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1 (* Title: HOL/SMT/SMT_Examples.thy |
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2 Author: Sascha Boehme, 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=true]] |
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12 |
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13 declare [[smt_certificates="$ISABELLE_SMT/Examples/SMT_Examples.certs"]] |
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14 |
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15 text {* |
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16 To avoid re-generation of certificates, |
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17 the following option is set to "false": |
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18 *} |
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19 |
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20 declare [[smt_fixed=true]] |
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21 |
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22 |
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23 |
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24 section {* Propositional and first-order logic *} |
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25 |
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26 lemma "True" by smt |
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27 |
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28 lemma "p \<or> \<not>p" by smt |
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29 |
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30 lemma "(p \<and> True) = p" by smt |
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31 |
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32 lemma "(p \<or> q) \<and> \<not>p \<Longrightarrow> q" by smt |
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33 |
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34 lemma "(a \<and> b) \<or> (c \<and> d) \<Longrightarrow> (a \<and> b) \<or> (c \<and> d)" |
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35 using [[z3_proofs=false]] (* no Z3 proof *) |
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36 by smt |
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37 |
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38 lemma "(p1 \<and> p2) \<or> p3 \<longrightarrow> (p1 \<longrightarrow> (p3 \<and> p2) \<or> (p1 \<and> p3)) \<or> p1" by smt |
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39 |
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40 lemma "P=P=P=P=P=P=P=P=P=P" by smt |
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41 |
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42 lemma |
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43 assumes "a | b | c | d" |
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44 and "e | f | (a & d)" |
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45 and "~(a | (c & ~c)) | b" |
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46 and "~(b & (x | ~x)) | c" |
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47 and "~(d | False) | c" |
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48 and "~(c | (~p & (p | (q & ~q))))" |
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49 shows False |
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50 using assms by smt |
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51 |
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52 axiomatization symm_f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where |
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53 symm_f: "symm_f x y = symm_f y x" |
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54 lemma "a = a \<and> symm_f a b = symm_f b a" by (smt symm_f) |
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55 |
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56 (* |
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57 Taken from ~~/src/HOL/ex/SAT_Examples.thy. |
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58 Translated from TPTP problem library: PUZ015-2.006.dimacs |
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59 *) |
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60 lemma |
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61 assumes "~x0" |
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62 and "~x30" |
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63 and "~x29" |
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64 and "~x59" |
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65 and "x1 | x31 | x0" |
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66 and "x2 | x32 | x1" |
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67 and "x3 | x33 | x2" |
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68 and "x4 | x34 | x3" |
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69 and "x35 | x4" |
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70 and "x5 | x36 | x30" |
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71 and "x6 | x37 | x5 | x31" |
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72 and "x7 | x38 | x6 | x32" |
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73 and "x8 | x39 | x7 | x33" |
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74 and "x9 | x40 | x8 | x34" |
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75 and "x41 | x9 | x35" |
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76 and "x10 | x42 | x36" |
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77 and "x11 | x43 | x10 | x37" |
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78 and "x12 | x44 | x11 | x38" |
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79 and "x13 | x45 | x12 | x39" |
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80 and "x14 | x46 | x13 | x40" |
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81 and "x47 | x14 | x41" |
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82 and "x15 | x48 | x42" |
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83 and "x16 | x49 | x15 | x43" |
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84 and "x17 | x50 | x16 | x44" |
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85 and "x18 | x51 | x17 | x45" |
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86 and "x19 | x52 | x18 | x46" |
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87 and "x53 | x19 | x47" |
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88 and "x20 | x54 | x48" |
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89 and "x21 | x55 | x20 | x49" |
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90 and "x22 | x56 | x21 | x50" |
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91 and "x23 | x57 | x22 | x51" |
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92 and "x24 | x58 | x23 | x52" |
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93 and "x59 | x24 | x53" |
