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1 (* Title: HOL/Real/Float.thy |
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2 Author: Steven Obua |
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3 *) |
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4 |
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5 header {* Floating Point Representation of the Reals *} |
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6 |
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7 theory Float |
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8 imports Complex_Main |
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9 uses "~~/src/Tools/float.ML" ("~~/src/HOL/Tools/float_arith.ML") |
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10 begin |
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11 |
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12 definition |
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13 pow2 :: "int \<Rightarrow> real" where |
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14 "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))" |
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15 |
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16 definition |
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17 float :: "int * int \<Rightarrow> real" where |
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18 "float x = real (fst x) * pow2 (snd x)" |
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19 |
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20 lemma pow2_0[simp]: "pow2 0 = 1" |
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21 by (simp add: pow2_def) |
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22 |
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23 lemma pow2_1[simp]: "pow2 1 = 2" |
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24 by (simp add: pow2_def) |
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25 |
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26 lemma pow2_neg: "pow2 x = inverse (pow2 (-x))" |
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27 by (simp add: pow2_def) |
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28 |
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29 lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)" |
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30 proof - |
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31 have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith |
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32 have g: "! a b. a - -1 = a + (1::int)" by arith |
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33 have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)" |
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34 apply (auto, induct_tac n) |
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35 apply (simp_all add: pow2_def) |
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36 apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if]) |
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37 by (auto simp add: h) |
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38 show ?thesis |
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39 proof (induct a) |
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40 case (1 n) |
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41 from pos show ?case by (simp add: ring_simps) |
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42 next |
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43 case (2 n) |
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44 show ?case |
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45 apply (auto) |
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46 apply (subst pow2_neg[of "- int n"]) |
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47 apply (subst pow2_neg[of "-1 - int n"]) |
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48 apply (auto simp add: g pos) |
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49 done |
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50 qed |
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51 qed |
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52 |
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53 lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)" |
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54 proof (induct b) |
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55 case (1 n) |
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56 show ?case |
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57 proof (induct n) |
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58 case 0 |
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59 show ?case by simp |
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60 next |
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61 case (Suc m) |
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62 show ?case by (auto simp add: ring_simps pow2_add1 prems) |
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63 qed |
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64 next |
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65 case (2 n) |
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66 show ?case |
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67 proof (induct n) |
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68 case 0 |
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69 show ?case |
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70 apply (auto) |
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71 apply (subst pow2_neg[of "a + -1"]) |
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72 apply (subst pow2_neg[of "-1"]) |
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73 apply (simp) |
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74 apply (insert pow2_add1[of "-a"]) |
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75 apply (simp add: ring_simps) |
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76 apply (subst pow2_neg[of "-a"]) |
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77 apply (simp) |
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78 done |
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79 case (Suc m) |
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80 have a: "int m - (a + -2) = 1 + (int m - a + 1)" by arith |
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81 have b: "int m - -2 = 1 + (int m + 1)" by arith |
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82 show ?case |
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83 apply (auto) |
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84 apply (subst pow2_neg[of "a + (-2 - int m)"]) |
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85 apply (subst pow2_neg[of "-2 - int m"]) |
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86 apply (auto simp add: ring_simps) |
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87 apply (subst a) |
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88 apply (subst b) |
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89 apply (simp only: pow2_add1) |
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90 apply (subst pow2_neg[of "int m - a + 1"]) |
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91 apply (subst pow2_neg[of "int m + 1"]) |
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92 apply auto |
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93 apply (insert prems) |
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94 apply (auto simp add: ring_simps) |
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95 done |
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96 qed |
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97 qed |
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98 |
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99 lemma "float (a, e) + float (b, e) = float (a + b, e)" |
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100 by (simp add: float_def ring_simps) |
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101 |
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102 definition |
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103 int_of_real :: "real \<Rightarrow> int" where |
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104 "int_of_real x = (SOME y. real y = x)" |
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105 |
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106 definition |
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107 real_is_int :: "real \<Rightarrow> bool" where |
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108 "real_is_int x = (EX (u::int). x = real u)" |
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109 |
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110 lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))" |
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111 by (auto simp add: real_is_int_def int_of_real_def) |
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112 |
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113 lemma float_transfer: "real_is_int ((real a)*(pow2 c)) \<Longrightarrow> float (a, b) = float (int_of_real ((real a)*(pow2 c)), b - c)" |
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114 by (simp add: float_def real_is_int_def2 pow2_add[symmetric]) |
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115 |
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116 lemma pow2_int: "pow2 (int c) = 2^c" |
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117 by (simp add: pow2_def) |
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118 |
