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1 |
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2 (* Authors: Jeremy Avigad and Amine Chaieb *) |
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3 |
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4 header {* Sets up transfer from nats to ints and back. *} |
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5 |
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6 theory Nat_Transfer |
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7 imports Main Parity |
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8 begin |
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9 |
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10 subsection {* Set up transfer from nat to int *} |
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11 |
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12 (* set up transfer direction *) |
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13 |
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14 lemma TransferMorphism_nat_int: "TransferMorphism nat (op <= (0::int))" |
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15 by (simp add: TransferMorphism_def) |
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16 |
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17 declare TransferMorphism_nat_int[transfer |
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18 add mode: manual |
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19 return: nat_0_le |
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20 labels: natint |
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21 ] |
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22 |
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23 (* basic functions and relations *) |
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24 |
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25 lemma transfer_nat_int_numerals: |
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26 "(0::nat) = nat 0" |
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27 "(1::nat) = nat 1" |
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28 "(2::nat) = nat 2" |
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29 "(3::nat) = nat 3" |
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30 by auto |
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31 |
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32 definition |
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33 tsub :: "int \<Rightarrow> int \<Rightarrow> int" |
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34 where |
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35 "tsub x y = (if x >= y then x - y else 0)" |
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36 |
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37 lemma tsub_eq: "x >= y \<Longrightarrow> tsub x y = x - y" |
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38 by (simp add: tsub_def) |
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39 |
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40 |
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41 lemma transfer_nat_int_functions: |
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42 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) + (nat y) = nat (x + y)" |
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43 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) * (nat y) = nat (x * y)" |
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44 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) - (nat y) = nat (tsub x y)" |
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45 "(x::int) >= 0 \<Longrightarrow> (nat x)^n = nat (x^n)" |
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46 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)" |
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47 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)" |
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48 by (auto simp add: eq_nat_nat_iff nat_mult_distrib |
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49 nat_power_eq nat_div_distrib nat_mod_distrib tsub_def) |
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50 |
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51 lemma transfer_nat_int_function_closures: |
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52 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x + y >= 0" |
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53 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x * y >= 0" |
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54 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> tsub x y >= 0" |
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55 "(x::int) >= 0 \<Longrightarrow> x^n >= 0" |
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56 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0" |
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57 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0" |
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58 "(0::int) >= 0" |
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59 "(1::int) >= 0" |
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60 "(2::int) >= 0" |
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61 "(3::int) >= 0" |
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62 "int z >= 0" |
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63 apply (auto simp add: zero_le_mult_iff tsub_def) |
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64 apply (case_tac "y = 0") |
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65 apply auto |
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66 apply (subst pos_imp_zdiv_nonneg_iff, auto) |
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67 apply (case_tac "y = 0") |
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68 apply force |
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69 apply (rule pos_mod_sign) |
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70 apply arith |
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71 done |
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72 |
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73 lemma transfer_nat_int_relations: |
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74 "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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75 (nat (x::int) = nat y) = (x = y)" |
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76 "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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77 (nat (x::int) < nat y) = (x < y)" |
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78 "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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79 (nat (x::int) <= nat y) = (x <= y)" |
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80 "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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81 (nat (x::int) dvd nat y) = (x dvd y)" |
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82 by (auto simp add: zdvd_int) |
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83 |
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84 declare TransferMorphism_nat_int[transfer add return: |
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85 transfer_nat_int_numerals |
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86 transfer_nat_int_functions |
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87 transfer_nat_int_function_closures |
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88 transfer_nat_int_relations |
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89 ] |
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90 |
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91 |
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92 (* first-order quantifiers *) |
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93 |
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94 lemma transfer_nat_int_quantifiers: |
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95 "(ALL (x::nat). P x) = (ALL (x::int). x >= 0 \<longrightarrow> P (nat x))" |
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96 "(EX (x::nat). P x) = (EX (x::int). x >= 0 & P (nat x))" |
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97 by (rule all_nat, rule ex_nat) |
