1 (* Title: HOL/Nat.ML |
1 (* Title: HOL/Nat.ML |
2 ID: $Id$ |
2 ID: $Id$ |
3 Author: Tobias Nipkow, Cambridge University Computer Laboratory |
3 Author: Tobias Nipkow |
4 Copyright 1991 University of Cambridge |
4 Copyright 1997 TU Muenchen |
5 |
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6 For Nat.thy. Type nat is defined as a set (Nat) over the type ind. |
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7 *) |
5 *) |
8 |
6 |
9 open Nat; |
7 goal thy "min 0 n = 0"; |
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8 br min_leastL 1; |
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9 by(trans_tac 1); |
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10 qed "min_0L"; |
10 |
11 |
11 goal Nat.thy "mono(%X. {Zero_Rep} Un (Suc_Rep``X))"; |
12 goal thy "min n 0 = 0"; |
12 by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1)); |
13 br min_leastR 1; |
13 qed "Nat_fun_mono"; |
14 by(trans_tac 1); |
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15 qed "min_0R"; |
14 |
16 |
15 val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski); |
17 goalw thy [min_def] "min (Suc m) (Suc n) = Suc(min m n)"; |
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18 by(split_tac [expand_if] 1); |
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19 by(Simp_tac 1); |
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20 qed "min_Suc_Suc"; |
16 |
21 |
17 (* Zero is a natural number -- this also justifies the type definition*) |
22 Addsimps [min_0L,min_0R,min_Suc_Suc]; |
18 goal Nat.thy "Zero_Rep: Nat"; |
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19 by (stac Nat_unfold 1); |
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20 by (rtac (singletonI RS UnI1) 1); |
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21 qed "Zero_RepI"; |
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22 |
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23 val prems = goal Nat.thy "i: Nat ==> Suc_Rep(i) : Nat"; |
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24 by (stac Nat_unfold 1); |
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25 by (rtac (imageI RS UnI2) 1); |
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26 by (resolve_tac prems 1); |
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27 qed "Suc_RepI"; |
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28 |
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29 (*** Induction ***) |
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30 |
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31 val major::prems = goal Nat.thy |
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32 "[| i: Nat; P(Zero_Rep); \ |
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33 \ !!j. [| j: Nat; P(j) |] ==> P(Suc_Rep(j)) |] ==> P(i)"; |
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34 by (rtac ([Nat_def, Nat_fun_mono, major] MRS def_induct) 1); |
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35 by (fast_tac (!claset addIs prems) 1); |
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36 qed "Nat_induct"; |
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37 |
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38 val prems = goalw Nat.thy [Zero_def,Suc_def] |
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39 "[| P(0); \ |
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40 \ !!k. P(k) ==> P(Suc(k)) |] ==> P(n)"; |
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41 by (rtac (Rep_Nat_inverse RS subst) 1); (*types force good instantiation*) |
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42 by (rtac (Rep_Nat RS Nat_induct) 1); |
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43 by (REPEAT (ares_tac prems 1 |
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44 ORELSE eresolve_tac [Abs_Nat_inverse RS subst] 1)); |
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45 qed "nat_induct"; |
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46 |
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47 (*Perform induction on n. *) |
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48 fun nat_ind_tac a i = |
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49 EVERY [res_inst_tac [("n",a)] nat_induct i, |
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50 rename_last_tac a ["1"] (i+1)]; |
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51 |
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52 (*A special form of induction for reasoning about m<n and m-n*) |
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53 val prems = goal Nat.thy |
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54 "[| !!x. P x 0; \ |
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55 \ !!y. P 0 (Suc y); \ |
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56 \ !!x y. [| P x y |] ==> P (Suc x) (Suc y) \ |
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57 \ |] ==> P m n"; |
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58 by (res_inst_tac [("x","m")] spec 1); |
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59 by (nat_ind_tac "n" 1); |
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60 by (rtac allI 2); |
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61 by (nat_ind_tac "x" 2); |
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62 by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1)); |
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63 qed "diff_induct"; |
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64 |
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65 (*Case analysis on the natural numbers*) |
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66 val prems = goal Nat.thy |
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67 "[| n=0 ==> P; !!x. n = Suc(x) ==> P |] ==> P"; |
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68 by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1); |
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69 by (fast_tac (!claset addSEs prems) 1); |
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70 by (nat_ind_tac "n" 1); |
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71 by (rtac (refl RS disjI1) 1); |
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72 by (Fast_tac 1); |
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73 qed "natE"; |
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74 |
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75 (*** Isomorphisms: Abs_Nat and Rep_Nat ***) |
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76 |
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77 (*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat), |
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78 since we assume the isomorphism equations will one day be given by Isabelle*) |
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79 |
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80 goal Nat.thy "inj(Rep_Nat)"; |
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81 by (rtac inj_inverseI 1); |
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82 by (rtac Rep_Nat_inverse 1); |
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83 qed "inj_Rep_Nat"; |
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84 |
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85 goal Nat.thy "inj_onto Abs_Nat Nat"; |
