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(* Title: HOL/UNITY/ELT
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1999 University of Cambridge
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leadsTo strengthened with a specification of the allowable sets transient parts
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*)
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Goalw [givenBy_def] "(givenBy v) = {A. ALL x:A. ALL y. v x = v y --> y: A}";
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by Safe_tac;
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by (res_inst_tac [("x", "v `` ?u")] image_eqI 2);
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by Auto_tac;
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qed "givenBy_eq_all";
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Goal "givenBy v = {A. EX P. A = {s. P(v s)}}";
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by (simp_tac (simpset() addsimps [givenBy_eq_all]) 1);
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by Safe_tac;
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by (res_inst_tac [("x", "%n. EX s. v s = n & s : ?A")] exI 1);
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by (Blast_tac 1);
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by Auto_tac;
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qed "givenBy_eq_Collect";
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val prems =
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Goal "(!!x y. [| x:A; v x = v y |] ==> y: A) ==> A: givenBy v";
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by (stac givenBy_eq_all 1);
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by (blast_tac (claset() addIs prems) 1);
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qed "givenByI";
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Goalw [givenBy_def] "[| A: givenBy v; x:A; v x = v y |] ==> y: A";
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by Auto_tac;
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qed "givenByD";
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Goal "{} : givenBy v";
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by (blast_tac (claset() addSIs [givenByI]) 1);
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qed "empty_mem_givenBy";
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AddIffs [empty_mem_givenBy];
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Goal "A: givenBy v ==> EX P. A = {s. P(v s)}";
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by (res_inst_tac [("x", "%n. EX s. v s = n & s : A")] exI 1);
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by (full_simp_tac (simpset() addsimps [givenBy_eq_all]) 1);
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by (Blast_tac 1);
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qed "givenBy_imp_eq_Collect";
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Goalw [givenBy_def] "EX P. A = {s. P(v s)} ==> A: givenBy v";
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by (Best_tac 1);
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qed "eq_Collect_imp_givenBy";
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Goal "givenBy v = {A. EX P. A = {s. P(v s)}}";
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by (blast_tac (claset() addIs [eq_Collect_imp_givenBy,
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givenBy_imp_eq_Collect]) 1);
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qed "givenBy_eq_eq_Collect";
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Goal "(funPair f g) o h = funPair (f o h) (g o h)";
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by (simp_tac (simpset() addsimps [funPair_def, o_def]) 1);
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qed "funPair_o_distrib";
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(** Standard leadsTo rules **)
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Goalw [leadsETo_def] "[| F: A ensures B; A-B: CC |] ==> F : A leadsTo[CC] B";
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by (blast_tac (claset() addIs [elt.Basis]) 1);
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qed "leadsETo_Basis";
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Goalw [leadsETo_def]
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"[| F : A leadsTo[CC] B; F : B leadsTo[CC] C |] ==> F : A leadsTo[CC] C";
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by (blast_tac (claset() addIs [elt.Trans]) 1);
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qed "leadsETo_Trans";
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(*Useful with cancellation, disjunction*)
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Goal "F : A leadsTo[CC] (A' Un A') ==> F : A leadsTo[CC] A'";
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by (asm_full_simp_tac (simpset() addsimps Un_ac) 1);
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qed "leadsETo_Un_duplicate";
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Goal "F : A leadsTo[CC] (A' Un C Un C) ==> F : A leadsTo[CC] (A' Un C)";
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by (asm_full_simp_tac (simpset() addsimps Un_ac) 1);
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qed "leadsETo_Un_duplicate2";
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(*The Union introduction rule as we should have liked to state it*)
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val prems = Goalw [leadsETo_def]
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"(!!A. A : S ==> F : A leadsTo[CC] B) ==> F : (Union S) leadsTo[CC] B";
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by (blast_tac (claset() addIs [elt.Union] addDs prems) 1);
