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1 (* Title: HOL/ex/Simult.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 Primitives for simultaneous recursive type definitions |
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7 includes worked example of trees & forests |
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8 |
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9 This is essentially the same data structure that on ex/term.ML, which is |
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10 simpler because it uses list as a new type former. The approach in this |
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11 file may be superior for other simultaneous recursions. |
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12 *) |
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13 |
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14 open Simult; |
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15 |
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16 (*** Monotonicity and unfolding of the function ***) |
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17 |
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18 goal Simult.thy "mono(%Z. A <*> Part Z In1 \ |
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19 \ <+> ({Numb(0)} <+> Part Z In0 <*> Part Z In1))"; |
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20 by (REPEAT (ares_tac [monoI, subset_refl, usum_mono, uprod_mono, |
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21 Part_mono] 1)); |
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22 qed "TF_fun_mono"; |
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23 |
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24 val TF_unfold = TF_fun_mono RS (TF_def RS def_lfp_Tarski); |
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25 |
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26 goalw Simult.thy [TF_def] "!!A B. A<=B ==> TF(A) <= TF(B)"; |
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27 by (REPEAT (ares_tac [lfp_mono, subset_refl, usum_mono, uprod_mono] 1)); |
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28 qed "TF_mono"; |
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29 |
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30 goalw Simult.thy [TF_def] "TF(sexp) <= sexp"; |
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31 by (rtac lfp_lowerbound 1); |
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32 by (blast_tac (!claset addIs sexp.intrs@[sexp_In0I, sexp_In1I] |
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33 addSEs [PartE]) 1); |
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34 qed "TF_sexp"; |
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35 |
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36 (* A <= sexp ==> TF(A) <= sexp *) |
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37 val TF_subset_sexp = standard |
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38 (TF_mono RS (TF_sexp RSN (2,subset_trans))); |
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39 |
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40 |
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41 (** Elimination -- structural induction on the set TF **) |
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42 |
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43 val TF_Rep_defs = [TCONS_def,FNIL_def,FCONS_def,NIL_def,CONS_def]; |
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44 |
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45 val major::prems = goalw Simult.thy TF_Rep_defs |
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46 "[| i: TF(A); \ |
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47 \ !!M N. [| M: A; N: Part (TF A) In1; R(N) |] ==> R(TCONS M N); \ |
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48 \ R(FNIL); \ |
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49 \ !!M N. [| M: Part (TF A) In0; N: Part (TF A) In1; R(M); R(N) \ |
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50 \ |] ==> R(FCONS M N) \ |
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51 \ |] ==> R(i)"; |
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52 by (rtac ([TF_def, TF_fun_mono, major] MRS def_induct) 1); |
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53 by (blast_tac (!claset addIs (prems@[PartI]) |
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54 addEs [usumE, uprodE, PartE]) 1); |
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55 qed "TF_induct"; |
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56 |
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57 (*This lemma replaces a use of subgoal_tac to prove tree_forest_induct*) |
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58 goalw Simult.thy [Part_def] |
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59 "!!A. ! M: TF(A). (M: Part (TF A) In0 --> P(M)) & \ |
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60 \ (M: Part (TF A) In1 --> Q(M)) \ |
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61 \ ==> (! M: Part (TF A) In0. P(M)) & (! M: Part (TF A) In1. Q(M))"; |
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62 by (Blast_tac 1); |
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63 qed "TF_induct_lemma"; |
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64 |
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65 (*Could prove ~ TCONS M N : Part (TF A) In1 etc. *) |
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66 |
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67 (*Induction on TF with separate predicates P, Q*) |
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68 val prems = goalw Simult.thy TF_Rep_defs |
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69 "[| !!M N. [| M: A; N: Part (TF A) In1; Q(N) |] ==> P(TCONS M N); \ |
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70 \ Q(FNIL); \ |
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71 \ !!M N. [| M: Part (TF A) In0; N: Part (TF A) In1; P(M); Q(N) \ |
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72 \ |] ==> Q(FCONS M N) \ |
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73 \ |] ==> (! M: Part (TF A) In0. P(M)) & (! N: Part (TF A) In1. Q(N))"; |
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74 by (rtac (ballI RS TF_induct_lemma) 1); |
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75 by (etac TF_induct 1); |
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76 by (rewrite_goals_tac TF_Rep_defs); |
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77 (*Blast_tac needs this type instantiation in order to preserve the |
