794 qed |
795 qed |
795 |
796 |
796 |
797 |
797 end |
798 end |
798 end |
799 end |
|
800 ======= |
|
801 theory Correctness |
|
802 imports PIPBasics |
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803 begin |
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804 |
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805 |
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806 text {* |
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807 The following two auxiliary lemmas are used to reason about @{term Max}. |
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808 *} |
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809 lemma image_Max_eqI: |
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810 assumes "finite B" |
|
811 and "b \<in> B" |
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812 and "\<forall> x \<in> B. f x \<le> f b" |
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813 shows "Max (f ` B) = f b" |
|
814 using assms |
|
815 using Max_eqI by blast |
|
816 |
|
817 lemma image_Max_subset: |
|
818 assumes "finite A" |
|
819 and "B \<subseteq> A" |
|
820 and "a \<in> B" |
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821 and "Max (f ` A) = f a" |
|
822 shows "Max (f ` B) = f a" |
|
823 proof(rule image_Max_eqI) |
|
824 show "finite B" |
|
825 using assms(1) assms(2) finite_subset by auto |
|
826 next |
|
827 show "a \<in> B" using assms by simp |
|
828 next |
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829 show "\<forall>x\<in>B. f x \<le> f a" |
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830 by (metis Max_ge assms(1) assms(2) assms(4) |
|
831 finite_imageI image_eqI subsetCE) |
|
832 qed |
|
833 |
|
834 text {* |
|
835 The following locale @{text "highest_gen"} sets the basic context for our |
|
836 investigation: supposing thread @{text th} holds the highest @{term cp}-value |
|
837 in state @{text s}, which means the task for @{text th} is the |
|
838 most urgent. We want to show that |
|
839 @{text th} is treated correctly by PIP, which means |
|
840 @{text th} will not be blocked unreasonably by other less urgent |
|
841 threads. |
|
842 *} |
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843 locale highest_gen = |
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844 fixes s th prio tm |
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845 assumes vt_s: "vt s" |
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846 and threads_s: "th \<in> threads s" |
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847 and highest: "preced th s = Max ((cp s)`threads s)" |
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848 -- {* The internal structure of @{term th}'s precedence is exposed:*} |
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849 and preced_th: "preced th s = Prc prio tm" |
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850 |
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851 -- {* @{term s} is a valid trace, so it will inherit all results derived for |
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852 a valid trace: *} |
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853 sublocale highest_gen < vat_s: valid_trace "s" |
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854 by (unfold_locales, insert vt_s, simp) |
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855 |
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856 context highest_gen |
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857 begin |
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858 |
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859 text {* |
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860 @{term tm} is the time when the precedence of @{term th} is set, so |
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861 @{term tm} must be a valid moment index into @{term s}. |
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862 *} |
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863 lemma lt_tm: "tm < length s" |
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864 by (insert preced_tm_lt[OF threads_s preced_th], simp) |
|
865 |
|
866 text {* |
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867 Since @{term th} holds the highest precedence and @{text "cp"} |
|
868 is the highest precedence of all threads in the sub-tree of |
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869 @{text "th"} and @{text th} is among these threads, |
