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1 (* Title: HOL/nat |
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2 ID: $Id$ |
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3 Author: Tobias Nipkow, Cambridge University Computer Laboratory |
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4 Copyright 1991 University of Cambridge |
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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 *) |
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8 |
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9 open Nat; |
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10 |
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11 goal Nat.thy "mono(%X. {Zero_Rep} Un (Suc_Rep``X))"; |
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12 by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1)); |
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13 val Nat_fun_mono = result(); |
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14 |
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15 val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski); |
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16 |
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17 (* Zero is a natural number -- this also justifies the type definition*) |
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18 goal Nat.thy "Zero_Rep: Nat"; |
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19 by (rtac (Nat_unfold RS ssubst) 1); |
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20 by (rtac (singletonI RS UnI1) 1); |
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21 val Zero_RepI = result(); |
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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 (rtac (Nat_unfold RS ssubst) 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 val Suc_RepI = result(); |
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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 (major RS (Nat_def RS def_induct)) 1); |
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35 by (rtac Nat_fun_mono 1); |
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36 by (fast_tac (set_cs addIs prems) 1); |
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37 val Nat_induct = result(); |
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38 |
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39 val prems = goalw Nat.thy [Zero_def,Suc_def] |
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40 "[| P(0); \ |
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41 \ !!k. P(k) ==> P(Suc(k)) |] ==> P(n)"; |
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42 by (rtac (Rep_Nat_inverse RS subst) 1); (*types force good instantiation*) |
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43 by (rtac (Rep_Nat RS Nat_induct) 1); |
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44 by (REPEAT (ares_tac prems 1 |
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45 ORELSE eresolve_tac [Abs_Nat_inverse RS subst] 1)); |
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46 val nat_induct = result(); |
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47 |
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48 (*Perform induction on n. *) |
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49 fun nat_ind_tac a i = |
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50 EVERY [res_inst_tac [("n",a)] nat_induct i, |
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51 rename_last_tac a ["1"] (i+1)]; |
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52 |
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53 (*A special form of induction for reasoning about m<n and m-n*) |
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54 val prems = goal Nat.thy |
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55 "[| !!x. P(x,0); \ |
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56 \ !!y. P(0,Suc(y)); \ |
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57 \ !!x y. [| P(x,y) |] ==> P(Suc(x),Suc(y)) \ |
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58 \ |] ==> P(m,n)"; |
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59 by (res_inst_tac [("x","m")] spec 1); |
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60 by (nat_ind_tac "n" 1); |
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61 by (rtac allI 2); |
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62 by (nat_ind_tac "x" 2); |
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63 by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1)); |
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64 val diff_induct = result(); |
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65 |
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66 (*Case analysis on the natural numbers*) |
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67 val prems = goal Nat.thy |
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68 "[| n=0 ==> P; !!x. n = Suc(x) ==> P |] ==> P"; |
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69 by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1); |
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70 by (fast_tac (HOL_cs addSEs prems) 1); |
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71 by (nat_ind_tac "n" 1); |
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72 by (rtac (refl RS disjI1) 1); |
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73 by (fast_tac HOL_cs 1); |
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74 val natE = result(); |
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75 |
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76 (*** Isomorphisms: Abs_Nat and Rep_Nat ***) |
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77 |
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78 (*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat), |
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79 since we assume the isomorphism equations will one day be given by Isabelle*) |
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80 |
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81 goal Nat.thy "inj(Rep_Nat)"; |
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82 by (rtac inj_inverseI 1); |
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83 by (rtac Rep_Nat_inverse 1); |
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84 val inj_Rep_Nat = result(); |
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85 |
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86 goal Nat.thy "inj_onto(Abs_Nat,Nat)"; |
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87 by (rtac inj_onto_inverseI 1); |
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88 by (etac Abs_Nat_inverse 1); |
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89 val inj_onto_Abs_Nat = result(); |
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90 |
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91 (*** Distinctness of constructors ***) |