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78 right overloading of equality. The injectiveness properties for |
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79 type 'a item hold only for set types.*) |
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80 val PartE' = read_instantiate [("'a", "?'c set")] PartE; |
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81 by (ALLGOALS (blast_tac (!claset addSEs [PartE'] addIs prems))); |
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82 qed "Tree_Forest_induct"; |
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83 |
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84 (*Induction for the abstract types 'a tree, 'a forest*) |
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85 val prems = goalw Simult.thy [Tcons_def,Fnil_def,Fcons_def] |
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86 "[| !!x ts. Q(ts) ==> P(Tcons x ts); \ |
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87 \ Q(Fnil); \ |
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88 \ !!t ts. [| P(t); Q(ts) |] ==> Q(Fcons t ts) \ |
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89 \ |] ==> (! t. P(t)) & (! ts. Q(ts))"; |
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90 by (res_inst_tac [("P1","%z.P(Abs_Tree(z))"), |
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91 ("Q1","%z.Q(Abs_Forest(z))")] |
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92 (Tree_Forest_induct RS conjE) 1); |
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93 (*Instantiates ?A1 to range(Leaf). *) |
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94 by (fast_tac (!claset addSEs [Rep_Tree_inverse RS subst, |
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95 Rep_Forest_inverse RS subst] |
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96 addSIs [Rep_Tree,Rep_Forest]) 4); |
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97 (*Cannot use simplifier: the rewrites work in the wrong direction!*) |
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98 by (ALLGOALS (fast_tac (!claset addSEs [Abs_Tree_inverse RS subst, |
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99 Abs_Forest_inverse RS subst] |
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100 addSIs prems))); |
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101 qed "tree_forest_induct"; |
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102 |
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103 |
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104 |
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105 (*** Isomorphisms ***) |
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106 |
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107 goal Simult.thy "inj(Rep_Tree)"; |
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108 by (rtac inj_inverseI 1); |
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109 by (rtac Rep_Tree_inverse 1); |
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110 qed "inj_Rep_Tree"; |
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111 |
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112 goal Simult.thy "inj_onto Abs_Tree (Part (TF(range Leaf)) In0)"; |
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113 by (rtac inj_onto_inverseI 1); |
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114 by (etac Abs_Tree_inverse 1); |
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115 qed "inj_onto_Abs_Tree"; |
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116 |
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117 goal Simult.thy "inj(Rep_Forest)"; |
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118 by (rtac inj_inverseI 1); |
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119 by (rtac Rep_Forest_inverse 1); |
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120 qed "inj_Rep_Forest"; |
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121 |
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122 goal Simult.thy "inj_onto Abs_Forest (Part (TF(range Leaf)) In1)"; |
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123 by (rtac inj_onto_inverseI 1); |
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124 by (etac Abs_Forest_inverse 1); |
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125 qed "inj_onto_Abs_Forest"; |
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126 |
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127 (** Introduction rules for constructors **) |
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128 |
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129 (* c : A <*> Part (TF A) In1 |
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130 <+> {Numb(0)} <+> Part (TF A) In0 <*> Part (TF A) In1 ==> c : TF(A) *) |
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131 val TF_I = TF_unfold RS equalityD2 RS subsetD; |
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132 |
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133 (*For reasoning about the representation*) |
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134 AddIs [TF_I, uprodI, usum_In0I, usum_In1I]; |
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135 AddSEs [Scons_inject]; |
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136 |
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137 goalw Simult.thy TF_Rep_defs |
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138 "!!A. [| a: A; M: Part (TF A) In1 |] ==> TCONS a M : Part (TF A) In0"; |
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139 by (Blast_tac 1); |
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140 qed "TCONS_I"; |
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141 |
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142 (* FNIL is a TF(A) -- this also justifies the type definition*) |
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143 goalw Simult.thy TF_Rep_defs "FNIL: Part (TF A) In1"; |
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144 by (Blast_tac 1); |
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145 qed "FNIL_I"; |
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146 |
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147 goalw Simult.thy TF_Rep_defs |
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148 "!!A. [| M: Part (TF A) In0; N: Part (TF A) In1 |] ==> \ |
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149 \ FCONS M N : Part (TF A) In1"; |
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150 by (Blast_tac 1); |
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151 qed "FCONS_I"; |
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152 |
