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 | notation "Ⓧ" with precedence 90 |
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15 | for @{ λabs.match abs in False with [ ] }. |
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16 | |
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17 | |
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18 | definition ltb ≝ |
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19 | λm, n: nat. |
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20 | leb (S m) n. |
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21 | |
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22 | definition geb ≝ |
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23 | λm, n: nat. |
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24 | leb n m. |
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25 | |
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26 | definition gtb ≝ |
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27 | λm, n: nat. |
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28 | ltb n m. |
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29 | |
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30 | (* dpm: unless I'm being stupid, this isn't defined in the stdlib? *) |
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31 | let rec eq_nat (n: nat) (m: nat) on n: bool ≝ |
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32 | match n with |
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33 | [ O ⇒ match m with [ O ⇒ true | _ ⇒ false ] |
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34 | | S n' ⇒ match m with [ S m' ⇒ eq_nat n' m' | _ ⇒ false ] |
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35 | ]. |
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36 | |
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37 | let rec forall |
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38 | (A: Type[0]) (f: A → bool) (l: list A) |
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39 | on l ≝ |
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40 | match l with |
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41 | [ nil ⇒ true |
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42 | | cons hd tl ⇒ f hd ∧ forall A f tl |
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43 | ]. |
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44 | |
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45 | let rec prefix |
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46 | (A: Type[0]) (k: nat) (l: list A) |
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47 | on l ≝ |
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48 | match l with |
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49 | [ nil ⇒ [ ] |
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50 | | cons hd tl ⇒ |
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51 | match k with |
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52 | [ O ⇒ [ ] |
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53 | | S k' ⇒ hd :: prefix A k' tl |
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54 | ] |
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55 | ]. |
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56 | |
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57 | let rec fold_left2 |
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58 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C → A) (accu: A) |
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59 | (left: list B) (right: list C) (proof: |left| = |right|) |
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60 | on left: A ≝ |
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61 | match left return λx. |x| = |right| → A with |
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62 | [ nil ⇒ λnil_prf. |
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63 | match right return λx. |[ ]| = |x| → A with |
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64 | [ nil ⇒ λnil_nil_prf. accu |
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65 | | cons hd tl ⇒ λcons_nil_absrd. ? |
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66 | ] nil_prf |
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67 | | cons hd tl ⇒ λcons_prf. |
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68 | match right return λx. |hd::tl| = |x| → A with |
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69 | [ nil ⇒ λcons_nil_absrd. ? |
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70 | | cons hd' tl' ⇒ λcons_cons_prf. |
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71 | fold_left2 … f (f accu hd hd') tl tl' ? |
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72 | ] cons_prf |
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73 | ] proof. |
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74 | [ 1: normalize in cons_nil_absrd; |
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75 | destruct(cons_nil_absrd) |
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76 | | 2: normalize in cons_nil_absrd; |
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77 | destruct(cons_nil_absrd) |
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78 | | 3: normalize in cons_cons_prf; |
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79 | @injective_S |
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80 | assumption |
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81 | ] |
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82 | qed. |
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83 | |
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84 | let rec remove_n_first_internal |
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85 | (i: nat) (A: Type[0]) (l: list A) (n: nat) |
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86 | on l ≝ |
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87 | match l with |
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88 | [ nil ⇒ [ ] |
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89 | | cons hd tl ⇒ |
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90 | match eq_nat i n with |
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91 | [ true ⇒ l |
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92 | | _ ⇒ remove_n_first_internal (S i) A tl n |
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93 | ] |
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94 | ]. |
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95 | |
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96 | definition remove_n_first ≝ |
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97 | λA: Type[0]. |
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98 | λn: nat. |
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99 | λl: list A. |
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100 | remove_n_first_internal 0 A l n. |
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101 | |
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102 | let rec foldi_from_until_internal |
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103 | (A: Type[0]) (i: nat) (res: ?) (rem: list A) (m: nat) (f: nat → list A → A → list A) |
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104 | on rem ≝ |
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105 | match rem with |
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106 | [ nil ⇒ res |
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107 | | cons e tl ⇒ |
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108 | match geb i m with |
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109 | [ true ⇒ res |
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110 | | _ ⇒ foldi_from_until_internal A (S i) (f i res e) tl m f |
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111 | ] |
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112 | ]. |
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113 | |
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114 | definition foldi_from_until ≝ |
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115 | λA: Type[0]. |
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116 | λn: nat. |
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117 | λm: nat. |
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118 | λf: ?. |
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119 | λa: ?. |
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120 | λl: ?. |
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121 | foldi_from_until_internal A 0 a (remove_n_first A n l) m f. |
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122 | |
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123 | definition foldi_from ≝ |
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124 | λA: Type[0]. |
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125 | λn. |
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126 | λf. |
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127 | λa. |
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128 | λl. |
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129 | foldi_from_until A n (|l|) f a l. |
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130 | |
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131 | definition foldi_until ≝ |
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132 | λA: Type[0]. |
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133 | λm. |
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134 | λf. |
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135 | λa. |
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136 | λl. |
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137 | foldi_from_until A 0 m f a l. |
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138 | |
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139 | definition foldi ≝ |
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140 | λA: Type[0]. |
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141 | λf. |
