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