(* Title: HOL/IOA/IOA.ML ID: $Id: IOA.ML,v 1.24 2005/09/06 17:03:39 wenzelm Exp $ Author: Tobias Nipkow & Konrad Slind Copyright 1994 TU Muenchen *) Addsimps [Let_def]; bind_thms ("ioa_projections", [asig_of_def, starts_of_def, trans_of_def]); bind_thms ("exec_rws", [executions_def,is_execution_fragment_def]); Goal "asig_of(x,y,z) = x & starts_of(x,y,z) = y & trans_of(x,y,z) = z"; by (simp_tac (simpset() addsimps ioa_projections) 1); qed "ioa_triple_proj"; Goalw [ioa_def,state_trans_def,actions_def, is_asig_def] "[| IOA(A); (s1,a,s2):trans_of(A) |] ==> a:actions(asig_of(A))"; by (REPEAT(etac conjE 1)); by (EVERY1[etac allE, etac impE, atac]); by (Asm_full_simp_tac 1); qed "trans_in_actions"; Goal "filter_oseq p (filter_oseq p s) = filter_oseq p s"; by (simp_tac (simpset() addsimps [filter_oseq_def]) 1); by (rtac ext 1); by (case_tac "s(i)" 1); by (Asm_simp_tac 1); by (Asm_simp_tac 1); qed "filter_oseq_idemp"; Goalw [mk_trace_def,filter_oseq_def] "(mk_trace A s n = None) = \ \ (s(n)=None | (? a. s(n)=Some(a) & a ~: externals(asig_of(A)))) \ \ & \ \ (mk_trace A s n = Some(a)) = \ \ (s(n)=Some(a) & a : externals(asig_of(A)))"; by (case_tac "s(n)" 1); by (ALLGOALS Asm_simp_tac); by (Fast_tac 1); qed "mk_trace_thm"; Goalw [reachable_def] "s:starts_of(A) ==> reachable A s"; by (res_inst_tac [("x","(%i. None,%i. s)")] bexI 1); by (Simp_tac 1); by (asm_simp_tac (simpset() addsimps exec_rws) 1); qed "reachable_0"; Goalw (reachable_def::exec_rws) "!!A. [| reachable A s; (s,a,t) : trans_of(A) |] ==> reachable A t"; by (asm_full_simp_tac (simpset() delsimps bex_simps) 1); by (split_all_tac 1); by Safe_tac; by (rename_tac "ex1 ex2 n" 1); by (res_inst_tac [("x","(%i. if i<n then ex1 i \ \ else (if i=n then Some a else None), \ \ %i. if i<Suc n then ex2 i else t)")] bexI 1); by (res_inst_tac [("x","Suc n")] exI 1); by (Simp_tac 1); by (Asm_full_simp_tac 1); by (rtac allI 1); by (cut_inst_tac [("m","na"),("n","n")] less_linear 1); by Auto_tac; qed "reachable_n"; val [p1,p2] = goalw (the_context ()) [invariant_def] "[| !!s. s:starts_of(A) ==> P(s); \ \ !!s t a. [|reachable A s; P(s)|] ==> (s,a,t): trans_of(A) --> P(t) |] \ \ ==> invariant A P"; by (rewrite_goals_tac(reachable_def::Let_def::exec_rws)); by Safe_tac; by (rename_tac "ex1 ex2 n" 1); by (res_inst_tac [("Q","reachable A (ex2 n)")] conjunct1 1); by (Asm_full_simp_tac 1); by (induct_tac "n" 1); by (fast_tac (claset() addIs [p1,reachable_0]) 1); by (eres_inst_tac[("x","na")] allE 1); by (case_tac "ex1 na" 1 THEN ALLGOALS Asm_full_simp_tac); by Safe_tac; by (etac (p2 RS mp) 1); by (ALLGOALS(fast_tac(claset() addDs [reachable_n]))); qed "invariantI"; val [p1,p2] = goal (the_context ()) "[| !!s. s : starts_of(A) ==> P(s); \ \ !!s t a. reachable A s ==> P(s) --> (s,a,t):trans_of(A) --> P(t) \ \ |] ==> invariant A P"; by (fast_tac (claset() addSIs [invariantI] addSDs [p1,p2]) 1); qed "invariantI1"; val [p1,p2] = goalw (the_context ()) [invariant_def] "[| invariant A P; reachable A s |] ==> P(s)"; by (rtac (p2 RS (p1 RS spec RS mp)) 1); qed "invariantE"; Goal "actions(asig_comp a b) = actions(a) Un actions(b)"; by (simp_tac (simpset() addsimps ([actions_def,asig_comp_def]@asig_projections)) 1); by (Fast_tac 1); qed "actions_asig_comp"; Goal "starts_of(A || B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}"; by (simp_tac (simpset() addsimps (par_def::ioa_projections)) 1); qed "starts_of_par"; (* Every state in an execution is reachable *) Goalw [reachable_def] "ex:executions(A) ==> !n. reachable A (snd ex n)"; by (Fast_tac 1); qed "states_of_exec_reachable"; Goal "(s,a,t) : trans_of(A || B || C || D) = \ \ ((a:actions(asig_of(A)) | a:actions(asig_of(B)) | a:actions(asig_of(C)) | \ \ a:actions(asig_of(D))) & \ \ (if a:actions(asig_of(A)) then (fst(s),a,fst(t)):trans_of(A) \ \ else fst t=fst s) & \ \ (if a:actions(asig_of(B)) then (fst(snd(s)),a,fst(snd(t))):trans_of(B) \ \ else fst(snd(t))=fst(snd(s))) & \ \ (if a:actions(asig_of(C)) then \ \ (fst(snd(snd(s))),a,fst(snd(snd(t)))):trans_of(C) \ \ else fst(snd(snd(t)))=fst(snd(snd(s)))) & \ \ (if a:actions(asig_of(D)) then \ \ (snd(snd(snd(s))),a,snd(snd(snd(t)))):trans_of(D) \ \ else snd(snd(snd(t)))=snd(snd(snd(s)))))"; (*SLOW*) by (simp_tac (simpset() addsimps ([par_def,actions_asig_comp,Pair_fst_snd_eq] @ ioa_projections)) 1); qed "trans_of_par4"; Goal "starts_of(restrict ioa acts) = starts_of(ioa) & \ \ trans_of(restrict ioa acts) = trans_of(ioa) & \ \ reachable (restrict ioa acts) s = reachable ioa s"; by (simp_tac (simpset() addsimps ([is_execution_fragment_def,executions_def, reachable_def,restrict_def]@ioa_projections)) 1); qed "cancel_restrict"; Goal "asig_of(A || B) = asig_comp (asig_of A) (asig_of B)"; by (simp_tac (simpset() addsimps (par_def::ioa_projections)) 1); qed "asig_of_par"; Goal "externals(asig_of(A1||A2)) = \ \ (externals(asig_of(A1)) Un externals(asig_of(A2)))"; by (asm_full_simp_tac (simpset() addsimps [externals_def,asig_of_par,asig_comp_def,asig_inputs_def,asig_outputs_def,Un_def,set_diff_def]) 1); by (rtac set_ext 1); by (Fast_tac 1); qed"externals_of_par"; Goalw [externals_def,actions_def,compat_ioas_def,compat_asigs_def] "[| compat_ioas A1 A2; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))"; by (Asm_full_simp_tac 1); by (Best_tac 1); qed"ext1_is_not_int2"; Goalw [externals_def,actions_def,compat_ioas_def,compat_asigs_def] "[| compat_ioas A2 A1 ; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))"; by (Asm_full_simp_tac 1); by (Best_tac 1); qed"ext2_is_not_int1"; val ext1_ext2_is_not_act2 = ext1_is_not_int2 RS int_and_ext_is_act; val ext1_ext2_is_not_act1 = ext2_is_not_int1 RS int_and_ext_is_act;