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94 and "x25 | x54" |
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95 and "x26 | x25 | x55" |
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96 and "x27 | x26 | x56" |
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97 and "x28 | x27 | x57" |
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98 and "x29 | x28 | x58" |
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99 and "~x1 | ~x31" |
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100 and "~x1 | ~x0" |
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101 and "~x31 | ~x0" |
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102 and "~x2 | ~x32" |
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103 and "~x2 | ~x1" |
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104 and "~x32 | ~x1" |
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105 and "~x3 | ~x33" |
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106 and "~x3 | ~x2" |
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107 and "~x33 | ~x2" |
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108 and "~x4 | ~x34" |
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109 and "~x4 | ~x3" |
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110 and "~x34 | ~x3" |
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111 and "~x35 | ~x4" |
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112 and "~x5 | ~x36" |
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113 and "~x5 | ~x30" |
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114 and "~x36 | ~x30" |
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115 and "~x6 | ~x37" |
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116 and "~x6 | ~x5" |
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117 and "~x6 | ~x31" |
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118 and "~x37 | ~x5" |
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119 and "~x37 | ~x31" |
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120 and "~x5 | ~x31" |
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121 and "~x7 | ~x38" |
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122 and "~x7 | ~x6" |
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123 and "~x7 | ~x32" |
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124 and "~x38 | ~x6" |
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125 and "~x38 | ~x32" |
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126 and "~x6 | ~x32" |
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127 and "~x8 | ~x39" |
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128 and "~x8 | ~x7" |
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129 and "~x8 | ~x33" |
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130 and "~x39 | ~x7" |
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131 and "~x39 | ~x33" |
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132 and "~x7 | ~x33" |
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133 and "~x9 | ~x40" |
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134 and "~x9 | ~x8" |
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135 and "~x9 | ~x34" |
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136 and "~x40 | ~x8" |
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137 and "~x40 | ~x34" |
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138 and "~x8 | ~x34" |
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139 and "~x41 | ~x9" |
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140 and "~x41 | ~x35" |
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141 and "~x9 | ~x35" |
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142 and "~x10 | ~x42" |
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143 and "~x10 | ~x36" |
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144 and "~x42 | ~x36" |
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145 and "~x11 | ~x43" |
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146 and "~x11 | ~x10" |
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147 and "~x11 | ~x37" |
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148 and "~x43 | ~x10" |
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149 and "~x43 | ~x37" |
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150 and "~x10 | ~x37" |
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151 and "~x12 | ~x44" |
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152 and "~x12 | ~x11" |
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153 and "~x12 | ~x38" |
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154 and "~x44 | ~x11" |
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155 and "~x44 | ~x38" |
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156 and "~x11 | ~x38" |
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157 and "~x13 | ~x45" |
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158 and "~x13 | ~x12" |
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159 and "~x13 | ~x39" |
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160 and "~x45 | ~x12" |
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161 and "~x45 | ~x39" |
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162 and "~x12 | ~x39" |
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163 and "~x14 | ~x46" |
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164 and "~x14 | ~x13" |
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165 and "~x14 | ~x40" |
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166 and "~x46 | ~x13" |
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167 and "~x46 | ~x40" |
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168 and "~x13 | ~x40" |
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169 and "~x47 | ~x14" |
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170 and "~x47 | ~x41" |
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171 and "~x14 | ~x41" |
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172 and "~x15 | ~x48" |
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173 and "~x15 | ~x42" |
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174 and "~x48 | ~x42" |
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175 and "~x16 | ~x49" |
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176 and "~x16 | ~x15" |
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177 and "~x16 | ~x43" |
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178 and "~x49 | ~x15" |
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179 and "~x49 | ~x43" |
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180 and "~x15 | ~x43" |
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181 and "~x17 | ~x50" |
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182 and "~x17 | ~x16" |
