fr(:)h(x)g Fz(#)g Fr(\000)31 b Fp(^)18 b Fr(P)35 b |
|
4897 Fz(\(\()p Fr(x)9 b Fz(:)g Fr(T)1730 4438 y Fy(1)1769 |
|
4898 4426 y Fz(\))g(::)g Fr(\000)j Fz(\))23 b Fr(M)32 b(T)2145 |
|
4899 4438 y Fy(2)2228 4426 y Fp(\))23 b Fr(P)35 b(\000)g Fz(\()p |
|
4900 Fs(lam)7 b Fz(\()p Fr(x:M)i Fz(\)\))24 b(\()p Fr(T)3039 |
|
4901 4438 y Fy(1)3099 4426 y Fp(!)f Fr(T)3254 4438 y Fy(2)3291 |
|
4902 4426 y Fz(\))p 593 4476 2742 4 v 1514 4550 a Fr(\000)35 |
|
4903 b Fp(`)23 b Fr(M)31 b Fz(:)23 b Fr(T)57 b Fp(\))47 b |
|
4904 Fr(P)35 b(\000)f(M)e(T)3299 4650 y Fz(\(3\))523 4742 |
|
4905 y(w)n(e)i(need)g(in)g(case)f(of)h(rule)f(T)n(yp)r(e)1597 |
|
4906 4754 y Fy(3)1669 4742 y Fz(to)g(b)r(e)i(able)e(to)h(rename)f(the)i(b)r |
|
4907 (ound)f(v)-5 b(ariable)33 b(to)h(b)r(e)523 4833 y(suitably)20 |
|
4908 b(fresh)g(for)g Fr(\000)1209 4803 y Fj(0)1232 4833 y |
|
4909 Fz(;)g(b)n(y)g(the)h(induction)g(w)n(e)f(only)f(kno)n(w)h(that)g |
|
4910 Fr(x)h Fz(is)f(fresh)h(for)e(the)i(smaller)523 4924 y(con)n(text)h |
|
4911 Fr(\000)12 b Fz(.)22 b(T)-7 b(o)22 b(b)r(e)h(able)f(to)g(do)g(this)h |
|
4912 (renaming)e(dep)r(ends)i(on)f(t)n(w)n(o)g(conditions:)g(\014rst,)g |
|
4913 (there)p Black Black eop end |
|
4914 %%Page: 4 4 |
|
4915 TeXDict begin 4 3 bop Black 523 232 a Fw(4)237 b(Christian)27 |
|
4916 b(Urban,)e(Stefan)h(Berghofer,)i(and)d(Mic)n(hael)i(Norrish)p |
|
4917 Black 523 448 a Fz(m)n(ust)d(exist)h(a)f(fresh)g(v)-5 |
|
4918 b(ariable)23 b(whic)n(h)h(w)n(e)g(can)g(c)n(ho)r(ose.)f(In)i(our)e |
|
4919 (example)h(this)h(means)f(that)523 540 y(the)k(con)n(text)g |
|
4920 Fr(\000)1025 510 y Fj(0)1076 540 y Fz(m)n(ust)g(not)g(con)n(tain)g(all) |
|
4921 f(p)r(ossible)h(free)g(v)-5 b(ariables.)27 b(Second,)h(the)g(relation) |
|
4922 523 631 y Fr(\000)42 b Fp(`)30 b Fr(M)39 b Fz(:)31 b |
|
4923 Fr(T)43 b Fz(m)n(ust)32 b(b)r(e)h(in)n(v)-5 b(arian)n(t)31 |
|
4924 b(under)g(suitable)h(renamings.)f(This)h(is)g(b)r(ecause)g(when)523 |
|
4925 722 y(w)n(e)j(c)n(hange)g(the)h(goal)f(from)g Fr(\000)1538 |
|
4926 692 y Fj(0)1598 722 y Fp(`)h Fs(lam)7 b Fz(\()p Fr(x:M)2006 |
|
4927 734 y Fy(1)2044 722 y Fz(\))37 b(:)g Fr(T)2222 734 y |
|
4928 Fy(1)2295 722 y Fp(!)g Fr(T)2464 734 y Fy(2)2537 722 |
|
4929 y Fz(to)f Fr(\000)2710 692 y Fj(0)2769 722 y Fp(`)h Fs(lam)7 |
|
4930 b Fz(\()p Fr(z)t(:M)3174 734 y Fy(1)3210 722 y Fz([)p |
|
4931 Fr(x)37 b Fz(:=)523 814 y Fr(z)t Fz(]\))g(:)h Fr(T)768 |
|
4932 826 y Fy(1)842 814 y Fp(!)g Fr(T)1012 826 y Fy(2)1049 |
|
4933 814 y Fz(,)e(w)n(e)g(m)n(ust)h(b)r(e)g(able)f(to)g(infer)g(from)g(\(\() |
|
4934 p Fr(x)25 b Fz(:)f Fr(T)2507 826 y Fy(1)2544 814 y Fz(\))g(::)g |
|
4935 Fr(\000)2733 783 y Fj(0)2756 814 y Fz(\))38 b Fp(`)f |
|
4936 Fr(M)2995 826 y Fy(1)3070 814 y Fz(:)h Fr(T)3180 826 |
|
4937 y Fy(2)3253 814 y Fz(that)523 905 y(\(\()p Fr(z)23 b |
|
4938 Fz(:)18 b Fr(T)739 917 y Fy(1)776 905 y Fz(\))h(::)g |
|
4939 Fr(\000)955 875 y Fj(0)978 905 y Fz(\))32 b Fp(`)h Fr(M)1207 |
|
4940 917 y Fy(1)1243 905 y Fz([)p Fr(x)g Fz(:=)g Fr(z)t Fz(])e(:)i |
|
4941 Fr(T)1669 917 y Fy(2)1739 905 y Fz(holds.)g(This)g(in)n(v)-5 |
|
4942 b(ariance)32 b(under)h(renamings)f(do)r(es,)523 996 y(ho)n(w)n(ev)n |
|
4943 (er,)21 b Fs(not)31 b Fz(hold)24 b(in)f(general,)f(not)h(ev)n(en)g |
|
4944 (under)h(renamings)e(with)h(fresh)h(v)-5 b(ariables.)22 |
|
4945 b(F)-7 b(or)523 1088 y(example)27 b(if)h(w)n(e)g(assume)e(that)i(v)-5 |
|
4946 b(ariables)27 b(are)f(linearly)h(ordered,)f(then)i(the)g(relation)1626 |
|
4947 1232 y Fn(v)c Fw(=)d Fk(min)5 b Fm(f)p Fn(v)1978 1240 |
|
4948 y Fx(0)2014 1232 y Fn(;)13 b(:)g(:)g(:)g(;)g(v)2221 1240 |
|
4949 y Fi(n)2264 1232 y Fm(g)p 1626 1270 677 4 v 1697 1343 |
|
4950 a Fw(\()p Fm(f)p Fn(v)1802 1351 y Fx(0)1838 1343 y Fn(;)g(:)g(:)g(:)g |
|
4951 (;)g(v)2045 1351 y Fi(n)2088 1343 y Fm(g)p Fn(;)h(v)s |
|
4952 Fw(\))523 1506 y Fz(that)24 b(asso)r(ciates)e(\014nite)i(subsets)f(of)g |
|
4953 (these)h(v)-5 b(ariables)22 b(to)h(the)h(smallest)f(v)-5 |
|
4954 b(ariable)22 b(o)r(ccurring)523 1606 y(in)k(it,)h(is)e |
|
4955 Fs(not)34 b Fz(in)n(v)-5 b(arian)n(t)25 b(\(apply)h(the)g(renaming)f([) |
|
4956 p Fr(v)h Fz(:=)d Fr(v)2308 1575 y Fj(0)2332 1606 y Fz(])j(where)f |
|
4957 Fr(v)2662 1575 y Fj(0)2712 1606 y Fz(is)g(a)h(v)-5 b(ariable)25 |
|
4958 b(that)h(is)523 1705 y(bigger)20 b(than)h(ev)n(ery)g(v)-5 |
|
4959 b(ariable)20 b(in)h Fp(f)p Fr(v)1644 1717 y Fy(0)1681 |
|
4960 1705 y Fr(;)14 b(:)g(:)g(:)g(;)g(v)1906 1717 y Fq(n)1951 |
|
4961 1705 y Fp(g)p Fz(\).)22 b(Other)e(examples)h(are)f(rules)h(that)h(in)n |
|
4962 (v)n(olv)n(e)523 1805 y(a)34 b(substitution)g(for)g(concrete)f(v)-5 |
|
4963 b(ariables)33 b(or)g(a)h(substitution)g(with)h(concrete)e(terms.)h(In) |
|
4964 523 1904 y(order)22 b(to)h(a)n(v)n(oid)e(suc)n(h)i(pathological)e |
|
4965 (cases,)h(w)n(e)h(require)f(that)h(the)h(relation)e(for)g(whic)n(h)h |
|
4966 (one)523 2004 y(w)n(an)n(ts)36 b(to)h(emplo)n(y)f(the)i(v)-5 |
|
4967 b(ariable)36 b(con)n(v)n(en)n(tion)f(m)n(ust)i(b)r(e)h(in)n(v)-5 |
|
4968 b(arian)n(t)35 b(under)i(renamings;)523 2104 y(from)28 |
|
4969 b(the)h(induction)g(w)n(e)g(require)e(that)i(the)g(v)-5 |
|
4970 b(ariable)27 b(con)n(v)n(en)n(tion)h(can)g(only)g(b)r(e)h(applied)523 |
|
4971 2203 y(in)f(con)n(texts)f(where)g(there)g(are)g(only)g(\014nitely)h |
|
4972 (man)n(y)f(free)h(names.)648 2295 y(Ho)n(w)n(ev)n(er,)g(these)j(t)n(w)n |
|
4973 (o)f(requiremen)n(ts)f(are)h Fs(not)38 b Fz(y)n(et)30 |
|
4974 b(su\016cien)n(t,)h(and)g(w)n(e)f(need)h(to)f(im-)523 |
|
4975 2386 y(p)r(ose)f(a)g(second)g(condition)h(that)g(inductiv)n(e)f |
|
4976 (de\014nitions)h(ha)n(v)n(e)e(to)i(satisfy)-7 b(.)29 |
|
4977 b(Consider)g(the)523 2477 y(function)f(that)g(tak)n(es)f(a)g(list)h(of) |
|
4978 f(v)-5 b(ariables)27 b(and)g(binds)h(them)g(in)g Fr(\025)p |
|
4979 Fz(-abstractions,)e(that)i(is)1034 2657 y Fk(bind)i Fn(t)21 |
|
4980 b Fw([)13 b(])1352 2614 y Fy(def)1365 2657 y Fw(=)56 |
|
4981 b Fn(t)154 b Fk(bind)29 b Fn(t)21 b Fw(\()p Fn(x)9 b |
|
4982 Fw(::)g Fk(xs)d Fw(\))2160 2614 y Fy(def)2174 2657 y |
|
4983 Fw(=)56 b Fk(lam)5 b Fw(\()p Fn(x:)p Fw(\()p Fk(bind)29 |
|
4984 b Fn(t)21 b Fk(xs)7 b Fw(\)\))523 2820 y Fz(F)-7 b(urther)24 |
|
4985 b(consider)f(the)h(relation)f Fr(,)-14 b Fp(!)p Fz(,)25 |
|
4986 b(whic)n(h)f(\\un)n(binds")f(the)h(outermost)g(abstractions)e(of)523 |
|
4987 2919 y(a)27 b Fr(\025)p Fz(-term)h(and)f(is)h(de\014ned)g(b)n(y:)p |
|
4988 641 3053 709 4 v 641 3132 a Fs(var)10 b Fz(\()p Fr(x)p |
|
4989 Fz(\))24 b Fr(,)-14 b Fp(!)23 b Fz([)14 b(])p Fr(;)g |
|
4990 Fs(var)c Fz(\()p Fr(x)p Fz(\))1391 3076 y(Un)n(bind)1659 |
|
4991 3088 y Fy(1)p 1862 3053 973 4 v 1862 3132 a Fs(app)d |
|
4992 Fz(\()p Fr(t)2057 3144 y Fy(1)2094 3132 y Fr(;)14 b(t)2161 |
|
4993 3144 y Fy(2)2198 3132 y Fz(\))24 b Fr(,)-14 b Fp(!)23 |
|
4994 b Fz([)14 b(])p Fr(;)g Fs(app)6 b Fz(\()p Fr(t)2660 3144 |
|
4995 y Fy(1)2698 3132 y Fr(;)14 b(t)2765 3144 y Fy(2)2802 |
|
4996 3132 y Fz(\))2876 3076 y(Un)n(bind)3143 3088 y Fy(2)1569 |
|
4997 3248 y Fr(t)23 b(,)-14 b Fp(!)23 b Fs(xs)7 b Fr(;)14 |
|
4998 b(t)1884 3218 y Fj(0)p 1376 3284 723 4 v 1376 3363 a |
|
4999 Fs(lam)7 b Fz(\()p Fr(x:t)p Fz(\))24 b Fr(,)-14 b Fp(!)24 |
|
5000 b Fr(x)9 b Fz(::)g Fs(xs)e Fr(;)14 b(t)2076 3333 y Fj(0)2141 |
|
5001 3306 y Fz(Un)n(bind)2408 3319 y Fy(3)3299 3226 y Fz(\(4\))523 |
|
5002 3533 y(Of)35 b(course,)f(this)h(relation)f(cannot)g(b)r(e)i(expressed)e |
|
5003 (as)g(a)g(function)i(b)r(ecause)e(the)i(b)r(ound)523 |
|
5004 3624 y(v)-5 b(ariables)23 b(do)g(not)h(ha)n(v)n(e)f(\\particular")e |
|
5005 (names.)j(Nonetheless)f(it)i(is)f(w)n(ell-de\014ned,)f(and)h(not)523 |
|
5006 3716 y(trivial.)j(F)-7 b(or)27 b(example,)g(w)n(e)h(ha)n(v)n(e)647 |
|
5007 3867 y Fk(lam)6 b Fw(\()p Fn(x:)p Fk(lam)f Fw(\()p Fn(y)s(:)p |
|
5008 Fk(app)g Fw(\()p Fk(var)j Fw(\()p Fn(x)p Fw(\))p Fn(;)13 |
|
5009 b Fk(app)5 b Fw(\()p Fk(var)k Fw(\()p Fn(y)s Fw(\))p |
|
5010 Fn(;)j Fk(var)d Fw(\()p Fn(z)s Fw(\)\)\)\)\))1734 3959 |
|
5011 y Fn(,)-13 b Fm(!)21 b Fw([)p Fn(x;)13 b(y)s Fw(])p Fn(;)g |
|
5012 Fk(app)6 b Fw(\()p Fk(var)i Fw(\()p Fn(x)p Fw(\))p Fn(;)13 |
|
5013 b Fk(app)5 b Fw(\()p Fk(var)k Fw(\()p Fn(y)s Fw(\))p |
|
5014 Fn(;)j Fk(var)d Fw(\()p Fn(z)s Fw(\)\)\))42 b(and)647 |
|
5015 4074 y Fk(lam)6 b Fw(\()p Fn(x:)p Fk(lam)f Fw(\()p Fn(y)s(:)p |
|
5016 Fk(app)g Fw(\()p Fk(var)j Fw(\()p Fn(x)p Fw(\))p Fn(;)13 |
|
5017 b Fk(app)5 b Fw(\()p Fk(var)k Fw(\()p Fn(y)s Fw(\))p |
|
5018 Fn(;)j Fk(var)d Fw(\()p Fn(z)s Fw(\)\)\)\)\))1734 4165 |
|
5019 y Fn(,)-13 b Fm(!)21 b Fw([)p Fn(y)s(;)13 b(x)p Fw(])p |
|
5020 Fn(;)g Fk(app)6 b Fw(\()p Fk(var)i Fw(\()p Fn(y)s Fw(\))p |
|
5021 Fn(;)13 b Fk(app)5 b Fw(\()p Fk(var)k Fw(\()p Fn(x)p |
|
5022 Fw(\))p Fn(;)j Fk(var)d Fw(\()p Fn(z)s Fw(\)\)\))523 |
|
5023 4327 y Fz(but)28 b(w)n(e)f(also)g(ha)n(v)n(e)f Fp(8)p |
|
5024 Fr(t)1232 4296 y Fj(0)1255 4327 y Fr(:)d Fs(lam)7 b Fz(\()p |
|
5025 Fr(x:)p Fs(lam)h Fz(\()p Fr(y)s(:)p Fs(app)e Fz(\()p |
|
5026 Fr(x;)14 b Fs(app)7 b Fz(\()p Fs(var)j Fz(\()p Fr(y)s |
|
5027 Fz(\))p Fr(;)k Fs(var)c Fz(\()p Fr(z)t Fz(\)\)\)\)\))24 |
|
5028 b Fp(6)p Fr(,)-14 b Fp(!)23 b Fz([)p Fr(x;)14 b(z)t Fz(])p |
|
5029 Fr(;)g(t)3201 4296 y Fj(0)3224 4327 y Fz(.)648 4426 y(F)-7 |
|
5030 b(urther,)31 b(one)g(can)g(also)f(easily)g(establish)h(\(b)n(y)h |
|
5031 (induction)f(on)g(the)h(term)f Fr(t)p Fz(\))h(that)g(for)523 |
|
5032 4526 y(ev)n(ery)f Fr(t)i Fz(there)g(exists)f(a)g Fr(t)1366 |
|
5033 4496 y Fj(0)1422 4526 y Fz(and)h(a)f(list)h Fs(xs)39 |
|
5034 b Fz(of)32 b(distinct)i(v)-5 b(ariables)31 b(suc)n(h)h(that)h |
|
5035 Fr(t)f(,)-14 b Fp(!)31 b Fs(xs)7 b Fr(;)14 b(t)3382 4496 |
|
5036 y Fj(0)523 4625 y Fz(holds,)29 b(demonstrating)g(that)h(the)g(relation) |
|
5037 f(is)h(\\total")e(if)i(the)h(last)e(t)n(w)n(o)g(parameters)f(are)523 |
|
5038 4725 y(view)n(ed)f(as)g(results.)648 4825 y(If)33 b(one)f(wished)h(to)f |
|
5039 (do)h(rule)f(inductions)h(o)n(v)n(er)e(the)i(de\014nition)g(of)g(this)g |
|
5040 (relation,)e(one)523 4924 y(migh)n(t)23 b(imagine)f(that)i(the)f(v)-5 |
|
5041 b(ariable)22 b(con)n(v)n(en)n(tion)f(allo)n(w)n(ed)h(us)h(to)g(assume)f |
|
5042 (that)h(the)h(b)r(ound)p Black Black eop end |
|
5043 %%Page: 5 5 |
|
5044 TeXDict begin 5 4 bop Black 1347 232 a Fw(Barendregt's)27 |
|
5045 b(V)-6 b(ariable)26 b(Con)n(v)n(en)n(tion)f(in)h(Rule)f(Inductions)237 |
|
5046 b(5)p Black 523 448 a Fz(name)33 b Fr(x)h Fz(w)n(as)f(distinct)h(from)f |
|
5047 (the)h(free)f(v)-5 b(ariables)33 b(of)g(the)h(conclusion)f(of)g(the)h |
|
5048 (rule,)f(and)523 548 y(in)j(particular)e(that)i Fr(x)h |
|
5049 Fz(could)e(not)h(app)r(ear)f(in)h(the)g(list)g Fs(xs)6 |
|
5050 b Fz(.)36 b(Ho)n(w)n(ev)n(er,)d(this)j(use)g(of)g(the)523 |
|
5051 648 y(v)-5 b(ariable)27 b(con)n(v)n(en)n(tion)f(quic)n(kly)h(leads)g |
|
5052 (to)g(the)h Fs(faulty)36 b Fz(lemma:)523 792 y Ft(Lemma)24 |
|
5053 b(\(F)-8 b(ault)m(y\).)78 b Fs(If)28 b Fr(t)c(,)-14 b |
|
5054 Fp(!)23 b Fz(\()p Fr(x)9 b Fz(::)g Fs(xs)e Fz(\))p Fr(;)14 |
|
5055 b(t)1873 762 y Fj(0)1925 792 y Fs(and)29 b Fr(x)23 b |
|
5056 Fp(2)g Fr(F)12 b(V)19 b Fz(\()p Fr(t)2427 762 y Fj(0)2451 |
|
5057 792 y Fz(\))28 b Fs(then)g Fr(x)c Fp(2)f Fr(F)12 b(V)19 |
|
5058 b Fz(\()p Fs(bind)33 b Fr(t)3222 762 y Fj(0)3268 792 |
|
5059 y Fs(xs)6 b Fz(\))p Fs(.)523 937 y Fz(The)26 b(\\pro)r(of)6 |
|
5060 b(")25 b(is)h(b)n(y)g(an)g(induction)h(o)n(v)n(er)d(the)i(rules)g(giv)n |
|
5061 (en)f(in)i(\(4\))f(and)g(assumes)f(that)i(the)523 1036 |
|
5062 y(binder)33 b Fr(x)g Fz(in)g(the)g(third)g(rule)g(is)f(fresh)h(with)g |
|
5063 (resp)r(ect)g(to)f Fs(xs)7 b Fz(.)32 b(This)h(lemma)g(is)g(of)f(course) |
|
5064 523 1136 y(false)h(as)g(witnessed)h(b)n(y)f(the)h(term)f |
|
5065 Fs(lam)7 b Fz(\()p Fr(x:)p Fs(lam)h Fz(\()p Fr(x:)p Fs(var)i |
|
5066 Fz(\()p Fr(x)p Fz(\)\)\).)36 b(Therefore,)c(including)i(the)523 |
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5067 1236 y(v)-5 b(ariable)29 b(con)n(v)n(en)n(tion)f(in)h(the)h(induction)g |
|
5068 (principle)g(that)f(comes)g(with)h(the)g(rules)f(in)h(\(4\),)523 |
|
5069 1335 y(w)n(ould)25 b(pro)r(duce)h(an)f(inconsistency)-7 |
|
5070 b(.)26 b(T)-7 b(o)25 b(prev)n(en)n(t)g(this)h(problem)f(w)n(e)h(in)n |
|
5071 (tro)r(duce)f(a)h(second)523 1435 y(condition)37 b(for)f(rules,)h(whic) |
|
5072 n(h)g(requires)e(that)j(all)f(v)-5 b(ariables)35 b(o)r(ccurring)h(as)g |
|
5073 (a)h(binder)g(in)523 1535 y(a)g(rule)f(m)n(ust)h(b)r(e)g(fresh)g(\(a)g |
|
5074 (notion)f(whic)n(h)h(w)n(e)g(shall)f(mak)n(e)g(precise)g(later)g(on\))h |
|
5075 (for)g(the)523 1634 y(conclusion)d(of)i(this)f(rule,)g(and)h(if)f(a)g |
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5076 (rule)g(has)g(sev)n(eral)f(suc)n(h)h(v)-5 b(ariables,)34 |
|
5077 b(they)h(m)n(ust)h(b)r(e)523 1734 y(m)n(utually)28 b(distinct.)523 |
|
5078 1878 y Ft(Our)44 b(Con)m(tribution:)37 b Fz(W)-7 b(e)39 |
|
5079 b(in)n(tro)r(duce)e(t)n(w)n(o)h(conditions)f(inductiv)n(e)i |
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5080 (de\014nitions)f(m)n(ust)523 1978 y(satisfy)28 b(in)h(order)e(to)h(mak) |
|
5081 n(e)g(sure)g(they)g(are)g(compatible)g(with)h(the)g(v)-5 |
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5082 b(ariable)27 b(con)n(v)n(en)n(tion.)523 2078 y(W)-7 b(e)22 |
|
5083 b(will)h(build)f(a)g(v)n(ersion)f(of)h(this)g(con)n(v)n(en)n(tion)f(in) |
|
5084 n(to)g(the)i(induction)f(principles)g(that)g(come)523 |
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5085 2177 y(with)31 b(the)g(inductiv)n(e)g(de\014nitions.)g(Moreo)n(v)n(er,) |
|
5086 d(it)j(will)g(b)r(e)g(sho)n(wn)f(ho)n(w)g(these)g(new)h(\(\\v)n(c-)523 |
|
5087 2277 y(compatible"\))g(induction)g(principles)g(can)f(b)r(e)i |
|
5088 (automatically)e(deriv)n(ed)g(in)h(the)g(nominal)523 |
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5089 2377 y(datat)n(yp)r(e)36 b(pac)n(k)-5 b(age)36 b([11,)12 |
|
5090 b(9].)37 b(The)g(presen)n(ted)f(results)g(ha)n(v)n(e)f(already)h(b)r |
|
5091 (een)h(extensiv)n(ely)523 2476 y(used)j(in)h(formalisations:)d(for)i |
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5092 (example)g(in)g(our)g(formalisations)e(of)i(the)h(CR)f(and)h(SN)523 |
|
5093 2576 y(prop)r(erties)f(in)h(the)g Fr(\025)p Fz(-calculus,)f(in)h(a)f |
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5094 (formalisation)f(b)n(y)i(Bengtson)e(and)i(P)n(arro)n(w)d(for)523 |
|
5095 2676 y(sev)n(eral)f(pro)r(ofs)g(in)i(the)g(pi-calculus)e([3],)h(in)h(a) |
|
5096 f(formalisation)f(of)h(Crary's)f(c)n(hapter)g(on)523 |
