--- a/src/ZF/UNITY/UNITY.ML Fri Jun 27 13:15:40 2003 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,707 +0,0 @@
-(* Title: ZF/UNITY/UNITY.ML
- ID: $Id$
- Author: Sidi O Ehmety, Computer Laboratory
- Copyright 2001 University of Cambridge
-
-The basic UNITY theory (revised version, based upon the "co" operator)
-From Misra, "A Logic for Concurrent Programming", 1994
-
-Proofs ported from HOL
-*)
-
-(** SKIP **)
-Goalw [SKIP_def] "SKIP:program";
-by (rewrite_goal_tac [program_def, mk_program_def] 1);
-by Auto_tac;
-qed "SKIP_in_program";
-AddIffs [SKIP_in_program];
-AddTCs [SKIP_in_program];
-
-(** programify: coersion from anything to program **)
-
-Goalw [programify_def]
-"F:program ==> programify(F)=F";
-by Auto_tac;
-qed "programify_program";
-Addsimps [programify_program];
-
-Goalw [programify_def]
-"programify(F):program";
-by Auto_tac;
-qed "programify_in_program";
-AddIffs [programify_in_program];
-AddTCs [programify_in_program];
-
-(** Collapsing rules: to remove programify from expressions **)
-Goalw [programify_def]
-"programify(programify(F))=programify(F)";
-by Auto_tac;
-qed "programify_idem";
-AddIffs [programify_idem];
-
-Goal
-"Init(programify(F)) = Init(F)";
-by (simp_tac (simpset() addsimps [Init_def]) 1);
-qed "Init_programify";
-AddIffs [Init_programify];
-
-Goal
-"Acts(programify(F)) = Acts(F)";
-by (simp_tac (simpset() addsimps [Acts_def]) 1);
-qed "Acts_programify";
-AddIffs [Acts_programify];
-
-Goal
-"AllowedActs(programify(F)) = AllowedActs(F)";
-by (simp_tac (simpset() addsimps [AllowedActs_def]) 1);
-qed "AllowedActs_programify";
-AddIffs [AllowedActs_programify];
-
-(** program's inspectors **)
-
-Goal "F:program ==>id(state):RawActs(F)";
-by (auto_tac (claset(), simpset()
- addsimps [program_def, RawActs_def]));
-qed "id_in_RawActs";
-
-Goal "id(state):Acts(F)";
-by (simp_tac (simpset()
- addsimps [id_in_RawActs, Acts_def]) 1);
-qed "id_in_Acts";
-
-Goal "F:program ==>id(state):RawAllowedActs(F)";
-by (auto_tac (claset(), simpset()
- addsimps [program_def, RawAllowedActs_def]));
-qed "id_in_RawAllowedActs";
-
-Goal "id(state):AllowedActs(F)";
-by (simp_tac (simpset()
- addsimps [id_in_RawAllowedActs, AllowedActs_def]) 1);
-qed "id_in_AllowedActs";
-
-AddIffs [id_in_Acts, id_in_AllowedActs];
-AddTCs [id_in_Acts, id_in_AllowedActs];
-
-Goal "cons(id(state), Acts(F)) = Acts(F)";
-by (simp_tac (simpset() addsimps [cons_absorb]) 1);
-qed "cons_id_Acts";
-
-Goal "cons(id(state), AllowedActs(F)) = AllowedActs(F)";
-by (simp_tac (simpset() addsimps [cons_absorb]) 1);
-qed "cons_id_AllowedActs";
-
-AddIffs [cons_id_Acts, cons_id_AllowedActs];
-
-(** inspectors's types **)
-Goal
-"F:program ==> RawInit(F)<=state";
-by (auto_tac (claset(), simpset()
- addsimps [program_def, RawInit_def]));
-qed "RawInit_type";
-
-Goal
-"F:program ==> RawActs(F)<=Pow(state*state)";
-by (auto_tac (claset(), simpset()
