src/HOL/IOA/IOA.ML
author paulson
Wed Nov 05 13:23:46 1997 +0100 (1997-11-05)
changeset 4153 e534c4c32d54
parent 4089 96fba19bcbe2
child 4530 ac1821645636
permissions -rw-r--r--
Ran expandshort, especially to introduce Safe_tac
     1 (*  Title:      HOL/IOA/meta_theory/IOA.ML
     2     ID:         $Id$
     3     Author:     Tobias Nipkow & Konrad Slind
     4     Copyright   1994  TU Muenchen
     5 
     6 The I/O automata of Lynch and Tuttle.
     7 *)
     8 
     9 Addsimps [Let_def];
    10 
    11 open IOA Asig;
    12 
    13 val ioa_projections = [asig_of_def, starts_of_def, trans_of_def];
    14 
    15 val exec_rws = [executions_def,is_execution_fragment_def];
    16 
    17 goal IOA.thy
    18 "asig_of((x,y,z)) = x & starts_of((x,y,z)) = y & trans_of((x,y,z)) = z";
    19   by (simp_tac (simpset() addsimps ioa_projections) 1);
    20   qed "ioa_triple_proj";
    21 
    22 goalw IOA.thy [ioa_def,state_trans_def,actions_def, is_asig_def]
    23   "!!A. [| IOA(A); (s1,a,s2):trans_of(A) |] ==> a:actions(asig_of(A))";
    24   by (REPEAT(etac conjE 1));
    25   by (EVERY1[etac allE, etac impE, atac]);
    26   by (Asm_full_simp_tac 1);
    27 qed "trans_in_actions";
    28 
    29 
    30 goal IOA.thy "filter_oseq p (filter_oseq p s) = filter_oseq p s";
    31   by (simp_tac (simpset() addsimps [filter_oseq_def]) 1);
    32   by (rtac ext 1);
    33   by (exhaust_tac "s(i)" 1);
    34   by (Asm_simp_tac 1);
    35   by (asm_simp_tac (simpset() addsplits [expand_if]) 1);
    36 qed "filter_oseq_idemp";
    37 
    38 goalw IOA.thy [mk_trace_def,filter_oseq_def]
    39 "(mk_trace A s n = None) =                                        \
    40 \  (s(n)=None | (? a. s(n)=Some(a) & a ~: externals(asig_of(A)))) \
    41 \  &                                                              \
    42 \  (mk_trace A s n = Some(a)) =                                   \
    43 \   (s(n)=Some(a) & a : externals(asig_of(A)))";
    44   by (exhaust_tac "s(n)" 1);
    45   by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if])));
    46   by (Fast_tac 1);
    47 qed "mk_trace_thm";
    48 
    49 goalw IOA.thy [reachable_def] "!!A. s:starts_of(A) ==> reachable A s";
    50   by (res_inst_tac [("x","(%i. None,%i. s)")] bexI 1);
    51   by (Simp_tac 1);
    52   by (asm_simp_tac (simpset() addsimps exec_rws) 1);
    53 qed "reachable_0";
    54 
    55 goalw IOA.thy (reachable_def::exec_rws)
    56 "!!A. [| reachable A s; (s,a,t) : trans_of(A) |] ==> reachable A t";
    57   by (asm_full_simp_tac (simpset() delsimps bex_simps) 1);
    58   by Safe_tac;
    59   by (res_inst_tac [("x","(%i. if i<n then fst ex i                    \
    60 \                            else (if i=n then Some a else None),    \
    61 \                         %i. if i<Suc n then snd ex i else t)")] bexI 1);
    62   by (res_inst_tac [("x","Suc(n)")] exI 1);
    63   by (Simp_tac 1);
    64   by (Asm_simp_tac 1);
    65   by (REPEAT(rtac allI 1));
    66   by (res_inst_tac [("m","na"),("n","n")] (make_elim less_linear) 1);
    67   by (etac disjE 1);
    68   by (asm_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
    69   by (etac disjE 1);
    70   by (Asm_simp_tac 1);
    71   by (forward_tac [less_not_sym] 1);
    72   by (asm_simp_tac (simpset() addsimps [less_not_refl2,less_Suc_eq]) 1);
    73 qed "reachable_n";
    74 
    75 val [p1,p2] = goalw IOA.thy [invariant_def]
    76   "[| !!s. s:starts_of(A) ==> P(s);                                          \
    77 \     !!s t a. [|reachable A s; P(s)|] ==> (s,a,t): trans_of(A) --> P(t) |] \
    78 \  ==> invariant A P";
    79   by (rewrite_goals_tac(reachable_def::Let_def::exec_rws));
    80   by Safe_tac;
    81   by (res_inst_tac [("Q","reachable A (snd ex n)")] conjunct1 1);
    82   by (nat_ind_tac "n" 1);
    83   by (fast_tac (claset() addIs [p1,reachable_0]) 1);
    84   by (eres_inst_tac[("x","n")]allE 1);
    85   by (exhaust_tac "fst ex n" 1 THEN ALLGOALS Asm_full_simp_tac);
    86   by Safe_tac;
    87   by (etac (p2 RS mp) 1);
    88   by (ALLGOALS(fast_tac(claset() addDs [reachable_n])));
    89 qed "invariantI";
    90 
    91 val [p1,p2] = goal IOA.thy