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119 lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)" |
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120 by (simp add: float_def pow2_int[symmetric] pow2_add[symmetric]) |
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121 |
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122 lemma real_is_int_real[simp]: "real_is_int (real (x::int))" |
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123 by (auto simp add: real_is_int_def int_of_real_def) |
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124 |
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125 lemma int_of_real_real[simp]: "int_of_real (real x) = x" |
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126 by (simp add: int_of_real_def) |
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127 |
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128 lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real (int_of_real x) = x" |
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129 by (auto simp add: int_of_real_def real_is_int_def) |
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130 |
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131 lemma real_is_int_add_int_of_real: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a+b)) = (int_of_real a) + (int_of_real b)" |
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132 by (auto simp add: int_of_real_def real_is_int_def) |
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133 |
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134 lemma real_is_int_add[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a+b)" |
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135 apply (subst real_is_int_def2) |
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136 apply (simp add: real_is_int_add_int_of_real real_int_of_real) |
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137 done |
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138 |
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139 lemma int_of_real_sub: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a-b)) = (int_of_real a) - (int_of_real b)" |
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140 by (auto simp add: int_of_real_def real_is_int_def) |
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141 |
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142 lemma real_is_int_sub[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a-b)" |
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143 apply (subst real_is_int_def2) |
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144 apply (simp add: int_of_real_sub real_int_of_real) |
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145 done |
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146 |
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147 lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x" |
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148 by (auto simp add: real_is_int_def) |
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149 |
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150 lemma int_of_real_mult: |
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151 assumes "real_is_int a" "real_is_int b" |
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152 shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)" |
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153 proof - |
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154 from prems have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto) |
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155 from prems have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto) |
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156 from a obtain a'::int where a':"a = real a'" by auto |
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157 from b obtain b'::int where b':"b = real b'" by auto |
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158 have r: "real a' * real b' = real (a' * b')" by auto |
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159 show ?thesis |
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160 apply (simp add: a' b') |
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161 apply (subst r) |
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162 apply (simp only: int_of_real_real) |
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163 done |
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164 qed |
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165 |
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166 lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)" |
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167 apply (subst real_is_int_def2) |
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168 apply (simp add: int_of_real_mult) |
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169 done |
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170 |
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171 lemma real_is_int_0[simp]: "real_is_int (0::real)" |
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172 by (simp add: real_is_int_def int_of_real_def) |
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173 |
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174 lemma real_is_int_1[simp]: "real_is_int (1::real)" |
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175 proof - |
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176 have "real_is_int (1::real) = real_is_int(real (1::int))" by auto |
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177 also have "\<dots> = True" by (simp only: real_is_int_real) |
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178 ultimately show ?thesis by auto |
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179 qed |
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180 |
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181 lemma real_is_int_n1: "real_is_int (-1::real)" |
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182 proof - |
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183 have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto |
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184 also have "\<dots> = True" by (simp only: real_is_int_real) |
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185 ultimately show ?thesis by auto |
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186 qed |
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187 |
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188 lemma real_is_int_number_of[simp]: "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)" |
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189 proof - |
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190 have neg1: "real_is_int (-1::real)" |
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191 proof - |
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192 have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto |
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193 also have "\<dots> = True" by (simp only: real_is_int_real) |
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194 ultimately show ?thesis by auto |
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195 qed |
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196 |
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197 { |
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198 fix x :: int |
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199 have "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)" |
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200 unfolding number_of_eq |
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201 apply (induct x) |
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202 apply (induct_tac n) |
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203 apply (simp) |
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204 apply (simp) |
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205 apply (induct_tac n) |
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206 apply (simp add: neg1) |
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207 proof - |
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208 fix n :: nat |
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209 assume rn: "(real_is_int (of_int (- (int (Suc n)))))" |
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210 have s: "-(int (Suc (Suc n))) = -1 + - (int (Suc n))" by simp |
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211 show "real_is_int (of_int (- (int (Suc (Suc n)))))" |
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212 apply (simp only: s of_int_add) |
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213 apply (rule real_is_int_add) |
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214 apply (simp add: neg1) |
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215 apply (simp only: rn) |
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216 done |
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217 qed |
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218 } |
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219 note Abs_Bin = this |
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220 { |
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221 fix x :: int |
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222 have "? u. x = u" |
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223 apply (rule exI[where x = "x"]) |
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224 apply (simp) |
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225 done |
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226 } |
