1 (* Title: HOL/Nat_Int_Bij.thy |
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2 ID: $Id$ |
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3 Author: Stefan Richter, Tobias Nipkow |
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4 *) |
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5 |
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6 header{* Bijections $\mathbb{N}\to\mathbb{N}^2$ and $\mathbb{N}\to\mathbb{Z}$*} |
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7 |
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8 theory Nat_Int_Bij |
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9 imports Hilbert_Choice Presburger |
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10 begin |
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11 |
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12 subsection{* A bijection between @{text "\<nat>"} and @{text "\<nat>\<twosuperior>"} *} |
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13 |
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14 text{* Definition and proofs are from \cite[page 85]{Oberschelp:1993}. *} |
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15 |
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16 definition nat2_to_nat:: "(nat * nat) \<Rightarrow> nat" where |
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17 "nat2_to_nat pair = (let (n,m) = pair in (n+m) * Suc (n+m) div 2 + n)" |
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18 definition nat_to_nat2:: "nat \<Rightarrow> (nat * nat)" where |
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19 "nat_to_nat2 = inv nat2_to_nat" |
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20 |
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21 lemma dvd2_a_x_suc_a: "2 dvd a * (Suc a)" |
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22 proof (cases "2 dvd a") |
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23 case True |
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24 then show ?thesis by (rule dvd_mult2) |
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25 next |
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26 case False |
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27 then have "Suc (a mod 2) = 2" by (simp add: dvd_eq_mod_eq_0) |
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28 then have "Suc a mod 2 = 0" by (simp add: mod_Suc) |
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29 then have "2 dvd Suc a" by (simp only:dvd_eq_mod_eq_0) |
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30 then show ?thesis by (rule dvd_mult) |
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31 qed |
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32 |
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33 lemma |
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34 assumes eq: "nat2_to_nat (u,v) = nat2_to_nat (x,y)" |
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35 shows nat2_to_nat_help: "u+v \<le> x+y" |
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36 proof (rule classical) |
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37 assume "\<not> ?thesis" |
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38 then have contrapos: "x+y < u+v" |
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39 by simp |
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40 have "nat2_to_nat (x,y) < (x+y) * Suc (x+y) div 2 + Suc (x + y)" |
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41 by (unfold nat2_to_nat_def) (simp add: Let_def) |
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42 also have "\<dots> = (x+y)*Suc(x+y) div 2 + 2 * Suc(x+y) div 2" |
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43 by (simp only: div_mult_self1_is_m) |
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44 also have "\<dots> = (x+y)*Suc(x+y) div 2 + 2 * Suc(x+y) div 2 |
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45 + ((x+y)*Suc(x+y) mod 2 + 2 * Suc(x+y) mod 2) div 2" |
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46 proof - |
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47 have "2 dvd (x+y)*Suc(x+y)" |
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48 by (rule dvd2_a_x_suc_a) |
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49 then have "(x+y)*Suc(x+y) mod 2 = 0" |
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50 by (simp only: dvd_eq_mod_eq_0) |
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51 also |
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52 have "2 * Suc(x+y) mod 2 = 0" |
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53 by (rule mod_mult_self1_is_0) |
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54 ultimately have |
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55 "((x+y)*Suc(x+y) mod 2 + 2 * Suc(x+y) mod 2) div 2 = 0" |
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56 by simp |
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57 then show ?thesis |
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58 by simp |
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59 qed |
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60 also have "\<dots> = ((x+y)*Suc(x+y) + 2*Suc(x+y)) div 2" |
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61 by (rule div_add1_eq [symmetric]) |
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62 also have "\<dots> = ((x+y+2)*Suc(x+y)) div 2" |
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63 by (simp only: add_mult_distrib [symmetric]) |
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64 also from contrapos have "\<dots> \<le> ((Suc(u+v))*(u+v)) div 2" |
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65 by (simp only: mult_le_mono div_le_mono) |
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66 also have "\<dots> \<le> nat2_to_nat (u,v)" |
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67 by (unfold nat2_to_nat_def) (simp add: Let_def) |
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68 finally show ?thesis |
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69 by (simp only: eq) |
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70 qed |
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71 |
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72 theorem nat2_to_nat_inj: "inj nat2_to_nat" |
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73 proof - |
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74 { |
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75 fix u v x y |
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76 assume eq1: "nat2_to_nat (u,v) = nat2_to_nat (x,y)" |
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77 then have "u+v \<le> x+y" by (rule nat2_to_nat_help) |
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78 also from eq1 [symmetric] have "x+y \<le> u+v" |
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79 by (rule nat2_to_nat_help) |
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80 finally have eq2: "u+v = x+y" . |
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81 with eq1 have ux: "u=x" |
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82 by (simp add: nat2_to_nat_def Let_def) |
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83 with eq2 have vy: "v=y" by simp |
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84 with ux have "(u,v) = (x,y)" by simp |
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85 } |
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86 then have "\<And>x y. nat2_to_nat x = nat2_to_nat y \<Longrightarrow> x=y" by fast |
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87 then show ?thesis unfolding inj_on_def by simp |
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88 qed |