|
5097 2775 y(logical)d(relation)h([4],)h(and)f(in)h(v)-5 b(arious)35 |
|
5098 b(formalised)f(pro)r(ofs)h(on)h(structural)f(op)r(erational)523 |
|
5099 2875 y(seman)n(tics.)523 3131 y Fu(2)112 b(Nominal)36 |
|
5100 b(Logic)523 3312 y Fz(Before)d(pro)r(ceeding,)g(w)n(e)g(brie\015y)h(in) |
|
5101 n(tro)r(duce)f(some)g(imp)r(ortan)n(t)h(notions)f(from)g(nominal)523 |
|
5102 3404 y(logic)25 b([8,)13 b(11].)25 b(In)h(particular,)e(w)n(e)h(will)h |
|
5103 (build)g(on)g(the)g(three)f(cen)n(tral)g(notions)g(of)g |
|
5104 Fs(p)l(ermuta-)523 3495 y(tions)p Fz(,)f Fs(supp)l(ort)31 |
|
5105 b Fz(and)23 b Fs(e)l(quivarianc)l(e)p Fz(.)i(P)n(erm)n(utations)d(are)g |
|
5106 (\014nite)i(bijectiv)n(e)g(mappings)e(from)523 3586 y(atoms)31 |
|
5107 b(to)g(atoms,)g(where)g(atoms)g(are)g(dra)n(wn)f(from)h(a)g(coun)n |
|
5108 (tably)g(in\014nite)h(set)g(denoted)523 3677 y(b)n(y)i |
|
5109 Fh(A)17 b Fz(.)40 b(W)-7 b(e)34 b(represen)n(t)f(p)r(erm)n(utations)g |
|
5110 (as)h(\014nite)g(lists)g(whose)f(elemen)n(ts)h(are)f(sw)n(appings)523 |
|
5111 3769 y(\(i.e.,)k(pairs)f(of)g(atoms\).)h(W)-7 b(e)37 |
|
5112 b(write)f(suc)n(h)g(p)r(erm)n(utations)g(as)g(\()p Fr(a)2620 |
|
5113 3781 y Fy(1)2671 3769 y Fr(b)2707 3781 y Fy(1)2744 3769 |
|
5114 y Fz(\)\()p Fr(a)2852 3781 y Fy(2)2904 3769 y Fr(b)2940 |
|
5115 3781 y Fy(2)2977 3769 y Fz(\))14 b Fp(\001)g(\001)g(\001)g |
|
5116 Fz(\()p Fr(a)3210 3781 y Fq(n)3269 3769 y Fr(b)3305 3781 |
|
5117 y Fq(n)3350 3769 y Fz(\);)523 3860 y(the)33 b(empt)n(y)g(list)g([)14 |
|
5118 b(])33 b(stands)f(for)h(the)g(iden)n(tit)n(y)g(p)r(erm)n(utation.)g(A)g |
|
5119 (p)r(erm)n(utation)f Fr(\031)k Fs(acting)523 3951 y Fz(on)27 |
|
5120 b(an)h(atom)f Fr(a)g Fz(is)h(de\014ned)g(as:)1111 4176 |
|
5121 y Fw([)13 b(])1166 4185 y Fg(\001)1205 4176 y Fn(a)1269 |
|
5122 4134 y Fy(def)1282 4176 y Fw(=)37 b Fn(a)176 b Fw(\(\()p |
|
5123 Fn(a)1697 4184 y Fx(1)1744 4176 y Fn(a)1785 4184 y Fx(2)1819 |
|
5124 4176 y Fw(\))9 b(::)g Fn(\031)s Fw(\))1986 4185 y Fg(\001)2023 |
|
5125 4176 y Fn(a)2087 4134 y Fy(def)2101 4176 y Fw(=)2197 |
|
5126 4023 y Ff(8)2197 4090 y(<)2197 4225 y(:)2277 4085 y Fn(a)2318 |
|
5127 4093 y Fx(2)2426 4085 y Fw(if)22 b Fn(\031)2539 4094 |
|
5128 y Fg(\001)2577 4085 y Fn(a)f Fw(=)g Fn(a)2761 4093 y |
|
5129 Fx(1)2277 4176 y Fn(a)2318 4184 y Fx(1)2426 4176 y Fw(if)h |
|
5130 Fn(\031)2539 4185 y Fg(\001)2577 4176 y Fn(a)f Fw(=)g |
|
5131 Fn(a)2761 4184 y Fx(2)2277 4267 y Fn(\031)2324 4276 y |
|
5132 Fg(\001)2362 4267 y Fn(a)i Fw(otherwise)523 4426 y Fz(where)34 |
|
5133 b(\()p Fr(a)14 b(b)p Fz(\))21 b(::)f Fr(\031)38 b Fz(is)c(the)h(comp)r |
|
5134 (osition)f(of)h(a)f(p)r(erm)n(utation)g(follo)n(w)n(ed)f(b)n(y)h(the)h |
|
5135 (sw)n(apping)523 4526 y(\()p Fr(a)14 b(b)p Fz(\).)27 |
|
5136 b(The)h(comp)r(osition)e(of)h Fr(\031)k Fz(follo)n(w)n(ed)26 |
|
5137 b(b)n(y)h(another)f(p)r(erm)n(utation)h Fr(\031)2801 |
|
5138 4496 y Fj(0)2852 4526 y Fz(is)g(giv)n(en)f(b)n(y)h(list-)523 |
|
5139 4625 y(concatenation,)c(written)h(as)f Fr(\031)1508 4595 |
|
5140 y Fj(0)1532 4625 y Fz(@)p Fr(\031)s Fz(,)g(and)h(the)g(in)n(v)n(erse)f |
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5141 (of)g(a)h(p)r(erm)n(utation)f(is)h(giv)n(en)f(b)n(y)h(list)523 |
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5142 4725 y(rev)n(ersal,)32 b(written)j(as)e Fr(\031)1314 |
|
5143 4695 y Fj(\000)p Fy(1)1404 4725 y Fz(.)h(Our)g(represen)n(tation)e(of)i |
|
5144 (p)r(erm)n(utations)g(as)g(lists)g(do)r(es)g(not)523 |
|
5145 4825 y(giv)n(e)23 b(unique)i(represen)n(tativ)n(es:)d(for)h(example,)h |
|
5146 (the)h(p)r(erm)n(utation)e(\()p Fr(a)14 b(a)p Fz(\))25 |
|
5147 b(is)f(\\equal")f(to)h(the)523 4924 y(iden)n(tit)n(y)k(p)r(erm)n |
|
5148 (utation.)f(W)-7 b(e)28 b(equate)f(p)r(erm)n(utations)g(with)h(a)g |
|
5149 (relation)e Fp(\030)p Fz(:)p Black Black eop end |
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5150 %%Page: 6 6 |
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5151 TeXDict begin 6 5 bop Black 523 232 a Fw(6)237 b(Christian)27 |
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5152 b(Urban,)e(Stefan)h(Berghofer,)i(and)d(Mic)n(hael)i(Norrish)p |
|
5153 Black Black 523 448 a Ft(De\014nition)k(1)p Black 41 |
|
5154 w(\(P)m(erm)m(utation)26 b(Equalit)m(y\).)h Fs(Two)f(p)l(ermutations)g |
|
5155 (ar)l(e)32 b Fz(equal)p Fs(,)25 b(written)523 548 y Fr(\031)570 |
|
5156 560 y Fy(1)631 548 y Fp(\030)d Fr(\031)765 560 y Fy(2)803 |
|
5157 548 y Fs(,)30 b(pr)l(ovide)l(d)i Fr(\031)1233 560 y Fy(1)1270 |
|
5158 557 y Fg(\001)1309 548 y Fr(a)23 b Fz(=)f Fr(\031)1510 |
|
5159 560 y Fy(2)1548 557 y Fg(\001)1586 548 y Fr(a)p Fs(,)30 |
|
5160 b(for)h(al)t(l)f Fr(a)23 b Fp(2)h Fh(A)17 b Fs(.)523 |
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5161 701 y Fz(The)28 b(p)r(erm)n(utation)f(action)g(on)g(atoms)g(can)g(b)r |
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5162 (e)h(lifted)h(to)e(other)g(t)n(yp)r(es.)p Black 523 855 |
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5163 a Ft(De\014nition)k(2)p Black 41 w(\(The)d(Action)g(of)g(a)g(P)m(erm)m |
|
5164 (utation\).)f Fs(A)f(p)l(ermutation)h(action)g Fr(\031)3237 |
|
5165 864 y Fg(\001)3276 855 y Fz(\()p Fp(\000)p Fz(\))523 |
|
5166 955 y Fs(lifts)e(to)g(a)g(typ)l(e)g Fr(T)36 b Fs(pr)l(ovide)l(d)27 |
|
5167 b(it)d(the)h(fol)t(lowing)i(thr)l(e)l(e)e(pr)l(op)l(erties)h(hold)g(on) |
|
5168 f(al)t(l)h(values)f Fr(x)e Fp(2)h Fr(T)p Black Black |
|
5169 1434 1097 a Fw(\()p Fn(i)p Fw(\))71 b([)13 b(])1646 1106 |
|
5170 y Fg(\001)1685 1097 y Fn(x)21 b Fw(=)g Fn(x)1408 1212 |
|
5171 y Fw(\()p Fn(ii)p Fw(\))71 b(\()p Fn(\031)1665 1220 y |
|
5172 Fx(1)1700 1212 y Fw(@)p Fn(\031)1804 1220 y Fx(2)1838 |
|
5173 1212 y Fw(\))1868 1221 y Fg(\001)1906 1212 y Fn(x)21 |
|
5174 b Fw(=)g Fn(\031)2096 1220 y Fx(1)2130 1221 y Fg(\001)2169 |
|
5175 1212 y Fw(\()p Fn(\031)2243 1220 y Fx(2)2277 1221 y Fg(\001)2315 |
|
5176 1212 y Fn(x)p Fw(\))1382 1327 y(\()p Fn(iii)p Fw(\))71 |
|
5177 b Fk(if)27 b Fn(\031)1710 1335 y Fx(1)1766 1327 y Fm(\030)21 |
|
5178 b Fn(\031)1891 1335 y Fx(2)1953 1327 y Fk(then)28 b Fn(\031)2167 |
|
5179 1335 y Fx(1)2202 1336 y Fg(\001)2240 1327 y Fn(x)21 b |
|
5180 Fw(=)g Fn(\031)2430 1335 y Fx(2)2464 1336 y Fg(\001)2503 |
|
5181 1327 y Fn(x)523 1474 y Fz(F)-7 b(or)27 b(example,)g(lists)h(and)f |
|
5182 (tuples)h(can)f(b)r(e)h(giv)n(en)f(the)h(follo)n(wing)f(p)r(erm)n |
|
5183 (utation)g(action:)1218 1656 y Fw(lists:)479 b Fn(\031)1897 |
|
5184 1665 y Fg(\001)1935 1656 y Fw([)13 b(])2038 1613 y Fy(def)2051 |
|
5185 1656 y Fw(=)61 b([)13 b(])1688 1775 y Fn(\031)1735 1784 |
|
5186 y Fg(\001)1773 1775 y Fw(\()p Fn(h)22 b Fw(::)g Fn(t)p |
|
5187 Fw(\))2038 1732 y Fy(def)2051 1775 y Fw(=)61 b(\()p Fn(\031)2249 |
|
5188 1784 y Fg(\001)2286 1775 y Fn(h)p Fw(\))22 b(::)g(\()p |
|
5189 Fn(\031)2523 1784 y Fg(\001)2560 1775 y Fn(t)p Fw(\))1218 |
|
5190 1894 y(tuples:)71 b Fn(\031)1558 1903 y Fg(\001)1596 |
|
5191 1894 y Fw(\()p Fn(x)1670 1902 y Fx(1)1704 1894 y Fn(;)13 |
|
5192 b(:)g(:)g(:)g(;)g(x)1918 1902 y Fi(n)1960 1894 y Fw(\))2038 |
|
5193 1851 y Fy(def)2051 1894 y Fw(=)61 b(\()p Fn(\031)2249 |
|
5194 1903 y Fg(\001)2286 1894 y Fn(x)2330 1902 y Fx(1)2364 |
|
5195 1894 y Fn(;)14 b(:)f(:)g(:)g(;)g(\031)2582 1903 y Fg(\001)2620 |
|
5196 1894 y Fn(x)2664 1902 y Fi(n)2706 1894 y Fw(\))3307 1762 |
|
5197 y(\(5\))523 2053 y Fz(F)-7 b(urther,)28 b(on)f Fr(\013)p |
|
5198 Fz(-equated)g Fr(\025)p Fz(-terms)h(w)n(e)f(can)g(de\014ne)h(the)g(p)r |
|
5199 (erm)n(utation)f(action:)1528 2236 y Fn(\031)1575 2245 |
|
5200 y Fg(\001)1613 2236 y Fk(var)9 b Fw(\()p Fn(x)p Fw(\))1878 |
|
5201 2193 y Fy(def)1892 2236 y Fw(=)60 b Fk(var)9 b Fw(\()p |
|
5202 Fn(\031)2204 2245 y Fg(\001)2242 2236 y Fn(x)p Fw(\))1312 |
|
5203 2378 y Fn(\031)1359 2387 y Fg(\001)1397 2378 y Fk(app)d |
|
5204 Fw(\()p Fn(M)1624 2386 y Fx(1)1658 2378 y Fn(;)13 b(M)1766 |
|
5205 2386 y Fx(2)1801 2378 y Fw(\))1878 2335 y Fy(def)1892 |
|
5206 2378 y Fw(=)60 b Fk(app)6 b Fw(\()p Fn(\031)2212 2387 |
|
5207 y Fg(\001)2249 2378 y Fn(M)2323 2386 y Fx(1)2358 2378 |
|
5208 y Fn(;)13 b(\031)2439 2387 y Fg(\001)2477 2378 y Fn(M)2551 |
|
5209 2386 y Fx(2)2586 2378 y Fw(\))1412 2520 y Fn(\031)1459 |
|
5210 2529 y Fg(\001)1497 2520 y Fk(lam)5 b Fw(\()p Fn(x:M)j |
|
5211 Fw(\))1878 2478 y Fy(def)1892 2520 y Fw(=)60 b Fk(lam)6 |
|
5212 b Fw(\()p Fn(\031)2217 2529 y Fg(\001)2254 2520 y Fn(x:\031)2366 |
|
5213 2529 y Fg(\001)2404 2520 y Fn(M)i Fw(\))3307 2365 y(\(6\))648 |
|
5214 2685 y Fz(The)39 b(second)g(notion)g(that)h(w)n(e)f(use)g(is)h(that)f |
|
5215 (of)h Fs(supp)l(ort)47 b Fz(\(roughly)39 b(sp)r(eaking,)g(the)523 |
|
5216 2784 y(supp)r(ort)32 b(of)g(an)g(elemen)n(t)g(is)g(its)g(set)g(of)g |
|
5217 (free)g(atoms\).)f(The)i(set)f(supp)r(orting)f(an)h(elemen)n(t)523 |
|
5218 2884 y(is)g(de\014ned)g(in)g(terms)f(of)h(p)r(erm)n(utation)f(actions)g |
|
5219 (on)h(that)g(elemen)n(t,)g(so)f(that)h(as)f(so)r(on)g(as)523 |
|
5220 2984 y(one)h(has)f(de\014ned)i(a)f(p)r(erm)n(utation)g(action)f(for)h |
|
5221 (a)g(t)n(yp)r(e,)g(one)g(automatically)f(deriv)n(es)g(its)523 |
|
5222 3083 y(accompan)n(ying)17 b(notion)h(of)h(supp)r(ort,)f(whic)n(h)h(in)g |
|
5223 (turn)f(determines)h(the)g(notion)f(of)h(freshness)523 |
|
5224 3183 y(\(see)28 b([11]\):)p Black 523 3336 a Ft(De\014nition)j(3)p |
|
5225 Black 41 w(\(Supp)s(ort)49 b(and)g(F)-8 b(reshness\).)47 |
|
5226 b Fs(The)j Fz(supp)r(ort)43 b Fs(of)h Fr(x)f Fs(is)h(de\014ne)l(d)f |
|
5227 (as:)523 3465 y Fe(supp)n Fz(\()p Fr(x)p Fz(\))849 3414 |
|
5228 y Fd(def)861 3465 y Fz(=)51 b Fp(f)p Fr(a)39 b Fp(j)h |
|
5229 Fe(infinite)m Fp(f)p Fr(b)f Fp(j)h Fz(\()p Fr(a)14 b(b)p |
|
5230 Fz(\))1852 3474 y Fg(\001)1890 3465 y Fr(x)40 b Fp(6)p |
|
5231 Fz(=)f Fr(x)p Fp(gg)p Fs(.)g(A)n(n)f(atom)i Fr(a)e Fs(is)i(said)g(to)f |
|
5232 (b)l(e)45 b Fz(fresh)523 3565 y Fs(for)31 b(an)e Fr(x)p |
|
5233 Fs(,)i(written)e Fr(a)23 b Fz(#)h Fr(x)p Fs(,)30 b(if)h |
|
5234 Fr(a)23 b Fp(62)g Fe(supp)o Fz(\()p Fr(x)p Fz(\))p Fs(.)523 |
|
5235 3718 y Fz(W)-7 b(e)23 b(will)f(also)f(use)h(the)g(auxiliary)f(notation) |
|
5236 h Fr(a)h Fz(#)g Fs(xs)6 b Fz(,)23 b(in)f(whic)n(h)g Fs(xs)29 |
|
5237 b Fz(stands)21 b(for)h(a)g(collection)523 3818 y(of)28 |
|
5238 b(ob)5 b(jects)27 b Fr(x)949 3830 y Fy(1)1001 3818 y |
|
5239 Fr(:)14 b(:)g(:)f(x)1158 3830 y Fq(n)1204 3818 y Fz(,)28 |
|
5240 b(to)g(mean)f Fr(a)d Fz(#)g Fr(x)1786 3830 y Fy(1)1837 |
|
5241 3818 y Fr(:)14 b(:)g(:)g(a)23 b Fz(#)h Fr(x)2155 3830 |
|
5242 y Fq(n)2200 3818 y Fz(.)k(W)-7 b(e)29 b(further)f(generalise)e(this)i |
|
5243 (nota-)523 3917 y(tion)d(to)g(a)g(collection)g(of)g(atoms,)f(namely)h |
|
5244 Fs(as)30 b Fz(#)24 b Fs(xs)6 b Fz(,)25 b(whic)n(h)g(means)g |
|
5245 Fr(a)2724 3929 y Fy(1)2784 3917 y Fz(#)f Fs(xs)c Fr(:)14 |
|
5246 b(:)g(:)f(a)3124 3929 y Fq(m)3210 3917 y Fz(#)24 b Fs(xs)6 |
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5247 b Fz(.)648 4017 y(Later)26 b(on)h(w)n(e)g(will)g(often)h(mak)n(e)e(use) |
|
5248 h(of)g(the)h(follo)n(wing)e(t)n(w)n(o)g(prop)r(erties)h(of)g |
|
5249 (freshness,)523 4117 y(whic)n(h)h(can)f(b)r(e)h(deriv)n(ed)f(from)g |
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5250 (the)h(de\014nition)g(of)f(supp)r(ort,)h(the)g(p)r(erm)n(utation)f |
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5251 (action)g(on)523 4216 y Fh(A)51 b Fz(and)27 b(the)h(requiremen)n(ts)f |
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5252 (of)g(p)r(erm)n(utation)g(actions)g(on)h(other)f(t)n(yp)r(es)g(\(see)h |
|
5253 ([11]\).)p Black 523 4370 a Ft(Lemma)i(1.)p Black Black |
|
5254 581 4465 a Fp(\017)p Black 41 w Fz(\()p Fr(a)p Fz(\))53 |
|
5255 b Fr(a)23 b Fz(#)h Fr(x)30 b Fs(implies)h Fr(\031)1396 |
|
5256 4474 y Fg(\001)1435 4465 y Fr(a)22 b Fz(#)i Fr(\031)1644 |
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5257 4474 y Fg(\001)1682 4465 y Fr(x)p Fs(;)31 b(and)p Black |
|
5258 581 4563 a Fp(\017)p Black 41 w Fz(\()p Fr(b)p Fz(\))53 |
|
5259 b Fs(if)31 b Fr(a)23 b Fz(#)g Fr(x)30 b Fs(and)g Fr(b)23 |
|
5260 b Fz(#)g Fr(x)p Fs(,)31 b(then)e Fz(\()p Fr(a)14 b(b)p |
|
5261 Fz(\))1891 4572 y Fg(\001)1930 4563 y Fr(x)23 b Fz(=)g |
|
5262 Fr(x)p Fs(.)523 4725 y Fz(Henceforth)h(w)n(e)g(will)h(only)e(b)r(e)i |
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5263 (in)n(terested)f(in)g(those)g(ob)5 b(jects)24 b(whic)n(h)g(ha)n(v)n(e)f |
|
5264 (\014nite)i(supp)r(ort,)523 4825 y(b)r(ecause)k(for)g(them)h(there)g |
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5265 (exists)f(alw)n(a)n(ys)e(a)j(fresh)f(atom)g(\(recall)g(that)h(the)g |
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5266 (set)f(of)h(atoms)523 4924 y Fh(A)51 b Fz(is)27 b(in\014nite\).)p |
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5267 Black Black eop end |
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5268 %%Page: 7 7 |
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5269 TeXDict begin 7 6 bop Black 1347 232 a Fw(Barendregt's)27 |
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5270 b(V)-6 b(ariable)26 b(Con)n(v)n(en)n(tion)f(in)h(Rule)f(Inductions)237 |
|
5271 b(7)p Black Black 523 448 a Ft(Lemma)30 b(2.)p Black |
|
5272 40 w Fs(If)22 b Fr(x)g Fs(is)g(\014nitely)f(supp)l(orte)l(d,)h(then)f |
|
5273 (ther)l(e)h(exists)f(an)g(atom)h Fr(a)f Fs(such)g(that)h |
|
5274 Fr(a)h Fz(#)g Fr(x)p Fs(.)523 601 y Fz(Un)n(winding)39 |
|
5275 b(the)g(de\014nitions)g(of)f(p)r(erm)n(utation)h(actions)e(and)i(supp)r |
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5276 (ort)f(one)g(can)h(often)523 701 y(easily)27 b(calculate)g(the)h(supp)r |
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5277 (ort)f(of)h(an)f(ob)5 b(ject:)p Black Black 833 850 a |
|
5278 Fw(atoms:)492 b Fc(supp)q Fw(\()p Fn(a)p Fw(\))21 b(=)g |
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5279 Fm(f)p Fn(a)p Fm(g)833 965 y Fw(tuples:)492 b Fc(supp)q |
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5280 Fw(\()p Fn(x)1778 973 y Fx(1)1812 965 y Fn(;)13 b(:)g(:)g(:)g(;)g(x) |
|
5281 2026 973 y Fi(n)2068 965 y Fw(\))22 b(=)f Fc(supp)q Fw(\()p |
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5282 Fn(x)2432 973 y Fx(1)2466 965 y Fw(\))c Fm([)g Fn(:)c(:)g(:)k |
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5283 Fm([)h Fc(supp)q Fw(\()p Fn(x)2987 973 y Fi(n)3028 965 |
|
5284 y Fw(\))833 1080 y(lists:)561 b Fc(supp)q Fw(\([)13 b(]\))21 |
