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1 (* Title: HOL/Library/NatTransfer.thy |
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2 Authors: Jeremy Avigad and Amine Chaieb |
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3 |
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4 Sets up transfer from nats to ints and |
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5 back. |
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6 *) |
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7 |
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8 |
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9 header {* NatTransfer *} |
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10 |
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11 theory NatTransfer |
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12 imports Main Parity |
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13 uses ("Tools/transfer_data.ML") |
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14 begin |
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15 |
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16 subsection {* A transfer Method between isomorphic domains*} |
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17 |
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18 definition TransferMorphism:: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> bool" |
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19 where "TransferMorphism a B = True" |
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20 |
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21 use "Tools/transfer_data.ML" |
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22 |
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23 setup TransferData.setup |
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24 |
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25 |
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26 subsection {* Set up transfer from nat to int *} |
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27 |
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28 (* set up transfer direction *) |
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29 |
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30 lemma TransferMorphism_nat_int: "TransferMorphism nat (op <= (0::int))" |
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31 by (simp add: TransferMorphism_def) |
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32 |
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33 declare TransferMorphism_nat_int[transfer |
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34 add mode: manual |
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35 return: nat_0_le |
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36 labels: natint |
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37 ] |
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38 |
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39 (* basic functions and relations *) |
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40 |
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41 lemma transfer_nat_int_numerals: |
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42 "(0::nat) = nat 0" |
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43 "(1::nat) = nat 1" |
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44 "(2::nat) = nat 2" |
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45 "(3::nat) = nat 3" |
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46 by auto |
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47 |
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48 definition |
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49 tsub :: "int \<Rightarrow> int \<Rightarrow> int" |
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50 where |
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51 "tsub x y = (if x >= y then x - y else 0)" |
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52 |
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53 lemma tsub_eq: "x >= y \<Longrightarrow> tsub x y = x - y" |
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54 by (simp add: tsub_def) |
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55 |
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56 |
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57 lemma transfer_nat_int_functions: |
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58 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) + (nat y) = nat (x + y)" |
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59 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) * (nat y) = nat (x * y)" |
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60 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) - (nat y) = nat (tsub x y)" |
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61 "(x::int) >= 0 \<Longrightarrow> (nat x)^n = nat (x^n)" |
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62 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)" |
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63 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)" |
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64 by (auto simp add: eq_nat_nat_iff nat_mult_distrib |
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65 nat_power_eq nat_div_distrib nat_mod_distrib tsub_def) |
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66 |
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67 lemma transfer_nat_int_function_closures: |
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68 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x + y >= 0" |
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69 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x * y >= 0" |
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70 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> tsub x y >= 0" |
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71 "(x::int) >= 0 \<Longrightarrow> x^n >= 0" |
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72 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0" |
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73 "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0" |
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74 "(0::int) >= 0" |
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75 "(1::int) >= 0" |
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76 "(2::int) >= 0" |
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77 "(3::int) >= 0" |
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78 "int z >= 0" |
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79 apply (auto simp add: zero_le_mult_iff tsub_def) |
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80 apply (case_tac "y = 0") |
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81 apply auto |
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82 apply (subst pos_imp_zdiv_nonneg_iff, auto) |
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83 apply (case_tac "y = 0") |
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84 apply force |
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85 apply (rule pos_mod_sign) |
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86 apply arith |
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87 done |
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88 |
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89 lemma transfer_nat_int_relations: |
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90 "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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91 (nat (x::int) = nat y) = (x = y)" |
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92 "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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93 (nat (x::int) < nat y) = (x < y)" |
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94 "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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95 (nat (x::int) <= nat y) = (x <= y)" |
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96 "x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
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97 (nat (x::int) dvd nat y) = (x dvd y)" |
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98 by (auto simp add: zdvd_int even_nat_def) |
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99 |
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100 declare TransferMorphism_nat_int[transfer add return: |
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101 transfer_nat_int_numerals |
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102 transfer_nat_int_functions |
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103 transfer_nat_int_function_closures |
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104 transfer_nat_int_relations |
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105 ] |
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106 |
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107 |
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108 (* first-order quantifiers *) |
