(* Title: HOLCF/IOA/meta_theory/Traces.ML
ID: $Id$
Author: Olaf M"uller
Copyright 1996 TU Muenchen
Theorems about Executions and Traces of I/O automata in HOLCF.
*)
Delsimps (ex_simps @ all_simps);
val exec_rws = [executions_def,is_exec_frag_def];
(* ----------------------------------------------------------------------------------- *)
section "recursive equations of operators";
(* ---------------------------------------------------------------- *)
(* filter_act *)
(* ---------------------------------------------------------------- *)
goal thy "filter_act`UU = UU";
by (simp_tac (!simpset addsimps [filter_act_def]) 1);
qed"filter_act_UU";
goal thy "filter_act`nil = nil";
by (simp_tac (!simpset addsimps [filter_act_def]) 1);
qed"filter_act_nil";
goal thy "filter_act`(x>>xs) = (fst x) >> filter_act`xs";
by (simp_tac (!simpset addsimps [filter_act_def]) 1);
qed"filter_act_cons";
Addsimps [filter_act_UU,filter_act_nil,filter_act_cons];
(* ---------------------------------------------------------------- *)
(* mk_trace *)
(* ---------------------------------------------------------------- *)
goal thy "mk_trace A`UU=UU";
by (simp_tac (!simpset addsimps [mk_trace_def]) 1);
qed"mk_trace_UU";
goal thy "mk_trace A`nil=nil";
by (simp_tac (!simpset addsimps [mk_trace_def]) 1);
qed"mk_trace_nil";
goal thy "mk_trace A`(at >> xs) = \
\ (if ((fst at):ext A) \
\ then (fst at) >> (mk_trace A`xs) \
\ else mk_trace A`xs)";
by (asm_full_simp_tac (!simpset addsimps [mk_trace_def]) 1);
qed"mk_trace_cons";
Addsimps [mk_trace_UU,mk_trace_nil,mk_trace_cons];
(* ---------------------------------------------------------------- *)
(* is_exec_fragC *)
(* ---------------------------------------------------------------- *)
goal thy "is_exec_fragC A = (LAM ex. (%s. case ex of \
\ nil => TT \
\ | x##xs => (flift1 \
\ (%p. Def ((s,p):trans_of A) andalso (is_exec_fragC A`xs) (snd p)) \
\ `x) \
\ ))";
by (rtac trans 1);
by (rtac fix_eq2 1);
by (rtac is_exec_fragC_def 1);
by (rtac beta_cfun 1);
by (simp_tac (!simpset addsimps [flift1_def]) 1);
qed"is_exec_fragC_unfold";
goal thy "(is_exec_fragC A`UU) s=UU";
by (stac is_exec_fragC_unfold 1);
by (Simp_tac 1);
qed"is_exec_fragC_UU";
goal thy "(is_exec_fragC A`nil) s = TT";
by (stac is_exec_fragC_unfold 1);
by (Simp_tac 1);
qed"is_exec_fragC_nil";
goal thy "(is_exec_fragC A`(pr>>xs)) s = \
\ (Def ((s,pr):trans_of A) \
\ andalso (is_exec_fragC A`xs)(snd pr))";
by (rtac trans 1);
by (stac is_exec_fragC_unfold 1);
by (asm_full_simp_tac (!simpset addsimps [Cons_def,flift1_def]) 1);
by (Simp_tac 1);
qed"is_exec_fragC_cons";
Addsimps [is_exec_fragC_UU,is_exec_fragC_nil,is_exec_fragC_cons];
(* ---------------------------------------------------------------- *)
(* is_exec_frag *)
(* ---------------------------------------------------------------- *)
goal thy "is_exec_frag A (s, UU)";
by (simp_tac (!simpset addsimps [is_exec_frag_def]) 1);
qed"is_exec_frag_UU";
goal thy "is_exec_frag A (s, nil)";
by (simp_tac (!simpset addsimps [is_exec_frag_def]) 1);
qed"is_exec_frag_nil";
goal thy "is_exec_frag A (s, (a,t)>>ex) = \
