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(* Title: HOL/UNITY/WFair
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1998 University of Cambridge
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Weak Fairness versions of transient, ensures, leadsTo.
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From Misra, "A Logic for Concurrent Programming", 1994
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*)
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(*** transient ***)
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Goalw [stable_def, constrains_def, transient_def]
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"[| stable acts A; transient acts A |] ==> A = {}";
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by (Blast_tac 1);
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qed "stable_transient_empty";
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Goalw [transient_def]
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"[| transient acts A; B<=A |] ==> transient acts B";
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by (Clarify_tac 1);
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by (rtac bexI 1 THEN assume_tac 2);
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by (Blast_tac 1);
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qed "transient_strengthen";
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Goalw [transient_def]
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"[| act:acts; A <= Domain act; act^^A <= Compl A |] \
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\ ==> transient acts A";
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by (Blast_tac 1);
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qed "transient_mem";
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(*** ensures ***)
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Goalw [ensures_def]
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"[| constrains acts (A-B) (A Un B); transient acts (A-B) |] \
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\ ==> ensures acts A B";
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by (Blast_tac 1);
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qed "ensuresI";
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Goalw [ensures_def]
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"ensures acts A B \
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\ ==> constrains acts (A-B) (A Un B) & transient acts (A-B)";
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by (Blast_tac 1);
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qed "ensuresD";
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(*The L-version (precondition strengthening) doesn't hold for ENSURES*)
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Goalw [ensures_def]
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"[| ensures acts A A'; A'<=B' |] ==> ensures acts A B'";
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by (blast_tac (claset() addIs [constrains_weaken, transient_strengthen]) 1);
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qed "ensures_weaken_R";
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Goalw [ensures_def, constrains_def, transient_def]
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"acts ~= {} ==> ensures acts A UNIV";
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by (Asm_simp_tac 1); (*omitting this causes PROOF FAILED, even with Safe_tac*)
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by (Blast_tac 1);
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qed "ensures_UNIV";
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Goalw [ensures_def]
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"[| stable acts C; \
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\ constrains acts (C Int (A - A')) (A Un A'); \
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\ transient acts (C Int (A-A')) |] \
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\ ==> ensures acts (C Int A) (C Int A')";
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by (asm_simp_tac (simpset() addsimps [Int_Un_distrib RS sym,
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Diff_Int_distrib RS sym,
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stable_constrains_Int]) 1);
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qed "stable_ensures_Int";
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(*** leadsTo ***)
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(*Synonyms for the theorems produced by the inductive defn package*)
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bind_thm ("leadsTo_Basis", leadsto.Basis);
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bind_thm ("leadsTo_Trans", leadsto.Trans);
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Goal "act: acts ==> leadsTo acts A UNIV";
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by (blast_tac (claset() addIs [ensures_UNIV RS leadsTo_Basis]) 1);
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qed "leadsTo_UNIV";
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Addsimps [leadsTo_UNIV];
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(*Useful with cancellation, disjunction*)
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Goal "leadsTo acts A (A' Un A') ==> leadsTo acts A A'";
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by (asm_full_simp_tac (simpset() addsimps Un_ac) 1);
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qed "leadsTo_Un_duplicate";
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Goal "leadsTo acts A (A' Un C Un C) ==> leadsTo acts A (A' Un C)";
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by (asm_full_simp_tac (simpset() addsimps Un_ac) 1);
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qed "leadsTo_Un_duplicate2";
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(*The Union introduction rule as we should have liked to state it*)
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val prems = goal thy
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"(!!A. A : S ==> leadsTo acts A B) ==> leadsTo acts (Union S) B";
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by (blast_tac (claset() addIs (leadsto.Union::prems)) 1);
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qed "leadsTo_Union";