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5285 b(=)g Fb(?)p Fw(,)26 b Fc(supp)q Fw(\()p Fn(h)21 b Fw(::)h |
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5286 Fn(t)p Fw(\))f(=)g Fc(supp)q Fw(\()p Fn(h)p Fw(\))c Fm([)h |
|
5287 Fc(supp)p Fw(\()p Fn(t)p Fw(\))833 1195 y Fn(\013)p Fw(-equated)25 |
|
5288 b Fn(\025)p Fw(-terms:)70 b Fc(supp)q Fw(\()p Fk(var)9 |
|
5289 b Fw(\()p Fn(x)p Fw(\)\))20 b(=)h Fm(f)p Fn(x)p Fm(g)1547 |
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5290 1287 y Fc(supp)q Fw(\()p Fk(app)5 b Fw(\()p Fn(M)t(;)13 |
|
5291 b(N)8 b Fw(\)\))21 b(=)h Fc(supp)p Fw(\()p Fn(M)8 b Fw(\))18 |
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5292 b Fm([)f Fc(supp)q Fw(\()p Fn(N)8 b Fw(\))1547 1378 y |
|
5293 Fc(supp)q Fw(\()p Fk(lam)d Fw(\()p Fn(x:M)j Fw(\)\))21 |
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5294 b(=)g Fc(supp)q Fw(\()p Fn(M)8 b Fw(\))17 b Fm(\000)g(f)p |
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5295 Fn(x)p Fm(g)523 1533 y Fz(W)-7 b(e)25 b(therefore)f(note)h(the)g(follo) |
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5296 n(wing:)e(all)i(elemen)n(ts)f(in)h Fh(A)48 b Fz(and)25 |
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5297 b(all)f Fr(\013)p Fz(-equated)g Fr(\025)p Fz(-terms)h(are)523 |
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5298 1632 y(\014nitely)d(supp)r(orted.)g(Lists)g(\(similarly)f(tuples\))i |
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5299 (con)n(taining)e(\014nitely)h(supp)r(orted)g(elemen)n(ts)523 |
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5300 1732 y(are)37 b(\014nitely)i(supp)r(orted.)f(The)g(last)g(three)g |
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5301 (equations)f(sho)n(w)h(that)g(the)h(supp)r(ort)f(of)g |
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5302 Fr(\013)p Fz(-)523 1832 y(equated)24 b Fr(\025)p Fz(-terms)g(coincides) |
|
5303 g(with)h(the)g(usual)f(notion)g(of)g(free)g(v)-5 b(ariables.)24 |
|
5304 b(Hence,)g Fr(a)f Fz(#)g Fr(M)523 1931 y Fz(with)k Fr(M)36 |
|
5305 b Fz(b)r(eing)27 b(an)f Fr(\013)p Fz(-equated)h Fr(\025)p |
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5306 Fz(-term)g(coincides)f(with)i Fr(a)e Fz(not)h(b)r(eing)g(free)g(in)g |
|
5307 Fr(M)9 b Fz(.)26 b(If)i Fr(b)e Fz(is)523 2031 y(an)h(atom,)h(then)g |
|
5308 Fr(a)23 b Fz(#)g Fr(b)k Fz(coincides)g(with)h Fr(a)23 |
|
5309 b Fp(6)p Fz(=)g Fr(b)p Fz(.)648 2130 y(The)k(last)h(notion)f(of)g |
|
5310 (nominal)h(logic)e(w)n(e)i(use)f(here)g(is)h(that)f(of)h |
|
5311 Fs(e)l(quivarianc)l(e)p Fz(.)p Black 523 2283 a Ft(De\014nition)j(4)p |
|
5312 Black 41 w(\(Equiv)-5 b(ariance\).)p Black 581 2377 a |
|
5313 Fp(\017)p Black 41 w Fs(A)30 b(r)l(elation)g Fr(R)g Fs(is)37 |
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5314 b Fz(equiv)-5 b(arian)n(t)29 b Fs(if)h Fr(R)15 b Fz(\()p |
|
5315 Fr(\031)1923 2386 y Fg(\001)1962 2377 y Fs(xs)6 b Fz(\))30 |
|
5316 b Fs(is)g(implie)l(d)i(by)e Fr(R)14 b Fs(xs)36 b(for)31 |
|
5317 b(al)t(l)g Fr(\031)s Fs(.)p Black 581 2476 a Fp(\017)p |
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5318 Black 41 w Fs(A)f(function)f Fr(f)39 b Fs(is)e Fz(equiv)-5 |
|
5319 b(arian)n(t)28 b Fs(pr)l(ovide)l(d)k Fr(\031)2069 2485 |
|
5320 y Fg(\001)2107 2476 y Fz(\()p Fr(f)23 b Fs(xs)6 b Fz(\))24 |
|
5321 b(=)e Fr(f)h Fz(\()p Fr(\031)2571 2485 y Fg(\001)2610 |
|
5322 2476 y Fs(xs)6 b Fz(\))30 b Fs(for)g(al)t(l)h Fr(\031)s |
|
5323 Fs(.)p Black 523 2690 a(R)l(emark)f(1.)p Black 42 w Fz(Note)35 |
|
5324 b(that)g(if)g(w)n(e)f(regard)f(the)i(term-constructors)d |
|
5325 Fs(var)10 b Fz(,)34 b Fs(app)42 b Fz(and)34 b Fs(lam)42 |
|
5326 b Fz(as)523 2789 y(functions,)35 b(then)g(they)g(are)e(equiv)-5 |
|
5327 b(arian)n(t)34 b(on)g(accoun)n(t)g(of)g(the)h(de\014nition)g(giv)n(en)f |
|
5328 (in)h(\(6\).)523 2889 y(Because)e(of)h(the)h(de\014nition)g(in)f |
|
5329 (\(5\),)g(the)h(cons-constructors)c(of)k(lists)f(are)f(equiv)-5 |
|
5330 b(arian)n(t.)523 2989 y(By)27 b(a)g(simple)h(structural)e(induction)i |
|
5331 (on)g(the)f(list)h(argumen)n(t)e(of)i Fs(valid)p Fz(,)h(w)n(e)e(can)g |
|
5332 (establish)523 3088 y(that)c(the)h(relation)e Fs(valid)34 |
|
5333 b Fz(is)23 b(equiv)-5 b(arian)n(t.)22 b(By)h(Lem.)g(1\()p |
|
5334 Fr(a)p Fz(\))g(freshness)f(is)h(equiv)-5 b(arian)n(t.)22 |
|
5335 b(Also)523 3188 y(list-mem)n(b)r(ership,)j(\()p Fp(\000)p |
|
5336 Fz(\))f Fp(2)f Fz(\()p Fp(\000)p Fz(\),)j(is)f(equiv)-5 |
|
5337 b(arian)n(t,)24 b(whic)n(h)i(can)f(b)r(e)h(sho)n(wn)e(b)n(y)h(an)h |
|
5338 (induction)523 3287 y(on)h(the)h(length)g(of)g(lists.)523 |
|
5339 3548 y Fu(3)112 b(Sc)m(hematic)36 b(T)-9 b(erms)37 b(and)h(Sc)m |
|
5340 (hematic)e(Rules)523 3734 y Fz(Inductiv)n(e)18 b(relations)g(are)f |
|
5341 (de\014ned)i(as)e(the)i(smallest)f(relation)g(closed)f(under)i(some)e |
|
5342 (sc)n(hematic)523 3825 y(rules.)31 b(In)h(this)g(section)f(w)n(e)g |
|
5343 (will)h(formally)f(sp)r(ecify)g(the)h(form)g(of)f(suc)n(h)g(rules.)h |
|
5344 (Diagram-)523 3917 y(matically)27 b(they)h(ha)n(v)n(e)e(the)i(form)1493 |
|
5345 4058 y Fk(pr)l(emises)78 b(side-c)l(onditions)p 1493 |
|
5346 4093 860 4 v 1750 4161 a(c)l(onclusion)2395 4102 y Fn(\045)3307 |
|
5347 4161 y Fw(\(7\))523 4327 y Fz(where)31 b(the)h(premises,)f |
|
5348 (side-conditions)g(and)h(conclusions)e(are)h(predicates)g(of)g(the)h |
|
5349 (form)523 4426 y Fr(R)15 b Fs(ts)28 b Fz(where)22 b(w)n(e)g(use)h(the)f |
|
5350 (letters)h Fr(R)q Fz(,)f Fr(S)5 b Fz(,)22 b Fr(P)34 b |
|
5351 Fz(and)23 b Fr(Q)f Fz(to)g(stand)g(for)g(predicates;)g |
|
5352 Fs(ts)29 b Fz(stands)22 b(for)523 4526 y(a)k(collection)g(of)h(sc)n |
|
5353 (hematic)f(terms)g(\(the)h(argumen)n(ts)f(of)g Fr(R)q |
|
5354 Fz(\).)h(They)f(are)g(either)g(v)-5 b(ariables,)523 4625 |
|
5355 y(abstractions)36 b(or)g(functions,)i(namely)77 b Fr(t)40 |
|
5356 b Fz(::=)f Fr(x)h Fp(j)f Fr(a:t)h Fp(j)f Fr(f)23 b Fs(ts)83 |
|
5357 b Fz(where)37 b Fr(a)h Fz(is)f(a)g(v)-5 b(ariable)523 |
|
5358 4725 y(standing)29 b(for)g(an)h(atom)f(and)g Fr(f)39 |
|
5359 b Fz(stands)29 b(for)g(a)g(function.)i(W)-7 b(e)30 b(call)f(the)h(v)-5 |
|
5360 b(ariable)28 b Fr(a)i Fz(in)g Fr(a:t)523 4825 y Fz(as)g(b)r(eing)h(in)g |
|
5361 Fs(binding)j(p)l(osition)p Fz(.)e(Note)e(that)i(a)e(sc)n(hematic)g |
|
5362 (rule)g(ma)n(y)g(con)n(tain)g(the)i(same)523 4924 y(v)-5 |
|
5363 b(ariable)27 b(in)g(binding)h(and)g(non-binding)f(p)r(ositions)g(\(One) |
|
5364 2383 4936 y Fy(4)2448 4924 y Fz(and)g(T)n(yp)r(e)2796 |
|
5365 4936 y Fy(3)2861 4924 y Fz(are)g(examples\).)p Black |
|
5366 Black eop end |
|
5367 %%Page: 8 8 |
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5368 TeXDict begin 8 7 bop Black 523 232 a Fw(8)237 b(Christian)27 |
|
5369 b(Urban,)e(Stefan)h(Berghofer,)i(and)d(Mic)n(hael)i(Norrish)p |
|
5370 Black 648 448 a Fz(Assuming)33 b(an)f(inductiv)n(e)i(de\014nition)f(of) |
|
5371 g(the)h(predicate)e Fr(R)q Fz(,)h(the)g(sc)n(hematic)g(rule)g(in)523 |
|
5372 540 y(\(7\))28 b(m)n(ust)g(b)r(e)g(of)f(the)h(form)1352 |
|
5373 692 y Fn(R)13 b Fk(ts)1486 700 y Fx(1)1555 692 y Fn(:)g(:)g(:)35 |
|
5374 b(R)13 b Fk(ts)1813 700 y Fi(n)1919 692 y Fn(S)1966 700 |
|
5375 y Fx(1)2001 692 y Fk(ss)2070 700 y Fx(1)2138 692 y Fn(:)h(:)f(:)34 |
|
5376 b(S)2309 700 y Fi(m)2368 692 y Fk(ss)2437 700 y Fi(m)p |
|
5377 1352 721 1144 4 v 1856 788 a Fn(R)14 b Fk(ts)2537 729 |
|
5378 y Fn(\045)3307 788 y Fw(\(8\))523 960 y Fz(where)k(the)h(predicates)e |
|
5379 Fr(S)1322 972 y Fq(i)1350 960 y Fs(ss)1424 972 y Fq(i)1471 |
|
5380 960 y Fz(\(the)i(ones)e(di\013eren)n(t)i(from)f Fr(R)q |
|
5381 Fz(\))g(stand)g(for)g(the)h(side-conditions)523 1060 |
|
5382 y(in)28 b(the)g(sc)n(hematic)f(rule.)648 1152 y(F)-7 |
|
5383 b(or)26 b(pro)n(ving)g(our)h(main)g(result)g(in)h(the)g(next)g(section) |
|
5384 f(it)g(is)h(con)n(v)n(enien)n(t)e(to)h(in)n(tro)r(duce)523 |
|
5385 1243 y(sev)n(eral)f(auxiliary)g(notions)g(for)h(sc)n(hematic)g(terms)g |
|
5386 (and)g(rules.)g(The)g(follo)n(wing)g(functions)523 1335 |
|
5387 y(calculate)j(for)h(a)f(sc)n(hematic)h(term)g(the)g(set)g(of)g(v)-5 |
|
5388 b(ariables)30 b(in)h(non-binding)f(p)r(osition)h(and)523 |
|
5389 1426 y(the)d(set)g(of)f(v)-5 b(ariables)26 b(in)i(binding)g(p)r |
|
5390 (osition,)g(resp)r(ectiv)n(ely:)995 1587 y Fk(vars)7 |
|
5391 b Fw(\()p Fn(x)p Fw(\))23 b(=)e Fm(f)p Fn(x)p Fm(g)948 |
|
5392 1678 y Fk(vars)8 b Fw(\()p Fn(a:t)p Fw(\))23 b(=)e Fk(vars)7 |
|
5393 b Fw(\()p Fn(t)p Fw(\))t Fm(\000)t(f)p Fn(a)p Fm(g)917 |
|
5394 1769 y Fk(vars)g Fw(\()p Fn(f)21 b Fk(ts)7 b Fw(\))24 |
|
5395 b(=)d Fk(vars)7 b Fw(\()p Fk(ts)g Fw(\))2078 1587 y Fk(varsbp)f |
|
5396 Fw(\()p Fn(x)p Fw(\))23 b(=)e Fb(?)2032 1678 y Fk(varsbp)6 |
|
5397 b Fw(\()p Fn(a:t)p Fw(\))23 b(=)e Fk(varsbp)7 b Fw(\()p |
|
5398 Fn(t)p Fw(\))16 b Fm([)i(f)p Fn(a)p Fm(g)2000 1769 y |
|
5399 Fk(varsbp)7 b Fw(\()p Fn(f)21 b Fk(ts)7 b Fw(\))23 b(=)e |
|
5400 Fk(varsbp)7 b Fw(\()p Fk(ts)g Fw(\))3307 1679 y(\(9\))523 |
|
5401 1941 y Fz(The)26 b(notation)f Fr(t)p Fz([)p Fs(as)7 b |
|
5402 Fz(;)14 b Fs(xs)6 b Fz(])26 b(will)g(b)r(e)g(used)g(for)f(sc)n(hematic) |
|
5403 g(terms)h(to)f(indicate)h(that)g(the)g(v)-5 b(ari-)523 |
|
5404 2040 y(ables)27 b(in)h(binding)g(p)r(osition)f(of)h Fr(t)g |
|
5405 Fz(are)f(included)h(in)g Fs(as)35 b Fz(and)27 b(the)h(other)f(v)-5 |
|
5406 b(ariables)27 b(of)g Fr(t)h Fz(are)523 2140 y(either)36 |
|
5407 b(in)g Fs(as)43 b Fz(or)35 b Fs(xs)7 b Fz(.)36 b(That)g(means)f(w)n(e)h |
|
5408 (ha)n(v)n(e)e(for)i Fr(t)p Fz([)p Fs(as)7 b Fz(;)14 b |
|
5409 Fs(xs)6 b Fz(])36 b(that)g Fs(varsbp)7 b Fz(\()p Fr(t)p |
|
5410 Fz(\))37 b Fp(\022)g Fs(as)43 b Fz(and)523 2239 y Fs(vars)7 |
|
5411 b Fz(\()p Fr(t)p Fz(\))24 b Fp(\022)f Fs(as)i Fp([)19 |
|
5412 b Fs(xs)34 b Fz(hold.)523 2331 y(W)-7 b(e)30 b(extend)h(this)f |
|
5413 (notation)f(also)g(to)h(sc)n(hematic)g(rules:)f(b)n(y)h(writing)f |
|
5414 Fr(\045)p Fz([)p Fs(as)7 b Fz(;)14 b Fs(xs)7 b Fz(])30 |
|
5415 b(for)f(\(8\))h(w)n(e)523 2423 y(mean)880 2568 y Fn(R)13 |
|
5416 b Fk(ts)1014 2576 y Fx(1)1049 2568 y Fw([)p Fk(as)7 b |
|
5417 Fw(;)13 b Fk(xs)7 b Fw(])34 b Fn(:)13 b(:)h(:)34 b(R)13 |
|
5418 b Fk(ts)1567 2576 y Fi(n)1610 2568 y Fw([)p Fk(as)7 b |
|
5419 Fw(;)13 b Fk(xs)7 b Fw(])77 b Fn(S)1960 2576 y Fx(1)2007 |
|
5420 2568 y Fk(ss)2076 2576 y Fx(1)2111 2568 y Fw([)p Fk(as)7 |
|
5421 b Fw(;)13 b Fk(xs)7 b Fw(])34 b Fn(:)13 b(:)g(:)35 b(S)2542 |
|
5422 2576 y Fi(m)2613 2568 y Fk(ss)2682 2576 y Fi(m)2741 2568 |
|
5423 y Fw([)p Fk(as)7 b Fw(;)13 b Fk(xs)7 b Fw(])p 868 2615 |
|
5424 2111 4 v 1743 2688 a Fn(R)14 b Fk(ts)6 b Fw([)p Fk(as)h |
|
5425 Fw(;)14 b Fk(xs)6 b Fw(])3020 2624 y Fn(\045)3268 2688 |
|
5426 y Fw(\(10\))523 2860 y Fz(Ho)n(w)n(ev)n(er,)18 b(unlik)n(e)h(in)g(the)h |
|
5427 (notation)f(for)g(sc)n(hematic)g(terms,)g(w)n(e)g(mean)g(in)h |
|
5428 Fr(\045)p Fz([)p Fs(as)7 b Fz(;)14 b Fs(xs)6 b Fz(])20 |
|
5429 b(that)g(the)523 2960 y Fs(as)36 b Fz(stand)28 b Fs(exactly)36 |
|
5430 b Fz(for)28 b(the)g(v)-5 b(ariables)27 b(o)r(ccurring)g(somewhere)g(in) |
|
5431 i Fr(\045)f Fz(in)h(binding)f(p)r(osition)523 3060 y(and)e(the)h |
|
5432 Fs(xs)32 b Fz(stand)26 b(for)g(the)h(rest)e(of)i(v)-5 |
|
5433 b(ariables.)25 b(That)h(means)f(w)n(e)h(ha)n(v)n(e)f(for)h |
|
5434 Fr(\045)p Fz([)p Fs(as)7 b Fz(;)14 b Fs(xs)6 b Fz(])27 |
|
5435 b(that)523 3159 y Fs(varsbp)6 b Fz(\()p Fr(\045)p Fz(\))27 |
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5436 b(=)f Fs(as)37 b Fz(and)29 b Fs(vars)7 b Fz(\()p Fr(\045)p |
|
5437 Fz(\))27 b(=)f Fs(xs)36 b Fz(hold,)29 b(assuming)g(suitable)g |
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5438 (generalisations)e(of)j(the)523 3259 y(functions)24 b |
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5439 Fs(vars)31 b Fz(and)24 b Fs(varsbp)30 b Fz(to)23 b(sc)n(hematic)h |
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5440 (rules.)f(T)-7 b(o)23 b(see)g(ho)n(w)g(the)h(sc)n(hematic)g(notation) |
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5441 523 3358 y(w)n(orks)h(out)i(in)g(examples,)f(reconsider)g(the)h |
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5442 (de\014nitions)g(for)f(the)h(relations)f(One,)h(giv)n(en)f(in)523 |
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5443 3458 y(\(1\),)h(and)f(T)n(yp)r(e,)g(giv)n(en)g(in)g(\(2\).)h(Using)f |
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5444 (our)g(sc)n(hematic)g(notation)g(for)f(the)i(rules,)f(w)n(e)g(ha)n(v)n |
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5445 (e)p Black Black 1079 3620 a Fw(One)1216 3628 y Fx(1)1250 |
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5446 3620 y Fw([)p Fm(\000)p Fw(;)13 b Fn(M)8 b Fw(])1079 |
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5447 3711 y(One)1216 3719 y Fx(2)1250 3711 y Fw([)p Fn(y)s |
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5448 Fw(;)13 b Fn(M)t(;)g(M)1540 3680 y Fl(0)1563 3711 y Fw(])1079 |
|
5449 3803 y(One)1216 3811 y Fx(3)1250 3803 y Fw([)p Fm(\000)p |
|
5450 Fw(;)g Fn(M)t(;)h(N)t(;)f(M)1659 3771 y Fl(0)1682 3803 |
|
5451 y Fn(;)g(N)1785 3771 y Fl(0)1807 3803 y Fw(])1079 3894 |
|
5452 y(One)1216 3902 y Fx(4)1250 3894 y Fw([)p Fn(y)s Fw(;)g |
|
5453 Fn(M)t(;)g(N)t(;)h(M)1640 3862 y Fl(0)1662 3894 y Fn(;)f(N)1765 |
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5454 3862 y Fl(0)1788 3894 y Fw(])2017 3620 y(T)n(yp)r(e)2190 |
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5455 3639 y Fx(1)2225 3620 y Fw([)p Fm(\000)p Fw(;)g Fn(\000)r(;)g(x;)g(T)e |
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5456 Fw(])2017 3711 y(T)n(yp)r(e)2190 3730 y Fx(2)2225 3711 |
|
5457 y Fw([)p Fm(\000)p Fw(;)i Fn(\000)r(;)g(M)t(;)g(N)t(;)h(T)2680 |
|
5458 3719 y Fx(1)2714 3711 y Fn(;)f(T)2793 3719 y Fx(2)2828 |
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5459 3711 y Fw(])2017 3803 y(T)n(yp)r(e)2190 3821 y Fx(3)2225 |
|
5460 3803 y Fw([)p Fn(x)p Fw(;)g Fn(\000)r(;)g(M)t(;)g(T)2564 |
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5461 3811 y Fx(1)2598 3803 y Fn(;)h(T)2678 3811 y Fx(2)2712 |
|
5462 3803 y Fw(])523 4056 y Fz(where)27 b(`)p Fp(\000)p Fz(')g(stands)h(for) |
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5463 f(no)g(v)-5 b(ariable)27 b(in)g(binding)h(p)r(osition.)648 |
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5464 4148 y(The)d(main)g(prop)r(ert)n(y)f(of)h(an)g(inductiv)n(e)g |