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109 |
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110 lemma transfer_nat_int_quantifiers: |
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111 "(ALL (x::nat). P x) = (ALL (x::int). x >= 0 \<longrightarrow> P (nat x))" |
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112 "(EX (x::nat). P x) = (EX (x::int). x >= 0 & P (nat x))" |
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113 by (rule all_nat, rule ex_nat) |
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114 |
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115 (* should we restrict these? *) |
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116 lemma all_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> |
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117 (ALL x. Q x \<longrightarrow> P x) = (ALL x. Q x \<longrightarrow> P' x)" |
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118 by auto |
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119 |
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120 lemma ex_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> |
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121 (EX x. Q x \<and> P x) = (EX x. Q x \<and> P' x)" |
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122 by auto |
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123 |
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124 declare TransferMorphism_nat_int[transfer add |
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125 return: transfer_nat_int_quantifiers |
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126 cong: all_cong ex_cong] |
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127 |
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128 |
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129 (* if *) |
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130 |
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131 lemma nat_if_cong: "(if P then (nat x) else (nat y)) = |
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132 nat (if P then x else y)" |
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133 by auto |
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134 |
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135 declare TransferMorphism_nat_int [transfer add return: nat_if_cong] |
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136 |
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137 |
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138 (* operations with sets *) |
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139 |
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140 definition |
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141 nat_set :: "int set \<Rightarrow> bool" |
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142 where |
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143 "nat_set S = (ALL x:S. x >= 0)" |
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144 |
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145 lemma transfer_nat_int_set_functions: |
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146 "card A = card (int ` A)" |
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147 "{} = nat ` ({}::int set)" |
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148 "A Un B = nat ` (int ` A Un int ` B)" |
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149 "A Int B = nat ` (int ` A Int int ` B)" |
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150 "{x. P x} = nat ` {x. x >= 0 & P(nat x)}" |
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151 "{..n} = nat ` {0..int n}" |
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152 "{m..n} = nat ` {int m..int n}" (* need all variants of these! *) |
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153 apply (rule card_image [symmetric]) |
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154 apply (auto simp add: inj_on_def image_def) |
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155 apply (rule_tac x = "int x" in bexI) |
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156 apply auto |
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157 apply (rule_tac x = "int x" in bexI) |
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158 apply auto |
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159 apply (rule_tac x = "int x" in bexI) |
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160 apply auto |
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161 apply (rule_tac x = "int x" in exI) |
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162 apply auto |
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163 apply (rule_tac x = "int x" in bexI) |
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164 apply auto |
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165 apply (rule_tac x = "int x" in bexI) |
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166 apply auto |
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167 done |
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168 |
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169 lemma transfer_nat_int_set_function_closures: |
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170 "nat_set {}" |
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171 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)" |
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172 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)" |
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173 "x >= 0 \<Longrightarrow> nat_set {x..y}" |
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174 "nat_set {x. x >= 0 & P x}" |
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175 "nat_set (int ` C)" |
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176 "nat_set A \<Longrightarrow> x : A \<Longrightarrow> x >= 0" (* does it hurt to turn this on? *) |
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177 unfolding nat_set_def apply auto |
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178 done |
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179 |
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180 lemma transfer_nat_int_set_relations: |
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181 "(finite A) = (finite (int ` A))" |
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182 "(x : A) = (int x : int ` A)" |
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183 "(A = B) = (int ` A = int ` B)" |
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184 "(A < B) = (int ` A < int ` B)" |
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185 "(A <= B) = (int ` A <= int ` B)" |
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186 |
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187 apply (rule iffI) |
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188 apply (erule finite_imageI) |
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189 apply (erule finite_imageD) |
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190 apply (auto simp add: image_def expand_set_eq inj_on_def) |
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191 apply (drule_tac x = "int x" in spec, auto) |
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192 apply (drule_tac x = "int x" in spec, auto) |
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193 apply (drule_tac x = "int x" in spec, auto) |
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194 done |
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195 |
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196 lemma transfer_nat_int_set_return_embed: "nat_set A \<Longrightarrow> |
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197 (int ` nat ` A = A)" |
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198 by (auto simp add: nat_set_def image_def) |
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199 |
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200 lemma transfer_nat_int_set_cong: "(!!x. x >= 0 \<Longrightarrow> P x = P' x) \<Longrightarrow> |
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201 {(x::int). x >= 0 & P x} = {x. x >= 0 & P' x}" |
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202 by auto |
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203 |
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204 declare TransferMorphism_nat_int[transfer add |
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205 return: transfer_nat_int_set_functions |
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206 transfer_nat_int_set_function_closures |
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207 transfer_nat_int_set_relations |
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208 transfer_nat_int_set_return_embed |
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209 cong: transfer_nat_int_set_cong |
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210 ] |
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211 |
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212 |
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213 (* setsum and setprod *) |