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153 (** Injectiveness of TCONS and FCONS **) |
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154 |
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155 goalw Simult.thy TF_Rep_defs "(TCONS K M=TCONS L N) = (K=L & M=N)"; |
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156 by (Blast_tac 1); |
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157 qed "TCONS_TCONS_eq"; |
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158 bind_thm ("TCONS_inject", (TCONS_TCONS_eq RS iffD1 RS conjE)); |
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159 |
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160 goalw Simult.thy TF_Rep_defs "(FCONS K M=FCONS L N) = (K=L & M=N)"; |
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161 by (Blast_tac 1); |
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162 qed "FCONS_FCONS_eq"; |
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163 bind_thm ("FCONS_inject", (FCONS_FCONS_eq RS iffD1 RS conjE)); |
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164 |
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165 (** Distinctness of TCONS, FNIL and FCONS **) |
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166 |
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167 goalw Simult.thy TF_Rep_defs "TCONS M N ~= FNIL"; |
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168 by (Blast_tac 1); |
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169 qed "TCONS_not_FNIL"; |
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170 bind_thm ("FNIL_not_TCONS", (TCONS_not_FNIL RS not_sym)); |
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171 |
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172 bind_thm ("TCONS_neq_FNIL", (TCONS_not_FNIL RS notE)); |
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173 val FNIL_neq_TCONS = sym RS TCONS_neq_FNIL; |
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174 |
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175 goalw Simult.thy TF_Rep_defs "FCONS M N ~= FNIL"; |
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176 by (Blast_tac 1); |
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177 qed "FCONS_not_FNIL"; |
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178 bind_thm ("FNIL_not_FCONS", (FCONS_not_FNIL RS not_sym)); |
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179 |
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180 bind_thm ("FCONS_neq_FNIL", (FCONS_not_FNIL RS notE)); |
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181 val FNIL_neq_FCONS = sym RS FCONS_neq_FNIL; |
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182 |
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183 goalw Simult.thy TF_Rep_defs "TCONS M N ~= FCONS K L"; |
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184 by (Blast_tac 1); |
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185 qed "TCONS_not_FCONS"; |
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186 bind_thm ("FCONS_not_TCONS", (TCONS_not_FCONS RS not_sym)); |
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187 |
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188 bind_thm ("TCONS_neq_FCONS", (TCONS_not_FCONS RS notE)); |
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189 val FCONS_neq_TCONS = sym RS TCONS_neq_FCONS; |
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190 |
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191 (*???? Too many derived rules ???? |
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192 Automatically generate symmetric forms? Always expand TF_Rep_defs? *) |
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193 |
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194 (** Injectiveness of Tcons and Fcons **) |
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195 |
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196 (*For reasoning about abstract constructors*) |
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197 AddSIs [Rep_Tree, Rep_Forest, TCONS_I, FNIL_I, FCONS_I]; |
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198 AddSEs [TCONS_inject, FCONS_inject, |
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199 TCONS_neq_FNIL, FNIL_neq_TCONS, |
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200 FCONS_neq_FNIL, FNIL_neq_FCONS, |
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201 TCONS_neq_FCONS, FCONS_neq_TCONS]; |
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202 AddSDs [inj_onto_Abs_Tree RS inj_ontoD, |
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203 inj_onto_Abs_Forest RS inj_ontoD, |
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204 inj_Rep_Tree RS injD, inj_Rep_Forest RS injD, |
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205 Leaf_inject]; |
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206 |
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207 goalw Simult.thy [Tcons_def] "(Tcons x xs=Tcons y ys) = (x=y & xs=ys)"; |
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208 by (Blast_tac 1); |
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209 qed "Tcons_Tcons_eq"; |
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210 bind_thm ("Tcons_inject", (Tcons_Tcons_eq RS iffD1 RS conjE)); |
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211 |
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212 goalw Simult.thy [Fcons_def,Fnil_def] "Fcons x xs ~= Fnil"; |
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213 by (Blast_tac 1); |
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214 qed "Fcons_not_Fnil"; |
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215 |
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216 bind_thm ("Fcons_neq_Fnil", Fcons_not_Fnil RS notE); |
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217 val Fnil_neq_Fcons = sym RS Fcons_neq_Fnil; |
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218 |
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219 |
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220 (** Injectiveness of Fcons **) |
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221 |
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222 goalw Simult.thy [Fcons_def] "(Fcons x xs=Fcons y ys) = (x=y & xs=ys)"; |
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223 by (Blast_tac 1); |
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224 qed "Fcons_Fcons_eq"; |
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225 bind_thm ("Fcons_inject", Fcons_Fcons_eq RS iffD1 RS conjE); |
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226 |