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227 then obtain u::int where "x = u" by auto |
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228 with Abs_Bin show ?thesis by auto |
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229 qed |
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230 |
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231 lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)" |
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232 by (simp add: int_of_real_def) |
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233 |
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234 lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)" |
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235 proof - |
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236 have 1: "(1::real) = real (1::int)" by auto |
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237 show ?thesis by (simp only: 1 int_of_real_real) |
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238 qed |
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239 |
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240 lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b" |
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241 proof - |
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242 have "real_is_int (number_of b)" by simp |
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243 then have uu: "?! u::int. number_of b = real u" by (auto simp add: real_is_int_rep) |
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244 then obtain u::int where u:"number_of b = real u" by auto |
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245 have "number_of b = real ((number_of b)::int)" |
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246 by (simp add: number_of_eq real_of_int_def) |
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247 have ub: "number_of b = real ((number_of b)::int)" |
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248 by (simp add: number_of_eq real_of_int_def) |
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249 from uu u ub have unb: "u = number_of b" |
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250 by blast |
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251 have "int_of_real (number_of b) = u" by (simp add: u) |
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252 with unb show ?thesis by simp |
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253 qed |
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254 |
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255 lemma float_transfer_even: "even a \<Longrightarrow> float (a, b) = float (a div 2, b+1)" |
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256 apply (subst float_transfer[where a="a" and b="b" and c="-1", simplified]) |
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257 apply (simp_all add: pow2_def even_def real_is_int_def ring_simps) |
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258 apply (auto) |
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259 proof - |
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260 fix q::int |
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261 have a:"b - (-1\<Colon>int) = (1\<Colon>int) + b" by arith |
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262 show "(float (q, (b - (-1\<Colon>int)))) = (float (q, ((1\<Colon>int) + b)))" |
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263 by (simp add: a) |
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264 qed |
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265 |
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266 lemma int_div_zdiv: "int (a div b) = (int a) div (int b)" |
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267 by (rule zdiv_int) |
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268 |
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269 lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)" |
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270 by (rule zmod_int) |
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271 |
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272 lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a" |
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273 by arith |
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274 |
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275 function norm_float :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where |
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276 "norm_float a b = (if a \<noteq> 0 \<and> even a then norm_float (a div 2) (b + 1) |
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277 else if a = 0 then (0, 0) else (a, b))" |
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278 by auto |
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279 |
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280 termination by (relation "measure (nat o abs o fst)") |
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281 (auto intro: abs_div_2_less) |
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282 |
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283 lemma norm_float: "float x = float (split norm_float x)" |
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284 proof - |
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285 { |
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286 fix a b :: int |
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287 have norm_float_pair: "float (a, b) = float (norm_float a b)" |
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288 proof (induct a b rule: norm_float.induct) |
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289 case (1 u v) |
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290 show ?case |
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291 proof cases |
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292 assume u: "u \<noteq> 0 \<and> even u" |
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293 with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2) (v + 1))" by auto |
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294 with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even) |
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295 then show ?thesis |
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296 apply (subst norm_float.simps) |
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297 apply (simp add: ind) |
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298 done |
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299 next |
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300 assume "~(u \<noteq> 0 \<and> even u)" |
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301 then show ?thesis |
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302 by (simp add: prems float_def) |
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303 qed |
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304 qed |
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305 } |
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306 note helper = this |
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307 have "? a b. x = (a,b)" by auto |
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308 then obtain a b where "x = (a, b)" by blast |
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309 then show ?thesis by (simp add: helper) |
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310 qed |
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311 |
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312 lemma float_add_l0: "float (0, e) + x = x" |
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313 by (simp add: float_def) |
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314 |
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315 lemma float_add_r0: "x + float (0, e) = x" |
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316 by (simp add: float_def) |
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317 |
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318 lemma float_add: |
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319 "float (a1, e1) + float (a2, e2) = |
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320 (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1) |
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321 else float (a1*2^(nat (e1-e2))+a2, e2))" |
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322 apply (simp add: float_def ring_simps) |
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323 apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric]) |
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324 done |
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325 |
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326 lemma float_add_assoc1: |
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327 "(x + float (y1, e1)) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x" |
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328 by simp |
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329 |
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330 lemma float_add_assoc2: |
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331 "(float (y1, e1) + x) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x" |
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332 by simp |
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333 |
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334 lemma float_add_assoc3: |
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335 "float (y1, e1) + (x + float (y2, e2)) = (float (y1, e1) + float (y2, e2)) + x" |