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qed "leadsETo_Union";
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val prems = Goal
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"(!!i. i : I ==> F : (A i) leadsTo[CC] B) \
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\ ==> F : (UN i:I. A i) leadsTo[CC] B";
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by (stac (Union_image_eq RS sym) 1);
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by (blast_tac (claset() addIs leadsETo_Union::prems) 1);
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qed "leadsETo_UN";
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(*The INDUCTION rule as we should have liked to state it*)
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val major::prems = Goalw [leadsETo_def]
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"[| F : za leadsTo[CC] zb; \
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\ !!A B. [| F : A ensures B; A-B : CC |] ==> P A B; \
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\ !!A B C. [| F : A leadsTo[CC] B; P A B; F : B leadsTo[CC] C; P B C |] \
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\ ==> P A C; \
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\ !!B S. ALL A:S. F : A leadsTo[CC] B & P A B ==> P (Union S) B \
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\ |] ==> P za zb";
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by (rtac (major RS CollectD RS elt.induct) 1);
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by (REPEAT (blast_tac (claset() addIs prems) 1));
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qed "leadsETo_induct";
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(** New facts involving leadsETo **)
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Goal "CC' <= CC ==> (A leadsTo[CC'] B) <= (A leadsTo[CC] B)";
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by Safe_tac;
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by (etac leadsETo_induct 1);
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by (blast_tac (claset() addIs [leadsETo_Union]) 3);
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by (blast_tac (claset() addIs [leadsETo_Trans]) 2);
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by (blast_tac (claset() addIs [leadsETo_Basis]) 1);
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qed "leadsETo_mono";
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val prems = Goalw [leadsETo_def]
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"(!!A. A : S ==> F : (A Int C) leadsTo[CC] B) ==> F : (Union S Int C) leadsTo[CC] B";
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by (simp_tac (HOL_ss addsimps [Int_Union_Union]) 1);
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by (blast_tac (claset() addIs [elt.Union] addDs prems) 1);
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qed "leadsETo_Union_Int";
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(*Binary union introduction rule*)
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Goal "[| F : A leadsTo[CC] C; F : B leadsTo[CC] C |] ==> F : (A Un B) leadsTo[CC] C";
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by (stac Un_eq_Union 1);
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by (blast_tac (claset() addIs [leadsETo_Union]) 1);
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qed "leadsETo_Un";
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val prems =
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Goal "(!!x. x : A ==> F : {x} leadsTo[CC] B) ==> F : A leadsTo[CC] B";
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by (stac (UN_singleton RS sym) 1 THEN rtac leadsETo_UN 1);
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by (blast_tac (claset() addIs prems) 1);
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qed "single_leadsETo_I";
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Goal "[| A<=B; {}:CC |] ==> F : A leadsTo[CC] B";
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by (asm_simp_tac (simpset() addsimps [subset_imp_ensures RS leadsETo_Basis,
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Diff_eq_empty_iff RS iffD2]) 1);
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qed "subset_imp_leadsETo";
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bind_thm ("empty_leadsETo", empty_subsetI RS subset_imp_leadsETo);
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Addsimps [empty_leadsETo];
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(** Weakening laws all require {}:CC **)
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Goal "[| F : A leadsTo[CC] A'; A'<=B'; {}:CC |] ==> F : A leadsTo[CC] B'";
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by (blast_tac (claset() addIs [subset_imp_leadsETo, leadsETo_Trans]) 1);
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qed "leadsETo_weaken_R";
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Goal "[| F : A leadsTo[CC] A'; B<=A; {}:CC |] ==> F : B leadsTo[CC] A'";
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by (blast_tac (claset() addIs [leadsETo_Trans, subset_imp_leadsETo]) 1);
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qed_spec_mp "leadsETo_weaken_L";
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(*Distributes over binary unions*)
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Goal "{} : CC ==> \
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\ F : (A Un B) leadsTo[CC] C = (F : A leadsTo[CC] C & F : B leadsTo[CC] C)";
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by (blast_tac (claset() addIs [leadsETo_Un, leadsETo_weaken_L]) 1);
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qed "leadsETo_Un_distrib";