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98 |
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99 (* should we restrict these? *) |
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100 lemma all_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> |
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101 (ALL x. Q x \<longrightarrow> P x) = (ALL x. Q x \<longrightarrow> P' x)" |
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102 by auto |
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103 |
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104 lemma ex_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> |
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105 (EX x. Q x \<and> P x) = (EX x. Q x \<and> P' x)" |
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106 by auto |
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107 |
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108 declare TransferMorphism_nat_int[transfer add |
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109 return: transfer_nat_int_quantifiers |
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110 cong: all_cong ex_cong] |
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111 |
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112 |
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113 (* if *) |
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114 |
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115 lemma nat_if_cong: "(if P then (nat x) else (nat y)) = |
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116 nat (if P then x else y)" |
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117 by auto |
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118 |
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119 declare TransferMorphism_nat_int [transfer add return: nat_if_cong] |
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120 |
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121 |
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122 (* operations with sets *) |
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123 |
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124 definition |
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125 nat_set :: "int set \<Rightarrow> bool" |
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126 where |
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127 "nat_set S = (ALL x:S. x >= 0)" |
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128 |
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129 lemma transfer_nat_int_set_functions: |
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130 "card A = card (int ` A)" |
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131 "{} = nat ` ({}::int set)" |
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132 "A Un B = nat ` (int ` A Un int ` B)" |
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133 "A Int B = nat ` (int ` A Int int ` B)" |
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134 "{x. P x} = nat ` {x. x >= 0 & P(nat x)}" |
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135 "{..n} = nat ` {0..int n}" |
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136 "{m..n} = nat ` {int m..int n}" (* need all variants of these! *) |
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137 apply (rule card_image [symmetric]) |
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138 apply (auto simp add: inj_on_def image_def) |
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139 apply (rule_tac x = "int x" in bexI) |
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140 apply auto |
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141 apply (rule_tac x = "int x" in bexI) |
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142 apply auto |
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143 apply (rule_tac x = "int x" in bexI) |
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144 apply auto |
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145 apply (rule_tac x = "int x" in exI) |
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146 apply auto |
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147 apply (rule_tac x = "int x" in bexI) |
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148 apply auto |
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149 apply (rule_tac x = "int x" in bexI) |
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150 apply auto |
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151 done |
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152 |
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153 lemma transfer_nat_int_set_function_closures: |
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154 "nat_set {}" |
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155 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)" |
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156 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)" |
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157 "x >= 0 \<Longrightarrow> nat_set {x..y}" |
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158 "nat_set {x. x >= 0 & P x}" |
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159 "nat_set (int ` C)" |
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160 "nat_set A \<Longrightarrow> x : A \<Longrightarrow> x >= 0" (* does it hurt to turn this on? *) |
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161 unfolding nat_set_def apply auto |
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162 done |
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163 |
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164 lemma transfer_nat_int_set_relations: |
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165 "(finite A) = (finite (int ` A))" |
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166 "(x : A) = (int x : int ` A)" |
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167 "(A = B) = (int ` A = int ` B)" |
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168 "(A < B) = (int ` A < int ` B)" |
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169 "(A <= B) = (int ` A <= int ` B)" |
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170 |
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171 apply (rule iffI) |
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172 apply (erule finite_imageI) |
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173 apply (erule finite_imageD) |
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174 apply (auto simp add: image_def expand_set_eq inj_on_def) |
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175 apply (drule_tac x = "int x" in spec, auto) |
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176 apply (drule_tac x = "int x" in spec, auto) |
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177 apply (drule_tac x = "int x" in spec, auto) |
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178 done |
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179 |
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180 lemma transfer_nat_int_set_return_embed: "nat_set A \<Longrightarrow> |
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181 (int ` nat ` A = A)" |
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182 by (auto simp add: nat_set_def image_def) |
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183 |
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184 lemma transfer_nat_int_set_cong: "(!!x. x >= 0 \<Longrightarrow> P x = P' x) \<Longrightarrow> |
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185 {(x::int). x >= 0 & P x} = {x. x >= 0 & P' x}" |
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186 by auto |
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187 |
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188 declare TransferMorphism_nat_int[transfer add |
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189 return: transfer_nat_int_set_functions |
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190 transfer_nat_int_set_function_closures |
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191 transfer_nat_int_set_relations |
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192 transfer_nat_int_set_return_embed |
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193 cong: transfer_nat_int_set_cong |
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194 ] |
