[697] | 1 | include "basics/types.ma". |
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[475] | 2 | |
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[761] | 3 | include "utilities/option.ma". |
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[698] | 4 | include "ASM/BitVector.ma". |
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[475] | 5 | |
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| 6 | inductive BitVectorTrie (A: Type[0]): nat → Type[0] ≝ |
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| 7 | Leaf: A → BitVectorTrie A O |
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| 8 | | Node: ∀n: nat. BitVectorTrie A n → BitVectorTrie A n → BitVectorTrie A (S n) |
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| 9 | | Stub: ∀n: nat. BitVectorTrie A n. |
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| 10 | |
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[1006] | 11 | let rec fold (A, B: Type[0]) (n: nat) (f: BitVector n → A → B → B) |
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| 12 | (t: BitVectorTrie A n) (b: B) on t: B ≝ |
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| 13 | (match t return λx.λ_.x = n → B with |
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| 14 | [ Leaf l ⇒ λ_.f (zero ?) l b |
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| 15 | | Node h l r ⇒ λK. |
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| 16 | fold A B h (λx.f ((VCons ? h false x)⌈(S h) ↦ n⌉)) l |
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| 17 | (fold A B h (λx.f ((VCons ? h true x)⌈(S h) ↦ n⌉)) r b) |
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| 18 | | Stub _ ⇒ λ_.b |
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| 19 | ]) (refl ? n). |
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| 20 | @K |
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| 21 | qed. |
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[782] | 22 | |
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[1424] | 23 | lemma Sm_leq_n_m_leq_n: |
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| 24 | ∀m, n: nat. |
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| 25 | S m ≤ n → m ≤ n. |
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| 26 | #m #n /2/ |
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| 27 | qed. |
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| 28 | |
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| 29 | let rec bvtfold_aux |
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| 30 | (a, b: Type[0]) (f: BitVector 16 → a → b → b) (seed: b) (n: nat) |
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| 31 | on n: n ≤ 16 → BitVectorTrie a n → BitVector (16 - n) → b ≝ |
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| 32 | match n return λn: nat. n ≤ 16 → BitVectorTrie a n → BitVector (16 - n) → b with |
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| 33 | [ O ⇒ λinvariant: 0 ≤ 16. λtrie: BitVectorTrie a 0. λpath: BitVector 16. |
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| 34 | match trie return λx: nat. λtrie': BitVectorTrie a x. ∀prf: x = 0. b with |
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| 35 | [ Leaf l ⇒ λproof. f path l seed |
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| 36 | | Stub s ⇒ λproof. seed |
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| 37 | | Node n' l r ⇒ λabsrd. ⊥ |
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| 38 | ] (refl … 0) |
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| 39 | | S n' ⇒ λinvariant: S n' ≤ 16. λtrie: BitVectorTrie a (S n'). λpath: BitVector (16 - S n'). |
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| 40 | match trie return λx: nat. λtrie': BitVectorTrie a x. ∀prf: x = S n'. b with |
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| 41 | [ Leaf l ⇒ λabsrd. ⊥ |
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| 42 | | Stub s ⇒ λproof. seed |
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| 43 | | Node n'' l r ⇒ λproof. |
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| 44 | bvtfold_aux a b f (bvtfold_aux a b f seed n' ? (l⌈BitVectorTrie a n'' ↦ BitVectorTrie a n'⌉) ((false:::path)⌈S (16 - S n') ↦ 16 - n'⌉)) n' ? (r⌈BitVectorTrie a n'' ↦ BitVectorTrie a n'⌉) ((true:::path)⌈S (16 - S n') ↦ 16 - n'⌉) |
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| 45 | ] (refl … (S n')) |
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| 46 | ]. |
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| 47 | [ 1, 2: destruct(absrd) |
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| 48 | | 3,8: >minus_S_S <minus_Sn_m // @le_S_S_to_le // |
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| 49 | | 4,7: destruct(proof) % |
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| 50 | | 5,6: @Sm_leq_n_m_leq_n // ] |
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| 51 | qed. |
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| 52 | |
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[1006] | 53 | (* these two can probably be generalized w/r/t the second type and |
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| 54 | * some sort of equality relationship *) |
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| 55 | lemma fold_eq: |
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| 56 | ∀A: Type[0]. |
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| 57 | ∀n: nat. |
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| 58 | ∀f. |
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| 59 | ∀t. |
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| 60 | ∀P, Q: Prop. |
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| 61 | (P → Q) → (∀a,t',P,Q.(P → Q) → f a t' P → f a t' Q) → fold A ? n f t P → fold A ? n f t Q. |
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| 62 | #A #n #f #t #P #Q #H |
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[1516] | 63 | generalize in match (refl ? n); generalize in match H; -H; generalize in match Q; -Q; generalize in match P; -P; |
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| 64 | elim t in f ⊢ (? → ? → ? → ???% → ? → ???%%%? → ???%%%?); |
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[1006] | 65 | [ #a #f #P #Q #HPQ #_ #Hf #HP whd in HP; whd @(Hf (zero 0) a P Q HPQ HP) |
