1 | include "basics/list.ma". |
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2 | include "basics/types.ma". |
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3 | include "arithmetics/nat.ma". |
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4 | |
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5 | let rec nub_by_internal (A: Type[0]) (f: A → A → bool) (l: list A) (n: nat) on n ≝ |
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6 | match n with |
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7 | [ O ⇒ |
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8 | match l with |
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9 | [ nil ⇒ [ ] |
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10 | | cons hd tl ⇒ l |
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11 | ] |
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12 | | S n ⇒ |
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13 | match l with |
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14 | [ nil ⇒ [ ] |
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15 | | cons hd tl ⇒ |
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16 | hd :: nub_by_internal A f (filter ? (λy. notb (f y hd)) tl) n |
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17 | ] |
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18 | ]. |
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19 | |
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20 | definition nub_by ≝ |
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21 | λA: Type[0]. |
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22 | λf: A → A → bool. |
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23 | λl: list A. |
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24 | nub_by_internal A f l (length ? l). |
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25 | |
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26 | let rec member (A: Type[0]) (eq: A → A → bool) (a: A) (l: list A) on l ≝ |
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27 | match l with |
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28 | [ nil ⇒ false |
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29 | | cons hd tl ⇒ orb (eq a hd) (member A eq a tl) |
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30 | ]. |
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31 | |
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32 | let rec take (A: Type[0]) (n: nat) (l: list A) on n: list A ≝ |
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33 | match n with |
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34 | [ O ⇒ [ ] |
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35 | | S n ⇒ |
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36 | match l with |
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37 | [ nil ⇒ [ ] |
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38 | | cons hd tl ⇒ hd :: take A n tl |
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39 | ] |
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40 | ]. |
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41 | |
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42 | let rec drop (A: Type[0]) (n: nat) (l: list A) on n ≝ |
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43 | match n with |
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44 | [ O ⇒ l |
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45 | | S n ⇒ |
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46 | match l with |
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47 | [ nil ⇒ [ ] |
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48 | | cons hd tl ⇒ drop A n tl |
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49 | ] |
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50 | ]. |
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51 | |
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52 | definition list_split ≝ |
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53 | λA: Type[0]. |
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54 | λn: nat. |
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55 | λl: list A. |
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56 | 〈take A n l, drop A n l〉. |
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57 | |
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58 | let rec mapi_internal (A: Type[0]) (B: Type[0]) (n: nat) (f: nat → A → B) |
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59 | (l: list A) on l: list B ≝ |
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60 | match l with |
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61 | [ nil ⇒ nil ? |
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62 | | cons hd tl ⇒ (f n hd) :: (mapi_internal A B (n + 1) f tl) |
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63 | ]. |
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64 | |
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65 | definition mapi ≝ |
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66 | λA, B: Type[0]. |
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67 | λf: nat → A → B. |
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68 | λl: list A. |
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69 | mapi_internal A B 0 f l. |
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70 | |
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71 | let rec zip (A: Type[0]) (B: Type[0]) (l: list A) (r: list B) on l: option (list (A × B)) ≝ |
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72 | match l with |
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73 | [ nil ⇒ Some ? (nil (A × B)) |
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74 | | cons hd tl ⇒ |
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75 | match r with |
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76 | [ nil ⇒ None ? |
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77 | | cons hd' tl' ⇒ |
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78 | match zip ? ? tl tl' with |
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79 | [ None ⇒ None ? |
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80 | | Some tail ⇒ Some ? (〈hd, hd'〉 :: tail) |
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81 | ] |
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82 | ] |
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83 | ]. |
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84 | |
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85 | let rec foldl (A: Type[0]) (B: Type[0]) (f: A → B → A) (a: A) (l: list B) on l ≝ |
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86 | match l with |
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87 | [ nil ⇒ a |
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88 | | cons hd tl ⇒ foldl A B f (f a hd) tl |
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89 | ]. |
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90 | |
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91 | definition flatten ≝ |
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92 | λA: Type[0]. |
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93 | λl: list (list A). |
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94 | foldl ? ? (append ?) [ ] l. |
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95 | |
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96 | definition if_then_else ≝ |
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97 | λA: Type[0]. |
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98 | λb: bool. |
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99 | λt: A. |
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100 | λf: A. |
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101 | match b with |
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102 | [ true ⇒ t |
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103 | | false ⇒ f |
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104 | ]. |
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105 | |
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106 | let rec rev (A: Type[0]) (l: list A) on l ≝ |
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107 | match l with |
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108 | [ nil ⇒ nil A |
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109 | | cons hd tl ⇒ (rev A tl) @ [ hd ] |
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110 | ]. |
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111 | |
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112 | (* |
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113 | notation "hvbox('if' b 'then' t 'else' f)" |
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114 | non associative with precedence 83 |
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115 | for @{ 'if_then_else $b $t $f }. |