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183 and "~x17 | ~x44" |
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184 and "~x50 | ~x16" |
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185 and "~x50 | ~x44" |
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186 and "~x16 | ~x44" |
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187 and "~x18 | ~x51" |
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188 and "~x18 | ~x17" |
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189 and "~x18 | ~x45" |
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190 and "~x51 | ~x17" |
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191 and "~x51 | ~x45" |
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192 and "~x17 | ~x45" |
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193 and "~x19 | ~x52" |
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194 and "~x19 | ~x18" |
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195 and "~x19 | ~x46" |
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196 and "~x52 | ~x18" |
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197 and "~x52 | ~x46" |
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198 and "~x18 | ~x46" |
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199 and "~x53 | ~x19" |
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200 and "~x53 | ~x47" |
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201 and "~x19 | ~x47" |
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202 and "~x20 | ~x54" |
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203 and "~x20 | ~x48" |
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204 and "~x54 | ~x48" |
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205 and "~x21 | ~x55" |
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206 and "~x21 | ~x20" |
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207 and "~x21 | ~x49" |
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208 and "~x55 | ~x20" |
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209 and "~x55 | ~x49" |
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210 and "~x20 | ~x49" |
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211 and "~x22 | ~x56" |
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212 and "~x22 | ~x21" |
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213 and "~x22 | ~x50" |
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214 and "~x56 | ~x21" |
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215 and "~x56 | ~x50" |
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216 and "~x21 | ~x50" |
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217 and "~x23 | ~x57" |
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218 and "~x23 | ~x22" |
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219 and "~x23 | ~x51" |
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220 and "~x57 | ~x22" |
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221 and "~x57 | ~x51" |
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222 and "~x22 | ~x51" |
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223 and "~x24 | ~x58" |
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224 and "~x24 | ~x23" |
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225 and "~x24 | ~x52" |
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226 and "~x58 | ~x23" |
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227 and "~x58 | ~x52" |
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228 and "~x23 | ~x52" |
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229 and "~x59 | ~x24" |
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230 and "~x59 | ~x53" |
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231 and "~x24 | ~x53" |
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232 and "~x25 | ~x54" |
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233 and "~x26 | ~x25" |
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234 and "~x26 | ~x55" |
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235 and "~x25 | ~x55" |
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236 and "~x27 | ~x26" |
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237 and "~x27 | ~x56" |
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238 and "~x26 | ~x56" |
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239 and "~x28 | ~x27" |
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240 and "~x28 | ~x57" |
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241 and "~x27 | ~x57" |
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242 and "~x29 | ~x28" |
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243 and "~x29 | ~x58" |
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244 and "~x28 | ~x58" |
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245 shows False |
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246 using assms by smt |
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247 |
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248 lemma "\<forall>x::int. P x \<longrightarrow> (\<forall>y::int. P x \<or> P y)" |
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249 by smt |
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250 |
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251 lemma |
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252 assumes "(\<forall>x y. P x y = x)" |
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253 shows "(\<exists>y. P x y) = P x c" |
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254 using assms by smt |
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255 |
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256 lemma |
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257 assumes "(\<forall>x y. P x y = x)" |
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258 and "(\<forall>x. \<exists>y. P x y) = (\<forall>x. P x c)" |
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259 shows "(EX y. P x y) = P x c" |
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260 using assms by smt |
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261 |
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262 lemma |
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263 assumes "if P x then \<not>(\<exists>y. P y) else (\<forall>y. \<not>P y)" |
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264 shows "P x \<longrightarrow> P y" |
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265 using assms by smt |
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266 |
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267 |
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268 section {* Arithmetic *} |
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269 |
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270 subsection {* Linear arithmetic over integers and reals *} |
