1 | include "basics/lists/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 | include "basics/russell.ma". |
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5 | |
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6 | (* let's implement a daemon not used by automation *) |
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7 | inductive DAEMONXXX: Type[0] ≝ K1DAEMONXXX: DAEMONXXX | K2DAEMONXXX: DAEMONXXX. |
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8 | axiom IMPOSSIBLE: K1DAEMONXXX = K2DAEMONXXX. |
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9 | example daemon: False. generalize in match IMPOSSIBLE; #IMPOSSIBLE destruct(IMPOSSIBLE) qed. |
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10 | example not_implemented: False. cases daemon qed. |
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11 | |
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12 | notation "⊥" with precedence 90 |
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13 | for @{ match ? in False with [ ] }. |
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14 | |
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15 | definition ltb ≝ |
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16 | λm, n: nat. |
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17 | leb (S m) n. |
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18 | |
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19 | definition geb ≝ |
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20 | λm, n: nat. |
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21 | ltb n m. |
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22 | |
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23 | definition gtb ≝ |
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24 | λm, n: nat. |
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25 | ltb n m. |
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26 | |
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27 | (* dpm: unless I'm being stupid, this isn't defined in the stdlib? *) |
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28 | let rec eq_nat (n: nat) (m: nat) on n: bool ≝ |
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29 | match n with |
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30 | [ O ⇒ match m with [ O ⇒ true | _ ⇒ false ] |
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31 | | S n' ⇒ match m with [ S m' ⇒ eq_nat n' m' | _ ⇒ false ] |
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32 | ]. |
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33 | |
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34 | let rec forall |
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35 | (A: Type[0]) (f: A → bool) (l: list A) |
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36 | on l ≝ |
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37 | match l with |
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38 | [ nil ⇒ true |
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39 | | cons hd tl ⇒ f hd ∧ forall A f tl |
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40 | ]. |
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41 | |
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42 | let rec prefix |
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43 | (A: Type[0]) (k: nat) (l: list A) |
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44 | on l ≝ |
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45 | match l with |
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46 | [ nil ⇒ [ ] |
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47 | | cons hd tl ⇒ |
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48 | match k with |
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49 | [ O ⇒ [ ] |
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50 | | S k' ⇒ hd :: prefix A k' tl |
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51 | ] |
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52 | ]. |
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53 | |
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54 | let rec fold_left2 |
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55 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C → A) (accu: A) |
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56 | (left: list B) (right: list C) (proof: |left| = |right|) |
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57 | on left: A ≝ |
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58 | match left return λx. |x| = |right| → A with |
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59 | [ nil ⇒ λnil_prf. |
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60 | match right return λx. |[ ]| = |x| → A with |
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61 | [ nil ⇒ λnil_nil_prf. accu |
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62 | | cons hd tl ⇒ λcons_nil_absrd. ? |
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63 | ] nil_prf |
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64 | | cons hd tl ⇒ λcons_prf. |
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65 | match right return λx. |hd::tl| = |x| → A with |
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66 | [ nil ⇒ λcons_nil_absrd. ? |
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67 | | cons hd' tl' ⇒ λcons_cons_prf. |
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68 | fold_left2 … f (f accu hd hd') tl tl' ? |
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69 | ] cons_prf |
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70 | ] proof. |
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71 | [ 1: normalize in cons_nil_absrd; |
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72 | destruct(cons_nil_absrd) |
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73 | | 2: normalize in cons_nil_absrd; |
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74 | destruct(cons_nil_absrd) |
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75 | | 3: normalize in cons_cons_prf; |
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76 | @injective_S |
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77 | assumption |
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78 | ] |
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79 | qed. |
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80 | |
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81 | let rec remove_n_first_internal |
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82 | (i: nat) (A: Type[0]) (l: list A) (n: nat) |
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83 | on l ≝ |
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84 | match l with |
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85 | [ nil ⇒ [ ] |
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86 | | cons hd tl ⇒ |