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336 by simp |
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337 |
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338 lemma float_add_assoc4: |
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339 "float (y1, e1) + (float (y2, e2) + x) = (float (y1, e1) + float (y2, e2)) + x" |
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340 by simp |
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341 |
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342 lemma float_mult_l0: "float (0, e) * x = float (0, 0)" |
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343 by (simp add: float_def) |
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344 |
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345 lemma float_mult_r0: "x * float (0, e) = float (0, 0)" |
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346 by (simp add: float_def) |
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347 |
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348 definition |
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349 lbound :: "real \<Rightarrow> real" |
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350 where |
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351 "lbound x = min 0 x" |
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352 |
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353 definition |
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354 ubound :: "real \<Rightarrow> real" |
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355 where |
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356 "ubound x = max 0 x" |
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357 |
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358 lemma lbound: "lbound x \<le> x" |
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359 by (simp add: lbound_def) |
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360 |
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361 lemma ubound: "x \<le> ubound x" |
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362 by (simp add: ubound_def) |
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363 |
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364 lemma float_mult: |
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365 "float (a1, e1) * float (a2, e2) = |
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366 (float (a1 * a2, e1 + e2))" |
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367 by (simp add: float_def pow2_add) |
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368 |
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369 lemma float_minus: |
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370 "- (float (a,b)) = float (-a, b)" |
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371 by (simp add: float_def) |
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372 |
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373 lemma zero_less_pow2: |
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374 "0 < pow2 x" |
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375 proof - |
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376 { |
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377 fix y |
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378 have "0 <= y \<Longrightarrow> 0 < pow2 y" |
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379 by (induct y, induct_tac n, simp_all add: pow2_add) |
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380 } |
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381 note helper=this |
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382 show ?thesis |
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383 apply (case_tac "0 <= x") |
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384 apply (simp add: helper) |
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385 apply (subst pow2_neg) |
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386 apply (simp add: helper) |
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387 done |
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388 qed |
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389 |
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390 lemma zero_le_float: |
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391 "(0 <= float (a,b)) = (0 <= a)" |
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392 apply (auto simp add: float_def) |
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393 apply (auto simp add: zero_le_mult_iff zero_less_pow2) |
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394 apply (insert zero_less_pow2[of b]) |
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395 apply (simp_all) |
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396 done |
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397 |
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398 lemma float_le_zero: |
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399 "(float (a,b) <= 0) = (a <= 0)" |
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400 apply (auto simp add: float_def) |
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401 apply (auto simp add: mult_le_0_iff) |
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402 apply (insert zero_less_pow2[of b]) |
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403 apply auto |
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404 done |
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405 |
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406 lemma float_abs: |
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407 "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))" |
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408 apply (auto simp add: abs_if) |
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409 apply (simp_all add: zero_le_float[symmetric, of a b] float_minus) |
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410 done |
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411 |
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412 lemma float_zero: |
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413 "float (0, b) = 0" |
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414 by (simp add: float_def) |
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415 |
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416 lemma float_pprt: |
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417 "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))" |
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418 by (auto simp add: zero_le_float float_le_zero float_zero) |
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419 |
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420 lemma pprt_lbound: "pprt (lbound x) = float (0, 0)" |
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421 apply (simp add: float_def) |
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422 apply (rule pprt_eq_0) |
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423 apply (simp add: lbound_def) |
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424 done |
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425 |
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426 lemma nprt_ubound: "nprt (ubound x) = float (0, 0)" |
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427 apply (simp add: float_def) |
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428 apply (rule nprt_eq_0) |
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429 apply (simp add: ubound_def) |
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430 done |
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431 |
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432 lemma float_nprt: |
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433 "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))" |
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434 by (auto simp add: zero_le_float float_le_zero float_zero) |
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435 |
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436 lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1" |
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437 by auto |
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438 |
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439 lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)" |
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440 by simp |
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441 |
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442 lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)" |
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443 by simp |
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444 |
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445 lemma mult_left_one: "1 * a = (a::'a::semiring_1)" |
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446 by simp |
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447 |
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448 lemma mult_right_one: "a * 1 = (a::'a::semiring_1)" |
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449 by simp |
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450 |
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451 lemma int_pow_0: "(a::int)^(Numeral0) = 1" |
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452 by simp |
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453 |
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454 lemma int_pow_1: "(a::int)^(Numeral1) = a" |
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455 by simp |
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456 |