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5465 (de\014nition,)g(sa)n(y)f(for)h(the)g(inductiv)n(e)h(predicate)523 |
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5466 4239 y Fr(R)q Fz(,)36 b(is)h(that)g(it)g(comes)f(with)h(an)f(induction) |
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5467 h(principle,)g(whic)n(h)f(establishes)g(a)g(prop)r(ert)n(y)523 |
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5468 4330 y Fr(P)26 b Fs(ts)41 b Fz(under)34 b(the)h(assumption)f(that)g |
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5469 Fr(R)15 b Fs(ts)41 b Fz(holds.)34 b(This)g(means)g(w)n(e)g(ha)n(v)n(e)f |
|
5470 (an)i(induction)523 4422 y(principle)28 b(diagrammatically)d(lo)r |
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5471 (oking)i(as)g(follo)n(ws)959 4583 y Fn(:)13 b(:)g(:)959 |
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5472 4674 y Fm(8)p Fk(as)19 b(xs)7 b Fn(:)22 b(P)11 b Fk(ts)1329 |
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5473 4682 y Fx(1)1364 4674 y Fw([)p Fk(as)c Fw(;)13 b Fk(xs)7 |
|
5474 b Fw(])39 b Fm(^)f Fn(:)13 b(:)h(:)38 b Fm(^)h Fn(P)11 |
|
5475 b Fk(ts)2059 4682 y Fi(n)2101 4674 y Fw([)p Fk(as)c Fw(;)13 |
|
5476 b Fk(xs)7 b Fw(])22 b Fm(^)1314 4765 y Fn(S)t Fk(ss)1434 |
|
5477 4773 y Fx(1)1468 4765 y Fw([)p Fk(as)7 b Fw(;)13 b Fk(xs)7 |
|
5478 b Fw(])39 b Fm(^)f Fn(:)14 b(:)f(:)38 b Fm(^)h Fn(S)t |
|
5479 Fk(ss)2160 4773 y Fi(m)2219 4765 y Fw([)p Fk(as)7 b Fw(;)13 |
|
5480 b Fk(xs)7 b Fw(])43 b Fm(\))f Fn(P)24 b Fk(ts)6 b Fw([)p |
|
5481 Fk(as)h Fw(;)14 b Fk(xs)6 b Fw(])959 4809 y Fn(:)13 b(:)g(:)p |
|
5482 948 4857 2033 4 v 1769 4924 a(R)h Fk(ts)28 b Fm(\))21 |
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5483 b Fn(P)j Fk(ts)1116 b Fw(\(11\))p Black Black eop end |
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5484 %%Page: 9 9 |
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5485 TeXDict begin 9 8 bop Black 1347 232 a Fw(Barendregt's)27 |
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5486 b(V)-6 b(ariable)26 b(Con)n(v)n(en)n(tion)f(in)h(Rule)f(Inductions)237 |
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5487 b(9)p Black 523 448 a Fz(where)26 b(for)f(ev)n(ery)g(sc)n(hematic)g |
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5488 (rule)h Fr(\045)g Fz(in)g(the)g(inductiv)n(e)h(de\014nition)f(w)n(e)g |
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5489 (ha)n(v)n(e)e(to)i(establish)523 548 y(an)34 b(implication.)h(These)f |
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5490 (implications)h(state)f(that)h(w)n(e)f(can)g(assume)g(the)h(prop)r(ert) |
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5491 n(y)f(for)523 648 y(all)25 b(premises)g(and)h(also)e(can)i(assume)e |
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5492 (that)i(the)g(side-conditions)f(hold;)h(w)n(e)f(ha)n(v)n(e)f(to)i(sho)n |
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5493 (w)523 747 y(that)i(the)g(prop)r(ert)n(y)e(holds)i(for)f(the)h |
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5494 (conclusion)e(of)i(the)g(sc)n(hematic)f(rule.)648 849 |
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5495 y(As)40 b(explained)f(in)h(the)h(in)n(tro)r(duction,)e(w)n(e)h(need)g |
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5496 (to)g(imp)r(ose)g(some)f(conditions)g(on)523 949 y(sc)n(hematic)33 |
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5497 b(rules)f(in)i(order)e(to)h(a)n(v)n(oid)f(fault)n(y)h(reasoning)e(and)i |
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5498 (to)g(p)r(ermit)h(an)f(argumen)n(t)523 1048 y(emplo)n(ying)28 |
|
5499 b(the)i(v)-5 b(ariable)28 b(con)n(v)n(en)n(tion.)g(A)h(rule)g |
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5500 Fr(\045)p Fz([)p Fs(as)7 b Fz(;)14 b Fs(xs)6 b Fz(],)30 |
|
5501 b(as)e(giv)n(en)g(in)i(\(10\),)f(is)g Fs(variable)523 |
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5502 1148 y(c)l(onvention)22 b(c)l(omp)l(atible)p Fz(,)f(short)d |
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5503 Fs(vc-c)l(omp)l(atible)p Fz(,)i(pro)n(vided)e(the)i(follo)n(wing)d(t)n |
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5504 (w)n(o)h(conditions)523 1247 y(are)27 b(satis\014ed.)p |
|
5505 Black 523 1432 a Ft(De\014nition)k(5)p Black 41 w(\(V)-8 |
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5506 b(ariable)37 b(Con)m(v)m(en)m(tion)g(Compatibilit)m(y\).)d |
|
5507 Fs(A)f(rule)g Fr(\045)p Fz([)p Fs(as)8 b Fz(;)14 b Fs(xs)6 |
|
5508 b Fz(])33 b Fs(with)523 1532 y(c)l(onclusion)d Fr(R)15 |
|
5509 b Fs(ts)36 b(is)h Fz(v)n(c-compatible)28 b Fs(pr)l(ovide)l(d)k(that:)p |
|
5510 Black 581 1706 a Fp(\017)p Black 41 w Fs(al)t(l)f(functions)f(and)g |
|
5511 (side-c)l(onditions)h(o)l(c)l(curring)f(in)g Fr(\045)g |
|
5512 Fs(ar)l(e)g(e)l(quivariant,)h(and)p Black 581 1808 a |
|
5513 Fp(\017)p Black 41 w Fs(the)f(side-c)l(onditions)i Fr(S)1420 |
|
5514 1820 y Fy(1)1457 1808 y Fs(ss)1531 1820 y Fy(1)1587 1808 |
|
5515 y Fp(^)19 b Fr(:)14 b(:)g(:)k Fp(^)h Fr(S)1901 1820 y |
|
5516 Fq(m)1964 1808 y Fs(ss)2039 1820 y Fq(m)2131 1808 y Fs(imply)31 |
|
5517 b(that)f(as)g Fz(#)24 b Fs(ts)36 b(holds)31 b(and)664 |
|
5518 1908 y(that)f(the)g(as)37 b(ar)l(e)30 b(distinct.)523 |
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5519 2090 y Fz(If)d(ev)n(ery)f(sc)n(hematic)g(rule)g(in)i(an)e(inductiv)n(e) |
|
5520 h(de\014nition)g(satis\014es)f(these)h(conditions,)f(then)523 |
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5521 2190 y(the)35 b(induction)g(principle)f(can)g(b)r(e)h(strengthened)f |
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5522 (suc)n(h)g(that)h(it)g(includes)f(a)g(v)n(ersion)f(of)523 |
|
5523 2289 y(the)28 b(v)-5 b(ariable)27 b(con)n(v)n(en)n(tion.)523 |
|
5524 2566 y Fu(4)112 b(Strengthening)37 b(of)h(the)f(Induction)g(Principle) |
|
5525 523 2767 y Fz(In)32 b(this)g(section)f(w)n(e)g(will)h(sho)n(w)f(ho)n(w) |
|
5526 g(to)g(obtain)g(a)h(stronger)d(induction)j(principle)g(than)523 |
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5527 2858 y(the)e(one)f(giv)n(en)f(in)i(\(11\).)f(By)g(stronger)e(w)n(e)i |
|
5528 (mean)g(that)h(it)g(has)f(the)g(v)-5 b(ariable)28 b(con)n(v)n(en)n |
|
5529 (tion)523 2950 y(already)i(built)j(in)f(\(this)g(will)g(then)h(enable)e |
|
5530 (us)h(to)g(giv)n(e)e(slic)n(k)i(pro)r(ofs)f(b)n(y)g(rule)h(induction) |
|
5531 523 3041 y(whic)n(h)f(do)g(not)g(need)g(an)n(y)f(renaming\).)h(F)-7 |
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5532 b(ormally)30 b(w)n(e)g(sho)n(w)h(that)g(induction)g(principles)523 |
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5533 3132 y(of)d(the)g(form)651 3281 y Fn(:)13 b(:)g(:)651 |
|
5534 3373 y Fm(8)p Fk(as)19 b(xs)h Fn(C)q(:)h Fw(\()p Fm(8)p |
|
5535 Fn(C)q(:P)i(C)18 b Fk(ts)1325 3381 y Fx(1)1360 3373 y |
|
5536 Fw([)p Fk(as)7 b Fw(;)13 b Fk(xs)7 b Fw(]\))17 b Fm(^)39 |
|
5537 b Fn(:)13 b(:)g(:)39 b Fm(^)17 b Fw(\()p Fm(8)p Fn(C)q(:P)23 |
|
5538 b(C)18 b Fk(ts)2277 3381 y Fi(n)2319 3373 y Fw([)p Fk(as)7 |
|
5539 b Fw(;)14 b Fk(xs)6 b Fw(]\))22 b Fm(^)1029 3477 y Fn(S)t |
|
5540 Fk(ss)1149 3485 y Fx(1)1183 3477 y Fw([)p Fk(as)7 b Fw(;)13 |
|
5541 b Fk(xs)7 b Fw(])39 b Fm(^)g Fn(:)13 b(:)g(:)39 b Fm(^)f |
|
5542 Fn(S)t Fk(ss)1875 3485 y Fi(n)1918 3477 y Fw([)p Fk(as)7 |
|
5543 b Fw(;)13 b Fk(xs)7 b Fw(])38 b Fm(^)p .80 .80 .80 TeXcolorrgb |
|
5544 .80 .80 .80 TeXcolorrgb 2272 3516 294 117 v .80 .80 .80 |
|
5545 TeXcolorrgb Black 64 w Fk(as)28 b Fw(#)21 b Fn(C)p .80 .80 .80 |
|
5546 TeXcolorrgb Black 73 w Fm(\))g Fn(P)i(C)18 b Fk(ts)7 |
|
5547 b Fw([)p Fk(as)g Fw(;)13 b Fk(xs)7 b Fw(])651 3533 y |
|
5548 Fn(:)13 b(:)g(:)p 639 3580 2514 4 v 1665 3648 a(R)g Fk(ts)28 |
|
5549 b Fm(\))21 b Fn(P)j(C)18 b Fk(ts)1148 b Fw(\(12\))523 |
|
5550 3826 y Fz(can)33 b(b)r(e)h(used,)f(where)g Fr(C)40 b |
|
5551 Fz(stands)33 b(for)f(an)h Fs(induction)j(c)l(ontext)p |
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5552 Fz(.)c(This)i(induction)f(con)n(text)523 3926 y(can)25 |
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5553 b(b)r(e)i(instan)n(tiated)e(appropriately)f(\(w)n(e)i(will)g(explain)g |
|
5554 (this)g(in)g(the)g(next)g(section\).)g(The)523 4026 y(only)f |
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5555 (requiremen)n(t)g(w)n(e)g(ha)n(v)n(e)f(ab)r(out)i Fr(C)32 |
|
5556 b Fz(is)25 b(that)h(it)g(needs)f(to)h(b)r(e)g(\014nitely)g(supp)r |
|
5557 (orted.)f(The)523 4125 y(main)h(di\013erence)g(b)r(et)n(w)n(een)g(the)g |
|
5558 (stronger)e(induction)i(principle)g(in)g(\(12\))g(and)g(the)g(w)n(eak)n |
|
5559 (er)523 4225 y(one)36 b(in)g(\(11\))g(is)g(that)g(in)h(a)f(pro)r(of)f |
|
5560 (using)h(the)g(stronger)f(w)n(e)h(can)f(assume)h(that)g(the)h |
|
5561 Fs(as)7 b Fz(,)523 4325 y(i.e.)37 b(the)g(v)-5 b(ariables)35 |
|
5562 b(in)h(binding-p)r(osition,)h(are)e(fresh)h(with)h(resp)r(ect)f(to)h |
|
5563 (the)g(con)n(text)f Fr(C)523 4424 y Fz(\(see)g(highligh)n(ted)f |
|
5564 (freshness-condition\).)g(This)h(additional)f(assumption)h(allo)n(ws)e |
|
5565 (us)i(to)523 4524 y(reason)25 b(as)g(in)i(informal)e(\\pap)r(er-and-p)r |
|
5566 (encil")f(pro)r(ofs)i(where)f(one)h(assumes)f(the)i(v)-5 |
|
5567 b(ariable)523 4623 y(con)n(v)n(en)n(tion)26 b(\(w)n(e)i(will)f(also)g |
|
5568 (sho)n(w)g(this)h(in)f(the)h(next)g(section\).)648 4725 |
|
5569 y(The)33 b(\014rst)g(condition)h(of)f(v)n(c-compatibilit)n(y)g(implies) |
|
5570 g(that)h(the)g(inductiv)n(ely)g(de\014ned)523 4825 y(predicate)25 |
|
5571 b Fr(R)i Fz(is)f(equiv)-5 b(arian)n(t)24 b(and)i(that)g(ev)n(ery)f(sc)n |
|
5572 (hematic)g(subterm)g(o)r(ccurring)g(in)h(a)f(rule)523 |
|
5573 4924 y(is)j(equiv)-5 b(arian)n(t.)p Black Black eop end |
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5574 %%Page: 10 10 |
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5575 TeXDict begin 10 9 bop Black 523 232 a Fw(10)199 b(Christian)27 |
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5576 b(Urban,)e(Stefan)h(Berghofer,)i(and)d(Mic)n(hael)i(Norrish)p |
|
5577 Black Black 523 448 a Ft(Lemma)j(3.)p Black 40 w Fs(\(a\))40 |
|
5578 b(If)f(al)t(l)h(functions)f(in)g(a)h(schematic)g(term)f |
|
5579 Fr(t)p Fz([)p Fs(as)7 b Fz(;)14 b Fs(xs)6 b Fz(])39 b |
|
5580 Fs(ar)l(e)h(e)l(quivariant,)523 548 y(then)32 b(\(viewe)l(d)h(as)g(a)f |
|
5581 (function\))g Fr(t)g Fs(is)h(e)l(quivariant,)g(that)f(is)h |
|
5582 Fr(\031)2480 557 y Fg(\001)2532 548 y Fr(t)p Fz([)p Fs(as)8 |
|
5583 b Fz(;)14 b Fs(xs)6 b Fz(])27 b(=)g Fr(t)p Fz([)p Fr(\031)3030 |
|
5584 557 y Fg(\001)3069 548 y Fs(as)7 b Fz(;)14 b Fr(\031)3239 |
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5585 557 y Fg(\001)3277 548 y Fs(xs)6 b Fz(])p Fs(.)523 648 |
|
5586 y(\(b\))27 b(If)g(al)t(l)h(functions)f(and)g(side-c)l(onditions)i(in)e |
|
5587 (the)g(rules)f(of)i(an)f(inductive)h(de\014nition)f(for)523 |
|
5588 747 y(the)j(pr)l(e)l(dic)l(ate)h Fr(R)f Fs(ar)l(e)g(e)l(quivariant,)h |
|
5589 (then)f Fr(R)g Fs(is)g(e)l(quivariant,)h(that)f(is)g(if)g |
|
5590 Fr(R)15 b Fs(ts)36 b(holds)31 b(than)523 847 y(also)g |
|
5591 Fr(R)14 b Fz(\()p Fr(\031)852 856 y Fg(\001)891 847 y |
|
5592 Fs(ts)6 b Fz(\))30 b Fs(holds.)p Black 523 1026 a(Pr)l(o)l(of.)p |
|
5593 Black 43 w Fz(The)c(\014rst)f(part)g(is)g(b)n(y)g(a)g(routine)g |
|
5594 (induction)h(on)f(the)h(structure)f(of)g(the)h(sc)n(hematic)523 |
|
5595 1125 y(term)j Fr(t)p Fz(.)f(The)h(second)f(part)g(is)g(b)n(y)g(a)g |
|
5596 (simple)h(rule)f(induction)h(using)f(the)h(w)n(eak)f(induction)523 |
|
5597 1225 y(principle)g(giv)n(en)e(in)i(\(11\).)523 1404 y(W)-7 |
|
5598 b(e)31 b(no)n(w)f(pro)n(v)n(e)e(our)i(main)g(theorem:)g(if)h(the)g |
|
5599 (rules)f(of)g(an)g(inductiv)n(e)h(de\014nition)f(are)g(v)n(c-)523 |
|
5600 1503 y(compatible,)d(then)i(the)f(strong)e(induction)i(principle)f(in)h |
|
5601 (\(12\))g(holds.)p Black 523 1683 a Ft(Theorem)i(1.)p |
|
5602 Black 41 w Fs(Given)42 b(an)f(inductive)h(de\014nition)g(for)g(the)f |
|
5603 (pr)l(e)l(dic)l(ate)h Fr(R)g Fs(involving)h(vc-)523 1783 |
|
5604 y(c)l(omp)l(atible)38 b(schematic)f(rules)g(only,)g(then)f(a)h(str)l |
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5605 (ong)f(induction)g(principle)j(is)d(available)523 1883 |
|
5606 y(for)c(this)g(de\014nition)f(establishing)i(the)f(implic)l(ation)h |
|
5607 Fr(R)14 b Fs(ts)32 b Fp(\))26 b Fr(P)g(C)20 b Fs(ts)37 |
|
5608 b(with)32 b(the)f(induction)523 1982 y(c)l(ontext)e Fr(C)36 |
|
5609 b Fs(b)l(eing)30 b(\014nitely)g(supp)l(orte)l(d.)p Black |
|
5610 523 2153 a(Pr)l(o)l(of.)p Black 43 w Fz(W)-7 b(e)33 b(need)f(to)g |
|
5611 (establish)g Fr(R)15 b Fs(ts)37 b Fp(\))31 b Fr(P)25 |
|
5612 b(C)20 b Fs(ts)39 b Fz(using)32 b(the)g(implications)g(indicated)h(in) |
|
5613 523 2244 y(\(12\).)c(T)-7 b(o)30 b(do)f(so)g(w)n(e)h(will)g(use)f(the)h |
|
5614 (w)n(eak)f(induction)h(from)f(\(11\))h(and)f(establish)h(that)g(the)523 |
|
5615 2335 y(prop)r(osition)d Fr(R)14 b Fs(ts)30 b Fp(\))23 |
|
5616 b(8)p Fr(\031)16 b(C)q(:P)26 b(C)20 b Fz(\()p Fr(\031)1668 |
|
5617 2344 y Fg(\001)1707 2335 y Fs(ts)6 b Fz(\))28 b(holds.)g(F)-7 |
|
5618 b(or)27 b(eac)n(h)g(sc)n(hematic)g(rule)g Fr(\045)p Fz([)p |
|
5619 Fs(as)7 b Fz(;)14 b Fs(xs)6 b Fz(])880 2503 y Fn(R)13 |
|
5620 b Fk(ts)1014 2511 y Fx(1)1049 2503 y Fw([)p Fk(as)7 b |
|
5621 Fw(;)13 b Fk(xs)7 b Fw(])34 b Fn(:)13 b(:)h(:)34 b(R)13 |
|
5622 b Fk(ts)1567 2511 y Fi(n)1610 2503 y Fw([)p Fk(as)7 b |
|
5623 Fw(;)13 b Fk(xs)7 b Fw(])77 b Fn(S)1960 2511 y Fx(1)2007 |
|
5624 2503 y Fk(ss)2076 2511 y Fx(1)2111 2503 y Fw([)p Fk(as)7 |
|
5625 b Fw(;)13 b Fk(xs)7 b Fw(])34 b Fn(:)13 b(:)g(:)35 b(S)2542 |
|
5626 2511 y Fi(m)2613 2503 y Fk(ss)2682 2511 y Fi(m)2741 2503 |
|
5627 y Fw([)p Fk(as)7 b Fw(;)13 b Fk(xs)7 b Fw(])p 868 2550 |
|
5628 2111 4 v 1743 2622 a Fn(R)14 b Fk(ts)6 b Fw([)p Fk(as)h |
|
5629 Fw(;)14 b Fk(xs)6 b Fw(])3020 2558 y Fn(\045)523 2789 |
|
5630 y Fz(in)28 b(the)g(inductiv)n(e)g(de\014nition)g(w)n(e)f(ha)n(v)n(e)g |
|
5631 (to)h(analyse)e(one)h(case.)g(The)h(reasoning)e(pro)r(ceeds)523 |
|
5632 2880 y(in)g(eac)n(h)e(of)i(them)g(as)f(follo)n(ws:)f(By)h(induction)h |
|
5633 (h)n(yp)r(othesis)f(and)h(side-conditions)e(w)n(e)h(ha)n(v)n(e)1005 |
|
5634 3045 y Fw(\()p Fm(8)p Fn(\031)15 b(C)q(:P)24 b(C)18 b |
|
5635 Fw(\()p Fn(\031)1437 3054 y Fg(\001)1474 3045 y Fk(ts)1537 |
|
5636 3053 y Fx(1)1572 3045 y Fw([)p Fk(as)7 b Fw(;)13 b Fk(xs)7 |
|
5637 b Fw(]\)\))34 b Fn(:)13 b(:)g(:)35 b Fw(\()p Fm(8)p Fn(\031)15 |
|
