author | paulson |
Thu, 26 Aug 1999 11:33:24 +0200 | |
changeset 7359 | 98a2afab3f86 |
parent 6909 | 21601bc4f3c2 |
child 7522 | d93b52bda2dd |
permissions | -rw-r--r-- |
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(* Title: HOL/UNITY/SubstAx |
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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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LeadsTo relation, restricted to the set of reachable states. |
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*) |
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overload_1st_set "SubstAx.op LeadsTo"; |
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(*Resembles the previous definition of LeadsTo*) |
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Goalw [LeadsTo_def] |
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"A LeadsTo B = {F. F : (reachable F Int A) leadsTo (reachable F Int B)}"; |
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by (blast_tac (claset() addDs [psp_stable2] |
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addIs [leadsTo_weaken, stable_reachable]) 1); |
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qed "LeadsTo_eq_leadsTo"; |
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(*** Specialized laws for handling invariants ***) |
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(** Conjoining an Always property **) |
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Goal "F : Always INV ==> (F : (INV Int A) LeadsTo A') = (F : A LeadsTo A')"; |
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by (asm_full_simp_tac |
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(simpset() addsimps [LeadsTo_def, Always_eq_includes_reachable, |
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Int_absorb2, Int_assoc RS sym]) 1); |
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qed "Always_LeadsTo_pre"; |
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Goal "F : Always INV ==> (F : A LeadsTo (INV Int A')) = (F : A LeadsTo A')"; |
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by (asm_full_simp_tac |
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(simpset() addsimps [LeadsTo_eq_leadsTo, Always_eq_includes_reachable, |
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Int_absorb2, Int_assoc RS sym]) 1); |
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qed "Always_LeadsTo_post"; |
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(* [| F : Always C; F : (C Int A) LeadsTo A' |] ==> F : A LeadsTo A' *) |
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bind_thm ("Always_LeadsToI", Always_LeadsTo_pre RS iffD1); |
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(* [| F : Always INV; F : A LeadsTo A' |] ==> F : A LeadsTo (INV Int A') *) |
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bind_thm ("Always_LeadsToD", Always_LeadsTo_post RS iffD2); |
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(*** Introduction rules: Basis, Trans, Union ***) |
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||
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Goal "F : A leadsTo B ==> F : A LeadsTo B"; |
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by (simp_tac (simpset() addsimps [LeadsTo_def]) 1); |
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by (blast_tac (claset() addIs [leadsTo_weaken_L]) 1); |
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qed "leadsTo_imp_LeadsTo"; |
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||
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Goal "[| F : A LeadsTo B; F : B LeadsTo C |] ==> F : A LeadsTo C"; |
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by (full_simp_tac (simpset() addsimps [LeadsTo_eq_leadsTo]) 1); |
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by (blast_tac (claset() addIs [leadsTo_Trans]) 1); |
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qed "LeadsTo_Trans"; |
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val prems = Goalw [LeadsTo_def] |
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"(!!A. A : S ==> F : A LeadsTo B) ==> F : (Union S) LeadsTo B"; |