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142 | λa. |
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143 | λl. |
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144 | foldi_from_until A 0 (|l|) f a l. |
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145 | |
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146 | definition hd_safe ≝ |
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147 | λA: Type[0]. |
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148 | λl: list A. |
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149 | λproof: 0 < |l|. |
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150 | match l return λx. 0 < |x| → A with |
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151 | [ nil ⇒ λnil_absrd. ? |
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152 | | cons hd tl ⇒ λcons_prf. hd |
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153 | ] proof. |
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154 | normalize in nil_absrd; |
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155 | cases(not_le_Sn_O 0) |
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156 | #HYP |
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157 | cases(HYP nil_absrd) |
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158 | qed. |
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159 | |
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160 | definition tail_safe ≝ |
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161 | λA: Type[0]. |
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162 | λl: list A. |
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163 | λproof: 0 < |l|. |
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164 | match l return λx. 0 < |x| → list A with |
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165 | [ nil ⇒ λnil_absrd. ? |
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166 | | cons hd tl ⇒ λcons_prf. tl |
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167 | ] proof. |
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168 | normalize in nil_absrd; |
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169 | cases(not_le_Sn_O 0) |
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170 | #HYP |
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171 | cases(HYP nil_absrd) |
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172 | qed. |
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173 | |
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174 | let rec safe_split |
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175 | (A: Type[0]) (l: list A) (index: nat) (proof: index ≤ |l|) |
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176 | on index ≝ |
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177 | match index return λx. x ≤ |l| → (list A) × (list A) with |
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178 | [ O ⇒ λzero_prf. 〈[], l〉 |
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179 | | S index' ⇒ λsucc_prf. |
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180 | match l return λx. S index' ≤ |x| → (list A) × (list A) with |
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181 | [ nil ⇒ λnil_absrd. ? |
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182 | | cons hd tl ⇒ λcons_prf. |
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183 | let 〈l1, l2〉 ≝ safe_split A tl index' ? in |
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184 | 〈hd :: l1, l2〉 |
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185 | ] succ_prf |
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186 | ] proof. |
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187 | [1: normalize in nil_absrd; |
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188 | cases(not_le_Sn_O index') |
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189 | #HYP |
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190 | cases(HYP nil_absrd) |
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191 | |2: normalize in cons_prf; |
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192 | @le_S_S_to_le |
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193 | assumption |
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194 | ] |
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195 | qed. |
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196 | |
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197 | let rec nth_safe |
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198 | (elt_type: Type[0]) (index: nat) (the_list: list elt_type) |
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199 | (proof: index < | the_list |) |
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200 | on index ≝ |
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201 | match index return λs. s < | the_list | → elt_type with |
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202 | [ O ⇒ |
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203 | match the_list return λt. 0 < | t | → elt_type with |
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204 | [ nil ⇒ λnil_absurd. ? |
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205 | | cons hd tl ⇒ λcons_proof. hd |
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206 | ] |
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207 | | S index' ⇒ |
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208 | match the_list return λt. S index' < | t | → elt_type with |
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209 | [ nil ⇒ λnil_absurd. ? |
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210 | | cons hd tl ⇒ |
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211 | λcons_proof. nth_safe elt_type index' tl ? |
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212 | ] |
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213 | ] proof. |
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214 | [ normalize in nil_absurd; |
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215 | cases (not_le_Sn_O 0) |
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216 | #ABSURD |
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217 | elim (ABSURD nil_absurd) |
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218 | | normalize in nil_absurd; |
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219 | cases (not_le_Sn_O (S index')) |
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220 | #ABSURD |
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221 | elim (ABSURD nil_absurd) |
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222 | | normalize in cons_proof; |
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223 | @le_S_S_to_le |
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224 | assumption |
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225 | ] |
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226 | qed. |
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227 | |
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228 | lemma shift_nth_safe: |
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229 | ∀T,i,l2,l1,K1,K2. |
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230 | nth_safe T i l1 K1 = nth_safe T (i+|l2|) (l2@l1) K2. |
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231 | #T #i #l2 elim l2 normalize |
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232 | [ #l1 #K1 <plus_n_O #K2 % |
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233 | | #hd #tl #IH #l1 #K1 <plus_n_Sm #K2 change with (? < ?) in K1; change with (? < ?) in K2; |
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234 | whd in ⊢ (???%); @IH ] |
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235 | qed. |
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236 | |
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237 | lemma shift_nth_prefix: |
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238 | ∀T,l1,i,l2,K1,K2. |
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239 | nth_safe T i l1 K1 = nth_safe T i (l1@l2) K2. |
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240 | #T #l1 elim l1 normalize |
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241 | [ |
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242 | #i #l1 #K1 cases(lt_to_not_zero … K1) |
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243 | | |
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244 | #hd #tl #IH #i #l2 |
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245 | cases i |
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246 | [ |
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247 | // |
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248 | | |
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249 | #i' #K1 #K2 whd in ⊢ (??%%); |
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250 | @IH |
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251 | ] |
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252 | ] |
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253 | qed. |
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254 | |
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255 | (*CSC: practically equal to shift_nth_safe but for commutation |
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256 | of addition. One of the two lemmas should disappear. *) |
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257 | lemma nth_safe_prepend: |
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258 | ∀A,l1,l2,j.∀H:j<|l2|.∀K:|l1|+j<|(l1@l2)|. |
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259 | nth_safe A j l2 H =nth_safe A (|l1|+j) (l1@l2) K. |
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260 | #A #l1 #l2 #j #H >commutative_plus @shift_nth_safe |
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261 | qed. |
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262 | |
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263 | lemma nth_safe_cons: |
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264 | ∀A,hd,tl,l2,j,j_ok,Sj_ok. |