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val prems = goal thy
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"(!!i. i : I ==> leadsTo acts (A i) B) ==> leadsTo acts (UN i:I. A i) B";
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by (simp_tac (simpset() addsimps [Union_image_eq RS sym]) 1);
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by (blast_tac (claset() addIs (leadsto.Union::prems)) 1);
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qed "leadsTo_UN";
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(*Binary union introduction rule*)
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Goal "[| leadsTo acts A C; leadsTo acts B C |] ==> leadsTo acts (A Un B) C";
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by (stac Un_eq_Union 1);
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by (blast_tac (claset() addIs [leadsTo_Union]) 1);
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qed "leadsTo_Un";
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(*The INDUCTION rule as we should have liked to state it*)
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val major::prems = goal thy
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"[| leadsTo acts za zb; \
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\ !!A B. ensures acts A B ==> P A B; \
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\ !!A B C. [| leadsTo acts A B; P A B; leadsTo acts B C; P B C |] \
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\ ==> P A C; \
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\ !!B S. ALL A:S. leadsTo acts A B & P A B ==> P (Union S) B \
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\ |] ==> P za zb";
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by (rtac (major RS leadsto.induct) 1);
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by (REPEAT (blast_tac (claset() addIs prems) 1));
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qed "leadsTo_induct";
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Goal "[| A<=B; id: acts |] ==> leadsTo acts A B";
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by (rtac leadsTo_Basis 1);
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by (rewrite_goals_tac [ensures_def, constrains_def, transient_def]);
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by (Blast_tac 1);
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qed "subset_imp_leadsTo";
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bind_thm ("empty_leadsTo", empty_subsetI RS subset_imp_leadsTo);
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Addsimps [empty_leadsTo];
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(*There's a direct proof by leadsTo_Trans and subset_imp_leadsTo, but it
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needs the extra premise id:acts*)
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Goal "leadsTo acts A A' ==> A'<=B' --> leadsTo acts A B'";
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by (etac leadsTo_induct 1);
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by (Clarify_tac 3);
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by (blast_tac (claset() addIs [leadsTo_Union]) 3);
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by (blast_tac (claset() addIs [leadsTo_Trans]) 2);
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by (blast_tac (claset() addIs [leadsTo_Basis, ensures_weaken_R]) 1);
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qed_spec_mp "leadsTo_weaken_R";
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Goal "[| leadsTo acts A A'; B<=A; id: acts |] ==> \
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\ leadsTo acts B A'";
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by (blast_tac (claset() addIs [leadsTo_Basis, leadsTo_Trans,
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subset_imp_leadsTo]) 1);
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qed_spec_mp "leadsTo_weaken_L";
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(*Distributes over binary unions*)
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Goal "id: acts ==> \
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\ leadsTo acts (A Un B) C = (leadsTo acts A C & leadsTo acts B C)";
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by (blast_tac (claset() addIs [leadsTo_Un, leadsTo_weaken_L]) 1);
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qed "leadsTo_Un_distrib";
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Goal "id: acts ==> \
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\ leadsTo acts (UN i:I. A i) B = (ALL i : I. leadsTo acts (A i) B)";
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by (blast_tac (claset() addIs [leadsTo_UN, leadsTo_weaken_L]) 1);
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qed "leadsTo_UN_distrib";
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Goal "id: acts ==> \
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\ leadsTo acts (Union S) B = (ALL A : S. leadsTo acts A B)";
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by (blast_tac (claset() addIs [leadsTo_Union, leadsTo_weaken_L]) 1);
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qed "leadsTo_Union_distrib";
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Goal "[| leadsTo acts A A'; id: acts; B<=A; A'<=B' |] \
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\ ==> leadsTo acts B B'";
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(*PROOF FAILED: why?*)
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by (blast_tac (claset() addIs [leadsTo_Trans, leadsTo_weaken_R,
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leadsTo_weaken_L]) 1);
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qed "leadsTo_weaken";
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(*Set difference: maybe combine with leadsTo_weaken_L??*)
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Goal "[| leadsTo acts (A-B) C; leadsTo acts B C; id: acts |] \
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\ ==> leadsTo acts A C";
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by (blast_tac (claset() addIs [leadsTo_Un, leadsTo_weaken]) 1);
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qed "leadsTo_Diff";
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(** Meta or object quantifier ???