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195 |
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196 |
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197 (* setsum and setprod *) |
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198 |
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199 (* this handles the case where the *domain* of f is nat *) |
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200 lemma transfer_nat_int_sum_prod: |
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201 "setsum f A = setsum (%x. f (nat x)) (int ` A)" |
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202 "setprod f A = setprod (%x. f (nat x)) (int ` A)" |
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203 apply (subst setsum_reindex) |
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204 apply (unfold inj_on_def, auto) |
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205 apply (subst setprod_reindex) |
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206 apply (unfold inj_on_def o_def, auto) |
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207 done |
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208 |
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209 (* this handles the case where the *range* of f is nat *) |
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210 lemma transfer_nat_int_sum_prod2: |
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211 "setsum f A = nat(setsum (%x. int (f x)) A)" |
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212 "setprod f A = nat(setprod (%x. int (f x)) A)" |
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213 apply (subst int_setsum [symmetric]) |
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214 apply auto |
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215 apply (subst int_setprod [symmetric]) |
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216 apply auto |
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217 done |
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218 |
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219 lemma transfer_nat_int_sum_prod_closure: |
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220 "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0" |
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221 "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0" |
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222 unfolding nat_set_def |
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223 apply (rule setsum_nonneg) |
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224 apply auto |
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225 apply (rule setprod_nonneg) |
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226 apply auto |
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227 done |
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228 |
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229 (* this version doesn't work, even with nat_set A \<Longrightarrow> |
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230 x : A \<Longrightarrow> x >= 0 turned on. Why not? |
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231 |
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232 also: what does =simp=> do? |
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233 |
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234 lemma transfer_nat_int_sum_prod_closure: |
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235 "(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0" |
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236 "(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0" |
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237 unfolding nat_set_def simp_implies_def |
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238 apply (rule setsum_nonneg) |
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239 apply auto |
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240 apply (rule setprod_nonneg) |
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241 apply auto |
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242 done |
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243 *) |
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244 |
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245 (* Making A = B in this lemma doesn't work. Why not? |
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246 Also, why aren't setsum_cong and setprod_cong enough, |
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247 with the previously mentioned rule turned on? *) |
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248 |
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249 lemma transfer_nat_int_sum_prod_cong: |
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250 "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
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251 setsum f A = setsum g B" |
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252 "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
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253 setprod f A = setprod g B" |
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254 unfolding nat_set_def |
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255 apply (subst setsum_cong, assumption) |
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256 apply auto [2] |
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257 apply (subst setprod_cong, assumption, auto) |
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258 done |
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259 |
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260 declare TransferMorphism_nat_int[transfer add |
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261 return: transfer_nat_int_sum_prod transfer_nat_int_sum_prod2 |
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262 transfer_nat_int_sum_prod_closure |
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263 cong: transfer_nat_int_sum_prod_cong] |
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264 |
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265 (* lists *) |
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266 |
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267 definition |
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268 embed_list :: "nat list \<Rightarrow> int list" |
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269 where |
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270 "embed_list l = map int l"; |
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271 |
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272 definition |
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273 nat_list :: "int list \<Rightarrow> bool" |
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274 where |
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275 "nat_list l = nat_set (set l)"; |
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276 |
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277 definition |
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278 return_list :: "int list \<Rightarrow> nat list" |
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279 where |
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280 "return_list l = map nat l"; |
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281 |
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282 thm nat_0_le; |
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283 |
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284 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow> |
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285 embed_list (return_list l) = l"; |
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286 unfolding embed_list_def return_list_def nat_list_def nat_set_def |
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287 apply (induct l); |
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288 apply auto; |
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289 done; |
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290 |
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291 lemma transfer_nat_int_list_functions: |
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292 "l @ m = return_list (embed_list l @ embed_list m)" |
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293 "[] = return_list []"; |
