author | paulson |
Mon, 17 May 1999 10:38:08 +0200 | |
changeset 6646 | 3ea726909fff |
parent 6575 | 70d758762c50 |
child 6826 | 02c4dd469ec0 |
permissions | -rw-r--r-- |
5899 | 1 |
(* Title: HOL/UNITY/PPROD.ML |
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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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*) |
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||
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|
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val rinst = read_instantiate_sg (sign_of thy); |
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||
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(**** PPROD ****) |
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||
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(*** Basic properties ***) |
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||
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Goalw [lift_set_def] "(f : lift_set i A) = (f i : A)"; |
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by Auto_tac; |
|
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qed "lift_set_iff"; |
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AddIffs [lift_set_iff]; |
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||
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removed the infernal States, eqStates, compatible, etc.
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Goalw [lift_act_def] "lift_act i Id = Id"; |
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by Auto_tac; |
6295
351b3c2b0d83
removed the infernal States, eqStates, compatible, etc.
paulson
parents:
6020
diff
changeset
|
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qed "lift_act_Id"; |
351b3c2b0d83
removed the infernal States, eqStates, compatible, etc.
paulson
parents:
6020
diff
changeset
|
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Addsimps [lift_act_Id]; |
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Goalw [lift_prog_def] "Init (lift_prog i F) = lift_set i (Init F)"; |
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by Auto_tac; |
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qed "Init_lift_prog"; |
|
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Addsimps [Init_lift_prog]; |
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||
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Goalw [lift_prog_def] "Acts (lift_prog i F) = lift_act i `` Acts F"; |
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6295
351b3c2b0d83
removed the infernal States, eqStates, compatible, etc.
paulson
parents:
6020
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changeset
|
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by (auto_tac (claset() addIs [Id_in_Acts RSN (2,image_eqI)], simpset())); |
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qed "Acts_lift_prog"; |
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||
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Goalw [PPROD_def] "Init (PPROD I F) = (INT i:I. lift_set i (Init (F i)))"; |
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by Auto_tac; |
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qed "Init_PPROD"; |
|
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Addsimps [Init_PPROD]; |
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||
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Goalw [lift_act_def] |
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"((f,f') : lift_act i act) = (EX s'. f' = f(i := s') & (f i, s') : act)"; |
|
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by (Blast_tac 1); |
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qed "lift_act_eq"; |
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AddIffs [lift_act_eq]; |
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||
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Goal "Acts (PPROD I F) = insert Id (UN i:I. lift_act i `` Acts (F i))"; |
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by (auto_tac (claset(), |
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simpset() addsimps [PPROD_def, Acts_lift_prog, Acts_JN])); |
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qed "Acts_PPROD"; |
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||
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Goal "PPROD {} F = SKIP"; |
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by (simp_tac (simpset() addsimps [PPROD_def]) 1); |
|
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qed "PPROD_empty"; |
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||
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Goal "(PPI i: I. SKIP) = SKIP"; |
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by (auto_tac (claset() addSIs [program_equalityI], |
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simpset() addsimps [Acts_lift_prog, SKIP_def, Acts_PPROD])); |
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qed "PPROD_SKIP"; |
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||
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Addsimps [PPROD_SKIP, PPROD_empty]; |