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457 lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0" |
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458 by simp |
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459 |
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460 lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1" |
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461 by simp |
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462 |
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463 lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0" |
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464 by simp |
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465 |
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466 lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1" |
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467 by simp |
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468 |
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469 lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1" |
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470 by simp |
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471 |
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472 lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1" |
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473 proof - |
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474 have 1:"((-1)::nat) = 0" |
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475 by simp |
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476 show ?thesis by (simp add: 1) |
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477 qed |
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478 |
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479 lemma fst_cong: "a=a' \<Longrightarrow> fst (a,b) = fst (a',b)" |
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480 by simp |
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481 |
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482 lemma snd_cong: "b=b' \<Longrightarrow> snd (a,b) = snd (a,b')" |
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483 by simp |
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484 |
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485 lemma lift_bool: "x \<Longrightarrow> x=True" |
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486 by simp |
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487 |
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488 lemma nlift_bool: "~x \<Longrightarrow> x=False" |
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489 by simp |
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490 |
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491 lemma not_false_eq_true: "(~ False) = True" by simp |
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492 |
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493 lemma not_true_eq_false: "(~ True) = False" by simp |
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494 |
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495 lemmas binarith = |
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496 normalize_bin_simps |
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497 pred_bin_simps succ_bin_simps |
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498 add_bin_simps minus_bin_simps mult_bin_simps |
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499 |
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500 lemma int_eq_number_of_eq: |
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501 "(((number_of v)::int)=(number_of w)) = iszero ((number_of (v + uminus w))::int)" |
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502 by simp |
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503 |
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504 lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)" |
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505 by (simp only: iszero_number_of_Pls) |
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506 |
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507 lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))" |
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508 by simp |
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509 |
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510 lemma int_iszero_number_of_Bit0: "iszero ((number_of (Int.Bit0 w))::int) = iszero ((number_of w)::int)" |
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511 by simp |
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512 |
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513 lemma int_iszero_number_of_Bit1: "\<not> iszero ((number_of (Int.Bit1 w))::int)" |
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514 by simp |
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515 |
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516 lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)" |
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517 by simp |
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518 |
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519 lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))" |
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520 by simp |
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521 |
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522 lemma int_neg_number_of_Min: "neg (-1::int)" |
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523 by simp |
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524 |
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525 lemma int_neg_number_of_Bit0: "neg ((number_of (Int.Bit0 w))::int) = neg ((number_of w)::int)" |
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526 by simp |
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527 |
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528 lemma int_neg_number_of_Bit1: "neg ((number_of (Int.Bit1 w))::int) = neg ((number_of w)::int)" |
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529 by simp |
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530 |
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531 lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (y + (uminus x)))::int))" |
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532 by simp |
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533 |
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534 lemmas intarithrel = |
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535 int_eq_number_of_eq |
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536 lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_Bit0 |
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537 lift_bool[OF int_iszero_number_of_Bit1] int_less_number_of_eq_neg nlift_bool[OF int_not_neg_number_of_Pls] lift_bool[OF int_neg_number_of_Min] |
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538 int_neg_number_of_Bit0 int_neg_number_of_Bit1 int_le_number_of_eq |
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539 |
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540 lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)" |
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541 by simp |
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542 |
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543 lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (v + (uminus w))" |
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544 by simp |
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545 |
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546 lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (v * w)" |
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547 by simp |
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548 |
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549 lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (uminus v)" |
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550 by simp |
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551 |
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552 lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym |
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553 |
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554 lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of |
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555 |
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556 lemmas powerarith = nat_number_of zpower_number_of_even |
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557 zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring] |
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558 zpower_Pls zpower_Min |
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559 |
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560 lemmas floatarith[simplified norm_0_1] = float_add float_add_l0 float_add_r0 float_mult float_mult_l0 float_mult_r0 |
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561 float_minus float_abs zero_le_float float_pprt float_nprt pprt_lbound nprt_ubound |
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562 |
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563 (* for use with the compute oracle *) |
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564 lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false |
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565 |
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566 use "~~/src/HOL/Tools/float_arith.ML" |
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567 |
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568 end |