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92 |
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93 goalw Nat.thy [Zero_def,Suc_def] "~ Suc(m)=0"; |
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94 by (rtac (inj_onto_Abs_Nat RS inj_onto_contraD) 1); |
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95 by (rtac Suc_Rep_not_Zero_Rep 1); |
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96 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1)); |
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97 val Suc_not_Zero = result(); |
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98 |
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99 val Zero_not_Suc = standard (Suc_not_Zero RS not_sym); |
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100 |
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101 val Suc_neq_Zero = standard (Suc_not_Zero RS notE); |
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102 val Zero_neq_Suc = sym RS Suc_neq_Zero; |
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103 |
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104 (** Injectiveness of Suc **) |
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105 |
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106 goalw Nat.thy [Suc_def] "inj(Suc)"; |
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107 by (rtac injI 1); |
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108 by (dtac (inj_onto_Abs_Nat RS inj_ontoD) 1); |
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109 by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1)); |
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110 by (dtac (inj_Suc_Rep RS injD) 1); |
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111 by (etac (inj_Rep_Nat RS injD) 1); |
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112 val inj_Suc = result(); |
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113 |
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114 val Suc_inject = inj_Suc RS injD;; |
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115 |
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116 goal Nat.thy "(Suc(m)=Suc(n)) = (m=n)"; |
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117 by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]); |
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118 val Suc_Suc_eq = result(); |
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119 |
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120 goal Nat.thy "~ n=Suc(n)"; |
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121 by (nat_ind_tac "n" 1); |
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122 by (ALLGOALS(asm_simp_tac (HOL_ss addsimps [Zero_not_Suc,Suc_Suc_eq]))); |
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123 val n_not_Suc_n = result(); |
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124 |
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125 (*** nat_case -- the selection operator for nat ***) |
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126 |
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127 goalw Nat.thy [nat_case_def] "nat_case(0, a, f) = a"; |
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128 by (fast_tac (set_cs addIs [select_equality] addEs [Zero_neq_Suc]) 1); |
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129 val nat_case_0 = result(); |
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130 |
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131 goalw Nat.thy [nat_case_def] "nat_case(Suc(k), a, f) = f(k)"; |
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132 by (fast_tac (set_cs addIs [select_equality] |
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133 addEs [make_elim Suc_inject, Suc_neq_Zero]) 1); |
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134 val nat_case_Suc = result(); |
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135 |
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136 (** Introduction rules for 'pred_nat' **) |
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137 |
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138 goalw Nat.thy [pred_nat_def] "<n, Suc(n)> : pred_nat"; |
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139 by (fast_tac set_cs 1); |
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140 val pred_natI = result(); |
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141 |
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142 val major::prems = goalw Nat.thy [pred_nat_def] |
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143 "[| p : pred_nat; !!x n. [| p = <n, Suc(n)> |] ==> R \ |
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144 \ |] ==> R"; |
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145 by (rtac (major RS CollectE) 1); |
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146 by (REPEAT (eresolve_tac ([asm_rl,exE]@prems) 1)); |
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147 val pred_natE = result(); |
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148 |
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149 goalw Nat.thy [wf_def] "wf(pred_nat)"; |
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150 by (strip_tac 1); |
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151 by (nat_ind_tac "x" 1); |
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152 by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, |
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153 make_elim Suc_inject]) 2); |
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154 by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, Zero_neq_Suc]) 1); |
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155 val wf_pred_nat = result(); |
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156 |
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157 |
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158 (*** nat_rec -- by wf recursion on pred_nat ***) |
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159 |
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160 val nat_rec_unfold = standard (wf_pred_nat RS (nat_rec_def RS def_wfrec)); |
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161 |
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162 (** conversion rules **) |
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163 |
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164 goal Nat.thy "nat_rec(0,c,h) = c"; |
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165 by (rtac (nat_rec_unfold RS trans) 1); |
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166 by (rtac nat_case_0 1); |
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167 val nat_rec_0 = result(); |
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168 |
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169 goal Nat.thy "nat_rec(Suc(n), c, h) = h(n, nat_rec(n,c,h))"; |
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170 by (rtac (nat_rec_unfold RS trans) 1); |
