(* Title: HOL/IOA/meta_theory/IOA.ML
ID: $Id$
Author: Tobias Nipkow & Konrad Slind
Copyright 1994 TU Muenchen
The I/O automata of Lynch and Tuttle.
*)
open IOA Asig;
val ioa_projections = [asig_of_def, starts_of_def, trans_of_def];
val exec_rws = [executions_def,is_execution_fragment_def];
goal IOA.thy
"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.thy [ioa_def,state_trans_def,actions_def, is_asig_def]
"!!A. [| 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 IOA.thy "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 (Option.option.induct_tac "s(i)" 1);
by (Simp_tac 1);
by (simp_tac (!simpset setloop (split_tac [expand_if])) 1);
qed "filter_oseq_idemp";
goalw IOA.thy [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 (Option.option.induct_tac "s(n)" 1);
by (ALLGOALS (simp_tac (!simpset setloop (split_tac [expand_if]))));
by (fast_tac HOL_cs 1);
qed "mk_trace_thm";
goalw IOA.thy [reachable_def] "!!A. 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 IOA.thy (reachable_def::exec_rws)
"!!A. [| reachable A s; (s,a,t) : trans_of(A) |] ==> reachable A t";
by(Asm_full_simp_tac 1);
by(safe_tac set_cs);
by(res_inst_tac [("x","(%i.if i<n then fst ex i \
\ else (if i=n then Some a else None), \
\ %i.if i<Suc n then snd ex i else t)")] bexI 1);
by(res_inst_tac [("x","Suc(n)")] exI 1);
by(Simp_tac 1);
by(asm_simp_tac (!simpset (*delsimps [less_Suc_eq]*)) 1);
by(REPEAT(rtac allI 1));
by(res_inst_tac [("m","na"),("n","n")] (make_elim less_linear) 1);
be disjE 1;
by(asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
be disjE 1;
by(Asm_simp_tac 1);
by(fast_tac HOL_cs 1);
by(forward_tac [less_not_sym] 1);
by(asm_simp_tac (!simpset addsimps [less_not_refl2,less_Suc_eq]) 1);
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 set_cs);
by (res_inst_tac [("Q","reachable A (snd ex n)")] conjunct1 1);
by (nat_ind_tac "n" 1);
by (fast_tac (set_cs addIs [p1,reachable_0]) 1);
by (eres_inst_tac[("x","n1")]allE 1);
by (eres_inst_tac[("P","%x.!a.?Q x a"), ("opt","fst ex n1")] optE 1);
by (Asm_simp_tac 1);
by (safe_tac HOL_cs);
by (etac (p2 RS mp) 1);
by (ALLGOALS(fast_tac(set_cs addDs [hd Option.option.inject RS iffD1,
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 (HOL_cs addSIs [invariantI] addSDs [p1,p2]) 1);
qed "invariantI1";
val [p1,p2] = goalw IOA.thy [invariant_def]
"[| invariant A P; reachable A s |] ==> P(s)";
br(p2 RS (p1 RS spec RS mp))1;
qed "invariantE";
goal IOA.thy
"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 IOA.thy
"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 IOA.thy [reachable_def]
"!!A. ex:executions(A) ==> !n. reachable A (snd ex n)";
by (fast_tac set_cs 1);
qed "states_of_exec_reachable";
goal IOA.thy
"(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)))))";
by(simp_tac (!simpset addsimps ([par_def,actions_asig_comp,Pair_fst_snd_eq]@
ioa_projections)
setloop (split_tac [expand_if])) 1);
qed "trans_of_par4";
goal IOA.thy "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 IOA.thy "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 IOA.thy "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 set_cs 1);
qed"externals_of_par";
goalw IOA.thy [externals_def,actions_def,compat_ioas_def,compat_asigs_def]
"!! a. [| compat_ioas A1 A2; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))";
by (Asm_full_simp_tac 1);
by (best_tac (set_cs addEs [equalityCE]) 1);
qed"ext1_is_not_int2";
goalw IOA.thy [externals_def,actions_def,compat_ioas_def,compat_asigs_def]
"!! a. [| compat_ioas A2 A1 ; a:externals(asig_of(A1))|] ==> a~:internals(asig_of(A2))";
by (Asm_full_simp_tac 1);
by (best_tac (set_cs 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;