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Goal "{} : CC ==> \
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\ F : (UN i:I. A i) leadsTo[CC] B = (ALL i : I. F : (A i) leadsTo[CC] B)";
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by (blast_tac (claset() addIs [leadsETo_UN, leadsETo_weaken_L]) 1);
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qed "leadsETo_UN_distrib";
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Goal "{} : CC \
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\ ==> F : (Union S) leadsTo[CC] B = (ALL A : S. F : A leadsTo[CC] B)";
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by (blast_tac (claset() addIs [leadsETo_Union, leadsETo_weaken_L]) 1);
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qed "leadsETo_Union_distrib";
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Goal "[| F : A leadsTo[CC'] A'; B<=A; A'<=B'; CC' <= CC; {}:CC |] \
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\ ==> F : B leadsTo[CC] B'";
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by (dtac (impOfSubs leadsETo_mono) 1);
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by (assume_tac 1);
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by (blast_tac (claset() addIs [leadsETo_weaken_R, leadsETo_weaken_L,
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leadsETo_Trans]) 1);
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qed "leadsETo_weaken";
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Goal "[| F : A leadsTo[CC] A'; CC <= givenBy v |] \
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\ ==> F : A leadsTo[givenBy v] A'";
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by (blast_tac (claset() addIs [empty_mem_givenBy, leadsETo_weaken]) 1);
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qed "leadsETo_givenBy";
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(*Set difference*)
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Goal "[| F : (A-B) leadsTo[CC] C; F : B leadsTo[CC] C; {}:CC |] \
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\ ==> F : A leadsTo[CC] C";
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by (blast_tac (claset() addIs [leadsETo_Un, leadsETo_weaken]) 1);
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qed "leadsETo_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 ==> F : (A i) leadsTo[CC] (A' i); {}:CC |] \
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\ ==> F : (UN i:I. A i) leadsTo[CC] (UN i:I. A' i)";
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by (simp_tac (HOL_ss addsimps [Union_image_eq RS sym]) 1);
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by (blast_tac (claset() addIs [leadsETo_Union, leadsETo_weaken_R]
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addIs prems) 1);
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qed "leadsETo_UN_UN";
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(*Binary union version*)
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Goal "[| F : A leadsTo[CC] A'; F : B leadsTo[CC] B'; {}:CC |] \
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\ ==> F : (A Un B) leadsTo[CC] (A' Un B')";
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by (blast_tac (claset() addIs [leadsETo_Un,
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leadsETo_weaken_R]) 1);
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qed "leadsETo_Un_Un";
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(** The cancellation law **)
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Goal "[| F : A leadsTo[CC] (A' Un B); F : B leadsTo[CC] B'; {}:CC |] \
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\ ==> F : A leadsTo[CC] (A' Un B')";
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by (blast_tac (claset() addIs [leadsETo_Un_Un,
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subset_imp_leadsETo, leadsETo_Trans]) 1);
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qed "leadsETo_cancel2";
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Goal "[| F : A leadsTo[CC] (A' Un B); F : (B-A') leadsTo[CC] B'; {}:CC |] \
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\ ==> F : A leadsTo[CC] (A' Un B')";
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by (rtac leadsETo_cancel2 1);
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by (assume_tac 2);
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by (ALLGOALS Asm_simp_tac);
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qed "leadsETo_cancel_Diff2";
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Goal "[| F : A leadsTo[CC] (B Un A'); F : B leadsTo[CC] B'; {}:CC |] \
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\ ==> F : A leadsTo[CC] (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 [leadsETo_cancel2]) 1);
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qed "leadsETo_cancel1";
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Goal "[| F : A leadsTo[CC] (B Un A'); F : (B-A') leadsTo[CC] B'; {}:CC |] \
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\ ==> F : A leadsTo[CC] (B' Un A')";
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by (rtac leadsETo_cancel1 1);
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by (assume_tac 2);
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by (ALLGOALS Asm_simp_tac);
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qed "leadsETo_cancel_Diff1";
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(** The impossibility law **)
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Goal "F : A leadsTo[CC] B ==> B={} --> A={}";
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by (etac leadsETo_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 "F : A leadsTo[CC] {} ==> A={}";