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870 its @{term cp} must equal to its precedence: |
|
871 *} |
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872 lemma eq_cp_s_th: "cp s th = preced th s" (is "?L = ?R") |
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873 proof - |
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874 have "?L \<le> ?R" |
|
875 by (unfold highest, rule Max_ge, |
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876 auto simp:threads_s finite_threads) |
|
877 moreover have "?R \<le> ?L" |
|
878 by (unfold vat_s.cp_rec, rule Max_ge, |
|
879 auto simp:the_preced_def vat_s.fsbttRAGs.finite_children) |
|
880 ultimately show ?thesis by auto |
|
881 qed |
|
882 |
|
883 lemma highest_cp_preced: "cp s th = Max (the_preced s ` threads s)" |
|
884 using eq_cp_s_th highest max_cp_eq the_preced_def by presburger |
|
885 |
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886 |
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887 lemma highest_preced_thread: "preced th s = Max (the_preced s ` threads s)" |
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888 by (fold eq_cp_s_th, unfold highest_cp_preced, simp) |
|
889 |
|
890 lemma highest': "cp s th = Max (cp s ` threads s)" |
|
891 by (simp add: eq_cp_s_th highest) |
|
892 |
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893 end |
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894 |
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895 locale extend_highest_gen = highest_gen + |
|
896 fixes t |
|
897 assumes vt_t: "vt (t@s)" |
|
898 and create_low: "Create th' prio' \<in> set t \<Longrightarrow> prio' \<le> prio" |
|
899 and set_diff_low: "Set th' prio' \<in> set t \<Longrightarrow> th' \<noteq> th \<and> prio' \<le> prio" |
|
900 and exit_diff: "Exit th' \<in> set t \<Longrightarrow> th' \<noteq> th" |
|
901 |
|
902 sublocale extend_highest_gen < vat_t: valid_trace "t@s" |
|
903 by (unfold_locales, insert vt_t, simp) |
|
904 |
|
905 lemma step_back_vt_app: |
|
906 assumes vt_ts: "vt (t@s)" |
|
907 shows "vt s" |
|
908 proof - |
|
909 from vt_ts show ?thesis |
|
910 proof(induct t) |
|
911 case Nil |
|
912 from Nil show ?case by auto |
|
913 next |
|
914 case (Cons e t) |
|
915 assume ih: " vt (t @ s) \<Longrightarrow> vt s" |
|
916 and vt_et: "vt ((e # t) @ s)" |
|
917 show ?case |
|
918 proof(rule ih) |
|
919 show "vt (t @ s)" |
|
920 proof(rule step_back_vt) |
|
921 from vt_et show "vt (e # t @ s)" by simp |
|
922 qed |
|
923 qed |
|
924 qed |
|
925 qed |
|
926 |
|
927 (* locale red_extend_highest_gen = extend_highest_gen + |
|
928 fixes i::nat |
|
929 *) |
|
930 |
|
931 (* |
|
932 sublocale red_extend_highest_gen < red_moment: extend_highest_gen "s" "th" "prio" "tm" "(moment i t)" |
|
933 apply (insert extend_highest_gen_axioms, subst (asm) (1) moment_restm_s [of i t, symmetric]) |
|
934 apply (unfold extend_highest_gen_def extend_highest_gen_axioms_def, clarsimp) |
|
935 by (unfold highest_gen_def, auto dest:step_back_vt_app) |
|
936 *) |
|
937 |
|
938 context extend_highest_gen |
|
939 begin |
|
940 |
|
941 lemma ind [consumes 0, case_names Nil Cons, induct type]: |
|
942 assumes |
|
943 h0: "R []" |
|
944 and h2: "\<And> e t. \<lbrakk>vt (t@s); step (t@s) e; |
|
945 extend_highest_gen s th prio tm t; |
|
946 extend_highest_gen s th prio tm (e#t); R t\<rbrakk> \<Longrightarrow> R (e#t)" |
|
947 shows "R t" |
|
948 proof - |
|
949 from vt_t extend_highest_gen_axioms show ?thesis |
|
950 proof(induct t) |
|
951 from h0 show "R []" . |
|
952 next |
|
953 case (Cons e t') |
|
954 assume ih: "\<lbrakk>vt (t' @ s); extend_highest_gen s th prio tm t'\<rbrakk> \<Longrightarrow> R t'" |
|
955 and vt_e: "vt ((e # t') @ s)" |
|
956 and et: "extend_highest_gen s th prio tm (e # t')" |
|
957 from vt_e and step_back_step have stp: "step (t'@s) e" by auto |
|
958 from vt_e and step_back_vt have vt_ts: "vt (t'@s)" by auto |
|
959 show ?case |
|
960 proof(rule h2 [OF vt_ts stp _ _ _ ]) |
|
961 show "R t'" |
|
962 proof(rule ih) |
|
963 from et show ext': "extend_highest_gen s th prio tm t'" |
|
964 by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt) |
|
965 next |
|
966 from vt_ts show "vt (t' @ s)" . |
|
967 qed |
|
968 next |
|
969 from et show "extend_highest_gen s th prio tm (e # t')" . |
|
970 next |
|
971 from et show ext': "extend_highest_gen s th prio tm t'" |
|
972 by (unfold extend_highest_gen_def extend_highest_gen_axioms_def, auto dest:step_back_vt) |
|
973 qed |
|
974 qed |
|
975 qed |
|
976 |
|
977 |
|
