1 (* Authors: Klaus Aehlig, Tobias Nipkow *) |
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2 |
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3 header {* Testing implementation of normalization by evaluation *} |
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4 |
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5 theory NormalForm |
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6 imports Complex_Main |
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7 begin |
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8 |
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9 lemma "True" by normalization |
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10 lemma "p \<longrightarrow> True" by normalization |
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11 declare disj_assoc [code nbe] |
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12 lemma "((P | Q) | R) = (P | (Q | R))" by normalization |
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13 lemma "0 + (n::nat) = n" by normalization |
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14 lemma "0 + Suc n = Suc n" by normalization |
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15 lemma "Suc n + Suc m = n + Suc (Suc m)" by normalization |
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16 lemma "~((0::nat) < (0::nat))" by normalization |
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17 |
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18 datatype n = Z | S n |
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19 |
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20 primrec add :: "n \<Rightarrow> n \<Rightarrow> n" where |
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21 "add Z = id" |
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22 | "add (S m) = S o add m" |
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23 |
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24 primrec add2 :: "n \<Rightarrow> n \<Rightarrow> n" where |
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25 "add2 Z n = n" |
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26 | "add2 (S m) n = S(add2 m n)" |
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27 |
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28 declare add2.simps [code] |
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29 lemma [code nbe]: "add2 (add2 n m) k = add2 n (add2 m k)" |
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30 by (induct n) auto |
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31 lemma [code]: "add2 n (S m) = S (add2 n m)" |
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32 by(induct n) auto |
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33 lemma [code]: "add2 n Z = n" |
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34 by(induct n) auto |
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35 |
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36 lemma "add2 (add2 n m) k = add2 n (add2 m k)" by normalization |
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37 lemma "add2 (add2 (S n) (S m)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization |
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38 lemma "add2 (add2 (S n) (add2 (S m) Z)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization |
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39 |
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40 primrec mul :: "n \<Rightarrow> n \<Rightarrow> n" where |
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41 "mul Z = (%n. Z)" |
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42 | "mul (S m) = (%n. add (mul m n) n)" |
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43 |
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44 primrec mul2 :: "n \<Rightarrow> n \<Rightarrow> n" where |
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45 "mul2 Z n = Z" |
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46 | "mul2 (S m) n = add2 n (mul2 m n)" |
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47 |
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48 primrec exp :: "n \<Rightarrow> n \<Rightarrow> n" where |
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49 "exp m Z = S Z" |
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50 | "exp m (S n) = mul (exp m n) m" |
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51 |
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52 lemma "mul2 (S(S(S(S(S Z))))) (S(S(S Z))) = S(S(S(S(S(S(S(S(S(S(S(S(S(S(S Z))))))))))))))" by normalization |
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53 lemma "mul (S(S(S(S(S Z))))) (S(S(S Z))) = S(S(S(S(S(S(S(S(S(S(S(S(S(S(S Z))))))))))))))" by normalization |
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54 lemma "exp (S(S Z)) (S(S(S(S Z)))) = exp (S(S(S(S Z)))) (S(S Z))" by normalization |
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55 |
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56 lemma "(let ((x,y),(u,v)) = ((Z,Z),(Z,Z)) in add (add x y) (add u v)) = Z" by normalization |
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57 lemma "split (%x y. x) (a, b) = a" by normalization |
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58 lemma "(%((x,y),(u,v)). add (add x y) (add u v)) ((Z,Z),(Z,Z)) = Z" by normalization |
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59 |
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60 lemma "case Z of Z \<Rightarrow> True | S x \<Rightarrow> False" by normalization |
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61 |
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62 lemma "[] @ [] = []" by normalization |
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63 lemma "map f [x,y,z::'x] = [f x, f y, f z]" by normalization |
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64 lemma "[a, b, c] @ xs = a # b # c # xs" by normalization |
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65 lemma "[] @ xs = xs" by normalization |
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66 lemma "map (%f. f True) [id, g, Not] = [True, g True, False]" by normalization |
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67 |
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68 lemma "map (%f. f True) ([id, g, Not] @ fs) = [True, g True, False] @ map (%f. f True) fs" |
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69 by normalization rule+ |
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70 lemma "rev [a, b, c] = [c, b, a]" by normalization |