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by (blast_tac (claset() addSIs [lemma]) 1);
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qed "leadsETo_empty";
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(** PSP: Progress-Safety-Progress **)
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(*Special case of PSP: Misra's "stable conjunction"*)
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Goalw [stable_def]
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"[| F : A leadsTo[CC] A'; F : stable B; ALL C:CC. C Int B : CC |] \
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\ ==> F : (A Int B) leadsTo[CC] (A' Int B)";
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by (etac leadsETo_induct 1);
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by (blast_tac (claset() addIs [leadsETo_Union_Int]) 3);
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by (blast_tac (claset() addIs [leadsETo_Trans]) 2);
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by (rtac leadsETo_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 (asm_simp_tac (simpset() addsimps [Diff_Int_distrib2 RS sym]) 2);
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by (blast_tac (claset() addIs [transient_strengthen, constrains_Int]) 1);
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qed "e_psp_stable";
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Goal "[| F : A leadsTo[CC] A'; F : stable B; ALL C:CC. C Int B : CC |] \
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\ ==> F : (B Int A) leadsTo[CC] (B Int A')";
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by (asm_simp_tac (simpset() addsimps e_psp_stable::Int_ac) 1);
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qed "e_psp_stable2";
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Goal "[| F : A leadsTo[CC] A'; F : B co B'; \
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\ ALL C:CC. C Int B Int B' : CC; {}:CC |] \
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\ ==> F : (A Int B') leadsTo[CC] ((A' Int B) Un (B' - B))";
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by (etac leadsETo_induct 1);
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by (blast_tac (claset() addIs [leadsETo_Union_Int]) 3);
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(*Transitivity case has a delicate argument involving "cancellation"*)
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by (rtac leadsETo_Un_duplicate2 2);
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by (etac leadsETo_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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by (blast_tac (claset() addIs [leadsETo_weaken_L]
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addDs [constrains_imp_subset]) 2);
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(*Basis case*)
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by (rtac leadsETo_Basis 1);
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by (blast_tac (claset() addIs [psp_ensures]) 1);
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by (subgoal_tac "A Int B' - (Ba Int B Un (B' - B)) = (A - Ba) Int B Int B'" 1);
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by Auto_tac;
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qed "e_psp";
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Goal "[| F : A leadsTo[CC] A'; F : B co B'; \
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\ ALL C:CC. C Int B Int B' : CC; {}:CC |] \
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\ ==> F : (B' Int A) leadsTo[CC] ((B Int A') Un (B' - B))";
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by (asm_full_simp_tac (simpset() addsimps e_psp::Int_ac) 1);
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qed "e_psp2";
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(*** Special properties involving the parameter [CC] ***)
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(*??IS THIS NEEDED?? or is it just an example of what's provable??*)
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Goal "[| F: (A leadsTo[givenBy v] B); F Join G : v localTo[C] F; \
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\ F Join G : stable C |] \
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\ ==> F Join G : ((C Int A) leadsTo[(%D. C Int D) `` givenBy v] B)";
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by (etac leadsETo_induct 1);
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by (stac Int_Union 3);
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by (blast_tac (claset() addIs [leadsETo_UN]) 3);
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by (blast_tac (claset() addIs [e_psp_stable2 RS leadsETo_weaken_L,
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leadsETo_Trans]) 2);
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by (rtac leadsETo_Basis 1);
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by (auto_tac (claset(),
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simpset() addsimps [Int_Diff, ensures_def, stable_def,
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|
315 |
givenBy_eq_Collect,
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|
316 |
Join_localTo,
|
|
317 |
Join_constrains, Join_transient]));
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|
318 |
by (blast_tac (claset() addIs [transient_strengthen]) 3);
|
|
319 |
by (blast_tac (claset() addDs [constrains_localTo_constrains]
|
|
320 |
addIs [constrains_Int RS constrains_weaken]) 2);
|
|
321 |
by (blast_tac (claset() addIs [constrains_Int RS constrains_weaken]) 1);
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|
322 |
qed "gen_leadsETo_localTo_imp_Join_leadsETo";
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|
323 |
|
|
324 |
(*USED???