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by (Simp_tac 1); |
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by (stac Int_Union 1); |
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by (blast_tac (claset() addIs [leadsTo_UN] addDs prems) 1); |
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qed "LeadsTo_Union"; |
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(*** Derived rules ***) |
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Goal "F : A LeadsTo UNIV"; |
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by (simp_tac (simpset() addsimps [LeadsTo_def]) 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 "F : A LeadsTo (A' Un A') ==> F : A LeadsTo 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 "F : A LeadsTo (A' Un C Un C) ==> F : A LeadsTo (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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val prems = |
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Goal "(!!i. i : I ==> F : (A i) LeadsTo B) ==> F : (UN i:I. A i) LeadsTo B"; |
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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 (LeadsTo_Union::prems)) 1); |
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qed "LeadsTo_UN"; |
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(*Binary union introduction rule*) |
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Goal "[| F : A LeadsTo C; F : B LeadsTo C |] ==> F : (A Un B) LeadsTo 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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(*Lets us look at the starting state*) |
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val prems = |
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Goal "(!!s. s : A ==> F : {s} LeadsTo B) ==> F : A LeadsTo B"; |
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by (stac (UN_singleton RS sym) 1 THEN rtac LeadsTo_UN 1); |
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by (blast_tac (claset() addIs prems) 1); |
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qed "single_LeadsTo_I"; |
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Goal "A <= B ==> F : A LeadsTo B"; |
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by (simp_tac (simpset() addsimps [LeadsTo_def]) 1); |
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by (blast_tac (claset() addIs [subset_imp_leadsTo]) 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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||
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Goal "[| F : A LeadsTo A'; A' <= B' |] ==> F : A LeadsTo B'"; |
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by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1); |
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by (blast_tac (claset() addIs [leadsTo_weaken_R]) 1); |
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qed_spec_mp "LeadsTo_weaken_R"; |
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Goal "[| F : A LeadsTo A'; B <= A |] \ |
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\ ==> F : B LeadsTo A'"; |
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by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1); |
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by (blast_tac (claset() addIs [leadsTo_weaken_L]) 1); |
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qed_spec_mp "LeadsTo_weaken_L"; |
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Goal "[| F : A LeadsTo A'; \ |
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\ B <= A; A' <= B' |] \ |
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\ ==> F : B LeadsTo B'"; |
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by (blast_tac (claset() addIs [LeadsTo_weaken_R, LeadsTo_weaken_L, |
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LeadsTo_Trans]) 1); |
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qed "LeadsTo_weaken"; |
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Goal "[| F : Always C; F : A LeadsTo A'; \ |
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\ C Int B <= A; C Int A' <= B' |] \ |