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86 by (rtac inj_onto_inverseI 1); |
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87 by (etac Abs_Nat_inverse 1); |
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88 qed "inj_onto_Abs_Nat"; |
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89 |
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90 (*** Distinctness of constructors ***) |
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91 |
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92 goalw Nat.thy [Zero_def,Suc_def] "Suc(m) ~= 0"; |
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93 by (rtac (inj_onto_Abs_Nat RS inj_onto_contraD) 1); |
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94 by (rtac Suc_Rep_not_Zero_Rep 1); |
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95 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1)); |
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96 qed "Suc_not_Zero"; |
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97 |
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98 bind_thm ("Zero_not_Suc", Suc_not_Zero RS not_sym); |
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99 |
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100 AddIffs [Suc_not_Zero,Zero_not_Suc]; |
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101 |
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102 bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE)); |
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103 val Zero_neq_Suc = sym RS Suc_neq_Zero; |
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104 |
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105 (** Injectiveness of Suc **) |
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106 |
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107 goalw Nat.thy [Suc_def] "inj(Suc)"; |
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108 by (rtac injI 1); |
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109 by (dtac (inj_onto_Abs_Nat RS inj_ontoD) 1); |
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110 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1)); |
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111 by (dtac (inj_Suc_Rep RS injD) 1); |
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112 by (etac (inj_Rep_Nat RS injD) 1); |
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113 qed "inj_Suc"; |
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114 |
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115 val Suc_inject = inj_Suc RS injD; |
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116 |
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117 goal Nat.thy "(Suc(m)=Suc(n)) = (m=n)"; |
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118 by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]); |
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119 qed "Suc_Suc_eq"; |
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120 |
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121 AddIffs [Suc_Suc_eq]; |
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122 |
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123 goal Nat.thy "n ~= Suc(n)"; |
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124 by (nat_ind_tac "n" 1); |
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125 by (ALLGOALS Asm_simp_tac); |
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126 qed "n_not_Suc_n"; |
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127 |
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128 bind_thm ("Suc_n_not_n", n_not_Suc_n RS not_sym); |
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129 |
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130 (*** nat_case -- the selection operator for nat ***) |
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131 |
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132 goalw Nat.thy [nat_case_def] "nat_case a f 0 = a"; |
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133 by (fast_tac (!claset addIs [select_equality]) 1); |
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134 qed "nat_case_0"; |
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135 |
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136 goalw Nat.thy [nat_case_def] "nat_case a f (Suc k) = f(k)"; |
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137 by (fast_tac (!claset addIs [select_equality]) 1); |
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138 qed "nat_case_Suc"; |
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139 |
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140 (** Introduction rules for 'pred_nat' **) |
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141 |
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142 goalw Nat.thy [pred_nat_def] "(n, Suc(n)) : pred_nat"; |
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143 by (Fast_tac 1); |
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144 qed "pred_natI"; |
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145 |
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146 val major::prems = goalw Nat.thy [pred_nat_def] |
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147 "[| p : pred_nat; !!x n. [| p = (n, Suc(n)) |] ==> R \ |
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148 \ |] ==> R"; |
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149 by (rtac (major RS CollectE) 1); |
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150 by (REPEAT (eresolve_tac ([asm_rl,exE]@prems) 1)); |
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151 qed "pred_natE"; |
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152 |
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153 goalw Nat.thy [wf_def] "wf(pred_nat)"; |
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154 by (strip_tac 1); |
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155 by (nat_ind_tac "x" 1); |
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156 by (fast_tac (!claset addSEs [mp, pred_natE]) 2); |
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157 by (fast_tac (!claset addSEs [mp, pred_natE]) 1); |
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158 qed "wf_pred_nat"; |
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159 |
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160 |
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161 (*** nat_rec -- by wf recursion on pred_nat ***) |
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162 |
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163 (* The unrolling rule for nat_rec *) |
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164 goal Nat.thy |
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165 "(%n. nat_rec c d n) = wfrec pred_nat (%f. nat_case ?c (%m. ?d m (f m)))"; |
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166 by (simp_tac (HOL_ss addsimps [nat_rec_def]) 1); |
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167 bind_thm("nat_rec_unfold", wf_pred_nat RS |
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168 ((result() RS eq_reflection) RS def_wfrec)); |
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169 |
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170 (*--------------------------------------------------------------------------- |
