author | wenzelm |
Fri, 04 Jun 1999 19:57:31 +0200 | |
changeset 6779 | 2912aff958bd |
parent 5184 | 9b8547a9496a |
child 7499 | 23e090051cb8 |
permissions | -rw-r--r-- |
4530 | 1 |
(* Title: HOL/IOA/IOA.ML |
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ID: $Id$ |
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Author: Tobias Nipkow & Konrad Slind |
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Copyright 1994 TU Muenchen |
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|
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The I/O automata of Lynch and Tuttle. |
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*) |
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|
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Addsimps [Let_def]; |
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|
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open IOA Asig; |
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|
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val ioa_projections = [asig_of_def, starts_of_def, trans_of_def]; |
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|
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val exec_rws = [executions_def,is_execution_fragment_def]; |
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Goal |
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"asig_of((x,y,z)) = x & starts_of((x,y,z)) = y & trans_of((x,y,z)) = z"; |
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by (simp_tac (simpset() addsimps ioa_projections) 1); |
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qed "ioa_triple_proj"; |
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Goalw [ioa_def,state_trans_def,actions_def, is_asig_def] |
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"[| IOA(A); (s1,a,s2):trans_of(A) |] ==> a:actions(asig_of(A))"; |
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by (REPEAT(etac conjE 1)); |
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by (EVERY1[etac allE, etac impE, atac]); |
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by (Asm_full_simp_tac 1); |
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qed "trans_in_actions"; |
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|
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Goal "filter_oseq p (filter_oseq p s) = filter_oseq p s"; |
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by (simp_tac (simpset() addsimps [filter_oseq_def]) 1); |
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by (rtac ext 1); |
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by (exhaust_tac "s(i)" 1); |
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by (Asm_simp_tac 1); |
|
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by (Asm_simp_tac 1); |
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qed "filter_oseq_idemp"; |
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Goalw [mk_trace_def,filter_oseq_def] |
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"(mk_trace A s n = None) = \ |
3346 | 40 |
\ (s(n)=None | (? a. s(n)=Some(a) & a ~: externals(asig_of(A)))) \ |
41 |
\ & \ |
|
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\ (mk_trace A s n = Some(a)) = \ |
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\ (s(n)=Some(a) & a : externals(asig_of(A)))"; |
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by (exhaust_tac "s(n)" 1); |
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by (ALLGOALS Asm_simp_tac); |
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by (Fast_tac 1); |
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qed "mk_trace_thm"; |
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|
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Goalw [reachable_def] "s:starts_of(A) ==> reachable A s"; |
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by (res_inst_tac [("x","(%i. None,%i. s)")] bexI 1); |
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by (Simp_tac 1); |
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by (asm_simp_tac (simpset() addsimps exec_rws) 1); |
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qed "reachable_0"; |
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Goalw (reachable_def::exec_rws) |
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"!!A. [| reachable A s; (s,a,t) : trans_of(A) |] ==> reachable A t"; |
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by (asm_full_simp_tac (simpset() delsimps bex_simps) 1); |
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by (split_all_tac 1); |
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by Safe_tac; |
4828 | 60 |
by (rename_tac "ex1 ex2 n" 1); |
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by (res_inst_tac [("x","(%i. if i<n then ex1 i \ |
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\ else (if i=n then Some a else None), \ |
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\ %i. if i<Suc n then ex2 i else t)")] bexI 1); |
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by (res_inst_tac [("x","Suc n")] exI 1); |
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by (Simp_tac 1); |
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by (Asm_full_simp_tac 1); |
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by (rtac allI 1); |
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by (res_inst_tac [("m","na"),("n","n")] (make_elim less_linear) 1); |
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by (etac disjE 1); |
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by (asm_simp_tac (simpset() addsimps [less_Suc_eq]) 1); |
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by (etac disjE 1); |
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by (Asm_simp_tac 1); |
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by (forward_tac [less_not_sym] 1); |
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by (asm_simp_tac (simpset() addsimps [less_not_refl2,less_Suc_eq]) 1); |
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qed "reachable_n"; |
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|
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val [p1,p2] = goalw IOA.thy [invariant_def] |
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"[| !!s. s:starts_of(A) ==> P(s); \ |
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\ !!s t a. [|reachable A s; P(s)|] ==> (s,a,t): trans_of(A) --> P(t) |] \ |
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\ ==> invariant A P"; |
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by (rewrite_goals_tac(reachable_def::Let_def::exec_rws)); |
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by Safe_tac; |
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by (rename_tac "ex1 ex2 n" 1); |
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by (res_inst_tac [("Q","reachable A (ex2 n)")] conjunct1 1); |
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by (Asm_full_simp_tac 1); |
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by (induct_tac "n" 1); |
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by (fast_tac (claset() addIs [p1,reachable_0]) 1); |
5184 | 88 |
by (eres_inst_tac[("x","na")] allE 1); |
89 |
by (exhaust_tac "ex1 na" 1 THEN ALLGOALS Asm_full_simp_tac); |
|
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by Safe_tac; |
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by (etac (p2 RS mp) 1); |
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by (ALLGOALS(fast_tac(claset() addDs [reachable_n]))); |