    92  "[| !!s. s : starts_of(A) ==> P(s); \
    93 \   !!s t a. reachable A s ==> P(s) --> (s,a,t):trans_of(A) --> P(t) \
    94 \ |] ==> invariant A P";
    95   by (fast_tac (claset() addSIs [invariantI] addSDs [p1,p2]) 1);
    96 qed "invariantI1";
    97 
    98 val [p1,p2] = goalw IOA.thy [invariant_def]
    99 "[| invariant A P; reachable A s |] ==> P(s)";
   100    by (rtac (p2 RS (p1 RS spec RS mp)) 1);
   101 qed "invariantE";
   102 
   103 goal IOA.thy 
   104 "actions(asig_comp a b) = actions(a) Un actions(b)";
   105   by (simp_tac (simpset() addsimps
   106                ([actions_def,asig_comp_def]@asig_projections)) 1);
   107   by (Fast_tac 1);
   108 qed "actions_asig_comp";
   109 
   110 goal IOA.thy
   111 "starts_of(A || B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}";
   112   by (simp_tac (simpset() addsimps (par_def::ioa_projections)) 1);
   113 qed "starts_of_par";
   114 
   115 (* Every state in an execution is reachable *)
   116 goalw IOA.thy [reachable_def] 
   117 "!!A. ex:executions(A) ==> !n. reachable A (snd ex n)";
   118   by (Fast_tac 1);
   119 qed "states_of_exec_reachable";
   120 
   121 
   122 goal IOA.thy 
   123 "(s,a,t) : trans_of(A || B || C || D) =                                      \
   124 \ ((a:actions(asig_of(A)) | a:actions(asig_of(B)) | a:actions(asig_of(C)) |  \
   125 \   a:actions(asig_of(D))) &                                                 \
   126 \  (if a:actions(asig_of(A)) then (fst(s),a,fst(t)):trans_of(A)              \
   127 \   else fst t=fst s) &                                                      \
   128 \  (if a:actions(asig_of(B)) then (fst(snd(s)),a,fst(snd(t))):trans_of(B)    \
   129 \   else fst(snd(t))=fst(snd(s))) &                                          \
   130 \  (if a:actions(asig_of(C)) then                                            \
   131 \     (fst(snd(snd(s))),a,fst(snd(snd(t)))):trans_of(C)                      \
   132 \   else fst(snd(snd(t)))=fst(snd(snd(s)))) &                                \
   133 \  (if a:actions(asig_of(D)) then                                            \
   134 \     (snd(snd(snd(s))),a,snd(snd(snd(t)))):trans_of(D)                      \
   135 \   else snd(snd(snd(t)))=snd(snd(snd(s)))))";
   136   by (simp_tac (simpset() addsimps ([par_def,actions_asig_comp,Pair_fst_snd_eq]@
   137                             ioa_projections)
   138                   addsplits [expand_if]) 1);
   139 qed "trans_of_par4";
   140 
   141 goal IOA.thy "starts_of(restrict ioa acts) = starts_of(ioa) &     \
   142 \             trans_of(restrict ioa acts) = trans_of(ioa) &       \
   143 \             reachable (restrict ioa acts) s = reachable ioa s";
   144 by (simp_tac (simpset() addsimps ([is_execution_fragment_def,executions_def,
   145                            reachable_def,restrict_def]@ioa_projections)) 1);
   146 qed "cancel_restrict";
   147 
   148 goal IOA.thy "asig_of(A || B) = asig_comp (asig_of A) (asig_of B)";
   149   by (simp_tac (simpset() addsimps (par_def::ioa_projections)) 1);
   150 qed "asig_of_par";
   151 
   152 
   153 goal IOA.thy "externals(asig_of(A1||A2)) =    \
   154 \  (externals(asig_of(A1)) Un externals(asig_of(A2)))";
   155 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);
   156 by (rtac set_ext 1); 
   157 by (Fast_tac 1);
   158 qed"externals_of_par"; 
   159 
   160 goalw IOA.thy [externals_def,actions_def,compat_ioas_def,compat_asigs_def]
   161  "!! a. [| compat_ioas A1 A2; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))";
   162 by (Asm_full_simp_tac 1);
   163 by (best_tac (claset() addEs [equalityCE]) 1);
   164 qed"ext1_is_not_int2";
   165 
   166 goalw IOA.thy [externals_def,actions_def,compat_ioas_def,compat_asigs_def]
   167  "!! a. [| compat_ioas A2 A1 ; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))";
   168 by (Asm_full_simp_tac 1);
   169 by (best_tac (claset() addEs [equalityCE]) 1);
   170 qed"ext2_is_not_int1";
   171 
   172 val ext1_ext2_is_not_act2 = ext1_is_not_int2 RS int_and_ext_is_act;
   173 val ext1_ext2_is_not_act1 = ext2_is_not_int1 RS int_and_ext_is_act;
   174