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171 * Old: |
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172 * bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec))); |
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173 *---------------------------------------------------------------------------*) |
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174 |
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175 (** conversion rules **) |
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176 |
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177 goal Nat.thy "nat_rec c h 0 = c"; |
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178 by (rtac (nat_rec_unfold RS trans) 1); |
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179 by (simp_tac (!simpset addsimps [nat_case_0]) 1); |
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180 qed "nat_rec_0"; |
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181 |
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182 goal Nat.thy "nat_rec c h (Suc n) = h n (nat_rec c h n)"; |
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183 by (rtac (nat_rec_unfold RS trans) 1); |
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184 by (simp_tac (!simpset addsimps [nat_case_Suc, pred_natI, cut_apply]) 1); |
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185 qed "nat_rec_Suc"; |
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186 |
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187 (*These 2 rules ease the use of primitive recursion. NOTE USE OF == *) |
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188 val [rew] = goal Nat.thy |
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189 "[| !!n. f(n) == nat_rec c h n |] ==> f(0) = c"; |
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190 by (rewtac rew); |
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191 by (rtac nat_rec_0 1); |
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192 qed "def_nat_rec_0"; |
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193 |
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194 val [rew] = goal Nat.thy |
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195 "[| !!n. f(n) == nat_rec c h n |] ==> f(Suc(n)) = h n (f n)"; |
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196 by (rewtac rew); |
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197 by (rtac nat_rec_Suc 1); |
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198 qed "def_nat_rec_Suc"; |
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199 |
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200 fun nat_recs def = |
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201 [standard (def RS def_nat_rec_0), |
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202 standard (def RS def_nat_rec_Suc)]; |
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203 |
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204 |
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205 (*** Basic properties of "less than" ***) |
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206 |
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207 (** Introduction properties **) |
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208 |
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209 val prems = goalw Nat.thy [less_def] "[| i<j; j<k |] ==> i<(k::nat)"; |
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210 by (rtac (trans_trancl RS transD) 1); |
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211 by (resolve_tac prems 1); |
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212 by (resolve_tac prems 1); |
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213 qed "less_trans"; |
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214 |
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215 goalw Nat.thy [less_def] "n < Suc(n)"; |
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216 by (rtac (pred_natI RS r_into_trancl) 1); |
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217 qed "lessI"; |
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218 AddIffs [lessI]; |
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219 |
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220 (* i<j ==> i<Suc(j) *) |
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221 val less_SucI = lessI RSN (2, less_trans); |
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222 |
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223 goal Nat.thy "0 < Suc(n)"; |
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224 by (nat_ind_tac "n" 1); |
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225 by (rtac lessI 1); |
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226 by (etac less_trans 1); |
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227 by (rtac lessI 1); |
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228 qed "zero_less_Suc"; |
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229 AddIffs [zero_less_Suc]; |
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230 |
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231 (** Elimination properties **) |
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232 |
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233 val prems = goalw Nat.thy [less_def] "n<m ==> ~ m<(n::nat)"; |
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234 by (fast_tac (!claset addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1); |
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235 qed "less_not_sym"; |
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236 |
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237 (* [| n<m; m<n |] ==> R *) |
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238 bind_thm ("less_asym", (less_not_sym RS notE)); |
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239 |
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240 goalw Nat.thy [less_def] "~ n<(n::nat)"; |
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241 by (rtac notI 1); |
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242 by (etac (wf_pred_nat RS wf_trancl RS wf_irrefl) 1); |
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243 qed "less_not_refl"; |
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244 |
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245 (* n<n ==> R *) |
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246 bind_thm ("less_irrefl", (less_not_refl RS notE)); |
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247 |
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248 goal Nat.thy "!!m. n<m ==> m ~= (n::nat)"; |
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249 by (fast_tac (!claset addEs [less_irrefl]) 1); |
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250 qed "less_not_refl2"; |
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251 |
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252 |
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253 val major::prems = goalw Nat.thy [less_def] |
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254 "[| i<k; k=Suc(i) ==> P; !!j. [| i<j; k=Suc(j) |] ==> P \ |
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255 \ |] ==> P"; |
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256 by (rtac (major RS tranclE) 1); |
