src/HOLCF/IOA/meta_theory/Traces.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/IOA/meta_theory/Traces.ML	Wed Apr 30 11:20:15 1997 +0200
@@ -0,0 +1,152 @@
+(*  Title:      HOLCF/IOA/meta_theory/Traces.ML
+    ID:        
+    Author:     Olaf M"uller
+    Copyright   1996  TU Muenchen
+
+Theorems about Executions and Traces of I/O automata in HOLCF.
+*)   
+
+
+
+val exec_rws = [executions_def,is_execution_fragment_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_ex_fr                             *)
+(* ---------------------------------------------------------------- *)
+
+
+goal thy "is_ex_fr A = (LAM ex. (%s. case ex of \
+\      nil => TT \
+\    | x##xs => (flift1 \ 
+\            (%p.Def ((s,p):trans_of A) andalso (is_ex_fr A`xs) (snd p)) \
+\             `x) \
+\   ))";
+by (rtac trans 1);
+br fix_eq2 1;
+br is_ex_fr_def 1;
+br beta_cfun 1;
+by (simp_tac (!simpset addsimps [flift1_def]) 1);
+qed"is_ex_fr_unfold";
+
+goal thy "(is_ex_fr A`UU) s=UU";
+by (stac is_ex_fr_unfold 1);
+by (Simp_tac 1);
+qed"is_ex_fr_UU";
+
+goal thy "(is_ex_fr A`nil) s = TT";
+by (stac is_ex_fr_unfold 1);
+by (Simp_tac 1);
+qed"is_ex_fr_nil";
+
+goal thy "(is_ex_fr A`(pr>>xs)) s = \
+\                        (Def ((s,pr):trans_of A) \
+\                andalso (is_ex_fr A`xs)(snd pr))";
+br trans 1;
+by (stac is_ex_fr_unfold 1);
+by (asm_full_simp_tac (!simpset addsimps [Cons_def,flift1_def]) 1);
+by (Simp_tac 1);
+qed"is_ex_fr_cons";
+
+
+Addsimps [is_ex_fr_UU,is_ex_fr_nil,is_ex_fr_cons];
+
+
+(* ---------------------------------------------------------------- *)
+(*                        is_execution_fragment                     *)
+(* ---------------------------------------------------------------- *)
+
+goal thy "is_execution_fragment A (s, UU)";
+by (simp_tac (!simpset addsimps [is_execution_fragment_def]) 1);
+qed"is_execution_fragment_UU";
+
+goal thy "is_execution_fragment A (s, nil)";
+by (simp_tac (!simpset addsimps [is_execution_fragment_def]) 1);
+qed"is_execution_fragment_nil";
+
+goal thy "is_execution_fragment A (s, (a,t)>>ex) = \
+\                               (((s,a,t):trans_of A) & \
+\                               is_execution_fragment A (t, ex))";
+by (simp_tac (!simpset addsimps [is_execution_fragment_def]) 1);
+qed"is_execution_fragment_cons";
+
+
+(* Delsimps [is_ex_fr_UU,is_ex_fr_nil,is_ex_fr_cons]; *)
+Addsimps [is_execution_fragment_UU,is_execution_fragment_nil, is_execution_fragment_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";
+