--- a/src/HOLCF/IOA/meta_theory/Traces.ML Sat May 27 21:18:51 2006 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,262 +0,0 @@
-(* Title: HOLCF/IOA/meta_theory/Traces.ML
- ID: $Id$
- Author: Olaf Mueller
-
-Theorems about Executions and Traces of I/O automata in HOLCF.
-*)
-
-(* global changes to simpset() and claset(), see also TLS.ML *)
-Delsimps (ex_simps @ all_simps);
-Delsimps [split_paired_Ex];
-Addsimps [Let_def];
-change_claset (fn cs => cs delSWrapper "split_all_tac");
-
-val exec_rws = [executions_def,is_exec_frag_def];
-
-
-
-(* ----------------------------------------------------------------------------------- *)
-
-section "recursive equations of operators";
-
-
-(* ---------------------------------------------------------------- *)
-(* filter_act *)
-(* ---------------------------------------------------------------- *)
-
-
-Goal "filter_act$UU = UU";
-by (simp_tac (simpset() addsimps [filter_act_def]) 1);
-qed"filter_act_UU";
-
-Goal "filter_act$nil = nil";
-by (simp_tac (simpset() addsimps [filter_act_def]) 1);
-qed"filter_act_nil";
-
-Goal "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 "mk_trace A$UU=UU";
-by (simp_tac (simpset() addsimps [mk_trace_def]) 1);
-qed"mk_trace_UU";
-
-Goal "mk_trace A$nil=nil";
-by (simp_tac (simpset() addsimps [mk_trace_def]) 1);
-qed"mk_trace_nil";
-
-Goal "mk_trace A$(at >> xs) = \
-\ (if ((fst at):ext A) \
-\ then (fst at) >> (mk_trace A$xs) \
-\ else mk_trace A$xs)";
-
-by (asm_full_simp_tac (simpset() addsimps [mk_trace_def]) 1);
-qed"mk_trace_cons";
-
-Addsimps [mk_trace_UU,mk_trace_nil,mk_trace_cons];
-
-(* ---------------------------------------------------------------- *)
-(* is_exec_fragC *)
-(* ---------------------------------------------------------------- *)
-
-
-Goal "is_exec_fragC A = (LAM ex. (%s. case ex of \
-\ nil => TT \
-\ | x##xs => (flift1 \
-\ (%p. Def ((s,p):trans_of A) andalso (is_exec_fragC A$xs) (snd p)) \
-\ $x) \
-\ ))";
-by (rtac trans 1);
-by (rtac fix_eq2 1);
-by (rtac is_exec_fragC_def 1);
-by (rtac beta_cfun 1);
-by (simp_tac (simpset() addsimps [flift1_def]) 1);
-qed"is_exec_fragC_unfold";
-
-Goal "(is_exec_fragC A$UU) s=UU";
-by (stac is_exec_fragC_unfold 1);
-by (Simp_tac 1);
-qed"is_exec_fragC_UU";
-
-Goal "(is_exec_fragC A$nil) s = TT";
-by (stac is_exec_fragC_unfold 1);
-by (Simp_tac 1);
-qed"is_exec_fragC_nil";
-
-Goal "(is_exec_fragC A$(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 [Consq_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 "is_exec_frag A (s, UU)";
-by (simp_tac (simpset() addsimps [is_exec_frag_def]) 1);
-qed"is_exec_frag_UU";
-
-Goal "is_exec_frag A (s, nil)";
-by (simp_tac (simpset() addsimps [is_exec_frag_def]) 1);
-qed"is_exec_frag_nil";
-
-Goal "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 "laststate";
-(* ---------------------------------------------------------------------------- *)
-
-Goal "laststate (s,UU) = s";
-by (simp_tac (simpset() addsimps [laststate_def]) 1);
-qed"laststate_UU";
-
-Goal "laststate (s,nil) = s";
-by (simp_tac (simpset() addsimps [laststate_def]) 1);
-qed"laststate_nil";
-
-Goal "!! ex. Finite ex ==> laststate (s,at>>ex) = laststate (snd at,ex)";
-by (simp_tac (simpset() addsimps [laststate_def]) 1);
-by (case_tac "ex=nil" 1);
-by (Asm_simp_tac 1);
-by (Asm_simp_tac 1);
-by (dtac (Finite_Last1 RS mp) 1);
-by (assume_tac 1);
-by (def_tac 1);
-qed"laststate_cons";
-
-Addsimps [laststate_UU,laststate_nil,laststate_cons];
-
-Goal "!!ex. Finite ex ==> (! s. ? u. laststate (s,ex)=u)";
-by (Seq_Finite_induct_tac 1);
-qed"exists_laststate";
-
-
-(* -------------------------------------------------------------------------------- *)
-
-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 [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$(snd ex)")] bexI 1);
-by (stac beta_cfun 1);
-by (cont_tacR 1);
-by (Simp_tac 1);
-by (safe_tac set_cs);
-by (res_inst_tac[("x","ex")] bexI 1);
-by (REPEAT (Asm_simp_tac 1));
-qed"has_trace_def2";
-
-
-(* -------------------------------------------------------------------------------- *)
-
-section "signatures and executions, schedules";
-
-
-(* All executions of A have only actions of A. This is only true because of the
- predicate state_trans (part of the predicate IOA): We have no dependent types.
- For executions of parallel automata this assumption is not needed, as in par_def
- this condition is included once more. (see Lemmas 1.1.1c in CompoExecs for example) *)
-
-Goal
- "!! A. is_trans_of A ==> \
-\ ! s. is_exec_frag A (s,xs) --> Forall (%a. a:act A) (filter_act$xs)";
-
-by (pair_induct_tac "xs" [is_exec_frag_def,Forall_def,sforall_def] 1);
-(* main case *)
-by (rename_tac "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
- "!! 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 [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";
-
-(*
-
-is ok but needs ForallQFilterP which has to been proven first (is trivial also)
-
-Goalw [traces_def,has_trace_def]
- "!! A.[| x:traces A |] ==> \
-\ Forall (%a. a:act A) x";
- by (safe_tac set_cs );
-by (rtac ForallQFilterP 1);
-by (fast_tac (!claset addSIs [ext_is_act]) 1);
-qed"traces_in_sig";
-*)
-
-
-(* -------------------------------------------------------------------------------- *)
-
-section "executions are prefix closed";
-
-(* only admissible in y, not if done in x !! *)
-Goal "!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 "! 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";
-