|
|
325 |
Could replace this proof by instantiation of the one above with C=UNIV*)
|
|
326 |
Goal "[| F: (A leadsTo[givenBy v] B); F Join G : v localTo[UNIV] F |] \
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|
327 |
\ ==> F Join G : (A leadsTo[givenBy v] B)";
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|
328 |
by (etac leadsETo_induct 1);
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|
329 |
by (blast_tac (claset() addIs [leadsETo_Union]) 3);
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|
330 |
by (blast_tac (claset() addIs [leadsETo_Trans]) 2);
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|
331 |
by (rtac leadsETo_Basis 1);
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|
332 |
by (auto_tac (claset(),
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|
333 |
simpset() addsimps [ensures_def, givenBy_eq_Collect,
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|
334 |
Join_localTo,
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|
335 |
Join_constrains, Join_transient]));
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|
336 |
by (force_tac (claset() addDs [constrains_localTo_constrains], simpset()) 1);
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|
337 |
qed "leadsETo_localTo_imp_Join_leadsETo";
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|
338 |
|
|
339 |
(*useful??*)
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|
340 |
Goal "[| F Join G : (A leadsTo[CC] B); ALL C:CC. G : stable C |] \
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|
341 |
\ ==> F: (A leadsTo[CC] B)";
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|
342 |
by (etac leadsETo_induct 1);
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|
343 |
by (blast_tac (claset() addIs [leadsETo_Union]) 3);
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|
344 |
by (blast_tac (claset() addIs [leadsETo_Trans]) 2);
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|
345 |
by (rtac leadsETo_Basis 1);
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|
346 |
by (case_tac "A <= B" 1);
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|
347 |
by (etac subset_imp_ensures 1);
|
|
348 |
by (auto_tac (claset() addIs [constrains_weaken],
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|
349 |
simpset() addsimps [stable_def, ensures_def,
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|
350 |
Join_constrains, Join_transient]));
|
|
351 |
by (REPEAT (thin_tac "?F : ?A co ?B" 1));
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|
352 |
by (etac transientE 1);
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|
353 |
by (rewtac constrains_def);
|
|
354 |
by (blast_tac (claset() addSDs [bspec]) 1);
|
|
355 |
qed "Join_leadsETo_stable_imp_leadsETo";
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|
356 |
|
|
357 |
|
|
358 |
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|
359 |
(**** EXTEND/PROJECT PROPERTIES ****)
|
|
360 |
|
|
361 |
Open_locale "Extend";
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|
362 |
|
|
363 |
(*Here h and f are locale constants*)
|
|
364 |
