[475] | 1 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 2 | (* Vector.ma: Fixed length polymorphic vectors, and routine operations on *) |
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| 3 | (* them. *) |
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| 4 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 5 | |
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[1599] | 6 | include "basics/lists/list.ma". |
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[475] | 7 | include "basics/bool.ma". |
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[697] | 8 | include "basics/types.ma". |
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[475] | 9 | |
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[698] | 10 | include "ASM/Util.ma". |
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[475] | 11 | |
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| 12 | include "arithmetics/nat.ma". |
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| 13 | |
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[744] | 14 | include "utilities/extranat.ma". |
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[475] | 15 | |
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| 16 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 17 | (* The datatype. *) |
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| 18 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 19 | |
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| 20 | inductive Vector (A: Type[0]): nat → Type[0] ≝ |
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| 21 | VEmpty: Vector A O |
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| 22 | | VCons: ∀n: nat. A → Vector A n → Vector A (S n). |
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| 23 | |
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| 24 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 25 | (* Syntax. *) |
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| 26 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 27 | |
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| 28 | notation "hvbox(hd break ::: tl)" |
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[1908] | 29 | right associative with precedence 57 |
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[475] | 30 | for @{ 'vcons $hd $tl }. |
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| 31 | |
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| 32 | notation "[[ list0 x sep ; ]]" |
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| 33 | non associative with precedence 90 |
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| 34 | for ${fold right @'vnil rec acc @{'vcons $x $acc}}. |
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| 35 | |
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| 36 | interpretation "Vector vnil" 'vnil = (VEmpty ?). |
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| 37 | interpretation "Vector vcons" 'vcons hd tl = (VCons ? ? hd tl). |
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| 38 | |
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| 39 | notation "hvbox(l break !!! break n)" |
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| 40 | non associative with precedence 90 |
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| 41 | for @{ 'get_index_v $l $n }. |
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| 42 | |
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[2032] | 43 | lemma dependent_rewrite_vectors: |
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| 44 | ∀A:Type[0]. |
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| 45 | ∀n, m: nat. |
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| 46 | ∀v1: Vector A n. |
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| 47 | ∀v2: Vector A m. |
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| 48 | ∀P: ∀m. Vector A m → Prop. |
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| 49 | n = m → v1 ≃ v2 → P n v1 → P m v2. |
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| 50 | #A #n #m #v1 #v2 #P #eq #jmeq |
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| 51 | destruct #assm assumption |
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| 52 | qed. |
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| 53 | |
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| 54 | lemma jmeq_cons_vector_monotone: |
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| 55 | ∀A: Type[0]. |
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| 56 | ∀m, n: nat. |
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| 57 | ∀v: Vector A m. |
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| 58 | ∀q: Vector A n. |
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| 59 | ∀prf: m = n. |
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| 60 | ∀hd: A. |
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| 61 | v ≃ q → hd:::v ≃ hd:::q. |
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| 62 | #A #m #n #v #q #prf #hd #E |
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| 63 | @(dependent_rewrite_vectors A … E) |
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| 64 | try assumption % |
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| 65 | qed. |
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| 66 | |
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[475] | 67 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 68 | (* Lookup. *) |
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| 69 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 70 | |
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| 71 | let rec get_index_v (A: Type[0]) (n: nat) |
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| 72 | (v: Vector A n) (m: nat) (lt: m < n) on m: A ≝ |
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| 73 | (match m with |
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| 74 | [ O ⇒ |
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| 75 | match v return λx.λ_. O < x → A with |
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| 76 | [ VEmpty ⇒ λabsd1: O < O. ? |
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| 77 | | VCons p hd tl ⇒ λprf1: O < S p. hd |
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| 78 | ] |
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| 79 | | S o ⇒ |
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| 80 | (match v return λx.λ_. S o < x → A with |
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| 81 | [ VEmpty ⇒ λprf: S o < O. ? |
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| 82 | | VCons p hd tl ⇒ λprf: S o < S p. get_index_v A p tl o ? |
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| 83 | ]) |
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| 84 | ]) lt. |
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| 85 | [ cases (not_le_Sn_O O) |