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171 by (rtac (nat_case_Suc RS trans) 1); |
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172 by(simp_tac (HOL_ss addsimps [pred_natI,cut_apply]) 1); |
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173 val nat_rec_Suc = result(); |
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174 |
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175 (*These 2 rules ease the use of primitive recursion. NOTE USE OF == *) |
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176 val [rew] = goal Nat.thy |
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177 "[| !!n. f(n) == nat_rec(n,c,h) |] ==> f(0) = c"; |
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178 by (rewtac rew); |
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179 by (rtac nat_rec_0 1); |
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180 val def_nat_rec_0 = result(); |
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181 |
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182 val [rew] = goal Nat.thy |
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183 "[| !!n. f(n) == nat_rec(n,c,h) |] ==> f(Suc(n)) = h(n,f(n))"; |
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184 by (rewtac rew); |
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185 by (rtac nat_rec_Suc 1); |
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186 val def_nat_rec_Suc = result(); |
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187 |
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188 fun nat_recs def = |
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189 [standard (def RS def_nat_rec_0), |
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190 standard (def RS def_nat_rec_Suc)]; |
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191 |
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192 |
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193 (*** Basic properties of "less than" ***) |
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194 |
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195 (** Introduction properties **) |
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196 |
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197 val prems = goalw Nat.thy [less_def] "[| i<j; j<k |] ==> i<k::nat"; |
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198 by (rtac (trans_trancl RS transD) 1); |
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199 by (resolve_tac prems 1); |
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200 by (resolve_tac prems 1); |
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201 val less_trans = result(); |
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202 |
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203 goalw Nat.thy [less_def] "n < Suc(n)"; |
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204 by (rtac (pred_natI RS r_into_trancl) 1); |
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205 val lessI = result(); |
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206 |
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207 (* i<j ==> i<Suc(j) *) |
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208 val less_SucI = lessI RSN (2, less_trans); |
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209 |
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210 goal Nat.thy "0 < Suc(n)"; |
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211 by (nat_ind_tac "n" 1); |
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212 by (rtac lessI 1); |
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213 by (etac less_trans 1); |
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214 by (rtac lessI 1); |
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215 val zero_less_Suc = result(); |
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216 |
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217 (** Elimination properties **) |
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218 |
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219 goalw Nat.thy [less_def] "n<m --> ~ m<n::nat"; |
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220 by (rtac (wf_pred_nat RS wf_trancl RS wf_anti_sym RS notI RS impI) 1); |
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221 by (assume_tac 1); |
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222 by (assume_tac 1); |
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223 val less_not_sym = result(); |
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224 |
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225 (* [| n<m; m<n |] ==> R *) |
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226 val less_anti_sym = standard (less_not_sym RS mp RS notE); |
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227 |
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228 |
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229 goalw Nat.thy [less_def] "~ n<n::nat"; |
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230 by (rtac notI 1); |
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231 by (etac (wf_pred_nat RS wf_trancl RS wf_anti_refl) 1); |
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232 val less_not_refl = result(); |
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233 |
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234 (* n<n ==> R *) |
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235 val less_anti_refl = standard (less_not_refl RS notE); |
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236 |
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237 |
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238 val major::prems = goalw Nat.thy [less_def] |
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239 "[| i<k; k=Suc(i) ==> P; !!j. [| i<j; k=Suc(j) |] ==> P \ |
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240 \ |] ==> P"; |
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241 by (rtac (major RS tranclE) 1); |
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242 by (fast_tac (HOL_cs addSEs (prems@[pred_natE, Pair_inject])) 1); |
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243 by (fast_tac (HOL_cs addSEs (prems@[pred_natE, Pair_inject])) 1); |
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244 val lessE = result(); |
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245 |
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246 goal Nat.thy "~ n<0"; |
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247 by (rtac notI 1); |
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248 by (etac lessE 1); |
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249 by (etac Zero_neq_Suc 1); |
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250 by (etac Zero_neq_Suc 1); |
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251 val not_less0 = result(); |
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252 |
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253 (* n<0 ==> R *) |