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\ ==> F : B LeadsTo B'"; |
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by (blast_tac (claset() addDs [Always_LeadsToI] addIs[LeadsTo_weaken] |
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addIs [Always_LeadsToD]) 1); |
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qed "Always_LeadsTo_weaken"; |
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(** Two theorems for "proof lattices" **) |
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Goal "[| F : A LeadsTo B |] ==> F : (A Un B) LeadsTo B"; |
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by (blast_tac (claset() addIs [LeadsTo_Un, subset_imp_LeadsTo]) 1); |
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qed "LeadsTo_Un_post"; |
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Goal "[| F : A LeadsTo B; F : B LeadsTo C |] \ |
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\ ==> F : (A Un B) LeadsTo C"; |
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by (blast_tac (claset() addIs [LeadsTo_Un, subset_imp_LeadsTo, |
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LeadsTo_weaken_L, LeadsTo_Trans]) 1); |
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qed "LeadsTo_Trans_Un"; |
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(** Distributive laws **) |
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Goal "(F : (A Un B) LeadsTo C) = (F : A LeadsTo C & F : B LeadsTo 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 "(F : (UN i:I. A i) LeadsTo B) = (ALL i : I. F : (A i) LeadsTo 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 "(F : (Union S) LeadsTo B) = (ALL A : S. F : A LeadsTo 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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(** More rules using the premise "Always INV" **) |
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Goal "[| F : (A-A') Co (A Un A'); F : transient (A-A') |] \ |
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\ ==> F : A LeadsTo A'"; |
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by (asm_full_simp_tac |
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(simpset() addsimps [LeadsTo_def, Constrains_eq_constrains]) 1); |
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by (rtac (ensuresI RS leadsTo_Basis) 1); |
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by (blast_tac (claset() addIs [transient_strengthen]) 2); |
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by (blast_tac (claset() addIs [constrains_weaken]) 1); |
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qed "LeadsTo_Basis"; |
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Goal "[| F : Always INV; \ |
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\ F : (INV Int (A-A')) Co (A Un A'); \ |
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\ F : transient (INV Int (A-A')) |] \ |
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\ ==> F : A LeadsTo A'"; |
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by (rtac Always_LeadsToI 1); |
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by (assume_tac 1); |
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by (rtac LeadsTo_Basis 1); |
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by (blast_tac (claset() addIs [transient_strengthen]) 2); |
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by (blast_tac (claset() addIs [Always_ConstrainsD RS Constrains_weaken]) 1); |
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qed "Always_LeadsTo_Basis"; |
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(*Set difference: maybe combine with leadsTo_weaken_L?? |
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This is the most useful form of the "disjunction" rule*) |
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Goal "[| F : (A-B) LeadsTo C; F : (A Int B) LeadsTo C |] \ |
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\ ==> F : A LeadsTo 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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val prems = |