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[1516] | 86 | normalize in absd1; |
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[475] | 87 | # H |
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| 88 | cases (H absd1) |
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| 89 | | cases (not_le_Sn_O (S o)) |
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[1516] | 90 | normalize in prf; |
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[475] | 91 | # H |
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| 92 | cases (H prf) |
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| 93 | | normalize |
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[1516] | 94 | normalize in prf; |
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[475] | 95 | @ le_S_S_to_le |
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| 96 | assumption |
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| 97 | ] |
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| 98 | qed. |
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| 99 | |
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| 100 | definition get_index' ≝ |
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| 101 | λA: Type[0]. |
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| 102 | λn, m: nat. |
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| 103 | λb: Vector A (S (n + m)). |
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| 104 | get_index_v A (S (n + m)) b n ?. |
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| 105 | normalize |
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[1063] | 106 | @le_S_S |
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| 107 | cases m // |
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[475] | 108 | qed. |
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| 109 | |
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| 110 | let rec get_index_weak_v (A: Type[0]) (n: nat) |
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| 111 | (v: Vector A n) (m: nat) on m ≝ |
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| 112 | match m with |
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| 113 | [ O ⇒ |
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| 114 | match v with |
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| 115 | [ VEmpty ⇒ None A |
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| 116 | | VCons p hd tl ⇒ Some A hd |
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| 117 | ] |
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| 118 | | S o ⇒ |
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| 119 | match v with |
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| 120 | [ VEmpty ⇒ None A |
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| 121 | | VCons p hd tl ⇒ get_index_weak_v A p tl o |
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| 122 | ] |
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| 123 | ]. |
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| 124 | |
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| 125 | interpretation "Vector get_index" 'get_index_v v n = (get_index_v ? ? v n). |
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| 126 | |
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| 127 | let rec set_index (A: Type[0]) (n: nat) (v: Vector A n) (m: nat) (a: A) (lt: m < n) on m: Vector A n ≝ |
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| 128 | (match m with |
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| 129 | [ O ⇒ |
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| 130 | match v return λx.λ_. O < x → Vector A x with |
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| 131 | [ VEmpty ⇒ λabsd1: O < O. [[ ]] |
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| 132 | | VCons p hd tl ⇒ λprf1: O < S p. (a ::: tl) |
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| 133 | ] |
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| 134 | | S o ⇒ |
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| 135 | (match v return λx.λ_. S o < x → Vector A x with |
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| 136 | [ VEmpty ⇒ λprf: S o < O. [[ ]] |
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| 137 | | VCons p hd tl ⇒ λprf: S o < S p. hd ::: (set_index A p tl o a ?) |
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| 138 | ]) |
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| 139 | ]) lt. |
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| 140 | normalize in prf ⊢ %; |
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| 141 | /2/; |
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| 142 | qed. |
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[873] | 143 | |
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[475] | 144 | let rec set_index_weak (A: Type[0]) (n: nat) |
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| 145 | (v: Vector A n) (m: nat) (a: A) on m ≝ |
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| 146 | match m with |
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| 147 | [ O ⇒ |
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| 148 | match v with |
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| 149 | [ VEmpty ⇒ None (Vector A n) |
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| 150 | | VCons o hd tl ⇒ Some (Vector A n) (? (VCons A o a tl)) |
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| 151 | ] |
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| 152 | | S o ⇒ |
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| 153 | match v with |
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| 154 | [ VEmpty ⇒ None (Vector A n) |
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| 155 | | VCons p hd tl ⇒ |
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| 156 | let settail ≝ set_index_weak A p tl o a in |
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| 157 | match settail with |
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| 158 | [ None ⇒ None (Vector A n) |
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| 159 | | Some j ⇒ Some (Vector A n) (? (VCons A p hd j)) |
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| 160 | ] |
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| 161 | ] |
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| 162 | ]. |
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| 163 | //. |
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| 164 | qed. |
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| 165 | |
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| 166 | let rec drop (A: Type[0]) (n: nat) |
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| 167 | (v: Vector A n) (m: nat) on m ≝ |
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| 168 | match m with |
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| 169 | [ O ⇒ Some (Vector A n) v |