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89 |
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90 lemma nat_to_nat2_surj: "surj nat_to_nat2" |
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91 by (simp only: nat_to_nat2_def nat2_to_nat_inj inj_imp_surj_inv) |
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92 |
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93 |
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94 lemma gauss_sum_nat_upto: "2 * (\<Sum>i\<le>n::nat. i) = n * (n + 1)" |
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95 using gauss_sum[where 'a = nat] |
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96 by (simp add:atLeast0AtMost setsum_shift_lb_Suc0_0 numeral_2_eq_2) |
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97 |
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98 lemma nat2_to_nat_surj: "surj nat2_to_nat" |
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99 proof (unfold surj_def) |
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100 { |
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101 fix z::nat |
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102 def r \<equiv> "Max {r. (\<Sum>i\<le>r. i) \<le> z}" |
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103 def x \<equiv> "z - (\<Sum>i\<le>r. i)" |
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104 |
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105 hence "finite {r. (\<Sum>i\<le>r. i) \<le> z}" |
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106 by (simp add: lessThan_Suc_atMost[symmetric] lessThan_Suc finite_less_ub) |
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107 also have "0 \<in> {r. (\<Sum>i\<le>r. i) \<le> z}" by simp |
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108 hence "{r::nat. (\<Sum>i\<le>r. i) \<le> z} \<noteq> {}" by fast |
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109 ultimately have a: "r \<in> {r. (\<Sum>i\<le>r. i) \<le> z} \<and> (\<forall>s \<in> {r. (\<Sum>i\<le>r. i) \<le> z}. s \<le> r)" |
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110 by (simp add: r_def del:mem_Collect_eq) |
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111 { |
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112 assume "r<x" |
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113 hence "r+1\<le>x" by simp |
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114 hence "(\<Sum>i\<le>r. i)+(r+1)\<le>z" using x_def by arith |
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115 hence "(r+1) \<in> {r. (\<Sum>i\<le>r. i) \<le> z}" by simp |
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116 with a have "(r+1)\<le>r" by simp |
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117 } |
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118 hence b: "x\<le>r" by force |
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119 |
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120 def y \<equiv> "r-x" |
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121 have "2*z=2*(\<Sum>i\<le>r. i)+2*x" using x_def a by simp arith |
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122 also have "\<dots> = r * (r+1) + 2*x" using gauss_sum_nat_upto by simp |
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123 also have "\<dots> = (x+y)*(x+y+1)+2*x" using y_def b by simp |
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124 also { have "2 dvd ((x+y)*(x+y+1))" using dvd2_a_x_suc_a by simp } |
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125 hence "\<dots> = 2 * nat2_to_nat(x,y)" |
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126 using nat2_to_nat_def by (simp add: Let_def dvd_mult_div_cancel) |
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127 finally have "z=nat2_to_nat (x, y)" by simp |
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128 } |
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129 thus "\<forall>y. \<exists>x. y = nat2_to_nat x" by fast |
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130 qed |
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131 |
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132 |
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133 subsection{* A bijection between @{text "\<nat>"} and @{text "\<int>"} *} |
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134 |
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135 definition nat_to_int_bij :: "nat \<Rightarrow> int" where |
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136 "nat_to_int_bij n = (if 2 dvd n then int(n div 2) else -int(Suc n div 2))" |
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137 |
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138 definition int_to_nat_bij :: "int \<Rightarrow> nat" where |
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139 "int_to_nat_bij i = (if 0<=i then 2*nat(i) else 2*nat(-i) - 1)" |
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140 |
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141 lemma i2n_n2i_id: "int_to_nat_bij (nat_to_int_bij n) = n" |
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142 by (simp add: int_to_nat_bij_def nat_to_int_bij_def) presburger |
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143 |
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144 lemma n2i_i2n_id: "nat_to_int_bij(int_to_nat_bij i) = i" |
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145 proof - |
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146 have "ALL m n::nat. m>0 \<longrightarrow> 2 * m - Suc 0 \<noteq> 2 * n" by presburger |
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147 thus ?thesis |
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148 by(simp add: nat_to_int_bij_def int_to_nat_bij_def, simp add:dvd_def) |
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149 qed |
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150 |
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151 lemma inv_nat_to_int_bij: "inv nat_to_int_bij = int_to_nat_bij" |
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152 by (simp add: i2n_n2i_id inv_equality n2i_i2n_id) |
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153 |
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154 lemma inv_int_to_nat_bij: "inv int_to_nat_bij = nat_to_int_bij" |
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155 by (simp add: i2n_n2i_id inv_equality n2i_i2n_id) |
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156 |
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157 lemma surj_nat_to_int_bij: "surj nat_to_int_bij" |
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158 by (blast intro: n2i_i2n_id surjI) |
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159 |
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160 lemma surj_int_to_nat_bij: "surj int_to_nat_bij" |
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161 by (blast intro: i2n_n2i_id surjI) |
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162 |
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163 lemma inj_nat_to_int_bij: "inj nat_to_int_bij" |
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164 by(simp add:inv_int_to_nat_bij[symmetric] surj_int_to_nat_bij surj_imp_inj_inv) |
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165 |
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166 lemma inj_int_to_nat_bij: "inj int_to_nat_bij" |
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167 by(simp add:inv_nat_to_int_bij[symmetric] surj_nat_to_int_bij surj_imp_inj_inv) |
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168 |
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169 |
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170 end |
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