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271 |
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272 lemma "(3::int) = 3" by smt |
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273 |
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274 lemma "(3::real) = 3" by smt |
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275 |
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276 lemma "(3 :: int) + 1 = 4" by smt |
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277 |
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278 lemma "x + (y + z) = y + (z + (x::int))" by smt |
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279 |
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280 lemma "max (3::int) 8 > 5" by smt |
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281 |
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282 lemma "abs (x :: real) + abs y \<ge> abs (x + y)" by smt |
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283 |
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284 lemma "P ((2::int) < 3) = P True" by smt |
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285 |
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286 lemma "x + 3 \<ge> 4 \<or> x < (1::int)" by smt |
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287 |
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288 lemma |
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289 assumes "x \<ge> (3::int)" and "y = x + 4" |
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290 shows "y - x > 0" |
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291 using assms by smt |
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292 |
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293 lemma "let x = (2 :: int) in x + x \<noteq> 5" by smt |
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294 |
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295 lemma |
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296 fixes x :: real |
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297 assumes "3 * x + 7 * a < 4" and "3 < 2 * x" |
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298 shows "a < 0" |
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299 using assms by smt |
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300 |
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301 lemma "(0 \<le> y + -1 * x \<or> \<not> 0 \<le> x \<or> 0 \<le> (x::int)) = (\<not> False)" by smt |
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302 |
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303 lemma "distinct [x < (3::int), 3 \<le> x]" by smt |
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304 |
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305 lemma |
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306 assumes "a > (0::int)" |
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307 shows "distinct [a, a * 2, a - a]" |
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308 using assms by smt |
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309 |
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310 lemma " |
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311 (n < m & m < n') | (n < m & m = n') | (n < n' & n' < m) | |
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312 (n = n' & n' < m) | (n = m & m < n') | |
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313 (n' < m & m < n) | (n' < m & m = n) | |
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314 (n' < n & n < m) | (n' = n & n < m) | (n' = m & m < n) | |
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315 (m < n & n < n') | (m < n & n' = n) | (m < n' & n' < n) | |
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316 (m = n & n < n') | (m = n' & n' < n) | |
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317 (n' = m & m = (n::int))" |
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318 by smt |
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319 |
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320 text{* |
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321 The following example was taken from HOL/ex/PresburgerEx.thy, where it says: |
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322 |
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323 This following theorem proves that all solutions to the |
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324 recurrence relation $x_{i+2} = |x_{i+1}| - x_i$ are periodic with |
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325 period 9. The example was brought to our attention by John |
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326 Harrison. It does does not require Presburger arithmetic but merely |
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327 quantifier-free linear arithmetic and holds for the rationals as well. |
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328 |
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329 Warning: it takes (in 2006) over 4.2 minutes! |
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330 |
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331 There, it is proved by "arith". SMT is able to prove this within a fraction |
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332 of one second. With proof reconstruction, it takes about 13 seconds on a Core2 |
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333 processor. |
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334 *} |
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335 |
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336 lemma "\<lbrakk> x3 = abs x2 - x1; x4 = abs x3 - x2; x5 = abs x4 - x3; |
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337 x6 = abs x5 - x4; x7 = abs x6 - x5; x8 = abs x7 - x6; |
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338 x9 = abs x8 - x7; x10 = abs x9 - x8; x11 = abs x10 - x9 \<rbrakk> |
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339 \<Longrightarrow> x1 = x10 & x2 = (x11::int)" |
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340 by smt |
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341 |
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342 |
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343 lemma "let P = 2 * x + 1 > x + (x::real) in P \<or> False \<or> P" by smt |
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344 |
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345 lemma "x + (let y = x mod 2 in 2 * y + 1) \<ge> x + (1::int)" by smt |
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346 |
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347 lemma "x + (let y = x mod 2 in y + y) < x + (3::int)" by smt |
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348 |
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349 lemma |