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87 | match eq_nat i n with |
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88 | [ true ⇒ l |
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89 | | _ ⇒ remove_n_first_internal (S i) A tl n |
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90 | ] |
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91 | ]. |
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92 | |
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93 | definition remove_n_first ≝ |
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94 | λA: Type[0]. |
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95 | λn: nat. |
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96 | λl: list A. |
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97 | remove_n_first_internal 0 A l n. |
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98 | |
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99 | let rec foldi_from_until_internal |
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100 | (A: Type[0]) (i: nat) (res: ?) (rem: list A) (m: nat) (f: nat → list A → A → list A) |
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101 | on rem ≝ |
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102 | match rem with |
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103 | [ nil ⇒ res |
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104 | | cons e tl ⇒ |
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105 | match geb i m with |
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106 | [ true ⇒ res |
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107 | | _ ⇒ foldi_from_until_internal A (S i) (f i res e) tl m f |
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108 | ] |
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109 | ]. |
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110 | |
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111 | definition foldi_from_until ≝ |
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112 | λA: Type[0]. |
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113 | λn: nat. |
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114 | λm: nat. |
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115 | λf: ?. |
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116 | λa: ?. |
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117 | λl: ?. |
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118 | foldi_from_until_internal A 0 a (remove_n_first A n l) m f. |
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119 | |
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120 | definition foldi_from ≝ |
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121 | λA: Type[0]. |
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122 | λn. |
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123 | λf. |
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124 | λa. |
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125 | λl. |
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126 | foldi_from_until A n (|l|) f a l. |
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127 | |
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128 | definition foldi_until ≝ |
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129 | λA: Type[0]. |
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130 | λm. |
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131 | λf. |
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132 | λa. |
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133 | λl. |
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134 | foldi_from_until A 0 m f a l. |
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135 | |
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136 | definition foldi ≝ |
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137 | λA: Type[0]. |
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138 | λf. |
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139 | λa. |
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140 | λl. |
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141 | foldi_from_until A 0 (|l|) f a l. |
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142 | |
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143 | definition hd_safe ≝ |
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144 | λA: Type[0]. |
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145 | λl: list A. |
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146 | λproof: 0 < |l|. |
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147 | match l return λx. 0 < |x| → A with |
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148 | [ nil ⇒ λnil_absrd. ? |
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149 | | cons hd tl ⇒ λcons_prf. hd |
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150 | ] proof. |
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151 | normalize in nil_absrd; |
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152 | cases(not_le_Sn_O 0) |
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153 | #HYP |
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154 | cases(HYP nil_absrd) |
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155 | qed. |
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156 | |
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157 | definition tail_safe ≝ |
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158 | λA: Type[0]. |
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159 | λl: list A. |
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160 | λproof: 0 < |l|. |
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161 | match l return λx. 0 < |x| → list A with |
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162 | [ nil ⇒ λnil_absrd. ? |
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163 | | cons hd tl ⇒ λcons_prf. tl |
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164 | ] proof. |
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165 | normalize in nil_absrd; |
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166 | cases(not_le_Sn_O 0) |
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167 | #HYP |
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168 | cases(HYP nil_absrd) |
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169 | qed. |
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170 | |
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171 | let rec split |
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172 | (A: Type[0]) (l: list A) (index: nat) (proof: index ≤ |l|) |
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173 | on index ≝ |
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174 | match index return λx. x ≤ |l| → (list A) × (list A) with |
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175 | [ O ⇒ λzero_prf. 〈[], l〉 |