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| 170 | | S o ⇒ |
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| 171 | match v with |
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| 172 | [ VEmpty ⇒ None (Vector A n) |
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| 173 | | VCons p hd tl ⇒ ? (drop A p tl o) |
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| 174 | ] |
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| 175 | ]. |
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| 176 | //. |
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| 177 | qed. |
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| 178 | |
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[744] | 179 | definition head' : ∀A:Type[0]. ∀n:nat. Vector A (S n) → A ≝ |
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| 180 | λA,n,v. match v return λx.λ_. match x with [ O ⇒ True | _ ⇒ A ] with |
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| 181 | [ VEmpty ⇒ I | VCons _ hd _ ⇒ hd ]. |
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| 182 | |
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| 183 | definition tail : ∀A:Type[0]. ∀n:nat. Vector A (S n) → Vector A n ≝ |
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| 184 | λA,n,v. match v return λx.λ_. match x with [ O ⇒ True | S m ⇒ Vector A m ] with |
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| 185 | [ VEmpty ⇒ I | VCons m hd tl ⇒ tl ]. |
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| 186 | |
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[2032] | 187 | let rec vsplit' (A: Type[0]) (m, n: nat) on m: Vector A (plus m n) → (Vector A m) × (Vector A n) ≝ |
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[475] | 188 | match m return λm. Vector A (plus m n) → (Vector A m) × (Vector A n) with |
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| 189 | [ O ⇒ λv. 〈[[ ]], v〉 |
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[2032] | 190 | | S m' ⇒ λv. let 〈l,r〉 ≝ vsplit' A m' n (tail ?? v) in 〈head' ?? v:::l, r〉 |
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[744] | 191 | ]. |
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| 192 | (* Prevent undesirable unfolding. *) |
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[2032] | 193 | let rec vsplit (A: Type[0]) (m, n: nat) (v:Vector A (plus m n)) on v : (Vector A m) × (Vector A n) ≝ |
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| 194 | vsplit' A m n v. |
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[475] | 195 | |
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| 196 | definition head: ∀A: Type[0]. ∀n: nat. Vector A (S n) → A × (Vector A n) ≝ |
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| 197 | λA: Type[0]. |
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| 198 | λn: nat. |
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| 199 | λv: Vector A (S n). |
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| 200 | match v return λl. λ_: Vector A l. l = S n → A × (Vector A n) with |
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| 201 | [ VEmpty ⇒ λK. ⊥ |
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| 202 | | VCons o he tl ⇒ λK. 〈he, (tl⌈Vector A o ↦ Vector A n⌉)〉 |
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| 203 | ] (? : S ? = S ?). |
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| 204 | // |
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[1069] | 205 | destruct |
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[475] | 206 | qed. |
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| 207 | |
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| 208 | definition from_singl: ∀A:Type[0]. Vector A (S O) → A ≝ |
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| 209 | λA: Type[0]. |
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| 210 | λv: Vector A (S 0). |
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| 211 | fst … (head … v). |
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| 212 | |
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| 213 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 214 | (* Folds and builds. *) |
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| 215 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 216 | |
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| 217 | let rec fold_right (A: Type[0]) (B: Type[0]) (n: nat) |
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| 218 | (f: A → B → B) (x: B) (v: Vector A n) on v ≝ |
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| 219 | match v with |
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| 220 | [ VEmpty ⇒ x |
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| 221 | | VCons n hd tl ⇒ f hd (fold_right A B n f x tl) |
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| 222 | ]. |
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| 223 | |
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[697] | 224 | let rec fold_right_i (A: Type[0]) (B: nat → Type[0]) (n: nat) |
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| 225 | (f: ∀n. A → B n → B (S n)) (x: B 0) (v: Vector A n) on v ≝ |
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| 226 | match v with |
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| 227 | [ VEmpty ⇒ x |
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| 228 | | VCons n hd tl ⇒ f ? hd (fold_right_i A B n f x tl) |
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| 229 | ]. |
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| 230 | |
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[475] | 231 | let rec fold_right2_i (A: Type[0]) (B: Type[0]) (C: nat → Type[0]) |
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| 232 | (f: ∀N. A → B → C N → C (S N)) (c: C O) (n: nat) |
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| 233 | (v: Vector A n) (q: Vector B n) on v : C n ≝ |
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| 234 | (match v return λx.λ_. x = n → C n with |
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| 235 | [ VEmpty ⇒ |
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| 236 | match q return λx.λ_. O = x → C x with |
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| 237 | [ VEmpty ⇒ λprf: O = O. c |
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| 238 | | VCons o hd tl ⇒ λabsd. ⊥ |
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| 239 | ] |
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| 240 | | VCons o hd tl ⇒ |
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| 241 | match q return λx.λ_. S o = x → C x with |
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| 242 | [ VEmpty ⇒ λabsd: S o = O. ⊥ |
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| 243 | | VCons p hd' tl' ⇒ λprf: S o = S p. |
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| 244 | (f ? hd hd' (fold_right2_i A B C f c ? tl (tl'⌈Vector B p ↦ Vector B o⌉)))⌈C (S o) ↦ C (S p)⌉ |