\ (((s,a,t):trans_of A) & \
\ is_exec_frag A (t, ex))";
by (simp_tac (!simpset addsimps [is_exec_frag_def]) 1);
qed"is_exec_frag_cons";
(* Delsimps [is_exec_fragC_UU,is_exec_fragC_nil,is_exec_fragC_cons]; *)
Addsimps [is_exec_frag_UU,is_exec_frag_nil, is_exec_frag_cons];
(* -------------------------------------------------------------------------------- *)
section "has_trace, mk_trace";
(* alternative definition of has_trace tailored for the refinement proof, as it does not
take the detour of schedules *)
goalw thy [executions_def,mk_trace_def,has_trace_def,schedules_def,has_schedule_def]
"has_trace A b = (? ex:executions A. b = mk_trace A`(snd ex))";
by (safe_tac set_cs);
(* 1 *)
by (res_inst_tac[("x","ex")] bexI 1);
by (stac beta_cfun 1);
by (cont_tacR 1);
by (Simp_tac 1);
by (Asm_simp_tac 1);
(* 2 *)
by (res_inst_tac[("x","filter_act`(snd ex)")] bexI 1);
by (stac beta_cfun 1);
by (cont_tacR 1);
by (Simp_tac 1);
by (safe_tac set_cs);
by (res_inst_tac[("x","ex")] bexI 1);
by (REPEAT (Asm_simp_tac 1));
qed"has_trace_def2";
(* -------------------------------------------------------------------------------- *)
section "signatures and executions, schedules";
(* All executions of A have only actions of A. This is only true because of the
predicate state_trans (part of the predicate IOA): We have no dependent types.
For executions of parallel automata this assumption is not needed, as in par_def
this condition is included once more. (see Lemmas 1.1.1c in CompoExecs for example) *)
goal thy
"!! A. is_trans_of A ==> \
\ ! s. is_exec_frag A (s,xs) --> Forall (%a. a:act A) (filter_act`xs)";
by (pair_induct_tac "xs" [is_exec_frag_def,Forall_def,sforall_def] 1);
(* main case *)
ren "ss a t" 1;
by (safe_tac set_cs);
by (REPEAT (asm_full_simp_tac (!simpset addsimps [is_trans_of_def]) 1));
qed"execfrag_in_sig";
goal thy
"!! A.[| is_trans_of A; x:executions A |] ==> \
\ Forall (%a. a:act A) (filter_act`(snd x))";
by (asm_full_simp_tac (!simpset addsimps [executions_def]) 1);
by (pair_tac "x" 1);
by (rtac (execfrag_in_sig RS spec RS mp) 1);
by (Auto_tac());
qed"exec_in_sig";
goalw thy [schedules_def,has_schedule_def]
"!! A.[| is_trans_of A; x:schedules A |] ==> \
\ Forall (%a. a:act A) x";
by (fast_tac (!claset addSIs [exec_in_sig]) 1);
qed"scheds_in_sig";
(* -------------------------------------------------------------------------------- *)
section "executions are prefix closed";
(* only admissible in y, not if done in x !! *)
goal thy "!x s. is_exec_frag A (s,x) & y<<x --> is_exec_frag A (s,y)";
by (pair_induct_tac "y" [is_exec_frag_def] 1);
by (strip_tac 1);
by (Seq_case_simp_tac "xa" 1);
by (pair_tac "a" 1);
by (Auto_tac());
qed"execfrag_prefixclosed";
bind_thm ("exec_prefixclosed",conjI RS (execfrag_prefixclosed RS spec RS spec RS mp));
(* second prefix notion for Finite x *)
goal thy "! y s. is_exec_frag A (s,x@@y) --> is_exec_frag A (s,x)";
by (pair_induct_tac "x" [is_exec_frag_def] 1);
by (strip_tac 1);
by (Seq_case_simp_tac "s" 1);
by (pair_tac "a" 1);
by (Auto_tac());
qed_spec_mp"exec_prefix2closed";