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265 | nth_safe A j (tl@l2) j_ok =nth_safe A (1+j) (hd::tl@l2) Sj_ok. |
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266 | // |
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267 | qed. |
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268 | |
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269 | lemma eq_nth_safe_proof_irrelevant: |
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270 | ∀A,l,i,i_ok,i_ok'. |
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271 | nth_safe A l i i_ok = nth_safe A l i i_ok'. |
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272 | #A #l #i #i_ok #i_ok' % |
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273 | qed. |
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274 | |
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275 | lemma nth_safe_append: |
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276 | ∀A,l,n,n_ok. |
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277 | ∃pre,suff. |
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278 | (pre @ [nth_safe A n l n_ok]) @ suff = l. |
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279 | #A #l elim l normalize |
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280 | [ #n #abs cases (absurd ? abs (not_le_Sn_O ?)) |
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281 | | #hd #tl #IH #n cases n normalize |
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282 | [ #_ %{[]} /2/ |
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283 | | #m #m_ok cases (IH m ?) |
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284 | [ #pre * #suff #EQ %{(hd::pre)} %{suff} <EQ in ⊢ (???%); % | skip ]] |
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285 | qed. |
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286 | |
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287 | definition last_safe ≝ |
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288 | λelt_type: Type[0]. |
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289 | λthe_list: list elt_type. |
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290 | λproof : 0 < | the_list |. |
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291 | nth_safe elt_type (|the_list| - 1) the_list ?. |
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292 | normalize /2 by lt_plus_to_minus/ |
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293 | qed. |
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294 | |
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295 | let rec reduce |
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296 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) on left ≝ |
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297 | match left with |
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298 | [ nil ⇒ 〈〈[ ], [ ]〉, 〈[ ], right〉〉 |
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299 | | cons hd tl ⇒ |
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300 | match right with |
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301 | [ nil ⇒ 〈〈[ ], left〉, 〈[ ], [ ]〉〉 |
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302 | | cons hd' tl' ⇒ |
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303 | let 〈cleft, cright〉 ≝ reduce A B tl tl' in |
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304 | let 〈commonl, restl〉 ≝ cleft in |
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305 | let 〈commonr, restr〉 ≝ cright in |
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306 | 〈〈hd :: commonl, restl〉, 〈hd' :: commonr, restr〉〉 |
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307 | ] |
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308 | ]. |
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309 | |
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310 | (* |
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311 | axiom reduce_strong: |
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312 | ∀A: Type[0]. |
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313 | ∀left: list A. |
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314 | ∀right: list A. |
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315 | Σret: ((list A) × (list A)) × ((list A) × (list A)). | \fst (\fst ret) | = | \fst (\snd ret) |. |
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316 | *) |
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317 | |
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318 | let rec reduce_strong |
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319 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) |
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320 | on left : Σret: ((list A) × (list A)) × ((list B) × (list B)). |\fst (\fst ret)| = |\fst (\snd ret)| ≝ |
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321 | match left with |
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322 | [ nil ⇒ 〈〈[ ], [ ]〉, 〈[ ], right〉〉 |
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323 | | cons hd tl ⇒ |
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324 | match right with |
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325 | [ nil ⇒ 〈〈[ ], left〉, 〈[ ], [ ]〉〉 |
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326 | | cons hd' tl' ⇒ |
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327 | let 〈cleft, cright〉 ≝ pi1 ?? (reduce_strong A B tl tl') in |
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328 | let 〈commonl, restl〉 ≝ cleft in |
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329 | let 〈commonr, restr〉 ≝ cright in |
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330 | 〈〈hd :: commonl, restl〉, 〈hd' :: commonr, restr〉〉 |
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331 | ] |
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332 | ]. |
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333 | [ 1: normalize % |
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334 | | 2: normalize % |
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335 | | 3: normalize >p3 in p2; >p4 cases (reduce_strong … tl tl1) normalize |
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336 | #X #H #EQ destruct // ] |
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337 | qed. |
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338 | |
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339 | let rec map2_opt |
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340 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C) |
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341 | (left: list A) (right: list B) on left ≝ |
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342 | match left with |
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343 | [ nil ⇒ |
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344 | match right with |
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345 | [ nil ⇒ Some ? (nil C) |
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346 | | _ ⇒ None ? |
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347 | ] |
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348 | | cons hd tl ⇒ |
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349 | match right with |
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350 | [ nil ⇒ None ? |
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351 | | cons hd' tl' ⇒ |
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352 | match map2_opt A B C f tl tl' with |
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353 | [ None ⇒ None ? |
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354 | | Some tail ⇒ Some ? (f hd hd' :: tail) |
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355 | ] |
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356 | ] |
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357 | ]. |
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358 | |
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359 | let rec map2 |
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360 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (f: A → B → C) |
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361 | (left: list A) (right: list B) (proof: | left | = | right |) on left ≝ |
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362 | match left return λx. | x | = | right | → list C with |
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363 | [ nil ⇒ |
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364 | match right return λy. | [] | = | y | → list C with |
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365 | [ nil ⇒ λnil_prf. nil C |
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366 | | _ ⇒ λcons_absrd. ? |
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367 | ] |
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368 | | cons hd tl ⇒ |
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369 | match right return λy. | hd::tl | = | y | → list C with |
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370 | [ nil ⇒ λnil_absrd. ? |
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371 | | cons hd' tl' ⇒ λcons_prf. (f hd hd') :: map2 A B C f tl tl' ? |
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372 | ] |
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373 | ] proof. |
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374 | [1: normalize in cons_absrd; |
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375 | destruct(cons_absrd) |
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376 | |2: normalize in nil_absrd; |
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377 | destruct(nil_absrd) |
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378 | |3: normalize in cons_prf; |
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379 | destruct(cons_prf) |
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380 | assumption |
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381 | ] |