978 lemma th_kept: "th \<in> threads (t @ s) \<and> |
|
979 preced th (t@s) = preced th s" (is "?Q t") |
|
980 proof - |
|
981 show ?thesis |
|
982 proof(induct rule:ind) |
|
983 case Nil |
|
984 from threads_s |
|
985 show ?case |
|
986 by auto |
|
987 next |
|
988 case (Cons e t) |
|
989 interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto |
|
990 interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto |
|
991 show ?case |
|
992 proof(cases e) |
|
993 case (Create thread prio) |
|
994 show ?thesis |
|
995 proof - |
|
996 from Cons and Create have "step (t@s) (Create thread prio)" by auto |
|
997 hence "th \<noteq> thread" |
|
998 proof(cases) |
|
999 case thread_create |
|
1000 with Cons show ?thesis by auto |
|
1001 qed |
|
1002 hence "preced th ((e # t) @ s) = preced th (t @ s)" |
|
1003 by (unfold Create, auto simp:preced_def) |
|
1004 moreover note Cons |
|
1005 ultimately show ?thesis |
|
1006 by (auto simp:Create) |
|
1007 qed |
|
1008 next |
|
1009 case (Exit thread) |
|
1010 from h_e.exit_diff and Exit |
|
1011 have neq_th: "thread \<noteq> th" by auto |
|
1012 with Cons |
|
1013 show ?thesis |
|
1014 by (unfold Exit, auto simp:preced_def) |
|
1015 next |
|
1016 case (P thread cs) |
|
1017 with Cons |
|
1018 show ?thesis |
|
1019 by (auto simp:P preced_def) |
|
1020 next |
|
1021 case (V thread cs) |
|
1022 with Cons |
|
1023 show ?thesis |
|
1024 by (auto simp:V preced_def) |
|
1025 next |
|
1026 case (Set thread prio') |
|
1027 show ?thesis |
|
1028 proof - |
|
1029 from h_e.set_diff_low and Set |
|
1030 have "th \<noteq> thread" by auto |
|
1031 hence "preced th ((e # t) @ s) = preced th (t @ s)" |
|
1032 by (unfold Set, auto simp:preced_def) |
|
1033 moreover note Cons |
|
1034 ultimately show ?thesis |
|
1035 by (auto simp:Set) |
|
1036 qed |
|
1037 qed |
|
1038 qed |
|
1039 qed |
|
1040 |
|
1041 text {* |
|
1042 According to @{thm th_kept}, thread @{text "th"} has its living status |
|
1043 and precedence kept along the way of @{text "t"}. The following lemma |
|
1044 shows that this preserved precedence of @{text "th"} remains as the highest |
|
1045 along the way of @{text "t"}. |
|
1046 |
|
1047 The proof goes by induction over @{text "t"} using the specialized |
|
1048 induction rule @{thm ind}, followed by case analysis of each possible |
|
1049 operations of PIP. All cases follow the same pattern rendered by the |
|
1050 generalized introduction rule @{thm "image_Max_eqI"}. |
|
1051 |
|
1052 The very essence is to show that precedences, no matter whether they |
|
1053 are newly introduced or modified, are always lower than the one held |
|
1054 by @{term "th"}, which by @{thm th_kept} is preserved along the way. |
|
1055 *} |
|
1056 lemma max_kept: "Max (the_preced (t @ s) ` (threads (t@s))) = preced th s" |
|
1057 proof(induct rule:ind) |
|
1058 case Nil |
|
1059 from highest_preced_thread |
|
1060 show ?case by simp |
|
1061 next |
|
1062 case (Cons e t) |
|
1063 interpret h_e: extend_highest_gen _ _ _ _ "(e # t)" using Cons by auto |
|
1064 interpret h_t: extend_highest_gen _ _ _ _ t using Cons by auto |
|
1065 show ?case |
|
1066 proof(cases e) |
|
1067 case (Create thread prio') |
|
1068 show ?thesis (is "Max (?f ` ?A) = ?t") |
|
1069 proof - |
|
1070 -- {* The following is the common pattern of each branch of the case analysis. *} |
|
1071 -- {* The major part is to show that @{text "th"} holds the highest precedence: *} |
|
1072 have "Max (?f ` ?A) = ?f th" |
|
1073 proof(rule image_Max_eqI) |
|
1074 show "finite ?A" using h_e.finite_threads by auto |
|
1075 next |
|
1076 show "th \<in> ?A" using h_e.th_kept by auto |
|
1077 next |
|
1078 show "\<forall>x\<in>?A. ?f x \<le> ?f th" |
|
1079 proof |
|
1080 fix x |
|
1081 assume "x \<in> ?A" |
|
1082 hence "x = thread \<or> x \<in> threads (t@s)" by (auto simp:Create) |
|
1083 thus "?f x \<le> ?f th" |
|
1084 proof |
|
1085 assume "x = thread" |
|
1086 thus ?thesis |
|
1087 apply (simp add:Create the_preced_def preced_def, fold preced_def) |
|
1088 using Create h_e.create_low h_t.th_kept lt_tm preced_leI2 |
|
1089 preced_th by force |
|
1090 next |
|
1091 assume h: "x \<in> threads (t @ s)" |
|
1092 from Cons(2)[unfolded Create] |
|
1093 have "x \<noteq> thread" using h by (cases, auto) |
|
1094 hence "?f x = the_preced (t@s) x" |
|
1095 by (simp add:Create the_preced_def preced_def) |
|
1096 hence "?f x \<le> Max (the_preced (t@s) ` threads (t@s))" |
|
1097 by (simp add: h_t.finite_threads h) |
|
1098 also have "... = ?f th" |
|