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257 by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE' |
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258 eresolve_tac (prems@[pred_natE, Pair_inject]))); |
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259 by (rtac refl 1); |
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260 qed "lessE"; |
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261 |
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262 goal Nat.thy "~ n<0"; |
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263 by (rtac notI 1); |
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264 by (etac lessE 1); |
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265 by (etac Zero_neq_Suc 1); |
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266 by (etac Zero_neq_Suc 1); |
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267 qed "not_less0"; |
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268 |
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269 AddIffs [not_less0]; |
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270 |
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271 (* n<0 ==> R *) |
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272 bind_thm ("less_zeroE", not_less0 RS notE); |
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273 |
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274 val [major,less,eq] = goal Nat.thy |
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275 "[| m < Suc(n); m<n ==> P; m=n ==> P |] ==> P"; |
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276 by (rtac (major RS lessE) 1); |
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277 by (rtac eq 1); |
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278 by (Fast_tac 1); |
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279 by (rtac less 1); |
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280 by (Fast_tac 1); |
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281 qed "less_SucE"; |
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282 |
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283 goal Nat.thy "(m < Suc(n)) = (m < n | m = n)"; |
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284 by (fast_tac (!claset addSIs [lessI] |
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285 addEs [less_trans, less_SucE]) 1); |
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286 qed "less_Suc_eq"; |
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287 |
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288 val prems = goal Nat.thy "m<n ==> n ~= 0"; |
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289 by (res_inst_tac [("n","n")] natE 1); |
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290 by (cut_facts_tac prems 1); |
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291 by (ALLGOALS Asm_full_simp_tac); |
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292 qed "gr_implies_not0"; |
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293 Addsimps [gr_implies_not0]; |
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294 |
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295 qed_goal "zero_less_eq" Nat.thy "0 < n = (n ~= 0)" (fn _ => [ |
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296 rtac iffI 1, |
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297 etac gr_implies_not0 1, |
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298 rtac natE 1, |
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299 contr_tac 1, |
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300 etac ssubst 1, |
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301 rtac zero_less_Suc 1]); |
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302 |
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303 (** Inductive (?) properties **) |
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304 |
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305 val [prem] = goal Nat.thy "Suc(m) < n ==> m<n"; |
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306 by (rtac (prem RS rev_mp) 1); |
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307 by (nat_ind_tac "n" 1); |
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308 by (ALLGOALS (fast_tac (!claset addSIs [lessI RS less_SucI] |
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309 addEs [less_trans, lessE]))); |
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310 qed "Suc_lessD"; |
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311 |
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312 val [major,minor] = goal Nat.thy |
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313 "[| Suc(i)<k; !!j. [| i<j; k=Suc(j) |] ==> P \ |
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314 \ |] ==> P"; |
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315 by (rtac (major RS lessE) 1); |
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316 by (etac (lessI RS minor) 1); |
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317 by (etac (Suc_lessD RS minor) 1); |
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318 by (assume_tac 1); |
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319 qed "Suc_lessE"; |
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320 |
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321 goal Nat.thy "!!m n. Suc(m) < Suc(n) ==> m<n"; |
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322 by (fast_tac (!claset addEs [lessE, Suc_lessD] addIs [lessI]) 1); |
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323 qed "Suc_less_SucD"; |
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324 |
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325 goal Nat.thy "!!m n. m<n ==> Suc(m) < Suc(n)"; |
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326 by (etac rev_mp 1); |
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327 by (nat_ind_tac "n" 1); |
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328 by (ALLGOALS (fast_tac (!claset addSIs [lessI] |
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329 addEs [less_trans, lessE]))); |
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330 qed "Suc_mono"; |
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331 |
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332 |
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333 goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)"; |
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334 by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]); |
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335 qed "Suc_less_eq"; |
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336 Addsimps [Suc_less_eq]; |
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337 |
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338 goal Nat.thy "~(Suc(n) < n)"; |
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339 by (fast_tac (!claset addEs [Suc_lessD RS less_irrefl]) 1); |
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340 qed "not_Suc_n_less_n"; |
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341 Addsimps [not_Suc_n_less_n]; |
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342 |
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343 goal Nat.thy "!!i. i<j ==> j<k --> Suc i < k"; |