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214 |
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215 (* this handles the case where the *domain* of f is nat *) |
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216 lemma transfer_nat_int_sum_prod: |
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217 "setsum f A = setsum (%x. f (nat x)) (int ` A)" |
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218 "setprod f A = setprod (%x. f (nat x)) (int ` A)" |
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219 apply (subst setsum_reindex) |
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220 apply (unfold inj_on_def, auto) |
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221 apply (subst setprod_reindex) |
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222 apply (unfold inj_on_def o_def, auto) |
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223 done |
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224 |
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225 (* this handles the case where the *range* of f is nat *) |
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226 lemma transfer_nat_int_sum_prod2: |
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227 "setsum f A = nat(setsum (%x. int (f x)) A)" |
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228 "setprod f A = nat(setprod (%x. int (f x)) A)" |
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229 apply (subst int_setsum [symmetric]) |
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230 apply auto |
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231 apply (subst int_setprod [symmetric]) |
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232 apply auto |
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233 done |
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234 |
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235 lemma transfer_nat_int_sum_prod_closure: |
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236 "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0" |
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237 "nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0" |
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238 unfolding nat_set_def |
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239 apply (rule setsum_nonneg) |
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240 apply auto |
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241 apply (rule setprod_nonneg) |
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242 apply auto |
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243 done |
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244 |
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245 (* this version doesn't work, even with nat_set A \<Longrightarrow> |
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246 x : A \<Longrightarrow> x >= 0 turned on. Why not? |
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247 |
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248 also: what does =simp=> do? |
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249 |
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250 lemma transfer_nat_int_sum_prod_closure: |
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251 "(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0" |
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252 "(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0" |
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253 unfolding nat_set_def simp_implies_def |
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254 apply (rule setsum_nonneg) |
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255 apply auto |
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256 apply (rule setprod_nonneg) |
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257 apply auto |
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258 done |
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259 *) |
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260 |
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261 (* Making A = B in this lemma doesn't work. Why not? |
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262 Also, why aren't setsum_cong and setprod_cong enough, |
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263 with the previously mentioned rule turned on? *) |
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264 |
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265 lemma transfer_nat_int_sum_prod_cong: |
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266 "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
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267 setsum f A = setsum g B" |
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268 "A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
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269 setprod f A = setprod g B" |
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270 unfolding nat_set_def |
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271 apply (subst setsum_cong, assumption) |
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272 apply auto [2] |
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273 apply (subst setprod_cong, assumption, auto) |
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274 done |
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275 |
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276 declare TransferMorphism_nat_int[transfer add |
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277 return: transfer_nat_int_sum_prod transfer_nat_int_sum_prod2 |
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278 transfer_nat_int_sum_prod_closure |
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279 cong: transfer_nat_int_sum_prod_cong] |
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280 |
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281 (* lists *) |
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282 |
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283 definition |
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284 embed_list :: "nat list \<Rightarrow> int list" |
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285 where |
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286 "embed_list l = map int l"; |
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287 |
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288 definition |
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289 nat_list :: "int list \<Rightarrow> bool" |
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290 where |
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291 "nat_list l = nat_set (set l)"; |
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292 |
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293 definition |
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294 return_list :: "int list \<Rightarrow> nat list" |
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295 where |
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296 "return_list l = map nat l"; |
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297 |
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298 thm nat_0_le; |
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299 |
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300 lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow> |
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301 embed_list (return_list l) = l"; |
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302 unfolding embed_list_def return_list_def nat_list_def nat_set_def |
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303 apply (induct l); |
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304 apply auto; |
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305 done; |
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306 |
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307 lemma transfer_nat_int_list_functions: |
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308 "l @ m = return_list (embed_list l @ embed_list m)" |
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309 "[] = return_list []"; |
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310 unfolding return_list_def embed_list_def; |
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311 apply auto; |
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312 apply (induct l, auto); |
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313 apply (induct m, auto); |
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314 done; |
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315 |
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316 (* |
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317 lemma transfer_nat_int_fold1: "fold f l x = |
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318 fold (%x. f (nat x)) (embed_list l) x"; |
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319 *) |
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320 |
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321 |
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322 |
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323 |