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see ball_constrains_UN in UNITY.ML***)
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val prems = goal thy
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"(!! i. i:I ==> leadsTo acts (A i) (A' i)) \
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\ ==> leadsTo acts (UN i:I. A i) (UN i:I. A' i)";
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by (simp_tac (simpset() addsimps [Union_image_eq RS sym]) 1);
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by (blast_tac (claset() addIs [leadsTo_Union, leadsTo_weaken_R]
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addIs prems) 1);
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qed "leadsTo_UN_UN";
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(*Version with no index set*)
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val prems = goal thy
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"(!! i. leadsTo acts (A i) (A' i)) \
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\ ==> leadsTo acts (UN i. A i) (UN i. A' i)";
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by (blast_tac (claset() addIs [leadsTo_UN_UN]
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addIs prems) 1);
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qed "leadsTo_UN_UN_noindex";
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(*Version with no index set*)
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Goal "ALL i. leadsTo acts (A i) (A' i) \
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\ ==> leadsTo acts (UN i. A i) (UN i. A' i)";
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by (blast_tac (claset() addIs [leadsTo_UN_UN]) 1);
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qed "all_leadsTo_UN_UN";
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(*Binary union version*)
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Goal "[| leadsTo acts A A'; leadsTo acts B B' |] \
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\ ==> leadsTo acts (A Un B) (A' Un B')";
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by (blast_tac (claset() addIs [leadsTo_Un,
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leadsTo_weaken_R]) 1);
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qed "leadsTo_Un_Un";
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(** The cancellation law **)
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Goal "[| leadsTo acts A (A' Un B); leadsTo acts B B'; id: acts |] \
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\ ==> leadsTo acts A (A' Un B')";
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by (blast_tac (claset() addIs [leadsTo_Un_Un,
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subset_imp_leadsTo, leadsTo_Trans]) 1);
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qed "leadsTo_cancel2";
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Goal "[| leadsTo acts A (A' Un B); leadsTo acts (B-A') B'; id: acts |] \
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\ ==> leadsTo acts A (A' Un B')";
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by (rtac leadsTo_cancel2 1);
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by (assume_tac 2);
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by (ALLGOALS Asm_simp_tac);
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qed "leadsTo_cancel_Diff2";
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Goal "[| leadsTo acts A (B Un A'); leadsTo acts B B'; id: acts |] \