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| 245 | ] |
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| 246 | ]) (refl ? n). |
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| 247 | [1,2: |
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| 248 | destruct |
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| 249 | |3,4: |
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| 250 | lapply (injective_S … prf) |
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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 | let rec fold_left (A: Type[0]) (B: Type[0]) (n: nat) |
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| 256 | (f: A → B → A) (x: A) (v: Vector B n) on v ≝ |
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| 257 | match v with |
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| 258 | [ VEmpty ⇒ x |
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[697] | 259 | | VCons n hd tl ⇒ fold_left A B n f (f x hd) tl |
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[475] | 260 | ]. |
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| 261 | |
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| 262 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 263 | (* Maps and zips. *) |
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| 264 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 265 | |
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| 266 | let rec map (A: Type[0]) (B: Type[0]) (n: nat) |
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| 267 | (f: A → B) (v: Vector A n) on v ≝ |
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| 268 | match v with |
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| 269 | [ VEmpty ⇒ [[ ]] |
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| 270 | | VCons n hd tl ⇒ (f hd) ::: (map A B n f tl) |
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| 271 | ]. |
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| 272 | |
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| 273 | let rec zip_with (A: Type[0]) (B: Type[0]) (C: Type[0]) (n: nat) |
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| 274 | (f: A → B → C) (v: Vector A n) (q: Vector B n) on v ≝ |
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| 275 | (match v return (λx.λr. x = n → Vector C x) with |
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| 276 | [ VEmpty ⇒ λ_. [[ ]] |
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| 277 | | VCons n hd tl ⇒ |
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| 278 | match q return (λy.λr. S n = y → Vector C (S n)) with |
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| 279 | [ VEmpty ⇒ ? |
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| 280 | | VCons m hd' tl' ⇒ |
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| 281 | λe: S n = S m. |
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| 282 | (f hd hd') ::: (zip_with A B C n f tl ?) |
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| 283 | ] |
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| 284 | ]) |
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| 285 | (refl ? n). |
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| 286 | [ #e |
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| 287 | destruct(e); |
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| 288 | | lapply (injective_S … e) |
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| 289 | # H |
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| 290 | > H |
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| 291 | @ tl' |
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| 292 | ] |
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| 293 | qed. |
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| 294 | |
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| 295 | definition zip ≝ |
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| 296 | λA, B: Type[0]. |
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| 297 | λn: nat. |
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| 298 | λv: Vector A n. |
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| 299 | λq: Vector B n. |
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[1598] | 300 | zip_with A B (A × B) n (mk_Prod A B) v q. |
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[475] | 301 | |
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| 302 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 303 | (* Building vectors from scratch *) |
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| 304 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 305 | |
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| 306 | let rec replicate (A: Type[0]) (n: nat) (h: A) on n ≝ |
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| 307 | match n return λn. Vector A n with |
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| 308 | [ O ⇒ [[ ]] |
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| 309 | | S m ⇒ h ::: (replicate A m h) |
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| 310 | ]. |
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| 311 | |
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| 312 | (* DPM: fixme. Weird matita bug in base case. *) |
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| 313 | let rec append (A: Type[0]) (n: nat) (m: nat) |
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| 314 | (v: Vector A n) (q: Vector A m) on v ≝ |
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| 315 | match v return (λn.λv. Vector A (n + m)) with |
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| 316 | [ VEmpty ⇒ (? q) |
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| 317 | | VCons o hd tl ⇒ hd ::: (append A o m tl q) |
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| 318 | ]. |
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| 319 | # H |
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| 320 | assumption |
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| 321 | qed. |
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| 322 | |
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| 323 | notation "hvbox(l break @@ r)" |
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| 324 | right associative with precedence 47 |
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| 325 | for @{ 'vappend $l $r }. |
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| 326 | |
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| 327 | interpretation "Vector append" 'vappend v1 v2 = (append ??? v1 v2). |
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[998] | 328 | |
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[2032] | 329 | lemma vector_cons_append: |
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[998] | 330 | ∀A: Type[0]. |