5638 b(C)q(:P)23 b(C)18 b Fw(\()p Fn(\031)2447 3054 y Fg(\001)2485 |
|
5639 3045 y Fk(ts)2548 3053 y Fi(n)2590 3045 y Fw([)p Fk(as)7 |
|
5640 b Fw(;)14 b Fk(xs)6 b Fw(]\)\))p Black 392 w(\(13\))p |
|
5641 Black 1350 3161 a Fn(S)1397 3169 y Fx(1)1444 3161 y Fk(ss)1513 |
|
5642 3169 y Fx(1)1548 3161 y Fw([)p Fk(as)h Fw(;)13 b Fk(xs)7 |
|
5643 b Fw(])34 b Fn(:)13 b(:)g(:)35 b(S)1979 3169 y Fi(m)2050 |
|
5644 3161 y Fk(ss)2119 3169 y Fi(m)2178 3161 y Fw([)p Fk(as)7 |
|
5645 b Fw(;)13 b Fk(xs)7 b Fw(])p Black 864 w(\(14\))p Black |
|
5646 523 3328 a Fz(hold.)43 b(Since)f Fr(\045)h Fz(is)f(assumed)g(to)h(b)r |
|
5647 (e)g(v)n(c-compatible,)e(w)n(e)h(ha)n(v)n(e)g(b)n(y)g(Lem.)h(3)f(that)h |
|
5648 (\(*\))523 3419 y Fr(\031)573 3428 y Fg(\001)612 3419 |
|
5649 y Fs(ts)680 3431 y Fq(i)708 3419 y Fz([)p Fs(as)7 b Fz(;)14 |
|
5650 b Fs(xs)6 b Fz(])39 b(is)g(equal)f(to)h Fs(ts)1499 3431 |
|
5651 y Fq(i)1526 3419 y Fz([)p Fr(\031)1599 3428 y Fg(\001)1638 |
|
5652 3419 y Fs(as)7 b Fz(;)14 b Fr(\031)1808 3428 y Fg(\001)1846 |
|
5653 3419 y Fs(xs)7 b Fz(])39 b(in)g(\(13\).)g(F)-7 b(or)38 |
|
5654 b(\(14\))h(w)n(e)f(can)h(further)g(infer)523 3511 y(from)27 |
|
5655 b(the)h(v)n(c-compatibilit)n(y)f(of)g Fr(\045)h Fz(that)1195 |
|
5656 3676 y Fw(\()p Fn(a)p Fw(\))20 b Fk(as)28 b Fw(#)22 b |
|
5657 Fk(ts)7 b Fw([)p Fk(as)g Fw(;)13 b Fk(xs)7 b Fw(])154 |
|
5658 b(and)f(\()p Fn(b)p Fw(\))21 b Fn(distinct)p Fw(\()p |
|
5659 Fk(as)6 b Fw(\))535 b(\(15\))523 3850 y Fz(hold.)27 b(W)-7 |
|
5660 b(e)27 b(ha)n(v)n(e)e(to)i(sho)n(w)e(that)i Fr(P)f(C)20 |
|
5661 b Fz(\()p Fr(\031)1786 3859 y Fg(\001)1825 3850 y Fs(ts)6 |
|
5662 b Fz([)p Fs(as)h Fz(;)14 b Fs(xs)6 b Fz(]\))28 b(holds,)e(whic)n(h)g(b) |
|
5663 r(ecause)h(of)f(Lem.)h(3)f(is)523 3950 y(equiv)-5 b(alen)n(t)27 |
|
5664 b(to)h Fr(P)e(C)20 b Fs(ts)6 b Fz([)p Fr(\031)1318 3959 |
|
5665 y Fg(\001)1357 3950 y Fs(as)h Fz(;)14 b Fr(\031)1527 |
|
5666 3959 y Fg(\001)1565 3950 y Fs(xs)6 b Fz(].)523 4095 y(The)25 |
|
5667 b(pro)r(of)g(pro)r(ceeds)g(b)n(y)g(using)g(Lem.)g(2)g(and)h(c)n(ho)r |
|
5668 (osing)e(for)h(ev)n(ery)f(atom)h Fr(a)g Fz(in)h Fs(as)32 |
|
5669 b Fz(a)25 b(fresh)523 4186 y(atom)i Fr(c)h Fz(suc)n(h)f(that)h(for)f |
|
5670 (all)g(the)h Fs(cs)35 b Fz(the)28 b(follo)n(wing)e(holds:)693 |
|
5671 4351 y Fw(\()p Fn(a)p Fw(\))21 b Fk(cs)28 b Fw(#)21 b |
|
5672 Fk(ts)7 b Fw([)p Fn(\031)1125 4360 y Fg(\001)1163 4351 |
|
5673 y Fk(as)g Fw(;)13 b Fn(\031)1321 4360 y Fg(\001)1359 |
|
5674 4351 y Fk(xs)7 b Fw(])107 b(\()p Fn(b)p Fw(\))21 b Fk(cs)28 |
|
5675 b Fw(#)21 b Fn(\031)1900 4360 y Fg(\001)1938 4351 y Fk(as)113 |
|
5676 b Fw(\()p Fn(c)p Fw(\))22 b Fk(cs)28 b Fw(#)21 b Fn(C)112 |
|
5677 b Fw(\()p Fn(d)p Fw(\))21 b Fn(distinct)p Fw(\()p Fk(cs)6 |
|
5678 b Fw(\))170 b(\(16\))523 4526 y Fz(Suc)n(h)31 b Fs(cs)39 |
|
5679 b Fz(alw)n(a)n(ys)29 b(exists:)i(the)h(\014rst)f(and)g(the)h(second)e |
|
5680 (prop)r(ert)n(y)h(can)f(b)r(e)i(obtained)f(since)523 |
|
5681 4625 y(the)23 b(sc)n(hematic)e(terms)h Fs(ts)6 b Fz([)p |
|
5682 Fr(\031)1403 4634 y Fg(\001)1442 4625 y Fs(as)h Fz(;)14 |
|
5683 b Fr(\031)1612 4634 y Fg(\001)1650 4625 y Fs(xs)7 b Fz(])22 |
|
5684 b(and)g Fr(\031)1981 4634 y Fg(\001)2019 4625 y Fs(as)29 |
|
5685 b Fz(stand)22 b(for)g(\014nitely)h(supp)r(orted)e(ob)5 |
|
5686 b(jects;)523 4725 y(the)25 b(third)h(can)e(also)g(b)r(e)i(obtained)e |
|
5687 (since)h(w)n(e)g(assumed)f(that)h(the)h(induction)f(con)n(text)g |
|
5688 Fr(C)31 b Fz(is)523 4825 y(\014nitely)25 b(supp)r(orted;)f(the)g(last)g |
|
5689 (can)g(b)r(e)g(obtained)g(b)n(y)g(c)n(ho)r(osing)f(the)h |
|
5690 Fs(cs)31 b Fz(one)24 b(after)g(another)523 4924 y(a)n(v)n(oiding)i(the) |
|
5691 i(ones)f(that)h(ha)n(v)n(e)e(already)g(b)r(een)i(c)n(hosen.)p |
|
5692 Black Black eop end |
|
5693 %%Page: 11 11 |
|
5694 TeXDict begin 11 10 bop Black 1347 232 a Fw(Barendregt's)27 |
|
5695 b(V)-6 b(ariable)26 b(Con)n(v)n(en)n(tion)f(in)h(Rule)f(Inductions)198 |
|
5696 b(11)p Black 648 461 a Fz(No)n(w)19 b(w)n(e)h(form)f(the)i(p)r(erm)n |
|
5697 (utation)e Fr(\031)1785 431 y Fj(0)1832 414 y Fy(def)1843 |
|
5698 461 y Fz(=)34 b(\()p Fr(\031)2024 470 y Fg(\001)2063 |
|
5699 461 y Fs(as)44 b(cs)7 b Fz(\))20 b(where)f(\()p Fr(\031)2628 |
|
5700 470 y Fg(\001)2667 461 y Fs(as)44 b(cs)7 b Fz(\))21 b(stands)e(for)h |
|
5701 (the)523 552 y(sequence)29 b(of)h(sw)n(appings)f(\()p |
|
5702 Fr(\031)1446 561 y Fg(\001)1484 552 y Fr(a)1528 564 y |
|
5703 Fy(1)1606 552 y Fr(c)1642 564 y Fy(1)1679 552 y Fz(\))14 |
|
5704 b Fr(:)g(:)g(:)g Fz(\()p Fr(\031)1918 561 y Fg(\001)1957 |
|
5705 552 y Fr(a)2001 564 y Fq(j)2076 552 y Fr(c)2112 564 y |
|
5706 Fq(j)2147 552 y Fz(\).)31 b(Since)f(p)r(erm)n(utations)f(are)g |
|
5707 (bijectiv)n(e)523 644 y(renamings,)24 b(w)n(e)i(can)f(infer)g(from)h |
|
5708 (\(15.)p Fr(b)p Fz(\))f(that)h Fs(distinct)8 b Fz(\()p |
|
5709 Fr(\031)2363 653 y Fg(\001)2402 644 y Fs(as)f Fz(\))26 |
|
5710 b(holds.)f(This)g(and)h(the)g(fact)523 735 y(in)i(\(16.)p |
|
5711 Fr(d)p Fz(\))f(implies)h(that)1516 826 y Fn(\031)1563 |
|
5712 790 y Fl(0)1585 826 y Fw(@)p Fn(\031)1692 835 y Fg(\001)1729 |
|
5713 826 y Fk(as)g Fw(=)22 b Fn(\031)1956 790 y Fl(0)1978 |
|
5714 835 y Fg(\001)2016 826 y Fw(\()p Fn(\031)2093 835 y Fg(\001)2130 |
|
5715 826 y Fk(as)7 b Fw(\))21 b(=)g Fk(cs)863 b Fw(\(17\))523 |
|
5716 947 y Fz(W)-7 b(e)29 b(then)h(instan)n(tiate)e(the)i |
|
5717 Fr(\031)i Fz(in)d(the)h(induction)f(h)n(yp)r(otheses)f(giv)n(en)g(in)h |
|
5718 (\(13\))g(with)g Fr(\031)3266 917 y Fj(0)3290 947 y Fz(@)p |
|
5719 Fr(\031)523 1039 y Fz(and)e(obtain)h(using)f(\(17\))g(and)h(\(*\))f(so) |
|
5720 g(that)993 1194 y Fw(\()p Fm(8)p Fn(C)q(:P)c(C)18 b Fk(ts)1351 |
|
5721 1202 y Fx(1)1385 1194 y Fw([)p Fk(cs)8 b Fw(;)13 b Fn(\031)1561 |
|
5722 1158 y Fl(0)1583 1194 y Fw(@)p Fn(\031)1690 1203 y Fg(\001)1727 |
|
5723 1194 y Fk(xs)7 b Fw(]\)\))34 b Fn(:)13 b(:)g(:)35 b Fw(\()p |
|
5724 Fm(8)p Fn(C)q(:P)23 b(C)18 b Fk(ts)2397 1202 y Fi(n)2440 |
|
5725 1194 y Fw([)p Fk(cs)7 b Fw(;)13 b Fn(\031)2615 1158 y |
|
5726 Fl(0)2637 1194 y Fw(@)p Fn(\031)2744 1203 y Fg(\001)2781 |
|
5727 1194 y Fk(xs)7 b Fw(]\)\))333 b(\(18\))523 1349 y Fz(hold.)27 |
|
5728 b(Since)f(the)h(rule)f Fr(\045)h Fz(is)f(v)n(c-compatible,)g(w)n(e)g |
|
5729 (can)g(infer)g(from)h(\(14\))f(and)g(the)h(equiv)-5 b(ari-)523 |
|
5730 1440 y(ance)27 b(of)h(the)g(side-conditions)e(that)1227 |
|
5731 1595 y Fn(S)1274 1603 y Fx(1)1322 1595 y Fk(ss)1390 1603 |
|
5732 y Fx(1)1425 1595 y Fw([)p Fk(cs)7 b Fw(;)13 b Fn(\031)1600 |
|
5733 1559 y Fl(0)1622 1595 y Fw(@)p Fn(\031)1729 1604 y Fg(\001)1767 |
|
5734 1595 y Fk(xs)7 b Fw(])34 b Fn(:)13 b(:)g(:)35 b(S)2066 |
|
5735 1603 y Fi(m)2137 1595 y Fk(ss)2206 1603 y Fi(m)2265 1595 |
|
5736 y Fw([)p Fk(cs)7 b Fw(;)13 b Fn(\031)2440 1559 y Fl(0)2462 |
|
5737 1595 y Fw(@)p Fn(\031)2569 1604 y Fg(\001)2606 1595 y |
|
5738 Fk(xs)7 b Fw(])568 b(\(19\))523 1750 y Fz(hold)31 b(\(w)n(e)g(use)f |
|
5739 (here)h(the)g(fact)g(that)g Fr(\031)1745 1720 y Fj(0)1769 |
|
5740 1750 y Fz(@)p Fr(\031)1884 1759 y Fg(\001)1922 1750 y |
|
5741 Fz(\()p Fs(ss)2029 1762 y Fq(i)2057 1750 y Fz([)p Fs(as)7 |
|
5742 b Fz(;)14 b Fs(xs)6 b Fz(]\))31 b(is)g(equal)g(to)f Fs(ss)2854 |
|
5743 1762 y Fq(i)2882 1750 y Fz([)p Fs(cs)7 b Fz(;)14 b Fr(\031)3071 |
|
5744 1720 y Fj(0)3094 1750 y Fz(@)p Fr(\031)3209 1759 y Fg(\001)3247 |
|
5745 1750 y Fs(xs)7 b Fz(]\).)523 1841 y(F)-7 b(rom)29 b(\(16.)p |
|
5746 Fr(c)p Fz(\),)g(\(18\),)g(\(19\))g(and)g(the)h(implication)f(from)g |
|
5747 (the)g(strong)f(induction)i(principle)523 1933 y(w)n(e)d(can)g(infer)h |
|
5748 Fr(P)35 b(C)29 b Fs(ts)7 b Fz([)p Fs(cs)g Fz(;)14 b Fr(\031)1423 |
|
5749 1902 y Fj(0)1446 1933 y Fz(@)p Fr(\031)1561 1942 y Fg(\001)1599 |
|
5750 1933 y Fs(xs)7 b Fz(])27 b(whic)n(h)h(b)n(y)f(Lem.)h(3)f(is)g(equiv)-5 |
|
5751 b(alen)n(t)28 b(to)1600 2088 y Fn(P)k(C)26 b(\031)1809 |
|
5752 2052 y Fl(0)1831 2097 y Fg(\001)1869 2088 y Fk(ts)7 b |
|
5753 Fw([)p Fn(\031)2000 2097 y Fg(\001)2038 2088 y Fk(as)g |
|
5754 Fw(;)13 b Fn(\031)2196 2097 y Fg(\001)2234 2088 y Fk(xs)7 |
|
5755 b Fw(])940 b(\(20\))523 2243 y Fz(F)-7 b(rom)32 b(\(15.)p |
|
5756 Fr(a)p Fz(\))g(w)n(e)g(can)f(b)n(y)h(Lem.)h(1\()p Fr(a)p |
|
5757 Fz(\))f(infer)g(that)h Fr(\031)2223 2252 y Fg(\001)2261 |
|
5758 2243 y Fs(as)38 b Fz(#)31 b Fs(ts)6 b Fz([)p Fr(\031)2616 |
|
5759 2252 y Fg(\001)2655 2243 y Fs(as)h Fz(;)14 b Fr(\031)2825 |
|
5760 2252 y Fg(\001)2863 2243 y Fs(xs)7 b Fz(])32 b(holds.)g(This)523 |
|
5761 2334 y(ho)n(w)n(ev)n(er)26 b(implies)h(b)n(y)h(\(16.)p |
|
5762 Fr(a)p Fz(\))f(and)g(b)n(y)h(rep)r(eated)f(application)g(of)g(Lem.)h |
|
5763 (1\()p Fr(b)p Fz(\))f(that)1400 2489 y Fn(\031)1447 2453 |
|
5764 y Fl(0)1469 2498 y Fg(\001)1508 2489 y Fk(ts)6 b Fw([)p |
|
5765 Fn(\031)1638 2498 y Fg(\001)1677 2489 y Fk(as)g Fw(;)13 |
|
5766 b Fn(\031)1834 2498 y Fg(\001)1872 2489 y Fk(xs)7 b Fw(])22 |
|
5767 b(=)f Fk(ts)7 b Fw([)p Fn(\031)2200 2498 y Fg(\001)2238 |
|
5768 2489 y Fk(as)g Fw(;)13 b Fn(\031)2396 2498 y Fg(\001)2433 |
|
5769 2489 y Fk(xs)7 b Fw(])741 b(\(21\))523 2652 y Fz(Substituting)28 |
|
5770 b(this)f(equation)g(in)n(to)g(\(20\))f(establishes)h(the)g(pro)r(of)g |
|
5771 (obligation)e(for)i(the)g(rule)523 2752 y Fr(\045)p Fz(.)h(Pro)n(vided) |
|
5772 e(w)n(e)i(analysed)e(all)i(suc)n(h)g(cases,)e(w)n(e)i(ha)n(v)n(e)f(sho) |
|
5773 n(wn)g Fr(R)14 b Fs(ts)30 b Fp(\))24 b(8)p Fr(\031)16 |
|
5774 b(C)q(:P)26 b(C)20 b Fz(\()p Fr(\031)3242 2761 y Fg(\001)3281 |
|
5775 2752 y Fs(ts)7 b Fz(\).)523 2851 y(W)-7 b(e)22 b(obtain)e(our)h |
|
5776 (original)e(goal)h(b)n(y)h(instan)n(tiating)g Fr(\031)j |
|
5777 Fz(with)e(the)g(iden)n(tit)n(y)f(p)r(erm)n(utation.)83 |
|
5778 b Fp(u)-55 b(t)523 3110 y Fu(5)112 b(Examples)523 3302 |
|
5779 y Fz(W)-7 b(e)28 b(can)f(no)n(w)g(apply)h(our)e(tec)n(hnique)i(to)f |
|
5780 (the)h(examples)f(from)h(the)g(In)n(tro)r(duction.)523 |
|
5781 3545 y Ft(5.1)95 b(Simple)29 b(T)m(yping)523 3704 y Fz(Giv)n(en)24 |
|
5782 b(the)h(t)n(yping)f(relation)g(de\014ned)h(in)f(\(2\))q(,)g(w)n(e)g(m)n |
|
5783 (ust)h(\014rst)f(c)n(hec)n(k)g(the)h(conditions)f(sp)r(elt)523 |
|
5784 3803 y(out)34 b(in)g(De\014nition)h(5.)e(The)h(\014rst)g(condition)f |
|
5785 (is)h(that)g(all)g(of)f(the)i(de\014nition's)f(functions)523 |
|
5786 3903 y(\(namely)23 b Fs(var)10 b Fz(,)24 b Fs(app)6 b |
|
5787 Fz(,)24 b Fr(l)r(am)f Fz(and)g(::\))h(and)f(side-conditions)g(\(namely) |
|
5788 g Fs(valid)r Fz(,)h Fp(2)g Fz(and)f(#\))h(m)n(ust)523 |
|
5789 4003 y(b)r(e)33 b(equiv)-5 b(arian)n(t.)31 b(This)h(is)g(easily)f |
|
5790 (con\014rmed)h(\(see)g(Remark)f(1\).)h(The)h(second)e(condition)523 |
|
5791 4102 y(requires)25 b(that)i(all)g(v)-5 b(ariables)25 |
|
5792 b(in)i(binding)g(p)r(ositions)f(b)r(e)h(distinct)h(\(there)e(is)h(just) |
|
5793 g(one,)g(the)523 4202 y Fr(x)k Fz(in)g(T)n(yp)r(e)888 |
|
5794 4214 y Fy(3)925 4202 y Fz(\);)h(and)e(that)h(it)h(b)r(e)f(fresh)f(for)h |
|
5795 (all)f(the)h(terms)g(app)r(earing)f(in)h(the)g(conclusion)523 |
|
5796 4301 y(of)h(that)f(rule,)h(namely)f Fr(\000)41 b Fp(`)29 |
|
5797 b Fs(lam)7 b Fz(\()p Fr(x:M)i Fz(\))30 b(:)g Fr(T)1957 |
|
5798 4313 y Fy(1)2023 4301 y Fp(!)g Fr(T)2185 4313 y Fy(2)2222 |
|
5799 4301 y Fz(,)i(under)f(the)h(assumption)f(that)h(the)523 |
|
5800 4401 y(side-condition,)27 b Fr(x)c Fz(#)h Fr(\000)12 |
|
5801 b Fz(,)27 b(of)h(this)g(rule)f(holds.)648 4501 y(In)41 |
|
5802 b(this)g(case,)g(therefore,)f(w)n(e)h(m)n(ust)g(c)n(hec)n(k)g(that)g |
|
5803 Fr(x)46 b Fz(#)g Fr(\000)12 b Fz(,)41 b Fr(x)46 b Fz(#)g |
|
5804 Fs(lam)7 b Fz(\()p Fr(x:M)i Fz(\))42 b(and)523 4600 y |
|
5805 Fr(x)c Fz(#)f Fr(T)763 4612 y Fy(1)837 4600 y Fp(!)g |
|
5806 Fr(T)1006 4612 y Fy(2)1079 4600 y Fz(hold.)f(The)h(\014rst)e(is)h |
|
5807 (immediate)h(giv)n(en)e(our)g(assumption;)h(the)h(second)523 |
|
5808 4700 y(follo)n(ws)24 b(from)h(the)g(de\014nition)g(of)g(supp)r(ort)g |
|
5809 (for)f(lam)n(b)r(da-terms)g(\()p Fr(x)g Fz(#)f Fs(lam)7 |
|
5810 b Fz(\()p Fr(x:M)i Fz(\))25 b(for)g(all)f Fr(x)523 4800 |
|
5811 y Fz(and)f Fr(M)9 b Fz(\);)23 b(and)h(the)f(third)h(follo)n(ws)e(from)h |
|
5812 (the)h(de\014nition)f(of)g(supp)r(ort)h(for)e(t)n(yp)r(es)i(\(w)n(e)f |
|
5813 (de\014ne)523 4924 y(p)r(erm)n(utation)k(on)h(t)n(yp)r(es)f |
|
5814 Fr(T)39 b Fz(as)27 b Fr(\031)1573 4933 y Fg(\001)1611 |
|
5815 4924 y Fr(T)1694 4877 y Fy(def)1706 4924 y Fz(=)33 b |
|
5816 Fr(T)39 b Fz(and)27 b(th)n(us)h(obtain)f(that)h Fe(supp)o |
|
5817 Fz(\()p Fr(T)12 b Fz(\))22 b(=)h Fh(?)p Fz(\).)p Black |
|
5818 Black eop end |
|
5819 %%Page: 12 12 |
|
5820 TeXDict begin 12 11 bop Black 523 232 a Fw(12)199 b(Christian)27 |
|
5821 b(Urban,)e(Stefan)h(Berghofer,)i(and)d(Mic)n(hael)i(Norrish)p |
|
5822 Black 648 448 a Fz(With)38 b(these)f(conditions)g(established,)g |
|
5823 (Theorem)g(1)g(tells)h(us)f(that)h(the)g(strong,)e(or)523 |
|
5824 540 y(v)n(c-compatible)26 b(principle)i(exists,)f(and)h(that)g(it)g(is) |
|
5825 772 688 y Fm(8)p Fn(\000)c(x)12 b(T)23 b(C)q(:)f(v)s(al)q(id)p |
|
5826 Fw(\()p Fn(\000)11 b Fw(\))16 b Fm(^)h Fw(\()p Fn(x)k |
|
5827 Fw(:)h Fn(T)11 b Fw(\))20 b Fm(2)i Fn(\000)32 b Fm(\))21 |
|
5828 b Fn(P)32 b(C)26 b(\000)32 b Fw(\()p Fk(var)9 b Fw(\()p |
|
5829 Fn(x)p Fw(\)\))20 b Fn(T)772 803 y Fm(8)p Fn(\000)k(M)d(N)g(T)1108 |
|
5830 811 y Fx(1)1155 803 y Fn(T)1200 811 y Fx(2)1247 803 y |
|
5831 Fn(C)q(:)33 b Fw(\()p Fm(8)p Fn(C)q(:)43 b(P)32 b(C)26 |
|
5832 b(\000)32 b(M)e Fw(\()p Fn(T)1970 811 y Fx(1)2025 803 |
|
5833 y Fm(!)21 b Fn(T)2168 811 y Fx(2)2203 803 y Fw(\)\))38 |
|
5834 b Fm(^)h Fw(\()p Fm(8)p Fn(C)q(:)j(P)32 b(C)26 b(\000)32 |
|
5835 b(N)e(T)2960 811 y Fx(1)2995 803 y Fw(\))42 b Fm(\))1357 |
|
5836 894 y Fn(P)32 b(C)27 b(\000)32 b Fw(\()p Fk(app)5 b Fw(\()p |
|
5837 Fn(M)t(;)13 b(N)8 b Fw(\)\))21 b Fn(T)2088 902 y Fx(2)772 |
|
5838 985 y Fm(8)p Fn(\000)j(x)12 b(M)21 b(T)1082 993 y Fx(1)1129 |
|
5839 985 y Fn(T)1174 993 y Fx(2)1221 985 y Fn(C)q(:)33 b(x)21 |
|
5840 b Fw(#)h Fn(\000)49 b Fm(^)38 b Fw(\()p Fm(8)p Fn(C)q(:)21 |
|
5841 b(P)32 b(C)27 b Fw(\(\()p Fn(x)20 b Fw(:)i Fn(T)2213 |
|
5842 993 y Fx(1)2247 985 y Fw(\))9 b(::)g Fn(\000)i Fw(\))21 |
|
5843 b Fn(M)29 b(T)2594 993 y Fx(2)2629 985 y Fw(\))38 b Fm(^)h |
|
5844 Fn(x)20 b Fw(#)i Fn(C)48 b Fm(\))1331 1077 y Fn(P)32 |
|
5845 b(C)27 b(\000)32 b Fw(\()p Fk(lam)5 b Fw(\()p Fn(x:M)j |
|
5846 Fw(\)\))21 b(\()p Fn(T)2063 1085 y Fx(1)2119 1077 y Fm(!)g |
|
5847 Fn(T)2262 1085 y Fx(2)2296 1077 y Fw(\))p 772 1124 2383 |