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\ ==> leadsTo acts A (B' Un A')";
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by (asm_full_simp_tac (simpset() addsimps [Un_commute]) 1);
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by (blast_tac (claset() addSIs [leadsTo_cancel2]) 1);
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qed "leadsTo_cancel1";
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Goal "[| leadsTo acts A (B Un A'); leadsTo acts (B-A') B'; id: acts |] \
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\ ==> leadsTo acts A (B' Un A')";
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by (rtac leadsTo_cancel1 1);
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by (assume_tac 2);
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by (ALLGOALS Asm_simp_tac);
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qed "leadsTo_cancel_Diff1";
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(** The impossibility law **)
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Goal "leadsTo acts A B ==> B={} --> A={}";
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by (etac leadsTo_induct 1);
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by (ALLGOALS Asm_simp_tac);
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by (rewrite_goals_tac [ensures_def, constrains_def, transient_def]);
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by (Blast_tac 1);
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val lemma = result() RS mp;
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Goal "leadsTo acts A {} ==> A={}";
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by (blast_tac (claset() addSIs [lemma]) 1);
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qed "leadsTo_empty";
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(** PSP: Progress-Safety-Progress **)
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(*Special case of PSP: Misra's "stable conjunction". Doesn't need id:acts. *)
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Goalw [stable_def]
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"[| leadsTo acts A A'; stable acts B |] \
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\ ==> leadsTo acts (A Int B) (A' Int B)";
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by (etac leadsTo_induct 1);
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by (simp_tac (simpset() addsimps [Int_Union_Union]) 3);
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by (blast_tac (claset() addIs [leadsTo_Union]) 3);
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by (blast_tac (claset() addIs [leadsTo_Trans]) 2);
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by (rtac leadsTo_Basis 1);
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by (asm_full_simp_tac
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(simpset() addsimps [ensures_def,
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Diff_Int_distrib2 RS sym, Int_Un_distrib2 RS sym]) 1);
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by (blast_tac (claset() addIs [transient_strengthen, constrains_Int]) 1);
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qed "PSP_stable";
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Goal "[| leadsTo acts A A'; stable acts B |] \
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\ ==> leadsTo acts (B Int A) (B Int A')";