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||
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Goalw [PPROD_def] |
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"PPROD (insert i I) F = (lift_prog i (F i)) Join (PPROD I F)"; |
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by Auto_tac; |
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qed "PPROD_insert"; |
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||
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Goalw [PPROD_def] "i : I ==> (lift_prog i (F i)) component (PPROD I F)"; |
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(*blast_tac doesn't use HO unification*) |
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by (fast_tac (claset() addIs [component_JN]) 1); |
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qed "component_PPROD"; |
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(*** Safety: co, stable, invariant ***) |
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(** 1st formulation of lifting **) |
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Goal "(lift_prog i F : (lift_set i A) co (lift_set i B)) = \ |
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\ (F : A co B)"; |
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by (auto_tac (claset(), |
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simpset() addsimps [constrains_def, Acts_lift_prog])); |
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by (Blast_tac 2); |
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by (Force_tac 1); |
|
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qed "lift_prog_constrains_eq"; |
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||
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Goal "(lift_prog i F : stable (lift_set i A)) = (F : stable A)"; |
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by (simp_tac (simpset() addsimps [stable_def, lift_prog_constrains_eq]) 1); |
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qed "lift_prog_stable_eq"; |
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||
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(*This one looks strange! Proof probably is by case analysis on i=j.*) |
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Goal "F i : A co B \ |
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\ ==> lift_prog j (F j) : (lift_set i A) co (lift_set i B)"; |
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by (auto_tac (claset(), |
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simpset() addsimps [constrains_def, Acts_lift_prog])); |
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by (REPEAT (Blast_tac 1)); |
|
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qed "constrains_imp_lift_prog_constrains"; |
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|
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Goal "i : I ==> \ |
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\ (PPROD I F : (lift_set i A) co (lift_set i B)) = \ |
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\ (F i : A co B)"; |
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by (asm_simp_tac (simpset() addsimps [PPROD_def, constrains_JN]) 1); |
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by (blast_tac (claset() addIs [lift_prog_constrains_eq RS iffD1, |
|
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constrains_imp_lift_prog_constrains]) 1); |
|
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qed "PPROD_constrains"; |
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||
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Goal "i : I ==> (PPROD I F : stable (lift_set i A)) = (F i : stable A)"; |
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by (asm_simp_tac (simpset() addsimps [stable_def, PPROD_constrains]) 1); |
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qed "PPROD_stable"; |
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||
107 |
||
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(** 2nd formulation of lifting **) |
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||
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Goal "[| lift_prog i F : AA co BB |] \ |
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\ ==> F : (Applyall AA i) co (Applyall BB i)"; |
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by (auto_tac (claset(), |
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simpset() addsimps [Applyall_def, constrains_def, |
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Acts_lift_prog])); |
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by (force_tac (claset() addSIs [rinst [("x", "?ff(i := ?u)")] image_eqI], |
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simpset()) 1); |
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qed "lift_prog_constrains_projection"; |
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||
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Goal "[| PPROD I F : AA co BB; i: I |] \ |
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\ ==> F i : (Applyall AA i) co (Applyall BB i)"; |
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by (rtac lift_prog_constrains_projection 1); |