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Goal "(!! i. i:I ==> F : (A i) LeadsTo (A' i)) \ |
191 |
\ ==> F : (UN i:I. A i) LeadsTo (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 [LeadsTo_Union, LeadsTo_weaken_R] |
194 |
addIs prems) 1); |
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qed "LeadsTo_UN_UN"; |
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197 |
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(*Version with no index set*) |
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val prems = |
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Goal "(!! i. F : (A i) LeadsTo (A' i)) \ |
201 |
\ ==> F : (UN i. A i) LeadsTo (UN i. A' i)"; |
|
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by (blast_tac (claset() addIs [LeadsTo_UN_UN] |
203 |
addIs prems) 1); |
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qed "LeadsTo_UN_UN_noindex"; |
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||
206 |
(*Version with no index set*) |
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Goal "ALL i. F : (A i) LeadsTo (A' i) \ |
208 |
\ ==> F : (UN i. A i) LeadsTo (UN i. A' i)"; |
|
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by (blast_tac (claset() addIs [LeadsTo_UN_UN]) 1); |
210 |
qed "all_LeadsTo_UN_UN"; |
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211 |
||
212 |
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213 |
(*Binary union version*) |
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Goal "[| F : A LeadsTo A'; F : B LeadsTo B' |] \ |
215 |
\ ==> F : (A Un B) LeadsTo (A' Un B')"; |
|
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by (blast_tac (claset() addIs [LeadsTo_Un, |
217 |
LeadsTo_weaken_R]) 1); |
|
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qed "LeadsTo_Un_Un"; |
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219 |
||
220 |
||
221 |
(** The cancellation law **) |
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222 |
||
6536 | 223 |
Goal "[| F : A LeadsTo (A' Un B); F : B LeadsTo B' |] \ |
224 |
\ ==> F : A LeadsTo (A' Un B')"; |
|
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by (blast_tac (claset() addIs [LeadsTo_Un_Un, |
226 |
subset_imp_LeadsTo, LeadsTo_Trans]) 1); |
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qed "LeadsTo_cancel2"; |
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228 |
||
6536 | 229 |
Goal "[| F : A LeadsTo (A' Un B); F : (B-A') LeadsTo B' |] \ |
230 |
\ ==> F : A LeadsTo (A' Un B')"; |
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by (rtac LeadsTo_cancel2 1); |
232 |
by (assume_tac 2); |
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233 |
by (ALLGOALS Asm_simp_tac); |
|
234 |
qed "LeadsTo_cancel_Diff2"; |
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||
6536 | 236 |
Goal "[| F : A LeadsTo (B Un A'); F : B LeadsTo B' |] \ |
237 |
\ ==> F : A LeadsTo (B' Un A')"; |
|
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by (asm_full_simp_tac (simpset() addsimps [Un_commute]) 1); |
239 |
by (blast_tac (claset() addSIs [LeadsTo_cancel2]) 1); |
|
240 |
qed "LeadsTo_cancel1"; |
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||
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Goal "[| F : A LeadsTo (B Un A'); F : (B-A') LeadsTo B' |] \ |
243 |
\ ==> F : A LeadsTo (B' Un A')"; |
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by (rtac LeadsTo_cancel1 1); |
245 |
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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250 |
(** The impossibility law **) |
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(*The set "A" may be non-empty, but it contains no reachable states*) |
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Goal "F : A LeadsTo {} ==> F : Always (-A)"; |
254 |
by (full_simp_tac (simpset() addsimps [LeadsTo_def, |
|
255 |
Always_eq_includes_reachable]) 1); |
|
256 |
by (dtac leadsTo_empty 1); |
|
257 |
by Auto_tac; |
|
4776 | 258 |
qed "LeadsTo_empty"; |
259 |
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260 |
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261 |
(** PSP: Progress-Safety-Progress **) |
|
262 |