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[2032] | 331 | ∀n: nat. |
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| 332 | ∀e: A. |
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| 333 | ∀v: Vector A n. |
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| 334 | e ::: v = [[ e ]] @@ v. |
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| 335 | #A #n #e #v |
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| 336 | cases v try % |
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| 337 | #n' #hd #tl % |
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| 338 | qed. |
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| 339 | |
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| 340 | lemma vector_cons_append2: |
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| 341 | ∀A: Type[0]. |
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| 342 | ∀n, m: nat. |
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| 343 | ∀v: Vector A n. |
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| 344 | ∀q: Vector A m. |
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| 345 | ∀hd: A. |
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| 346 | hd:::(v@@q) = (hd:::v)@@q. |
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| 347 | #A #n #m #v #q |
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| 348 | elim v try (#hd %) |
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| 349 | #n' #hd' #tl' #ih #hd' |
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| 350 | <ih % |
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| 351 | qed. |
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| 352 | |
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| 353 | lemma vector_associative_append: |
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| 354 | ∀A: Type[0]. |
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| 355 | ∀n, m, o: nat. |
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| 356 | ∀v: Vector A n. |
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| 357 | ∀q: Vector A m. |
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| 358 | ∀r: Vector A o. |
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| 359 | (v @@ q) @@ r ≃ v @@ (q @@ r). |
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| 360 | #A #n #m #o #v #q #r |
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| 361 | elim v try % |
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| 362 | #n' #hd #tl #ih |
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| 363 | <(vector_cons_append2 A … hd) |
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| 364 | @jmeq_cons_vector_monotone |
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| 365 | try assumption |
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| 366 | @associative_plus |
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| 367 | qed. |
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| 368 | |
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| 369 | axiom vsplit_elim': |
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| 370 | ∀A: Type[0]. |
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[998] | 371 | ∀B: Type[1]. |
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| 372 | ∀l, m, v. |
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| 373 | ∀T: Vector A l → Vector A m → B. |
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| 374 | ∀P: B → Prop. |
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| 375 | (∀lft, rgt. v = lft @@ rgt → P (T lft rgt)) → |
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[2032] | 376 | P (let 〈lft, rgt〉 ≝ vsplit A l m v in T lft rgt). |
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[998] | 377 | |
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[2032] | 378 | axiom vsplit_elim'': |
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[998] | 379 | ∀A: Type[0]. |
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| 380 | ∀B,B': Type[1]. |
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| 381 | ∀l, m, v. |
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| 382 | ∀T: Vector A l → Vector A m → B. |
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| 383 | ∀T': Vector A l → Vector A m → B'. |
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| 384 | ∀P: B → B' → Prop. |
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| 385 | (∀lft, rgt. v = lft @@ rgt → P (T lft rgt) (T' lft rgt)) → |
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[2032] | 386 | P (let 〈lft, rgt〉 ≝ vsplit A l m v in T lft rgt) |
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| 387 | (let 〈lft, rgt〉 ≝ vsplit A l m v in T' lft rgt). |
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[475] | 388 | |
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| 389 | let rec scan_left (A: Type[0]) (B: Type[0]) (n: nat) |
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| 390 | (f: A → B → A) (a: A) (v: Vector B n) on v ≝ |
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| 391 | a ::: |
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| 392 | (match v with |
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| 393 | [ VEmpty ⇒ VEmpty A |
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| 394 | | VCons o hd tl ⇒ scan_left A B o f (f a hd) tl |
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| 395 | ]). |
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| 396 | |
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| 397 | let rec scan_right (A: Type[0]) (B: Type[0]) (n: nat) |
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| 398 | (f: A → B → A) (b: B) (v: Vector A n) on v ≝ |
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| 399 | match v with |
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| 400 | [ VEmpty ⇒ ? |
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| 401 | | VCons o hd tl ⇒ f hd b :: (scan_right A B o f b tl) |
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| 402 | ]. |
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| 403 | // |
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| 404 | qed. |
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| 405 | |
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| 406 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 407 | (* Other manipulations. *) |
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| 408 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 409 | |
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[697] | 410 | (* At some points matita will attempt to reduce reverse with a known vector, |
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| 411 | which reduces the equality proof for the cast. Normalising this proof needs |
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[744] | 412 | to be fast enough to keep matita usable, so use plus_n_Sm_fast. *) |