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344 by (nat_ind_tac "k" 1); |
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345 by (ALLGOALS (asm_simp_tac (!simpset))); |
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346 by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
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347 by (fast_tac (!claset addDs [Suc_lessD]) 1); |
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348 qed_spec_mp "less_trans_Suc"; |
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349 |
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350 (*"Less than" is a linear ordering*) |
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351 goal Nat.thy "m<n | m=n | n<(m::nat)"; |
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352 by (nat_ind_tac "m" 1); |
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353 by (nat_ind_tac "n" 1); |
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354 by (rtac (refl RS disjI1 RS disjI2) 1); |
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355 by (rtac (zero_less_Suc RS disjI1) 1); |
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356 by (fast_tac (!claset addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1); |
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357 qed "less_linear"; |
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358 |
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359 qed_goal "nat_less_cases" Nat.thy |
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360 "[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m" |
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361 ( fn prems => |
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362 [ |
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363 (res_inst_tac [("m1","n"),("n1","m")] (less_linear RS disjE) 1), |
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364 (etac disjE 2), |
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365 (etac (hd (tl (tl prems))) 1), |
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366 (etac (sym RS hd (tl prems)) 1), |
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367 (etac (hd prems) 1) |
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368 ]); |
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369 |
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370 (*Can be used with less_Suc_eq to get n=m | n<m *) |
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371 goal Nat.thy "(~ m < n) = (n < Suc(m))"; |
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372 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
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373 by (ALLGOALS Asm_simp_tac); |
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374 qed "not_less_eq"; |
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375 |
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376 (*Complete induction, aka course-of-values induction*) |
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377 val prems = goalw Nat.thy [less_def] |
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378 "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |] ==> P(n)"; |
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379 by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1); |
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380 by (eresolve_tac prems 1); |
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381 qed "less_induct"; |
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382 |
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383 qed_goal "nat_induct2" Nat.thy |
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384 "[| P 0; P 1; !!k. P k ==> P (Suc (Suc k)) |] ==> P n" (fn prems => [ |
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385 cut_facts_tac prems 1, |
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386 rtac less_induct 1, |
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387 res_inst_tac [("n","n")] natE 1, |
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388 hyp_subst_tac 1, |
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389 atac 1, |
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390 hyp_subst_tac 1, |
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391 res_inst_tac [("n","x")] natE 1, |
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392 hyp_subst_tac 1, |
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393 atac 1, |
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394 hyp_subst_tac 1, |
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395 resolve_tac prems 1, |
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396 dtac spec 1, |
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397 etac mp 1, |
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398 rtac (lessI RS less_trans) 1, |
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399 rtac (lessI RS Suc_mono) 1]); |
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400 |
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401 (*** Properties of <= ***) |
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402 |
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403 goalw Nat.thy [le_def] "(m <= n) = (m < Suc n)"; |
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404 by (rtac not_less_eq 1); |
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405 qed "le_eq_less_Suc"; |
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406 |
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407 goalw Nat.thy [le_def] "0 <= n"; |
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408 by (rtac not_less0 1); |
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409 qed "le0"; |
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410 |
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411 goalw Nat.thy [le_def] "~ Suc n <= n"; |
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412 by (Simp_tac 1); |
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413 qed "Suc_n_not_le_n"; |
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414 |
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415 goalw Nat.thy [le_def] "(i <= 0) = (i = 0)"; |
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416 by (nat_ind_tac "i" 1); |
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417 by (ALLGOALS Asm_simp_tac); |
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418 qed "le_0_eq"; |
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419 |
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420 Addsimps [less_not_refl, |
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421 (*less_Suc_eq, makes simpset non-confluent*) le0, le_0_eq, |
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422 Suc_n_not_le_n, |
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423 n_not_Suc_n, Suc_n_not_n, |
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424 nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc]; |
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425 |
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426 (* |
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427 goal Nat.thy "(Suc m < n | Suc m = n) = (m < n)"; |
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428 by (stac (less_Suc_eq RS sym) 1); |
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429 by (rtac Suc_less_eq 1); |