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by (asm_simp_tac (simpset() addsimps (PSP_stable::Int_ac)) 1);
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qed "PSP_stable2";
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Goalw [ensures_def]
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"[| ensures acts A A'; constrains acts B B' |] \
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\ ==> ensures acts (A Int B) ((A' Int B) Un (B' - B))";
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by Safe_tac;
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by (blast_tac (claset() addIs [constrainsI] addDs [constrainsD]) 1);
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by (etac transient_strengthen 1);
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by (Blast_tac 1);
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qed "PSP_ensures";
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Goal "[| leadsTo acts A A'; constrains acts B B'; id: acts |] \
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\ ==> leadsTo acts (A Int B) ((A' Int B) Un (B' - B))";
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by (etac leadsTo_induct 1);
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by (simp_tac (simpset() addsimps [Int_Union_Union]) 3);
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by (blast_tac (claset() addIs [leadsTo_Union]) 3);
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(*Transitivity case has a delicate argument involving "cancellation"*)
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by (rtac leadsTo_Un_duplicate2 2);
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by (etac leadsTo_cancel_Diff1 2);
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by (assume_tac 3);
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by (asm_full_simp_tac (simpset() addsimps [Int_Diff, Diff_triv]) 2);
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(*Basis case*)
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by (blast_tac (claset() addIs [leadsTo_Basis, PSP_ensures]) 1);
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qed "PSP";
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Goal "[| leadsTo acts A A'; constrains acts B B'; id: acts |] \
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\ ==> leadsTo acts (B Int A) ((B Int A') Un (B' - B))";
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by (asm_simp_tac (simpset() addsimps (PSP::Int_ac)) 1);
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qed "PSP2";
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Goalw [unless_def]
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"[| leadsTo acts A A'; unless acts B B'; id: acts |] \
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\ ==> leadsTo acts (A Int B) ((A' Int B) Un B')";
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by (dtac PSP 1);
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by (assume_tac 1);
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by (asm_full_simp_tac (simpset() addsimps [Un_Diff_Diff, Int_Diff_Un]) 2);
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by (asm_full_simp_tac (simpset() addsimps [Diff_Int_distrib]) 2);
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by (etac leadsTo_Diff 2);
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by (blast_tac (claset() addIs [subset_imp_leadsTo]) 2);
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by Auto_tac;
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qed "PSP_unless";