- addsimps [program_def, RawActs_def]));
-qed "RawActs_type";
-
-Goal
-"F:program ==> RawAllowedActs(F)<=Pow(state*state)";
-by (auto_tac (claset(), simpset()
- addsimps [program_def, RawAllowedActs_def]));
-qed "RawAllowedActs_type";
-
-Goal "Init(F)<=state";
-by (simp_tac (simpset()
- addsimps [RawInit_type, Init_def]) 1);
-qed "Init_type";
-
-bind_thm("InitD", Init_type RS subsetD);
-
-Goalw [st_set_def] "st_set(Init(F))";
-by (rtac Init_type 1);
-qed "st_set_Init";
-AddIffs [st_set_Init];
-
-Goal
-"Acts(F)<=Pow(state*state)";
-by (simp_tac (simpset()
- addsimps [RawActs_type, Acts_def]) 1);
-qed "Acts_type";
-
-Goal
-"AllowedActs(F)<=Pow(state*state)";
-by (simp_tac (simpset()
- addsimps [RawAllowedActs_type, AllowedActs_def]) 1);
-qed "AllowedActs_type";
-
-(* Needed in Behaviors *)
-Goal "[| act:Acts(F); <s,s'>:act |] ==> s:state & s':state";
-by (blast_tac (claset() addDs [Acts_type RS subsetD]) 1);
-qed "ActsD";
-
-Goal "[| act:AllowedActs(F); <s,s'>:act |] ==> s:state & s':state";
-by (blast_tac (claset() addDs [AllowedActs_type RS subsetD]) 1);
-qed "AllowedActsD";
-
-(** More simplification rules involving state
- and Init, Acts, and AllowedActs **)
-
-Goal "state <= Init(F) <-> Init(F)=state";
-by (cut_inst_tac [("F", "F")] Init_type 1);
-by Auto_tac;
-qed "state_subset_is_Init_iff";
-AddIffs [state_subset_is_Init_iff];
-
-Goal "Pow(state*state) <= Acts(F) <-> Acts(F)=Pow(state*state)";
-by (cut_inst_tac [("F", "F")] Acts_type 1);
-by Auto_tac;
-qed "Pow_state_times_state_is_subset_Acts_iff";
-AddIffs [Pow_state_times_state_is_subset_Acts_iff];
-
-Goal "Pow(state*state) <= AllowedActs(F) <-> AllowedActs(F)=Pow(state*state)";
-by (cut_inst_tac [("F", "F")] AllowedActs_type 1);
-by Auto_tac;
-qed "Pow_state_times_state_is_subset_AllowedActs_iff";
-AddIffs [Pow_state_times_state_is_subset_AllowedActs_iff];
-
-(** Eliminating `Int state' from expressions **)
-Goal "Init(F) Int state = Init(F)";
-by (cut_inst_tac [("F", "F")] Init_type 1);
-by (Blast_tac 1);
-qed "Init_Int_state";
-AddIffs [Init_Int_state];
-
-Goal "state Int Init(F) = Init(F)";
-by (cut_inst_tac [("F", "F")] Init_type 1);
-by (Blast_tac 1);
-qed "state_Int_Init";
-AddIffs [state_Int_Init];
-
-Goal "Acts(F) Int Pow(state*state) = Acts(F)";
-by (cut_inst_tac [("F", "F")] Acts_type 1);
-by (Blast_tac 1);
-qed "Acts_Int_Pow_state_times_state";
-AddIffs [Acts_Int_Pow_state_times_state];
-
-Goal "Pow(state*state) Int Acts(F) = Acts(F)";
-by (cut_inst_tac [("F", "F")] Acts_type 1);
-by (Blast_tac 1);
-qed "state_times_state_Int_Acts";
-AddIffs [state_times_state_Int_Acts];
-
-Goal "AllowedActs(F) Int Pow(state*state) = AllowedActs(F)";
-by (cut_inst_tac [("F", "F")] AllowedActs_type 1);
-by (Blast_tac 1);
-qed "AllowedActs_Int_Pow_state_times_state";
-AddIffs [AllowedActs_Int_Pow_state_times_state];
-
-Goal "Pow(state*state) Int AllowedActs(F) = AllowedActs(F)";