|
5848 4 v 1508 1192 a Fn(\000)32 b Fm(`)21 b Fn(M)30 b Fw(:)22 |
|
5849 b Fn(T)53 b Fm(\))42 b Fn(P)32 b(C)27 b(\000)32 b(M)d(T)523 |
|
5850 1344 y Fz(This)24 b(principle)g(can)f(no)n(w)g(b)r(e)h(used)g(to)g |
|
5851 (establish)f(the)h(w)n(eak)n(ening)f(result.)g(The)h(statemen)n(t)523 |
|
5852 1435 y(is)1089 1526 y Fn(\000)32 b Fm(`)21 b Fn(M)29 |
|
5853 b Fw(:)22 b Fn(T)53 b Fm(\))43 b Fn(\000)32 b Fm(\022)21 |
|
5854 b Fn(\000)1818 1490 y Fl(0)1882 1526 y Fm(\))43 b Fk(valid)7 |
|
5855 b Fw(\()p Fn(\000)2254 1490 y Fl(0)2276 1526 y Fw(\))43 |
|
5856 b Fm(\))f Fn(\000)2526 1490 y Fl(0)2569 1526 y Fm(`)22 |
|
5857 b Fn(M)29 b Fw(:)22 b Fn(T)439 b Fw(\(22\))523 1645 y |
|
5858 Fz(With)25 b(the)g(strong)d(induction)j(principle,)f(the)g(formal)g |
|
5859 (pro)r(of)f(of)h(this)h(statemen)n(t)f(pro)r(ceeds)523 |
|
5860 1737 y(lik)n(e)35 b(the)g(informal)f(one)h(giv)n(en)f(in)h(the)g(In)n |
|
5861 (tro)r(duction.)g(There,)f(in)h(the)g(T)n(yp)r(e)3032 |
|
5862 1749 y Fy(3)3104 1737 y Fz(case,)f(w)n(e)523 1828 y(used)24 |
|
5863 b(the)f(v)-5 b(ariable)23 b(con)n(v)n(en)n(tion)f(to)h(assume)g(that)h |
|
5864 (the)g(b)r(ound)g Fr(x)g Fz(w)n(as)e(fresh)i(for)f Fr(\000)3123 |
|
5865 1798 y Fj(0)3146 1828 y Fz(.)g(Giv)n(en)523 1919 y(this)j(information,) |
|
5866 f(w)n(e)g(instan)n(tiate)g(the)g(induction)h(con)n(text)f |
|
5867 Fr(C)32 b Fz(in)26 b(the)f(strong)g(induction)523 2011 |
|
5868 y(principle)d(with)h Fr(\000)1106 1980 y Fj(0)1151 2011 |
|
5869 y Fz(\(whic)n(h)g(is)f(\014nitely)h(supp)r(orted\).)f(The)g(complete)h |
|
5870 (instan)n(tiation)e(of)i(the)523 2102 y(v)n(c-compatible)j(induction)i |
|
5871 (principle)g(is)1095 2250 y Fn(P)k Fw(=)21 b Fn(\025\000)i(M)e(T)i |
|
5872 (\000)1593 2218 y Fl(0)1615 2250 y Fn(:)43 b(\000)32 |
|
5873 b Fm(\022)21 b Fn(\000)1897 2218 y Fl(0)1940 2250 y Fm(\))g |
|
5874 Fk(valid)8 b Fw(\()p Fn(\000)2291 2218 y Fl(0)2313 2250 |
|
5875 y Fw(\))21 b Fm(\))g Fn(\000)2520 2218 y Fl(0)2563 2250 |
|
5876 y Fm(`)h Fn(M)29 b Fw(:)22 b Fn(T)1033 2341 y(C)27 b |
|
5877 Fw(=)21 b Fn(\000)1254 2310 y Fl(0)1583 2341 y Fn(\000)32 |
|
5878 b Fw(=)21 b Fn(\000)317 b(M)30 b Fw(=)21 b Fn(M)316 b(T)31 |
|
5879 b Fw(=)21 b Fn(T)523 2499 y Fz(whic)n(h)h(after)g(some)f(b)r(eta-con)n |
|
5880 (tractions)f(giv)n(es)h(us)h(the)g(statemen)n(t)g(in)h(\(22\).)f(The)g |
|
5881 (induction)523 2599 y(cases)27 b(are)f(then)i(as)f(follo)n(ws)g |
|
5882 (\(stripping)g(o\013)h(the)g(outermost)f(quan)n(ti\014ers\):)632 |
|
5883 2767 y(\(1\))48 b Fs(valid)10 b Fz(\()p Fr(\000)i Fz(\))42 |
|
5884 b Fp(^)g Fz(\()p Fr(x)23 b Fz(:)h Fr(T)12 b Fz(\))22 |
|
5885 b Fp(2)h Fr(\000)58 b Fp(\))46 b Fr(\000)35 b Fp(\022)23 |
|
5886 b Fr(\000)2047 2737 y Fj(0)2116 2767 y Fp(\))46 b Fs(valid)10 |
|
5887 b Fz(\()p Fr(\000)2518 2737 y Fj(0)2541 2767 y Fz(\))24 |
|
5888 b Fp(\))f Fr(\000)2766 2737 y Fj(0)2812 2767 y Fp(`)f |
|
5889 Fs(var)10 b Fz(\()p Fr(x)p Fz(\))24 b(:)f Fr(T)632 2914 |
|
5890 y Fz(\(2\))48 b(\()p Fp(8)p Fr(\000)928 2884 y Fj(00)969 |
|
5891 2914 y Fr(:)23 b(\000)35 b Fp(\022)23 b Fr(\000)1252 |
|
5892 2884 y Fj(00)1317 2914 y Fp(\))g Fs(valid)10 b Fz(\()p |
|
5893 Fr(\000)1696 2884 y Fj(00)1738 2914 y Fz(\))24 b Fp(\))f |
|
5894 Fr(\000)1963 2884 y Fj(00)2028 2914 y Fp(`)g Fr(M)2183 |
|
5895 2926 y Fy(1)2243 2914 y Fz(:)g Fr(T)2338 2926 y Fy(1)2398 |
|
5896 2914 y Fp(!)g Fr(T)2553 2926 y Fy(2)2589 2914 y Fz(\))h |
|
5897 Fp(^)786 3013 y Fz(\()p Fp(8)p Fr(\000)928 2983 y Fj(00)969 |
|
5898 3013 y Fr(:)f(\000)35 b Fp(\022)23 b Fr(\000)1252 2983 |
|
5899 y Fj(00)1317 3013 y Fp(\))g Fs(valid)10 b Fz(\()p Fr(\000)1696 |
|
5900 2983 y Fj(00)1738 3013 y Fz(\))24 b Fp(\))f Fr(\000)1963 |
|
5901 2983 y Fj(00)2028 3013 y Fp(`)g Fr(M)2183 3025 y Fy(2)2243 |
|
5902 3013 y Fz(:)g Fr(T)2338 3025 y Fy(1)2375 3013 y Fz(\))46 |
|
5903 b Fp(\))786 3113 y Fr(\000)34 b Fp(\022)23 b Fr(\000)1022 |
|
5904 3083 y Fj(0)1091 3113 y Fp(\))46 b Fs(valid)11 b Fz(\()p |
|
5905 Fr(\000)1494 3083 y Fj(0)1517 3113 y Fz(\))46 b Fp(\))g |
|
5906 Fr(\000)1787 3083 y Fj(0)1833 3113 y Fp(`)23 b Fs(app)7 |
|
5907 b Fz(\()p Fr(M)2153 3125 y Fy(1)2190 3113 y Fr(;)14 b(M)2308 |
|
5908 3125 y Fy(2)2344 3113 y Fz(\))24 b(:)f Fr(T)2495 3125 |
|
5909 y Fy(2)632 3260 y Fz(\(3\))48 b(\()p Fp(8)p Fr(\000)928 |
|
5910 3230 y Fj(00)969 3260 y Fr(:)23 b Fz(\()p Fr(x)h Fz(:)f |
|
5911 Fr(T)1213 3272 y Fy(1)1250 3260 y Fz(\))9 b(::)g Fr(\000)36 |
|
5912 b Fp(\022)22 b Fr(\000)1583 3230 y Fj(00)1671 3260 y |
|
5913 Fp(\))h Fs(valid)11 b Fz(\()p Fr(\000)2051 3230 y Fj(00)2093 |
|
5914 3260 y Fz(\))46 b Fp(\))h Fr(\000)2364 3230 y Fj(00)2429 |
|
5915 3260 y Fp(`)23 b Fr(M)31 b Fz(:)23 b Fr(T)2710 3272 y |
|
5916 Fy(2)2747 3260 y Fz(\))42 b Fp(^)g Fr(x)23 b Fz(#)g Fr(\000)3143 |
|
5917 3230 y Fj(0)3213 3260 y Fp(\))786 3360 y Fr(\000)34 b |
|
5918 Fp(\022)23 b Fr(\000)1022 3329 y Fj(0)1091 3360 y Fp(\))46 |
|
5919 b Fs(valid)11 b Fz(\()p Fr(\000)1494 3329 y Fj(0)1517 |
|
5920 3360 y Fz(\))46 b Fp(\))g Fr(\000)1787 3329 y Fj(0)1833 |
|
5921 3360 y Fp(`)23 b Fs(lam)7 b Fz(\()p Fr(x:M)i Fz(\))24 |
|
5922 b(:)f Fr(T)2388 3372 y Fy(1)2448 3360 y Fp(!)g Fr(T)2603 |
|
5923 3372 y Fy(2)523 3530 y Fz(The)44 b(\014rst)f(t)n(w)n(o)h(cases)e(are)h |
|
5924 (trivial.)g(F)-7 b(or)43 b(\(3\),)h(w)n(e)g(instan)n(tiate)f |
|
5925 Fr(\000)2705 3499 y Fj(00)2791 3530 y Fz(in)h(the)h(induction)523 |
|
5926 3629 y(h)n(yp)r(othesis)39 b(to)h(b)r(e)g(\()p Fr(x)k |
|
5927 Fz(:)g Fr(T)1422 3641 y Fy(1)1458 3629 y Fz(\))30 b(::)g |
|
5928 Fr(\000)1659 3599 y Fj(0)1682 3629 y Fz(.)40 b(F)-7 b(rom)39 |
|
5929 b(the)h(assumption)f Fr(\000)55 b Fp(\022)43 b Fr(\000)2858 |
|
5930 3599 y Fj(0)2921 3629 y Fz(w)n(e)c(ha)n(v)n(e)g(\()p |
|
5931 Fr(x)44 b Fz(:)523 3729 y Fr(T)572 3741 y Fy(1)609 3729 |
|
5932 y Fz(\))18 b(::)f Fr(\000)43 b Fp(\022)31 b Fz(\()p Fr(x)h |
|
5933 Fz(:)f Fr(T)1126 3741 y Fy(1)1163 3729 y Fz(\))18 b(::)g |
|
5934 Fr(\000)1340 3699 y Fj(0)1362 3729 y Fz(.)33 b(Moreo)n(v)n(er)d(from)i |
|
5935 (the)h(assumption)f Fs(valid)10 b Fz(\()p Fr(\000)2854 |
|
5936 3699 y Fj(0)2878 3729 y Fz(\))33 b(w)n(e)f(also)f(ha)n(v)n(e)523 |
|
5937 3828 y Fs(valid)10 b Fz(\(\()p Fr(x)26 b Fz(:)e Fr(T)934 |
|
5938 3840 y Fy(1)971 3828 y Fz(\))10 b(::)g Fr(\000)1132 3798 |
|
5939 y Fj(0)1156 3828 y Fz(\))29 b(using)e(the)i(v)-5 b(ariable)27 |
|
5940 b(con)n(v)n(en)n(tion's)g Fr(x)e Fz(#)f Fr(\000)2590 |
|
5941 3798 y Fj(0)2613 3828 y Fz(.)29 b(Hence)f(w)n(e)g(can)g(deriv)n(e)523 |
|
5942 3928 y(\()p Fr(x)d Fz(:)f Fr(T)723 3940 y Fy(1)760 3928 |
|
5943 y Fz(\))10 b(::)g Fr(\000)921 3898 y Fj(0)970 3928 y |
|
5944 Fp(`)24 b Fr(M)33 b Fz(:)24 b Fr(T)1255 3940 y Fy(2)1320 |
|
5945 3928 y Fz(using)k(the)h(induction)f(h)n(yp)r(othesis.)g(No)n(w)g |
|
5946 (applying)g(rule)g(T)n(yp)r(e)3368 3940 y Fy(3)523 4028 |
|
5947 y Fz(w)n(e)i(can)g(obtain)g Fr(\000)1127 3998 y Fj(0)1177 |
|
5948 4028 y Fp(`)d Fs(lam)7 b Fz(\()p Fr(x:M)i Fz(\))28 b(:)f |
|
5949 Fr(T)1744 4040 y Fy(1)1809 4028 y Fp(!)g Fr(T)1968 4040 |
|
5950 y Fy(2)2005 4028 y Fz(,)j(again)f(using)h(the)h(v)-5 |
|
5951 b(ariable)29 b(con)n(v)n(en)n(tion's)523 4127 y Fr(x)36 |
|
5952 b Fz(#)g Fr(\000)774 4097 y Fj(0)797 4127 y Fz(.)f(This)g(completes)g |
|
5953 (the)h(pro)r(of.)e(Its)h Fs(r)l(e)l(adable)44 b Fz(v)n(ersion)33 |
|
5954 b(expressed)h(in)i(Isab)r(elle's)523 4227 y(Isar-language)19 |
|
5955 b([12])j(and)h(using)f(the)h(nominal)g(datat)n(yp)r(e)f(pac)n(k)-5 |
|
5956 b(age)21 b([9])h(is)h(sho)n(wn)f(in)h(Fig.)f(1.)648 4327 |
|
5957 y(By)h(w)n(a)n(y)f(of)i(con)n(trast,)e(recall)h(that)h(a)f(pro)r(of)g |
|
5958 (without)h(the)g(stronger)d(induction)j(princi-)523 4426 |
|
5959 y(ple)i(w)n(ould)f(not)g(b)r(e)h(able)f(to)h(assume)f(an)n(ything)g(ab) |
|
5960 r(out)g(the)h(relationship)f(b)r(et)n(w)n(een)g Fr(x)h |
|
5961 Fz(and)523 4526 y Fr(\000)586 4496 y Fj(0)609 4526 y |
|
5962 Fz(,)f(forcing)f(the)h(pro)n(v)n(er)d(to)j Fr(\013)p |
|
5963 Fz(-con)n(v)n(ert)e Fs(lam)7 b Fz(\()p Fr(x:M)i Fz(\))25 |
|
5964 b(to)g(a)f(form)g(with)i(a)e(new)h(and)f(suitably)523 |
|
5965 4625 y(fresh)j(b)r(ound)h(v)-5 b(ariable,)26 b Fs(lam)7 |
|
5966 b Fz(\()p Fr(z)t(:)p Fz(\(\()p Fr(z)18 b(x)p Fz(\))1750 |
|
5967 4634 y Fg(\001)1789 4625 y Fr(M)9 b Fz(\)\),)27 b(sa)n(y)-7 |
|
5968 b(.)27 b(A)n(t)h(this)f(p)r(oin)n(t,)h(the)g(simplicit)n(y)f(of)h(the) |
|
5969 523 4725 y(pro)r(of)23 b(using)g(the)g(v)-5 b(ariable)22 |
|
5970 b(con)n(v)n(en)n(tion)g(disapp)r(ears:)g(the)i(inductiv)n(e)f(h)n(yp)r |
|
5971 (othesis)g(is)g(m)n(uc)n(h)523 4825 y(harder)36 b(to)i(sho)n(w)f |
|
5972 (applicable)g(b)r(ecause)g(it)h(men)n(tions)f Fr(M)9 |
|
5973 b Fz(,)38 b(but)g(the)g(desired)f(goal)f(is)i(in)523 |
|
5974 4924 y(terms)27 b(of)h(\()p Fr(z)17 b(x)p Fz(\))1016 |
|
5975 4933 y Fg(\001)1055 4924 y Fr(M)9 b Fz(.)p Black Black |
|
5976 eop end |
|
5977 %%Page: 13 13 |
|
5978 TeXDict begin 13 12 bop Black 1347 232 a Fw(Barendregt's)27 |
|
5979 b(V)-6 b(ariable)26 b(Con)n(v)n(en)n(tion)f(in)h(Rule)f(Inductions)198 |
|
5980 b(13)p Black Black Black Black 552 417 a Fv(lemma)27 |
|
5981 b Fk(we)l(akening)7 b Fw(:)662 509 y Fv(assumes)26 b |
|
5982 Fk(a)1045 517 y Fx(1)1080 509 y Fw(:)i Fn(\000)38 b Fm(`)27 |
|
5983 b Fk(M)12 b Fw(:)p Fk(T)37 b Fv(and)28 b Fk(a)1696 517 |
|
5984 y Fx(2)1730 509 y Fw(:)g Fn(\000)38 b Fm(\022)27 b Fn(\000)2018 |
|
5985 477 y Fl(0)2059 509 y Fv(and)h Fk(a)2273 517 y Fx(3)2307 |
|
5986 509 y Fw(:)g Fk(valid)35 b Fn(\000)2614 477 y Fl(0)662 |
|
5987 600 y Fv(sho)n(ws)27 b Fn(\000)980 568 y Fl(0)1021 600 |
|
5988 y Fm(`)g Fk(M)12 b Fw(:)p Fk(T)552 691 y Fv(using)28 |
|
5989 b Fk(a)826 699 y Fx(1)888 691 y Fk(a)933 699 y Fx(2)995 |
|
5990 691 y Fk(a)1040 699 y Fx(3)552 782 y Fv(pro)r(of)37 b |
|
5991 Fw(\()p Fk(nominal-induct)e Fn(\000)j Fk(M)28 b(T)f(avoiding)7 |
|
5992 b Fw(:)28 b Fn(\000)2025 751 y Fl(0)2066 782 y Fk(rule)6 |
|
5993 b Fw(:)28 b Fk(str)l(ong-)r(typing-)r(induct)7 b Fw(\))662 |
|
5994 874 y Fv(case)27 b Fw(\()p Fk(T)-6 b(yp)l(e)1039 882 |
|
5995 y Fx(3)1101 874 y Fk(x)37 b Fn(\000)h Fk(T)1323 882 y |
|
5996 Fx(1)1385 874 y Fk(T)1450 882 y Fx(2)1512 874 y Fk(M)12 |
|
5997 b Fw(\))662 965 y Fv(ha)n(v)n(e)26 b Fk(vc)t Fw(:)i Fk(x)9 |
|
5998 b Fw(#)p Fn(\000)1161 933 y Fl(0)1203 965 y Fv(b)n(y)27 |
|
5999 b Fk(fact)g Fw(|)f(v)l(ariable)g(con)n(v)n(en)n(tion)639 |
|
6000 1056 y Fv(ha)n(v)n(e)20 b Fk(ih)6 b Fw(:)22 b(\()p Fk(x)9 |
|
6001 b Fw(:)p Fk(T)1107 1064 y Fx(1)1142 1056 y Fw(\)::)p |
|
6002 Fn(\000)i Fm(\022)p Fw(\()p Fk(x)e Fw(:)p Fk(T)1492 1064 |
|
6003 y Fx(1)1527 1056 y Fw(\)::)p Fn(\000)1665 1025 y Fl(0)1679 |
|
6004 1056 y Fw(=)-13 b Fm(\))p Fk(valid)29 b Fw(\(\()p Fk(x)9 |
|
6005 b Fw(:)p Fk(T)2179 1064 y Fx(1)2214 1056 y Fw(\)::)p |
|
6006 Fn(\000)2353 1025 y Fl(0)2366 1056 y Fw(\)=)-13 b Fm(\))p |
|
6007 Fw(\()p Fk(x)9 b Fw(:)p Fk(T)2680 1064 y Fx(1)2715 1056 |
|
6008 y Fw(\)::)p Fn(\000)2854 1025 y Fl(0)2867 1056 y Fm(`)p |
|
6009 Fk(M)j Fw(:)p Fk(T)3081 1064 y Fx(2)3137 1056 y Fv(b)n(y)22 |
|
6010 b Fk(fact)662 1148 y Fv(ha)n(v)n(e)k Fn(\000)38 b Fm(\022)27 |
|
6011 b Fn(\000)1102 1116 y Fl(0)1143 1148 y Fv(b)n(y)g Fk(fact)662 |
|
6012 1239 y Fv(then)g(ha)n(v)n(e)f Fw(\()p Fk(x)9 b Fw(:)p |
|
6013 Fk(T)1224 1247 y Fx(1)1259 1239 y Fw(\)::)p Fn(\000)39 |
|
6014 b Fm(\022)27 b Fw(\()p Fk(x)9 b Fw(:)p Fk(T)1664 1247 |
|
6015 y Fx(1)1699 1239 y Fw(\)::)p Fn(\000)1837 1207 y Fl(0)1879 |
|
6016 1239 y Fv(b)n(y)26 b Fk(simp)662 1330 y Fv(moreo)n(v)n(er)662 |
|
6017 1422 y(ha)n(v)n(e)g Fk(valid)34 b Fn(\000)1122 1390 y |
|
6018 Fl(0)1163 1422 y Fv(b)n(y)27 b Fk(fact)662 1513 y Fv(then)g(ha)n(v)n(e) |
|
6019 f Fk(valid)35 b Fw(\(\()p Fk(x)9 b Fw(:)p Fk(T)1446 1521 |
|
6020 y Fx(1)1481 1513 y Fw(\)::)p Fn(\000)1619 1481 y Fl(0)1633 |
|
6021 1513 y Fw(\))28 b Fv(using)g Fk(vc)k Fv(b)n(y)c Fw(\()p |
|
6022 Fk(simp)f(add)8 b Fw(:)28 b Fk(valid-)q(c)l(ons)6 b Fw(\))662 |
|
6023 1604 y Fv(ultimately)30 b(ha)n(v)n(e)c Fw(\()p Fk(x)9 |
|
6024 b Fw(:)p Fk(T)1448 1612 y Fx(1)1483 1604 y Fw(\)::)p |
|
6025 Fn(\000)1621 1573 y Fl(0)1662 1604 y Fm(`)28 b Fk(M)12 |
|
6026 b Fw(:)p Fk(T)1904 1612 y Fx(2)1966 1604 y Fv(using)28 |
|
6027 b Fk(ih)33 b Fv(b)n(y)27 b Fk(simp)662 1696 y Fv(with)h |
|
6028 Fk(vc)k Fv(sho)n(w)c Fn(\000)1248 1664 y Fl(0)1289 1696 |
|
6029 y Fm(`)f Fk(lam)6 b Fw(\()p Fk(x)j Fn(:)p Fk(M)j Fw(\))27 |
|
6030 b(:)h Fk(T)1838 1704 y Fx(1)1900 1696 y Fm(!)f Fk(T)2069 |
|
6031 1704 y Fx(2)2131 1696 y Fv(b)n(y)g Fk(auto)552 1787 y |
|
6032 Fv(qed)g Fw(\()p Fk(auto)5 b Fw(\))28 b(|)d(cases)i(T)n(yp)r(e)1421 |
|
6033 1795 y Fx(1)1481 1787 y Fw(and)e(T)n(yp)r(e)1803 1795 |
|
6034 y Fx(2)523 1927 y Fv(Fig.)15 b(1.)32 b Fw(A)f(readable)i(Isab)r |
|
6035 (elle-Isar)h(pro)r(of)f(for)g(the)f(w)n(eak)n(ening)g(lemma)f(using)i |
|
6036 (the)f(strong)g(in-)523 2018 y(duction)j(principle)h(of)h(the)e(t)n |
|
6037 (yping)g(relation.)i(The)f(stronger)h(induction)e(principle)h(allo)n |
|
6038 (ws)i(us)523 2109 y(to)31 b(assume)g(a)h(v)l(ariable)f(con)n(v)n(en)n |
|
6039 (tion,)g(in)g(this)g(pro)r(of)h Fn(x)e Fw(#)g Fn(\000)2349 |
|
6040 2077 y Fl(0)2371 2109 y Fw(,)i(whic)n(h)f(mak)n(es)f(the)h(pro)r(of)h |
|
6041 (to)f(go)523 2200 y(through)25 b(without)h(di\016culties.)p |
|
6042 Black 523 2508 a Ft(5.2)95 b(P)m(arallel)31 b(Reduction)523 |
|
6043 2728 y Fz(In)h([2],)f(the)h(cen)n(tral)f(lemma)h(of)f(the)h(pro)r(of)g |
|
6044 (for)f(the)h(Ch)n(urc)n(h-Rosser)d(prop)r(ert)n(y)h(of)i(b)r(eta-)523 |
|
6045 2827 y(reduction)27 b(is)h(the)g(substitutivit)n(y)h(prop)r(ert)n(y)d |
|
6046 (of)i(the)g Fp(\000)-14 b(!)2286 2845 y Fo(1)2241 2827 |
|
6047 y Fp(\000)f(!)p Fz(-reduction.)28 b(T)-7 b(o)27 b(formalise)g(this)523 |
|
6048 2927 y(pro)r(of)g(while)h(preserving)e(the)i(informal)f(v)n(ersion's)f |
|
6049 (simplicit)n(y)-7 b(,)27 b(w)n(e)h(will)g(need)f(the)h(strong)523 |
|
6050 3027 y(induction)g(principle)g(for)f Fp(\000)-15 b(!)1434 |
|
6051 3045 y Fo(1)1389 3027 y Fp(\000)g(!)q Fz(.)648 3129 y(Before)18 |
|
6052 b(pro)r(ceeding,)h(w)n(e)g(need)h(t)n(w)n(o)e(imp)r(ortan)n(t)i(prop)r |
|
6053 (erties)e(of)i(the)g(substitution)g(func-)523 3220 y(tion,)30 |
|
6054 b(whic)n(h)f(o)r(ccurs)g(in)h(the)g(redex)f(rule)g(One)2003 |
|
6055 3232 y Fy(4)2040 3220 y Fz(.)h(W)-7 b(e)30 b(c)n(haracterise)d(the)j |
|
6056 (action)g(of)f(a)g(p)r(er-)523 3311 y(m)n(utation)41 |
|
6057 b(o)n(v)n(er)e(a)i(substitution)g(\(sho)n(wing)f(that)i(substitution)f |
|