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324 subsection {* Set up transfer from int to nat *} |
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325 |
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326 (* set up transfer direction *) |
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327 |
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328 lemma TransferMorphism_int_nat: "TransferMorphism int (UNIV :: nat set)" |
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329 by (simp add: TransferMorphism_def) |
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330 |
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331 declare TransferMorphism_int_nat[transfer add |
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332 mode: manual |
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333 (* labels: int-nat *) |
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334 return: nat_int |
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335 ] |
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336 |
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337 |
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338 (* basic functions and relations *) |
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339 |
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340 definition |
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341 is_nat :: "int \<Rightarrow> bool" |
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342 where |
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343 "is_nat x = (x >= 0)" |
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344 |
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345 lemma transfer_int_nat_numerals: |
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346 "0 = int 0" |
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347 "1 = int 1" |
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348 "2 = int 2" |
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349 "3 = int 3" |
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350 by auto |
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351 |
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352 lemma transfer_int_nat_functions: |
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353 "(int x) + (int y) = int (x + y)" |
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354 "(int x) * (int y) = int (x * y)" |
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355 "tsub (int x) (int y) = int (x - y)" |
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356 "(int x)^n = int (x^n)" |
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357 "(int x) div (int y) = int (x div y)" |
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358 "(int x) mod (int y) = int (x mod y)" |
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359 by (auto simp add: int_mult tsub_def int_power zdiv_int zmod_int) |
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360 |
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361 lemma transfer_int_nat_function_closures: |
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362 "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x + y)" |
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363 "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x * y)" |
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364 "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (tsub x y)" |
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365 "is_nat x \<Longrightarrow> is_nat (x^n)" |
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366 "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)" |
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367 "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)" |
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368 "is_nat 0" |
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369 "is_nat 1" |
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370 "is_nat 2" |
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371 "is_nat 3" |
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372 "is_nat (int z)" |
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373 by (simp_all only: is_nat_def transfer_nat_int_function_closures) |
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374 |
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375 lemma transfer_int_nat_relations: |
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376 "(int x = int y) = (x = y)" |
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377 "(int x < int y) = (x < y)" |
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378 "(int x <= int y) = (x <= y)" |
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379 "(int x dvd int y) = (x dvd y)" |
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380 "(even (int x)) = (even x)" |
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381 by (auto simp add: zdvd_int even_nat_def) |
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382 |
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383 declare TransferMorphism_int_nat[transfer add return: |
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384 transfer_int_nat_numerals |
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385 transfer_int_nat_functions |
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386 transfer_int_nat_function_closures |
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387 transfer_int_nat_relations |
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388 UNIV_code |
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389 ] |
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390 |
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391 |
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392 (* first-order quantifiers *) |
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393 |
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394 lemma transfer_int_nat_quantifiers: |
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395 "(ALL (x::int) >= 0. P x) = (ALL (x::nat). P (int x))" |
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396 "(EX (x::int) >= 0. P x) = (EX (x::nat). P (int x))" |
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397 apply (subst all_nat) |
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398 apply auto [1] |
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399 apply (subst ex_nat) |
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400 apply auto |
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401 done |
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402 |
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403 declare TransferMorphism_int_nat[transfer add |
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404 return: transfer_int_nat_quantifiers] |
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405 |
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406 |
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407 (* if *) |
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408 |
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409 lemma int_if_cong: "(if P then (int x) else (int y)) = |
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410 int (if P then x else y)" |
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411 by auto |
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412 |
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413 declare TransferMorphism_int_nat [transfer add return: int_if_cong] |
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414 |
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415 |
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416 |
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417 (* operations with sets *) |
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418 |
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419 lemma transfer_int_nat_set_functions: |
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420 "nat_set A \<Longrightarrow> card A = card (nat ` A)" |
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421 "{} = int ` ({}::nat set)" |
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422 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Un B = int ` (nat ` A Un nat ` B)" |
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423 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Int B = int ` (nat ` A Int nat ` B)" |
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424 "{x. x >= 0 & P x} = int ` {x. P(int x)}" |
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425 "is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}" |
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426 (* need all variants of these! *) |
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427 by (simp_all only: is_nat_def transfer_nat_int_set_functions |
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428 transfer_nat_int_set_function_closures |
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429 transfer_nat_int_set_return_embed nat_0_le |
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430 cong: transfer_nat_int_set_cong) |
