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(* Title: HOL/IMP/Transition.ML
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ID: $Id$
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Author: Tobias Nipkow & Robert Sandner, TUM
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Copyright 1996 TUM
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Equivalence of Natural and Transition semantics
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*)
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open Transition;
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val relpow_cs = rel_cs addSEs [rel_pow_0_E];
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val evalc1_elim_cases = map (evalc1.mk_cases com.simps)
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["(SKIP,s) -1-> t", "(x:=a,s) -1-> t", "(c1;c2, s) -1-> t",
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"(IF b THEN c1 ELSE c2, s) -1-> t", "(WHILE b DO c,s) -1-> t"];
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val evalc1_cs = relpow_cs addIs (evalc.intrs@evalc1.intrs);
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goal Transition.thy "!!c. (c,s) -(0)-> (SKIP,u) ==> c = SKIP & s = u";
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by(fast_tac evalc1_cs 1);
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val hlemma1 = result();
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goal Transition.thy "!!s. (SKIP,s) -(m)-> (SKIP,t) ==> s = t & m = 0";
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be rel_pow_E2 1;
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by (Asm_full_simp_tac 1);
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by (eresolve_tac evalc1_elim_cases 1);
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val hlemma2 = result();
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goal Transition.thy
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"!s t u c d. (c,s) -(n)-> (SKIP,t) --> (d,t) -*-> (SKIP,u) --> \
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\ (c;d, s) -*-> (SKIP, u)";
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by(nat_ind_tac "n" 1);
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(* case n = 0 *)
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by(fast_tac (evalc1_cs addIs [rtrancl_into_rtrancl2])1);
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(* induction step *)
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by (safe_tac (HOL_cs addSDs [rel_pow_Suc_D2]));
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by(split_all_tac 1);
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by(fast_tac (evalc1_cs addIs [rtrancl_into_rtrancl2]) 1);
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qed_spec_mp "lemma1";
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goal Transition.thy "!c s s1. <c,s> -c-> s1 --> (c,s) -*-> (SKIP,s1)";
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br evalc.mutual_induct 1;
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(* SKIP *)
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br rtrancl_refl 1;
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(* ASSIGN *)
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by (fast_tac (evalc1_cs addSIs [r_into_rtrancl]) 1);
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(* SEMI *)
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by (fast_tac (set_cs addDs [rtrancl_imp_UN_rel_pow] addIs [lemma1]) 1);
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(* IF *)
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by (fast_tac (evalc1_cs addIs [rtrancl_into_rtrancl2]) 1);
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by (fast_tac (evalc1_cs addIs [rtrancl_into_rtrancl2]) 1);
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(* WHILE *)
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by (fast_tac (evalc1_cs addSIs [r_into_rtrancl]) 1);
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by (fast_tac (evalc1_cs addDs [rtrancl_imp_UN_rel_pow]
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addIs [rtrancl_into_rtrancl2,lemma1]) 1);
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qed_spec_mp "evalc_impl_evalc1";
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goal Transition.thy
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"!c d s u. (c;d,s) -(n)-> (SKIP,u) --> \
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\ (? t m. (c,s) -*-> (SKIP,t) & (d,t) -(m)-> (SKIP,u) & m <= n)";
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by(nat_ind_tac "n" 1);
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(* case n = 0 *)
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by (fast_tac (HOL_cs addSDs [hlemma1] addss !simpset) 1);
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(* induction step *)
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by (fast_tac (HOL_cs addSIs [rtrancl_refl,le_SucI,le_refl]
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addSDs [rel_pow_Suc_D2]
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addSEs (evalc1_elim_cases@
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[rel_pow_imp_rtrancl,rtrancl_into_rtrancl2])) 1);
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qed_spec_mp "lemma2";
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goal Transition.thy "!s t. (c,s) -*-> (SKIP,t) --> <c,s> -c-> t";
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by (com.induct_tac "c" 1);
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by (safe_tac (evalc1_cs addSDs [rtrancl_imp_UN_rel_pow]));
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(* SKIP *)
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by (fast_tac (evalc1_cs addSEs rel_pow_E2::evalc1_elim_cases) 1);
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(* ASSIGN *)
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by (fast_tac (evalc1_cs addSDs [hlemma2]
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addSEs rel_pow_E2::evalc1_elim_cases
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addss !simpset) 1);
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(* SEMI *)
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by (fast_tac (evalc1_cs addSDs [lemma2,rel_pow_imp_rtrancl]) 1);
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(* IF *)
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be rel_pow_E2 1;
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by (Asm_full_simp_tac 1);
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by (fast_tac (evalc1_cs addSDs[rel_pow_imp_rtrancl]addEs evalc1_elim_cases) 1);
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(* WHILE, induction on the length of the computation *)
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by(rotate_tac 1 1);
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by (etac rev_mp 1);
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by (res_inst_tac [("x","s")] spec 1);
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by(res_inst_tac [("n","n")] less_induct 1);
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by (strip_tac 1);
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be rel_pow_E2 1;
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by (Asm_full_simp_tac 1);
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by (eresolve_tac evalc1_elim_cases 1);
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(* WhileFalse *)
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by (fast_tac (evalc1_cs addSDs [hlemma2]) 1);
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(* WhileTrue *)
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by(fast_tac(evalc1_cs addSDs[lemma2,le_imp_less_or_eq,less_Suc_eq RS iffD2])1);
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qed_spec_mp "evalc1_impl_evalc";
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(**** proof of the equivalence of evalc and evalc1 ****)
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goal Transition.thy "((c, s) -*-> (SKIP, t)) = (<c,s> -c-> t)";
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by (fast_tac (HOL_cs addSEs [evalc1_impl_evalc,evalc_impl_evalc1]) 1);
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qed "evalc1_eq_evalc";
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