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1 (* Title: HOL/IMP/Equiv.ML |
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2 ID: $Id$ |
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3 Author: Heiko Loetzbeyer & Robert Sandner, TUM |
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4 Copyright 1994 TUM |
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5 *) |
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6 |
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7 goal Equiv.thy "!n. (<a,s> -a-> n) = (n = A a s)"; |
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8 by (aexp.induct_tac "a" 1); (* struct. ind. *) |
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9 by (ALLGOALS(simp_tac (HOL_ss addsimps A_simps))); (* rewr. Den. *) |
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10 by (TRYALL (fast_tac (set_cs addSIs (evala.intrs@prems) |
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11 addSEs evala_elim_cases))); |
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12 bind_thm("aexp_iff", result() RS spec); |
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13 |
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14 goal Equiv.thy "!w. (<b,s> -b-> w) = (w = B b s)"; |
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15 by (bexp.induct_tac "b" 1); |
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16 by (ALLGOALS(asm_simp_tac (HOL_ss addcongs [conj_cong] |
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17 addsimps (aexp_iff::B_simps@evalb_simps)))); |
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18 bind_thm("bexp_iff", result() RS spec); |
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19 |
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20 val bexp1 = bexp_iff RS iffD1; |
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21 val bexp2 = bexp_iff RS iffD2; |
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22 |
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23 val BfstI = read_instantiate_sg (sign_of Equiv.thy) |
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24 [("P","%x.B ?b x")] (fst_conv RS ssubst); |
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25 val BfstD = read_instantiate_sg (sign_of Equiv.thy) |
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26 [("P","%x.B ?b x")] (fst_conv RS subst); |
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27 |
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28 goal Equiv.thy "!!c. <c,s> -c-> t ==> <s,t> : C(c)"; |
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29 |
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30 (* start with rule induction *) |
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31 be (evalc.mutual_induct RS spec RS spec RS spec RSN (2,rev_mp)) 1; |
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32 by (rewrite_tac (Gamma_def::C_simps)); |
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33 (* simplification with HOL_ss again too agressive *) |
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34 (* skip *) |
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35 by (fast_tac comp_cs 1); |
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36 (* assign *) |
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37 by (asm_full_simp_tac (prod_ss addsimps [aexp_iff]) 1); |
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38 (* comp *) |
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39 by (fast_tac comp_cs 1); |
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40 (* if *) |
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41 by(fast_tac (set_cs addSIs [BfstI] addSDs [BfstD,bexp1]) 1); |
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42 by(fast_tac (set_cs addSIs [BfstI] addSDs [BfstD,bexp1]) 1); |
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43 (* while *) |
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44 by (rtac (rewrite_rule [Gamma_def] (Gamma_mono RS lfp_Tarski RS ssubst)) 1); |
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45 by (fast_tac (comp_cs addSIs [bexp1,BfstI] addSDs [BfstD,bexp1]) 1); |
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46 by (rtac (rewrite_rule [Gamma_def] (Gamma_mono RS lfp_Tarski RS ssubst)) 1); |
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47 by (fast_tac (comp_cs addSIs [bexp1,BfstI] addSDs [BfstD,bexp1]) 1); |
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48 |
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49 qed "com1"; |
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50 |
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51 |
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52 val com_ss = prod_ss addsimps (aexp_iff::bexp_iff::evalc.intrs); |
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53 |
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54 goal Equiv.thy "!io:C(c). <c,fst(io)> -c-> snd(io)"; |
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55 by (com.induct_tac "c" 1); |
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56 by (rewrite_tac C_simps); |
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57 by (safe_tac comp_cs); |
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58 by (ALLGOALS (asm_full_simp_tac com_ss)); |
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59 |
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60 (* comp *) |
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61 by (REPEAT (EVERY [(dtac bspec 1),(atac 1)])); |
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62 by (asm_full_simp_tac com_ss 1); |
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63 |
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64 (* while *) |
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65 by (etac (Gamma_mono RSN (2,induct)) 1); |
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66 by (rewrite_goals_tac [Gamma_def]); |
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67 by (safe_tac comp_cs); |
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68 by (EVERY1 [dtac bspec, atac]); |
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69 by (ALLGOALS (asm_full_simp_tac com_ss)); |
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70 |
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71 qed "com2"; |
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72 |
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73 |
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74 (**** Proof of Equivalence ****) |
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75 |
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76 val com_iff_cs = set_cs addEs [com2 RS bspec] addDs [com1]; |
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77 |
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78 goal Equiv.thy "C(c) = {io . <c,fst(io)> -c-> snd(io)}"; |
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79 by (rtac equalityI 1); |
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80 (* => *) |
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81 by (fast_tac com_iff_cs 1); |
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82 (* <= *) |
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83 by (REPEAT (step_tac com_iff_cs 1)); |
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84 by (asm_full_simp_tac (prod_ss addsimps [surjective_pairing RS sym])1); |
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85 qed "com_equivalence"; |