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(*** Proving the induction rules ***)
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(** The most general rule: r is any wf relation; f is any variant function **)
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Goal "[| wf r; \
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\ ALL m. leadsTo acts (A Int f-``{m}) \
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5239
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332 |
\ ((A Int f-``(r^-1 ^^ {m})) Un B); \
|
5253
|
333 |
\ id: acts |] \
|
|
334 |
\ ==> leadsTo acts (A Int f-``{m}) B";
|
4776
|
335 |
by (eres_inst_tac [("a","m")] wf_induct 1);
|
5253
|
336 |
by (subgoal_tac "leadsTo acts (A Int (f -`` (r^-1 ^^ {x}))) B" 1);
|
4776
|
337 |
by (stac vimage_eq_UN 2);
|
|
338 |
by (asm_simp_tac (HOL_ss addsimps (UN_simps RL [sym])) 2);
|
|
339 |
by (blast_tac (claset() addIs [leadsTo_UN]) 2);
|
|
340 |
by (blast_tac (claset() addIs [leadsTo_cancel1, leadsTo_Un_duplicate]) 1);
|
|
341 |
val lemma = result();
|
|
342 |
|
|
343 |
|
|
344 |
(** Meta or object quantifier ????? **)
|
5239
|
345 |
Goal "[| wf r; \
|
5253
|
346 |
\ ALL m. leadsTo acts (A Int f-``{m}) \
|
5239
|
347 |
\ ((A Int f-``(r^-1 ^^ {m})) Un B); \
|
5253
|
348 |
\ id: acts |] \
|
|
349 |
\ ==> leadsTo acts A B";
|
4776
|
350 |
by (res_inst_tac [("t", "A")] subst 1);
|
|
351 |
by (rtac leadsTo_UN 2);
|
|
352 |
by (etac lemma 2);
|
|
353 |
by (REPEAT (assume_tac 2));
|
|
354 |
by (Fast_tac 1); (*Blast_tac: Function unknown's argument not a parameter*)
|
|
355 |
qed "leadsTo_wf_induct";
|
|
356 |
|
|
357 |
|
5239
|
358 |
Goal "[| wf r; \
|
5253
|
359 |
\ ALL m:I. leadsTo acts (A Int f-``{m}) \
|
5257
|
360 |
\ ((A Int f-``(r^-1 ^^ {m})) Un B); \
|
5253
|
361 |
\ id: acts |] \
|
|
362 |
\ ==> leadsTo acts A ((A - (f-``I)) Un B)";
|
4776
|
363 |
by (etac leadsTo_wf_induct 1);
|
|
364 |
by Safe_tac;
|
|
365 |
by (case_tac "m:I" 1);
|
|
366 |
by (blast_tac (claset() addIs [leadsTo_weaken]) 1);
|
|
367 |
by (blast_tac (claset() addIs [subset_imp_leadsTo]) 1);
|
|
368 |
qed "bounded_induct";
|
|
369 |
|
|
370 |
|
5253
|
371 |
(*Alternative proof is via the lemma leadsTo acts (A Int f-``(lessThan m)) B*)
|
|
372 |
Goal "[| ALL m. leadsTo acts (A Int f-``{m}) \
|
5239
|
373 |
\ ((A Int f-``(lessThan m)) Un B); \
|
5253
|
374 |
\ id: acts |] \
|
|
375 |
\ ==> leadsTo acts A B";
|
4776
|
376 |
by (rtac (wf_less_than RS leadsTo_wf_induct) 1);
|
|
377 |
by (assume_tac 2);
|
|
378 |
by (Asm_simp_tac 1);
|
|
379 |
qed "lessThan_induct";
|
|
380 |
|
5253
|
381 |
Goal "[| ALL m:(greaterThan l). leadsTo acts (A Int f-``{m}) \
|
5239
|
382 |
\ ((A Int f-``(lessThan m)) Un B); \
|
5253
|
383 |
\ id: acts |] \
|
|
384 |
\ ==> leadsTo acts A ((A Int (f-``(atMost l))) Un B)";
|
4776
|
385 |
by (simp_tac (HOL_ss addsimps [Diff_eq RS sym, vimage_Compl, Compl_greaterThan RS sym]) 1);
|
|
386 |
by (rtac (wf_less_than RS bounded_induct) 1);
|
|
387 |
by (assume_tac 2);
|
|
388 |
by (Asm_simp_tac 1);
|
|
389 |
qed "lessThan_bounded_induct";
|
|
390 |
|
5253
|
391 |
Goal "[| ALL m:(lessThan l). leadsTo acts (A Int f-``{m}) \
|
5239
|
392 |
\ ((A Int f-``(greaterThan m)) Un B); \
|
5253
|
393 |
\ id: acts |] \
|
|
394 |
\ ==> leadsTo acts A ((A Int (f-``(atLeast l))) Un B)";
|
4776
|
395 |
by (res_inst_tac [("f","f"),("f1", "%k. l - k")]
|
|
396 |
(wf_less_than RS wf_inv_image RS leadsTo_wf_induct) 1);
|
|
397 |
by (assume_tac 2);
|
|
398 |