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294 unfolding return_list_def embed_list_def; |
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295 apply auto; |
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296 apply (induct l, auto); |
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297 apply (induct m, auto); |
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298 done; |
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299 |
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300 (* |
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301 lemma transfer_nat_int_fold1: "fold f l x = |
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302 fold (%x. f (nat x)) (embed_list l) x"; |
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303 *) |
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304 |
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305 |
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306 |
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307 |
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308 subsection {* Set up transfer from int to nat *} |
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309 |
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310 (* set up transfer direction *) |
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311 |
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312 lemma TransferMorphism_int_nat: "TransferMorphism int (UNIV :: nat set)" |
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313 by (simp add: TransferMorphism_def) |
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314 |
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315 declare TransferMorphism_int_nat[transfer add |
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316 mode: manual |
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317 (* labels: int-nat *) |
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318 return: nat_int |
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319 ] |
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320 |
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321 |
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322 (* basic functions and relations *) |
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323 |
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324 definition |
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325 is_nat :: "int \<Rightarrow> bool" |
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326 where |
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327 "is_nat x = (x >= 0)" |
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328 |
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329 lemma transfer_int_nat_numerals: |
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330 "0 = int 0" |
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331 "1 = int 1" |
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332 "2 = int 2" |
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333 "3 = int 3" |
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334 by auto |
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335 |
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336 lemma transfer_int_nat_functions: |
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337 "(int x) + (int y) = int (x + y)" |
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338 "(int x) * (int y) = int (x * y)" |
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339 "tsub (int x) (int y) = int (x - y)" |
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340 "(int x)^n = int (x^n)" |
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341 "(int x) div (int y) = int (x div y)" |
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342 "(int x) mod (int y) = int (x mod y)" |
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343 by (auto simp add: int_mult tsub_def int_power zdiv_int zmod_int) |
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344 |
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345 lemma transfer_int_nat_function_closures: |
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346 "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x + y)" |
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347 "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x * y)" |
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348 "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (tsub x y)" |
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349 "is_nat x \<Longrightarrow> is_nat (x^n)" |
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350 "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)" |
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351 "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)" |
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352 "is_nat 0" |
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353 "is_nat 1" |
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354 "is_nat 2" |
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355 "is_nat 3" |
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356 "is_nat (int z)" |
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357 by (simp_all only: is_nat_def transfer_nat_int_function_closures) |
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358 |
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359 lemma transfer_int_nat_relations: |
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360 "(int x = int y) = (x = y)" |
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361 "(int x < int y) = (x < y)" |
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362 "(int x <= int y) = (x <= y)" |
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363 "(int x dvd int y) = (x dvd y)" |
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364 "(even (int x)) = (even x)" |
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365 by (auto simp add: zdvd_int even_nat_def) |
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366 |
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367 lemma UNIV_apply: |
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368 "UNIV x = True" |
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369 by (simp add: top_fun_eq top_bool_eq) |
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370 |
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371 declare TransferMorphism_int_nat[transfer add return: |
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372 transfer_int_nat_numerals |
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373 transfer_int_nat_functions |
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374 transfer_int_nat_function_closures |
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375 transfer_int_nat_relations |
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376 UNIV_apply |
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377 ] |
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378 |
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379 |
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380 (* first-order quantifiers *) |
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381 |
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382 lemma transfer_int_nat_quantifiers: |
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383 "(ALL (x::int) >= 0. P x) = (ALL (x::nat). P (int x))" |
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384 "(EX (x::int) >= 0. P x) = (EX (x::nat). P (int x))" |
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385 apply (subst all_nat) |
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386 apply auto [1] |
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387 apply (subst ex_nat) |
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388 apply auto |
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389 done |
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390 |
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391 declare TransferMorphism_int_nat[transfer add |
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392 return: transfer_int_nat_quantifiers] |
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393 |
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394 |
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395 (* if *) |
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396 |