6058 (is)g(equiv)-5 b(arian)n(t\),)40 b(and)523 3403 y(the)27 |
|
6059 b(supp)r(ort)e(of)i(a)e(substitution.)i(Both)f(pro)r(ofs)f(are)g(b)n(y) |
|
6060 h(straigh)n(tforw)n(ard)d(v)n(c-compatible)523 3494 y |
|
6061 Fs(structur)l(al)35 b Fz(induction)28 b(o)n(v)n(er)e |
|
6062 Fr(M)9 b Fz(:)1184 3688 y Fn(\031)1231 3697 y Fg(\001)1269 |
|
6063 3688 y Fw(\()p Fn(M)f Fw([)p Fn(x)21 b Fw(:=)g Fn(N)8 |
|
6064 b Fw(]\))22 b(=)g(\()p Fn(\031)1870 3697 y Fg(\001)1907 |
|
6065 3688 y Fn(M)8 b Fw(\)[\()p Fn(\031)2117 3697 y Fg(\001)2155 |
|
6066 3688 y Fn(x)p Fw(\))21 b(:=)g(\()p Fn(\031)2429 3697 |
|
6067 y Fg(\001)2467 3688 y Fn(N)8 b Fw(\)])p Black 681 w(\(23\))p |
|
6068 Black 1128 3804 a Fc(supp)q Fw(\()p Fn(M)g Fw([)p Fn(x)22 |
|
6069 b Fw(:=)f Fn(N)8 b Fw(]\))22 b Fm(\022)f Fw(\()p Fc(supp)q |
|
6070 Fw(\()p Fn(M)8 b Fw(\))17 b Fm(\000)g(f)p Fn(x)p Fm(g)p |
|
6071 Fw(\))g Fm([)g Fc(supp)q Fw(\()p Fn(N)8 b Fw(\))p Black |
|
6072 515 w(\(24\))p Black 648 4017 a Fz(With)21 b(this)g(w)n(e)g(can)f |
|
6073 (start)h(to)g(c)n(hec)n(k)f(the)h(v)n(c-compatibilit)n(y)f(conditions:) |
|
6074 g(the)h(condition)523 4117 y(ab)r(out)d(equiv)-5 b(ariance)18 |
|
6075 b(of)g(functions)h(and)f(side-conditions)g(is)g(again)f(easily)h |
|
6076 (con\014rmed.)g(The)523 4216 y(second)28 b(condition)g(is)g(that)g(b)r |
|
6077 (ound)h(v)-5 b(ariables)27 b(are)g(free)h(in)g(the)h(relation's)e |
|
6078 (rules')h(conclu-)523 4316 y(sions.)33 b(In)h(rule)f(One)1186 |
|
6079 4328 y Fy(2)1223 4316 y Fz(,)h(this)g(is)g(trivial)f(b)r(ecause)g |
|
6080 Fr(y)j Fz(#)d Fs(lam)7 b Fz(\()p Fr(y)s(:M)i Fz(\))34 |
|
6081 b(and)f Fr(y)j Fz(#)e Fs(lam)7 b Fz(\()p Fr(y)s(:M)3350 |
|
6082 4286 y Fj(0)3373 4316 y Fz(\))523 4416 y(hold.)36 b(A)g(problem)g |
|
6083 (arises,)e(ho)n(w)n(ev)n(er,)g(with)j(rule)f(One)2298 |
|
6084 4428 y Fy(4)2334 4416 y Fz(.)h(Here)e(w)n(e)h(ha)n(v)n(e)f(to)h(sho)n |
|
6085 (w)f(that)523 4515 y Fr(y)f Fz(#)d Fs(app)6 b Fz(\()p |
|
6086 Fs(lam)h Fz(\()p Fr(y)s(:M)i Fz(\))p Fr(;)14 b(N)9 b |
|
6087 Fz(\))33 b(and)f Fr(y)h Fz(#)e Fr(M)1829 4485 y Fj(0)1852 |
|
6088 4515 y Fz([)p Fr(y)j Fz(:=)c Fr(N)2144 4485 y Fj(0)2167 |
|
6089 4515 y Fz(],)j(and)f(w)n(e)g(ha)n(v)n(e)f(no)h(assumptions)f(to)523 |
|
6090 4615 y(hand)d(ab)r(out)f Fr(y)s Fz(.)648 4725 y(It)22 |
|
6091 b(is)g(certainly)g(true)g(that)g Fr(y)j Fz(is)d(fresh)g(for)f |
|
6092 Fs(lam)7 b Fz(\()p Fr(y)s(:M)i Fz(\),)23 b(but)f(it)h(ma)n(y)e(o)r |
|
6093 (ccur)h(in)g Fr(N)9 b Fz(.)22 b(As)h(for)523 4825 y(the)k(term)g |
|
6094 Fr(M)953 4795 y Fj(0)976 4825 y Fz([)p Fr(y)f Fz(:=)d |
|
6095 Fr(N)1253 4795 y Fj(0)1276 4825 y Fz(],)k(w)n(e)g(kno)n(w)f(that)h(an)n |
|
6096 (y)f(o)r(ccurrences)g(of)g Fr(y)k Fz(in)d Fr(M)2818 4795 |
|
6097 y Fj(0)2868 4825 y Fz(will)h(b)r(e)f(mask)n(ed)523 4924 |
|
6098 y(b)n(y)g(the)h(substitution)g(\(see)g(\(24\)\),)g(but)g |
|
6099 Fr(y)i Fz(ma)n(y)d(still)h(b)r(e)g(free)f(in)h Fr(N)2651 |
|
6100 4894 y Fj(0)2674 4924 y Fz(.)p Black Black eop end |
|
6101 %%Page: 14 14 |
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6102 TeXDict begin 14 13 bop Black 523 232 a Fw(14)199 b(Christian)27 |
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6103 b(Urban,)e(Stefan)h(Berghofer,)i(and)d(Mic)n(hael)i(Norrish)p |
|
6104 Black 648 448 a Fz(W)-7 b(e)28 b(need)f(to)h(reform)n(ulate)e(One)1677 |
|
6105 460 y Fy(4)1742 448 y Fz(to)h(read)1143 584 y Fn(y)d |
|
6106 Fw(#)d Fn(N)85 b(y)24 b Fw(#)e Fn(N)1653 552 y Fl(0)1752 |
|
6107 584 y Fn(M)30 b Fm(\000)-13 b(!)1925 600 y Fo(1)1884 |
|
6108 584 y Fm(\000)g(!)21 b Fn(M)2111 552 y Fl(0)2210 584 |
|
6109 y Fn(N)30 b Fm(\000)-13 b(!)2370 600 y Fo(1)2329 584 |
|
6110 y Fm(\000)g(!)22 b Fn(N)2544 552 y Fl(0)p 1143 618 1424 |
|
6111 4 v 1247 694 a Fk(app)6 b Fw(\()p Fk(lam)f Fw(\()p Fn(y)s(:M)j |
|
6112 Fw(\))p Fn(;)13 b(N)8 b Fw(\))21 b Fm(\000)-13 b(!)1954 |
|
6113 710 y Fo(1)1914 694 y Fm(\000)f(!)22 b Fn(M)2141 663 |
|
6114 y Fl(0)2163 694 y Fw([)p Fn(y)j Fw(:=)c Fn(N)2418 663 |
|
6115 y Fl(0)2441 694 y Fw(])2608 642 y(One)2744 610 y Fl(00)2744 |
|
6116 655 y Fx(4)523 849 y Fz(so)27 b(that)g(the)h(v)n(c-compatibilit)n(y)e |
|
6117 (conditions)h(can)g(b)r(e)h(disc)n(harged.)d(In)j(other)f(w)n(ords,)f |
|
6118 (if)h(w)n(e)523 949 y(ha)n(v)n(e)19 b(rule)h(One)1014 |
|
6119 919 y Fj(00)1014 970 y Fy(4)1077 949 y Fz(w)n(e)f(can)h(apply)g |
|
6120 (Theorem)g(1,)g(but)h(not)f(if)h(w)n(e)f(use)g(One)2739 |
|
6121 961 y Fy(4)2776 949 y Fz(.)g(This)g(is)g(anno)n(ying)523 |
|
6122 1049 y(b)r(ecause)34 b(b)r(oth)h(v)n(ersions)d(can)j(b)r(e)f(sho)n(wn)g |
|
6123 (to)g(de\014ne)h(the)g(same)f(relation,)f(but)i(w)n(e)f(ha)n(v)n(e)523 |
|
6124 1148 y(no)26 b(general,)f(and)i(automatable,)e(metho)r(d)i(for)f |
|
6125 (determining)g(this.)h(F)-7 b(or)26 b(the)h(momen)n(t,)f(w)n(e)523 |
|
6126 1248 y(reject)34 b(rule)g(One)1079 1260 y Fy(4)1150 1248 |
|
6127 y Fz(and)g(require)f(the)h(user)g(of)g(the)g(nominal)g(datat)n(yp)r(e)g |
|
6128 (pac)n(k)-5 b(age)32 b(to)i(use)523 1348 y(One)671 1317 |
|
6129 y Fj(00)671 1368 y Fy(4)713 1348 y Fz(.)28 b(If)g(this)g(is)f(done,)h |
|
6130 (the)g(substitutivit)n(y)g(lemma)f(is)h(almost)f(automatic:)p |
|
6131 Black Black 566 1480 a Fv(lemma)g Fk(substitutivity-)s(aux)9 |
|
6132 b Fw(:)621 1571 y Fv(assumes)27 b Fk(a)6 b Fw(:)27 b |
|
6133 Fk(N)12 b Fm(\000)-13 b(!)1246 1579 y Fx(1)1281 1571 |
|
6134 y Fk(N)1358 1539 y Fl(0)621 1662 y Fv(sho)n(ws)27 b Fk(M)12 |
|
6135 b Fw([)p Fk(x)d Fw(:=)p Fk(N)j Fw(])28 b Fm(\000)-13 |
|
6136 b(!)1342 1670 y Fx(1)1404 1662 y Fk(M)12 b Fw([)p Fk(x)d |
|
6137 Fw(:=)p Fk(N)1709 1631 y Fl(0)1723 1662 y Fw(])566 1754 |
|
6138 y Fv(using)29 b Fk(a)k Fv(b)n(y)28 b Fw(\()p Fk(nominal-induct)g(M)f |
|
6139 (avoiding)7 b Fw(:)28 b Fk(x)g(N)g(N)2208 1722 y Fl(0)2249 |
|
6140 1754 y Fk(rule)6 b Fw(:)28 b Fk(str)l(ong-)r(lam-induct)7 |
|
6141 b Fw(\))29 b(\()p Fk(auto)5 b Fw(\))566 1886 y Fv(lemma)27 |
|
6142 b Fk(subtitutivity)7 b Fw(:)621 1977 y Fv(assumes)27 |
|
6143 b Fk(a)1005 1985 y Fx(1)1039 1977 y Fw(:)h Fk(M)12 b |
|
6144 Fm(\000)-13 b(!)1293 1985 y Fx(1)1327 1977 y Fk(M)1416 |
|
6145 1945 y Fl(0)1458 1977 y Fv(and)27 b Fk(a)1671 1985 y |
|
6146 Fx(2)1706 1977 y Fw(:)h Fk(N)12 b Fm(\000)-13 b(!)1948 |
|
6147 1985 y Fx(1)1982 1977 y Fk(N)2059 1945 y Fl(0)621 2068 |
|
6148 y Fv(sho)n(ws)27 b Fk(M)12 b Fw([)p Fk(x)d Fw(:=)p Fk(N)j |
|
6149 Fw(])p Fm(\000)-12 b(!)1315 2076 y Fx(1)1349 2068 y Fk(M)1438 |
|
6150 2037 y Fl(0)1452 2068 y Fw([)p Fk(x)9 b Fw(:=)p Fk(N)1677 |
|
6151 2037 y Fl(0)1690 2068 y Fw(])566 2160 y Fv(using)21 b |
|
6152 Fk(a)833 2168 y Fx(1)889 2160 y Fk(a)934 2168 y Fx(2)989 |
|
6153 2160 y Fv(b)n(y)f Fw(\()p Fk(nominal-induct)h(M)g(M)1838 |
|
6154 2128 y Fl(0)1872 2160 y Fk(avoiding)7 b Fw(:)21 b Fk(N)f(N)2353 |
|
6155 2128 y Fl(0)2388 2160 y Fk(x)g(rule)6 b Fw(:)21 b Fk(str)l(ong-)r(p)l |
|
6156 (ar)l(al)t(lel-)q(induct)7 b Fw(\))1205 2251 y(\()p Fk(auto)27 |
|
6157 b(simp)e(add)8 b Fw(:)26 b Fk(substitutivity-)s(aux)h(substitution-)r |
|
6158 (lemma)e(fr)l(esh-)r(atm)6 b Fw(\))523 2387 y Fz(The)29 |
|
6159 b(\014rst)g(lemma)g(is)h(pro)n(v)n(ed)d(b)n(y)i(a)g(v)n(c-compatible)f |
|
6160 Fs(structur)l(al)37 b Fz(induction)29 b(o)n(v)n(er)f |
|
6161 Fr(M)9 b Fz(;)29 b(the)523 2487 y(second,)39 b(the)h(actual)f |
|
6162 (substitutivit)n(y)i(prop)r(ert)n(y)-7 b(,)38 b(is)i(pro)n(v)n(ed)e(b)n |
|
6163 (y)h(a)g(v)n(c-compatible)g Fs(rule)523 2586 y Fz(induction)34 |
|
6164 b(relying)e(on)h(the)g(substitution)h(lemma,)f(and)h(the)f(lemma)g |
|
6165 Fs(fr)l(esh-atm)p Fz(,)h(whic)n(h)523 2686 y(states)27 |
|
6166 b(that)h Fr(x)c Fz(#)f Fr(y)30 b Fz(is)e(the)g(same)f(as)g |
|
6167 Fr(x)c Fp(6)p Fz(=)g Fr(y)30 b Fz(when)e Fr(y)i Fz(is)e(an)f(atom.)523 |
|
6168 2936 y Fu(6)112 b(Related)37 b(W)-9 b(ork)523 3121 y |
|
6169 Fz(Apart)28 b(from)g(our)g(o)n(wn)g(preliminary)f(w)n(ork)g(in)i(this)f |
|
6170 (area)f([10],)h(w)n(e)g(b)r(eliev)n(e)g(the)h(prettiest)523 |
|
6171 3220 y(formal)35 b(pro)r(of)g(of)g(the)h(w)n(eak)n(ening)e(lemma)h(to)h |
|
6172 (b)r(e)g(that)f(in)h(Pitts)g([8].)f(This)g(pro)r(of)g(uses)523 |
|
6173 3320 y(the)d(equiv)-5 b(ariance)29 b(prop)r(ert)n(y)h(of)h(the)h(t)n |
|
6174 (yping)f(relation,)f(and)h(includes)g(a)g(renaming)f(step)523 |
|
6175 3419 y(using)25 b(p)r(erm)n(utations.)h(Because)e(of)i(the)g(pleasan)n |
|
6176 (t)f(prop)r(erties)f(that)i(p)r(erm)n(utations)g(enjo)n(y)523 |
|
6177 3519 y(\(they)j(are)f(bijectiv)n(e)h(renamings,)e(in)i(con)n(trast)e |
|
6178 (to)i(substitutions)f(whic)n(h)h(migh)n(t)f(iden)n(tify)523 |
|
6179 3619 y(t)n(w)n(o)k(names\),)g(the)h(renaming)f(can)g(b)r(e)h(done)g |
|
6180 (with)g(relativ)n(ely)e(minimal)i(o)n(v)n(erhead.)e(Our)523 |
|
6181 3718 y(con)n(tribution)20 b(is)g(that)g(w)n(e)g(ha)n(v)n(e)f(built)i |
|
6182 (this)g(renaming)e(in)n(to)h(our)g(v)n(c-compatible)f(induction)523 |
|
6183 3818 y(principles)k(once)f(and)g(for)h(all.)f(Pro)r(ofs)g(using)g(the)h |
|
6184 (v)n(c-compatible)f(principles)g(then)i(do)e(not)523 |
|
6185 3918 y(need)28 b(to)f(p)r(erform)g(an)n(y)g(explicit)h(renaming)f |
|
6186 (steps.)648 4009 y(Somewhat)i(similar)g(to)h(our)f(approac)n(h)g(is)g |
|
6187 (the)i(w)n(ork)d(of)i(P)n(ollac)n(k)e(and)i(McKinna)f([6].)523 |
|
6188 4100 y(Starting)c(from)g(the)g(standard)f(induction)i(principle)f(that) |
|
6189 g(is)h(asso)r(ciated)d(with)j(an)f(induc-)523 4192 y(tiv)n(e)d |
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6190 (de\014nition,)g(w)n(e)g(deriv)n(ed)f(an)h(induction)h(principle)f |
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6191 (that)g(allo)n(ws)f(em)n(ulation)g(of)h(Baren-)523 4283 |
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6192 y(dregt's)33 b(v)-5 b(ariable)33 b(con)n(v)n(en)n(tion.)g(P)n(ollac)n |
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6193 (k)e(and)j(McKinna,)g(in)g(con)n(trast,)e(ga)n(v)n(e)h(a)g(\\w)n(eak") |
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6194 523 4374 y(and)26 b(\\strong")e(v)n(ersion)h(of)i(the)f(t)n(yping)g |
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6195 (relation.)g(These)g(v)n(ersions)f(di\013er)h(in)h(the)f(w)n(a)n(y)g |
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6196 (the)523 4466 y(rule)h(for)g(abstractions)f(is)i(stated:)1228 |
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6197 4585 y Fn(x)21 b Fw(#)g Fn(M)85 b Fw(\()p Fn(x)21 b Fw(:)g |
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6198 Fn(T)1719 4593 y Fx(1)1754 4585 y Fw(\))g(::)h Fn(\000)32 |
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6199 b Fm(`)21 b Fn(M)8 b Fw([)p Fn(y)25 b Fw(:=)c Fn(x)p |
|
6200 Fw(])g(:)h Fn(T)2458 4593 y Fx(2)p 1228 4624 1265 4 v |
|
6201 1450 4697 a Fn(\000)32 b Fm(`)21 b Fk(lam)5 b Fw(\()p |
|
6202 Fn(y)s(:M)j Fw(\))21 b(:)h Fn(T)2037 4705 y Fx(1)2093 |
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6203 4697 y Fm(!)f Fn(T)2236 4705 y Fx(2)2534 4650 y Fw(w)n(eak)1133 |
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6204 4827 y Fm(8)p Fn(x:)g(x)f Fw(#)i Fn(\000)32 b Fm(\))21 |
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6205 b Fw(\()p Fn(x)g Fw(:)g Fn(T)1771 4835 y Fx(1)1806 4827 |
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6206 y Fw(\))g(::)h Fn(\000)32 b Fm(`)21 b Fn(M)8 b Fw([)p |
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6207 Fn(y)24 b Fw(:=)e Fn(x)p Fw(])f(:)h Fn(T)2510 4835 y |
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6208 Fx(2)p 1133 4866 1412 4 v 1428 4938 a Fn(\000)32 b Fm(`)21 |
|
6209 b Fk(lam)6 b Fw(\()p Fn(y)s(:M)i Fw(\))21 b(:)g Fn(T)2015 |
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6210 4946 y Fx(1)2071 4938 y Fm(!)g Fn(T)2214 4946 y Fx(2)2586 |
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6211 4882 y Fw(strong)p Black Black eop end |
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6212 %%Page: 15 15 |
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6213 TeXDict begin 15 14 bop Black 1347 232 a Fw(Barendregt's)27 |
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6214 b(V)-6 b(ariable)26 b(Con)n(v)n(en)n(tion)f(in)h(Rule)f(Inductions)198 |
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6215 b(15)p Black 523 448 a Fz(They)27 b(then)h(sho)n(w)n(ed)f(that)g(b)r |
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6216 (oth)h(v)n(ersions)e(deriv)n(e)h(the)g(same)g(t)n(yping)h(judgemen)n |
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6217 (ts.)f(With)523 548 y(this)h(they)g(pro)n(v)n(ed)e(the)i(w)n(eak)n |
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6218 (ening)e(lemma)i(using)f(the)h(\\strong")e(v)n(ersion)g(of)h(the)h |
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6219 (princi-)523 648 y(ple,)23 b(while)h(kno)n(wing)e(that)h(the)h(result)f |
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6220 (held)g(for)g(the)g(\\w)n(eak")e(relation)h(as)h(w)n(ell.)g(The)g(main) |
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6221 523 747 y(di\013erence)32 b(b)r(et)n(w)n(een)g(this)g(and)g(our)f(w)n |
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6222 (ork)g(seems)g(to)h(b)r(e)g(of)g(con)n(v)n(enience:)f(w)n(e)h(can)f |
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6223 (rela-)523 847 y(tiv)n(ely)g(easily)f(deriv)n(e,)h(in)g(a)g(uniform)g |
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6224 (w)n(a)n(y)-7 b(,)30 b(an)h(induction)g(principle)h(for)e(v)n |
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6225 (c-compatible)523 946 y(relations)22 b(\(w)n(e)i(ha)n(v)n(e)e |
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6226 (illustrated)h(this)h(p)r(oin)n(t)g(with)g(t)n(w)n(o)f(examples\).)g |
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6227 (Ac)n(hieving)g(the)h(same)523 1046 y(uniformit)n(y)c(in)h(the)f(st)n |
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6228 (yle)g(of)g(McKinna)g(and)g(P)n(ollac)n(k)e(do)r(es)i(not)g(seem)g(as)g |
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6229 (straigh)n(tforw)n(ard.)523 1306 y Fu(7)112 b(F)-9 b(uture)38 |
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6230 b(W)-9 b(ork)523 1492 y Fz(Our)24 b(future)g(w)n(ork)f(will)i(concen)n |
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6231 (trate)e(on)h(t)n(w)n(o)g(asp)r(ects:)g(\014rst)g(on)g(generalising)e |
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6232 (our)i(de\014ni-)523 1583 y(tion)e(of)f(sc)n(hematic)g(rules)g(so)g |