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[697] | 413 | |
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| 414 | let rec revapp (A: Type[0]) (n: nat) (m:nat) |
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| 415 | (v: Vector A n) (acc: Vector A m) on v : Vector A (n + m) ≝ |
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| 416 | match v return λn'.λ_. Vector A (n' + m) with |
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| 417 | [ VEmpty ⇒ acc |
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| 418 | | VCons o hd tl ⇒ (revapp ??? tl (hd:::acc))⌈Vector A (o+S m) ↦ Vector A (S o + m)⌉ |
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[475] | 419 | ]. |
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[889] | 420 | < plus_n_Sm_fast @refl qed. |
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[475] | 421 | |
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[697] | 422 | let rec reverse (A: Type[0]) (n: nat) (v: Vector A n) on v : Vector A n ≝ |
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| 423 | (revapp A n 0 v [[ ]])⌈Vector A (n+0) ↦ Vector A n⌉. |
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| 424 | < plus_n_O @refl qed. |
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| 425 | |
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[744] | 426 | let rec pad_vector (A:Type[0]) (a:A) (n,m:nat) (v:Vector A m) on n : Vector A (n+m) ≝ |
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| 427 | match n return λn.Vector A (n+m) with |
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| 428 | [ O ⇒ v |
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| 429 | | S n' ⇒ a:::(pad_vector A a n' m v) |
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| 430 | ]. |
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| 431 | |
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[475] | 432 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 433 | (* Conversions to and from lists. *) |
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| 434 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 435 | |
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| 436 | let rec list_of_vector (A: Type[0]) (n: nat) |
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| 437 | (v: Vector A n) on v ≝ |
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| 438 | match v return λn.λv. list A with |
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| 439 | [ VEmpty ⇒ [] |
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| 440 | | VCons o hd tl ⇒ hd :: (list_of_vector A o tl) |
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| 441 | ]. |
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| 442 | |
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| 443 | let rec vector_of_list (A: Type[0]) (l: list A) on l ≝ |
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| 444 | match l return λl. Vector A (length A l) with |
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| 445 | [ nil ⇒ ? |
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| 446 | | cons hd tl ⇒ hd ::: (vector_of_list A tl) |
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| 447 | ]. |
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| 448 | normalize |
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| 449 | @ VEmpty |
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| 450 | qed. |
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| 451 | |
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| 452 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 453 | (* Rotates and shifts. *) |
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| 454 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 455 | |
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| 456 | let rec rotate_left (A: Type[0]) (n: nat) |
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| 457 | (m: nat) (v: Vector A n) on m: Vector A n ≝ |
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| 458 | match m with |
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| 459 | [ O ⇒ v |
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| 460 | | S o ⇒ |
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| 461 | match v with |
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| 462 | [ VEmpty ⇒ [[ ]] |
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| 463 | | VCons p hd tl ⇒ |
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| 464 | rotate_left A (S p) o ((append A p ? tl [[hd]])⌈Vector A (p + S O) ↦ Vector A (S p)⌉) |
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| 465 | ] |
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| 466 | ]. |
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[1063] | 467 | /2/ |
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[475] | 468 | qed. |
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| 469 | |
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| 470 | definition rotate_right ≝ |
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| 471 | λA: Type[0]. |
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| 472 | λn, m: nat. |
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| 473 | λv: Vector A n. |
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| 474 | reverse A n (rotate_left A n m (reverse A n v)). |
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| 475 | |
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| 476 | definition shift_left_1 ≝ |
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| 477 | λA: Type[0]. |
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| 478 | λn: nat. |
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| 479 | λv: Vector A (S n). |
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| 480 | λa: A. |
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| 481 | match v return λy.λ_. y = S n → Vector A y with |
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| 482 | [ VEmpty ⇒ λH.⊥ |
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| 483 | | VCons o hd tl ⇒ λH.reverse … (a::: reverse … tl) |
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| 484 | ] (refl ? (S n)). |
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| 485 | destruct. |
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| 486 | qed. |
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| 487 | |
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[744] | 488 | |
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| 489 | (* XXX this is horrible - but useful to ensure that we can normalise in the proof assistant. *) |
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| 490 | definition switch_bv_plus : ∀A:Type[0]. ∀n,m. Vector A (n+m) → Vector A (m+n) ≝ |
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[749] | 491 | λA,n,m. match commutative_plus_faster n m return λx.λ_.Vector A (n+m) → Vector A x with [ refl ⇒ λi.i ]. |
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[744] | 492 | |