-by (cut_inst_tac [("F", "F")] AllowedActs_type 1);
-by (Blast_tac 1);
-qed "state_times_state_Int_AllowedActs";
-AddIffs [state_times_state_Int_AllowedActs];
-
-(** mk_program **)
-
-Goalw [mk_program_def, program_def] "mk_program(init, acts, allowed):program";
-by Auto_tac;
-qed "mk_program_in_program";
-AddIffs [mk_program_in_program];
-AddTCs [mk_program_in_program];
-
-Goalw [RawInit_def, mk_program_def]
- "RawInit(mk_program(init, acts, allowed)) = init Int state";
-by Auto_tac;
-qed "RawInit_eq";
-AddIffs [RawInit_eq];
-
-Goalw [RawActs_def, mk_program_def]
-"RawActs(mk_program(init, acts, allowed)) = cons(id(state), acts Int Pow(state*state))";
-by Auto_tac;
-qed "RawActs_eq";
-AddIffs [RawActs_eq];
-
-Goalw [RawAllowedActs_def, mk_program_def]
-"RawAllowedActs(mk_program(init, acts, allowed)) = \
-\ cons(id(state), allowed Int Pow(state*state))";
-by Auto_tac;
-qed "RawAllowedActs_eq";
-AddIffs [RawAllowedActs_eq];
-
-Goalw [Init_def] "Init(mk_program(init, acts, allowed)) = init Int state";
-by (Simp_tac 1);
-qed "Init_eq";
-AddIffs [Init_eq];
-
-Goalw [Acts_def]
-"Acts(mk_program(init, acts, allowed)) = cons(id(state), acts Int Pow(state*state))";
-by (Simp_tac 1);
-qed "Acts_eq";
-AddIffs [Acts_eq];
-
-Goalw [AllowedActs_def]
-"AllowedActs(mk_program(init, acts, allowed))= \
-\ cons(id(state), allowed Int Pow(state*state))";
-by (Simp_tac 1);
-qed "AllowedActs_eq";
-AddIffs [AllowedActs_eq];
-
-(**Init, Acts, and AlowedActs of SKIP **)
-
-Goalw [SKIP_def] "RawInit(SKIP) = state";
-by Auto_tac;
-qed "RawInit_SKIP";
-AddIffs [RawInit_SKIP];
-
-Goalw [SKIP_def] "RawAllowedActs(SKIP) = Pow(state*state)";
-by Auto_tac;
-qed "RawAllowedActs_SKIP";
-AddIffs [RawAllowedActs_SKIP];
-
-Goalw [SKIP_def] "RawActs(SKIP) = {id(state)}";
-by Auto_tac;
-qed "RawActs_SKIP";
-AddIffs [RawActs_SKIP];
-
-Goalw [Init_def] "Init(SKIP) = state";
-by Auto_tac;
-qed "Init_SKIP";
-AddIffs [Init_SKIP];
-
-Goalw [Acts_def] "Acts(SKIP) = {id(state)}";
-by Auto_tac;
-qed "Acts_SKIP";
-AddIffs [Acts_SKIP];
-
-Goalw [AllowedActs_def] "AllowedActs(SKIP) = Pow(state*state)";
-by Auto_tac;
-qed "AllowedActs_SKIP";
-AddIffs [AllowedActs_SKIP];
-
-(** Equality of UNITY programs **)
-
-Goal
-"F:program ==> mk_program(RawInit(F), RawActs(F), RawAllowedActs(F))=F";
-by (rewrite_goal_tac [program_def, mk_program_def,RawInit_def,
- RawActs_def, RawAllowedActs_def] 1);
-by Auto_tac;
-by (REPEAT(Blast_tac 1));
-qed "raw_surjective_mk_program";
-Addsimps [raw_surjective_mk_program];
-
-Goalw [Init_def, Acts_def, AllowedActs_def]
- "mk_program(Init(F), Acts(F), AllowedActs(F)) = programify(F)";
-by Auto_tac;
-qed "surjective_mk_program";
-AddIffs [surjective_mk_program];
-
-Goal "[|Init(F) = Init(G); Acts(F) = Acts(G); \
-\ AllowedActs(F) = AllowedActs(G); F:program; G:program |] ==> F = G";
-by (stac (programify_program RS sym) 1);
-by (rtac sym 2);
-by (stac (programify_program RS sym) 2);