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430 qed "Suc_le_eq"; |
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431 |
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432 this could make the simpset (with less_Suc_eq added again) more confluent, |
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433 but less_Suc_eq makes additional problems with terms of the form 0 < Suc (...) |
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434 *) |
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435 |
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436 (*Prevents simplification of f and g: much faster*) |
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437 qed_goal "nat_case_weak_cong" Nat.thy |
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438 "m=n ==> nat_case a f m = nat_case a f n" |
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439 (fn [prem] => [rtac (prem RS arg_cong) 1]); |
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440 |
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441 qed_goal "nat_rec_weak_cong" Nat.thy |
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442 "m=n ==> nat_rec a f m = nat_rec a f n" |
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443 (fn [prem] => [rtac (prem RS arg_cong) 1]); |
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444 |
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445 qed_goal "expand_nat_case" Nat.thy |
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446 "P(nat_case z s n) = ((n=0 --> P z) & (!m. n = Suc m --> P(s m)))" |
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447 (fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]); |
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448 |
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449 val prems = goalw Nat.thy [le_def] "~n<m ==> m<=(n::nat)"; |
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450 by (resolve_tac prems 1); |
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451 qed "leI"; |
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452 |
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453 val prems = goalw Nat.thy [le_def] "m<=n ==> ~ n < (m::nat)"; |
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454 by (resolve_tac prems 1); |
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455 qed "leD"; |
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456 |
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457 val leE = make_elim leD; |
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458 |
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459 goal Nat.thy "(~n<m) = (m<=(n::nat))"; |
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460 by (fast_tac (!claset addIs [leI] addEs [leE]) 1); |
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461 qed "not_less_iff_le"; |
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462 |
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463 goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)"; |
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464 by (Fast_tac 1); |
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465 qed "not_leE"; |
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466 |
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467 goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n"; |
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468 by (simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
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469 by (fast_tac (!claset addEs [less_irrefl,less_asym]) 1); |
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470 qed "lessD"; |
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471 |
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472 goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n"; |
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473 by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
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474 qed "Suc_leD"; |
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475 |
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476 (* stronger version of Suc_leD *) |
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477 goalw Nat.thy [le_def] |
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478 "!!m. Suc m <= n ==> m < n"; |
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479 by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
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480 by (cut_facts_tac [less_linear] 1); |
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481 by (Fast_tac 1); |
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482 qed "Suc_le_lessD"; |
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483 |
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484 goal Nat.thy "(Suc m <= n) = (m < n)"; |
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485 by (fast_tac (!claset addIs [lessD, Suc_le_lessD]) 1); |
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486 qed "Suc_le_eq"; |
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487 |
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488 goalw Nat.thy [le_def] "!!m. m <= n ==> m <= Suc n"; |
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489 by (fast_tac (!claset addDs [Suc_lessD]) 1); |
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490 qed "le_SucI"; |
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491 Addsimps[le_SucI]; |
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492 |
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493 bind_thm ("le_Suc", not_Suc_n_less_n RS leI); |
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494 |
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495 goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)"; |
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496 by (fast_tac (!claset addEs [less_asym]) 1); |
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497 qed "less_imp_le"; |
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498 |
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499 goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)"; |
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500 by (cut_facts_tac [less_linear] 1); |
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501 by (fast_tac (!claset addEs [less_irrefl,less_asym]) 1); |
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502 qed "le_imp_less_or_eq"; |
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503 |
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504 goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)"; |
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505 by (cut_facts_tac [less_linear] 1); |
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506 by (fast_tac (!claset addEs [less_irrefl,less_asym]) 1); |
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507 by (flexflex_tac); |
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508 qed "less_or_eq_imp_le"; |
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509 |
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510 goal Nat.thy "(x <= (y::nat)) = (x < y | x=y)"; |
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511 by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1)); |