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176 | | S index' ⇒ λsucc_prf. |
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177 | match l return λx. S index' ≤ |x| → (list A) × (list A) with |
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178 | [ nil ⇒ λnil_absrd. ? |
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179 | | cons hd tl ⇒ λcons_prf. |
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180 | let 〈l1, l2〉 ≝ split A tl index' ? in |
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181 | 〈hd :: l1, l2〉 |
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182 | ] succ_prf |
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183 | ] proof. |
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184 | [1: normalize in nil_absrd; |
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185 | cases(not_le_Sn_O index') |
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186 | #HYP |
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187 | cases(HYP nil_absrd) |
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188 | |2: normalize in cons_prf; |
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189 | @le_S_S_to_le |
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190 | assumption |
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191 | ] |
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192 | qed. |
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193 | |
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194 | let rec nth_safe |
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195 | (elt_type: Type[0]) (index: nat) (the_list: list elt_type) |
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196 | (proof: index < | the_list |) |
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197 | on index ≝ |
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198 | match index return λs. s < | the_list | → elt_type with |
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199 | [ O ⇒ |
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200 | match the_list return λt. 0 < | t | → elt_type with |
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201 | [ nil ⇒ λnil_absurd. ? |
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202 | | cons hd tl ⇒ λcons_proof. hd |
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203 | ] |
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204 | | S index' ⇒ |
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205 | match the_list return λt. S index' < | t | → elt_type with |
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206 | [ nil ⇒ λnil_absurd. ? |
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207 | | cons hd tl ⇒ |
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208 | λcons_proof. nth_safe elt_type index' tl ? |
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209 | ] |
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210 | ] proof. |
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211 | [ normalize in nil_absurd; |
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212 | cases (not_le_Sn_O 0) |
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213 | #ABSURD |
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214 | elim (ABSURD nil_absurd) |
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215 | | normalize in nil_absurd; |
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216 | cases (not_le_Sn_O (S index')) |
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217 | #ABSURD |
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218 | elim (ABSURD nil_absurd) |
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219 | | normalize in cons_proof; |
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220 | @le_S_S_to_le |
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221 | assumption |
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222 | ] |
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223 | qed. |
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224 | |
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225 | definition last_safe ≝ |
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226 | λelt_type: Type[0]. |
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227 | λthe_list: list elt_type. |
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228 | λproof : 0 < | the_list |. |
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229 | nth_safe elt_type (|the_list| - 1) the_list ?. |
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230 | normalize /2/ |
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231 | qed. |
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232 | |
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233 | let rec reduce |
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234 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) on left ≝ |
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235 | match left with |
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236 | [ nil ⇒ 〈〈[ ], [ ]〉, 〈[ ], right〉〉 |
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237 | | cons hd tl ⇒ |
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238 | match right with |
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239 | [ nil ⇒ 〈〈[ ], left〉, 〈[ ], [ ]〉〉 |
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240 | | cons hd' tl' ⇒ |
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241 | let 〈cleft, cright〉 ≝ reduce A B tl tl' in |
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242 | let 〈commonl, restl〉 ≝ cleft in |
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243 | let 〈commonr, restr〉 ≝ cright in |
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244 | 〈〈hd :: commonl, restl〉, 〈hd' :: commonr, restr〉〉 |
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245 | ] |
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246 | ]. |
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247 | |
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248 | (* |
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249 | axiom reduce_strong: |
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250 | ∀A: Type[0]. |
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251 | ∀left: list A. |
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252 | ∀right: list A. |
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253 | Σret: ((list A) × (list A)) × ((list A) × (list A)). | \fst (\fst ret) | = | \fst (\snd ret) |. |
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254 | *) |
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255 | |
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256 | let rec reduce_strong |
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257 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) |
