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1 (* Title: HOL/IOA/meta_theory/IOA.ML |
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2 ID: $Id$ |
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3 Author: Tobias Nipkow & Konrad Slind |
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4 Copyright 1994 TU Muenchen |
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5 |
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6 The I/O automata of Lynch and Tuttle. |
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7 *) |
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8 |
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9 open IOA Asig; |
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10 |
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11 val ioa_projections = [asig_of_def, starts_of_def, trans_of_def]; |
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12 |
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13 val exec_rws = [executions_def,is_execution_fragment_def]; |
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14 |
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15 goal IOA.thy |
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16 "asig_of(<x,y,z>) = x & starts_of(<x,y,z>) = y & trans_of(<x,y,z>) = z"; |
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17 by (simp_tac (SS addsimps ioa_projections) 1); |
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18 qed "ioa_triple_proj"; |
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19 |
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20 goalw IOA.thy [ioa_def,state_trans_def,actions_def, is_asig_def] |
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21 "!!A. [| IOA(A); <s1,a,s2>:trans_of(A) |] ==> a:actions(asig_of(A))"; |
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22 by (REPEAT(etac conjE 1)); |
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23 by (EVERY1[etac allE, etac impE, atac]); |
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24 by (asm_full_simp_tac SS 1); |
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25 qed "trans_in_actions"; |
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26 |
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27 |
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28 goal IOA.thy "filter_oseq p (filter_oseq p s) = filter_oseq p s"; |
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29 by (simp_tac (SS addsimps [filter_oseq_def]) 1); |
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30 by (rtac ext 1); |
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31 by (Option.option.induct_tac "s(i)" 1); |
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32 by (simp_tac SS 1); |
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33 by (simp_tac (SS setloop (split_tac [expand_if])) 1); |
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34 qed "filter_oseq_idemp"; |
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35 |
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36 goalw IOA.thy [mk_behaviour_def,filter_oseq_def] |
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37 "(mk_behaviour A s n = None) = \ |
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38 \ (s(n)=None | (? a. s(n)=Some(a) & a ~: externals(asig_of(A)))) \ |
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39 \ & \ |
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40 \ (mk_behaviour A s n = Some(a)) = \ |
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41 \ (s(n)=Some(a) & a : externals(asig_of(A)))"; |
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42 by (Option.option.induct_tac "s(n)" 1); |
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43 by (ALLGOALS (simp_tac (SS setloop (split_tac [expand_if])))); |
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44 by (fast_tac HOL_cs 1); |
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45 qed "mk_behaviour_thm"; |
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46 |
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47 goalw IOA.thy [reachable_def] "!!A. s:starts_of(A) ==> reachable A s"; |
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48 by (res_inst_tac [("x","<%i.None,%i.s>")] bexI 1); |
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49 by (simp_tac SS 1); |
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50 by (asm_simp_tac (SS addsimps exec_rws) 1); |
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51 qed "reachable_0"; |
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52 |
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53 goalw IOA.thy (reachable_def::exec_rws) |
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54 "!!A. [| reachable A s; <s,a,t> : trans_of(A) |] ==> reachable A t"; |
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55 by(asm_full_simp_tac SS 1); |
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56 by(safe_tac set_cs); |
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57 by(res_inst_tac [("x","<%i.if i<n then fst ex i \ |
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58 \ else (if i=n then Some a else None), \ |
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59 \ %i.if i<Suc n then snd ex i else t>")] bexI 1); |
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60 by(res_inst_tac [("x","Suc(n)")] exI 1); |
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61 by(simp_tac SS 1); |
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62 by(asm_simp_tac (SS delsimps [less_Suc_eq]) 1); |
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63 by(REPEAT(rtac allI 1)); |
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64 by(res_inst_tac [("m","na"),("n","n")] (make_elim less_linear) 1); |
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65 be disjE 1; |
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66 by(asm_simp_tac SS 1); |
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67 be disjE 1; |
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68 by(asm_simp_tac SS 1); |
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69 by(fast_tac HOL_cs 1); |
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70 by(forward_tac [less_not_sym] 1); |
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71 by(asm_simp_tac (SS addsimps [less_not_refl2]) 1); |
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72 qed "reachable_n"; |
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73 |
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74 val [p1,p2] = goalw IOA.thy [invariant_def] |
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75 "[| !!s. s:starts_of(A) ==> P(s); \ |
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76 \ !!s t a. [|reachable A s; P(s)|] ==> <s,a,t>: trans_of(A) --> P(t) |] \ |
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77 \ ==> invariant A P"; |
