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1 (* Title: SMT_Examples.thy |
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2 Author: Sascha Böhme, TU Muenchen |
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3 *) |
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4 |
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5 header {* Examples for the 'smt' tactic. *} |
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6 |
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7 theory SMT_Examples |
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8 imports "../SMT" |
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9 begin |
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10 |
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11 declare [[smt_solver=z3, z3_proofs=false]] |
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12 declare [[smt_trace=true]] (*FIXME*) |
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13 |
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14 |
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15 section {* Propositional and first-order logic *} |
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16 |
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17 lemma "True" by smt |
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18 lemma "p \<or> \<not>p" by smt |
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19 lemma "(p \<and> True) = p" by smt |
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20 lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q" by smt |
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21 lemma "(a \<and> b) \<or> (c \<and> d) \<Longrightarrow> (a \<and> b) \<or> (c \<and> d)" by smt |
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22 lemma "P=P=P=P=P=P=P=P=P=P" by smt |
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23 |
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24 axiomatization symm_f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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25 symm_f: "symm_f x y = symm_f y x" |
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26 lemma "a = a \<and> symm_f a b = symm_f b a" by (smt add: symm_f) |
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27 |
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28 |
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29 section {* Linear arithmetic *} |
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30 |
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31 lemma "(3::int) = 3" by smt |
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32 lemma "(3::real) = 3" by smt |
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33 lemma "(3 :: int) + 1 = 4" by smt |
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34 lemma "max (3::int) 8 > 5" by smt |
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35 lemma "abs (x :: real) + abs y \<ge> abs (x + y)" by smt |
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36 lemma "let x = (2 :: int) in x + x \<noteq> 5" by smt |
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37 |
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38 text{* |
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39 The following example was taken from HOL/ex/PresburgerEx.thy, where it says: |
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40 |
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41 This following theorem proves that all solutions to the |
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42 recurrence relation $x_{i+2} = |x_{i+1}| - x_i$ are periodic with |
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43 period 9. The example was brought to our attention by John |
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44 Harrison. It does does not require Presburger arithmetic but merely |
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45 quantifier-free linear arithmetic and holds for the rationals as well. |
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46 |
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47 Warning: it takes (in 2006) over 4.2 minutes! |
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48 |
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49 There, it is proved by "arith". SMT is able to prove this within a fraction |
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50 of one second. |
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51 *} |
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52 |
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53 lemma "\<lbrakk> x3 = abs x2 - x1; x4 = abs x3 - x2; x5 = abs x4 - x3; |
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54 x6 = abs x5 - x4; x7 = abs x6 - x5; x8 = abs x7 - x6; |
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55 x9 = abs x8 - x7; x10 = abs x9 - x8; x11 = abs x10 - x9 \<rbrakk> |
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56 \<Longrightarrow> x1 = x10 & x2 = (x11::int)" |
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57 by smt |
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58 |
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59 lemma "\<exists>x::int. 0 < x" by smt |
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60 lemma "\<exists>x::real. 0 < x" by smt |
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61 lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)" by smt |
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62 lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)" by smt |
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63 lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))" by smt |
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64 lemma "~ (\<exists>x::int. False)" by smt |
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65 |
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66 |
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67 section {* Non-linear arithmetic *} |
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68 |
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69 lemma "((x::int) * (1 + y) - x * (1 - y)) = (2 * x * y)" by smt |
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70 lemma |
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71 "(U::int) + (1 + p) * (b + e) + p * d = |
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72 U + (2 * (1 + p) * (b + e) + (1 + p) * d + d * p) - (1 + p) * (b + d + e)" |
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73 by smt |
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74 |
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75 |
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76 section {* Linear arithmetic for natural numbers *} |
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77 |
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78 lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a" by smt |
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79 lemma "let x = (1::nat) + y in x - y > 0 * x" by smt |
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80 lemma |
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81 "let x = (1::nat) + y in |
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82 let P = (if x > 0 then True else False) in |
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83 False \<or> P = (x - 1 = y) \<or> (\<not>P \<longrightarrow> False)" |
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84 by smt |
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85 |
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86 |
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87 section {* Bitvectors *} |
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88 |
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89 locale bv |
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90 begin |
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91 |
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92 declare [[smt_solver=z3]] |
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93 |
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94 lemma "(27 :: 4 word) = -5" by smt |