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512 qed "le_eq_less_or_eq"; |
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513 |
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514 goal Nat.thy "n <= (n::nat)"; |
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515 by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1); |
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516 qed "le_refl"; |
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517 |
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518 val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)"; |
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519 by (dtac le_imp_less_or_eq 1); |
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520 by (fast_tac (!claset addIs [less_trans]) 1); |
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521 qed "le_less_trans"; |
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522 |
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523 goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)"; |
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524 by (dtac le_imp_less_or_eq 1); |
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525 by (fast_tac (!claset addIs [less_trans]) 1); |
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526 qed "less_le_trans"; |
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527 |
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528 goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)"; |
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529 by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
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530 rtac less_or_eq_imp_le, fast_tac (!claset addIs [less_trans])]); |
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531 qed "le_trans"; |
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532 |
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533 val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)"; |
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534 by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
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535 fast_tac (!claset addEs [less_irrefl,less_asym])]); |
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536 qed "le_anti_sym"; |
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537 |
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538 goal Nat.thy "(Suc(n) <= Suc(m)) = (n <= m)"; |
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539 by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1); |
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540 qed "Suc_le_mono"; |
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541 |
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542 AddIffs [le_refl,Suc_le_mono]; |
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543 |
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544 |
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545 (** LEAST -- the least number operator **) |
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546 |
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547 val [prem1,prem2] = goalw Nat.thy [Least_def] |
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548 "[| P(k); !!x. x<k ==> ~P(x) |] ==> (LEAST x.P(x)) = k"; |
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549 by (rtac select_equality 1); |
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550 by (fast_tac (!claset addSIs [prem1,prem2]) 1); |
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551 by (cut_facts_tac [less_linear] 1); |
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552 by (fast_tac (!claset addSIs [prem1] addSDs [prem2]) 1); |
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553 qed "Least_equality"; |
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554 |
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555 val [prem] = goal Nat.thy "P(k) ==> P(LEAST x.P(x))"; |
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556 by (rtac (prem RS rev_mp) 1); |
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557 by (res_inst_tac [("n","k")] less_induct 1); |
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558 by (rtac impI 1); |
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559 by (rtac classical 1); |
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560 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1); |
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561 by (assume_tac 1); |
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562 by (assume_tac 2); |
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563 by (Fast_tac 1); |
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564 qed "LeastI"; |
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565 |
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566 (*Proof is almost identical to the one above!*) |
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567 val [prem] = goal Nat.thy "P(k) ==> (LEAST x.P(x)) <= k"; |
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568 by (rtac (prem RS rev_mp) 1); |
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569 by (res_inst_tac [("n","k")] less_induct 1); |
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570 by (rtac impI 1); |
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571 by (rtac classical 1); |
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572 by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1); |
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573 by (assume_tac 1); |
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574 by (rtac le_refl 2); |
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575 by (fast_tac (!claset addIs [less_imp_le,le_trans]) 1); |
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576 qed "Least_le"; |
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577 |
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578 val [prem] = goal Nat.thy "k < (LEAST x.P(x)) ==> ~P(k)"; |
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579 by (rtac notI 1); |
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580 by (etac (rewrite_rule [le_def] Least_le RS notE) 1); |
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581 by (rtac prem 1); |
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582 qed "not_less_Least"; |
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583 |
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584 qed_goalw "Least_Suc" Nat.thy [Least_def] |
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585 "!!P. [| ? n. P(Suc n); ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))" |
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586 (fn _ => [ |
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587 rtac select_equality 1, |
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588 fold_goals_tac [Least_def], |
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589 safe_tac (!claset addSEs [LeastI]), |
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590 res_inst_tac [("n","j")] natE 1, |
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591 Fast_tac 1, |
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592 fast_tac (!claset addDs [Suc_less_SucD, not_less_Least]) 1, |
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593 res_inst_tac [("n","k")] natE 1, |