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(*Special case of PSP: Misra's "stable conjunction"*) |
6536 | 264 |
Goal "[| F : A LeadsTo A'; F : Stable B |] \ |
265 |
\ ==> F : (A Int B) LeadsTo (A' Int B)"; |
|
6575 | 266 |
by (full_simp_tac |
267 |
(simpset() addsimps [LeadsTo_eq_leadsTo, Stable_eq_stable]) 1); |
|
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268 |
by (dtac psp_stable 1); |
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|
269 |
by (assume_tac 1); |
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|
270 |
by (asm_full_simp_tac (simpset() addsimps Int_ac) 1); |
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271 |
qed "PSP_Stable"; |
4776 | 272 |
|
6536 | 273 |
Goal "[| F : A LeadsTo A'; F : Stable B |] \ |
274 |
\ ==> F : (B Int A) LeadsTo (B Int A')"; |
|
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|
275 |
by (asm_simp_tac (simpset() addsimps PSP_Stable::Int_ac) 1); |
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|
276 |
qed "PSP_Stable2"; |
4776 | 277 |
|
6575 | 278 |
Goal "[| F : A LeadsTo A'; F : B Co B' |] \ |
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|
279 |
\ ==> F : (A Int B') LeadsTo ((A' Int B) Un (B' - B))"; |
6575 | 280 |
by (full_simp_tac |
281 |
(simpset() addsimps [LeadsTo_def, Constrains_eq_constrains]) 1); |
|
5313
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|
282 |
by (blast_tac (claset() addDs [psp] addIs [leadsTo_weaken]) 1); |
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|
283 |
qed "PSP"; |
4776 | 284 |
|
6536 | 285 |
Goal "[| F : A LeadsTo A'; F : B Co B' |] \ |
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|
286 |
\ ==> F : (B' Int A) LeadsTo ((B Int A') Un (B' - B))"; |
5536 | 287 |
by (asm_simp_tac (simpset() addsimps PSP::Int_ac) 1); |
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|
288 |
qed "PSP2"; |
4776 | 289 |
|
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|
290 |
Goalw [Unless_def] |
6536 | 291 |
"[| F : A LeadsTo A'; F : B Unless B' |] \ |
292 |
\ ==> F : (A Int B) LeadsTo ((A' Int B) Un B')"; |
|
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|
293 |
by (dtac PSP 1); |
4776 | 294 |
by (assume_tac 1); |
5313
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|
295 |
by (blast_tac (claset() addIs [LeadsTo_Diff, LeadsTo_weaken, |
5584 | 296 |
subset_imp_LeadsTo]) 1); |
5313
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|
297 |
qed "PSP_Unless"; |
4776 | 298 |
|
299 |
||
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Revising the Client proof as suggested by Michel Charpentier. New lemmas
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|
300 |
Goal "[| F : Stable A; F : transient C; \ |
6570 | 301 |
\ F : Always (-A Un B Un C) |] ==> F : A LeadsTo B"; |
302 |
by (etac Always_LeadsTo_weaken 1); |
|
5804
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Revising the Client proof as suggested by Michel Charpentier. New lemmas
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changeset
|
303 |
by (rtac LeadsTo_Diff 1); |
6710
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|
304 |
by (etac (transient_imp_leadsTo RS leadsTo_imp_LeadsTo RS PSP_Stable2) 2); |
5804
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
paulson
parents:
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diff
changeset
|
305 |
by (ALLGOALS (blast_tac (claset() addIs [subset_imp_LeadsTo]))); |
6570 | 306 |
qed "Stable_transient_Always_LeadsTo"; |
5804
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Revising the Client proof as suggested by Michel Charpentier. New lemmas
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|
307 |
|
8e0a4c4fd67b
Revising the Client proof as suggested by Michel Charpentier. New lemmas
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changeset
|
308 |
|
4776 | 309 |
(*** Induction rules ***) |
310 |
||
311 |
(** Meta or object quantifier ????? **) |
|
5232 | 312 |
Goal "[| wf r; \ |
6536 | 313 |
\ ALL m. F : (A Int f-``{m}) LeadsTo \ |
5584 | 314 |
\ ((A Int f-``(r^-1 ^^ {m})) Un B) |] \ |
6536 | 315 |
\ ==> F : A LeadsTo B"; |
6575 | 316 |