-by (stac (surjective_mk_program RS sym) 3);
-by (stac (surjective_mk_program RS sym) 3);
-by (ALLGOALS(Asm_simp_tac));
-qed "program_equalityI";
-
-val [major,minor] =
-Goal "[| F = G; \
-\ [| Init(F) = Init(G); Acts(F) = Acts(G); AllowedActs(F) = AllowedActs(G) |]\
-\ ==> P |] ==> P";
-by (rtac minor 1);
-by (auto_tac (claset(), simpset() addsimps [major]));
-qed "program_equalityE";
-
-
-Goal "[| F:program; G:program |] ==>(F=G) <-> \
-\ (Init(F) = Init(G) & Acts(F) = Acts(G) & AllowedActs(F) = AllowedActs(G))";
-by (blast_tac (claset() addIs [program_equalityI, program_equalityE]) 1);
-qed "program_equality_iff";
-
-(*** These rules allow "lazy" definition expansion
-
-...skipping 1 line
-
-***)
-
-Goal "F == mk_program (init,acts,allowed) ==> Init(F) = init Int state";
-by Auto_tac;
-qed "def_prg_Init";
-
-
-Goal "F == mk_program (init,acts,allowed) ==> \
-\ Acts(F) = cons(id(state), acts Int Pow(state*state))";
-by Auto_tac;
-qed "def_prg_Acts";
-
-
-Goal "F == mk_program (init,acts,allowed) ==> \
-\ AllowedActs(F) = cons(id(state), allowed Int Pow(state*state))";
-by Auto_tac;
-qed "def_prg_AllowedActs";
-
-
-val [rew] = goal thy
- "[| F == mk_program (init,acts,allowed) |] \
-\ ==> Init(F) = init Int state & Acts(F) = cons(id(state), acts Int Pow(state*state)) & \
-\ AllowedActs(F) = cons(id(state), allowed Int Pow(state*state)) ";
-by (rewtac rew);
-by Auto_tac;
-qed "def_prg_simps";
-
-
-(*An action is expanded only if a pair of states is being tested against it*)
-Goal "[| act == {<s,s'>:A*B. P(s, s')} |] ==> \
-\ (<s,s'>:act) <-> (<s,s'>:A*B & P(s, s'))";
-by Auto_tac;
-qed "def_act_simp";
-
-fun simp_of_act def = def RS def_act_simp;
-
-(*A set is expanded only if an element is being tested against it*)
-Goal "A == B ==> (x : A) <-> (x : B)";
-by Auto_tac;
-qed "def_set_simp";
-
-fun simp_of_set def = def RS def_set_simp;
-
-(*** co ***)
-
-Goalw [constrains_def]
-"A co B <= program";
-by Auto_tac;
-qed "constrains_type";
-
-
-val prems = Goalw [constrains_def]
- "[|(!!act s s'. [| act: Acts(F); <s,s'>:act; s:A|] ==> s':A'); \
- \ F:program; st_set(A) |] ==> F:A co A'";
-by (auto_tac (claset() delrules [subsetI], simpset()));
-by (ALLGOALS(asm_full_simp_tac (simpset() addsimps prems)));
-by (Clarify_tac 1);
-by (blast_tac(claset() addIs prems) 1);
-qed "constrainsI";
-
-Goalw [constrains_def]
- "F:A co B ==> ALL act:Acts(F). act``A<=B";
-by (Blast_tac 1);
-qed "constrainsD";
-
-Goalw [constrains_def]
- "F:A co B ==> F:program & st_set(A)";
-by (Blast_tac 1);
-qed "constrainsD2";
-
-Goalw [constrains_def, st_set_def] "F : 0 co B <-> F:program";
-by (Blast_tac 1);
-qed "constrains_empty";
-
-Goalw [constrains_def, st_set_def]
- "(F : A co 0) <-> (A=0 & F:program)";
-by (cut_inst_tac [("F", "F")] Acts_type 1);
-by Auto_tac;
-by (Blast_tac 1);
-qed "constrains_empty2";
-
-Goalw [constrains_def, st_set_def]