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350 assumes "x \<noteq> (0::real)" |
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351 shows "x + x \<noteq> (let P = (abs x > 1) in if P \<or> \<not>P then 4 else 2) * x" |
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352 using assms by smt |
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353 |
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354 lemma |
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355 assumes "(n + m) mod 2 = 0" and "n mod 4 = 3" |
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356 shows "n mod 2 = 1 & m mod 2 = (1::int)" |
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357 using assms by smt |
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358 |
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359 |
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360 subsection {* Linear arithmetic with quantifiers *} |
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361 |
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362 lemma "~ (\<exists>x::int. False)" by smt |
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363 |
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364 lemma "~ (\<exists>x::real. False)" by smt |
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365 |
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366 lemma "\<exists>x::int. 0 < x" |
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367 using [[z3_proofs=false]] (* no Z3 proof *) |
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368 by smt |
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369 |
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370 lemma "\<exists>x::real. 0 < x" |
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371 using [[z3_proofs=false]] (* no Z3 proof *) |
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372 by smt |
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373 |
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374 lemma "\<forall>x::int. \<exists>y. y > x" |
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375 using [[z3_proofs=false]] (* no Z3 proof *) |
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376 by smt |
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377 |
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378 lemma "\<forall>x y::int. (x = 0 \<and> y = 1) \<longrightarrow> x \<noteq> y" by smt |
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379 |
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380 lemma "\<exists>x::int. \<forall>y. x < y \<longrightarrow> y < 0 \<or> y >= 0" by smt |
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381 |
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382 lemma "\<forall>x y::int. x < y \<longrightarrow> (2 * x + 1) < (2 * y)" by smt |
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383 |
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384 lemma "\<forall>x y::int. (2 * x + 1) \<noteq> (2 * y)" by smt |
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385 |
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386 lemma "\<forall>x y::int. x + y > 2 \<or> x + y = 2 \<or> x + y < 2" by smt |
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387 |
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388 lemma "\<forall>x::int. if x > 0 then x + 1 > 0 else 1 > x" by smt |
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389 |
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390 lemma "if (ALL x::int. x < 0 \<or> x > 0) then False else True" by smt |
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391 |
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392 lemma "(if (ALL x::int. x < 0 \<or> x > 0) then -1 else 3) > (0::int)" by smt |
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393 |
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394 lemma "~ (\<exists>x y z::int. 4 * x + -6 * y = (1::int))" by smt |
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395 |
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396 lemma "\<exists>x::int. \<forall>x y. 0 < x \<and> 0 < y \<longrightarrow> (0::int) < x + y" by smt |
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397 |
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398 lemma "\<exists>u::int. \<forall>(x::int) y::real. 0 < x \<and> 0 < y \<longrightarrow> -1 < x" by smt |
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399 |
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400 lemma "\<exists>x::int. (\<forall>y. y \<ge> x \<longrightarrow> y > 0) \<longrightarrow> x > 0" by smt |
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401 |
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402 lemma "\<forall>x::int. trigger [pat x] (x < a \<longrightarrow> 2 * x < 2 * a)" by smt |
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403 |
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404 |
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405 subsection {* Non-linear arithmetic over integers and reals *} |
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406 |
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407 lemma "a > (0::int) \<Longrightarrow> a*b > 0 \<Longrightarrow> b > 0" |
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408 using [[z3_proofs=false]] -- {* Isabelle's arithmetic decision procedures |
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409 are too weak to automatically prove @{thm zero_less_mult_pos}. *} |
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410 by smt |
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411 |
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412 lemma "(a::int) * (x + 1 + y) = a * x + a * (y + 1)" by smt |
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413 |
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414 lemma "((x::real) * (1 + y) - x * (1 - y)) = (2 * x * y)" by smt |
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415 |
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416 lemma |
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417 "(U::int) + (1 + p) * (b + e) + p * d = |
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418 U + (2 * (1 + p) * (b + e) + (1 + p) * d + d * p) - (1 + p) * (b + d + e)" |
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419 by smt |
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420 |
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421 |
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422 subsection {* Linear arithmetic for natural numbers *} |
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423 |
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424 lemma "2 * (x::nat) ~= 1" by smt |
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425 |
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426 lemma "a < 3 \<Longrightarrow> (7::nat) > 2 * a" by smt |
