src/HOL/IOA/IOA.ML

converted theory Transitive_Closure;

(*  Title:      HOL/IOA/IOA.ML
    ID:         $Id$
    Author:     Tobias Nipkow & Konrad Slind
    Copyright   1994  TU Muenchen

The I/O automata of Lynch and Tuttle.
*)

Addsimps [Let_def];

val ioa_projections = [asig_of_def, starts_of_def, trans_of_def];

val 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 IOA.thy [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 IOA.thy
 "[| !!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 IOA.thy [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 (claset() addEs [equalityCE]) 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 (claset() addEs [equalityCE]) 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;