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382 | qed. |
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383 | |
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384 | let rec map3 |
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385 | (A: Type[0]) (B: Type[0]) (C: Type[0]) (D: Type[0]) (f: A → B → C → D) |
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386 | (left: list A) (centre: list B) (right: list C) |
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387 | (prflc: |left| = |centre|) (prfcr: |centre| = |right|) on left ≝ |
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388 | match left return λx. |x| = |centre| → list D with |
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389 | [ nil ⇒ λnil_prf. |
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390 | match centre return λx. |x| = |right| → list D with |
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391 | [ nil ⇒ λnil_nil_prf. |
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392 | match right return λx. |nil ?| = |x| → list D with |
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393 | [ nil ⇒ λnil_nil_nil_prf. nil D |
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394 | | cons hd tl ⇒ λcons_nil_nil_absrd. ? |
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395 | ] nil_nil_prf |
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396 | | cons hd tl ⇒ λnil_cons_absrd. ? |
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397 | ] prfcr |
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398 | | cons hd tl ⇒ λcons_prf. |
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399 | match centre return λx. |x| = |right| → list D with |
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400 | [ nil ⇒ λcons_nil_absrd. ? |
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401 | | cons hd' tl' ⇒ λcons_cons_prf. |
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402 | match right return λx. |right| = |x| → |cons ? hd' tl'| = |x| → list D with |
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403 | [ nil ⇒ λrefl_prf. λcons_cons_nil_absrd. ? |
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404 | | cons hd'' tl'' ⇒ λrefl_prf. λcons_cons_cons_prf. |
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405 | (f hd hd' hd'') :: (map3 A B C D f tl tl' tl'' ? ?) |
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406 | ] (refl ? (|right|)) cons_cons_prf |
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407 | ] prfcr |
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408 | ] prflc. |
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409 | [ 1: normalize in cons_nil_nil_absrd; |
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410 | destruct(cons_nil_nil_absrd) |
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411 | | 2: generalize in match nil_cons_absrd; |
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412 | <prfcr <nil_prf #HYP |
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413 | normalize in HYP; |
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414 | destruct(HYP) |
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415 | | 3: generalize in match cons_nil_absrd; |
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416 | <prfcr <cons_prf #HYP |
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417 | normalize in HYP; |
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418 | destruct(HYP) |
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419 | | 4: normalize in cons_cons_nil_absrd; |
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420 | destruct(cons_cons_nil_absrd) |
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421 | | 5: normalize in cons_cons_cons_prf; |
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422 | destruct(cons_cons_cons_prf) |
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423 | assumption |
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424 | | 6: generalize in match cons_cons_cons_prf; |
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425 | <refl_prf <prfcr <cons_prf #HYP |
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426 | normalize in HYP; |
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427 | destruct(HYP) |
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428 | @sym_eq assumption |
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429 | ] |
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430 | qed. |
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431 | |
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432 | lemma eq_rect_Type0_r : |
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433 | ∀A: Type[0]. |
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434 | ∀a:A. |
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435 | ∀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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436 | #A #a #P #H #x #p lapply H lapply P cases p // |
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437 | qed. |
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438 | |
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439 | let rec safe_nth (A: Type[0]) (n: nat) (l: list A) (p: n < length A l) on n: A ≝ |
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440 | match n return λo. o < length A l → A with |
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441 | [ O ⇒ |
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442 | match l return λm. 0 < length A m → A with |
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443 | [ nil ⇒ λabsd1. ? |
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444 | | cons hd tl ⇒ λprf1. hd |
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445 | ] |
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446 | | S n' ⇒ |
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447 | match l return λm. S n' < length A m → A with |
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448 | [ nil ⇒ λabsd2. ? |
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449 | | cons hd tl ⇒ λprf2. safe_nth A n' tl ? |
---|
450 | ] |
---|
451 | ] ?. |
---|
452 | [ 1: |
---|
453 | @ p |
---|
454 | | 4: |
---|
455 | normalize in prf2; |
---|
456 | normalize |
---|
457 | @ le_S_S_to_le |
---|
458 | assumption |
---|
459 | | 2: |
---|
460 | normalize in absd1; |
---|
461 | cases (not_le_Sn_O O) |
---|
462 | # H |
---|
463 | elim (H absd1) |
---|
464 | | 3: |
---|
465 | normalize in absd2; |
---|
466 | cases (not_le_Sn_O (S n')) |
---|
467 | # H |
---|
468 | elim (H absd2) |
---|
469 | ] |
---|
470 | qed. |
---|
471 | |
---|
472 | let rec nub_by_internal (A: Type[0]) (f: A → A → bool) (l: list A) (n: nat) on n ≝ |
---|
473 | match n with |
---|
474 | [ O ⇒ |
---|
475 | match l with |
---|
476 | [ nil ⇒ [ ] |
---|
477 | | cons hd tl ⇒ l |
---|
478 | ] |
---|
479 | | S n ⇒ |
---|
480 | match l with |
---|
481 | [ nil ⇒ [ ] |
---|
482 | | cons hd tl ⇒ |
---|
483 | hd :: nub_by_internal A f (filter ? (λy. notb (f y hd)) tl) n |
---|
484 | ] |
---|
485 | ]. |
---|
486 | |
---|
487 | definition nub_by ≝ |
---|
488 | λA: Type[0]. |
---|
489 | λf: A → A → bool. |
---|
490 | λl: list A. |
---|
491 | nub_by_internal A f l (length ? l). |
---|
492 | |
---|
493 | let rec member (A: Type[0]) (eq: A → A → bool) (a: A) (l: list A) on l ≝ |
---|
494 | match l with |
---|
495 | [ nil ⇒ false |
---|
496 | | cons hd tl ⇒ orb (eq a hd) (member A eq a tl) |
---|
497 | ]. |
---|
498 | |
---|
499 | let rec take (A: Type[0]) (n: nat) (l: list A) on n: list A ≝ |
---|
500 | match n with |
---|
501 | [ O ⇒ [ ] |
---|
502 | | S n ⇒ |
---|
503 | match l with |
---|
504 | [ nil ⇒ [ ] |
---|
505 | | cons hd tl ⇒ hd :: take A n tl |
---|
506 | ] |
---|
507 | ]. |
---|
508 | |
---|
509 | let rec drop (A: Type[0]) (n: nat) (l: list A) on n ≝ |
---|
510 | match n with |
---|
511 | [ O ⇒ l |
---|
512 | | S n ⇒ |
---|
513 | match l with |
---|
514 | [ nil ⇒ [ ] |
---|
515 | | cons hd tl ⇒ drop A n tl |
---|
516 | ] |
---|
517 | ]. |
---|
518 | |
---|
519 | definition list_split ≝ |
---|
520 | λA: Type[0]. |
---|
521 | λn: nat. |
---|
522 | λl: list A. |
---|
523 | 〈take A n l, drop A n l〉. |
---|
524 | |
---|
525 | let rec mapi_internal (A: Type[0]) (B: Type[0]) (n: nat) (f: nat → A → B) |
---|
526 | (l: list A) on l: list B ≝ |
---|
527 | match l with |
---|
528 | [ nil ⇒ nil ? |
---|
529 | | cons hd tl ⇒ (f n hd) :: (mapi_internal A B (n + 1) f tl) |
---|
530 | ]. |
---|
531 | |
---|
532 | definition mapi ≝ |
---|
533 | λA, B: Type[0]. |
---|
534 | λf: nat → A → B. |
---|
535 | λl: list A. |
---|
536 | mapi_internal A B 0 f l. |
---|
537 | |
---|
538 | let rec zip_pottier |
---|
539 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) |
---|
540 | on left ≝ |
---|
541 | match left with |
---|
542 | [ nil ⇒ [ ] |
---|
543 | | cons hd tl ⇒ |
---|
544 | match right with |
---|
545 | [ nil ⇒ [ ] |
---|
546 | | cons hd' tl' ⇒ 〈hd, hd'〉 :: zip_pottier A B tl tl' |
---|
547 | ] |
---|
548 | ]. |
---|
549 | |
---|
550 | let rec zip_safe |
---|
551 | (A: Type[0]) (B: Type[0]) (left: list A) (right: list B) (prf: |left| = |right|) |
---|
552 | on left ≝ |
---|
553 | match left return λx. |x| = |right| → list (A × B) with |
---|
554 | [ nil ⇒ λnil_prf. |
---|
555 | match right return λx. |[ ]| = |x| → list (A × B) with |
---|
556 | [ nil ⇒ λnil_nil_prf. [ ] |
---|
557 | | cons hd tl ⇒ λnil_cons_absrd. ? |
---|
558 | ] nil_prf |
---|
559 | | cons hd tl ⇒ λcons_prf. |
---|
560 | match right return λx. |hd::tl| = |x| → list (A × B) with |
---|
561 | [ nil ⇒ λcons_nil_absrd. ? |
---|
562 | | cons hd' tl' ⇒ λcons_cons_prf. 〈hd, hd'〉 :: zip_safe A B tl tl' ? |
---|
563 | ] cons_prf |
---|
564 | ] prf. |
---|
565 | [ 1: normalize in nil_cons_absrd; |
---|
566 | destruct(nil_cons_absrd) |
---|
567 | | 2: normalize in cons_nil_absrd; |
---|
568 | destruct(cons_nil_absrd) |
---|
569 | | 3: normalize in cons_cons_prf; |
---|
570 | @injective_S |
---|
571 | assumption |
---|
572 | ] |
---|
573 | qed. |
---|
574 | |
---|