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(*rotate this assumption to be last*) |
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by (dres_inst_tac [("psi", "PPROD I F : ?C")] asm_rl 1); |
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by (asm_full_simp_tac (simpset() addsimps [PPROD_def, constrains_JN]) 1); |
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qed "PPROD_constrains_projection"; |
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(** invariant **) |
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||
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(*UNUSED*) |
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Goal "(lift_prog i F : invariant (lift_set i A)) = (F : invariant A)"; |
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by (auto_tac (claset(), |
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simpset() addsimps [invariant_def, lift_prog_stable_eq])); |
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qed "lift_prog_invariant_eq"; |
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|
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Goal "[| F i : invariant A; i : I |] \ |
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\ ==> PPROD I F : invariant (lift_set i A)"; |
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by (auto_tac (claset(), |
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simpset() addsimps [invariant_def, PPROD_stable])); |
|
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qed "invariant_imp_PPROD_invariant"; |
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(*The f0 premise ensures that the product is well-defined.*) |
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Goal "[| PPROD I F : invariant (lift_set i A); i : I; \ |
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\ f0: Init (PPROD I F) |] ==> F i : invariant A"; |
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by (auto_tac (claset(), |
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simpset() addsimps [invariant_def, PPROD_stable])); |
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by (dres_inst_tac [("c", "f0(i:=x)")] subsetD 1); |
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by Auto_tac; |
|
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qed "PPROD_invariant_imp_invariant"; |
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||
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Goal "[| i : I; f0: Init (PPROD I F) |] \ |
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\ ==> (PPROD I F : invariant (lift_set i A)) = (F i : invariant A)"; |
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by (blast_tac (claset() addIs [invariant_imp_PPROD_invariant, |
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PPROD_invariant_imp_invariant]) 1); |
|
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qed "PPROD_invariant"; |
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||
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(*The f0 premise isn't needed if F is a constant program because then |
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we get an initial state by replicating that of F*) |
|
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Goal "i : I \ |
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\ ==> ((PPI x:I. F) : invariant (lift_set i A)) = (F : invariant A)"; |
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by (auto_tac (claset(), |
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simpset() addsimps [invariant_def, PPROD_stable])); |
|
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qed "PFUN_invariant"; |
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||
5899 | 165 |
|
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(*** Substitution Axiom versions: Co, Stable ***) |
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|
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(** Reachability **) |
169 |
||
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Goal "[| f : reachable (PPROD I F); i : I |] ==> f i : reachable (F i)"; |
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by (etac reachable.induct 1); |
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by (auto_tac |
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(claset() addIs reachable.intrs, |
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simpset() addsimps [Acts_PPROD])); |
|
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qed "reachable_PPROD"; |
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||
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Goal "reachable (PPROD I F) <= {f. ALL i:I. f i : reachable (F i)}"; |
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by (force_tac (claset() addSDs [reachable_PPROD], simpset()) 1); |
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qed "reachable_PPROD_subset1"; |
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||
5972 | 181 |
Goal "[| i ~: I; A : reachable (F i) |] \ |
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\ ==> ALL f. f : reachable (PPROD I F) \ |