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[475] | 493 | definition shift_right_1 ≝ |
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| 494 | λA: Type[0]. |
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| 495 | λn: nat. |
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| 496 | λv: Vector A (S n). |
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| 497 | λa: A. |
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[2032] | 498 | let 〈v',dropped〉 ≝ vsplit ? n 1 (switch_bv_plus ? 1 n v) in a:::v'. |
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[744] | 499 | (* reverse … (shift_left_1 … (reverse … v) a).*) |
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| 500 | |
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| 501 | definition shift_left : ∀A:Type[0]. ∀n,m:nat. Vector A n → A → Vector A n ≝ |
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[475] | 502 | λA: Type[0]. |
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| 503 | λn, m: nat. |
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[744] | 504 | match nat_compare n m return λx,y.λ_. Vector A x → A → Vector A x with |
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| 505 | [ nat_lt _ _ ⇒ λv,a. replicate … a |
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| 506 | | nat_eq _ ⇒ λv,a. replicate … a |
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[2032] | 507 | | nat_gt d m ⇒ λv,a. let 〈v0,v'〉 ≝ vsplit … v in switch_bv_plus … (v' @@ (replicate … a)) |
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[744] | 508 | ]. |
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| 509 | |
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| 510 | (* iterate … (λx. shift_left_1 … x a) v m.*) |
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[475] | 511 | |
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| 512 | definition shift_right ≝ |
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| 513 | λA: Type[0]. |
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| 514 | λn, m: nat. |
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| 515 | λv: Vector A (S n). |
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| 516 | λa: A. |
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| 517 | iterate … (λx. shift_right_1 … x a) v m. |
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| 518 | |
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| 519 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 520 | (* Decidable equality. *) |
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| 521 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 522 | |
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| 523 | let rec eq_v (A: Type[0]) (n: nat) (f: A → A → bool) (b: Vector A n) (c: Vector A n) on b : bool ≝ |
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[726] | 524 | (match b return λx.λ_. Vector A x → bool with |
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| 525 | [ VEmpty ⇒ λc. |
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| 526 | match c return λx.λ_. match x return λ_.Type[0] with [ O ⇒ bool | _ ⇒ True ] with |
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| 527 | [ VEmpty ⇒ true |
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| 528 | | VCons p hd tl ⇒ I |
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| 529 | ] |
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| 530 | | VCons m hd tl ⇒ λc. andb (f hd (head' A m c)) (eq_v A m f tl (tail A m c)) |
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| 531 | ] |
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| 532 | ) c. |
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[475] | 533 | |
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[726] | 534 | lemma vector_inv_n: ∀A,n. ∀P:Vector A n → Type[0]. ∀v:Vector A n. |
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| 535 | match n return λn'. (Vector A n' → Type[0]) → Vector A n' → Type[0] with |
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[697] | 536 | [ O ⇒ λP.λv.P [[ ]] → P v |
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| 537 | | S m ⇒ λP.λv.(∀h,t. P (VCons A m h t)) → P v |
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| 538 | ] P v. |
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[1516] | 539 | #A #n #P #v lapply P cases v normalize // |
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[873] | 540 | qed. |
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[697] | 541 | |
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[726] | 542 | lemma eq_v_elim: ∀P:bool → Type[0]. ∀A,f. |
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| 543 | (∀Q:bool → Type[0]. ∀a,b. (a = b → Q true) → (a ≠ b → Q false) → Q (f a b)) → |
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[697] | 544 | ∀n,x,y. |
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| 545 | (x = y → P true) → |
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| 546 | (x ≠ y → P false) → |
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| 547 | P (eq_v A n f x y). |
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| 548 | #P #A #f #f_elim #n #x elim x |
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| 549 | [ #y @(vector_inv_n … y) |
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| 550 | normalize /2/ |
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| 551 | | #m #h #t #IH #y @(vector_inv_n … y) |
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[1516] | 552 | #h' #t' #Ht #Hf whd in ⊢ (?%); |
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[697] | 553 | @(f_elim ? h h') #Eh |
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| 554 | [ @IH [ #Et @Ht >Eh >Et @refl | #NEt @Hf % #E' destruct (E') elim NEt /2/ ] |
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| 555 | | @Hf % #E' destruct (E') elim Eh /2/ |
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| 556 | ] |
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| 557 | ] qed. |
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| 558 | |
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[961] | 559 | lemma eq_v_true : ∀A,f. (∀a. f a a = true) → ∀n,v. eq_v A n f v v = true. |
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| 560 | #A #f #f_true #n #v elim v |
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| 561 | [ // |
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[1516] | 562 | | #m #h #t #IH whd in ⊢ (??%%); >f_true >IH @refl |
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[961] | 563 | ] qed. |
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| 564 | |
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| 565 | lemma vector_neq_tail : ∀A,n,h. ∀t,t':Vector A n. h:::t≠h:::t' → t ≠ t'. |
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| 566 | #A #n #h #t #t' * #NE % #E @NE >E @refl |