1099 by (metis Cons.hyps(5) h_e.th_kept the_preced_def) |
|
1100 finally show ?thesis . |
|
1101 qed |
|
1102 qed |
|
1103 qed |
|
1104 -- {* The minor part is to show that the precedence of @{text "th"} |
|
1105 equals to preserved one, given by the foregoing lemma @{thm th_kept} *} |
|
1106 also have "... = ?t" using h_e.th_kept the_preced_def by auto |
|
1107 -- {* Then it follows trivially that the precedence preserved |
|
1108 for @{term "th"} remains the maximum of all living threads along the way. *} |
|
1109 finally show ?thesis . |
|
1110 qed |
|
1111 next |
|
1112 case (Exit thread) |
|
1113 show ?thesis (is "Max (?f ` ?A) = ?t") |
|
1114 proof - |
|
1115 have "Max (?f ` ?A) = ?f th" |
|
1116 proof(rule image_Max_eqI) |
|
1117 show "finite ?A" using h_e.finite_threads by auto |
|
1118 next |
|
1119 show "th \<in> ?A" using h_e.th_kept by auto |
|
1120 next |
|
1121 show "\<forall>x\<in>?A. ?f x \<le> ?f th" |
|
1122 proof |
|
1123 fix x |
|
1124 assume "x \<in> ?A" |
|
1125 hence "x \<in> threads (t@s)" by (simp add: Exit) |
|
1126 hence "?f x \<le> Max (?f ` threads (t@s))" |
|
1127 by (simp add: h_t.finite_threads) |
|
1128 also have "... \<le> ?f th" |
|
1129 apply (simp add:Exit the_preced_def preced_def, fold preced_def) |
|
1130 using Cons.hyps(5) h_t.th_kept the_preced_def by auto |
|
1131 finally show "?f x \<le> ?f th" . |
|
1132 qed |
|
1133 qed |
|
1134 also have "... = ?t" using h_e.th_kept the_preced_def by auto |
|
1135 finally show ?thesis . |
|
1136 qed |
|
1137 next |
|
1138 case (P thread cs) |
|
1139 with Cons |
|
1140 show ?thesis by (auto simp:preced_def the_preced_def) |
|
1141 next |
|
1142 case (V thread cs) |
|
1143 with Cons |
|
1144 show ?thesis by (auto simp:preced_def the_preced_def) |
|
1145 next |
|
1146 case (Set thread prio') |
|
1147 show ?thesis (is "Max (?f ` ?A) = ?t") |
|
1148 proof - |
|
1149 have "Max (?f ` ?A) = ?f th" |
|
1150 proof(rule image_Max_eqI) |
|
1151 show "finite ?A" using h_e.finite_threads by auto |
|
1152 next |
|
1153 show "th \<in> ?A" using h_e.th_kept by auto |
|
1154 next |
|
1155 show "\<forall>x\<in>?A. ?f x \<le> ?f th" |
|
1156 proof |
|
1157 fix x |
|
1158 assume h: "x \<in> ?A" |
|
1159 show "?f x \<le> ?f th" |
|
1160 proof(cases "x = thread") |
|
1161 case True |
|
1162 moreover have "the_preced (Set thread prio' # t @ s) thread \<le> the_preced (t @ s) th" |
|
1163 proof - |
|
1164 have "the_preced (t @ s) th = Prc prio tm" |
|
1165 using h_t.th_kept preced_th by (simp add:the_preced_def) |
|
1166 moreover have "prio' \<le> prio" using Set h_e.set_diff_low by auto |
|
1167 ultimately show ?thesis by (insert lt_tm, auto simp:the_preced_def preced_def) |
|
1168 qed |
|
1169 ultimately show ?thesis |
|
1170 by (unfold Set, simp add:the_preced_def preced_def) |
|
1171 next |
|
1172 case False |
|
1173 then have "?f x = the_preced (t@s) x" |
|
1174 by (simp add:the_preced_def preced_def Set) |
|
1175 also have "... \<le> Max (the_preced (t@s) ` threads (t@s))" |
|
1176 using Set h h_t.finite_threads by auto |
|
1177 also have "... = ?f th" by (metis Cons.hyps(5) h_e.th_kept the_preced_def) |
|
1178 finally show ?thesis . |
|
1179 qed |
|
1180 qed |
|
1181 qed |
|
1182 also have "... = ?t" using h_e.th_kept the_preced_def by auto |
|
1183 finally show ?thesis . |
|
1184 qed |
|
1185 qed |
|
1186 qed |
|
1187 |
|
1188 lemma max_preced: "preced th (t@s) = Max (the_preced (t@s) ` (threads (t@s)))" |
|
1189 by (insert th_kept max_kept, auto) |
|
1190 |
|
1191 text {* |
|
1192 The reason behind the following lemma is that: |
|
1193 Since @{term "cp"} is defined as the maximum precedence |
|
1194 of those threads contained in the sub-tree of node @{term "Th th"} |
|
1195 in @{term "RAG (t@s)"}, and all these threads are living threads, and |
|
1196 @{term "th"} is also among them, the maximum precedence of |
|
1197 them all must be the one for @{text "th"}. |
|
1198 *} |
|
1199 lemma th_cp_max_preced: |
|
1200 "cp (t@s) th = Max (the_preced (t@s) ` (threads (t@s)))" (is "?L = ?R") |
|
1201 proof - |
|
1202 let ?f = "the_preced (t@s)" |
|
1203 have "?L = ?f th" |
|
1204 proof(unfold cp_alt_def, rule image_Max_eqI) |
|
1205 show "finite {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}" |
|
1206 proof - |
|
1207 have "{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)} = |
|
1208 the_thread ` {n . n \<in> subtree (RAG (t @ s)) (Th th) \<and> |
|
1209 (\<exists> th'. n = Th th')}" |
|
1210 by (smt Collect_cong Setcompr_eq_image mem_Collect_eq the_thread.simps) |