-"(F: state co B) <-> (state<=B & F:program)";
-by (cut_inst_tac [("F", "F")] Acts_type 1);
-by (Blast_tac 1);
-qed "constrains_state";
-
-Goalw [constrains_def, st_set_def] "F:A co state <-> (F:program & st_set(A))";
-by (cut_inst_tac [("F", "F")] Acts_type 1);
-by (Blast_tac 1);
-qed "constrains_state2";
-
-AddIffs [constrains_empty, constrains_empty2,
- constrains_state, constrains_state2];
-
-(*monotonic in 2nd argument*)
-Goalw [constrains_def]
- "[| F:A co A'; A'<=B' |] ==> F : A co B'";
-by (Blast_tac 1);
-qed "constrains_weaken_R";
-
-(*anti-monotonic in 1st argument*)
-Goalw [constrains_def, st_set_def]
- "[| F : A co A'; B<=A |] ==> F : B co A'";
-by (Blast_tac 1);
-qed "constrains_weaken_L";
-
-Goal
- "[| F : A co A'; B<=A; A'<=B' |] ==> F : B co B'";
-by (dtac constrains_weaken_R 1);
-by (dtac constrains_weaken_L 2);
-by (REPEAT(Blast_tac 1));
-qed "constrains_weaken";
-
-(** Union **)
-
-Goalw [constrains_def, st_set_def]
- "[| F : A co A'; F:B co B' |] ==> F:(A Un B) co (A' Un B')";
-by Auto_tac;
-by (Force_tac 1);
-qed "constrains_Un";
-
-val major::minor::_ = Goalw [constrains_def, st_set_def]
-"[|(!!i. i:I ==> F:A(i) co A'(i)); F:program |]==> F:(UN i:I. A(i)) co (UN i:I. A'(i))";
-by (cut_facts_tac [minor] 1);
-by Safe_tac;
-by (ALLGOALS(ftac major ));
-by (ALLGOALS(Asm_full_simp_tac));
-by (REPEAT(Blast_tac 1));
-qed "constrains_UN";
-
-Goalw [constrains_def, st_set_def]
- "(A Un B) co C = (A co C) Int (B co C)";
-by Auto_tac;
-by (Force_tac 1);
-qed "constrains_Un_distrib";
-
-Goalw [constrains_def, st_set_def]
- "i:I ==> (UN i:I. A(i)) co B = (INT i:I. A(i) co B)";
-by (rtac equalityI 1);
-by (REPEAT(Force_tac 1));
-qed "constrains_UN_distrib";
-
-(** Intersection **)
-Goalw [constrains_def, st_set_def]
- "C co (A Int B) = (C co A) Int (C co B)";
-by (rtac equalityI 1);
-by (ALLGOALS(Clarify_tac)); (* to speed up the proof *)
-by (REPEAT(Blast_tac 1));
-qed "constrains_Int_distrib";
-
-Goalw [constrains_def, st_set_def]
-"x:I ==> A co (INT i:I. B(i)) = (INT i:I. A co B(i))";
-by (rtac equalityI 1);
-by Safe_tac;
-by (REPEAT(Blast_tac 1));
-qed "constrains_INT_distrib";
-
-Goalw [constrains_def, st_set_def]
- "[| F : A co A'; F : B co B' |] ==> F : (A Int B) co (A' Int B')";
-by (Clarify_tac 1);
-by (Blast_tac 1);
-qed "constrains_Int";
-
-val major::minor::_ = Goalw [constrains_def, st_set_def]
-"[| (!!i. i:I==>F:A(i) co A'(i)); F:program|]==> F:(INT i:I. A(i)) co (INT i:I. A'(i))";
-by (cut_facts_tac [minor] 1);
-by (cut_inst_tac [("F", "F")] Acts_type 1);
-by (case_tac "I=0" 1);
-by (asm_full_simp_tac (simpset() addsimps [Inter_def]) 1);
-by (etac not_emptyE 1);
-by Safe_tac;
-by (forw_inst_tac [("i", "xd")] major 1);
-by (ftac major 2);
-by (ftac major 3);
-by (REPEAT(Force_tac 1));
-qed "constrains_INT";
-
-(* The rule below simulates the HOL's one for (INT z. A i) co (INT z. B i) *)
-Goalw [constrains_def]