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258 | on left : Σret: ((list A) × (list A)) × ((list B) × (list B)). |\fst (\fst ret)| = |\fst (\snd ret)| ≝ |
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259 | match left with |
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260 | [ nil ⇒ 〈〈[ ], [ ]〉, 〈[ ], right〉〉 |
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261 | | cons hd tl ⇒ |
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262 | match right with |
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263 | [ nil ⇒ 〈〈[ ], left〉, 〈[ ], [ ]〉〉 |
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264 | | cons hd' tl' ⇒ |
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265 | let 〈cleft, cright〉 ≝ reduce_strong A B tl tl' in |
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266 | let 〈commonl, restl〉 ≝ cleft in |
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267 | let 〈commonr, restr〉 ≝ cright in |
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268 | 〈〈hd :: commonl, restl〉, 〈hd' :: commonr, restr〉〉 |
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269 | ] |
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270 | ]. |
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271 | [ 1: normalize % |
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272 | | 2: normalize % |
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273 | | 3: normalize >p3 in p2; >p4 cases (reduce_strong … tl tl1) normalize |
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274 | #X #H #EQ destruct // ] |
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275 | qed. |
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276 | |
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277 | let rec map2_opt |
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278 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C) |
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279 | (left: list A) (right: list B) on left ≝ |
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280 | match left with |
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281 | [ nil ⇒ |
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282 | match right with |
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283 | [ nil ⇒ Some ? (nil C) |
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284 | | _ ⇒ None ? |
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285 | ] |
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286 | | cons hd tl ⇒ |
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287 | match right with |
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288 | [ nil ⇒ None ? |
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289 | | cons hd' tl' ⇒ |
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290 | match map2_opt A B C f tl tl' with |
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291 | [ None ⇒ None ? |
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292 | | Some tail ⇒ Some ? (f hd hd' :: tail) |
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293 | ] |
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294 | ] |
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295 | ]. |
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296 | |
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297 | let rec map2 |
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298 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C) |
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299 | (left: list A) (right: list B) (proof: | left | = | right |) on left ≝ |
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300 | match left return λx. | x | = | right | → list C with |
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301 | [ nil ⇒ |
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302 | match right return λy. | [] | = | y | → list C with |
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303 | [ nil ⇒ λnil_prf. nil C |
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304 | | _ ⇒ λcons_absrd. ? |
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305 | ] |
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306 | | cons hd tl ⇒ |
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307 | match right return λy. | hd::tl | = | y | → list C with |
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308 | [ nil ⇒ λnil_absrd. ? |
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309 | | cons hd' tl' ⇒ λcons_prf. (f hd hd') :: map2 A B C f tl tl' ? |
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310 | ] |
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311 | ] proof. |
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312 | [1: normalize in cons_absrd; |
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313 | destruct(cons_absrd) |
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314 | |2: normalize in nil_absrd; |
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315 | destruct(nil_absrd) |
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316 | |3: normalize in cons_prf; |
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317 | destruct(cons_prf) |
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318 | assumption |
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319 | ] |
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320 | qed. |
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321 | |
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322 | let rec map3 |
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323 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (D: Type[0]) (f: A → B → C → D) |
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324 | (left: list A) (centre: list B) (right: list C) |
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325 | (prflc: |left| = |centre|) (prfcr: |centre| = |right|) on left ≝ |
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326 | match left return λx. |x| = |centre| → list D with |
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327 | [ nil ⇒ λnil_prf. |
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328 | match centre return λx. |x| = |right| → list D with |
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329 | [ nil ⇒ λnil_nil_prf. |
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330 | match right return λx. |nil ?| = |x| → list D with |
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331 | [ nil ⇒ λnil_nil_nil_prf. nil D |
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332 | | cons hd tl ⇒ λcons_nil_nil_absrd. ? |
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333 | ] nil_nil_prf |
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334 | | cons hd tl ⇒ λnil_cons_absrd. ? |
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335 | ] prfcr |
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336 | | cons hd tl ⇒ λcons_prf. |
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337 | match centre return λx. |x| = |right| → list D with |