|
1211 moreover have "finite ..." by (simp add: vat_t.fsbtRAGs.finite_subtree) |
|
1212 ultimately show ?thesis by simp |
|
1213 qed |
|
1214 next |
|
1215 show "th \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}" |
|
1216 by (auto simp:subtree_def) |
|
1217 next |
|
1218 show "\<forall>x\<in>{th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}. |
|
1219 the_preced (t @ s) x \<le> the_preced (t @ s) th" |
|
1220 proof |
|
1221 fix th' |
|
1222 assume "th' \<in> {th'. Th th' \<in> subtree (RAG (t @ s)) (Th th)}" |
|
1223 hence "Th th' \<in> subtree (RAG (t @ s)) (Th th)" by auto |
|
1224 moreover have "... \<subseteq> Field (RAG (t @ s)) \<union> {Th th}" |
|
1225 by (meson subtree_Field) |
|
1226 ultimately have "Th th' \<in> ..." by auto |
|
1227 hence "th' \<in> threads (t@s)" |
|
1228 proof |
|
1229 assume "Th th' \<in> {Th th}" |
|
1230 thus ?thesis using th_kept by auto |
|
1231 next |
|
1232 assume "Th th' \<in> Field (RAG (t @ s))" |
|
1233 thus ?thesis using vat_t.not_in_thread_isolated by blast |
|
1234 qed |
|
1235 thus "the_preced (t @ s) th' \<le> the_preced (t @ s) th" |
|
1236 by (metis Max_ge finite_imageI finite_threads image_eqI |
|
1237 max_kept th_kept the_preced_def) |
|
1238 qed |
|
1239 qed |
|
1240 also have "... = ?R" by (simp add: max_preced the_preced_def) |
|
1241 finally show ?thesis . |
|
1242 qed |
|
1243 |
|
1244 lemma th_cp_max[simp]: "Max (cp (t@s) ` threads (t@s)) = cp (t@s) th" |
|
1245 using max_cp_eq th_cp_max_preced the_preced_def vt_t by presburger |
|
1246 |
|
1247 lemma [simp]: "Max (cp (t@s) ` threads (t@s)) = Max (the_preced (t@s) ` threads (t@s))" |
|
1248 by (simp add: th_cp_max_preced) |
|
1249 |
|
1250 lemma [simp]: "Max (the_preced (t@s) ` threads (t@s)) = the_preced (t@s) th" |
|
1251 using max_kept th_kept the_preced_def by auto |
|
1252 |
|
1253 lemma [simp]: "the_preced (t@s) th = preced th (t@s)" |
|
1254 using the_preced_def by auto |
|
1255 |
|
1256 lemma [simp]: "preced th (t@s) = preced th s" |
|
1257 by (simp add: th_kept) |
|
1258 |
|
1259 lemma [simp]: "cp s th = preced th s" |
|
1260 by (simp add: eq_cp_s_th) |
|
1261 |
|
1262 lemma th_cp_preced [simp]: "cp (t@s) th = preced th s" |
|
1263 by (fold max_kept, unfold th_cp_max_preced, simp) |
|
1264 |
|
1265 lemma preced_less: |
|
1266 assumes th'_in: "th' \<in> threads s" |
|
1267 and neq_th': "th' \<noteq> th" |
|
1268 shows "preced th' s < preced th s" |
|
1269 using assms |
|
1270 by (metis Max.coboundedI finite_imageI highest not_le order.trans |
|
1271 preced_linorder rev_image_eqI threads_s vat_s.finite_threads |
|
1272 vat_s.le_cp) |
|
1273 |
|
1274 section {* The `blocking thread` *} |
|
1275 |
|
1276 text {* |
|
1277 |
|
1278 The purpose of PIP is to ensure that the most urgent thread @{term |
|
1279 th} is not blocked unreasonably. Therefore, below, we will derive |
|
1280 properties of the blocking thread. By blocking thread, we mean a |
|
1281 thread in running state t @ s, but is different from thread @{term |
|
1282 th}. |
|
1283 |
|
1284 The first lemmas shows that the @{term cp}-value of the blocking |
|
1285 thread @{text th'} equals to the highest precedence in the whole |
|
1286 system. |
|
1287 |
|
1288 *} |
|
1289 |
|
1290 lemma runing_preced_inversion: |
|
1291 assumes runing': "th' \<in> runing (t @ s)" |
|
1292 shows "cp (t @ s) th' = preced th s" |
|
1293 proof - |
|
1294 have "cp (t @ s) th' = Max (cp (t @ s) ` readys (t @ s))" |
|
1295 using assms by (unfold runing_def, auto) |
|
1296 also have "\<dots> = preced th s" |
|
1297 by (metis th_cp_max th_cp_preced vat_t.max_cp_readys_threads) |
|
1298 finally show ?thesis . |
|
1299 qed |
|
1300 |
|
1301 text {* |
|
1302 |
|
1303 The next lemma shows how the counters for @{term "P"} and @{term |
|
1304 "V"} operations relate to the running threads in the states @{term |
|
1305 s} and @{term "t @ s"}: if a thread's @{term "P"}-count equals its |
|
1306 @{term "V"}-count (which means it no longer has any resource in its |
|
1307 possession), it cannot be a running thread. |
|
1308 |
|
1309 The proof is by contraction with the assumption @{text "th' \<noteq> th"}. |
|
1310 The key is the use of @{thm count_eq_dependants} to derive the |
|
1311 emptiness of @{text th'}s @{term dependants}-set from the balance of |
|
1312 its @{term P} and @{term V} counts. From this, it can be shown |
|
1313 @{text th'}s @{term cp}-value equals to its own precedence. |
|
1314 |
|
1315 On the other hand, since @{text th'} is running, by @{thm |