575 | let rec zip (A: Type[0]) (B: Type[0]) (l: list A) (r: list B) on l: option (list (A × B)) ≝ |
---|
576 | match l with |
---|
577 | [ nil ⇒ Some ? (nil (A × B)) |
---|
578 | | cons hd tl ⇒ |
---|
579 | match r with |
---|
580 | [ nil ⇒ None ? |
---|
581 | | cons hd' tl' ⇒ |
---|
582 | match zip ? ? tl tl' with |
---|
583 | [ None ⇒ None ? |
---|
584 | | Some tail ⇒ Some ? (〈hd, hd'〉 :: tail) |
---|
585 | ] |
---|
586 | ] |
---|
587 | ]. |
---|
588 | |
---|
589 | let rec foldl (A: Type[0]) (B: Type[0]) (f: A → B → A) (a: A) (l: list B) on l ≝ |
---|
590 | match l with |
---|
591 | [ nil ⇒ a |
---|
592 | | cons hd tl ⇒ foldl A B f (f a hd) tl |
---|
593 | ]. |
---|
594 | |
---|
595 | lemma foldl_step: |
---|
596 | ∀A:Type[0]. |
---|
597 | ∀B: Type[0]. |
---|
598 | ∀H: A → B → A. |
---|
599 | ∀acc: A. |
---|
600 | ∀pre: list B. |
---|
601 | ∀hd:B. |
---|
602 | foldl A B H acc (pre@[hd]) = (H (foldl A B H acc pre) hd). |
---|
603 | #A #B #H #acc #pre generalize in match acc; -acc; elim pre |
---|
604 | [ normalize; // |
---|
605 | | #hd #tl #IH #acc #X normalize; @IH ] |
---|
606 | qed. |
---|
607 | |
---|
608 | lemma foldl_append: |
---|
609 | ∀A:Type[0]. |
---|
610 | ∀B: Type[0]. |
---|
611 | ∀H: A → B → A. |
---|
612 | ∀acc: A. |
---|
613 | ∀suff,pre: list B. |
---|
614 | foldl A B H acc (pre@suff) = (foldl A B H (foldl A B H acc pre) suff). |
---|
615 | #A #B #H #acc #suff elim suff |
---|
616 | [ #pre >append_nil % |
---|
617 | | #hd #tl #IH #pre whd in ⊢ (???%); <(foldl_step … H ??) applyS (IH (pre@[hd])) ] |
---|
618 | qed. |
---|
619 | |
---|
620 | (* redirecting to library reverse *) |
---|
621 | definition rev ≝ reverse. |
---|
622 | |
---|
623 | lemma append_length: |
---|
624 | ∀A: Type[0]. |
---|
625 | ∀l, r: list A. |
---|
626 | |(l @ r)| = |l| + |r|. |
---|
627 | #A #L #R |
---|
628 | elim L |
---|
629 | [ % |
---|
630 | | #HD #TL #IH |
---|
631 | normalize >IH % |
---|
632 | ] |
---|
633 | qed. |
---|
634 | |
---|
635 | lemma rev_append: |
---|
636 | ∀A: Type[0]. |
---|
637 | ∀l, r: list A. |
---|
638 | rev A (l @ r) = rev A r @ rev A l. |
---|
639 | #A #L #R |
---|
640 | elim L |
---|
641 | [ normalize >append_nil % |
---|
642 | | #HD #TL normalize #IH |
---|
643 | >rev_append_def |
---|
644 | >rev_append_def |
---|
645 | >rev_append_def |
---|
646 | >append_nil |
---|
647 | normalize |
---|
648 | >IH |
---|
649 | @associative_append |
---|
650 | ] |
---|
651 | qed. |
---|
652 | |
---|
653 | lemma rev_length: |
---|
654 | ∀A: Type[0]. |
---|
655 | ∀l: list A. |
---|
656 | |rev A l| = |l|. |
---|
657 | #A #L |
---|
658 | elim L |
---|
659 | [ % |
---|
660 | | #HD #TL normalize #IH |
---|
661 | >rev_append_def |
---|
662 | >(append_length A (rev A TL) [HD]) |
---|
663 | normalize /2 by / |
---|
664 | ] |
---|
665 | qed. |
---|
666 | |
---|
667 | lemma nth_append_first: |
---|
668 | ∀A:Type[0]. |
---|
669 | ∀n:nat.∀l1,l2:list A.∀d:A. |
---|
670 | n < |l1| → nth n A (l1@l2) d = nth n A l1 d. |
---|
671 | #A #n #l1 #l2 #d |
---|
672 | generalize in match n; -n; elim l1 |
---|
673 | [ normalize #k #Hk @⊥ @(absurd … Hk) @not_le_Sn_O |
---|
674 | | #h #t #Hind #k normalize |
---|
675 | cases k -k |
---|
676 | [ #Hk normalize @refl |
---|
677 | | #k #Hk normalize @(Hind k) @le_S_S_to_le @Hk |
---|
678 | ] |
---|
679 | ] |
---|
680 | qed. |
---|
681 | |
---|
682 | lemma nth_append_second: |
---|
683 | ∀A: Type[0].∀n.∀l1,l2:list A.∀d.n ≥ length A l1 -> |
---|
684 | nth n A (l1@l2) d = nth (n - length A l1) A l2 d. |
---|
685 | #A #n #l1 #l2 #d |
---|
686 | generalize in match n; -n; elim l1 |
---|
687 | [ normalize #k #Hk <(minus_n_O) @refl |
---|
688 | | #h #t #Hind #k normalize |
---|
689 | cases k -k; |
---|
690 | [ #Hk @⊥ @(absurd (S (|t|) ≤ 0)) [ @Hk | @not_le_Sn_O ] |
---|
691 | | #k #Hk normalize @(Hind k) @le_S_S_to_le @Hk |
---|
692 | ] |
---|
693 | ] |
---|
694 | qed. |
---|
695 | |
---|
696 | |
---|
697 | let rec fold_left_i_aux (A: Type[0]) (B: Type[0]) |
---|
698 | (f: nat → A → B → A) (x: A) (i: nat) (l: list B) on l ≝ |
---|
699 | match l with |
---|
700 | [ nil ⇒ x |
---|
701 | | cons hd tl ⇒ fold_left_i_aux A B f (f i x hd) (S i) tl |
---|
702 | ]. |
---|
703 | |
---|
704 | definition fold_left_i ≝ λA,B,f,x. fold_left_i_aux A B f x O. |
---|
705 | |
---|
706 | notation "hvbox(t⌈o ↦ h⌉)" |
---|
707 | with precedence 45 |
---|
708 | for @{ match (? : $o=$h) with [ refl ⇒ $t ] }. |
---|
709 | |
---|
710 | definition function_apply ≝ |
---|
711 | λA, B: Type[0]. |
---|
712 | λf: A → B. |
---|
713 | λa: A. |
---|
714 | f a. |
---|
715 | |
---|
716 | notation "f break $ x" |
---|
717 | left associative with precedence 99 |
---|
718 | for @{ 'function_apply $f $x }. |
---|
719 | |
---|
720 | interpretation "Function application" 'function_apply f x = (function_apply ? ? f x). |
---|
721 | |
---|
722 | let rec iterate (A: Type[0]) (f: A → A) (a: A) (n: nat) on n ≝ |
---|
723 | match n with |
---|
724 | [ O ⇒ a |
---|
725 | | S o ⇒ f (iterate A f a o) |
---|
726 | ]. |
---|
727 | |
---|
728 | let rec division_aux (m: nat) (n : nat) (p: nat) ≝ |
---|
729 | match ltb n (S p) with |
---|
730 | [ true ⇒ O |
---|
731 | | false ⇒ |
---|
732 | match m with |
---|
733 | [ O ⇒ O |
---|
734 | | (S q) ⇒ S (division_aux q (n - (S p)) p) |
---|
735 | ] |
---|
736 | ]. |
---|
737 | |
---|
738 | definition division ≝ |
---|
739 | λm, n: nat. |
---|
740 | match n with |
---|
741 | [ O ⇒ S m |
---|
742 | | S o ⇒ division_aux m m o |
---|
743 | ]. |
---|
744 | |
---|
745 | notation "hvbox(n break ÷ m)" |
---|
746 | right associative with precedence 47 |
---|
747 | for @{ 'division $n $m }. |
---|
748 | |
---|
749 | interpretation "Nat division" 'division n m = (division n m). |
---|
750 | |
---|
751 | let rec modulus_aux (m: nat) (n: nat) (p: nat) ≝ |
---|
752 | match leb n p with |
---|
753 | [ true ⇒ n |
---|
754 | | false ⇒ |
---|
755 | match m with |
---|
756 | [ O ⇒ n |
---|
757 | | S o ⇒ modulus_aux o (n - (S p)) p |
---|
758 | ] |
---|
759 | ]. |
---|
760 | |
---|
761 | definition modulus ≝ |
---|
762 | λm, n: nat. |
---|
763 | match n with |
---|
764 | [ O ⇒ m |
---|
765 | | S o ⇒ modulus_aux m m o |
---|
766 | ]. |
---|
767 | |
---|
768 | notation "hvbox(n break 'mod' m)" |
---|
769 | right associative with precedence 47 |
---|
770 | for @{ 'modulus $n $m }. |
---|
771 | |
---|
772 | interpretation "Nat modulus" 'modulus m n = (modulus m n). |
---|
773 | |
---|
774 | definition divide_with_remainder ≝ |
---|
775 | λm, n: nat. |
---|
776 | mk_Prod … (m ÷ n) (modulus m n). |
---|
777 | |
---|
778 | notation "hvbox(a break ⊎ b)" |
---|
779 | left associative with precedence 55 |
---|
780 | for @{ 'disjoint_union $a $b }. |
---|
781 | interpretation "sum" 'disjoint_union A B = (Sum A B). |
---|
782 | |
---|
783 | theorem less_than_or_equal_monotone: |
---|
784 | ∀m, n: nat. |
---|
785 | m ≤ n → (S m) ≤ (S n). |
---|
786 | #m #n #H |
---|
787 | elim H |
---|
788 | /2 by le_n, le_S/ |
---|
789 | qed. |
---|
790 | |
---|
791 | theorem less_than_or_equal_b_complete: |
---|
792 | ∀m, n: nat. |
---|
793 | leb m n = false → ¬(m ≤ n). |
---|
794 | #m; |
---|
795 | elim m; |
---|
796 | normalize |
---|
797 | [ #n #H |
---|
798 | destruct |
---|
799 | | #y #H1 #z |
---|
800 | cases z |
---|
801 | normalize |
---|
802 | [ #H |
---|
803 | /2 by / |
---|
804 | | /3 by not_le_to_not_le_S_S/ |
---|
805 | ] |
---|
806 | ] |
---|
807 | qed. |
---|
808 | |
---|
809 | theorem less_than_or_equal_b_correct: |
---|
810 | ∀m, n: nat. |
---|
811 | leb m n = true → m ≤ n. |
---|
812 | #m |
---|
813 | elim m |
---|
814 | // |
---|
815 | #y #H1 #z |
---|
816 | cases z |
---|
817 | normalize |
---|
818 | [ #H |
---|
819 | destruct |
---|
820 | | #n #H lapply (H1 … H) /2 by le_S_S/ |
---|
821 | ] |
---|
822 | qed. |
---|
823 | |
---|
824 | definition less_than_or_equal_b_elim: |
---|
825 | ∀m, n: nat. |
---|
826 | ∀P: bool → Type[0]. |
---|
827 | (m ≤ n → P true) → (¬(m ≤ n) → P false) → P (leb m n). |
---|
828 | #m #n #P #H1 #H2; |
---|
829 | lapply (less_than_or_equal_b_correct m n) |
---|
830 | lapply (less_than_or_equal_b_complete m n) |
---|
831 | cases (leb m n) |
---|
832 | /3 by / |
---|
833 | qed. |
---|
834 | |
---|
835 | lemma lt_m_n_to_exists_o_plus_m_n: |
---|
836 | ∀m, n: nat. |
---|
837 | m < n → ∃o: nat. m + o = n. |
---|
838 | #m #n |
---|
839 | cases m |
---|
840 | [1: |
---|
841 | #irrelevant |
---|
842 | %{n} % |
---|
843 | |2: |
---|
844 | #m' #lt_hyp |
---|
845 | %{(n - S m')} |
---|
846 | >commutative_plus in ⊢ (??%?); |
---|
847 | <plus_minus_m_m |
---|
848 | [1: |
---|
849 | % |
---|
850 | |2: |
---|
851 | @lt_S_to_lt |
---|
852 | assumption |
---|
853 | ] |
---|
854 | ] |
---|
855 | qed. |
---|
856 | |
---|
857 | lemma lt_O_n_to_S_pred_n_n: |
---|
858 | ∀n: nat. |
---|
859 | 0 < n → S (pred n) = n. |
---|
860 | #n |
---|
861 | cases n |
---|
862 | [1: |
---|
863 | #absurd |
---|
864 | cases(lt_to_not_zero 0 0 absurd) |
---|
865 | |2: |
---|
866 | #n' #lt_hyp % |
---|
867 | ] |
---|
868 | qed. |
---|
869 | |
---|
870 | lemma exists_plus_m_n_to_exists_Sn: |
---|
871 | ∀m, n: nat. |
---|
872 | m < n → ∃p: nat. S p = n. |
---|
873 | #m #n |
---|
874 | cases m |
---|
875 | [1: |
---|
876 | #lt_hyp %{(pred n)} |
---|
877 | @(lt_O_n_to_S_pred_n_n … lt_hyp) |
---|
878 | |2: |
---|
879 | #m' #lt_hyp %{(pred n)} |
---|
880 | @(lt_O_n_to_S_pred_n_n) |
---|
881 | @(transitive_le … (S m') …) |
---|
882 | [1: |
---|
883 | // |
---|
884 | |2: |
---|
885 | @lt_S_to_lt |
---|
886 | assumption |
---|
887 | ] |
---|
888 | ] |
---|
889 | qed. |
---|
890 | |
---|
891 | lemma plus_right_monotone: |
---|
892 | ∀m, n, o: nat. |
---|
893 | m = n → m + o = n + o. |
---|
894 | #m #n #o #refl >refl % |
---|
895 | qed. |
---|
896 | |
---|
897 | lemma plus_left_monotone: |
---|
898 | ∀m, n, o: nat. |
---|
899 | m = n → o + m = o + n. |
---|
900 | #m #n #o #refl destruct % |
---|
901 | qed. |
---|
902 | |
---|
903 | lemma minus_plus_cancel: |
---|
904 | ∀m, n : nat. |
---|
905 | ∀proof: n ≤ m. |
---|
906 | (m - n) + n = m. |
---|
907 | #m #n #proof /2 by plus_minus/ |
---|
908 | qed. |
---|
909 | |