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338 | [ nil ⇒ λcons_nil_absrd. ? |
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339 | | cons hd' tl' ⇒ λcons_cons_prf. |
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340 | match right return λx. |right| = |x| → |cons ? hd' tl'| = |x| → list D with |
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341 | [ nil ⇒ λrefl_prf. λcons_cons_nil_absrd. ? |
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342 | | cons hd'' tl'' ⇒ λrefl_prf. λcons_cons_cons_prf. |
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343 | (f hd hd' hd'') :: (map3 A B C D f tl tl' tl'' ? ?) |
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344 | ] (refl ? (|right|)) cons_cons_prf |
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345 | ] prfcr |
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346 | ] prflc. |
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347 | [ 1: normalize in cons_nil_nil_absrd; |
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348 | destruct(cons_nil_nil_absrd) |
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349 | | 2: generalize in match nil_cons_absrd; |
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350 | <prfcr <nil_prf #HYP |
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351 | normalize in HYP; |
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352 | destruct(HYP) |
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353 | | 3: generalize in match cons_nil_absrd; |
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354 | <prfcr <cons_prf #HYP |
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355 | normalize in HYP; |
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356 | destruct(HYP) |
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357 | | 4: normalize in cons_cons_nil_absrd; |
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358 | destruct(cons_cons_nil_absrd) |
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359 | | 5: normalize in cons_cons_cons_prf; |
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360 | destruct(cons_cons_cons_prf) |
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361 | assumption |
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362 | | 6: generalize in match cons_cons_cons_prf; |
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363 | <refl_prf <prfcr <cons_prf #HYP |
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364 | normalize in HYP; |
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365 | destruct(HYP) |
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366 | @sym_eq assumption |
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367 | ] |
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368 | qed. |
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369 | |
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370 | lemma eq_rect_Type0_r : |
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371 | ∀A: Type[0]. |
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372 | ∀a:A. |
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373 | ∀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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374 | #A #a #P #H #x #p lapply H lapply P cases p // |
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375 | qed. |
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376 | |
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377 | let rec safe_nth (A: Type[0]) (n: nat) (l: list A) (p: n < length A l) on n: A ≝ |
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378 | match n return λo. o < length A l → A with |
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379 | [ O ⇒ |
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380 | match l return λm. 0 < length A m → A with |
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381 | [ nil ⇒ λabsd1. ? |
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382 | | cons hd tl ⇒ λprf1. hd |
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383 | ] |
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384 | | S n' ⇒ |
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385 | match l return λm. S n' < length A m → A with |
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386 | [ nil ⇒ λabsd2. ? |
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387 | | cons hd tl ⇒ λprf2. safe_nth A n' tl ? |
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388 | ] |
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389 | ] ?. |
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390 | [ 1: |
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391 | @ p |
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392 | | 4: |
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393 | normalize in prf2; |
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394 | normalize |
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395 | @ le_S_S_to_le |
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396 | assumption |
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397 | | 2: |
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398 | normalize in absd1; |
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399 | cases (not_le_Sn_O O) |
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400 | # H |
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401 | elim (H absd1) |
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402 | | 3: |
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403 | normalize in absd2; |
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404 | cases (not_le_Sn_O (S n')) |
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405 | # H |
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406 | elim (H absd2) |
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407 | ] |
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408 | qed. |
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409 | |
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410 | let rec nub_by_internal (A: Type[0]) (f: A → A → bool) (l: list A) (n: nat) on n ≝ |
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411 | match n with |
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412 | [ O ⇒ |
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413 | match l with |
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414 | [ nil ⇒ [ ] |
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415 | | cons hd tl ⇒ l |
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416 | ] |
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417 | | S n ⇒ |
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418 | match l with |
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419 | [ nil ⇒ [ ] |
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420 | | cons hd tl ⇒ |