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95 lemma "(27 :: 4 word) = 11" by smt |
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96 lemma "23 < (27::8 word)" by smt |
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97 lemma "27 + 11 = (6::5 word)" by smt |
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98 lemma "7 * 3 = (21::8 word)" by smt |
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99 lemma "11 - 27 = (-16::8 word)" by smt |
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100 lemma "- -11 = (11::5 word)" by smt |
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101 lemma "-40 + 1 = (-39::7 word)" by smt |
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102 lemma "a + 2 * b + c - b = (b + c) + (a :: 32 word)" by smt |
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103 |
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104 lemma "0b110 AND 0b101 = (0b100 :: 32 word)" by smt |
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105 lemma "0b110 OR 0b011 = (0b111 :: 8 word)" by smt |
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106 lemma "0xF0 XOR 0xFF = (0x0F :: 8 word)" by smt |
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107 lemma "NOT (0xF0 :: 16 word) = 0xFF0F" by smt |
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108 |
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109 lemma "word_cat (27::4 word) (27::8 word) = (2843::12 word)" by smt |
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110 lemma "word_cat (0b0011::4 word) (0b1111::6word) = (0b0011001111 :: 10 word)" |
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111 by smt |
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112 |
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113 lemma "slice 1 (0b10110 :: 4 word) = (0b11 :: 2 word)" by smt |
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114 |
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115 lemma "ucast (0b1010 :: 4 word) = (0b1010 :: 10 word)" by smt |
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116 lemma "scast (0b1010 :: 4 word) = (0b111010 :: 6 word)" by smt |
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117 |
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118 lemma "bv_lshr 0b10011 2 = (0b100::8 word)" by smt |
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119 lemma "bv_ashr 0b10011 2 = (0b100::8 word)" by smt |
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120 |
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121 lemma "word_rotr 2 0b0110 = (0b1001::4 word)" by smt |
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122 lemma "word_rotl 1 0b1110 = (0b1101::4 word)" by smt |
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123 |
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124 lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)" by smt |
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125 |
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126 lemma "w < 256 \<Longrightarrow> (w :: 16 word) AND 0x00FF = w" by smt |
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127 |
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128 end |
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129 |
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130 |
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131 section {* Pairs *} |
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132 |
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133 lemma "fst (x, y) = a \<Longrightarrow> x = a" by smt |
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134 lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2" by smt |
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135 |
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136 |
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137 section {* Higher-order problems and recursion *} |
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138 |
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139 lemma "(f g x = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)" by smt |
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140 lemma "P ((2::int) < 3) = P True" by smt |
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141 lemma "P ((2::int) < 3) = (P True :: bool)" by smt |
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142 lemma "P (0 \<le> (a :: 4 word)) = P True" using [[smt_solver=z3]] by smt |
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143 lemma "id 3 = 3 \<and> id True = True" by (smt add: id_def) |
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144 lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> ((f (i1 := v1)) (i2 := v2)) i = f i" by smt |
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145 lemma "map (\<lambda>i::nat. i + 1) [0, 1] = [1, 2]" by (smt add: map.simps) |
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146 lemma "(ALL x. P x) | ~ All P" by smt |
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147 |
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148 fun dec_10 :: "nat \<Rightarrow> nat" where |
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149 "dec_10 n = (if n < 10 then n else dec_10 (n - 10))" |
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150 lemma "dec_10 (4 * dec_10 4) = 6" by (smt add: dec_10.simps) |
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151 |
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152 axiomatization |
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153 eval_dioph :: "int list \<Rightarrow> nat list \<Rightarrow> int" |
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154 where |
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155 eval_dioph_mod: |
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156 "eval_dioph ks xs mod int n = eval_dioph ks (map (\<lambda>x. x mod n) xs) mod int n" |
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157 and |
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158 eval_dioph_div_mult: |
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159 "eval_dioph ks (map (\<lambda>x. x div n) xs) * int n + |
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160 eval_dioph ks (map (\<lambda>x. x mod n) xs) = eval_dioph ks xs" |
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161 lemma |
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162 "(eval_dioph ks xs = l) = |
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163 (eval_dioph ks (map (\<lambda>x. x mod 2) xs) mod 2 = l mod 2 \<and> |
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164 eval_dioph ks (map (\<lambda>x. x div 2) xs) = |
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165 (l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2)" |
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166 by (smt add: eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2]) |
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167 |
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168 |
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169 section {* Monomorphization examples *} |
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170 |
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171 definition P :: "'a \<Rightarrow> bool" where "P x = True" |
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172 lemma poly_P: "P x \<and> (P [x] \<or> \<not>P[x])" by (simp add: P_def) |
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173 lemma "P (1::int)" by (smt add: poly_P) |
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174 |
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175 consts g :: "'a \<Rightarrow> nat" |
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176 axioms |
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177 g1: "g (Some x) = g [x]" |
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178 g2: "g None = g []" |
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179 g3: "g xs = length xs" |
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180 lemma "g (Some (3::int)) = g (Some True)" by (smt add: g1 g2 g3 list.size) |
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181 |
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182 end |