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427 |
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428 lemma "let x = (1::nat) + y in x - y > 0 * x" by smt |
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429 |
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430 lemma |
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431 "let x = (1::nat) + y in |
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432 let P = (if x > 0 then True else False) in |
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433 False \<or> P = (x - 1 = y) \<or> (\<not>P \<longrightarrow> False)" |
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434 by smt |
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435 |
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436 lemma "distinct [a + (1::nat), a * 2 + 3, a - a]" by smt |
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437 |
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438 lemma "int (nat \<bar>x::int\<bar>) = \<bar>x\<bar>" by smt |
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439 |
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440 definition prime_nat :: "nat \<Rightarrow> bool" where |
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441 "prime_nat p = (1 < p \<and> (\<forall>m. m dvd p --> m = 1 \<or> m = p))" |
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442 lemma "prime_nat (4*m + 1) \<Longrightarrow> m \<ge> (1::nat)" by (smt prime_nat_def) |
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443 |
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444 |
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445 section {* Bitvectors *} |
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446 |
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447 locale z3_bv_test |
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448 begin |
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449 |
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450 text {* |
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451 The following examples only work for Z3, and only without proof reconstruction. |
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452 *} |
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453 |
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454 declare [[smt_solver=z3, z3_proofs=false]] |
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455 |
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456 |
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457 subsection {* Bitvector arithmetic *} |
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458 |
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459 lemma "(27 :: 4 word) = -5" by smt |
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460 |
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461 lemma "(27 :: 4 word) = 11" by smt |
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462 |
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463 lemma "23 < (27::8 word)" by smt |
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464 |
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465 lemma "27 + 11 = (6::5 word)" by smt |
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466 |
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467 lemma "7 * 3 = (21::8 word)" by smt |
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468 |
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469 lemma "11 - 27 = (-16::8 word)" by smt |
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470 |
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471 lemma "- -11 = (11::5 word)" by smt |
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472 |
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473 lemma "-40 + 1 = (-39::7 word)" by smt |
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474 |
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475 lemma "a + 2 * b + c - b = (b + c) + (a :: 32 word)" by smt |
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476 |
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477 lemma "x = (5 :: 4 word) \<Longrightarrow> 4 * x = 4" by smt |
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478 |
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479 |
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480 subsection {* Bit-level logic *} |
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481 |
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482 lemma "0b110 AND 0b101 = (0b100 :: 32 word)" by smt |
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483 |
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484 lemma "0b110 OR 0b011 = (0b111 :: 8 word)" by smt |
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485 |
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486 lemma "0xF0 XOR 0xFF = (0x0F :: 8 word)" by smt |
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487 |
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488 lemma "NOT (0xF0 :: 16 word) = 0xFF0F" by smt |
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489 |
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490 lemma "word_cat (27::4 word) (27::8 word) = (2843::12 word)" by smt |
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491 |
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492 lemma "word_cat (0b0011::4 word) (0b1111::6word) = (0b0011001111 :: 10 word)" |
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493 by smt |
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494 |
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495 lemma "slice 1 (0b10110 :: 4 word) = (0b11 :: 2 word)" by smt |
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496 |
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497 lemma "ucast (0b1010 :: 4 word) = (0b1010 :: 10 word)" by smt |
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498 |
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499 lemma "scast (0b1010 :: 4 word) = (0b111010 :: 6 word)" by smt |
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500 |
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501 lemma "bv_lshr 0b10011 2 = (0b100::8 word)" by smt |
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502 |
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503 lemma "bv_ashr 0b10011 2 = (0b100::8 word)" by smt |
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504 |
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505 lemma "word_rotr 2 0b0110 = (0b1001::4 word)" by smt |
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506 |
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507 lemma "word_rotl 1 0b1110 = (0b1101::4 word)" by smt |
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508 |
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509 lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)" by smt |