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421 | hd :: nub_by_internal A f (filter ? (λy. notb (f y hd)) tl) n |
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422 | ] |
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423 | ]. |
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424 | |
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425 | definition nub_by ≝ |
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426 | λA: Type[0]. |
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427 | λf: A → A → bool. |
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428 | λl: list A. |
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429 | nub_by_internal A f l (length ? l). |
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430 | |
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431 | let rec member (A: Type[0]) (eq: A → A → bool) (a: A) (l: list A) on l ≝ |
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432 | match l with |
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433 | [ nil ⇒ false |
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434 | | cons hd tl ⇒ orb (eq a hd) (member A eq a tl) |
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435 | ]. |
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436 | |
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437 | let rec take (A: Type[0]) (n: nat) (l: list A) on n: list A ≝ |
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438 | match n with |
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439 | [ O ⇒ [ ] |
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440 | | S n ⇒ |
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441 | match l with |
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442 | [ nil ⇒ [ ] |
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443 | | cons hd tl ⇒ hd :: take A n tl |
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444 | ] |
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445 | ]. |
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446 | |
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447 | let rec drop (A: Type[0]) (n: nat) (l: list A) on n ≝ |
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448 | match n with |
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449 | [ O ⇒ l |
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450 | | S n ⇒ |
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451 | match l with |
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452 | [ nil ⇒ [ ] |
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453 | | cons hd tl ⇒ drop A n tl |
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454 | ] |
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455 | ]. |
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456 | |
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457 | definition list_split ≝ |
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458 | λA: Type[0]. |
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459 | λn: nat. |
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460 | λl: list A. |
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461 | 〈take A n l, drop A n l〉. |
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462 | |
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463 | let rec mapi_internal (A: Type[0]) (B: Type[0]) (n: nat) (f: nat → A → B) |
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464 | (l: list A) on l: list B ≝ |
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465 | match l with |
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466 | [ nil ⇒ nil ? |
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467 | | cons hd tl ⇒ (f n hd) :: (mapi_internal A B (n + 1) f tl) |
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468 | ]. |
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469 | |
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470 | definition mapi ≝ |
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471 | λA, B: Type[0]. |
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472 | λf: nat → A → B. |
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473 | λl: list A. |
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474 | mapi_internal A B 0 f l. |
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475 | |
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476 | let rec zip_pottier |
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477 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) |
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478 | on left ≝ |
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479 | match left with |
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480 | [ nil ⇒ [ ] |
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481 | | cons hd tl ⇒ |
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482 | match right with |
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483 | [ nil ⇒ [ ] |
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484 | | cons hd' tl' ⇒ 〈hd, hd'〉 :: zip_pottier A B tl tl' |
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485 | ] |
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486 | ]. |
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487 | |
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488 | let rec zip_safe |
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489 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) (prf: |left| = |right|) |
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490 | on left ≝ |
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491 | match left return λx. |x| = |right| → list (A × B) with |
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492 | [ nil ⇒ λnil_prf. |
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493 | match right return λx. |[ ]| = |x| → list (A × B) with |
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494 | [ nil ⇒ λnil_nil_prf. [ ] |
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495 | | cons hd tl ⇒ λnil_cons_absrd. ? |
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496 | ] nil_prf |
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497 | | cons hd tl ⇒ λcons_prf. |
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498 | match right return λx. |hd::tl| = |x| → list (A × B) with |
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499 | [ nil ⇒ λcons_nil_absrd. ? |
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500 | | cons hd' tl' ⇒ λcons_cons_prf. 〈hd, hd'〉 :: zip_safe A B tl tl' ? |
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501 | ] cons_prf |
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502 | ] prf. |
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503 | [ 1: normalize in nil_cons_absrd; |
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504 | destruct(nil_cons_absrd) |