-"[| ALL z. F:{s:state. P(s, z)} co {s:state. Q(s, z)}; F:program |]==>\
-\ F:{s:state. ALL z. P(s, z)} co {s:state. ALL z. Q(s, z)}";
-by (Blast_tac 1);
-qed "constrains_All";
-
-Goalw [constrains_def, st_set_def]
- "[| F:A co A' |] ==> A <= A'";
-by (Force_tac 1);
-qed "constrains_imp_subset";
-
-(*The reasoning is by subsets since "co" refers to single actions
- only. So this rule isn't that useful.*)
-
-Goalw [constrains_def, st_set_def]
- "[| F : A co B; F : B co C |] ==> F : A co C";
-by Auto_tac;
-by (Blast_tac 1);
-qed "constrains_trans";
-
-Goal
-"[| F : A co (A' Un B); F : B co B' |] ==> F:A co (A' Un B')";
-by (dres_inst_tac [("A", "B")] constrains_imp_subset 1);
-by (blast_tac (claset() addIs [constrains_weaken_R]) 1);
-qed "constrains_cancel";
-
-(*** unless ***)
-
-Goalw [unless_def, constrains_def]
- "A unless B <= program";
-by Auto_tac;
-qed "unless_type";
-
-Goalw [unless_def] "[| F:(A-B) co (A Un B) |] ==> F : A unless B";
-by (blast_tac (claset() addDs [constrainsD2]) 1);
-qed "unlessI";
-
-Goalw [unless_def] "F :A unless B ==> F : (A-B) co (A Un B)";
-by Auto_tac;
-qed "unlessD";
-
-(*** initially ***)
-
-Goalw [initially_def]
-"initially(A) <= program";
-by (Blast_tac 1);
-qed "initially_type";
-
-Goalw [initially_def]
-"[| F:program; Init(F)<=A |] ==> F:initially(A)";
-by (Blast_tac 1);
-qed "initiallyI";
-
-Goalw [initially_def]
-"F:initially(A) ==> Init(F)<=A";
-by (Blast_tac 1);
-qed "initiallyD";
-
-(*** stable ***)
-
-Goalw [stable_def, constrains_def]
- "stable(A)<=program";
-by (Blast_tac 1);
-qed "stable_type";
-
-Goalw [stable_def]
- "F : A co A ==> F : stable(A)";
-by (assume_tac 1);
-qed "stableI";
-
-Goalw [stable_def] "F:stable(A) ==> F : A co A";
-by (assume_tac 1);
-qed "stableD";
-
-Goalw [stable_def, constrains_def] "F:stable(A) ==> F:program & st_set(A)";
-by Auto_tac;
-qed "stableD2";
-
-Goalw [stable_def, constrains_def] "stable(state) = program";
-by (auto_tac (claset() addDs [Acts_type RS subsetD], simpset()));
-qed "stable_state";
-AddIffs [stable_state];
-
-Goalw [unless_def, stable_def]
- "stable(A)= A unless 0";
-by Auto_tac;
-qed "stable_unless";
-
-
-(** Union **)
-
-Goalw [stable_def]
- "[| F : stable(A); F:stable(A') |] ==> F : stable(A Un A')";
-by (blast_tac (claset() addIs [constrains_Un]) 1);
-qed "stable_Un";
-
-val [major, minor] = Goalw [stable_def]
-"[|(!!i. i:I ==> F : stable(A(i))); F:program |] ==> F:stable (UN i:I. A(i))";
-by (cut_facts_tac [minor] 1);
-by (blast_tac (claset() addIs [constrains_UN, major]) 1);
-qed "stable_UN";
-
-Goalw [stable_def]
- "[| F : stable(A); F : stable(A') |] ==> F : stable (A Int A')";
-by (blast_tac (claset() addIs [constrains_Int]) 1);
-qed "stable_Int";
-
-val [major, minor] = Goalw [stable_def]
-"[| (!!i. i:I ==> F:stable(A(i))); F:program |] ==> F : stable (INT i:I. A(i))";
-by (cut_facts_tac [minor] 1);
-by (blast_tac (claset() addIs [constrains_INT, major]) 1);
