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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 co-datatype definitions, one of which goes wrong! |
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7 |
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8 Need to find sufficient conditions for co-datatypes 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! |
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13 *) |
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14 structure CoUnit = Co_Datatype_Fun |
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15 (val thy = QUniv.thy; |
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16 val rec_specs = |
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17 [("counit", "quniv(0)", |
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18 [(["Con"], "i=>i")])]; |
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19 val rec_styp = "i"; |
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20 val ext = None |
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21 val sintrs = ["x: counit ==> Con(x) : counit"]; |
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22 val monos = []; |
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23 val type_intrs = co_datatype_intrs |
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24 val type_elims = co_datatype_elims); |
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25 |
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26 val [ConI] = CoUnit.intrs; |
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27 |
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28 (*USELESS because folding on Con(?xa) == ?xa fails*) |
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29 val ConE = CoUnit.mk_cases CoUnit.con_defs "Con(x) : counit"; |
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30 |
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31 (*Proving freeness results*) |
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32 val Con_iff = CoUnit.mk_free "Con(x)=Con(y) <-> x=y"; |
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33 |
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34 (*Should be a singleton, not everything!*) |
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35 goal CoUnit.thy "counit = quniv(0)"; |
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36 by (rtac (CoUnit.dom_subset RS equalityI) 1); |
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37 by (rtac subsetI 1); |
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38 by (etac CoUnit.co_induct 1); |
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39 by (rtac subset_refl 1); |
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40 by (rewrite_goals_tac CoUnit.con_defs); |
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41 by (fast_tac ZF_cs 1); |
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42 val counit_eq_univ = result(); |
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43 |
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44 |
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45 (*****************************************************************) |
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46 |
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47 (*A similar example, but the constructor is non-degenerate and it works! |
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48 The resulting set is a singleton. |
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49 *) |
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50 |
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51 structure CoUnit2 = Co_Datatype_Fun |
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52 (val thy = QUniv.thy; |
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53 val rec_specs = |
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54 [("counit2", "quniv(0)", |
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55 [(["Con2"], "[i,i]=>i")])]; |
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56 val rec_styp = "i"; |
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57 val ext = None |
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58 val sintrs = ["[| x: counit2; y: counit2 |] ==> Con2(x,y) : counit2"]; |
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59 val monos = []; |
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60 val type_intrs = co_datatype_intrs |
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61 val type_elims = co_datatype_elims); |
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62 |
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63 val [Con2I] = CoUnit2.intrs; |
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64 |
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65 val Con2E = CoUnit2.mk_cases CoUnit2.con_defs "Con2(x,y) : counit2"; |
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66 |
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67 (*Proving freeness results*) |
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68 val Con2_iff = CoUnit2.mk_free "Con2(x,y)=Con2(x',y') <-> x=x' & y=y'"; |
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69 |
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70 goalw CoUnit2.thy CoUnit2.con_defs "bnd_mono(univ(0), %x. Con2(x,x))"; |
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71 by (rtac bnd_monoI 1); |
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72 by (REPEAT (ares_tac [subset_refl, QPair_subset_univ, QPair_mono] 1)); |
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73 val Con2_bnd_mono = result(); |
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74 |
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75 goal CoUnit2.thy "lfp(univ(0), %x. Con2(x,x)) : counit2"; |
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76 by (rtac (singletonI RS CoUnit2.co_induct) 1); |
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77 by (rtac (qunivI RS singleton_subsetI) 1); |
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78 by (rtac ([lfp_subset, empty_subsetI RS univ_mono] MRS subset_trans) 1); |
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79 by (fast_tac (ZF_cs addSIs [Con2_bnd_mono RS lfp_Tarski]) 1); |
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80 val lfp_Con2_in_counit2 = result(); |
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81 |
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82 (*borrowed from ex/llist_eq.ML! the proofs are almost identical!*) |
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83 val lleq_cs = subset_cs |
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84 addSIs [succI1, Int_Vset_0_subset, |
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85 QPair_Int_Vset_succ_subset_trans, |
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86 QPair_Int_Vset_subset_trans]; |
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87 |
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88 goal CoUnit2.thy |
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89 "!!i. Ord(i) ==> ALL x y. x: counit2 & y: counit2 --> x Int Vset(i) <= y"; |
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90 by (etac trans_induct 1); |
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91 by (safe_tac subset_cs); |
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92 by (etac CoUnit2.elim 1); |
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93 by (etac CoUnit2.elim 1); |
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94 by (safe_tac subset_cs); |
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95 by (rewrite_goals_tac CoUnit2.con_defs); |
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96 by (etac Ord_cases 1 THEN REPEAT_FIRST hyp_subst_tac); |
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97 (*0 case*) |
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98 by (fast_tac lleq_cs 1); |
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99 (*succ(j) case*) |
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100 by (fast_tac lleq_cs 1); |
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101 (*Limit(i) case*) |
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102 by (etac (Limit_Vfrom_eq RS ssubst) 1); |
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103 by (rtac (Int_UN_distrib RS ssubst) 1); |
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104 by (fast_tac lleq_cs 1); |
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105 val counit2_Int_Vset_subset_lemma = result(); |
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106 |
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107 val counit2_Int_Vset_subset = standard |
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108 (counit2_Int_Vset_subset_lemma RS spec RS spec RS mp); |
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109 |
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110 goal CoUnit2.thy "!!x y. [| x: counit2; y: counit2 |] ==> x=y"; |
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111 by (rtac equalityI 1); |
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112 by (REPEAT (ares_tac [conjI, counit2_Int_Vset_subset RS Int_Vset_subset] 1)); |
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113 val counit2_implies_equal = result(); |
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114 |
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115 goal CoUnit2.thy "counit2 = {lfp(univ(0), %x. Con2(x,x))}"; |
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116 by (rtac equalityI 1); |
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117 by (rtac (lfp_Con2_in_counit2 RS singleton_subsetI) 2); |
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118 by (rtac subsetI 1); |
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119 by (dtac (lfp_Con2_in_counit2 RS counit2_implies_equal) 1); |
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120 by (etac subst 1); |
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121 by (rtac singletonI 1); |
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122 val counit2_eq_univ = result(); |