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254 val less_zeroE = standard (not_less0 RS notE); |
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255 |
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256 val [major,less,eq] = goal Nat.thy |
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257 "[| m < Suc(n); m<n ==> P; m=n ==> P |] ==> P"; |
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258 by (rtac (major RS lessE) 1); |
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259 by (rtac eq 1); |
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260 by (fast_tac (HOL_cs addSDs [Suc_inject]) 1); |
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261 by (rtac less 1); |
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262 by (fast_tac (HOL_cs addSDs [Suc_inject]) 1); |
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263 val less_SucE = result(); |
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264 |
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265 goal Nat.thy "(m < Suc(n)) = (m < n | m = n)"; |
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266 by (fast_tac (HOL_cs addSIs [lessI] |
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267 addEs [less_trans, less_SucE]) 1); |
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268 val less_Suc_eq = result(); |
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269 |
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270 |
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271 (** Inductive (?) properties **) |
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272 |
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273 val [prem] = goal Nat.thy "Suc(m) < n ==> m<n"; |
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274 by (rtac (prem RS rev_mp) 1); |
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275 by (nat_ind_tac "n" 1); |
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276 by (rtac impI 1); |
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277 by (etac less_zeroE 1); |
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278 by (fast_tac (HOL_cs addSIs [lessI RS less_SucI] |
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279 addSDs [Suc_inject] |
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280 addEs [less_trans, lessE]) 1); |
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281 val Suc_lessD = result(); |
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282 |
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283 val [major,minor] = goal Nat.thy |
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284 "[| Suc(i)<k; !!j. [| i<j; k=Suc(j) |] ==> P \ |
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285 \ |] ==> P"; |
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286 by (rtac (major RS lessE) 1); |
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287 by (etac (lessI RS minor) 1); |
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288 by (etac (Suc_lessD RS minor) 1); |
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289 by (assume_tac 1); |
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290 val Suc_lessE = result(); |
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291 |
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292 val [major] = goal Nat.thy "Suc(m) < Suc(n) ==> m<n"; |
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293 by (rtac (major RS lessE) 1); |
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294 by (REPEAT (rtac lessI 1 |
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295 ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1)); |
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296 val Suc_less_SucD = result(); |
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297 |
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298 val prems = goal Nat.thy "m<n ==> Suc(m) < Suc(n)"; |
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299 by (subgoal_tac "m<n --> Suc(m) < Suc(n)" 1); |
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300 by (fast_tac (HOL_cs addIs prems) 1); |
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301 by (nat_ind_tac "n" 1); |
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302 by (rtac impI 1); |
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303 by (etac less_zeroE 1); |
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304 by (fast_tac (HOL_cs addSIs [lessI] |
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305 addSDs [Suc_inject] |
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306 addEs [less_trans, lessE]) 1); |
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307 val Suc_mono = result(); |
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308 |
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309 goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)"; |
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310 by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]); |
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311 val Suc_less_eq = result(); |
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312 |
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313 val [major] = goal Nat.thy "Suc(n)<n ==> P"; |
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314 by (rtac (major RS Suc_lessD RS less_anti_refl) 1); |
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315 val not_Suc_n_less_n = result(); |
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316 |
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317 (*"Less than" is a linear ordering*) |
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318 goal Nat.thy "m<n | m=n | n<m::nat"; |
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319 by (nat_ind_tac "m" 1); |
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320 by (nat_ind_tac "n" 1); |
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321 by (rtac (refl RS disjI1 RS disjI2) 1); |
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322 by (rtac (zero_less_Suc RS disjI1) 1); |
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323 by (fast_tac (HOL_cs addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1); |
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324 val less_linear = result(); |
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325 |
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326 (*Can be used with less_Suc_eq to get n=m | n<m *) |
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327 goal Nat.thy "(~ m < n) = (n < Suc(m))"; |
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328 by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
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329 by(ALLGOALS(asm_simp_tac (HOL_ss addsimps |
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330 [not_less0,zero_less_Suc,Suc_less_eq]))); |
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331 val not_less_eq = result(); |
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332 |