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505 | | 2: normalize in cons_nil_absrd; |
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506 | destruct(cons_nil_absrd) |
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507 | | 3: normalize in cons_cons_prf; |
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508 | @injective_S |
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509 | assumption |
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510 | ] |
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511 | qed. |
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512 | |
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513 | 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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514 | match l with |
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515 | [ nil ⇒ Some ? (nil (A × B)) |
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516 | | cons hd tl ⇒ |
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517 | match r with |
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518 | [ nil ⇒ None ? |
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519 | | cons hd' tl' ⇒ |
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520 | match zip ? ? tl tl' with |
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521 | [ None ⇒ None ? |
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522 | | Some tail ⇒ Some ? (〈hd, hd'〉 :: tail) |
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523 | ] |
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524 | ] |
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525 | ]. |
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526 | |
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527 | let rec foldl (A: Type[0]) (B: Type[0]) (f: A → B → A) (a: A) (l: list B) on l ≝ |
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528 | match l with |
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529 | [ nil ⇒ a |
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530 | | cons hd tl ⇒ foldl A B f (f a hd) tl |
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531 | ]. |
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532 | |
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533 | lemma foldl_step: |
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534 | ∀A:Type[0]. |
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535 | ∀B: Type[0]. |
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536 | ∀H: A → B → A. |
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537 | ∀acc: A. |
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538 | ∀pre: list B. |
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539 | ∀hd:B. |
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540 | foldl A B H acc (pre@[hd]) = (H (foldl A B H acc pre) hd). |
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541 | #A #B #H #acc #pre generalize in match acc; -acc; elim pre |
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542 | [ normalize; // |
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543 | | #hd #tl #IH #acc #X normalize; @IH ] |
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544 | qed. |
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545 | |
---|
546 | lemma foldl_append: |
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547 | ∀A:Type[0]. |
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548 | ∀B: Type[0]. |
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549 | ∀H: A → B → A. |
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550 | ∀acc: A. |
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551 | ∀suff,pre: list B. |
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552 | foldl A B H acc (pre@suff) = (foldl A B H (foldl A B H acc pre) suff). |
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553 | #A #B #H #acc #suff elim suff |
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554 | [ #pre >append_nil % |
---|
555 | | #hd #tl #IH #pre whd in ⊢ (???%); <(foldl_step … H ??) applyS (IH (pre@[hd])) ] |
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556 | qed. |
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557 | |
---|
558 | definition flatten ≝ |
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559 | λA: Type[0]. |
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560 | λl: list (list A). |
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561 | foldr ? ? (append ?) [ ] l. |
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562 | |
---|
563 | let rec rev (A: Type[0]) (l: list A) on l ≝ |
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564 | match l with |
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565 | [ nil ⇒ nil A |
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566 | | cons hd tl ⇒ (rev A tl) @ [ hd ] |
---|
567 | ]. |
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568 | |
---|
569 | lemma append_length: |
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570 | ∀A: Type[0]. |
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571 | ∀l, r: list A. |
---|
572 | |(l @ r)| = |l| + |r|. |
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573 | #A #L #R |
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574 | elim L |
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575 | [ % |
---|
576 | | #HD #TL #IH |
---|
577 | normalize >IH % |
---|
578 | ] |
---|
579 | qed. |
---|
580 | |
---|
581 | lemma append_nil: |
---|
582 | ∀A: Type[0]. |
---|
583 | ∀l: list A. |
---|
584 | l @ [ ] = l. |
---|
585 | #A #L |
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586 | elim L // |
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587 | qed. |
---|
588 | |
---|
589 | lemma rev_append: |
---|
590 | ∀A: Type[0]. |
---|
591 | ∀l, r: list A. |
---|
592 | rev A (l @ r) = rev A r @ rev A l. |
---|
593 | #A #L #R |
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594 | elim L |
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595 | [ normalize >append_nil % |
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596 | | #HD #TL #IH |
---|
597 | normalize >IH |
---|
598 | @associative_append |
---|
599 | ] |
---|
600 | qed. |
---|
601 | |
---|
602 | lemma rev_length: |
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603 | ∀A: Type[0]. |
---|
604 | ∀l: list A. |
---|
605 | |rev A l| = |l|. |
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606 | #A #L |
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607 | elim L |
---|
608 | [ % |
---|
609 | | #HD #TL #IH |
---|
610 | normalize |
---|
611 | >(append_length A (rev A TL) [HD]) |
---|
612 | normalize /2/ |
---|
613 | ] |
---|
614 | qed. |
---|
615 | |
---|
616 | lemma nth_append_first: |
---|
617 | ∀A:Type[0]. |
---|
618 | ∀n:nat.∀l1,l2:list A.∀d:A. |
---|
619 | n < |l1| → nth n A (l1@l2) d = nth n A l1 d. |
---|
620 | #A #n #l1 #l2 #d |
---|