|
1316 runing_preced_inversion}, its @{term cp}-value equals to the |
|
1317 precedence of @{term th}. |
|
1318 |
|
1319 Combining the above two results we have that @{text th'} and @{term |
|
1320 th} have the same precedence. By uniqueness of precedences, we have |
|
1321 @{text "th' = th"}, which is in contradiction with the assumption |
|
1322 @{text "th' \<noteq> th"}. |
|
1323 |
|
1324 *} |
|
1325 |
|
1326 lemma eq_pv_blocked: (* ddd *) |
|
1327 assumes neq_th': "th' \<noteq> th" |
|
1328 and eq_pv: "cntP (t @ s) th' = cntV (t @ s) th'" |
|
1329 shows "th' \<notin> runing (t @ s)" |
|
1330 proof |
|
1331 assume otherwise: "th' \<in> runing (t @ s)" |
|
1332 show False |
|
1333 proof - |
|
1334 have th'_in: "th' \<in> threads (t @ s)" |
|
1335 using otherwise readys_threads runing_def by auto |
|
1336 have "th' = th" |
|
1337 proof(rule preced_unique) |
|
1338 -- {* The proof goes like this: |
|
1339 it is first shown that the @{term preced}-value of @{term th'} |
|
1340 equals to that of @{term th}, then by uniqueness |
|
1341 of @{term preced}-values (given by lemma @{thm preced_unique}), |
|
1342 @{term th'} equals to @{term th}: *} |
|
1343 show "preced th' (t @ s) = preced th (t @ s)" (is "?L = ?R") |
|
1344 proof - |
|
1345 -- {* Since the counts of @{term th'} are balanced, the subtree |
|
1346 of it contains only itself, so, its @{term cp}-value |
|
1347 equals its @{term preced}-value: *} |
|
1348 have "?L = cp (t @ s) th'" |
|
1349 by (unfold cp_eq_cpreced cpreced_def count_eq_dependants[OF eq_pv], simp) |
|
1350 -- {* Since @{term "th'"} is running, by @{thm runing_preced_inversion}, |
|
1351 its @{term cp}-value equals @{term "preced th s"}, |
|
1352 which equals to @{term "?R"} by simplification: *} |
|
1353 also have "... = ?R" |
|
1354 using runing_preced_inversion[OF otherwise] by simp |
|
1355 finally show ?thesis . |
|
1356 qed |
|
1357 qed (auto simp: th'_in th_kept) |
|
1358 with `th' \<noteq> th` show ?thesis by simp |
|
1359 qed |
|
1360 qed |
|
1361 |
|
1362 text {* |
|
1363 The following lemma is the extrapolation of @{thm eq_pv_blocked}. |
|
1364 It says if a thread, different from @{term th}, |
|
1365 does not hold any resource at the very beginning, |
|
1366 it will keep hand-emptied in the future @{term "t@s"}. |
|
1367 *} |
|
1368 lemma eq_pv_persist: (* ddd *) |
|
1369 assumes neq_th': "th' \<noteq> th" |
|
1370 and eq_pv: "cntP s th' = cntV s th'" |
|
1371 shows "cntP (t @ s) th' = cntV (t @ s) th'" |
|
1372 proof(induction rule: ind) |
|
1373 -- {* The nontrivial case is for the @{term Cons}: *} |
|
1374 case (Cons e t) |
|
1375 -- {* All results derived so far hold for both @{term s} and @{term "t@s"}: *} |
|
1376 interpret vat_t: extend_highest_gen s th prio tm t using Cons by simp |
|
1377 interpret vat_e: extend_highest_gen s th prio tm "(e # t)" using Cons by simp |
|
1378 show ?case |
|
1379 proof - |
|
1380 -- {* It can be proved that @{term cntP}-value of @{term th'} does not change |
|
1381 by the happening of event @{term e}: *} |
|
1382 have "cntP ((e#t)@s) th' = cntP (t@s) th'" |
|
1383 proof(rule ccontr) -- {* Proof by contradiction. *} |
|
1384 -- {* Suppose @{term cntP}-value of @{term th'} is changed by @{term e}: *} |
|
1385 assume otherwise: "cntP ((e # t) @ s) th' \<noteq> cntP (t @ s) th'" |
|
1386 -- {* Then the actor of @{term e} must be @{term th'} and @{term e} |
|
1387 must be a @{term P}-event: *} |
|
1388 hence "isP e" "actor e = th'" by (auto simp:cntP_diff_inv) |
|
1389 with vat_t.actor_inv[OF Cons(2)] |
|
1390 -- {* According to @{thm actor_inv}, @{term th'} must be running at |
|
1391 the moment @{term "t@s"}: *} |
|
1392 have "th' \<in> runing (t@s)" by (cases e, auto) |
|
1393 -- {* However, an application of @{thm eq_pv_blocked} to induction hypothesis |
|
1394 shows @{term th'} can not be running at moment @{term "t@s"}: *} |
|
1395 moreover have "th' \<notin> runing (t@s)" |
|
1396 using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] . |
|
1397 -- {* Contradiction is finally derived: *} |
|
1398 ultimately show False by simp |
|
1399 qed |
|
1400 -- {* It can also be proved that @{term cntV}-value of @{term th'} does not change |
|
1401 by the happening of event @{term e}: *} |
|
1402 -- {* The proof follows exactly the same pattern as the case for @{term cntP}-value: *} |
|
1403 moreover have "cntV ((e#t)@s) th' = cntV (t@s) th'" |
|
1404 proof(rule ccontr) -- {* Proof by contradiction. *} |
|
1405 assume otherwise: "cntV ((e # t) @ s) th' \<noteq> cntV (t @ s) th'" |
