1 (* Title: ZF/ex/counit.ML |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1993 University of Cambridge |
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5 |
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6 Trivial codatatype definitions, one of which goes wrong! |
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7 |
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8 Need to find sufficient conditions for codatatypes to work correctly! |
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9 *) |
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10 |
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11 (*This degenerate definition does not work well because the one constructor's |
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12 definition is trivial! The same thing occurs with Aczel's Special Final |
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13 Coalgebra Theorem |
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14 *) |
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15 structure CoUnit = CoDatatype_Fun |
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16 (val thy = QUniv.thy; |
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17 val rec_specs = |
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18 [("counit", "quniv(0)", |
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19 [(["Con"], "i=>i")])]; |
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20 val rec_styp = "i"; |
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21 val ext = None |
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22 val sintrs = ["x: counit ==> Con(x) : counit"]; |
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23 val monos = []; |
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24 val type_intrs = codatatype_intrs |
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25 val type_elims = codatatype_elims); |
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26 |
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27 val [ConI] = CoUnit.intrs; |
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28 |
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29 (*USELESS because folding on Con(?xa) == ?xa fails*) |
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30 val ConE = CoUnit.mk_cases CoUnit.con_defs "Con(x) : counit"; |
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31 |
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32 (*Proving freeness results*) |
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33 val Con_iff = CoUnit.mk_free "Con(x)=Con(y) <-> x=y"; |
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34 |
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35 (*Should be a singleton, not everything!*) |
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36 goal CoUnit.thy "counit = quniv(0)"; |
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37 by (rtac (CoUnit.dom_subset RS equalityI) 1); |
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38 by (rtac subsetI 1); |
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39 by (etac CoUnit.coinduct 1); |
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40 by (rtac subset_refl 1); |
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41 by (rewrite_goals_tac CoUnit.con_defs); |
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42 by (fast_tac ZF_cs 1); |
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43 val counit_eq_univ = result(); |
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44 |
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45 |
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46 (*****************************************************************) |
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47 |
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48 (*A similar example, but the constructor is non-degenerate and it works! |
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49 The resulting set is a singleton. |
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50 *) |
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51 |
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52 structure CoUnit2 = CoDatatype_Fun |
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53 (val thy = QUniv.thy; |
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54 val rec_specs = |
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55 [("counit2", "quniv(0)", |
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56 [(["Con2"], "[i,i]=>i")])]; |
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57 val rec_styp = "i"; |
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58 val ext = None |
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59 val sintrs = ["[| x: counit2; y: counit2 |] ==> Con2(x,y) : counit2"]; |
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60 val monos = []; |
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61 val type_intrs = codatatype_intrs |
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62 val type_elims = codatatype_elims); |
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63 |
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64 val [Con2I] = CoUnit2.intrs; |
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65 |
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66 val Con2E = CoUnit2.mk_cases CoUnit2.con_defs "Con2(x,y) : counit2"; |
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67 |
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68 (*Proving freeness results*) |
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69 val Con2_iff = CoUnit2.mk_free "Con2(x,y)=Con2(x',y') <-> x=x' & y=y'"; |
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70 |
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71 goalw CoUnit2.thy CoUnit2.con_defs "bnd_mono(univ(0), %x. Con2(x,x))"; |
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72 by (rtac bnd_monoI 1); |
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73 by (REPEAT (ares_tac [subset_refl, QPair_subset_univ, QPair_mono] 1)); |
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74 val Con2_bnd_mono = result(); |
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75 |
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76 goal CoUnit2.thy "lfp(univ(0), %x. Con2(x,x)) : counit2"; |
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77 by (rtac (singletonI RS CoUnit2.coinduct) 1); |
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78 by (rtac (qunivI RS singleton_subsetI) 1); |
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79 by (rtac ([lfp_subset, empty_subsetI RS univ_mono] MRS subset_trans) 1); |
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80 by (fast_tac (ZF_cs addSIs [Con2_bnd_mono RS lfp_Tarski]) 1); |
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81 val lfp_Con2_in_counit2 = result(); |
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82 |
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83 (*Lemma for proving finality. Borrowed from ex/llist_eq.ML!*) |
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84 goal CoUnit2.thy |
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85 "!!i. Ord(i) ==> ALL x y. x: counit2 & y: counit2 --> x Int Vset(i) <= y"; |
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86 by (etac trans_induct 1); |
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87 by (safe_tac subset_cs); |
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88 by (etac CoUnit2.elim 1); |
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89 by (etac CoUnit2.elim 1); |
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90 by (rewrite_goals_tac CoUnit2.con_defs); |
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91 by (fast_tac lleq_cs 1); |
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92 val counit2_Int_Vset_subset_lemma = result(); |
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93 |
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94 val counit2_Int_Vset_subset = standard |
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95 (counit2_Int_Vset_subset_lemma RS spec RS spec RS mp); |
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96 |
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97 goal CoUnit2.thy "!!x y. [| x: counit2; y: counit2 |] ==> x=y"; |
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98 by (rtac equalityI 1); |
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99 by (REPEAT (ares_tac [conjI, counit2_Int_Vset_subset RS Int_Vset_subset] 1)); |
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100 val counit2_implies_equal = result(); |
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101 |
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102 goal CoUnit2.thy "counit2 = {lfp(univ(0), %x. Con2(x,x))}"; |
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103 by (rtac equalityI 1); |
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104 by (rtac (lfp_Con2_in_counit2 RS singleton_subsetI) 2); |
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105 by (rtac subsetI 1); |
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106 by (dtac (lfp_Con2_in_counit2 RS counit2_implies_equal) 1); |
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107 by (etac subst 1); |
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108 by (rtac singletonI 1); |
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109 val counit2_eq_univ = result(); |
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