by (full_simp_tac (simpset() addsimps [LeadsTo_eq_leadsTo]) 1); |
4776 | 317 |
by (etac leadsTo_wf_induct 1); |
318 |
by (blast_tac (claset() addIs [leadsTo_weaken]) 1); |
|
319 |
qed "LeadsTo_wf_induct"; |
|
320 |
||
321 |
||
5232 | 322 |
Goal "[| wf r; \ |
6536 | 323 |
\ ALL m:I. F : (A Int f-``{m}) LeadsTo \ |
5584 | 324 |
\ ((A Int f-``(r^-1 ^^ {m})) Un B) |] \ |
6536 | 325 |
\ ==> F : A LeadsTo ((A - (f-``I)) Un B)"; |
4776 | 326 |
by (etac LeadsTo_wf_induct 1); |
327 |
by Safe_tac; |
|
328 |
by (case_tac "m:I" 1); |
|
329 |
by (blast_tac (claset() addIs [LeadsTo_weaken]) 1); |
|
330 |
by (blast_tac (claset() addIs [subset_imp_LeadsTo]) 1); |
|
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|
331 |
qed "Bounded_induct"; |
4776 | 332 |
|
333 |
||
6536 | 334 |
Goal "[| ALL m. F : (A Int f-``{m}) LeadsTo \ |
5584 | 335 |
\ ((A Int f-``(lessThan m)) Un B) |] \ |
6536 | 336 |
\ ==> F : A LeadsTo B"; |
4776 | 337 |
by (rtac (wf_less_than RS LeadsTo_wf_induct) 1); |
338 |
by (Asm_simp_tac 1); |
|
5277
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|
339 |
qed "LessThan_induct"; |
4776 | 340 |
|
5544 | 341 |
(*Integer version. Could generalize from #0 to any lower bound*) |
5584 | 342 |
val [reach, prem] = |
6909 | 343 |
Goal "[| F : Always {s. (#0::int) <= f s}; \ |
6536 | 344 |
\ !! z. F : (A Int {s. f s = z}) LeadsTo \ |
5584 | 345 |
\ ((A Int {s. f s < z}) Un B) |] \ |
6536 | 346 |
\ ==> F : A LeadsTo B"; |
5569
8c7e1190e789
Renaming of Integ/Integ.* to Integ/Int.*, and renaming of related constants
paulson
parents:
5544
diff
changeset
|
347 |
by (res_inst_tac [("f", "nat o f")] (allI RS LessThan_induct) 1); |
5544 | 348 |
by (simp_tac (simpset() addsimps [vimage_def]) 1); |
6570 | 349 |
by (rtac ([reach, prem] MRS Always_LeadsTo_weaken) 1); |
5584 | 350 |
by (auto_tac (claset(), simpset() addsimps [nat_eq_iff, nat_less_iff])); |
5544 | 351 |
qed "integ_0_le_induct"; |
352 |
||
6536 | 353 |
Goal "[| ALL m:(greaterThan l). F : (A Int f-``{m}) LeadsTo \ |
5584 | 354 |
\ ((A Int f-``(lessThan m)) Un B) |] \ |
6536 | 355 |
\ ==> F : A LeadsTo ((A Int (f-``(atMost l))) Un B)"; |
4776 | 356 |
by (simp_tac (HOL_ss addsimps [Diff_eq RS sym, vimage_Compl, Compl_greaterThan RS sym]) 1); |
5277
e4297d03e5d2
A higher-level treatment of LeadsTo, minimizing use of "reachable"
paulson
parents:
5257
diff
changeset
|
357 |
by (rtac (wf_less_than RS Bounded_induct) 1); |
4776 | 358 |
by (Asm_simp_tac 1); |
5277
e4297d03e5d2
A higher-level treatment of LeadsTo, minimizing use of "reachable"
paulson
parents:
5257
diff
changeset
|
359 |
qed "LessThan_bounded_induct"; |
4776 | 360 |
|
6536 | 361 |
Goal "[| ALL m:(lessThan l). F : (A Int f-``{m}) LeadsTo \ |
5584 | 362 |
\ ((A Int f-``(greaterThan m)) Un B) |] \ |
6536 | 363 |
\ ==> F : A LeadsTo ((A Int (f-``(atLeast l))) Un B)"; |
4776 | 364 |
by (res_inst_tac [("f","f"),("f1", "%k. l - k")] |
365 |
(wf_less_than RS wf_inv_image RS LeadsTo_wf_induct) 1); |
|
366 |
by (simp_tac (simpset() addsimps [inv_image_def, Image_singleton]) 1); |
|
367 |
by (Clarify_tac 1); |
|
368 |
by (case_tac "m<l" 1); |
|
369 |
by (blast_tac (claset() addIs [not_leE, subset_imp_LeadsTo]) 2); |
|
370 |
by (blast_tac (claset() addIs [LeadsTo_weaken_R, diff_less_mono2]) 1); |
|
5277
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A higher-level treatment of LeadsTo, minimizing use of "reachable"
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diff
changeset
|
371 |
qed "GreaterThan_bounded_induct"; |
4776 | 372 |
|
373 |
||
374 |
(*** Completion: Binary and General Finite versions ***) |
|
375 |
||
6536 | 376 |
Goal "[| F : A LeadsTo A'; F : Stable A'; \ |
377 |
\ F : B LeadsTo B'; F : Stable B' |] \ |
|
378 |
\ ==> F : (A Int B) LeadsTo (A' Int B')"; |
|
6575 | 379 |
by (full_simp_tac |
380 |
(simpset() addsimps [LeadsTo_eq_leadsTo, Stable_eq_stable]) 1); |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
5277
diff
changeset
|
381 |
by (blast_tac (claset() addIs [stable_completion, leadsTo_weaken]) 1); |
5277
e4297d03e5d2
A higher-level treatment of LeadsTo, minimizing use of "reachable"
paulson
parents:
5257
diff
changeset
|
382 |
qed "Stable_completion"; |