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\ --> f(i:=A) : reachable (lift_prog i (F i) Join PPROD I F)"; |
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by (etac reachable.induct 1); |
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by (ALLGOALS Clarify_tac); |
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by (etac reachable.induct 1); |
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(*Init, Init case*) |
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by (force_tac (claset() addIs reachable.intrs, |
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simpset() addsimps [Acts_lift_prog]) 1); |
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(*Init of F, action of PPROD F case*) |
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by (rtac reachable.Acts 1); |
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by (force_tac (claset(), simpset() addsimps [Acts_Join]) 1); |
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by (assume_tac 1); |
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by (force_tac (claset() addIs [ext], simpset() addsimps [Acts_PPROD]) 1); |
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(*induction over the 2nd "reachable" assumption*) |
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by (eres_inst_tac [("xa","f")] reachable.induct 1); |
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(*Init of PPROD F, action of F case*) |
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by (res_inst_tac [("act","lift_act i act")] reachable.Acts 1); |
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by (force_tac (claset(), simpset() addsimps [Acts_lift_prog, Acts_Join]) 1); |
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by (force_tac (claset() addIs [reachable.Init], simpset()) 1); |
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by (force_tac (claset() addIs [ext], simpset() addsimps [lift_act_def]) 1); |
|
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(*last case: an action of PPROD I F*) |
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by (rtac reachable.Acts 1); |
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by (force_tac (claset(), simpset() addsimps [Acts_Join]) 1); |
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by (assume_tac 1); |
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by (force_tac (claset() addIs [ext], simpset() addsimps [Acts_PPROD]) 1); |
207 |
qed_spec_mp "reachable_lift_Join_PPROD"; |
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208 |
||
209 |
||
210 |
(*The index set must be finite: otherwise infinitely many copies of F can |
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perform actions, and PPROD can never catch up in finite time.*) |
|
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Goal "finite I \ |
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\ ==> {f. ALL i:I. f i : reachable (F i)} <= reachable (PPROD I F)"; |
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by (etac finite_induct 1); |
215 |
by (Simp_tac 1); |
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by (force_tac (claset() addDs [reachable_lift_Join_PPROD], |
|
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simpset() addsimps [PPROD_insert]) 1); |
|
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qed "reachable_PPROD_subset2"; |
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||
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Goal "finite I ==> \ |
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\ reachable (PPROD I F) = {f. ALL i:I. f i : reachable (F i)}"; |
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by (REPEAT_FIRST (ares_tac [equalityI, |
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reachable_PPROD_subset1, |
|
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reachable_PPROD_subset2])); |
|
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qed "reachable_PPROD_eq"; |
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||
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||
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(** Co **) |
5899 | 229 |
|
6536 | 230 |
Goal "[| F i : A Co B; i: I; finite I |] \ |
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\ ==> PPROD I F : (lift_set i A) Co (lift_set i B)"; |
|
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by (auto_tac |
233 |
(claset(), |
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simpset() addsimps [Constrains_def, Collect_conj_eq RS sym, |
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reachable_PPROD_eq])); |
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by (auto_tac (claset(), |
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simpset() addsimps [constrains_def, Acts_lift_prog, PPROD_def, |
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Acts_JN])); |
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by (REPEAT (blast_tac (claset() addIs reachable.intrs) 1)); |
|
5972 | 240 |
qed "Constrains_imp_PPROD_Constrains"; |
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||
6299 | 242 |
Goal "[| ALL i:I. f0 i : R i; i: I |] \ |
243 |
\ ==> Applyall ({f. (ALL i:I. f i : R i)} Int lift_set i A) i = R i Int A"; |
