1 (* Title: HOL/ex/SML_Quickcheck_Examples.thy |
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2 Author: Stefan Berghofer, Lukas Bulwahn |
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3 Copyright 2011 TU Muenchen |
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4 *) |
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5 |
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6 theory SML_Quickcheck_Examples |
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7 imports "~~/src/HOL/Library/SML_Quickcheck" |
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8 begin |
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9 |
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10 text {* |
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11 This is a regression test for the 'quickcheck' counterexample generator based on the |
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12 SML code generator. |
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13 |
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14 The counterexample generator has been superseded by counterexample generators based on |
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15 the generic code generator. |
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16 *} |
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17 |
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18 declare [[quickcheck_finite_types = false]] |
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19 declare [[quickcheck_timeout = 600.0]] |
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20 |
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21 subsection {* Lists *} |
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22 |
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23 theorem "map g (map f xs) = map (g o f) xs" |
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24 quickcheck[SML, expect = no_counterexample] |
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25 oops |
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26 |
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27 theorem "map g (map f xs) = map (f o g) xs" |
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28 quickcheck[SML, expect = counterexample] |
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29 oops |
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30 |
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31 theorem "rev (xs @ ys) = rev ys @ rev xs" |
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32 quickcheck[SML, expect = no_counterexample] |
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33 oops |
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34 |
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35 theorem "rev (xs @ ys) = rev xs @ rev ys" |
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36 quickcheck[SML, expect = counterexample] |
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37 oops |
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38 |
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39 theorem "rev (rev xs) = xs" |
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40 quickcheck[SML, expect = no_counterexample] |
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41 oops |
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42 |
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43 theorem "rev xs = xs" |
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44 quickcheck[tester = SML, expect = counterexample] |
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45 oops |
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46 |
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47 |
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48 text {* An example involving functions inside other data structures *} |
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49 |
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50 primrec app :: "('a \<Rightarrow> 'a) list \<Rightarrow> 'a \<Rightarrow> 'a" where |
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51 "app [] x = x" |
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52 | "app (f # fs) x = app fs (f x)" |
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53 |
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54 lemma "app (fs @ gs) x = app gs (app fs x)" |
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55 quickcheck[SML, expect = no_counterexample] |
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56 by (induct fs arbitrary: x) simp_all |
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57 |
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58 lemma "app (fs @ gs) x = app fs (app gs x)" |
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59 quickcheck[SML, expect = counterexample] |
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60 oops |
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61 |
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62 primrec occurs :: "'a \<Rightarrow> 'a list \<Rightarrow> nat" where |
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63 "occurs a [] = 0" |
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64 | "occurs a (x#xs) = (if (x=a) then Suc(occurs a xs) else occurs a xs)" |
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65 |
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66 primrec del1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
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67 "del1 a [] = []" |
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68 | "del1 a (x#xs) = (if (x=a) then xs else (x#del1 a xs))" |
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69 |
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70 text {* A lemma, you'd think to be true from our experience with delAll *} |
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71 lemma "Suc (occurs a (del1 a xs)) = occurs a xs" |
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72 -- {* Wrong. Precondition needed.*} |
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73 quickcheck[SML, expect = counterexample] |
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74 oops |
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75 |
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76 lemma "xs ~= [] \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs" |
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77 quickcheck[SML, expect = counterexample] |
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78 -- {* Also wrong.*} |
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79 oops |
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80 |
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81 lemma "0 < occurs a xs \<longrightarrow> Suc (occurs a (del1 a xs)) = occurs a xs" |
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82 quickcheck[SML, expect = no_counterexample] |
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83 by (induct xs) auto |
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84 |
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85 primrec replace :: "'a \<Rightarrow> 'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where |
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86 "replace a b [] = []" |
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87 | "replace a b (x#xs) = (if (x=a) then (b#(replace a b xs)) |
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88 else (x#(replace a b xs)))" |
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89 |
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90 lemma "occurs a xs = occurs b (replace a b xs)" |
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91 quickcheck[SML, expect = counterexample] |
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92 -- {* Wrong. Precondition needed.*} |
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93 oops |
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94 |
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95 lemma "occurs b xs = 0 \<or> a=b \<longrightarrow> occurs a xs = occurs b (replace a b xs)" |
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96 quickcheck[SML, expect = no_counterexample] |
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97 by (induct xs) simp_all |
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98 |
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99 |
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100 subsection {* Trees *} |
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101 |
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102 datatype 'a tree = Twig | Leaf 'a | Branch "'a tree" "'a tree" |
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103 |
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104 primrec leaves :: "'a tree \<Rightarrow> 'a list" where |
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105 "leaves Twig = []" |
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106 | "leaves (Leaf a) = [a]" |
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107 | "leaves (Branch l r) = (leaves l) @ (leaves r)" |
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108 |
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109 primrec plant :: "'a list \<Rightarrow> 'a tree" where |
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110 "plant [] = Twig " |
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111 | "plant (x#xs) = Branch (Leaf x) (plant xs)" |
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112 |
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113 primrec mirror :: "'a tree \<Rightarrow> 'a tree" where |
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114 "mirror (Twig) = Twig " |
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115 | "mirror (Leaf a) = Leaf a " |
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116 | "mirror (Branch l r) = Branch (mirror r) (mirror l)" |
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117 |
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118 theorem "plant (rev (leaves xt)) = mirror xt" |
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119 quickcheck[SML, expect = counterexample] |
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120 --{* Wrong! *} |
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121 oops |
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122 |
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123 theorem "plant((leaves xt) @ (leaves yt)) = Branch xt yt" |
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124 quickcheck[SML, expect = counterexample] |
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125 --{* Wrong! *} |
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126 oops |
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127 |
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128 datatype 'a ntree = Tip "'a" | Node "'a" "'a ntree" "'a ntree" |
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129 |
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130 primrec inOrder :: "'a ntree \<Rightarrow> 'a list" where |
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131 "inOrder (Tip a)= [a]" |
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132 | "inOrder (Node f x y) = (inOrder x)@[f]@(inOrder y)" |
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133 |
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134 primrec root :: "'a ntree \<Rightarrow> 'a" where |
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135 "root (Tip a) = a" |
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136 | "root (Node f x y) = f" |
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137 |
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138 theorem "hd (inOrder xt) = root xt" |
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139 quickcheck[SML, expect = counterexample] |
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140 --{* Wrong! *} |
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141 oops |
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142 |
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143 end |
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