4776 | 383 |
|
384 |
||
5584 | 385 |
Goal "finite I \ |
6536 | 386 |
\ ==> (ALL i:I. F : (A i) LeadsTo (A' i)) --> \ |
5648 | 387 |
\ (ALL i:I. F : Stable (A' i)) --> \ |
6536 | 388 |
\ F : (INT i:I. A i) LeadsTo (INT i:I. A' i)"; |
4776 | 389 |
by (etac finite_induct 1); |
390 |
by (Asm_simp_tac 1); |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
5277
diff
changeset
|
391 |
by (asm_simp_tac (simpset() addsimps [Stable_completion, ball_Stable_INT]) 1); |
5277
e4297d03e5d2
A higher-level treatment of LeadsTo, minimizing use of "reachable"
paulson
parents:
5257
diff
changeset
|
392 |
qed_spec_mp "Finite_stable_completion"; |
4776 | 393 |
|
394 |
||
6536 | 395 |
Goal "[| F : A LeadsTo (A' Un C); F : A' Co (A' Un C); \ |
396 |
\ F : B LeadsTo (B' Un C); F : B' Co (B' Un C) |] \ |
|
397 |
\ ==> F : (A Int B) LeadsTo ((A' Int B') Un C)"; |
|
6575 | 398 |
by (full_simp_tac |
399 |
(simpset() addsimps [LeadsTo_eq_leadsTo, Constrains_eq_constrains, |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
5277
diff
changeset
|
400 |
Int_Un_distrib]) 1); |
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
5277
diff
changeset
|
401 |
by (blast_tac (claset() addIs [completion, leadsTo_weaken]) 1); |
5277
e4297d03e5d2
A higher-level treatment of LeadsTo, minimizing use of "reachable"
paulson
parents:
5257
diff
changeset
|
402 |
qed "Completion"; |
4776 | 403 |
|
404 |
||
6564 | 405 |
Goal "finite I \ |
6536 | 406 |
\ ==> (ALL i:I. F : (A i) LeadsTo (A' i Un C)) --> \ |
407 |
\ (ALL i:I. F : (A' i) Co (A' i Un C)) --> \ |
|
408 |
\ F : (INT i:I. A i) LeadsTo ((INT i:I. A' i) Un C)"; |
|
4776 | 409 |
by (etac finite_induct 1); |
410 |
by (ALLGOALS Asm_simp_tac); |
|
411 |
by (Clarify_tac 1); |
|
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
5277
diff
changeset
|
412 |
by (dtac ball_Constrains_INT 1); |
5277
e4297d03e5d2
A higher-level treatment of LeadsTo, minimizing use of "reachable"
paulson
parents:
5257
diff
changeset
|
413 |
by (asm_full_simp_tac (simpset() addsimps [Completion]) 1); |
6564 | 414 |
qed_spec_mp "Finite_completion"; |
5232 | 415 |
|
416 |
||
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
5277
diff
changeset
|
417 |
(*proves "ensures/leadsTo" properties when the program is specified*) |
5426
566f47250bd0
A new approach, using simp_of_act and simp_of_set to activate definitions when
paulson
parents:
5422
diff
changeset
|
418 |
fun ensures_tac sact = |
5240
bbcd79ef7cf2
Constant "invariant" and new constrains_tac, ensures_tac
paulson
parents:
5232
diff
changeset
|
419 |
SELECT_GOAL |
6570 | 420 |
(EVERY [REPEAT (Always_Int_tac 1), |
421 |
etac Always_LeadsTo_Basis 1 |
|
5240
bbcd79ef7cf2
Constant "invariant" and new constrains_tac, ensures_tac
paulson
parents:
5232
diff
changeset
|
422 |
ORELSE (*subgoal may involve LeadsTo, leadsTo or ensures*) |
5313
1861a564d7e2
Constrains, Stable, Invariant...more of the substitution axiom, but Union
paulson
parents:
5277
diff
changeset
|
423 |
REPEAT (ares_tac [LeadsTo_Basis, ensuresI] 1), |
6536 | 424 |
(*now there are two subgoals: co & transient*) |
5648 | 425 |
simp_tac (simpset() addsimps !program_defs_ref) 2, |
5240
bbcd79ef7cf2
Constant "invariant" and new constrains_tac, ensures_tac
paulson
parents:
5232
diff
changeset
|
426 |
res_inst_tac [("act", sact)] transient_mem 2, |
5340 | 427 |
(*simplify the command's domain*) |
5426
566f47250bd0
A new approach, using simp_of_act and simp_of_set to activate definitions when
paulson
parents:
5422
diff
changeset
|
428 |
simp_tac (simpset() addsimps [Domain_def]) 3, |
566f47250bd0
A new approach, using simp_of_act and simp_of_set to activate definitions when
paulson
parents:
5422
diff
changeset
|
429 |
constrains_tac 1, |
5240
bbcd79ef7cf2
Constant "invariant" and new constrains_tac, ensures_tac
paulson
parents:
5232
diff
changeset
|
430 |
ALLGOALS Clarify_tac, |
5422 | 431 |
ALLGOALS Asm_full_simp_tac]); |
5240
bbcd79ef7cf2
Constant "invariant" and new constrains_tac, ensures_tac
paulson
parents:
5232
diff
changeset
|
432 |
|
bbcd79ef7cf2
Constant "invariant" and new constrains_tac, ensures_tac
paulson
parents:
5232
diff
changeset
|
433 |