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71 normal_form "rev (a#b#cs) = rev cs @ [b, a]" |
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72 normal_form "map (%F. F [a,b,c::'x]) (map map [f,g,h])" |
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73 normal_form "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))" |
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74 normal_form "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])" |
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75 lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" |
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76 by normalization |
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77 normal_form "case xs of [] \<Rightarrow> True | x#xs \<Rightarrow> False" |
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78 normal_form "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) xs = P" |
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79 lemma "let x = y in [x, x] = [y, y]" by normalization |
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80 lemma "Let y (%x. [x,x]) = [y, y]" by normalization |
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81 normal_form "case n of Z \<Rightarrow> True | S x \<Rightarrow> False" |
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82 lemma "(%(x,y). add x y) (S z,S z) = S (add z (S z))" by normalization |
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83 normal_form "filter (%x. x) ([True,False,x]@xs)" |
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84 normal_form "filter Not ([True,False,x]@xs)" |
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85 |
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86 lemma "[x,y,z] @ [a,b,c] = [x, y, z, a, b, c]" by normalization |
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87 lemma "(%(xs, ys). xs @ ys) ([a, b, c], [d, e, f]) = [a, b, c, d, e, f]" by normalization |
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88 lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" by normalization |
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89 |
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90 lemma "last [a, b, c] = c" by normalization |
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91 lemma "last ([a, b, c] @ xs) = last (c # xs)" by normalization |
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92 |
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93 lemma "(2::int) + 3 - 1 + (- k) * 2 = 4 + - k * 2" by normalization |
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94 lemma "(-4::int) * 2 = -8" by normalization |
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95 lemma "abs ((-4::int) + 2 * 1) = 2" by normalization |
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96 lemma "(2::int) + 3 = 5" by normalization |
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97 lemma "(2::int) + 3 * (- 4) * (- 1) = 14" by normalization |
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98 lemma "(2::int) + 3 * (- 4) * 1 + 0 = -10" by normalization |
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99 lemma "(2::int) < 3" by normalization |
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100 lemma "(2::int) <= 3" by normalization |
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101 lemma "abs ((-4::int) + 2 * 1) = 2" by normalization |
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102 lemma "4 - 42 * abs (3 + (-7\<Colon>int)) = -164" by normalization |
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103 lemma "(if (0\<Colon>nat) \<le> (x\<Colon>nat) then 0\<Colon>nat else x) = 0" by normalization |
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104 lemma "4 = Suc (Suc (Suc (Suc 0)))" by normalization |
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105 lemma "nat 4 = Suc (Suc (Suc (Suc 0)))" by normalization |
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106 lemma "[Suc 0, 0] = [Suc 0, 0]" by normalization |
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107 lemma "max (Suc 0) 0 = Suc 0" by normalization |
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108 lemma "(42::rat) / 1704 = 1 / 284 + 3 / 142" by normalization |
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109 normal_form "Suc 0 \<in> set ms" |
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110 |
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111 lemma "f = f" by normalization |
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112 lemma "f x = f x" by normalization |
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113 lemma "(f o g) x = f (g x)" by normalization |
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114 lemma "(f o id) x = f x" by normalization |
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115 normal_form "(\<lambda>x. x)" |
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116 |
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117 (* Church numerals: *) |
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118 |
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119 normal_form "(%m n f x. m f (n f x)) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))" |
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120 normal_form "(%m n f x. m (n f) x) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))" |
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121 normal_form "(%m n. n m) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))" |
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122 |
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123 (* handling of type classes in connection with equality *) |
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124 |
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125 lemma "map f [x, y] = [f x, f y]" by normalization |
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126 lemma "(map f [x, y], w) = ([f x, f y], w)" by normalization |
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127 lemma "map f [x, y] = [f x \<Colon> 'a\<Colon>semigroup_add, f y]" by normalization |
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128 lemma "map f [x \<Colon> 'a\<Colon>semigroup_add, y] = [f x, f y]" by normalization |
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129 lemma "(map f [x \<Colon> 'a\<Colon>semigroup_add, y], w \<Colon> 'b\<Colon>finite) = ([f x, f y], w)" by normalization |
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130 |
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131 end |
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