open IOA Asig;
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 (SS addsimps [ioa_projections_def]) 1);
val ioa_triple_proj = result();
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 SS 1);
val trans_in_actions = result();
goal IOA.thy "filter_oseq(p,filter_oseq(p,s)) = filter_oseq(p,s)";
by (simp_tac (SS addsimps [filter_oseq_def]) 1);
by (rtac ext 1);
by (Option.option.induct_tac "s(i)" 1);
by (simp_tac SS 1);
by (simp_tac (SS setloop (split_tac [expand_if])) 1);
val filter_oseq_idemp = result();
goalw IOA.thy [mk_behaviour_def,filter_oseq_def]
"(mk_behaviour(A, s, n) = None) = \
\ (s(n)=None | (? a. s(n)=Some(a) & a ~: externals(asig_of(A)))) \
\ & \
\ (mk_behaviour(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 (SS setloop (split_tac [expand_if]))));
by (fast_tac HOL_cs 1);
val mk_behaviour_thm = result();
goalw IOA.thy [reachable_def] "!!A. s:starts_of(A) ==> reachable(A,s)";
by (res_inst_tac [("x","<(%i.None),(%i.s)>::('action,'a)execution")] bexI 1);
by (simp_tac SS 1);
by (asm_simp_tac (SS addsimps exec_rws) 1);
val reachable_0 = result();
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 SS 1);
by(safe_tac set_cs);
by(res_inst_tac [("x","<%i.if(i<n,fst(ex,i),if(i=n,Some(a),None)), \
\ %i.if(i<Suc(n),snd(ex,i),t)>")] bexI 1);
by(res_inst_tac [("x","Suc(n)")] exI 1);
by(simp_tac SS 1);
by(asm_simp_tac (SS 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 SS 1);
be disjE 1;
by(asm_simp_tac SS 1);
by(fast_tac HOL_cs 1);
by(forward_tac [less_not_sym] 1);
by(asm_simp_tac (SS addsimps [less_not_refl2]) 1);
val reachable_n = result();
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::'action.?Q(x,a)"),
("opt","fst(ex,n1)")] optE 1);
by (asm_simp_tac HOL_ss 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])));
val invariantI = result();
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);
val invariantI1 = result();
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;
val invariantE = result();
goal IOA.thy
"actions(asig_comp(a,b)) = actions(a) Un actions(b)";
by(simp_tac (pair_ss addsimps
[actions_def,asig_projections_def,asig_comp_def]) 1);
by(fast_tac eq_cs 1);
val actions_asig_comp = result();
goal IOA.thy
"starts_of(A || B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}";
by(simp_tac (SS addsimps [par_def,ioa_projections_def]) 1);
val starts_of_par = result();
(* 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);
val states_of_exec_reachable = result();
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)), <fst(s),a,fst(t)>:trans_of(A), fst(t)=fst(s)) & \
\ if(a:actions(asig_of(B)), <fst(snd(s)),a,fst(snd(t))>:trans_of(B), \
\ fst(snd(t))=fst(snd(s))) & \
\ if(a:actions(asig_of(C)), \
\ <fst(snd(snd(s))),a,fst(snd(snd(t)))>:trans_of(C), \
\ fst(snd(snd(t)))=fst(snd(snd(s)))) & \
\ if(a:actions(asig_of(D)), \
\ <snd(snd(snd(s))),a,snd(snd(snd(t)))>:trans_of(D), \
\ snd(snd(snd(t)))=snd(snd(snd(s)))))";
by(simp_tac (SS addsimps [par_def,ioa_projections_def,actions_asig_comp,
Pair_fst_snd_eq (* old pair_eq *)]
setloop (split_tac [expand_if])) 1);
val trans_of_par4 = result();
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 (SS addsimps [ioa_projections_def,is_execution_fragment_def,
executions_def,reachable_def,restrict_def]) 1);
val cancel_restrict = result();
goal IOA.thy "asig_of(A || B) = asig_comp(asig_of(A),asig_of(B))";
by(simp_tac (SS addsimps [ioa_projections_def,par_def]) 1);
val asig_of_par = result();