1 theory ComputeNumeral |
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2 imports ComputeHOL ComputeFloat |
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3 begin |
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4 |
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5 (* normalization of bit strings *) |
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6 lemmas bitnorm = normalize_bin_simps |
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7 |
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8 (* neg for bit strings *) |
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9 lemma neg1: "neg Int.Pls = False" by (simp add: Int.Pls_def) |
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10 lemma neg2: "neg Int.Min = True" apply (subst Int.Min_def) by auto |
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11 lemma neg3: "neg (Int.Bit0 x) = neg x" apply (simp add: neg_def) apply (subst Bit0_def) by auto |
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12 lemma neg4: "neg (Int.Bit1 x) = neg x" apply (simp add: neg_def) apply (subst Bit1_def) by auto |
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13 lemmas bitneg = neg1 neg2 neg3 neg4 |
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14 |
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15 (* iszero for bit strings *) |
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16 lemma iszero1: "iszero Int.Pls = True" by (simp add: Int.Pls_def iszero_def) |
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17 lemma iszero2: "iszero Int.Min = False" apply (subst Int.Min_def) apply (subst iszero_def) by simp |
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18 lemma iszero3: "iszero (Int.Bit0 x) = iszero x" apply (subst Int.Bit0_def) apply (subst iszero_def)+ by auto |
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19 lemma iszero4: "iszero (Int.Bit1 x) = False" apply (subst Int.Bit1_def) apply (subst iszero_def)+ apply simp by arith |
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20 lemmas bitiszero = iszero1 iszero2 iszero3 iszero4 |
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21 |
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22 (* lezero for bit strings *) |
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23 constdefs |
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24 "lezero x == (x \<le> 0)" |
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25 lemma lezero1: "lezero Int.Pls = True" unfolding Int.Pls_def lezero_def by auto |
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26 lemma lezero2: "lezero Int.Min = True" unfolding Int.Min_def lezero_def by auto |
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27 lemma lezero3: "lezero (Int.Bit0 x) = lezero x" unfolding Int.Bit0_def lezero_def by auto |
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28 lemma lezero4: "lezero (Int.Bit1 x) = neg x" unfolding Int.Bit1_def lezero_def neg_def by auto |
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29 lemmas bitlezero = lezero1 lezero2 lezero3 lezero4 |
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30 |
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31 (* equality for bit strings *) |
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32 lemmas biteq = eq_bin_simps |
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33 |
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34 (* x < y for bit strings *) |
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35 lemmas bitless = less_bin_simps |
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36 |
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37 (* x \<le> y for bit strings *) |
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38 lemmas bitle = le_bin_simps |
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39 |
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40 (* succ for bit strings *) |
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41 lemmas bitsucc = succ_bin_simps |
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42 |
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43 (* pred for bit strings *) |
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44 lemmas bitpred = pred_bin_simps |
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45 |
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46 (* unary minus for bit strings *) |
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47 lemmas bituminus = minus_bin_simps |
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48 |
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49 (* addition for bit strings *) |
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50 lemmas bitadd = add_bin_simps |
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51 |
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52 (* multiplication for bit strings *) |
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53 lemma mult_Pls_right: "x * Int.Pls = Int.Pls" by (simp add: Pls_def) |
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54 lemma mult_Min_right: "x * Int.Min = - x" by (subst mult_commute, simp add: mult_Min) |
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55 lemma multb0x: "(Int.Bit0 x) * y = Int.Bit0 (x * y)" by (rule mult_Bit0) |
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56 lemma multxb0: "x * (Int.Bit0 y) = Int.Bit0 (x * y)" unfolding Bit0_def by simp |
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57 lemma multb1: "(Int.Bit1 x) * (Int.Bit1 y) = Int.Bit1 (Int.Bit0 (x * y) + x + y)" |
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58 unfolding Bit0_def Bit1_def by (simp add: algebra_simps) |
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59 lemmas bitmul = mult_Pls mult_Min mult_Pls_right mult_Min_right multb0x multxb0 multb1 |
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60 |
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61 lemmas bitarith = bitnorm bitiszero bitneg bitlezero biteq bitless bitle bitsucc bitpred bituminus bitadd bitmul |
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62 |
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63 constdefs |
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64 "nat_norm_number_of (x::nat) == x" |
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65 |
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66 lemma nat_norm_number_of: "nat_norm_number_of (number_of w) = (if lezero w then 0 else number_of w)" |
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67 apply (simp add: nat_norm_number_of_def) |
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68 unfolding lezero_def iszero_def neg_def |
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69 apply (simp add: numeral_simps) |
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70 done |
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71 |
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72 (* Normalization of nat literals *) |
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73 lemma natnorm0: "(0::nat) = number_of (Int.Pls)" by auto |
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74 lemma natnorm1: "(1 :: nat) = number_of (Int.Bit1 Int.Pls)" by auto |
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75 lemmas natnorm = natnorm0 natnorm1 nat_norm_number_of |
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76 |
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77 (* Suc *) |
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78 lemma natsuc: "Suc (number_of x) = (if neg x then 1 else number_of (Int.succ x))" by (auto simp add: number_of_is_id) |
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79 |
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80 (* Addition for nat *) |
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81 lemma natadd: "number_of x + ((number_of y)::nat) = (if neg x then (number_of y) else (if neg y then number_of x else (number_of (x + y))))" |
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82 unfolding nat_number_of_def number_of_is_id neg_def |
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83 by auto |
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84 |
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85 (* Subtraction for nat *) |
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86 lemma natsub: "(number_of x) - ((number_of y)::nat) = |
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87 (if neg x then 0 else (if neg y then number_of x else (nat_norm_number_of (number_of (x + (- y))))))" |
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88 unfolding nat_norm_number_of |
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89 by (auto simp add: number_of_is_id neg_def lezero_def iszero_def Let_def nat_number_of_def) |
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90 |
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91 (* Multiplication for nat *) |
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92 lemma natmul: "(number_of x) * ((number_of y)::nat) = |
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93 (if neg x then 0 else (if neg y then 0 else number_of (x * y)))" |
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94 unfolding nat_number_of_def number_of_is_id neg_def |