|
1406 hence "isV e" "actor e = th'" by (auto simp:cntV_diff_inv) |
|
1407 with vat_t.actor_inv[OF Cons(2)] |
|
1408 have "th' \<in> runing (t@s)" by (cases e, auto) |
|
1409 moreover have "th' \<notin> runing (t@s)" |
|
1410 using vat_t.eq_pv_blocked[OF neq_th' Cons(5)] . |
|
1411 ultimately show False by simp |
|
1412 qed |
|
1413 -- {* Finally, it can be shown that the @{term cntP} and @{term cntV} |
|
1414 value for @{term th'} are still in balance, so @{term th'} |
|
1415 is still hand-emptied after the execution of event @{term e}: *} |
|
1416 ultimately show ?thesis using Cons(5) by metis |
|
1417 qed |
|
1418 qed (auto simp:eq_pv) |
|
1419 |
|
1420 text {* |
|
1421 |
|
1422 By combining @{thm eq_pv_blocked} and @{thm eq_pv_persist}, it can |
|
1423 be derived easily that @{term th'} can not be running in the future: |
|
1424 |
|
1425 *} |
|
1426 |
|
1427 lemma eq_pv_blocked_persist: |
|
1428 assumes neq_th': "th' \<noteq> th" |
|
1429 and eq_pv: "cntP s th' = cntV s th'" |
|
1430 shows "th' \<notin> runing (t @ s)" |
|
1431 using assms |
|
1432 by (simp add: eq_pv_blocked eq_pv_persist) |
|
1433 |
|
1434 text {* |
|
1435 |
|
1436 The following lemma shows the blocking thread @{term th'} must hold |
|
1437 some resource in the very beginning. |
|
1438 |
|
1439 *} |
|
1440 |
|
1441 lemma runing_cntP_cntV_inv: (* ddd *) |
|
1442 assumes is_runing: "th' \<in> runing (t @ s)" |
|
1443 and neq_th': "th' \<noteq> th" |
|
1444 shows "cntP s th' > cntV s th'" |
|
1445 using assms |
|
1446 proof - |
|
1447 -- {* First, it can be shown that the number of @{term P} and |
|
1448 @{term V} operations can not be equal for thred @{term th'} *} |
|
1449 have "cntP s th' \<noteq> cntV s th'" |
|
1450 proof |
|
1451 -- {* The proof goes by contradiction, suppose otherwise: *} |
|
1452 assume otherwise: "cntP s th' = cntV s th'" |
|
1453 -- {* By applying @{thm eq_pv_blocked_persist} to this: *} |
|
1454 from eq_pv_blocked_persist[OF neq_th' otherwise] |
|
1455 -- {* we have that @{term th'} can not be running at moment @{term "t@s"}: *} |
|
1456 have "th' \<notin> runing (t@s)" . |
|
1457 -- {* This is obvious in contradiction with assumption @{thm is_runing} *} |
|
1458 thus False using is_runing by simp |
|
1459 qed |
|
1460 -- {* However, the number of @{term V} is always less or equal to @{term P}: *} |
|
1461 moreover have "cntV s th' \<le> cntP s th'" using vat_s.cnp_cnv_cncs by auto |
|
1462 -- {* Thesis is finally derived by combining the these two results: *} |
|
1463 ultimately show ?thesis by auto |
|
1464 qed |
|
1465 |
|
1466 |
|
1467 text {* |
|
1468 |
|
1469 The following lemmas shows the blocking thread @{text th'} must be |
|
1470 live at the very beginning, i.e. the moment (or state) @{term s}. |
|
1471 The proof is a simple combination of the results above: |
|
1472 |
|
1473 *} |
|
1474 |
|
1475 lemma runing_threads_inv: |
|
1476 assumes runing': "th' \<in> runing (t@s)" |
|
1477 and neq_th': "th' \<noteq> th" |
|
1478 shows "th' \<in> threads s" |
|
1479 proof(rule ccontr) -- {* Proof by contradiction: *} |
|
1480 assume otherwise: "th' \<notin> threads s" |
|
1481 have "th' \<notin> runing (t @ s)" |
|
1482 proof - |
|
1483 from vat_s.cnp_cnv_eq[OF otherwise] |
|
1484 have "cntP s th' = cntV s th'" . |
|
1485 from eq_pv_blocked_persist[OF neq_th' this] |
|
1486 show ?thesis . |
|
1487 qed |
|
1488 with runing' show False by simp |
|
1489 qed |
|
1490 |
|
1491 text {* |
|
1492 |
|
1493 The following lemma summarises the above lemmas to give an overall |
|
1494 characterisationof the blocking thread @{text "th'"}: |
|
1495 |
|
1496 *} |
|
1497 |
|
1498 lemma runing_inversion: (* ddd, one of the main lemmas to present *) |
|
1499 assumes runing': "th' \<in> runing (t@s)" |
|
1500 and neq_th: "th' \<noteq> th" |
|
1501 shows "th' \<in> threads s" |
|
1502 and "\<not>detached s th'" |
|
1503 and "cp (t@s) th' = preced th s" |
|
1504 proof - |
|
1505 from runing_threads_inv[OF assms] |
|
1506 show "th' \<in> threads s" . |
|
1507 next |
|
1508 from runing_cntP_cntV_inv[OF runing' neq_th] |
|
1509 show "\<not>detached s th'" using vat_s.detached_eq by simp |
|
1510 next |
|
1511 from runing_preced_inversion[OF runing'] |
|
1512 show "cp (t@s) th' = preced th s" . |
|
1513 qed |
|
1514 |
|
1515 |
|
1516 section {* The existence of `blocking thread` *} |
|
1517 |
|
1518 text {* |
|
1519 |
|
1520 Suppose @{term th} is not running, it is first shown that there is a |
|