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227 |
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228 (*** TF_rec -- by wf recursion on pred_sexp ***) |
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229 |
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230 goal Simult.thy |
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231 "(%M. TF_rec M b c d) = wfrec (trancl pred_sexp) \ |
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232 \ (%g. Case (Split(%x y. b x y (g y))) \ |
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233 \ (List_case c (%x y. d x y (g x) (g y))))"; |
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234 by (simp_tac (HOL_ss addsimps [TF_rec_def]) 1); |
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235 val TF_rec_unfold = (wf_pred_sexp RS wf_trancl) RS |
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236 ((result() RS eq_reflection) RS def_wfrec); |
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237 |
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238 (*--------------------------------------------------------------------------- |
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239 * Old: |
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240 * val TF_rec_unfold = |
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241 * wf_pred_sexp RS wf_trancl RS (TF_rec_def RS def_wfrec); |
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242 *---------------------------------------------------------------------------*) |
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243 |
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244 (** conversion rules for TF_rec **) |
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245 |
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246 goalw Simult.thy [TCONS_def] |
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247 "!!M N. [| M: sexp; N: sexp |] ==> \ |
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248 \ TF_rec (TCONS M N) b c d = b M N (TF_rec N b c d)"; |
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249 by (rtac (TF_rec_unfold RS trans) 1); |
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250 by (simp_tac (!simpset addsimps [Case_In0, Split]) 1); |
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251 by (asm_simp_tac (!simpset addsimps [In0_def]) 1); |
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252 qed "TF_rec_TCONS"; |
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253 |
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254 goalw Simult.thy [FNIL_def] "TF_rec FNIL b c d = c"; |
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255 by (rtac (TF_rec_unfold RS trans) 1); |
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256 by (simp_tac (HOL_ss addsimps [Case_In1, List_case_NIL]) 1); |
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257 qed "TF_rec_FNIL"; |
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258 |
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259 goalw Simult.thy [FCONS_def] |
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260 "!!M N. [| M: sexp; N: sexp |] ==> \ |
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261 \ TF_rec (FCONS M N) b c d = d M N (TF_rec M b c d) (TF_rec N b c d)"; |
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262 by (rtac (TF_rec_unfold RS trans) 1); |
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263 by (simp_tac (HOL_ss addsimps [Case_In1, List_case_CONS]) 1); |
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264 by (asm_simp_tac (!simpset addsimps [CONS_def,In1_def]) 1); |
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265 qed "TF_rec_FCONS"; |
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266 |
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267 |
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268 (*** tree_rec, forest_rec -- by TF_rec ***) |
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269 |
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270 val Rep_Tree_in_sexp = |
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271 [range_Leaf_subset_sexp RS TF_subset_sexp RS (Part_subset RS subset_trans), |
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272 Rep_Tree] MRS subsetD; |
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273 val Rep_Forest_in_sexp = |
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274 [range_Leaf_subset_sexp RS TF_subset_sexp RS (Part_subset RS subset_trans), |
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275 Rep_Forest] MRS subsetD; |
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276 |
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277 val tf_rec_ss = HOL_ss addsimps |
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278 [TF_rec_TCONS, TF_rec_FNIL, TF_rec_FCONS, |
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279 TCONS_I, FNIL_I, FCONS_I, Rep_Tree, Rep_Forest, |
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280 Rep_Tree_inverse, Rep_Forest_inverse, |
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281 Abs_Tree_inverse, Abs_Forest_inverse, |
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282 inj_Leaf, inv_f_f, sexp.LeafI, range_eqI, |
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283 Rep_Tree_in_sexp, Rep_Forest_in_sexp]; |
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284 |
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285 goalw Simult.thy [tree_rec_def, forest_rec_def, Tcons_def] |
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286 "tree_rec (Tcons a tf) b c d = b a tf (forest_rec tf b c d)"; |
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287 by (simp_tac tf_rec_ss 1); |
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288 qed "tree_rec_Tcons"; |
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289 |
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290 goalw Simult.thy [forest_rec_def, Fnil_def] "forest_rec Fnil b c d = c"; |
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291 by (simp_tac tf_rec_ss 1); |
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292 qed "forest_rec_Fnil"; |
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293 |
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294 goalw Simult.thy [tree_rec_def, forest_rec_def, Fcons_def] |
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295 "forest_rec (Fcons t tf) b c d = \ |
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296 \ d t tf (tree_rec t b c d) (forest_rec tf b c d)"; |
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297 by (simp_tac tf_rec_ss 1); |
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298 qed "forest_rec_Cons"; |