|
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by (force_tac (claset() addSIs [rinst [("x", "?ff(i := ?u)")] image_eqI], |
|
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simpset() addsimps [Applyall_def, lift_set_def]) 1); |
|
5972 | 246 |
qed "Applyall_Int_depend"; |
247 |
||
6575 | 248 |
(*Again, we need the f0 premise so that PPROD I F has an initial state; |
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otherwise its Co-property is vacuous.*) |
|
6536 | 250 |
Goal "[| PPROD I F : (lift_set i A) Co (lift_set i B); \ |
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\ i: I; finite I; f0: Init (PPROD I F) |] \ |
6536 | 252 |
\ ==> F i : A Co B"; |
6575 | 253 |
by (full_simp_tac (simpset() addsimps [Constrains_eq_constrains]) 1); |
5972 | 254 |
by (subgoal_tac "ALL i:I. f0 i : reachable (F i)" 1); |
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by (blast_tac (claset() addIs [reachable.Init]) 2); |
|
256 |
by (dtac PPROD_constrains_projection 1); |
|
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by (assume_tac 1); |
|
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by (asm_full_simp_tac |
|
6299 | 259 |
(simpset() addsimps [Applyall_Int_depend, reachable_PPROD_eq]) 1); |
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qed "PPROD_Constrains_imp_Constrains"; |
5899 | 261 |
|
262 |
||
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Goal "[| i: I; finite I; f0: Init (PPROD I F) |] \ |
6536 | 264 |
\ ==> (PPROD I F : (lift_set i A) Co (lift_set i B)) = \ |
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\ (F i : A Co B)"; |
|
5972 | 266 |
by (blast_tac (claset() addIs [Constrains_imp_PPROD_Constrains, |
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PPROD_Constrains_imp_Constrains]) 1); |
|
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qed "PPROD_Constrains"; |
|
269 |
||
270 |
Goal "[| i: I; finite I; f0: Init (PPROD I F) |] \ |
|
6020 | 271 |
\ ==> (PPROD I F : Stable (lift_set i A)) = (F i : Stable A)"; |
5972 | 272 |
by (asm_simp_tac (simpset() delsimps [Init_PPROD] |
273 |
addsimps [Stable_def, PPROD_Constrains]) 1); |
|
5899 | 274 |
qed "PPROD_Stable"; |
275 |
||
276 |
||
5972 | 277 |
(** PFUN (no dependence on i) doesn't require the f0 premise **) |
5899 | 278 |
|
6299 | 279 |
Goal "i: I \ |
280 |
\ ==> Applyall ({f. (ALL i:I. f i : R)} Int lift_set i A) i = R Int A"; |
|
5972 | 281 |
by (force_tac (claset(), simpset() addsimps [Applyall_def]) 1); |
282 |
qed "Applyall_Int"; |
|
283 |
||
6536 | 284 |
Goal "[| (PPI x:I. F) : (lift_set i A) Co (lift_set i B); \ |
5972 | 285 |
\ i: I; finite I |] \ |
6536 | 286 |
\ ==> F : A Co B"; |
6575 | 287 |
by (full_simp_tac (simpset() addsimps [Constrains_eq_constrains]) 1); |
5972 | 288 |
by (dtac PPROD_constrains_projection 1); |
289 |
by (assume_tac 1); |
|
290 |
by (asm_full_simp_tac |
|
291 |
(simpset() addsimps [Applyall_Int, Collect_conj_eq RS sym, |
|
292 |
reachable_PPROD_eq]) 1); |
|
293 |
qed "PFUN_Constrains_imp_Constrains"; |
|
294 |
||
295 |
Goal "[| i: I; finite I |] \ |
|
6536 | 296 |
\ ==> ((PPI x:I. F) : (lift_set i A) Co (lift_set i B)) = \ |
297 |
\ (F : A Co B)"; |
|
5972 | 298 |
by (blast_tac (claset() addIs [Constrains_imp_PPROD_Constrains, |
299 |
PFUN_Constrains_imp_Constrains]) 1); |
|
300 |
qed "PFUN_Constrains"; |
|
301 |
||
302 |
Goal "[| i: I; finite I |] \ |
|
6020 | 303 |
\ ==> ((PPI x:I. F) : Stable (lift_set i A)) = (F : Stable A)"; |
5972 | 304 |
by (asm_simp_tac (simpset() addsimps [Stable_def, PFUN_Constrains]) 1); |
305 |
qed "PFUN_Stable"; |
|
306 |
||
307 |
||
308 |
||
309 |
(*** guarantees properties ***) |
|
310 |
||
311 |
||
312 |
Goal "drop_act i (lift_act i act) = act"; |
|
313 |
by (force_tac (claset() addSIs [rinst [("x", "?ff(i := ?u)")] exI], |
|
314 |
simpset() addsimps [drop_act_def, lift_act_def]) 1); |
|
315 |
qed "lift_act_inverse"; |
|
316 |
Addsimps [lift_act_inverse]; |
|
317 |
||
318 |
||
319 |
Goal "(lift_prog i F) Join G = lift_prog i H \ |
|
320 |
\ ==> EX J. H = F Join J"; |
|
321 |
by (etac program_equalityE 1); |
|
322 |
by (auto_tac (claset(), simpset() addsimps [Acts_lift_prog, Acts_Join])); |
|
323 |
by (res_inst_tac [("x", |
|
324 |
"mk_program(Applyall(Init G) i, drop_act i `` Acts G)")] |
|
325 |
exI 1); |
|
326 |
by (rtac program_equalityI 1); |
|
327 |
(*Init*) |
|
328 |
by (simp_tac (simpset() addsimps [Applyall_def]) 1); |
|
329 |
(*Blast_tac can't do HO unification, needed to invent function states*) |
|
330 |
by (fast_tac (claset() addEs [equalityE]) 1); |
|
331 |
(*Now for the Actions*) |
|
332 |
by (dres_inst_tac [("f", "op `` (drop_act i)")] arg_cong 1); |
|
333 |
by (asm_full_simp_tac |
|
334 |
(simpset() addsimps [insert_absorb, Acts_Join, |
|
335 |
image_Un, image_compose RS sym, o_def]) 1); |
|
336 |
qed "lift_prog_Join_eq_lift_prog_D"; |
|
337 |
||
338 |
||
339 |
Goal "F : X guarantees Y \ |
|
340 |
\ ==> lift_prog i F : (lift_prog i `` X) guarantees (lift_prog i `` Y)"; |
|
341 |
by (rtac guaranteesI 1); |
|
342 |
by Auto_tac; |
|
343 |
by (blast_tac (claset() addDs [lift_prog_Join_eq_lift_prog_D, guaranteesD]) 1); |
|
344 |
qed "lift_prog_guarantees"; |
|
345 |
||
346 |