1521 path in RAG leading from node @{term th} to another thread @{text |
|
1522 "th'"} in the @{term readys}-set (So @{text "th'"} is an ancestor of |
|
1523 @{term th}}). |
|
1524 |
|
1525 Now, since @{term readys}-set is non-empty, there must be one in it |
|
1526 which holds the highest @{term cp}-value, which, by definition, is |
|
1527 the @{term runing}-thread. However, we are going to show more: this |
|
1528 running thread is exactly @{term "th'"}. |
|
1529 |
|
1530 *} |
|
1531 |
|
1532 lemma th_blockedE: (* ddd, the other main lemma to be presented: *) |
|
1533 assumes "th \<notin> runing (t@s)" |
|
1534 obtains th' where "Th th' \<in> ancestors (RAG (t @ s)) (Th th)" |
|
1535 "th' \<in> runing (t@s)" |
|
1536 proof - |
|
1537 -- {* According to @{thm vat_t.th_chain_to_ready}, either |
|
1538 @{term "th"} is in @{term "readys"} or there is path leading from it to |
|
1539 one thread in @{term "readys"}. *} |
|
1540 have "th \<in> readys (t @ s) \<or> (\<exists>th'. th' \<in> readys (t @ s) \<and> (Th th, Th th') \<in> (RAG (t @ s))\<^sup>+)" |
|
1541 using th_kept vat_t.th_chain_to_ready by auto |
|
1542 -- {* However, @{term th} can not be in @{term readys}, because otherwise, since |
|
1543 @{term th} holds the highest @{term cp}-value, it must be @{term "runing"}. *} |
|
1544 moreover have "th \<notin> readys (t@s)" |
|
1545 using assms runing_def th_cp_max vat_t.max_cp_readys_threads by auto |
|
1546 -- {* So, there must be a path from @{term th} to another thread @{text "th'"} in |
|
1547 term @{term readys}: *} |
|
1548 ultimately obtain th' where th'_in: "th' \<in> readys (t@s)" |
|
1549 and dp: "(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+" by auto |
|
1550 -- {* We are going to show that this @{term th'} is running. *} |
|
1551 have "th' \<in> runing (t@s)" |
|
1552 proof - |
|
1553 -- {* We only need to show that this @{term th'} holds the highest @{term cp}-value: *} |
|
1554 have "cp (t@s) th' = Max (cp (t@s) ` readys (t@s))" (is "?L = ?R") |
|
1555 proof - |
|
1556 have "?L = Max ((the_preced (t @ s) \<circ> the_thread) ` subtree (tRAG (t @ s)) (Th th'))" |
|
1557 by (unfold cp_alt_def1, simp) |
|
1558 also have "... = (the_preced (t @ s) \<circ> the_thread) (Th th)" |
|
1559 proof(rule image_Max_subset) |
|
1560 show "finite (Th ` (threads (t@s)))" by (simp add: vat_t.finite_threads) |
|
1561 next |
|
1562 show "subtree (tRAG (t @ s)) (Th th') \<subseteq> Th ` threads (t @ s)" |
|
1563 by (metis Range.intros dp trancl_range vat_t.range_in vat_t.subtree_tRAG_thread) |
|
1564 next |
|
1565 show "Th th \<in> subtree (tRAG (t @ s)) (Th th')" using dp |
|
1566 by (unfold tRAG_subtree_eq, auto simp:subtree_def) |
|
1567 next |
|
1568 show "Max ((the_preced (t @ s) \<circ> the_thread) ` Th ` threads (t @ s)) = |
|
1569 (the_preced (t @ s) \<circ> the_thread) (Th th)" (is "Max ?L = _") |
|
1570 proof - |
|
1571 have "?L = the_preced (t @ s) ` threads (t @ s)" |
|
1572 by (unfold image_comp, rule image_cong, auto) |
|
1573 thus ?thesis using max_preced the_preced_def by auto |
|
1574 qed |
|
1575 qed |
|
1576 also have "... = ?R" |
|
1577 using th_cp_max th_cp_preced th_kept |
|
1578 the_preced_def vat_t.max_cp_readys_threads by auto |
|
1579 finally show ?thesis . |
|
1580 qed |
|
1581 -- {* Now, since @{term th'} holds the highest @{term cp} |
|
1582 and we have already show it is in @{term readys}, |
|
1583 it is @{term runing} by definition. *} |
|
1584 with `th' \<in> readys (t@s)` show ?thesis by (simp add: runing_def) |
|
1585 qed |
|
1586 -- {* It is easy to show @{term th'} is an ancestor of @{term th}: *} |
|
1587 moreover have "Th th' \<in> ancestors (RAG (t @ s)) (Th th)" |
|
1588 using `(Th th, Th th') \<in> (RAG (t @ s))\<^sup>+` by (auto simp:ancestors_def) |
|
1589 ultimately show ?thesis using that by metis |
|
1590 qed |
|
1591 |
|
1592 text {* |
|
1593 |
|
1594 Now it is easy to see there is always a thread to run by case |
|
1595 analysis on whether thread @{term th} is running: if the answer is |
|
1596 yes, the the running thread is obviously @{term th} itself; |
|
1597 otherwise, the running thread is the @{text th'} given by lemma |
|
1598 @{thm th_blockedE}. |
|
1599 |
|
1600 *} |
|
1601 |
|
1602 lemma live: "runing (t@s) \<noteq> {}" |
|
1603 proof(cases "th \<in> runing (t@s)") |
|
1604 case True thus ?thesis by auto |
|
1605 next |
|
1606 case False |
|
1607 thus ?thesis using th_blockedE by auto |
|
1608 qed |
|
1609 |
|
1610 |
|
1611 end |
|
1612 end |
|
1613 >>>>>>> other |