Goal "extend_set h `` (givenBy v) <= (givenBy (v o f))";
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|
365 |
by (simp_tac (simpset() addsimps [givenBy_eq_all]) 1);
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|
366 |
by (Blast_tac 1);
|
|
367 |
qed "extend_set_givenBy_subset";
|
|
368 |
|
|
369 |
Goal "D : givenBy v ==> extend_set h D : givenBy (v o f)";
|
|
370 |
by (full_simp_tac (simpset() addsimps [givenBy_eq_all]) 1);
|
|
371 |
by (Blast_tac 1);
|
|
372 |
qed "extend_set_givenBy_I";
|
|
373 |
|
|
374 |
|
|
375 |
Goal "F : A leadsTo[CC] B \
|
|
376 |
\ ==> extend h F : (extend_set h A) leadsTo[extend_set h `` CC] \
|
|
377 |
\ (extend_set h B)";
|
|
378 |
by (etac leadsETo_induct 1);
|
|
379 |
by (asm_simp_tac (simpset() addsimps [leadsETo_UN, extend_set_Union]) 3);
|
|
380 |
by (blast_tac (claset() addIs [leadsETo_Trans]) 2);
|
|
381 |
by (asm_simp_tac (simpset() addsimps [leadsETo_Basis, extend_ensures,
|
|
382 |
extend_set_Diff_distrib RS sym]) 1);
|
|
383 |
qed "leadsETo_imp_extend_leadsETo";
|
|
384 |
|
|
385 |
(*NOW OBSOLETE: SEE BELOW !! Generalizes the version proved in Project.ML*)
|
|
386 |
Goalw [LOCALTO_def, transient_def, Diff_def]
|
|
387 |
"[| G : (v o f) localTo[C] extend h F; project h C G : transient D; \
|
|
388 |
\ D : givenBy v |] \
|
|
389 |
\ ==> F : transient D";
|
|
390 |
by (auto_tac (claset(),
|
|
391 |
simpset() addsimps [givenBy_eq_Collect]));
|
|
392 |
by (case_tac "Restrict C act : Restrict C ``extend_act h `` Acts F" 1);
|
|
393 |
by Auto_tac;
|
|
394 |
by (rtac bexI 1);
|
|
395 |
by (assume_tac 2);
|
|
396 |
by (Blast_tac 1);
|
|
397 |
by (case_tac "{s. P (v s)} = {}" 1);
|
|
398 |
by (auto_tac (claset(),
|
|
399 |
simpset() addsimps [stable_def, constrains_def]));
|
|
400 |
by (subgoal_tac
|
|
401 |
"ALL z. Restrict C act ^^ {s. v (f s) = z} <= {s. v (f s) = z}" 1);
|
|
402 |
by (blast_tac (claset() addSDs [bspec]) 2);
|
|
403 |
by (thin_tac "ALL z. ?P z" 1);
|
|
404 |
by (subgoal_tac "project_act h (Restrict C act) ^^ {s. P (v s)} <= {s. P (v s)}" 1);
|
|
405 |
by (Clarify_tac 2);
|
|
406 |
by (asm_full_simp_tac (simpset() addsimps [project_act_def]) 2);
|
|
407 |
by (force_tac (claset() addSDs [spec, ImageI RSN (2, subsetD)], simpset()) 2);
|
|
408 |
by (blast_tac (claset() addSDs [subsetD]) 1);
|
|
409 |
qed "localTo_project_transient_transient";
|
|
410 |
|
|
411 |
|
|
412 |
Goal "A Int extend_set h ((project_set h A) Int B) = A Int extend_set h B";
|
|
413 |
by (auto_tac (claset() addIs [project_set_I],
|
|
414 |
simpset()));
|
|
415 |
qed "Int_extend_set_lemma";
|
|
416 |
|
|
417 |
Goal "G : C co B ==> project h C G : project_set h C co project_set h B";
|
|
418 |
by (full_simp_tac (simpset() addsimps [constrains_def, project_def,
|
|
419 |
project_act_def, project_set_def]) 1);
|
|
420 |
by (Blast_tac 1);
|
|
421 |
qed "project_constrains_project_set";
|
|
422 |
|
|
423 |
Goal "G : stable C ==> project h C G : stable (project_set h C)";
|
|
424 |
by (asm_full_simp_tac (simpset() addsimps [stable_def,
|
|
425 |
project_constrains_project_set]) 1);
|
|
426 |
qed "project_stable_project_set";
|
|
427 |
|