by (simp_tac (simpset() addsimps [inv_image_def, Image_singleton]) 1);
|
|
399 |
by (Clarify_tac 1);
|
|
400 |
by (case_tac "m<l" 1);
|
|
401 |
by (blast_tac (claset() addIs [not_leE, subset_imp_leadsTo]) 2);
|
|
402 |
by (blast_tac (claset() addIs [leadsTo_weaken_R, diff_less_mono2]) 1);
|
|
403 |
qed "greaterThan_bounded_induct";
|
|
404 |
|
|
405 |
|
|
406 |
|
|
407 |
(*** wlt ****)
|
|
408 |
|
|
409 |
(*Misra's property W3*)
|
5253
|
410 |
Goalw [wlt_def] "leadsTo acts (wlt acts B) B";
|
4776
|
411 |
by (blast_tac (claset() addSIs [leadsTo_Union]) 1);
|
|
412 |
qed "wlt_leadsTo";
|
|
413 |
|
5253
|
414 |
Goalw [wlt_def] "leadsTo acts A B ==> A <= wlt acts B";
|
4776
|
415 |
by (blast_tac (claset() addSIs [leadsTo_Union]) 1);
|
|
416 |
qed "leadsTo_subset";
|
|
417 |
|
|
418 |
(*Misra's property W2*)
|
5253
|
419 |
Goal "id: acts ==> leadsTo acts A B = (A <= wlt acts B)";
|
4776
|
420 |
by (blast_tac (claset() addSIs [leadsTo_subset,
|
|
421 |
wlt_leadsTo RS leadsTo_weaken_L]) 1);
|
|
422 |
qed "leadsTo_eq_subset_wlt";
|
|
423 |
|
|
424 |
(*Misra's property W4*)
|
5253
|
425 |
Goal "id: acts ==> B <= wlt acts B";
|
4776
|
426 |
by (asm_simp_tac (simpset() addsimps [leadsTo_eq_subset_wlt RS sym,
|
|
427 |
subset_imp_leadsTo]) 1);
|
|
428 |
qed "wlt_increasing";
|
|
429 |
|
|
430 |
|
|
431 |
(*Used in the Trans case below*)
|
5069
|
432 |
Goalw [constrains_def]
|
5111
|
433 |
"[| B <= A2; \
|
5253
|
434 |
\ constrains acts (A1 - B) (A1 Un B); \
|
|
435 |
\ constrains acts (A2 - C) (A2 Un C) |] \
|
|
436 |
\ ==> constrains acts (A1 Un A2 - C) (A1 Un A2 Un C)";
|
4776
|
437 |
by (Clarify_tac 1);
|
|
438 |
by (blast_tac (claset() addSDs [bspec]) 1);
|
|
439 |
val lemma1 = result();
|
|
440 |
|
|
441 |
|
|
442 |
(*Lemma (1,2,3) of Misra's draft book, Chapter 4, "Progress"*)
|
5253
|
443 |
Goal "[| leadsTo acts A A'; id: acts |] ==> \
|
|
444 |
\ EX B. A<=B & leadsTo acts B A' & constrains acts (B-A') (B Un A')";
|
4776
|
445 |
by (etac leadsTo_induct 1);
|
|
446 |
(*Basis*)
|
|
447 |
by (blast_tac (claset() addIs [leadsTo_Basis]
|
|
448 |
addDs [ensuresD]) 1);
|
|
449 |
(*Trans*)
|
|
450 |
by (Clarify_tac 1);
|
|
451 |
by (res_inst_tac [("x", "Ba Un Bb")] exI 1);
|
|
452 |
by (blast_tac (claset() addIs [lemma1, leadsTo_Un_Un, leadsTo_cancel1,
|
|
453 |
leadsTo_Un_duplicate]) 1);
|
|
454 |
(*Union*)
|
|
455 |
by (clarify_tac (claset() addSDs [ball_conj_distrib RS iffD1,
|
|
456 |
bchoice, ball_constrains_UN]) 1);;
|
|
457 |
by (res_inst_tac [("x", "UN A:S. f A")] exI 1);
|
|
458 |
by (blast_tac (claset() addIs [leadsTo_UN, constrains_weaken]) 1);
|
|
459 |
qed "leadsTo_123";
|
|
460 |
|
|
461 |
|
|
462 |
(*Misra's property W5*)
|
5253
|
463 |
Goal "id: acts ==> constrains acts (wlt acts B - B) (wlt acts B)";
|
4776
|
464 |
by (forward_tac [wlt_leadsTo RS leadsTo_123] 1);
|
|
465 |
by (Clarify_tac 1);
|
5253
|
466 |
by (subgoal_tac "Ba = wlt acts B" 1);
|
4776
|
467 |
by (blast_tac (claset() addDs [leadsTo_eq_subset_wlt]) 2);
|
|
468 |
by (Clarify_tac 1);
|
|
469 |
by (asm_full_simp_tac (simpset() addsimps [wlt_increasing, Un_absorb2]) 1);
|
|
470 |
qed "wlt_constrains_wlt";
|
|
471 |
|
|
472 |
|
|
473 |
(*** Completion: Binary and General Finite versions ***)
|
|
474 |
|
5253
|
475 |
Goal "[| leadsTo acts A A'; stable acts A'; \
|
|
476 |
\ leadsTo acts B B'; stable acts B'; id: acts |] \
|
|
477 |
\ ==> leadsTo acts (A Int B) (A' Int B')";
|
|
478 |
by (subgoal_tac "stable acts (wlt acts B')" 1);
|
4776
|
479 |