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qed "invariantI"; |
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|
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val [p1,p2] = goal IOA.thy |
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"[| !!s. s : starts_of(A) ==> P(s); \ |
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\ !!s t a. reachable A s ==> P(s) --> (s,a,t):trans_of(A) --> P(t) \ |
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\ |] ==> invariant A P"; |
4089 | 99 |
by (fast_tac (claset() addSIs [invariantI] addSDs [p1,p2]) 1); |
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qed "invariantI1"; |
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101 |
|
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val [p1,p2] = goalw IOA.thy [invariant_def] |
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103 |
"[| invariant A P; reachable A s |] ==> P(s)"; |
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104 |
by (rtac (p2 RS (p1 RS spec RS mp)) 1); |
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105 |
qed "invariantE"; |
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106 |
|
5069 | 107 |
Goal |
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108 |
"actions(asig_comp a b) = actions(a) Un actions(b)"; |
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by (simp_tac (simpset() addsimps |
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([actions_def,asig_comp_def]@asig_projections)) 1); |
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111 |
by (Fast_tac 1); |
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112 |
qed "actions_asig_comp"; |
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113 |
|
5069 | 114 |
Goal |
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115 |
"starts_of(A || B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}"; |
4089 | 116 |
by (simp_tac (simpset() addsimps (par_def::ioa_projections)) 1); |
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117 |
qed "starts_of_par"; |
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118 |
|
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119 |
(* Every state in an execution is reachable *) |
5069 | 120 |
Goalw [reachable_def] |
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121 |
"ex:executions(A) ==> !n. reachable A (snd ex n)"; |
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122 |
by (Fast_tac 1); |
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123 |
qed "states_of_exec_reachable"; |
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124 |
|
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125 |
|
5069 | 126 |
Goal |
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127 |
"(s,a,t) : trans_of(A || B || C || D) = \ |
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128 |
\ ((a:actions(asig_of(A)) | a:actions(asig_of(B)) | a:actions(asig_of(C)) | \ |
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129 |
\ a:actions(asig_of(D))) & \ |
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130 |
\ (if a:actions(asig_of(A)) then (fst(s),a,fst(t)):trans_of(A) \ |
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131 |
\ else fst t=fst s) & \ |
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132 |
\ (if a:actions(asig_of(B)) then (fst(snd(s)),a,fst(snd(t))):trans_of(B) \ |
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133 |
\ else fst(snd(t))=fst(snd(s))) & \ |
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\ (if a:actions(asig_of(C)) then \ |
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\ (fst(snd(snd(s))),a,fst(snd(snd(t)))):trans_of(C) \ |
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\ else fst(snd(snd(t)))=fst(snd(snd(s)))) & \ |
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\ (if a:actions(asig_of(D)) then \ |
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\ (snd(snd(snd(s))),a,snd(snd(snd(t)))):trans_of(D) \ |
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\ else snd(snd(snd(t)))=snd(snd(snd(s)))))"; |
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by (simp_tac (simpset() addsimps ([par_def,actions_asig_comp,Pair_fst_snd_eq]@ |
4831 | 141 |
ioa_projections)) 1); |
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qed "trans_of_par4"; |
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|
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Goal "starts_of(restrict ioa acts) = starts_of(ioa) & \ |
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\ trans_of(restrict ioa acts) = trans_of(ioa) & \ |
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\ reachable (restrict ioa acts) s = reachable ioa s"; |
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by (simp_tac (simpset() addsimps ([is_execution_fragment_def,executions_def, |
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reachable_def,restrict_def]@ioa_projections)) 1); |
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qed "cancel_restrict"; |
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|
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Goal "asig_of(A || B) = asig_comp (asig_of A) (asig_of B)"; |
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by (simp_tac (simpset() addsimps (par_def::ioa_projections)) 1); |
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qed "asig_of_par"; |
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154 |
|
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|
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Goal "externals(asig_of(A1||A2)) = \ |
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\ (externals(asig_of(A1)) Un externals(asig_of(A2)))"; |
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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); |
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159 |
by (rtac set_ext 1); |
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160 |
by (Fast_tac 1); |
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qed"externals_of_par"; |
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162 |
|
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Goalw [externals_def,actions_def,compat_ioas_def,compat_asigs_def] |
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"[| compat_ioas A1 A2; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))"; |
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by (Asm_full_simp_tac 1); |
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by (best_tac (claset() addEs [equalityCE]) 1); |
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167 |
qed"ext1_is_not_int2"; |
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168 |
|
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Goalw [externals_def,actions_def,compat_ioas_def,compat_asigs_def] |
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"[| compat_ioas A2 A1 ; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))"; |
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by (Asm_full_simp_tac 1); |
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by (best_tac (claset() addEs [equalityCE]) 1); |
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qed"ext2_is_not_int1"; |
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174 |
|
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val ext1_ext2_is_not_act2 = ext1_is_not_int2 RS int_and_ext_is_act; |
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val ext1_ext2_is_not_act1 = ext2_is_not_int1 RS int_and_ext_is_act; |
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177 |