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| 567 | qed. |
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| 568 | |
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| 569 | lemma eq_v_false : ∀A,f. (∀a,a'. f a a' = true → a = a') → ∀n,v,v'. v≠v' → eq_v A n f v v' = false. |
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| 570 | #A #f #f_true #n elim n |
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| 571 | [ #v #v' @(vector_inv_n ??? v) @(vector_inv_n ??? v') * #H @False_ind @H @refl |
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| 572 | | #m #IH #v #v' @(vector_inv_n ??? v) #h #t @(vector_inv_n ??? v') #h' #t' |
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[1516] | 573 | #NE normalize lapply (f_true h h') cases (f h h') // #E @IH >E in NE; /2/ |
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[961] | 574 | ] qed. |
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| 575 | |
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[475] | 576 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 577 | (* Subvectors. *) |
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| 578 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 579 | |
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| 580 | definition mem ≝ |
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| 581 | λA: Type[0]. |
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| 582 | λeq_a : A → A → bool. |
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| 583 | λn: nat. |
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| 584 | λl: Vector A n. |
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| 585 | λx: A. |
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| 586 | fold_right … (λy,v. (eq_a x y) ∨ v) false l. |
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| 587 | |
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[2032] | 588 | lemma mem_append: ∀A,eq_a. (∀x. eq_a x x = true) → ∀n,m.∀v: Vector A n. ∀w: Vector A m. ∀x. mem A eq_a … (v@@x:::w) x. |
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| 589 | #A #eq_a #refl #n #m #v elim v |
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| 590 | [ #w #x whd whd in match (mem ?????); >refl // |
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| 591 | | /2/ |
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| 592 | ] |
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| 593 | qed. |
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| 594 | |
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[1646] | 595 | let rec subvector_with |
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| 596 | (a: Type[0]) (n: nat) (m: nat) (eq: a → a → bool) (sub: Vector a n) (sup: Vector a m) |
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| 597 | on sub: bool ≝ |
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| 598 | match sub with |
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| 599 | [ VEmpty ⇒ true |
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| 600 | | VCons n' hd tl ⇒ |
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| 601 | if mem … eq … sup hd then |
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| 602 | subvector_with … eq tl sup |
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| 603 | else |
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| 604 | false |
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| 605 | ]. |
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[2032] | 606 | |
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| 607 | lemma subvector_with_refl0: |
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| 608 | ∀A:Type[0]. ∀n. ∀eq: A → A → bool. (∀a. eq a a = true) → ∀v: Vector A n. |
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| 609 | ∀m. ∀w: Vector A m. subvector_with A … eq v (w@@v). |
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| 610 | #A #n #eq #refl #v elim v |
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| 611 | [ // |
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| 612 | | #m #hd #tl #IH #m #w whd in match (subvector_with ??????); >mem_append // |
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| 613 | change with (bool_to_Prop (subvector_with ??????)) lapply (IH … (w@@[[hd]])) |
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| 614 | lapply (vector_associative_append ???? w [[hd]] tl) #EQ |
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| 615 | @(dependent_rewrite_vectors … EQ) // |
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| 616 | ] |
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| 617 | qed. |
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| 618 | |
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| 619 | lemma subvector_with_refl: |
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| 620 | ∀A:Type[0]. ∀n. ∀eq: A → A → bool. (∀a. eq a a = true) → ∀v: Vector A n. |
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| 621 | subvector_with A … eq v v. |
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| 622 | #A #n #eq #refl #v @(subvector_with_refl0 … v … [[]]) // |
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| 623 | qed. |
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| 624 | |
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[475] | 625 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 626 | (* Lemmas. *) |
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| 627 | (* -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-= *) |
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| 628 | |
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| 629 | lemma map_fusion: |
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| 630 | ∀A, B, C: Type[0]. |
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| 631 | ∀n: nat. |
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| 632 | ∀v: Vector A n. |
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| 633 | ∀f: A → B. |
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| 634 | ∀g: B → C. |
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| 635 | map B C n g (map A B n f v) = map A C n (λx. g (f x)) v. |
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| 636 | #A #B #C #n #v #f #g |
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| 637 | elim v |
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| 638 | [ normalize |
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| 639 | % |
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| 640 | | #N #H #V #H2 |
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| 641 | normalize |
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| 642 | > H2 |
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| 643 | % |
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| 644 | ] |
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[2032] | 645 | qed. |
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