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78 by (rewrite_goals_tac(reachable_def::Let_def::exec_rws)); |
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79 by (safe_tac set_cs); |
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80 by (res_inst_tac [("Q","reachable A (snd ex n)")] conjunct1 1); |
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81 by (nat_ind_tac "n" 1); |
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82 by (fast_tac (set_cs addIs [p1,reachable_0]) 1); |
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83 by (eres_inst_tac[("x","n1")]allE 1); |
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84 by (eres_inst_tac[("P","%x.!a.?Q x a"), ("opt","fst ex n1")] optE 1); |
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85 by (asm_simp_tac HOL_ss 1); |
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86 by (safe_tac HOL_cs); |
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87 by (etac (p2 RS mp) 1); |
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88 by (ALLGOALS(fast_tac(set_cs addDs [hd Option.option.inject RS iffD1, |
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89 reachable_n]))); |
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90 qed "invariantI"; |
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91 |
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92 val [p1,p2] = goal IOA.thy |
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93 "[| !!s. s : starts_of(A) ==> P(s); \ |
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94 \ !!s t a. reachable A s ==> P(s) --> <s,a,t>:trans_of(A) --> P(t) \ |
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95 \ |] ==> invariant A P"; |
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96 by (fast_tac (HOL_cs addSIs [invariantI] addSDs [p1,p2]) 1); |
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97 qed "invariantI1"; |
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98 |
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99 val [p1,p2] = goalw IOA.thy [invariant_def] |
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100 "[| invariant A P; reachable A s |] ==> P(s)"; |
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101 br(p2 RS (p1 RS spec RS mp))1; |
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102 qed "invariantE"; |
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103 |
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104 goal IOA.thy |
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105 "actions(asig_comp a b) = actions(a) Un actions(b)"; |
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106 by(simp_tac (prod_ss addsimps |
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107 ([actions_def,asig_comp_def]@asig_projections)) 1); |
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108 by(fast_tac eq_cs 1); |
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109 qed "actions_asig_comp"; |
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110 |
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111 goal IOA.thy |
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112 "starts_of(A || B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}"; |
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113 by(simp_tac (SS addsimps (par_def::ioa_projections)) 1); |
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114 qed "starts_of_par"; |
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115 |
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116 (* Every state in an execution is reachable *) |
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117 goalw IOA.thy [reachable_def] |
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118 "!!A. ex:executions(A) ==> !n. reachable A (snd ex n)"; |
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119 by (fast_tac set_cs 1); |
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120 qed "states_of_exec_reachable"; |
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121 |
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122 |
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123 goal IOA.thy |
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124 "<s,a,t> : trans_of(A || B || C || D) = \ |
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125 \ ((a:actions(asig_of(A)) | a:actions(asig_of(B)) | a:actions(asig_of(C)) | \ |
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126 \ a:actions(asig_of(D))) & \ |
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127 \ (if a:actions(asig_of(A)) then <fst(s),a,fst(t)>:trans_of(A) \ |
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128 \ else fst t=fst s) & \ |
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129 \ (if a:actions(asig_of(B)) then <fst(snd(s)),a,fst(snd(t))>:trans_of(B) \ |
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130 \ else fst(snd(t))=fst(snd(s))) & \ |
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131 \ (if a:actions(asig_of(C)) then \ |
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132 \ <fst(snd(snd(s))),a,fst(snd(snd(t)))>:trans_of(C) \ |
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133 \ else fst(snd(snd(t)))=fst(snd(snd(s)))) & \ |
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134 \ (if a:actions(asig_of(D)) then \ |
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135 \ <snd(snd(snd(s))),a,snd(snd(snd(t)))>:trans_of(D) \ |
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136 \ else snd(snd(snd(t)))=snd(snd(snd(s)))))"; |
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137 by(simp_tac (SS addsimps ([par_def,actions_asig_comp,Pair_fst_snd_eq]@ |
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138 ioa_projections) |
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139 setloop (split_tac [expand_if])) 1); |
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140 qed "trans_of_par4"; |
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141 |
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142 goal IOA.thy "starts_of(restrict ioa acts) = starts_of(ioa) & \ |
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143 \ trans_of(restrict ioa acts) = trans_of(ioa) & \ |
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144 \ reachable (restrict ioa acts) s = reachable ioa s"; |
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145 by(simp_tac (SS addsimps ([is_execution_fragment_def,executions_def, |
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146 reachable_def,restrict_def]@ioa_projections)) 1); |
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147 qed "cancel_restrict"; |
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148 |
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149 goal IOA.thy "asig_of(A || B) = asig_comp (asig_of A) (asig_of B)"; |
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150 by(simp_tac (SS addsimps (par_def::ioa_projections)) 1); |
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151 qed "asig_of_par"; |