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6233 (that)h(they)g(ma)n(y)-7 b(,)21 b(for)g(example,)g(include)h(quan)n |
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6234 (ti\014ers.)e(Second)523 1675 y(on)34 b(b)r(eing)h(more)e(lib)r(eral)h |
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6235 (ab)r(out)g(whic)n(h)h(v)-5 b(ariables)33 b(can)h(b)r(e)g(included)h |
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6236 (in)g(the)g(induction)523 1766 y(con)n(text.)d(T)-7 b(o)32 |
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6237 b(see)g(what)g(w)n(e)g(ha)n(v)n(e)g(in)g(mind)h(with)g(this,)g(recall)e |
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6238 (that)i(w)n(e)f(allo)n(w)n(ed)f(in)i(the)523 1857 y(induction)26 |
|
6239 b(con)n(text)f(only)h(v)-5 b(ariables)24 b(that)i(are)f(in)g(binding)h |
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6240 (p)r(osition.)g(Ho)n(w)n(ev)n(er)e(there)h(are)523 1949 |
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6241 y(examples)k(where)g(this)h(is)g(to)r(o)f(restrictiv)n(e:)f(for)i |
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6242 (example)f(Crary)f(giv)n(es)g(in)i([4,)f(P)n(age)f(231])523 |
|
6243 2040 y(the)38 b(follo)n(wing)f(m)n(utual)g(inductiv)n(e)h(de\014nition) |
|
6244 g(for)f(the)h(judgemen)n(ts)g Fr(\000)51 b Fp(`)40 b |
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6245 Fr(s)f Fp(,)h Fr(t)g Fz(:)g Fr(T)523 2131 y Fz(and)e |
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6246 Fr(\000)52 b Fp(`)41 b Fr(p)f Fp($)h Fr(q)i Fz(:)e Fr(T)49 |
|
6247 b Fz(\(they)39 b(represen)n(t)e(a)h(t)n(yp)r(e-driv)n(en)f(equiv)-5 |
|
6248 b(alence)38 b(algorithm)f(for)523 2223 y(lam)n(b)r(da-terms)27 |
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6249 b(with)h(constan)n(ts\):)523 2377 y Fn(s)21 b Fm(+)g |
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6250 Fn(p)77 b(t)21 b Fm(+)g Fn(q)80 b(\000)32 b Fm(`)21 b |
|
6251 Fn(p)g Fm($)g Fn(q)j Fw(:)e Fn(T)p 523 2411 934 4 v 776 |
|
6252 2480 a(\000)32 b Fm(`)21 b Fn(s)h Fm(,)f Fn(t)g Fw(:)g |
|
6253 Fn(b)1472 2433 y Fw(Ae)1564 2441 y Fx(1)1641 2373 y Fw(\()p |
|
6254 Fn(x)g Fw(:)h Fn(T)1824 2381 y Fx(1)1858 2373 y Fw(\))9 |
|
6255 b(::)g Fn(\000)32 b Fm(`)21 b Fn(s)g(x)g Fm(,)g Fn(t)g(x)g |
|
6256 Fw(:)h Fn(T)2517 2381 y Fx(2)p 1641 2411 910 4 v 1760 |
|
6257 2480 a Fn(\000)32 b Fm(`)21 b Fn(s)g Fm(,)g Fn(t)g Fw(:)h |
|
6258 Fn(T)2199 2488 y Fx(1)2255 2480 y Fm(!)f Fn(T)2398 2488 |
|
6259 y Fx(2)2567 2433 y Fw(Ae)2659 2441 y Fx(2)p 2736 2411 |
|
6260 539 4 v 2736 2480 a Fn(\000)32 b Fm(`)21 b Fn(s)g Fm(,)g |
|
6261 Fn(t)g Fw(:)h Fn(unit)3290 2433 y Fw(Ae)3382 2441 y Fx(3)675 |
|
6262 2611 y Fw(\()p Fn(x)f Fw(:)h Fn(T)11 b Fw(\))20 b Fm(2)i |
|
6263 Fn(\000)p 626 2649 474 4 v 626 2718 a(\000)32 b Fm(`)21 |
|
6264 b Fn(x)g Fm($)g Fn(x)g Fw(:)h Fn(T)1115 2671 y Fw(P)n(e)1199 |
|
6265 2679 y Fx(1)1298 2615 y Fn(\000)32 b Fm(`)21 b Fn(p)g |
|
6266 Fm($)g Fn(q)j Fw(:)e Fn(T)1749 2623 y Fx(1)1805 2615 |
|
6267 y Fm(!)f Fn(T)1948 2623 y Fx(2)2059 2615 y Fn(\000)32 |
|
6268 b Fm(`)21 b Fn(s)g Fm($)g Fn(t)g Fw(:)h Fn(T)2498 2623 |
|
6269 y Fx(1)p 1298 2649 1235 4 v 1619 2718 a Fn(\000)32 b |
|
6270 Fm(`)21 b Fn(p)g(s)g Fm($)h Fn(q)i(t)d Fw(:)h Fn(T)2177 |
|
6271 2726 y Fx(2)2548 2671 y Fw(P)n(e)2633 2679 y Fx(2)p 2731 |
|
6272 2649 448 4 v 2731 2718 a Fn(\000)32 b Fm(`)21 b Fn(k)j |
|
6273 Fm($)d Fn(k)j Fw(:)d Fn(b)3195 2671 y Fw(P)n(e)3279 2679 |
|
6274 y Fx(3)523 2876 y Fz(What)35 b(is)f(in)n(teresting)g(is)g(that)h(these) |
|
6275 f(rules)g(do)g(not)g(con)n(tain)g(an)n(y)g(v)-5 b(ariable)33 |
|
6276 b(in)i(binding)523 2976 y(p)r(osition.)j(Still,)h(in)f(some)g(pro)r |
|
6277 (ofs)f(b)n(y)h(induction)g(o)n(v)n(er)f(those)g(rules)h(one)f(w)n(an)n |
|
6278 (ts)h(to)g(b)r(e)523 3075 y(able)g(to)f(assume)g(that)i(the)f(v)-5 |
|
6279 b(ariable)37 b Fr(x)h Fz(in)g(the)h(rule)e(Ae)2404 3087 |
|
6280 y Fy(2)2479 3075 y Fz(satis\014es)h(certain)f(freshness)523 |
|
6281 3175 y(conditions.)c(Our)g(implemen)n(tation)g(already)f(deals)h(with)h |
|
6282 (this)g(situation)f(b)n(y)g(explicitly)523 3275 y(giving)26 |
|
6283 b(the)h(information)g(that)g Fr(x)g Fz(should)g(app)r(ear)f(in)h(the)g |
|
6284 (induction)g(con)n(text.)g(Ho)n(w)n(ev)n(er,)523 3374 |
|
6285 y(w)n(e)g(ha)n(v)n(e)g(not)g(y)n(et)h(w)n(ork)n(ed)e(out)h(the)h |
|
6286 (theory)-7 b(.)523 3635 y Fu(8)112 b(Conclusion)523 3828 |
|
6287 y Fz(In)22 b(the)h Fa(POPLmark)d Fz(Challenge)i([1],)g(the)g(pro)r(of)f |
|
6288 (of)h(the)h(w)n(eak)n(ening)d(lemma)i(is)g(describ)r(ed)523 |
|
6289 3928 y(as)j(a)h(\\straigh)n(tforw)n(ard)d(induction".)j(In)g(fact,)g |
|
6290 (mec)n(hanising)f(this)h(informal)g(pro)r(of)f(is)h Fs(not)523 |
|
6291 4028 y Fz(straigh)n(tforw)n(ard)38 b(at)j(all)g(\(see)g(for)f(example)h |
|
6292 ([6,)13 b(5,)g(8]\).)41 b(W)-7 b(e)42 b(ha)n(v)n(e)e(giv)n(en)g(a)g(no) |
|
6293 n(v)n(el)g(rule)523 4127 y(induction)d(principle)h(for)e(the)h(t)n |
|
6294 (yping)g(relation)f(that)i(mak)n(es)e(pro)n(ving)f(the)j(w)n(eak)n |
|
6295 (ening)523 4227 y(lemma)28 b(mec)n(hanically)e(as)h(simple)h(as)f(p)r |
|
6296 (erforming)f(the)i(informal)f(pro)r(of.)648 4327 y(Imp)r(ortan)n(tly)-7 |
|
6297 b(,)34 b(this)h(new)g(principle)g(can)g(b)r(e)g(deriv)n(ed)f(from)h |
|
6298 (the)g(original)f(inductiv)n(e)523 4426 y(de\014nition)c(of)f(the)h(t)n |
|
6299 (yping)f(relation)f(in)i(a)f(mec)n(hanical)f(w)n(a)n(y)-7 |
|
6300 b(.)28 b(This)i(metho)r(d)f(extends)h(our)523 4526 y(earlier)24 |
|
6301 b(w)n(ork)g([10,)13 b(7],)25 b(where)g(w)n(e)g(constructed)g(our)g(new) |
|
6302 g(induction)h(principles)f(b)n(y)g(hand.)523 4625 y(By)j(formally)f |
|
6303 (deriving)g(principles)h(that)g(a)n(v)n(oid)f(the)h(need)h(to)e(rename) |
|
6304 h(b)r(ound)g(v)-5 b(ariables,)523 4725 y(w)n(e)24 b(adv)-5 |
|
6305 b(ance)24 b(the)i(state-of-the-art)d(in)i(mec)n(hanical)e(theorem-pro)n |
|
6306 (ving)g(o)n(v)n(er)g(syn)n(tax)g(with)523 4825 y(binders.)36 |
|
6307 b(The)g(results)g(of)g(this)h(pap)r(er)f(ha)n(v)n(e)f(already)g(b)r |
|
6308 (een)i(used)f(man)n(y)g(times)g(in)h(the)523 4924 y(nominal)19 |
|
6309 b(datat)n(yp)r(e)g(pac)n(k)-5 b(age:)18 b(for)h(example)g(in)h(the)g |
|
6310 (pro)r(ofs)f(of)g(the)h(CR)g(and)f(SN)h(prop)r(erties)p |
|
6311 Black Black eop end |
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6312 %%Page: 16 16 |
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6313 TeXDict begin 16 15 bop Black 523 232 a Fw(16)199 b(Christian)27 |
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6314 b(Urban,)e(Stefan)h(Berghofer,)i(and)d(Mic)n(hael)i(Norrish)p |
|
6315 Black 523 448 a Fz(in)21 b(the)g Fr(\025)p Fz(-calculus,)f(in)h(pro)r |
|
6316 (ofs)f(ab)r(out)g(the)h(pi-calculus,)f(in)h(pro)r(ofs)e(ab)r(out)i |
|
6317 (logical)e(relations)523 548 y(and)27 b(in)h(sev)n(eral)e(pro)r(ofs)h |
|
6318 (from)g(structural)g(op)r(erational)f(seman)n(tics.)648 |
|
6319 648 y(The)32 b(fact)g(that)h(our)e(tec)n(hnique)i(ma)n(y)e(require)h |
|
6320 (users)f(to)h(cast)g(some)g(inductiv)n(e)g(de\014-)523 |
|
6321 747 y(nitions)g(in)g(alternativ)n(e)f(forms)h(is)f(unfortunate.)i(In)f |
|
6322 (the)g(earlier)f([10],)g(our)g(hand-pro)r(ofs)523 847 |
|
6323 y(correctly)d(deriv)n(ed)h(a)g(v)n(c-compatible)f(principle)h(from)g |
|
6324 (the)h(original)e(de\014nition)i(of)f Fp(\000)-14 b(!)3294 |
|
6325 865 y Fo(1)3248 847 y Fp(\000)g(!)p Fz(;)523 946 y(w)n(e)27 |
|
6326 b(hop)r(e)h(that)g(future)g(w)n(ork)e(will)i(automatically)e(justify)j |
|
6327 (comparable)d(deriv)-5 b(ations.)523 1071 y Ft(Ac)m(kno)m(wledgemen)m |
|
6328 (ts)31 b Fz(W)-7 b(e)32 b(are)f(v)n(ery)g(grateful)g(to)h(Andrew)g |
|
6329 (Pitts)f(for)h(the)g(man)n(y)f(dis-)523 1171 y(cussions)c(with)h(him)g |
|
6330 (on)f(the)h(sub)5 b(ject)28 b(of)f(this)h(pap)r(er.)523 |
|
6331 1436 y Fu(References)p Black 561 1627 a Fw(1.)p Black |
|
6332 43 w(B.)f(E.)h(Aydemir,)d(A.)i(Bohannon,)g(M.)h(F)-6 |
|
6333 b(airbairn,)28 b(J.)g(N.)e(F)-6 b(oster,)28 b(B.)g(C.)f(Pierce,)i(P)-6 |
|
6334 b(.)27 b(Sew)n(ell,)663 1719 y(D.)f(Vytiniotis,)g(G.)h(W)-6 |
|
6335 b(ash)n(burn,)25 b(S.)h(W)-6 b(eiric)n(h,)26 b(and)g(S.)g(Zdancewic.)36 |
|
6336 b(Mec)n(hanized)27 b(Metathe-)663 1810 y(ory)i(for)i(the)e(Masses:)i |
|
6337 (The)f(P)n(opl)q(Mark)g(Challenge.)47 b(In)29 b Fk(Pr)l(o)l(c.)j(of)e |
|
6338 (the)j(18th)f(International)663 1901 y(Confer)l(enc)l(e)d(on)f(The)l |
|
6339 (or)l(em)h(Pr)l(oving)g(in)f(Higher-Or)l(der)i(L)l(o)l(gics)f |
|
6340 (\(TPHOLs\))p Fw(,)e(v)n(olume)e(3603)663 1993 y(of)h |
|
6341 Fk(LNCS)p Fw(,)g(pages)h(50{65,)h(2005.)p Black 561 2084 |
|
6342 a(2.)p Black 43 w(H.)j(Barendregt.)53 b Fk(The)33 b(L)l(amb)l(da)h |
|
6343 (Calculus:)e(its)h(Syntax)i(and)e(Semantics)p Fw(,)g(v)n(olume)d(103)j |
|
6344 (of)663 2175 y Fk(Studies)28 b(in)f(L)l(o)l(gic)h(and)g(the)h(F)-6 |
|
6345 b(oundations)29 b(of)e(Mathematics)p Fw(.)36 b(North-Holland,)26 |
|
6346 b(1981.)p Black 561 2267 a(3.)p Black 43 w(J.)i(Bengtson)g(and)e(J.)i |
|
6347 (P)n(arro)n(w.)40 b(F)-6 b(ormalising)28 b(the)f(pi-Calculus)h(using)g |
|
6348 (Nominal)e(Logic.)41 b(In)663 2358 y Fk(Pr)l(o)l(c.)23 |
|
6349 b(of)f(the)h(10th)h(International)f(Confer)l(enc)l(e)h(on)f(F)-6 |
|
6350 b(oundations)24 b(of)e(Softwar)l(e)i(Scienc)l(e)f(and)663 |
|
6351 2449 y(Computation)28 b(Structur)l(es)j(\(F)n(OSSA)n(CS\))p |
|
6352 Fw(,)c(v)n(olume)d(4423)k(of)e Fk(LNCS)p Fw(,)g(pages)h(63{77,)g(2007.) |
|
6353 p Black 561 2540 a(4.)p Black 43 w(K.)22 b(Crary)-6 b(.)30 |
|
6354 b Fk(A)l(dvanc)l(e)l(d)c(T)-6 b(opics)26 b(in)e(T)-6 |
|
6355 b(yp)l(es)26 b(and)f(Pr)l(o)l(gr)l(amming)h(L)l(anguages)p |
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6356 Fw(,)f(c)n(hapter)d(Logical)663 2632 y(Relations)27 b(and)e(a)h(Case)i |
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6357 (Study)c(in)i(Equiv)l(alence)g(Chec)n(king,)g(pages)h(223{244.)38 |
|
6358 b(MIT)27 b(Press,)663 2723 y(2005.)p Black 561 2814 a(5.)p |
|
6359 Black 43 w(J.)e(Gallier.)35 b Fk(L)l(o)l(gic)28 b(for)e(Computer)j |
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6360 (Scienc)l(e:)e(F)-6 b(oundations)28 b(of)f(A)n(utomatic)h(The)l(or)l |
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6361 (em)g(Pr)l(ov-)663 2906 y(ing)p Fw(.)34 b(Harp)r(er)26 |
|
6362 b(&)f(Ro)n(w,)h(1986.)p Black 561 2997 a(6.)p Black 43 |
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6363 w(J.)32 b(McKinna)f(and)g(R.)g(P)n(ollac)n(k.)53 b(Some)30 |
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6364 b(t)n(yp)r(e)h(theory)g(and)g(lam)n(b)r(da)f(calculus)j(formalised.)663 |
|
6365 3088 y Fk(Journal)28 b(of)f(A)n(utomate)l(d)j(R)l(e)l(asoning)p |
|
6366 Fw(,)d(23\(1-4\),)g(1999.)p Black 561 3180 a(7.)p Black |
|
6367 43 w(M.)32 b(Norrish.)54 b(Mec)n(hanising)33 b Fn(\025)p |
|
6368 Fw(-calculus)f(using)g(a)h(classical)h(\014rst)e(order)g(theory)g(of)h |
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6369 (terms)663 3271 y(with)26 b(p)r(erm)n(utation.)33 b Fk(Higher-Or)l(der) |
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6370 d(and)e(Symb)l(olic)f(Computation)p Fw(,)g(19:169{195,)j(2006.)p |
|
6371 Black 561 3362 a(8.)p Black 43 w(A.)25 b(M.)i(Pitts.)36 |
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6372 b(Nominal)25 b(Logic,)j(A)d(First)i(Order)e(Theory)h(of)h(Names)e(and)h |
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6373 (Binding.)36 b Fk(Infor-)663 3454 y(mation)27 b(and)h(Computation)p |
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6374 Fw(,)g(186:165{193,)i(2003.)p Black 561 3545 a(9.)p Black |
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6375 43 w(C.)39 b(Urban)e(and)h(S.)g(Berghofer.)73 b(A)38 |
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6376 b(Recursion)g(Com)n(binator)g(for)h(Nominal)f(Datat)n(yp)r(es)663 |
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6377 3636 y(Implemen)n(ted)22 b(in)j(Isab)r(elle/HOL.)35 b(In)25 |
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6378 b Fk(Pr)l(o)l(c.)i(of)g(the)h(3r)l(d)g(International)h(Joint)e(Confer)l |
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6379 (enc)l(e)663 3728 y(on)g(A)n(utomate)l(d)j(R)l(e)l(asoning)e(\(IJCAR\)) |
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6380 p Fw(,)f(v)n(olume)d(4130)k(of)e Fk(LNAI)p Fw(,)g(pages)g(498{512,)j |
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6381 (2006.)p Black 523 3819 a(10.)p Black 43 w(C.)22 b(Urban)f(and)h(M.)g |
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6382 (Norrish.)29 b(A)21 b(formal)h(treatmen)n(t)f(of)i(the)e(Barendregt)h |
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6383 (V)-6 b(ariable)23 b(Con)n(v)n(en-)663 3910 y(tion)28 |
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6384 b(in)h(rule)f(inductions.)43 b(In)28 b Fk(MERLIN)h('05:)h(Pr)l(o)l(c)l |
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6385 (e)l(e)l(dings)i(of)e(the)h(3r)l(d)g(A)n(CM)f(SIGPLAN)663 |
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6387 (out)h(languages)f(with)f(variable)h(binding)p Fw(,)e(pages)663 |
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6388 4093 y(25{32,)28 b(New)d(Y)-6 b(ork,)25 b(NY,)h(USA,)e(2005.)k(A)n(CM)e |
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6391 53 b(In)31 b Fk(Pr)l(o)l(c.)i(of)g(the)663 4276 y(20th)d(International) |
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6392 g(Confer)l(enc)l(e)g(on)f(A)n(utomate)l(d)i(De)l(duction)f(\(CADE\))p |
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6393 Fw(,)d(v)n(olume)f(3632)j(of)663 4367 y Fk(L)l(e)l(ctur)l(e)g(Notes)g |
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6394 (in)e(Computer)i(Scienc)l(e)p Fw(,)e(pages)f(38{53,)i(2005.)p |
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6398 b(In)24 b Fk(Pr)l(o)l(c.)j(of)e(the)j(12th)f(International)g(Confer)l |
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6399 (enc)l(e)h(on)e(The)l(or)l(em)h(Pr)l(oving)g(in)663 4641 |
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6400 y(Higher)h(Or)l(der)g(L)l(o)l(gics)h(\(TPHOLs\))p Fw(,)e(n)n(um)n(b)r |
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6401 (er)c(1690)k(in)f(LNCS,)g(pages)g(167{184,)j(1999.)p |
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6402 Black Black eop end |
|
6403 %%Trailer |
|
6404 |
|
6405 userdict /end-hook known{end-hook}if |
|
6406 %%EOF |