-qed "stable_INT";
-
-Goalw [stable_def]
-"[|ALL z. F:stable({s:state. P(s, z)}); F:program|]==>F:stable({s:state. ALL z. P(s, z)})";
-by (rtac constrains_All 1);
-by Auto_tac;
-qed "stable_All";
-
-Goalw [stable_def, constrains_def, st_set_def]
-"[| F : stable(C); F : A co (C Un A') |] ==> F : (C Un A) co (C Un A')";
-by Auto_tac;
-by (blast_tac (claset() addSDs [bspec]) 1);
-qed "stable_constrains_Un";
-
-Goalw [stable_def, constrains_def, st_set_def]
- "[| F : stable(C); F : (C Int A) co A' |] ==> F : (C Int A) co (C Int A')";
-by (Clarify_tac 1);
-by (Blast_tac 1);
-qed "stable_constrains_Int";
-
-(* [| F:stable(C); F :(C Int A) co A |] ==> F:stable(C Int A) *)
-bind_thm ("stable_constrains_stable", stable_constrains_Int RS stableI);
-
-(** invariant **)
-
-Goalw [invariant_def]
- "invariant(A) <= program";
-by (blast_tac (claset() addDs [stable_type RS subsetD]) 1);
-qed "invariant_type";
-
-Goalw [invariant_def, initially_def]
- "[| Init(F)<=A; F:stable(A) |] ==> F : invariant(A)";
-by (forward_tac [stable_type RS subsetD] 1);
-by Auto_tac;
-qed "invariantI";
-
-Goalw [invariant_def, initially_def]
-"F:invariant(A) ==> Init(F)<=A & F:stable(A)";
-by Auto_tac;
-qed "invariantD";
-
-Goalw [invariant_def]
- "F:invariant(A) ==> F:program & st_set(A)";
-by (blast_tac (claset() addDs [stableD2]) 1);
-qed "invariantD2";
-
-(*Could also say "invariant A Int invariant B <= invariant (A Int B)"*)
-Goalw [invariant_def, initially_def]
- "[| F : invariant(A); F : invariant(B) |] ==> F : invariant(A Int B)";
-by (asm_full_simp_tac (simpset() addsimps [stable_Int]) 1);
-by (Blast_tac 1);
-qed "invariant_Int";
-
-(** The Elimination Theorem. The "free" m has become universally quantified!
- Should the premise be !!m instead of ALL m ? Would make it harder
- to use in forward proof. **)
-
-(* The general case easier to prove that le special case! *)
-Goalw [constrains_def, st_set_def]
- "[| ALL m:M. F : {s:A. x(s) = m} co B(m); F:program |] \
-\ ==> F:{s:A. x(s):M} co (UN m:M. B(m))";
-by Safe_tac;
-by Auto_tac;
-by (Blast_tac 1);
-qed "elimination";
-
-(* As above, but for the special case of A=state *)
-Goal "[| ALL m:M. F : {s:state. x(s) = m} co B(m); F:program |] \
-\ ==> F:{s:state. x(s):M} co (UN m:M. B(m))";
-by (rtac elimination 1);
-by (ALLGOALS(Clarify_tac));
-qed "eliminiation2";
-
-(** strongest_rhs **)
-
-Goalw [constrains_def, strongest_rhs_def, st_set_def]
- "[| F:program; st_set(A) |] ==> F:A co (strongest_rhs(F,A))";
-by (auto_tac (claset() addDs [Acts_type RS subsetD], simpset()));
-qed "constrains_strongest_rhs";
-
-Goalw [constrains_def, strongest_rhs_def, st_set_def]
-"[| F:A co B; st_set(B) |] ==> strongest_rhs(F,A) <= B";
-by Safe_tac;
-by (dtac InterD 1);
-by Auto_tac;
-qed "strongest_rhs_is_strongest";
-
-(* Used in WFair.thy *)
-Goal "A:Pow(Pow(B)) ==> Union(A):Pow(B)";
-by Auto_tac;
-qed "Union_PowI";