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116 | *) |
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117 | notation > "'if' term 19 e 'then' term 19 t 'else' term 48 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }. |
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118 | notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 48 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }. |
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119 | |
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120 | interpretation "Bool if_then_else" 'if_then_else b t f = (if_then_else ? b t f). |
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121 | |
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122 | let rec fold_left_i_aux (A: Type[0]) (B: Type[0]) |
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123 | (f: nat → A → B → A) (x: A) (i: nat) (l: list B) on l ≝ |
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124 | match l with |
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125 | [ nil ⇒ x |
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126 | | cons hd tl ⇒ fold_left_i_aux A B f (f i x hd) (S i) tl |
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127 | ]. |
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128 | |
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129 | definition fold_left_i ≝ λA,B,f,x. fold_left_i_aux A B f x O. |
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130 | |
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131 | lemma eq_rect_Type0_r : |
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132 | ∀A: Type[0]. |
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133 | ∀a:A. |
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134 | ∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p. |
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135 | #A #a #P #H #x #p |
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136 | generalize in match H |
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137 | generalize in match P |
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138 | cases p |
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139 | // |
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140 | qed. |
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141 | |
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142 | |
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143 | notation "hvbox(t⌈o ↦ h⌉)" |
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144 | with precedence 45 |
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145 | for @{ match (? : $o=$h) with [ refl ⇒ $t ] }. |
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146 | |
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147 | definition function_apply ≝ |
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148 | λA, B: Type[0]. |
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149 | λf: A → B. |
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150 | λa: A. |
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151 | f a. |
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152 | |
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153 | notation "f break $ x" |
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154 | left associative with precedence 99 |
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155 | for @{ 'function_apply $f $x }. |
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156 | |
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157 | interpretation "Function application" 'function_apply f x = (function_apply ? ? f x). |
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158 | |
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159 | let rec iterate (A: Type[0]) (f: A → A) (a: A) (n: nat) on n ≝ |
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160 | match n with |
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161 | [ O ⇒ a |
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162 | | S o ⇒ f (iterate A f a o) |
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163 | ]. |
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164 | |
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165 | notation > "hvbox('let' 〈ident x,ident y〉 ≝ t 'in' s)" |
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166 | with precedence 10 |
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167 | for @{ match $t with [ pair ${ident x} ${ident y} ⇒ $s ] }. |
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168 | |
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169 | (* Yeah, I probably ought to do something more general... *) |
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170 | notation "hvbox(\langle term 19 a, break term 19 b, break term 19 c\rangle)" |
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171 | with precedence 90 for @{ 'triple $a $b $c}. |
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172 | interpretation "Triple construction" 'triple x y z = (pair ? ? (pair ? ? x y) z). |
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173 | |
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174 | notation > "hvbox('let' 〈ident x,ident y,ident z〉 ≝ t 'in' s)" |
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175 | with precedence 10 |
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176 | for @{ match $t with [ pair ${fresh xy} ${ident z} ⇒ match ${fresh xy} with [ pair ${ident x} ${ident y} ⇒ $s ] ] }. |
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177 | |
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178 | notation "⊥" with precedence 90 |
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179 | for @{ match ? in False with [ ] }. |
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180 | |
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181 | let rec exclusive_disjunction (b: bool) (c: bool) on b ≝ |
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182 | match b with |
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183 | [ true ⇒ |
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184 | match c with |
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185 | [ false ⇒ true |
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186 | | true ⇒ false |
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187 | ] |
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188 | | false ⇒ |
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189 | match c with |
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190 | [ false ⇒ false |
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191 | | true ⇒ true |
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192 | ] |
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193 | ]. |
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194 | |
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195 | definition ltb ≝ |
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196 | λm, n: nat. |
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197 | leb (S m) n. |
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198 | |
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199 | definition geb ≝ |
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200 | λm, n: nat. |
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201 | ltb n m. |
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202 | |
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203 | definition gtb ≝ |
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204 | λm, n: nat. |
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205 | leb n m. |
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206 | |
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207 | (* dpm: unless I'm being stupid, this isn't defined in the stdlib? *) |
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208 | let rec eq_nat (n: nat) (m: nat) on n: bool ≝ |
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209 | match n with |
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210 | [ O ⇒ match m with [ O ⇒ true | _ ⇒ false ] |
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211 | | S n' ⇒ match m with [ S m' ⇒ eq_nat n' m' | _ ⇒ false ] |
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212 | ]. |
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213 | |
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214 | (* dpm: conflicts with library definitions |
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215 | interpretation "Nat less than" 'lt m n = (ltb m n). |
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216 | interpretation "Nat greater than" 'gt m n = (gtb m n). |
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217 | interpretation "Nat greater than eq" 'geq m n = (geb m n). |
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218 | *) |