|
428 |
(*!! Generalizes the version proved in Project.ML*)
|
|
429 |
Goalw [LOCALTO_def, transient_def, Diff_def]
|
|
430 |
"[| G : (v o f) localTo[C] extend h F; \
|
|
431 |
\ project h C G : transient (C' Int D); \
|
|
432 |
\ project h C G : stable C'; \
|
|
433 |
\ D : givenBy v; (C' Int D) <= D |] \
|
|
434 |
\ ==> F : transient (C' Int D)";
|
|
435 |
by (auto_tac (claset(),
|
|
436 |
simpset() addsimps [givenBy_eq_Collect]));
|
|
437 |
by (case_tac "Restrict C act : Restrict C ``extend_act h `` Acts F" 1);
|
|
438 |
by Auto_tac;
|
|
439 |
by (rtac bexI 1);
|
|
440 |
by (assume_tac 2);
|
|
441 |
by (Blast_tac 1);
|
|
442 |
by (case_tac "(C' Int {s. P (v s)}) = {}" 1);
|
|
443 |
by (auto_tac (claset(),
|
|
444 |
simpset() addsimps [stable_def, constrains_def]));
|
|
445 |
by (subgoal_tac
|
|
446 |
"ALL z. Restrict C act ^^ {s. v (f s) = z} <= {s. v (f s) = z}" 1);
|
|
447 |
by (blast_tac (claset() addSDs [bspec]) 2);
|
|
448 |
by (thin_tac "ALL z. ?P z" 1);
|
|
449 |
by (subgoal_tac "project_act h (Restrict C act) ^^ (C' Int {s. P (v s)}) <= (C' Int {s. P (v s)})" 1);
|
|
450 |
by (Clarify_tac 2);
|
|
451 |
by (asm_full_simp_tac (simpset() addsimps [project_act_def]) 2);
|
|
452 |
by (thin_tac "(C' Int {s. P (v s)}) <= Domain ?A" 2);
|
|
453 |
by (thin_tac "?A <= -C' Un ?B" 2);
|
|
454 |
by (rtac conjI 2);
|
|
455 |
by (force_tac (claset() addSDs [spec, ImageI RSN (2, subsetD)], simpset()) 3);
|
|
456 |
by (Blast_tac 2);
|
|
457 |
by (blast_tac (claset() addSDs [subsetD]) 1);
|
|
458 |
qed "localTo_project_transient_transient";
|
|
459 |
|
|
460 |
(*This version's stronger in the "ensures" precondition
|
|
461 |
BUT there's no ensures_weaken_L*)
|
|
462 |
Goal "[| project h C G : transient (project_set h C Int (A-B)) --> \
|
|
463 |
\ F : transient (project_set h C Int (A-B)); \
|
|
464 |
\ extend h F Join G : stable C; \
|
|
465 |
\ F Join project h C G : (project_set h C Int A) ensures B |] \
|
|
466 |
\ ==> extend h F Join G : (C Int extend_set h A) ensures (extend_set h B)";
|
|
467 |
by (stac (Int_extend_set_lemma RS sym) 1);
|
|
468 |
by (rtac Join_project_ensures 1);
|
|
469 |
by (auto_tac (claset(), simpset() addsimps [Int_Diff]));
|
|
470 |
qed "Join_project_ensures_strong";
|
|
471 |
|
|
472 |
Goal "[| extend h F Join G : stable C; \
|
|
473 |
\ F Join project h C G : (project_set h C Int A) leadsTo[(%D. project_set h C Int D)``givenBy v] B; \
|
|
474 |
\ G : (v o f) localTo[C] extend h F |] \
|
|
475 |
\ ==> extend h F Join G : \
|
|
476 |
\ (C Int extend_set h (project_set h C Int A)) \
|
|
477 |
\ leadsTo[(%D. C Int extend_set h D)``givenBy v] (extend_set h B)";
|
|
478 |
by (etac leadsETo_induct 1);
|
|
479 |
by (asm_simp_tac (simpset() delsimps UN_simps
|
|
480 |
addsimps [Int_UN_distrib, leadsETo_UN, extend_set_Union]) 3);
|
|
481 |
by (blast_tac (claset() addIs [e_psp_stable2 RS leadsETo_weaken_L,
|
|
482 |
leadsETo_Trans]) 2);
|
|
483 |
by (Clarify_tac 1);
|
|
484 |
by (rtac leadsETo_Basis 1);
|
|
485 |
by (etac rev_image_eqI 2);
|
|
486 |
by (asm_simp_tac (simpset() addsimps [Int_Diff, Int_extend_set_lemma,
|
|
487 |