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594 Fast_tac 1, |
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595 hyp_subst_tac 1, |
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596 rewtac Least_def, |
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597 rtac (select_equality RS arg_cong RS sym) 1, |
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598 safe_tac (!claset), |
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599 dtac Suc_mono 1, |
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600 Fast_tac 1, |
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601 cut_facts_tac [less_linear] 1, |
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602 safe_tac (!claset), |
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603 atac 2, |
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604 Fast_tac 2, |
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605 dtac Suc_mono 1, |
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606 Fast_tac 1]); |
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607 |
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608 |
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609 (*** Instantiation of transitivity prover ***) |
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610 |
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611 structure Less_Arith = |
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612 struct |
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613 val nat_leI = leI; |
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614 val nat_leD = leD; |
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615 val lessI = lessI; |
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616 val zero_less_Suc = zero_less_Suc; |
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617 val less_reflE = less_irrefl; |
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618 val less_zeroE = less_zeroE; |
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619 val less_incr = Suc_mono; |
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620 val less_decr = Suc_less_SucD; |
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621 val less_incr_rhs = Suc_mono RS Suc_lessD; |
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622 val less_decr_lhs = Suc_lessD; |
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623 val less_trans_Suc = less_trans_Suc; |
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624 val leI = lessD RS (Suc_le_mono RS iffD1); |
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625 val not_lessI = leI RS leD |
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626 val not_leI = prove_goal Nat.thy "!!m::nat. n < m ==> ~ m <= n" |
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627 (fn _ => [etac swap2 1, etac leD 1]); |
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628 val eqI = prove_goal Nat.thy "!!m. [| m < Suc n; n < Suc m |] ==> m=n" |
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629 (fn _ => [etac less_SucE 1, |
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630 fast_tac (HOL_cs addSDs [Suc_less_SucD] addSEs [less_irrefl] |
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631 addDs [less_trans_Suc]) 1, |
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632 atac 1]); |
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633 val leD = le_eq_less_Suc RS iffD1; |
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634 val not_lessD = nat_leI RS leD; |
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635 val not_leD = not_leE |
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636 val eqD1 = prove_goal Nat.thy "!!n. m = n ==> m < Suc n" |
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637 (fn _ => [etac subst 1, rtac lessI 1]); |
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638 val eqD2 = sym RS eqD1; |
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639 |
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640 fun is_zero(t) = t = Const("0",Type("nat",[])); |
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641 |
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642 fun nnb T = T = Type("fun",[Type("nat",[]), |
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643 Type("fun",[Type("nat",[]), |
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644 Type("bool",[])])]) |
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645 |
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646 fun decomp_Suc(Const("Suc",_)$t) = let val (a,i) = decomp_Suc t in (a,i+1) end |
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647 | decomp_Suc t = (t,0); |
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648 |
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649 fun decomp2(rel,T,lhs,rhs) = |
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650 if not(nnb T) then None else |
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651 let val (x,i) = decomp_Suc lhs |
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652 val (y,j) = decomp_Suc rhs |
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653 in case rel of |
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654 "op <" => Some(x,i,"<",y,j) |
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655 | "op <=" => Some(x,i,"<=",y,j) |
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656 | "op =" => Some(x,i,"=",y,j) |
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657 | _ => None |
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658 end; |
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659 |
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660 fun negate(Some(x,i,rel,y,j)) = Some(x,i,"~"^rel,y,j) |
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661 | negate None = None; |
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662 |
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663 fun decomp(_$(Const(rel,T)$lhs$rhs)) = decomp2(rel,T,lhs,rhs) |
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664 | decomp(_$(Const("not",_)$(Const(rel,T)$lhs$rhs))) = |
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665 negate(decomp2(rel,T,lhs,rhs)) |
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666 | decomp _ = None |
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667 |
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668 end; |
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669 |
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670 structure Trans_Tac = Trans_Tac_Fun(Less_Arith); |
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671 |
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672 open Trans_Tac; |
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673 |
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674 (*** eliminates ~= in premises, which trans_tac cannot deal with ***) |
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675 qed_goal "nat_neqE" Nat.thy |
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676 "[| (m::nat) ~= n; m < n ==> P; n < m ==> P |] ==> P" |
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677 (fn major::prems => [cut_facts_tac [less_linear] 1, |
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678 REPEAT(eresolve_tac ([disjE,major RS notE]@prems) 1)]); |
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