---|
910 | lemma lt_to_le_to_le: |
---|
911 | ∀n, m, p: nat. |
---|
912 | n < m → m ≤ p → n ≤ p. |
---|
913 | #n #m #p #H #H1 |
---|
914 | elim H |
---|
915 | [1: |
---|
916 | @(transitive_le n m p) /2/ |
---|
917 | |2: |
---|
918 | /2/ |
---|
919 | ] |
---|
920 | qed. |
---|
921 | |
---|
922 | lemma eqb_decidable: |
---|
923 | ∀l, r: nat. |
---|
924 | (eqb l r = true) ∨ (eqb l r = false). |
---|
925 | #l #r // |
---|
926 | qed. |
---|
927 | |
---|
928 | lemma r_Sr_and_l_r_to_Sl_r: |
---|
929 | ∀r, l: nat. |
---|
930 | (∃r': nat. r = S r' ∧ l = r') → S l = r. |
---|
931 | #r #l #exists_hyp |
---|
932 | cases exists_hyp #r' |
---|
933 | #and_hyp cases and_hyp |
---|
934 | #left_hyp #right_hyp |
---|
935 | destruct % |
---|
936 | qed. |
---|
937 | |
---|
938 | lemma eqb_Sn_to_exists_n': |
---|
939 | ∀m, n: nat. |
---|
940 | eqb (S m) n = true → ∃n': nat. n = S n'. |
---|
941 | #m #n |
---|
942 | cases n |
---|
943 | [1: |
---|
944 | normalize #absurd |
---|
945 | destruct(absurd) |
---|
946 | |2: |
---|
947 | #n' #_ %{n'} % |
---|
948 | ] |
---|
949 | qed. |
---|
950 | |
---|
951 | lemma eqb_true_to_eqb_S_S_true: |
---|
952 | ∀m, n: nat. |
---|
953 | eqb m n = true → eqb (S m) (S n) = true. |
---|
954 | #m #n normalize #assm assumption |
---|
955 | qed. |
---|
956 | |
---|
957 | lemma eqb_S_S_true_to_eqb_true: |
---|
958 | ∀m, n: nat. |
---|
959 | eqb (S m) (S n) = true → eqb m n = true. |
---|
960 | #m #n normalize #assm assumption |
---|
961 | qed. |
---|
962 | |
---|
963 | lemma eqb_true_to_refl: |
---|
964 | ∀l, r: nat. |
---|
965 | eqb l r = true → l = r. |
---|
966 | #l |
---|
967 | elim l |
---|
968 | [1: |
---|
969 | #r cases r |
---|
970 | [1: |
---|
971 | #_ % |
---|
972 | |2: |
---|
973 | #l' normalize |
---|
974 | #absurd destruct(absurd) |
---|
975 | ] |
---|
976 | |2: |
---|
977 | #l' #inductive_hypothesis #r |
---|
978 | #eqb_refl @r_Sr_and_l_r_to_Sl_r |
---|
979 | %{(pred r)} @conj |
---|
980 | [1: |
---|
981 | cases (eqb_Sn_to_exists_n' … eqb_refl) |
---|
982 | #r' #S_assm >S_assm % |
---|
983 | |2: |
---|
984 | cases (eqb_Sn_to_exists_n' … eqb_refl) |
---|
985 | #r' #refl_assm destruct normalize |
---|
986 | @inductive_hypothesis |
---|
987 | normalize in eqb_refl; assumption |
---|
988 | ] |
---|
989 | ] |
---|
990 | qed. |
---|
991 | |
---|
992 | lemma r_O_or_exists_r_r_Sr_and_l_neq_r_to_Sl_neq_r: |
---|
993 | ∀r, l: nat. |
---|
994 | r = O ∨ (∃r': nat. r = S r' ∧ l ≠ r') → S l ≠ r. |
---|
995 | #r #l #disj_hyp |
---|
996 | cases disj_hyp |
---|
997 | [1: |
---|
998 | #r_O_refl destruct @nmk |
---|
999 | #absurd destruct(absurd) |
---|
1000 | |2: |
---|
1001 | #exists_hyp cases exists_hyp #r' |
---|
1002 | #conj_hyp cases conj_hyp #left_conj #right_conj |
---|
1003 | destruct @nmk #S_S_refl_hyp |
---|
1004 | elim right_conj #hyp @hyp // |
---|
1005 | ] |
---|
1006 | qed. |
---|
1007 | |
---|
1008 | lemma neq_l_r_to_neq_Sl_Sr: |
---|
1009 | ∀l, r: nat. |
---|
1010 | l ≠ r → S l ≠ S r. |
---|
1011 | #l #r #l_neq_r_assm |
---|
1012 | @nmk #Sl_Sr_assm cases l_neq_r_assm |
---|
1013 | #assm @assm // |
---|
1014 | qed. |
---|
1015 | |
---|
1016 | lemma eqb_false_to_not_refl: |
---|
1017 | ∀l, r: nat. |
---|
1018 | eqb l r = false → l ≠ r. |
---|
1019 | #l |
---|
1020 | elim l |
---|
1021 | [1: |
---|
1022 | #r cases r |
---|
1023 | [1: |
---|
1024 | normalize #absurd destruct(absurd) |
---|
1025 | |2: |
---|
1026 | #r' #_ @nmk |
---|
1027 | #absurd destruct(absurd) |
---|
1028 | ] |
---|
1029 | |2: |
---|
1030 | #l' #inductive_hypothesis #r |
---|
1031 | cases r |
---|
1032 | [1: |
---|
1033 | #eqb_false_assm |
---|
1034 | @r_O_or_exists_r_r_Sr_and_l_neq_r_to_Sl_neq_r |
---|
1035 | @or_introl % |
---|
1036 | |2: |
---|
1037 | #r' #eqb_false_assm |
---|
1038 | @neq_l_r_to_neq_Sl_Sr |
---|
1039 | @inductive_hypothesis |
---|
1040 | assumption |
---|
1041 | ] |
---|
1042 | ] |
---|
1043 | qed. |
---|
1044 | |
---|
1045 | lemma le_to_lt_or_eq: |
---|
1046 | ∀m, n: nat. |
---|
1047 | m ≤ n → m = n ∨ m < n. |
---|
1048 | #m #n #le_hyp |
---|
1049 | cases le_hyp |
---|
1050 | [1: |
---|
1051 | @or_introl % |
---|
1052 | |2: |
---|
1053 | #m' #le_hyp' |
---|
1054 | @or_intror |
---|
1055 | normalize |
---|
1056 | @le_S_S assumption |
---|
1057 | ] |
---|
1058 | qed. |
---|
1059 | |
---|
1060 | lemma le_neq_to_lt: |
---|
1061 | ∀m, n: nat. |
---|
1062 | m ≤ n → m ≠ n → m < n. |
---|
1063 | #m #n #le_hyp #neq_hyp |
---|
1064 | cases neq_hyp |
---|
1065 | #eq_absurd_hyp |
---|
1066 | generalize in match (le_to_lt_or_eq m n le_hyp); |
---|
1067 | #disj_assm cases disj_assm |
---|
1068 | [1: |
---|
1069 | #absurd cases (eq_absurd_hyp absurd) |
---|
1070 | |2: |
---|
1071 | #assm assumption |
---|
1072 | ] |
---|
1073 | qed. |
---|
1074 | |
---|
1075 | inverter nat_jmdiscr for nat. |
---|
1076 | |
---|
1077 | lemma plus_lt_to_lt: |
---|
1078 | ∀m, n, o: nat. |
---|
1079 | m + n < o → m < o. |
---|
1080 | #m #n #o |
---|
1081 | elim n |
---|
1082 | [1: |
---|
1083 | <(plus_n_O m) in ⊢ (% → ?); |
---|
1084 | #assumption assumption |
---|
1085 | |2: |
---|
1086 | #n' #inductive_hypothesis |
---|
1087 | <(plus_n_Sm m n') in ⊢ (% → ?); |
---|
1088 | #assm @inductive_hypothesis |
---|
1089 | normalize in assm; normalize |
---|
1090 | /2 by lt_S_to_lt/ |
---|
1091 | ] |
---|
1092 | qed. |
---|
1093 | |
---|
1094 | include "arithmetics/div_and_mod.ma". |
---|
1095 | |
---|
1096 | lemma n_plus_1_n_to_False: |
---|
1097 | ∀n: nat. |
---|
1098 | n + 1 = n → False. |
---|
1099 | #n elim n |
---|
1100 | [1: |
---|
1101 | normalize #absurd destruct(absurd) |
---|
1102 | |2: |
---|
1103 | #n' #inductive_hypothesis normalize |
---|
1104 | #absurd @inductive_hypothesis /2 by injective_S/ |
---|
1105 | ] |
---|
1106 | qed. |
---|
1107 | |
---|
1108 | lemma one_two_times_n_to_False: |
---|
1109 | ∀n: nat. |
---|
1110 | 1=2*n→False. |
---|
1111 | #n cases n |
---|
1112 | [1: |
---|
1113 | normalize #absurd destruct(absurd) |
---|
1114 | |2: |
---|
1115 | #n' normalize #absurd |
---|
1116 | lapply (injective_S … absurd) -absurd #absurd |
---|
1117 | /2/ |
---|
1118 | ] |
---|
1119 | qed. |
---|
1120 | |
---|
1121 | (* |
---|
1122 | let rec odd_p |
---|
1123 | (n: nat) |
---|
1124 | on n ≝ |
---|
1125 | match n with |
---|
1126 | [ O ⇒ False |
---|
1127 | | S n' ⇒ even_p n' |
---|
1128 | ] |
---|
1129 | and even_p |
---|
1130 | (n: nat) |
---|
1131 | on n ≝ |
---|
1132 | match n with |
---|
1133 | [ O ⇒ True |
---|
1134 | | S n' ⇒ odd_p n' |
---|
1135 | ]. |
---|
1136 | |
---|
1137 | let rec n_even_p_to_n_plus_2_even_p |
---|
1138 | (n: nat) |
---|
1139 | on n: even_p n → even_p (n + 2) ≝ |
---|
1140 | match n with |
---|
1141 | [ O ⇒ ? |
---|
1142 | | S n' ⇒ ? |
---|
1143 | ] |
---|
1144 | and n_odd_p_to_n_plus_2_odd_p |
---|
1145 | (n: nat) |
---|
1146 | on n: odd_p n → odd_p (n + 2) ≝ |
---|
1147 | match n with |
---|
1148 | [ O ⇒ ? |
---|
1149 | | S n' ⇒ ? |
---|
1150 | ]. |
---|
1151 | [1,3: |
---|
1152 | normalize #assm assumption |
---|
1153 | |2: |
---|
1154 | normalize @n_odd_p_to_n_plus_2_odd_p |
---|
1155 | |4: |
---|
1156 | normalize @n_even_p_to_n_plus_2_even_p |
---|
1157 | ] |
---|
1158 | qed. |
---|
1159 | |
---|
1160 | let rec two_times_n_even_p |
---|
1161 | (n: nat) |
---|
1162 | on n: even_p (2 * n) ≝ |
---|
1163 | match n with |
---|
1164 | [ O ⇒ ? |
---|
1165 | | S n' ⇒ ? |
---|
1166 | ] |
---|
1167 | and two_times_n_plus_one_odd_p |
---|
1168 | (n: nat) |
---|
1169 | on n: odd_p ((2 * n) + 1) ≝ |
---|
1170 | match n with |
---|
1171 | [ O ⇒ ? |
---|
1172 | | S n' ⇒ ? |
---|
1173 | ]. |
---|
1174 | [1,3: |
---|
1175 | normalize @I |
---|
1176 | |2: |
---|
1177 | normalize |
---|
1178 | >plus_n_Sm |
---|
1179 | <(associative_plus n' n' 1) |
---|
1180 | >(plus_n_O (n' + n')) |
---|
1181 | cut(n' + n' + 0 + 1 = 2 * n' + 1) |
---|
1182 | [1: |
---|
1183 | // |
---|
1184 | |2: |
---|
1185 | #refl_assm >refl_assm |
---|
1186 | @two_times_n_plus_one_odd_p |
---|
1187 | ] |
---|
1188 | |4: |
---|
1189 | normalize |
---|
1190 | >plus_n_Sm |
---|
1191 | cut(n' + (n' + 1) + 1 = (2 * n') + 2) |
---|
1192 | [1: |
---|
1193 | normalize /2/ |
---|
1194 | |2: |
---|
1195 | #refl_assm >refl_assm |
---|
1196 | @n_even_p_to_n_plus_2_even_p |
---|
1197 | @two_times_n_even_p |
---|
1198 | ] |
---|
1199 | ] |
---|
1200 | qed. |
---|
1201 | |
---|
1202 | include alias "basics/logic.ma". |
---|
1203 | |
---|
1204 | let rec even_p_to_not_odd_p |
---|
1205 | (n: nat) |
---|
1206 | on n: even_p n → ¬ odd_p n ≝ |
---|
1207 | match n with |
---|
1208 | [ O ⇒ ? |
---|
1209 | | S n' ⇒ ? |
---|
1210 | ] |
---|
1211 | and odd_p_to_not_even_p |
---|
1212 | (n: nat) |
---|
1213 | on n: odd_p n → ¬ even_p n ≝ |
---|
1214 | match n with |
---|
1215 | [ O ⇒ ? |
---|
1216 | | S n' ⇒ ? |
---|
1217 | ]. |
---|
1218 | [1: |
---|
1219 | normalize #_ |
---|
1220 | @nmk #assm assumption |
---|
1221 | |3: |
---|
1222 | normalize #absurd |
---|
1223 | cases absurd |
---|
1224 | |2: |
---|
1225 | normalize |
---|
1226 | @odd_p_to_not_even_p |
---|
1227 | |4: |
---|
1228 | normalize |
---|
1229 | @even_p_to_not_odd_p |
---|
1230 | ] |
---|
1231 | qed. |
---|
1232 | |
---|
1233 | lemma even_p_odd_p_cases: |
---|
1234 | ∀n: nat. |
---|
1235 | even_p n ∨ odd_p n. |
---|
1236 | #n elim n |
---|
1237 | [1: |
---|
1238 | normalize @or_introl @I |
---|
1239 | |2: |
---|
1240 | #n' #inductive_hypothesis |
---|
1241 | normalize |
---|
1242 | cases inductive_hypothesis |
---|
1243 | #assm |
---|
1244 | try (@or_introl assumption) |
---|
1245 | try (@or_intror assumption) |
---|
1246 | ] |
---|
1247 | qed. |
---|
1248 | *) |
---|
1249 | |
---|
1250 | lemma two_times_n_plus_one_refl_two_times_n_to_False: |
---|
1251 | ∀m, n: nat. |
---|
1252 | 2 * m + 1 = 2 * n → False. |
---|
1253 | @nat_elim2 |
---|
1254 | [ * [2: #n ] normalize [ <plus_n_Sm ] |
---|
1255 | | #n normalize |
---|
1256 | | #n #m #IH >commutative_times >commutative_times in ⊢ (???%→?); normalize |
---|
1257 | ] #EQ destruct @IH |
---|
1258 | >commutative_times >commutative_times in ⊢ (???%); assumption |
---|
1259 | qed. |
---|
1260 | |