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95 by (simp add: nat_mult_distrib) |
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96 |
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97 lemma nateq: "(((number_of x)::nat) = (number_of y)) = ((lezero x \<and> lezero y) \<or> (x = y))" |
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98 by (auto simp add: iszero_def lezero_def neg_def number_of_is_id) |
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99 |
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100 lemma natless: "(((number_of x)::nat) < (number_of y)) = ((x < y) \<and> (\<not> (lezero y)))" |
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101 by (simp add: lezero_def numeral_simps not_le) |
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102 |
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103 lemma natle: "(((number_of x)::nat) \<le> (number_of y)) = (y < x \<longrightarrow> lezero x)" |
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104 by (auto simp add: number_of_is_id lezero_def nat_number_of_def) |
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105 |
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106 fun natfac :: "nat \<Rightarrow> nat" |
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107 where |
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108 "natfac n = (if n = 0 then 1 else n * (natfac (n - 1)))" |
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109 |
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110 lemmas compute_natarith = bitarith natnorm natsuc natadd natsub natmul nateq natless natle natfac.simps |
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111 |
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112 lemma number_eq: "(((number_of x)::'a::{number_ring, ordered_idom}) = (number_of y)) = (x = y)" |
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113 unfolding number_of_eq |
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114 apply simp |
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115 done |
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116 |
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117 lemma number_le: "(((number_of x)::'a::{number_ring, ordered_idom}) \<le> (number_of y)) = (x \<le> y)" |
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118 unfolding number_of_eq |
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119 apply simp |
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120 done |
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121 |
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122 lemma number_less: "(((number_of x)::'a::{number_ring, ordered_idom}) < (number_of y)) = (x < y)" |
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123 unfolding number_of_eq |
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124 apply simp |
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125 done |
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126 |
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127 lemma number_diff: "((number_of x)::'a::{number_ring, ordered_idom}) - number_of y = number_of (x + (- y))" |
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128 apply (subst diff_number_of_eq) |
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129 apply simp |
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130 done |
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131 |
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132 lemmas number_norm = number_of_Pls[symmetric] numeral_1_eq_1[symmetric] |
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133 |
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134 lemmas compute_numberarith = number_of_minus[symmetric] number_of_add[symmetric] number_diff number_of_mult[symmetric] number_norm number_eq number_le number_less |
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135 |
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136 lemma compute_real_of_nat_number_of: "real ((number_of v)::nat) = (if neg v then 0 else number_of v)" |
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137 by (simp only: real_of_nat_number_of number_of_is_id) |
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138 |
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139 lemma compute_nat_of_int_number_of: "nat ((number_of v)::int) = (number_of v)" |
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140 by simp |
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141 |
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142 lemmas compute_num_conversions = compute_real_of_nat_number_of compute_nat_of_int_number_of real_number_of |
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143 |
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144 lemmas zpowerarith = zpower_number_of_even |
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145 zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring] |
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146 zpower_Pls zpower_Min |
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147 |
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148 (* div, mod *) |
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149 |
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150 lemma adjust: "adjust b (q, r) = (if 0 \<le> r - b then (2 * q + 1, r - b) else (2 * q, r))" |
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151 by (auto simp only: adjust_def) |
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152 |
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153 lemma negateSnd: "negateSnd (q, r) = (q, -r)" |
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154 by (simp add: negateSnd_def) |
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155 |
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156 lemma divmod: "IntDiv.divmod a b = (if 0\<le>a then |
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157 if 0\<le>b then posDivAlg a b |
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158 else if a=0 then (0, 0) |
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159 else negateSnd (negDivAlg (-a) (-b)) |
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160 else |
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161 if 0<b then negDivAlg a b |
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162 else negateSnd (posDivAlg (-a) (-b)))" |
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163 by (auto simp only: IntDiv.divmod_def) |
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164 |
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165 lemmas compute_div_mod = div_def mod_def divmod adjust negateSnd posDivAlg.simps negDivAlg.simps |
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166 |
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167 |
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168 |
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169 (* collecting all the theorems *) |
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170 |
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171 lemma even_Pls: "even (Int.Pls) = True" |
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172 apply (unfold Pls_def even_def) |
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173 by simp |
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174 |
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175 lemma even_Min: "even (Int.Min) = False" |
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176 apply (unfold Min_def even_def) |
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177 by simp |
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178 |
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179 lemma even_B0: "even (Int.Bit0 x) = True" |
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180 apply (unfold Bit0_def) |
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181 by simp |
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182 |
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183 lemma even_B1: "even (Int.Bit1 x) = False" |
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184 apply (unfold Bit1_def) |
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185 by simp |
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186 |
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187 lemma even_number_of: "even ((number_of w)::int) = even w" |
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188 by (simp only: number_of_is_id) |
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189 |
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190 lemmas compute_even = even_Pls even_Min even_B0 even_B1 even_number_of |
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191 |
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192 lemmas compute_numeral = compute_if compute_let compute_pair compute_bool |
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193 compute_natarith compute_numberarith max_def min_def compute_num_conversions zpowerarith compute_div_mod compute_even |
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194 |
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195 end |
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