621 | generalize in match n; -n; elim l1 |
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622 | [ normalize #k #Hk @⊥ @(absurd … Hk) @not_le_Sn_O |
---|
623 | | #h #t #Hind #k normalize |
---|
624 | cases k -k |
---|
625 | [ #Hk normalize @refl |
---|
626 | | #k #Hk normalize @(Hind k) @le_S_S_to_le @Hk |
---|
627 | ] |
---|
628 | ] |
---|
629 | qed. |
---|
630 | |
---|
631 | lemma nth_append_second: |
---|
632 | ∀A: Type[0].∀n.∀l1,l2:list A.∀d.n ≥ length A l1 -> |
---|
633 | nth n A (l1@l2) d = nth (n - length A l1) A l2 d. |
---|
634 | #A #n #l1 #l2 #d |
---|
635 | generalize in match n; -n; elim l1 |
---|
636 | [ normalize #k #Hk <(minus_n_O) @refl |
---|
637 | | #h #t #Hind #k normalize |
---|
638 | cases k -k; |
---|
639 | [ #Hk @⊥ @(absurd (S (|t|) ≤ 0)) [ @Hk | @not_le_Sn_O ] |
---|
640 | | #k #Hk normalize @(Hind k) @le_S_S_to_le @Hk |
---|
641 | ] |
---|
642 | ] |
---|
643 | qed. |
---|
644 | |
---|
645 | |
---|
646 | let rec fold_left_i_aux (A: Type[0]) (B: Type[0]) |
---|
647 | (f: nat → A → B → A) (x: A) (i: nat) (l: list B) on l ≝ |
---|
648 | match l with |
---|
649 | [ nil ⇒ x |
---|
650 | | cons hd tl ⇒ fold_left_i_aux A B f (f i x hd) (S i) tl |
---|
651 | ]. |
---|
652 | |
---|
653 | definition fold_left_i ≝ λA,B,f,x. fold_left_i_aux A B f x O. |
---|
654 | |
---|
655 | notation "hvbox(t⌈o ↦ h⌉)" |
---|
656 | with precedence 45 |
---|
657 | for @{ match (? : $o=$h) with [ refl ⇒ $t ] }. |
---|
658 | |
---|
659 | definition function_apply ≝ |
---|
660 | λA, B: Type[0]. |
---|
661 | λf: A → B. |
---|
662 | λa: A. |
---|
663 | f a. |
---|
664 | |
---|
665 | notation "f break $ x" |
---|
666 | left associative with precedence 99 |
---|
667 | for @{ 'function_apply $f $x }. |
---|
668 | |
---|
669 | interpretation "Function application" 'function_apply f x = (function_apply ? ? f x). |
---|
670 | |
---|
671 | let rec iterate (A: Type[0]) (f: A → A) (a: A) (n: nat) on n ≝ |
---|
672 | match n with |
---|
673 | [ O ⇒ a |
---|
674 | | S o ⇒ f (iterate A f a o) |
---|
675 | ]. |
---|
676 | |
---|
677 | let rec division_aux (m: nat) (n : nat) (p: nat) ≝ |
---|
678 | match ltb n (S p) with |
---|
679 | [ true ⇒ O |
---|
680 | | false ⇒ |
---|
681 | match m with |
---|
682 | [ O ⇒ O |
---|
683 | | (S q) ⇒ S (division_aux q (n - (S p)) p) |
---|
684 | ] |
---|
685 | ]. |
---|
686 | |
---|
687 | definition division ≝ |
---|
688 | λm, n: nat. |
---|
689 | match n with |
---|
690 | [ O ⇒ S m |
---|
691 | | S o ⇒ division_aux m m o |
---|
692 | ]. |
---|
693 | |
---|
694 | notation "hvbox(n break ÷ m)" |
---|
695 | right associative with precedence 47 |
---|
696 | for @{ 'division $n $m }. |
---|
697 | |
---|
698 | interpretation "Nat division" 'division n m = (division n m). |
---|
699 | |
---|
700 | let rec modulus_aux (m: nat) (n: nat) (p: nat) ≝ |
---|
701 | match leb n p with |
---|
702 | [ true ⇒ n |
---|
703 | | false ⇒ |
---|
704 | match m with |
---|
705 | [ O ⇒ n |
---|
706 | | S o ⇒ modulus_aux o (n - (S p)) p |
---|
707 | ] |
---|
708 | ]. |
---|
709 | |
---|
710 | definition modulus ≝ |
---|
711 | λm, n: nat. |
---|
712 | match n with |
---|
713 | [ O ⇒ m |
---|
714 | | S o ⇒ modulus_aux m m o |
---|
715 | ]. |
---|
716 | |
---|
717 | notation "hvbox(n break 'mod' m)" |
---|
718 | right associative with precedence 47 |
---|
719 | for @{ 'modulus $n $m }. |
---|
720 | |
---|
721 | interpretation "Nat modulus" 'modulus m n = (modulus m n). |
---|
722 | |
---|
723 | definition divide_with_remainder ≝ |
---|
724 | λm, n: nat. |
---|
725 | mk_Prod … (m ÷ n) (modulus m n). |
---|
726 | |
---|
727 | let rec exponential (m: nat) (n: nat) on n ≝ |
---|
728 | match n with |
---|
729 | [ O ⇒ S O |
---|
730 | | S o ⇒ m * exponential m o |
---|
731 | ]. |
---|
732 | |
---|
733 | interpretation "Nat exponential" 'exp n m = (exponential n m). |
---|
734 | |
---|
735 | notation "hvbox(a break ⊎ b)" |
---|
736 | left associative with precedence 50 |
---|
737 | for @{ 'disjoint_union $a $b }. |
---|
738 | interpretation "sum" 'disjoint_union A B = (Sum A B). |
---|
739 | |
---|
740 | theorem less_than_or_equal_monotone: |
---|
741 | ∀m, n: nat. |
---|
742 | m ≤ n → (S m) ≤ (S n). |
---|
743 | #m #n #H |
---|
744 | elim H |
---|
745 | /2/ |
---|
746 | qed. |
---|
747 | |
---|
748 | theorem less_than_or_equal_b_complete: |
---|
749 | ∀m, n: nat. |
---|
750 | leb m n = false → ¬(m ≤ n). |
---|
751 | #m; |
---|
752 | elim m; |
---|
753 | normalize |
---|
754 | [ #n #H |
---|
755 | destruct |
---|
756 | | #y #H1 #z |
---|
757 | cases z |
---|
758 | normalize |
---|
759 | [ #H |
---|
760 | /2/ |
---|
761 | | /3/ |
---|
762 | ] |
---|
763 | ] |
---|
764 | qed. |
---|
765 | |
---|
766 | theorem less_than_or_equal_b_correct: |
---|
767 | ∀m, n: nat. |
---|
768 | leb m n = true → m ≤ n. |
---|
769 | #m |
---|
770 | elim m |
---|
771 | // |
---|
772 | #y #H1 #z |
---|
773 | cases z |
---|
774 | normalize |
---|
775 | [ #H |
---|
776 | destruct |
---|
777 | | #n #H lapply (H1 … H) /2/ |
---|
778 | ] |
---|
779 | qed. |
---|
780 | |
---|
781 | definition less_than_or_equal_b_elim: |
---|
782 | ∀m, n: nat. |
---|
783 | ∀P: bool → Type[0]. |
---|
784 | (m ≤ n → P true) → (¬(m ≤ n) → P false) → P (leb m n). |
---|
785 | #m #n #P #H1 #H2; |
---|
786 | lapply (less_than_or_equal_b_correct m n) |
---|
787 | lapply (less_than_or_equal_b_complete m n) |
---|
788 | cases (leb m n) |
---|
789 | /3/ |
---|
790 | qed. |
---|
791 | |
---|
792 | lemma inclusive_disjunction_true: |
---|
793 | ∀b, c: bool. |
---|
794 | (orb b c) = true → b = true ∨ c = true. |
---|
795 | # b |
---|
796 | # c |
---|
797 | elim b |
---|
798 | [ normalize |
---|
799 | # H |
---|
800 | @ or_introl |
---|
801 | % |
---|
802 | | normalize |
---|
803 | /3 by trans_eq, orb_true_l/ |
---|
804 | ] |
---|
805 | qed. |
---|
806 | |
---|
807 | lemma conjunction_true: |
---|
808 | ∀b, c: bool. |
---|
809 | andb b c = true → b = true ∧ c = true. |
---|
810 | # b |
---|
811 | # c |
---|
812 | elim b |
---|
813 | normalize |
---|
814 | [ /2 by conj/ |
---|
815 | | # K |
---|
816 | destruct |
---|
817 | ] |
---|
818 | qed. |
---|
819 | |
---|
820 | lemma eq_true_false: false=true → False. |
---|
821 | # K |
---|
822 | destruct |
---|
823 | qed. |
---|
824 | |
---|
825 | lemma inclusive_disjunction_b_true: ∀b. orb b true = true. |
---|
826 | # b |
---|
827 | cases b |
---|
828 | % |
---|
829 | qed. |
---|
830 | |
---|
831 | definition bool_to_Prop ≝ |
---|
832 | λb. match b with [ true ⇒ True | false ⇒ False ]. |
---|
833 | |
---|
834 | coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0]. |
---|
835 | |
---|
836 | lemma eq_false_to_notb: ∀b. b = false → ¬ b. |
---|
837 | *; /2/ |
---|
838 | qed. |
---|
839 | |
---|
840 | lemma length_append: |
---|
841 | ∀A.∀l1,l2:list A. |
---|
842 | |l1 @ l2| = |l1| + |l2|. |
---|
843 | #A #l1 elim l1 |
---|
844 | [ // |
---|
845 | | #hd #tl #IH #l2 normalize <IH //] |
---|
846 | qed. |
---|