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510 |
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511 lemma "w < 256 \<Longrightarrow> (w :: 16 word) AND 0x00FF = w" by smt |
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512 |
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513 end |
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514 |
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515 lemma |
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516 assumes "bv2int 0 = 0" |
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517 and "bv2int 1 = 1" |
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518 and "bv2int 2 = 2" |
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519 and "bv2int 3 = 3" |
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520 and "\<forall>x::2 word. bv2int x > 0" |
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521 shows "\<forall>i::int. i < 0 \<longrightarrow> (\<forall>x::2 word. bv2int x > i)" |
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522 using assms |
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523 using [[smt_solver=z3]] |
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524 by smt |
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525 |
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526 lemma "P (0 \<le> (a :: 4 word)) = P True" |
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527 using [[smt_solver=z3, z3_proofs=false]] |
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528 by smt |
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529 |
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530 |
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531 section {* Pairs *} |
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532 |
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533 lemma "fst (x, y) = a \<Longrightarrow> x = a" by smt |
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534 |
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535 lemma "p1 = (x, y) \<and> p2 = (y, x) \<Longrightarrow> fst p1 = snd p2" by smt |
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536 |
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537 |
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538 section {* Higher-order problems and recursion *} |
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539 |
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540 lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> (f (i1 := v1, i2 := v2)) i = f i" by smt |
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541 |
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542 lemma "(f g (x::'a::type) = (g x \<and> True)) \<or> (f g x = True) \<or> (g x = True)" |
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543 by smt |
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544 |
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545 lemma "id 3 = 3 \<and> id True = True" by (smt id_def) |
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546 |
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547 lemma "i \<noteq> i1 \<and> i \<noteq> i2 \<Longrightarrow> ((f (i1 := v1)) (i2 := v2)) i = f i" by smt |
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548 |
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549 lemma "map (\<lambda>i::nat. i + 1) [0, 1] = [1, 2]" by (smt map.simps) |
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550 |
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551 lemma "(ALL x. P x) | ~ All P" by smt |
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552 |
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553 fun dec_10 :: "nat \<Rightarrow> nat" where |
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554 "dec_10 n = (if n < 10 then n else dec_10 (n - 10))" |
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555 lemma "dec_10 (4 * dec_10 4) = 6" by (smt dec_10.simps) |
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556 |
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557 axiomatization |
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558 eval_dioph :: "int list \<Rightarrow> nat list \<Rightarrow> int" |
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559 where |
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560 eval_dioph_mod: |
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561 "eval_dioph ks xs mod int n = eval_dioph ks (map (\<lambda>x. x mod n) xs) mod int n" |
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562 and |
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563 eval_dioph_div_mult: |
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564 "eval_dioph ks (map (\<lambda>x. x div n) xs) * int n + |
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565 eval_dioph ks (map (\<lambda>x. x mod n) xs) = eval_dioph ks xs" |
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566 lemma |
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567 "(eval_dioph ks xs = l) = |
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568 (eval_dioph ks (map (\<lambda>x. x mod 2) xs) mod 2 = l mod 2 \<and> |
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569 eval_dioph ks (map (\<lambda>x. x div 2) xs) = |
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570 (l - eval_dioph ks (map (\<lambda>x. x mod 2) xs)) div 2)" |
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571 by (smt eval_dioph_mod[where n=2] eval_dioph_div_mult[where n=2]) |
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572 |
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573 |
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574 section {* Monomorphization examples *} |
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575 |
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576 definition P :: "'a \<Rightarrow> bool" where "P x = True" |
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577 lemma poly_P: "P x \<and> (P [x] \<or> \<not>P[x])" by (simp add: P_def) |
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578 lemma "P (1::int)" by (smt poly_P) |
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579 |
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580 consts g :: "'a \<Rightarrow> nat" |
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581 axioms |
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582 g1: "g (Some x) = g [x]" |
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583 g2: "g None = g []" |
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584 g3: "g xs = length xs" |
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585 lemma "g (Some (3::int)) = g (Some True)" by (smt g1 g2 g3 list.size) |
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586 |
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587 end |