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219 | |
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220 | let rec division_aux (m: nat) (n : nat) (p: nat) ≝ |
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221 | match ltb n (S p) with |
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222 | [ true ⇒ O |
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223 | | false ⇒ |
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224 | match m with |
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225 | [ O ⇒ O |
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226 | | (S q) ⇒ S (division_aux q (n - (S p)) p) |
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227 | ] |
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228 | ]. |
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229 | |
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230 | definition division ≝ |
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231 | λm, n: nat. |
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232 | match n with |
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233 | [ O ⇒ S m |
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234 | | S o ⇒ division_aux m m o |
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235 | ]. |
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236 | |
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237 | notation "hvbox(n break ÷ m)" |
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238 | right associative with precedence 47 |
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239 | for @{ 'division $n $m }. |
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240 | |
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241 | interpretation "Nat division" 'division n m = (division n m). |
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242 | |
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243 | let rec modulus_aux (m: nat) (n: nat) (p: nat) ≝ |
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244 | match leb n p with |
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245 | [ true ⇒ n |
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246 | | false ⇒ |
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247 | match m with |
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248 | [ O ⇒ n |
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249 | | S o ⇒ modulus_aux o (n - (S p)) p |
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250 | ] |
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251 | ]. |
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252 | |
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253 | definition modulus ≝ |
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254 | λm, n: nat. |
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255 | match n with |
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256 | [ O ⇒ m |
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257 | | S o ⇒ modulus_aux m m o |
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258 | ]. |
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259 | |
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260 | notation "hvbox(n break 'mod' m)" |
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261 | right associative with precedence 47 |
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262 | for @{ 'modulus $n $m }. |
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263 | |
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264 | interpretation "Nat modulus" 'modulus m n = (modulus m n). |
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265 | |
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266 | definition divide_with_remainder ≝ |
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267 | λm, n: nat. |
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268 | pair ? ? (m ÷ n) (modulus m n). |
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269 | |
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270 | let rec exponential (m: nat) (n: nat) on n ≝ |
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271 | match n with |
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272 | [ O ⇒ S O |
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273 | | S o ⇒ m * exponential m o |
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274 | ]. |
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275 | |
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276 | interpretation "Nat exponential" 'exp n m = (exponential n m). |
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277 | |
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278 | notation "hvbox(a break ⊎ b)" |
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279 | left associative with precedence 50 |
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280 | for @{ 'disjoint_union $a $b }. |
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281 | interpretation "sum" 'disjoint_union A B = (Sum A B). |
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282 | |
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283 | theorem less_than_or_equal_monotone: |
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284 | ∀m, n: nat. |
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285 | m ≤ n → (S m) ≤ (S n). |
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286 | #m #n #H |
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287 | elim H |
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288 | /2/ |
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289 | qed. |
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290 | |
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291 | theorem less_than_or_equal_b_complete: |
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292 | ∀m, n: nat. |
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293 | leb m n = false → ¬(m ≤ n). |
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294 | #m; |
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295 | elim m; |
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296 | normalize |
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297 | [ #n #H |
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298 | destruct |
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299 | | #y #H1 #z |
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300 | cases z |
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301 | normalize |
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302 | [ #H |
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303 | /2/ |
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304 | | /3/ |
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305 | ] |
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306 | ] |
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307 | qed. |
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308 | |
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309 | theorem less_than_or_equal_b_correct: |
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310 | ∀m, n: nat. |
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311 | leb m n = true → m ≤ n. |
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312 | #m |
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313 | elim m |
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314 | // |
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315 | #y #H1 #z |
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316 | cases z |
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317 | normalize |
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318 | [ #H |
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319 | destruct |
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320 | | #n #H lapply (H1 … H) /2/ |
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321 | ] |
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322 | qed. |
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323 | |
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324 | definition less_than_or_equal_b_elim: |
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325 | ∀m, n: nat. |
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326 | ∀P: bool → Type[0]. |
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327 | (m ≤ n → P true) → (¬(m ≤ n) → P false) → P (leb m n). |
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328 | #m #n #P #H1 #H2; |
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329 | lapply (less_than_or_equal_b_correct m n) |
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330 | lapply (less_than_or_equal_b_complete m n) |
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331 | cases (leb m n) |
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332 | /3/ |
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333 | qed. |
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