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397 lemma int_if_cong: "(if P then (int x) else (int y)) = |
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398 int (if P then x else y)" |
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399 by auto |
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400 |
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401 declare TransferMorphism_int_nat [transfer add return: int_if_cong] |
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402 |
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403 |
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404 |
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405 (* operations with sets *) |
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406 |
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407 lemma transfer_int_nat_set_functions: |
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408 "nat_set A \<Longrightarrow> card A = card (nat ` A)" |
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409 "{} = int ` ({}::nat set)" |
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410 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Un B = int ` (nat ` A Un nat ` B)" |
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411 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Int B = int ` (nat ` A Int nat ` B)" |
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412 "{x. x >= 0 & P x} = int ` {x. P(int x)}" |
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413 "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}" |
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414 (* need all variants of these! *) |
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415 by (simp_all only: is_nat_def transfer_nat_int_set_functions |
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416 transfer_nat_int_set_function_closures |
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417 transfer_nat_int_set_return_embed nat_0_le |
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418 cong: transfer_nat_int_set_cong) |
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419 |
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420 lemma transfer_int_nat_set_function_closures: |
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421 "nat_set {}" |
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422 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)" |
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423 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)" |
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424 "is_nat x \<Longrightarrow> nat_set {x..y}" |
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425 "nat_set {x. x >= 0 & P x}" |
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426 "nat_set (int ` C)" |
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427 "nat_set A \<Longrightarrow> x : A \<Longrightarrow> is_nat x" |
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428 by (simp_all only: transfer_nat_int_set_function_closures is_nat_def) |
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429 |
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430 lemma transfer_int_nat_set_relations: |
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431 "nat_set A \<Longrightarrow> finite A = finite (nat ` A)" |
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432 "is_nat x \<Longrightarrow> nat_set A \<Longrightarrow> (x : A) = (nat x : nat ` A)" |
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433 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A = B) = (nat ` A = nat ` B)" |
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434 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A < B) = (nat ` A < nat ` B)" |
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435 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A <= B) = (nat ` A <= nat ` B)" |
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436 by (simp_all only: is_nat_def transfer_nat_int_set_relations |
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437 transfer_nat_int_set_return_embed nat_0_le) |
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438 |
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439 lemma transfer_int_nat_set_return_embed: "nat ` int ` A = A" |
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440 by (simp only: transfer_nat_int_set_relations |
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441 transfer_nat_int_set_function_closures |
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442 transfer_nat_int_set_return_embed nat_0_le) |
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443 |
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444 lemma transfer_int_nat_set_cong: "(!!x. P x = P' x) \<Longrightarrow> |
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445 {(x::nat). P x} = {x. P' x}" |
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446 by auto |
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447 |
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448 declare TransferMorphism_int_nat[transfer add |
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449 return: transfer_int_nat_set_functions |
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450 transfer_int_nat_set_function_closures |
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451 transfer_int_nat_set_relations |
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452 transfer_int_nat_set_return_embed |
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453 cong: transfer_int_nat_set_cong |
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454 ] |
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455 |
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456 |
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457 (* setsum and setprod *) |
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458 |
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459 (* this handles the case where the *domain* of f is int *) |
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460 lemma transfer_int_nat_sum_prod: |
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461 "nat_set A \<Longrightarrow> setsum f A = setsum (%x. f (int x)) (nat ` A)" |
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462 "nat_set A \<Longrightarrow> setprod f A = setprod (%x. f (int x)) (nat ` A)" |
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463 apply (subst setsum_reindex) |
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464 apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff) |
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465 apply (subst setprod_reindex) |
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466 apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff |
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467 cong: setprod_cong) |
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468 done |
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469 |
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470 (* this handles the case where the *range* of f is int *) |
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471 lemma transfer_int_nat_sum_prod2: |
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472 "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> setsum f A = int(setsum (%x. nat (f x)) A)" |
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473 "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> |
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474 setprod f A = int(setprod (%x. nat (f x)) A)" |
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475 unfolding is_nat_def |
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476 apply (subst int_setsum, auto) |
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477 apply (subst int_setprod, auto simp add: cong: setprod_cong) |
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478 done |
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479 |
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480 declare TransferMorphism_int_nat[transfer add |
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481 return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2 |
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482 cong: setsum_cong setprod_cong] |
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483 |
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484 end |