by (asm_full_simp_tac (simpset() addsimps [stable_def]) 2);
|
|
480 |
by (EVERY [etac (constrains_Un RS constrains_weaken) 2,
|
|
481 |
etac wlt_constrains_wlt 2,
|
|
482 |
fast_tac (claset() addEs [wlt_increasing RSN (2,rev_subsetD)]) 3,
|
|
483 |
Blast_tac 2]);
|
5253
|
484 |
by (subgoal_tac "leadsTo acts (A Int wlt acts B') (A' Int wlt acts B')" 1);
|
4776
|
485 |
by (blast_tac (claset() addIs [PSP_stable]) 2);
|
5253
|
486 |
by (subgoal_tac "leadsTo acts (A' Int wlt acts B') (A' Int B')" 1);
|
4776
|
487 |
by (blast_tac (claset() addIs [wlt_leadsTo, PSP_stable2]) 2);
|
5253
|
488 |
by (subgoal_tac "leadsTo acts (A Int B) (A Int wlt acts B')" 1);
|
4776
|
489 |
by (blast_tac (claset() addIs [leadsTo_subset RS subsetD,
|
|
490 |
subset_imp_leadsTo]) 2);
|
|
491 |
(*Blast_tac gives PROOF FAILED*)
|
|
492 |
by (best_tac (claset() addIs [leadsTo_Trans]) 1);
|
|
493 |
qed "stable_completion";
|
|
494 |
|
|
495 |
|
5253
|
496 |
Goal "[| finite I; id: acts |] \
|
|
497 |
\ ==> (ALL i:I. leadsTo acts (A i) (A' i)) --> \
|
|
498 |
\ (ALL i:I. stable acts (A' i)) --> \
|
|
499 |
\ leadsTo acts (INT i:I. A i) (INT i:I. A' i)";
|
4776
|
500 |
by (etac finite_induct 1);
|
|
501 |
by (Asm_simp_tac 1);
|
|
502 |
by (asm_simp_tac
|
|
503 |
(simpset() addsimps [stable_completion, stable_def,
|
|
504 |
ball_constrains_INT]) 1);
|
|
505 |
qed_spec_mp "finite_stable_completion";
|
|
506 |
|
|
507 |
|
5253
|
508 |
Goal "[| W = wlt acts (B' Un C); \
|
|
509 |
\ leadsTo acts A (A' Un C); constrains acts A' (A' Un C); \
|
|
510 |
\ leadsTo acts B (B' Un C); constrains acts B' (B' Un C); \
|
|
511 |
\ id: acts |] \
|
|
512 |
\ ==> leadsTo acts (A Int B) ((A' Int B') Un C)";
|
|
513 |
by (subgoal_tac "constrains acts (W-C) (W Un B' Un C)" 1);
|
4776
|
514 |
by (blast_tac (claset() addIs [[asm_rl, wlt_constrains_wlt]
|
|
515 |
MRS constrains_Un RS constrains_weaken]) 2);
|
5253
|
516 |
by (subgoal_tac "constrains acts (W-C) W" 1);
|
4776
|
517 |
by (asm_full_simp_tac
|
|
518 |
(simpset() addsimps [wlt_increasing, Un_assoc, Un_absorb2]) 2);
|
5253
|
519 |
by (subgoal_tac "leadsTo acts (A Int W - C) (A' Int W Un C)" 1);
|
4776
|
520 |
by (simp_tac (simpset() addsimps [Int_Diff]) 2);
|
|
521 |
by (blast_tac (claset() addIs [wlt_leadsTo, PSP RS leadsTo_weaken_R]) 2);
|
5253
|
522 |
by (subgoal_tac "leadsTo acts (A' Int W Un C) (A' Int B' Un C)" 1);
|
4776
|
523 |
by (blast_tac (claset() addIs [wlt_leadsTo, leadsTo_Un_Un,
|
|
524 |
PSP2 RS leadsTo_weaken_R,
|
|
525 |
subset_refl RS subset_imp_leadsTo,
|
|
526 |
leadsTo_Un_duplicate2]) 2);
|
|
527 |
by (dtac leadsTo_Diff 1);
|
|
528 |
by (assume_tac 2);
|
|
529 |
by (blast_tac (claset() addIs [subset_imp_leadsTo]) 1);
|
|
530 |
by (subgoal_tac "A Int B <= A Int W" 1);
|
|
531 |
by (blast_tac (claset() addIs [leadsTo_subset, Int_mono]
|
|
532 |
delrules [subsetI]) 2);
|
|
533 |
by (blast_tac (claset() addIs [leadsTo_Trans, subset_imp_leadsTo]) 1);
|
|
534 |
bind_thm("completion", refl RS result());
|
|
535 |
|
|
536 |
|
5253
|
537 |
Goal "[| finite I; id: acts |] \
|
|
538 |
\ ==> (ALL i:I. leadsTo acts (A i) (A' i Un C)) --> \
|
|
539 |
\ (ALL i:I. constrains acts (A' i) (A' i Un C)) --> \
|
|
540 |
\ leadsTo acts (INT i:I. A i) ((INT i:I. A' i) Un C)";
|
4776
|
541 |
by (etac finite_induct 1);
|
|
542 |
by (ALLGOALS Asm_simp_tac);
|
|
543 |
by (Clarify_tac 1);
|
|
544 |
by (dtac ball_constrains_INT 1);
|
|
545 |
by (asm_full_simp_tac (simpset() addsimps [completion]) 1);
|
|
546 |
qed "finite_completion";
|
|
547 |
|