---|
1261 | lemma not_Some_neq_None_to_False: |
---|
1262 | ∀a: Type[0]. |
---|
1263 | ∀e: a. |
---|
1264 | ¬ (Some a e ≠ None a) → False. |
---|
1265 | #a #e #absurd cases absurd -absurd |
---|
1266 | #absurd @absurd @nmk -absurd |
---|
1267 | #absurd destruct(absurd) |
---|
1268 | qed. |
---|
1269 | |
---|
1270 | lemma not_None_to_Some: |
---|
1271 | ∀A: Type[0]. |
---|
1272 | ∀o: option A. |
---|
1273 | o ≠ None A → ∃v: A. o = Some A v. |
---|
1274 | #A #o cases o |
---|
1275 | [1: |
---|
1276 | #absurd cases absurd #absurd' cases (absurd' (refl …)) |
---|
1277 | |2: |
---|
1278 | #v' #ignore /2/ |
---|
1279 | ] |
---|
1280 | qed. |
---|
1281 | |
---|
1282 | lemma inclusive_disjunction_true: |
---|
1283 | ∀b, c: bool. |
---|
1284 | (orb b c) = true → b = true ∨ c = true. |
---|
1285 | # b |
---|
1286 | # c |
---|
1287 | elim b |
---|
1288 | [ normalize |
---|
1289 | # H |
---|
1290 | @ or_introl |
---|
1291 | % |
---|
1292 | | normalize |
---|
1293 | /3 by trans_eq, orb_true_l/ |
---|
1294 | ] |
---|
1295 | qed. |
---|
1296 | |
---|
1297 | lemma conjunction_true: |
---|
1298 | ∀b, c: bool. |
---|
1299 | andb b c = true → b = true ∧ c = true. |
---|
1300 | # b |
---|
1301 | # c |
---|
1302 | elim b |
---|
1303 | normalize |
---|
1304 | [ /2 by conj/ |
---|
1305 | | # K |
---|
1306 | destruct |
---|
1307 | ] |
---|
1308 | qed. |
---|
1309 | |
---|
1310 | lemma eq_true_false: false=true → False. |
---|
1311 | # K |
---|
1312 | destruct |
---|
1313 | qed. |
---|
1314 | |
---|
1315 | lemma inclusive_disjunction_b_true: ∀b. orb b true = true. |
---|
1316 | # b |
---|
1317 | cases b |
---|
1318 | % |
---|
1319 | qed. |
---|
1320 | |
---|
1321 | (* XXX: to be moved into logic/basics.ma *) |
---|
1322 | lemma and_intro_right: |
---|
1323 | ∀a, b: Prop. |
---|
1324 | a → (a → b) → a ∧ b. |
---|
1325 | #a #b /3/ |
---|
1326 | qed. |
---|
1327 | |
---|
1328 | definition bool_to_Prop ≝ |
---|
1329 | λb. match b with [ true ⇒ True | false ⇒ False ]. |
---|
1330 | |
---|
1331 | coercion bool_to_Prop: ∀b:bool. Prop ≝ bool_to_Prop on _b:bool to Type[0]. |
---|
1332 | |
---|
1333 | lemma bool_as_Prop_to_eq : ∀b : bool. b → b = true. |
---|
1334 | **% |
---|
1335 | qed. |
---|
1336 | |
---|
1337 | (* with this you can use prf : b with b : bool with rewriting |
---|
1338 | >prf rewrites b as true *) |
---|
1339 | coercion bool_to_Prop_to_eq : ∀b : bool.∀prf : b.b = true |
---|
1340 | ≝ bool_as_Prop_to_eq on _prf : bool_to_Prop ? to (? = true). |
---|
1341 | |
---|
1342 | lemma andb_Prop : ∀b,d : bool.b → d → b∧d. |
---|
1343 | #b #d #btrue #dtrue >btrue >dtrue % |
---|
1344 | qed. |
---|
1345 | |
---|
1346 | lemma andb_Prop_true : ∀b,d : bool. (b∧d) → And (bool_to_Prop b) (bool_to_Prop d). |
---|
1347 | #b #d #bdtrue elim (andb_true … bdtrue) #btrue #dtrue >btrue >dtrue % % |
---|
1348 | qed. |
---|
1349 | |
---|
1350 | lemma orb_Prop_l : ∀b,d : bool.b → b∨d. |
---|
1351 | #b #d #btrue >btrue % |
---|
1352 | qed. |
---|
1353 | |
---|
1354 | lemma orb_Prop_r : ∀b,d : bool.d → b∨d. |
---|
1355 | #b #d #dtrue >dtrue elim b % |
---|
1356 | qed. |
---|
1357 | |
---|
1358 | lemma orb_Prop_true : ∀b,d : bool. (b∨d) → Or (bool_to_Prop b) (bool_to_Prop d). |
---|
1359 | #b #d #bdtrue elim (orb_true_l … bdtrue) #xtrue >xtrue [%1 | %2] % |
---|
1360 | qed. |
---|
1361 | |
---|
1362 | lemma notb_Prop : ∀b : bool. Not (bool_to_Prop b) → notb b. |
---|
1363 | * * #H [@H % | %] |
---|
1364 | qed. |
---|
1365 | |
---|
1366 | lemma Prop_notb : ∀b:bool. notb b → Not (bool_to_Prop b). |
---|
1367 | * /2/ |
---|
1368 | qed. |
---|
1369 | |
---|
1370 | lemma not_orb : ∀b,c:bool. |
---|
1371 | ¬ (b∨c) → |
---|
1372 | (bool_to_Prop (¬b))∧(bool_to_Prop (¬c)). |
---|
1373 | * * normalize /2/ |
---|
1374 | qed. |
---|
1375 | |
---|
1376 | lemma eq_false_to_notb: ∀b. b = false → ¬ b. |
---|
1377 | *; /2 by eq_true_false, I/ |
---|
1378 | qed. |
---|
1379 | |
---|
1380 | lemma not_b_to_eq_false : ∀b : bool. Not (bool_to_Prop b) → b = false. |
---|
1381 | ** #H [elim (H ?) % | %] |
---|
1382 | qed. |
---|
1383 | |
---|
1384 | (* with this you can use prf : ¬b with b : bool with rewriting |
---|
1385 | >prf rewrites b as false *) |
---|
1386 | coercion not_bool_to_Prop_to_eq : ∀b : bool.∀prf : Not (bool_to_Prop b).b = false |
---|
1387 | ≝ not_b_to_eq_false on _prf : Not (bool_to_Prop ?) to (? = false). |
---|
1388 | |
---|
1389 | |
---|
1390 | lemma true_or_false_Prop : ∀b : bool.Or (bool_to_Prop b) (¬(bool_to_Prop b)). |
---|
1391 | * [%1 % | %2 % *] |
---|
1392 | qed. |
---|
1393 | |
---|
1394 | lemma eq_true_to_b : ∀b. b = true → b. |
---|
1395 | #b #btrue >btrue % |
---|
1396 | qed. |
---|
1397 | |
---|
1398 | definition if_then_else_safe : ∀A : Type[0].∀b : bool.(b → A) → (¬(bool_to_Prop b) → A) → A ≝ |
---|
1399 | λA,b,f,g. |
---|
1400 | match b return λx.match x with [true ⇒ bool_to_Prop b | false ⇒ ¬bool_to_Prop b] → A with |
---|
1401 | [ true ⇒ f |
---|
1402 | | false ⇒ g |
---|
1403 | ] ?. elim b % * |
---|
1404 | qed. |
---|
1405 | |
---|
1406 | notation > "'If' b 'then' 'with' ident prf1 'do' f 'else' 'with' ident prf2 'do' g" with precedence 46 for |
---|
1407 | @{'if_then_else_safe $b (λ${ident prf1}.$f) (λ${ident prf2}.$g)}. |
---|
1408 | notation > "'If' b 'then' 'with' ident prf1 : ty1 'do' f 'else' 'with' ident prf2 : ty2 'do' g" with precedence 46 for |
---|
1409 | @{'if_then_else_safe $b (λ${ident prf1}:$ty1.$f) (λ${ident prf2}:$ty2.$g)}. |
---|
1410 | notation > "'If' b 'then' f 'else' 'with' ident prf2 'do' g" with precedence 46 for |
---|
1411 | @{'if_then_else_safe $b (λ_.$f) (λ${ident prf2}.$g)}. |
---|
1412 | notation > "'If' b 'then' f 'else' 'with' ident prf2 : ty2 'do' g" with precedence 46 for |
---|
1413 | @{'if_then_else_safe $b (λ_.$f) (λ${ident prf2}:$ty2.$g)}. |
---|
1414 | notation > "'If' b 'then' 'with' ident prf1 'do' f 'else' g" with precedence 46 for |
---|
1415 | @{'if_then_else_safe $b (λ${ident prf1}.$f) (λ_.$g)}. |
---|
1416 | notation > "'If' b 'then' 'with' ident prf1 : ty1 'do' f 'else' g" with precedence 46 for |
---|
1417 | @{'if_then_else_safe $b (λ${ident prf1}:$ty1.$f) (λ_.$g)}. |
---|
1418 | |
---|
1419 | notation < "hvbox('If' \nbsp b \nbsp 'then' \nbsp break 'with' \nbsp ident prf1 : ty1 \nbsp 'do' \nbsp break f \nbsp break 'else' \nbsp break 'with' \nbsp ident prf2 : ty2 \nbsp 'do' \nbsp break g)" with precedence 46 for |
---|
1420 | @{'if_then_else_safe $b (λ${ident prf1}:$ty1.$f) (λ${ident prf2}:$ty2.$g)}. |
---|
1421 | notation < "hvbox('If' \nbsp b \nbsp 'then' \nbsp break f \nbsp break 'else' \nbsp break 'with' \nbsp ident prf2 : ty2 \nbsp 'do' \nbsp break g)" with precedence 46 for |
---|
1422 | @{'if_then_else_safe $b (λ_:$ty1.$f) (λ${ident prf2}:$ty2.$g)}. |
---|
1423 | notation < "hvbox('If' \nbsp b \nbsp 'then' \nbsp break 'with' \nbsp ident prf1 : ty1 \nbsp 'do' \nbsp break f \nbsp break 'else' \nbsp break g)" with precedence 46 for |
---|
1424 | @{'if_then_else_safe $b (λ${ident prf1}:$ty1.$f) (λ_:$ty2.$g)}. |
---|
1425 | |
---|
1426 | interpretation "dependent if then else" 'if_then_else_safe b f g = (if_then_else_safe ? b f g). |
---|
1427 | |
---|
1428 | lemma If_elim : ∀A.∀P : A → Prop.∀b : bool.∀f : b → A.∀g : Not (bool_to_Prop b) → A. |
---|
1429 | (∀prf.P (f prf)) → (∀prf.P (g prf)) → P (If b then with prf do f prf else with prf do g prf). |
---|
1430 | #A #P * #f #g #H1 #H2 normalize // qed. |
---|
1431 | |
---|
1432 | (* Also extracts an equality proof (useful when not using Russell). *) |
---|
1433 | notation > "hvbox('let' 〈ident x,ident y〉 'as' ident E 'return' ty ≝ t 'in' s)" |
---|
1434 | with precedence 10 |
---|
1435 | for @{ match $t return λx.∀${ident E}: x = $t. $ty with [ mk_Prod ${ident x} ${ident y} ⇒ |
---|
1436 | λ${ident E}.$s ] (refl ? $t) }. |
---|
1437 | |
---|
1438 | notation > "hvbox('deplet' 〈ident x,ident y〉 'as' ident E ≝ t 'in' s)" |
---|
1439 | with precedence 10 |
---|
1440 | for @{ match $t return λx.∀${ident E}: x = $t. Σz: ?. ? with [ mk_Prod ${ident x} ${ident y} ⇒ |
---|
1441 | λ${ident E}.$s ] (refl ? $t) }. |
---|
1442 | |
---|
1443 | notation > "hvbox('deplet' 〈ident x,ident y,ident z〉 'as' ident E ≝ t 'in' s)" |
---|
1444 | with precedence 10 |
---|
1445 | for @{ match $t return λx.∀${fresh w}:x = $t. Σq: ?. ? with [ mk_Prod ${fresh xy} ${ident z} ⇒ |
---|
1446 | match ${fresh xy} return λx.∀${ident E}:? = $t. Σu: ?. ? with [ mk_Prod ${ident x} ${ident y} ⇒ |
---|
1447 | λ${ident E}.$s ] ] (refl ? $t) }. |
---|
1448 | |
---|
1449 | notation > "hvbox('let' 〈ident x,ident y,ident z〉 'as' ident E 'return' ty ≝ t 'in' s)" |
---|
1450 | with precedence 10 |
---|
1451 | for @{ match $t return λx.∀${fresh w}:x = $t. Σq: ?. ? with [ mk_Prod ${fresh xy} ${ident z} ⇒ |
---|
1452 | match ${fresh xy} return λx.∀${ident E}:? = $t. $ty with [ mk_Prod ${ident x} ${ident y} ⇒ |
---|
1453 | λ${ident E}.$s ] ] (refl ? $t) }. |
---|
1454 | |
---|
1455 | lemma length_append: |
---|
1456 | ∀A.∀l1,l2:list A. |
---|
1457 | |l1 @ l2| = |l1| + |l2|. |
---|
1458 | #A #l1 elim l1 |
---|
1459 | [ // |
---|
1460 | | #hd #tl #IH #l2 normalize <IH //] |
---|
1461 | qed. |
---|
1462 | |
---|
1463 | lemma nth_cons: |
---|
1464 | ∀n,A,h,t,y. |
---|
1465 | nth (S n) A (h::t) y = nth n A t y. |
---|
1466 | #n #A #h #t #y /2 by refl/ |
---|
1467 | qed. |
---|
1468 | |
---|
1469 | lemma option_destruct_Some: ∀A,a,b. Some A a = Some A b → a=b. |
---|
1470 | #A #a #b #EQ destruct // |
---|
1471 | qed. |
---|
1472 | |
---|
1473 | (*CSC: just a synonim, get rid of it!*) |
---|
1474 | lemma Some_eq: |
---|
1475 | ∀T:Type[0].∀x,y:T. Some T x = Some T y → x = y ≝ option_destruct_Some. |
---|
1476 | |
---|
1477 | lemma pi1_eq: ∀A:Type[0].∀P:A->Prop.∀s1:Σa1:A.P a1.∀s2:Σa2:A.P a2. |
---|
1478 | s1 = s2 → (pi1 ?? s1) = (pi1 ?? s2). |
---|
1479 | #A #P #s1 #s2 / by / |
---|
1480 | qed. |
---|
1481 | |
---|
1482 | lemma not_neq_None_to_eq : ∀A.∀a : option A.¬a≠None ? → a = None ?. |
---|
1483 | #A * [2: #a] * // #ABS elim (ABS ?) % #ABS' destruct(ABS') |
---|
1484 | qed. |
---|
1485 | |
---|