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431 |
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432 lemma transfer_int_nat_set_function_closures: |
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433 "nat_set {}" |
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434 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)" |
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435 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)" |
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436 "is_nat x \<Longrightarrow> nat_set {x..y}" |
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437 "nat_set {x. x >= 0 & P x}" |
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438 "nat_set (int ` C)" |
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439 "nat_set A \<Longrightarrow> x : A \<Longrightarrow> is_nat x" |
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440 by (simp_all only: transfer_nat_int_set_function_closures is_nat_def) |
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441 |
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442 lemma transfer_int_nat_set_relations: |
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443 "nat_set A \<Longrightarrow> finite A = finite (nat ` A)" |
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444 "is_nat x \<Longrightarrow> nat_set A \<Longrightarrow> (x : A) = (nat x : nat ` A)" |
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445 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A = B) = (nat ` A = nat ` B)" |
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446 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A < B) = (nat ` A < nat ` B)" |
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447 "nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A <= B) = (nat ` A <= nat ` B)" |
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448 by (simp_all only: is_nat_def transfer_nat_int_set_relations |
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449 transfer_nat_int_set_return_embed nat_0_le) |
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450 |
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451 lemma transfer_int_nat_set_return_embed: "nat ` int ` A = A" |
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452 by (simp only: transfer_nat_int_set_relations |
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453 transfer_nat_int_set_function_closures |
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454 transfer_nat_int_set_return_embed nat_0_le) |
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455 |
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456 lemma transfer_int_nat_set_cong: "(!!x. P x = P' x) \<Longrightarrow> |
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457 {(x::nat). P x} = {x. P' x}" |
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458 by auto |
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459 |
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460 declare TransferMorphism_int_nat[transfer add |
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461 return: transfer_int_nat_set_functions |
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462 transfer_int_nat_set_function_closures |
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463 transfer_int_nat_set_relations |
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464 transfer_int_nat_set_return_embed |
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465 cong: transfer_int_nat_set_cong |
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466 ] |
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467 |
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468 |
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469 (* setsum and setprod *) |
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470 |
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471 (* this handles the case where the *domain* of f is int *) |
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472 lemma transfer_int_nat_sum_prod: |
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473 "nat_set A \<Longrightarrow> setsum f A = setsum (%x. f (int x)) (nat ` A)" |
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474 "nat_set A \<Longrightarrow> setprod f A = setprod (%x. f (int x)) (nat ` A)" |
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475 apply (subst setsum_reindex) |
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476 apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff) |
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477 apply (subst setprod_reindex) |
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478 apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff |
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479 cong: setprod_cong) |
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480 done |
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481 |
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482 (* this handles the case where the *range* of f is int *) |
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483 lemma transfer_int_nat_sum_prod2: |
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484 "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> setsum f A = int(setsum (%x. nat (f x)) A)" |
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485 "(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> |
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486 setprod f A = int(setprod (%x. nat (f x)) A)" |
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487 unfolding is_nat_def |
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488 apply (subst int_setsum, auto) |
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489 apply (subst int_setprod, auto simp add: cong: setprod_cong) |
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490 done |
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491 |
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492 declare TransferMorphism_int_nat[transfer add |
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493 return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2 |
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494 cong: setsum_cong setprod_cong] |
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495 |
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496 |
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497 subsection {* Test it out *} |
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498 |
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499 (* nat to int *) |
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500 |
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501 lemma ex1: "(x::nat) + y = y + x" |
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502 by auto |
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503 |
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504 thm ex1 [transferred] |
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505 |
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506 lemma ex2: "(a::nat) div b * b + a mod b = a" |
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507 by (rule mod_div_equality) |
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508 |
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509 thm ex2 [transferred] |
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510 |
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511 lemma ex3: "ALL (x::nat). ALL y. EX z. z >= x + y" |
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512 by auto |
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513 |
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514 thm ex3 [transferred natint] |
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515 |
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516 lemma ex4: "(x::nat) >= y \<Longrightarrow> (x - y) + y = x" |
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517 by auto |
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518 |
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519 thm ex4 [transferred] |
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520 |
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521 lemma ex5: "(2::nat) * (SUM i <= n. i) = n * (n + 1)" |
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522 by (induct n rule: nat_induct, auto) |
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523 |
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524 thm ex5 [transferred] |
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525 |
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526 theorem ex6: "0 <= (n::int) \<Longrightarrow> 2 * \<Sum>{0..n} = n * (n + 1)" |
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527 by (rule ex5 [transferred]) |
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528 |
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529 thm ex6 [transferred] |
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530 |
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531 thm ex5 [transferred, transferred] |
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532 |
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533 end |