extend_set_Diff_distrib RS sym]) 2);
|
|
488 |
by (rtac Join_project_ensures_strong 1);
|
|
489 |
by (auto_tac (claset() addIs [localTo_project_transient_transient,
|
|
490 |
project_stable_project_set],
|
|
491 |
simpset() addsimps [Int_left_absorb, Join_stable]));
|
|
492 |
by (asm_simp_tac
|
|
493 |
(simpset() addsimps [stable_ensures_Int RS ensures_weaken_R,
|
|
494 |
Int_lower2, project_stable_project_set,
|
|
495 |
Join_stable, extend_stable_project_set]) 1);
|
|
496 |
val lemma = result();
|
|
497 |
|
|
498 |
Goal "[| extend h F Join G : stable C; \
|
|
499 |
\ F Join project h C G : (project_set h C Int A) leadsTo[(%D. project_set h C Int D)``givenBy v] B; \
|
|
500 |
\ G : (v o f) localTo[C] extend h F |] \
|
|
501 |
\ ==> extend h F Join G : (C Int extend_set h A) \
|
|
502 |
\ leadsTo[(%D. C Int extend_set h D)``givenBy v] (extend_set h B)";
|
|
503 |
by (rtac (lemma RS leadsETo_weaken) 1);
|
|
504 |
by (auto_tac (claset() addIs [project_set_I], simpset()));
|
|
505 |
qed "project_leadsETo_lemma";
|
|
506 |
|
|
507 |
Goal "[| F Join project h UNIV G : A leadsTo[givenBy v] B; \
|
|
508 |
\ G : (v o f) localTo[UNIV] extend h F |] \
|
|
509 |
\ ==> extend h F Join G : (extend_set h A) \
|
|
510 |
\ leadsTo[givenBy (v o f)] (extend_set h B)";
|
|
511 |
by (rtac (make_elim project_leadsETo_lemma) 1);
|
|
512 |
by Auto_tac;
|
|
513 |
by (etac leadsETo_givenBy 1);
|
|
514 |
by (rtac extend_set_givenBy_subset 1);
|
|
515 |
qed "project_leadsETo_D";
|
|
516 |
|
|
517 |
Goal "[| F Join project h (reachable (extend h F Join G)) G \
|
|
518 |
\ : A LeadsTo[givenBy v] B; \
|
|
519 |
\ G : (v o f) LocalTo extend h F |] \
|
|
520 |
\ ==> extend h F Join G : \
|
|
521 |
\ (extend_set h A) LeadsTo[givenBy (v o f)] (extend_set h B)";
|
|
522 |
by (rtac (make_elim (subset_refl RS stable_reachable RS
|
|
523 |
project_leadsETo_lemma)) 1);
|
|
524 |
by (auto_tac (claset(),
|
|
525 |
simpset() addsimps [LeadsETo_def, LocalTo_def]));
|
|
526 |
by (asm_full_simp_tac
|
|
527 |
(simpset() addsimps [project_set_reachable_extend_eq RS sym]) 1);
|
|
528 |
by (etac (impOfSubs leadsETo_mono) 1);
|
|
529 |
by (blast_tac (claset() addIs [extend_set_givenBy_I]) 1);
|
|
530 |
qed "project_LeadsETo_D";
|
|
531 |
|
|
532 |
Goalw [extending_def]
|
|
533 |
"extending (%G. UNIV) h F \
|
|
534 |
\ ((v o f) localTo[UNIV] extend h F) \
|
|
535 |
\ (extend_set h A leadsTo[givenBy (v o f)] extend_set h B) \
|
|
536 |
\ (A leadsTo[givenBy v] B)";
|
|
537 |
by (auto_tac (claset(),
|
|
538 |
simpset() addsimps [project_leadsETo_D, Join_localTo]));
|
|
539 |
qed "extending_leadsETo";
|
|
540 |
|
|
541 |
|
|
542 |
Goalw [extending_def]
|
|
543 |
"extending (%G. reachable (extend h F Join G)) h F \
|
|
544 |
\ ((v o f) LocalTo extend h F) \
|
|
545 |
\ (extend_set h A LeadsTo[givenBy (v o f)] extend_set h B) \
|
|
546 |
\ (A LeadsTo[givenBy v] B)";
|
|
547 |
|
|
548 |
by (force_tac (claset() addIs [project_LeadsETo_D],
|
|
549 |
simpset()addsimps [LocalTo_def, Join_assoc RS sym,
|
|
550 |
Join_localTo]) 1);
|
|
551 |
qed "extending_LeadsETo";
|
|
552 |
|
|
553 |
|
|
554 |
Close_locale "Extend";
|
|
555 |
|
|
556 |
|