1486 | coercion not_neq_None : ∀A.∀a : option A.∀prf : ¬a≠None ?.a = None ? ≝ |
---|
1487 | not_neq_None_to_eq on _prf : ¬?≠None ? to ? = None ?. |
---|
1488 | |
---|
1489 | lemma None_Some_elim: ∀P:Prop. ∀A,a. None A = Some A a → P. |
---|
1490 | #P #A #a #abs destruct |
---|
1491 | qed. |
---|
1492 | |
---|
1493 | discriminator Prod. |
---|
1494 | |
---|
1495 | lemma pair_replace: |
---|
1496 | ∀A,B,T:Type[0].∀P:T → Prop. ∀t. ∀a,b.∀c,c': A × B. c'=c → c ≃ 〈a,b〉 → |
---|
1497 | P (t a b) → P (let 〈a',b'〉 ≝ c' in t a' b'). |
---|
1498 | #A #B #T #P #t #a #b * #x #y * #x' #y' #H1 #H2 destruct // |
---|
1499 | qed. |
---|
1500 | |
---|
1501 | lemma jmpair_as_replace: |
---|
1502 | ∀A,B,T:Type[0].∀P:T → Prop. ∀a:A. ∀b:B.∀c,c': A × B. ∀t: ∀a,b. c' ≃ 〈a, b〉 → T. ∀EQc: c'= c. |
---|
1503 | ∀EQ:c ≃ 〈a,b〉. |
---|
1504 | P (t a b ?) → P ((let 〈a',b'〉 (*as H*) ≝ c' in λH. t a' b' H) (refl_jmeq ? c')). |
---|
1505 | [2: |
---|
1506 | >EQc @EQ |
---|
1507 | |1: |
---|
1508 | #A #B #T #P #a #b |
---|
1509 | #c #c' #t #EQc >EQc in t; -c' normalize #f #EQ |
---|
1510 | letin eta ≝ (? : ∀ab:A × B.∀P:A × B → Type[0]. P ab → P 〈\fst ab,\snd ab〉) [2: * // ] |
---|
1511 | change with (eta 〈a,b〉 (λab.c ≃ ab) EQ) in match EQ; |
---|
1512 | @(jmeq_elim ?? (λab.λH.P (f (\fst ab) (\snd ab) ?) → ?) ?? EQ) normalize -eta |
---|
1513 | -EQ cases c in f; normalize // |
---|
1514 | ] |
---|
1515 | qed. |
---|
1516 | |
---|
1517 | lemma if_then_else_replace: |
---|
1518 | ∀T: Type[0]. |
---|
1519 | ∀P: T → Prop. |
---|
1520 | ∀t1, t2: T. |
---|
1521 | ∀c, c', c'': bool. |
---|
1522 | c' = c → c ≃ c'' → P (if c'' then t1 else t2) → P (if c' then t1 else t2). |
---|
1523 | #T #P #t1 #t2 #c #c' #c'' #c_refl #c_refl' destruct #assm |
---|
1524 | assumption |
---|
1525 | qed. |
---|
1526 | |
---|
1527 | lemma refl_to_jmrefl: |
---|
1528 | ∀a: Type[0]. |
---|
1529 | ∀l, r: a. |
---|
1530 | l = r → l ≃ r. |
---|
1531 | #a #l #r #refl destruct % |
---|
1532 | qed. |
---|
1533 | |
---|
1534 | lemma prod_eq_left: |
---|
1535 | ∀A: Type[0]. |
---|
1536 | ∀p, q, r: A. |
---|
1537 | p = q → 〈p, r〉 = 〈q, r〉. |
---|
1538 | #A #p #q #r #hyp |
---|
1539 | destruct % |
---|
1540 | qed. |
---|
1541 | |
---|
1542 | lemma prod_eq_right: |
---|
1543 | ∀A: Type[0]. |
---|
1544 | ∀p, q, r: A. |
---|
1545 | p = q → 〈r, p〉 = 〈r, q〉. |
---|
1546 | #A #p #q #r #hyp |
---|
1547 | destruct % |
---|
1548 | qed. |
---|
1549 | |
---|
1550 | lemma destruct_bug_fix_1: |
---|
1551 | ∀n: nat. |
---|
1552 | S n = 0 → False. |
---|
1553 | #n #absurd destruct(absurd) |
---|
1554 | qed. |
---|
1555 | |
---|
1556 | lemma destruct_bug_fix_2: |
---|
1557 | ∀m, n: nat. |
---|
1558 | S m = S n → m = n. |
---|
1559 | #m #n #refl destruct % |
---|
1560 | qed. |
---|
1561 | |
---|
1562 | lemma eq_rect_Type1_r: |
---|
1563 | ∀A: Type[1]. |
---|
1564 | ∀a: A. |
---|
1565 | ∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p. |
---|
1566 | #A #a #P #H #x #p |
---|
1567 | generalize in match H; |
---|
1568 | generalize in match P; |
---|
1569 | cases p // |
---|
1570 | qed. |
---|
1571 | |
---|
1572 | lemma Some_Some_elim: |
---|
1573 | ∀T:Type[0].∀x,y:T.∀P:Type[2]. (x=y → P) → Some T x = Some T y → P. |
---|
1574 | #T #x #y #P #H #K @H @option_destruct_Some // |
---|
1575 | qed. |
---|
1576 | |
---|
1577 | lemma pair_destruct_right: |
---|
1578 | ∀A: Type[0]. |
---|
1579 | ∀B: Type[0]. |
---|
1580 | ∀a, c: A. |
---|
1581 | ∀b, d: B. |
---|
1582 | 〈a, b〉 = 〈c, d〉 → b = d. |
---|
1583 | #A #B #a #b #c #d // |
---|
1584 | qed. |
---|
1585 | |
---|
1586 | lemma pose: ∀A:Type[0].∀B:A → Type[0].∀a:A. (∀a':A. a'=a → B a') → B a. |
---|
1587 | /2/ |
---|
1588 | qed. |
---|
1589 | |
---|
1590 | definition eq_sum: |
---|
1591 | ∀A, B. (A → A → bool) → (B → B → bool) → (A ⊎ B) → (A ⊎ B) → bool ≝ |
---|
1592 | λlt, rt, leq, req, left, right. |
---|
1593 | match left with |
---|
1594 | [ inl l ⇒ |
---|
1595 | match right with |
---|
1596 | [ inl l' ⇒ leq l l' |
---|
1597 | | _ ⇒ false |
---|
1598 | ] |
---|
1599 | | inr r ⇒ |
---|
1600 | match right with |
---|
1601 | [ inr r' ⇒ req r r' |
---|
1602 | | _ ⇒ false |
---|
1603 | ] |
---|
1604 | ]. |
---|
1605 | |
---|
1606 | lemma eq_sum_refl: |
---|
1607 | ∀A, B: Type[0]. |
---|
1608 | ∀leq, req. |
---|
1609 | ∀s. |
---|
1610 | ∀leq_refl: (∀t. leq t t = true). |
---|
1611 | ∀req_refl: (∀u. req u u = true). |
---|
1612 | eq_sum A B leq req s s = true. |
---|
1613 | #A #B #leq #req #s #leq_refl #req_refl |
---|
1614 | cases s assumption |
---|
1615 | qed. |
---|
1616 | |
---|
1617 | definition eq_prod: ∀A, B. (A → A → bool) → (B → B → bool) → (A × B) → (A × B) → bool ≝ |
---|
1618 | λlt, rt, leq, req, left, right. |
---|
1619 | let 〈l, r〉 ≝ left in |
---|
1620 | let 〈l', r'〉 ≝ right in |
---|
1621 | leq l l' ∧ req r r'. |
---|
1622 | |
---|
1623 | lemma eq_prod_refl: |
---|
1624 | ∀A, B: Type[0]. |
---|
1625 | ∀leq, req. |
---|
1626 | ∀s. |
---|
1627 | ∀leq_refl: (∀t. leq t t = true). |
---|
1628 | ∀req_refl: (∀u. req u u = true). |
---|
1629 | eq_prod A B leq req s s = true. |
---|
1630 | #A #B #leq #req #s #leq_refl #req_refl |
---|
1631 | cases s |
---|
1632 | whd in ⊢ (? → ? → ??%?); |
---|
1633 | #l #r |
---|
1634 | >leq_refl @req_refl |
---|
1635 | qed. |
---|
1636 | |
---|
1637 | lemma geb_to_leb: ∀a,b:ℕ.geb a b = leb b a. |
---|
1638 | #a #b / by refl/ |
---|
1639 | qed. |
---|
1640 | |
---|
1641 | lemma nth_last: ∀A,a,l. |
---|
1642 | nth (|l|) A l a = a. |
---|
1643 | #A #a #l elim l |
---|
1644 | [ / by / |
---|
1645 | | #h #t #Hind whd in match (nth ????); whd in match (tail ??); @Hind |
---|
1646 | ] |
---|
1647 | qed. |
---|
1648 | |
---|
1649 | |
---|
1650 | lemma minus_zero_to_le: ∀n,m:ℕ.n - m = 0 → n ≤ m. |
---|
1651 | #n |
---|
1652 | elim n |
---|
1653 | [ #m #_ @le_O_n |
---|
1654 | | #n' #Hind #m cases m |
---|
1655 | [ #H -n whd in match (minus ??) in H; >H @le_n |
---|
1656 | | #m' -m #H whd in match (minus ??) in H; @le_S_S @Hind @H |
---|
1657 | ] |
---|
1658 | ] |
---|
1659 | qed. |
---|
1660 | |
---|
1661 | lemma plus_zero_zero: ∀n,m:ℕ.n + m = 0 → m = 0. |
---|
1662 | #n #m #Hn @sym_eq @le_n_O_to_eq <Hn >commutative_plus @le_plus_n_r |
---|
1663 | qed. |
---|
1664 | |
---|
1665 | (* this can probably be done more simply somewhere *) |
---|
1666 | lemma not_true_is_false: |
---|
1667 | ∀b:bool.b ≠ true → b = false. |
---|
1668 | #b cases b |
---|
1669 | [ #H @⊥ @(absurd ?? H) @refl |
---|
1670 | | #H @refl |
---|
1671 | ] |
---|
1672 | qed. |
---|
1673 | |
---|
1674 | (* this is in the stdlib, but commented out, why? *) |
---|
1675 | theorem plus_Sn_m1: ∀n,m:nat. S m + n = m + S n. |
---|
1676 | #n (elim n) normalize /2 by S_pred/ qed. |
---|
1677 | |
---|
1678 | (* these lemmas seem superfluous, but not sure how to replace them *) |
---|
1679 | lemma double_plus_equal: ∀a,b:ℕ.a + a = b + b → a = b. |
---|
1680 | #a elim a |
---|
1681 | [ normalize #b // |
---|
1682 | | -a #a #Hind #b cases b [ /2 by le_n_O_to_eq/ | -b #b normalize |
---|
1683 | <plus_n_Sm <plus_n_Sm #H |
---|
1684 | >(Hind b (injective_S ?? (injective_S ?? H))) // ] |
---|
1685 | ] |
---|
1686 | qed. |
---|
1687 | |
---|
1688 | lemma sth_not_s: ∀x.x ≠ S x. |
---|
1689 | #x cases x |
---|
1690 | [ // | #y // ] |
---|
1691 | qed. |
---|
1692 | |
---|
1693 | lemma le_to_eq_plus: ∀n,z. |
---|
1694 | n ≤ z → ∃k.z = n + k. |
---|
1695 | #n #z elim z |
---|
1696 | [ #H cases (le_to_or_lt_eq … H) |
---|
1697 | [ #H2 @⊥ @(absurd … H2) @not_le_Sn_O |
---|
1698 | | #H2 @(ex_intro … 0) >H2 // |
---|
1699 | ] |
---|
1700 | | #z' #Hind #H cases (le_to_or_lt_eq … H) |
---|
1701 | [ #H' elim (Hind (le_S_S_to_le … H')) #k' #H2 @(ex_intro … (S k')) |
---|
1702 | >H2 >plus_n_Sm // |
---|
1703 | | #H' @(ex_intro … 0) >H' // |
---|
1704 | ] |
---|
1705 | ] |
---|
1706 | qed. |
---|
1707 | |
---|
1708 | lemma nth_safe_nth: |
---|
1709 | ∀A:Type[0].∀l:list A.∀i.∀proof:i < |l|.∀x.nth_safe A i l proof = nth i A l x. |
---|
1710 | #A #l elim l |
---|
1711 | [ #i #prf @⊥ @(absurd ? prf) @le_to_not_lt @le_O_n |
---|
1712 | | -l #h #t #Hind #i elim i |
---|
1713 | [ #prf #x normalize @refl |
---|
1714 | | -i #i #Hind2 #prf #x whd in match (nth ????); whd in match (tail ??); |
---|
1715 | whd in match (nth_safe ????); @Hind |
---|
1716 | ] |
---|
1717 | ] |
---|
1718 | qed. |
---|
1719 | |
---|
1720 | lemma flatten_singleton: |
---|
1721 | ∀A,a. flatten A [a] = a. |
---|
1722 | #A #a normalize // |
---|
1723 | qed. |
---|
1724 | |
---|
1725 | lemma length_flatten_cons: |
---|
1726 | ∀A,hd,tl. |
---|
1727 | |flatten A (hd::tl)| = |hd| + |flatten A tl|. |
---|
1728 | #A #hd #tl normalize // |
---|
1729 | qed. |
---|
1730 | |
---|
1731 | lemma tech_transitive_lt_3: |
---|
1732 | ∀n1,n2,n3,b. |
---|
1733 | n1 < b → n2 < b → n3 < b → |
---|
1734 | n1 + n2 + n3 < 3 * b. |
---|
1735 | #n1 #n2 #n3 #b #H1 #H2 #H3 |
---|
1736 | @(transitive_lt … (b + n2 + n3)) [ @monotonic_lt_plus_l @monotonic_lt_plus_l // ] |
---|
1737 | @(transitive_lt … (b + b + n3)) [ @monotonic_lt_plus_l @monotonic_lt_plus_r // ] |
---|
1738 | @(lt_to_le_to_lt … (b + b + b)) [ @monotonic_lt_plus_r // ] // |
---|
1739 | qed. |
---|
1740 | |
---|
1741 | lemma m_lt_1_to_m_refl_0: |
---|
1742 | ∀m: nat. |
---|
1743 | m < 1 → m = 0. |
---|
1744 | #m cases m try (#irrelevant %) |
---|
1745 | #m' whd in ⊢ (% → ?); #relevant |
---|
1746 | lapply (le_S_S_to_le … relevant) |
---|
1747 | change with (? < ? → ?) -relevant #relevant |
---|
1748 | cases (lt_to_not_zero … relevant) |
---|
1749 | qed. |
---|
1750 | |
---|
1751 | lemma ltb_false_to_not_lt: ∀p,q:ℕ.ltb p q = false → p ≮ q. |
---|
1752 | #p #q #H @leb_false_to_not_le @H |
---|
1753 | qed. |
---|
1754 | |
---|
1755 | lemma ltb_true_to_lt: ∀p,q:ℕ.ltb p q = true → p < q. |
---|
1756 | #p #q #H @leb_true_to_le @H |
---|
1757 | qed. |
---|
1758 | |
---|
1759 | lemma plus_equals_zero: ∀x,y:ℕ.x + y = x → y = 0. |
---|
1760 | #x #y cases y |
---|
1761 | [ #_ @refl |
---|
1762 | | -y #y elim x |
---|
1763 | [ <plus_O_n / by / |
---|
1764 | | -x #x #Hind #H2 @Hind normalize in H2; @injective_S @H2 |
---|
1765 | ] |
---|
1766 | ] |
---|
1767 | qed. |
---|