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(* Title: HOLCF/IOA/meta_theory/TLS.ML
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ID: $Id$
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Author: Olaf M"uller
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Copyright 1997 TU Muenchen
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Temporal Logic of Steps -- tailored for I/O automata
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*)
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(* ---------------------------------------------------------------- *)
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(* ex2seqC *)
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(* ---------------------------------------------------------------- *)
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goal thy "ex2seqC = (LAM ex. (%s. case ex of \
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\ nil => (s,None,s)>>nil \
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\ | x##xs => (flift1 (%pr. \
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\ (s,Some (fst pr), snd pr)>> (ex2seqC`xs) (snd pr)) \
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\ `x) \
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\ ))";
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by (rtac trans 1);
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by (rtac fix_eq2 1);
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by (rtac ex2seqC_def 1);
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by (rtac beta_cfun 1);
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by (simp_tac (simpset() addsimps [flift1_def]) 1);
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qed"ex2seqC_unfold";
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goal thy "(ex2seqC `UU) s=UU";
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by (stac ex2seqC_unfold 1);
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by (Simp_tac 1);
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qed"ex2seqC_UU";
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goal thy "(ex2seqC `nil) s = (s,None,s)>>nil";
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by (stac ex2seqC_unfold 1);
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by (Simp_tac 1);
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qed"ex2seqC_nil";
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goal thy "(ex2seqC `((a,t)>>xs)) s = \
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\ (s,Some a,t)>> ((ex2seqC`xs) t)";
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by (rtac trans 1);
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by (stac ex2seqC_unfold 1);
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by (asm_full_simp_tac (simpset() addsimps [Cons_def,flift1_def]) 1);
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by (asm_full_simp_tac (simpset() addsimps [Cons_def,flift1_def]) 1);
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qed"ex2seqC_cons";
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Addsimps [ex2seqC_UU,ex2seqC_nil,ex2seqC_cons];
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goal thy "ex2seq (s, UU) = UU";
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by (simp_tac (simpset() addsimps [ex2seq_def]) 1);
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qed"ex2seq_UU";
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goal thy "ex2seq (s, nil) = (s,None,s)>>nil";
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by (simp_tac (simpset() addsimps [ex2seq_def]) 1);
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qed"ex2seq_nil";
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goal thy "ex2seq (s, (a,t)>>ex) = (s,Some a,t) >> ex2seq (t, ex)";
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by (simp_tac (simpset() addsimps [ex2seq_def]) 1);
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qed"ex2seq_cons";
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Delsimps [ex2seqC_UU,ex2seqC_nil,ex2seqC_cons];
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Addsimps [ex2seq_UU,ex2seq_nil, ex2seq_cons];
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(* FIX: Not true for UU, as ex2seq is defined continously !!!!! *)
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goal thy "ex2seq exec ~= UU & ex2seq exec ~= nil";
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goal thy "ex |== [] P .--> P";
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(* ----------------------------------------------------------- *)
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(* Interface TL -- TLS *)
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(* ---------------------------------------------------------- *)
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goalw thy [Init_def,Next_def,temp_sat_def,satisfies_def,IMPLIES_def,AND_def]
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"!! s. (P s) & s-a--A-> t --> (Q t) \
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\ ==> ex |== (Init (%(s,a,t). P s) .& Init (%(s,a,t). s -a--A-> t) \
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\ .--> (Next (Init (%(s,a,t).Q s))))";
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by (asm_full_simp_tac (simpset() setloop split_tac [expand_if]) 1);
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by (pair_tac "ex" 1);
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by (Seq_case_simp_tac "y" 1);
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