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333 (*Complete induction, aka course-of-values induction*) |
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334 val prems = goalw Nat.thy [less_def] |
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335 "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |] ==> P(n)"; |
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336 by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1); |
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337 by (eresolve_tac prems 1); |
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338 val less_induct = result(); |
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339 |
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340 |
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341 (*** Properties of <= ***) |
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342 |
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343 goalw Nat.thy [le_def] "0 <= n"; |
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344 by (rtac not_less0 1); |
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345 val le0 = result(); |
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346 |
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347 val nat_simps = [not_less0, less_not_refl, zero_less_Suc, lessI, |
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348 Suc_less_eq, less_Suc_eq, le0, |
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349 Suc_not_Zero, Zero_not_Suc, Suc_Suc_eq, |
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350 nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc]; |
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351 |
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352 val nat_ss = pair_ss addsimps nat_simps; |
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353 |
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354 val prems = goalw Nat.thy [le_def] "~(n<m) ==> m<=n::nat"; |
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355 by (resolve_tac prems 1); |
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356 val leI = result(); |
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357 |
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358 val prems = goalw Nat.thy [le_def] "m<=n ==> ~(n<m::nat)"; |
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359 by (resolve_tac prems 1); |
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360 val leD = result(); |
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361 |
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362 val leE = make_elim leD; |
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363 |
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364 goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<m::nat"; |
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365 by (fast_tac HOL_cs 1); |
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366 val not_leE = result(); |
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367 |
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368 goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n"; |
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369 by(simp_tac (HOL_ss addsimps [less_Suc_eq]) 1); |
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370 by (fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym]) 1); |
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371 val lessD = result(); |
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372 |
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373 goalw Nat.thy [le_def] "!!m. m < n ==> m <= n::nat"; |
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374 by (fast_tac (HOL_cs addEs [less_anti_sym]) 1); |
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375 val less_imp_le = result(); |
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376 |
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377 goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=n::nat"; |
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378 by (cut_facts_tac [less_linear] 1); |
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379 by (fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym]) 1); |
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380 val le_imp_less_or_eq = result(); |
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381 |
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382 goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <= n::nat"; |
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383 by (cut_facts_tac [less_linear] 1); |
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384 by (fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym]) 1); |
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385 by (flexflex_tac); |
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386 val less_or_eq_imp_le = result(); |
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387 |
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388 goal Nat.thy "(x <= y::nat) = (x < y | x=y)"; |
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389 by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1)); |
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390 val le_eq_less_or_eq = result(); |
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391 |
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392 goal Nat.thy "n <= n::nat"; |
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393 by(simp_tac (HOL_ss addsimps [le_eq_less_or_eq]) 1); |
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394 val le_refl = result(); |
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395 |
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396 val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < k::nat"; |
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397 by (dtac le_imp_less_or_eq 1); |
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398 by (fast_tac (HOL_cs addIs [less_trans]) 1); |
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399 val le_less_trans = result(); |
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400 |
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401 goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= k::nat"; |
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402 by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
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403 rtac less_or_eq_imp_le, fast_tac (HOL_cs addIs [less_trans])]); |
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404 val le_trans = result(); |
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405 |
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406 val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = n::nat"; |
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407 by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
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408 fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym])]); |
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409 val le_anti_sym = result(); |