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| 66 | | #h #l #r #Hl #Hr #f #P #Q #HPQ #_ #Hf #HP normalize normalize in HP; @(Hl ? (fold A Prop h (λx.f (true:::x)) r P) (fold A Prop h (λx.f (true:::x)) r Q) ? (refl ? h) ?) |
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| 67 | [ @(Hr ? P Q HPQ (refl ? h) ?) |
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| 68 | #a #t' #X #Y #HXY #Hff @(Hf (true:::a) t' X Y HXY Hff) |
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| 69 | | #a #t' #X #Y #HXY #Hff @(Hf (false:::a) t' X Y HXY Hff) ] |
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| 70 | | #h #f #P #Q #HPQ #_ #Hf #HP whd in HP; whd @(HPQ HP) ] |
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| 71 | @HP |
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| 72 | qed. |
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| 73 | |
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| 74 | lemma fold_init: |
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| 75 | ∀A:Type[0]. |
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| 76 | ∀n:nat. |
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| 77 | ∀f. |
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| 78 | ∀t. |
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| 79 | ∀P: Prop. |
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| 80 | (∀a,t',P.f a t' P → P) → fold A Prop n f t P → P. |
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[1516] | 81 | #A #n #f #t #P #H generalize in match (refl ? n); generalize in match H; -H; generalize in match P; -P; |
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| 82 | elim t in f ⊢ (? → ? → ???% → ???%%%? → ?); -t |
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[1006] | 83 | [ #a #f #P #Hf #_ normalize @(Hf [[]]) |
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| 84 | | #h #l #r #Hl #Hr #f #P #Hf #_ normalize #HP @(Hr (λx.f (true:::x))) |
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| 85 | [ #a #t' #X @(Hf (true:::a) t' X) | @(refl ? h) | @(Hl (λx.f (false:::x))) |
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| 86 | [ #a #t' #X @(Hf (false:::a) t' X) | @(refl ? h) | @HP ] |
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| 87 | ] |
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| 88 | | #h #f #P #Hf #_ normalize // |
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[1424] | 89 | |
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| 90 | |
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[1006] | 91 | ] |
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| 92 | qed. |
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| 93 | |
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| 94 | definition forall |
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| 95 | ≝ |
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| 96 | λA.λn.λt:BitVectorTrie A n.λP.fold ? ? ? (λk.λa.λacc.(P k a) ∧ acc) t True. |
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| 97 | |
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| 98 | lemma forall_nodel: |
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| 99 | ∀A:Type[0]. |
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| 100 | ∀n:nat. |
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| 101 | ∀l,r. |
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| 102 | ∀P:BitVector (S n) → A → Prop. |
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| 103 | forall A (S n) (Node ? n l r) P → forall A n l (λx.λa.P (false:::x) a). |
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| 104 | #A #n #l #r #P #Hl |
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| 105 | whd @(fold_eq A n ? ? (fold A ? n (λk.λa.λacc.P (true:::k) a∧acc) r True) True) |
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| 106 | [ // |
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| 107 | | #n #t' #X #Y #HXY #HX %1 |
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| 108 | [ @(proj1 ? ? HX) | @HXY @(proj2 ? ? HX) ] |
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[1516] | 109 | | whd in Hl; @Hl ] |
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[1006] | 110 | qed. |
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| 111 | |
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| 112 | lemma forall_noder: |
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| 113 | ∀A:Type[0]. |
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| 114 | ∀n:nat. |
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| 115 | ∀l,r. |
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| 116 | ∀P:BitVector (S n) → A → Prop. |
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| 117 | forall A (S n) (Node ? n l r) P → forall A n r (λx.λa.P (true:::x) a). |
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| 118 | #A #n #l #r #P #Hr |
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| 119 | whd @(fold_init A n (λk.λa.λacc.P (false:::k) a∧acc) l) |
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| 120 | [ #n #t' #P #HP @(proj2 ? ? HP) |
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| 121 | | @Hr |
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| 122 | ] |
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| 123 | qed. |
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| 124 | |
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[1044] | 125 | lemma forall_node: |
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| 126 | ∀A.∀n.∀l,r.∀P:BitVector (S n) → A → Prop. |
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| 127 | forall A n l (λx.λa.P (false:::x) a) → forall A n r (λx.λa.P (true:::x) a) → |
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| 128 | forall A (S n) (Node ? n l r) P. |
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| 129 | #A #n #l #r #P #Hl #Hr |
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| 130 | normalize @(fold_eq … True) |
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| 131 | [ #_ @Hr |
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| 132 | | #x #t' #X #Y #HXY #HP %1 [ @(proj1 … HP) | @HXY @(proj2 … HP) ] |
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| 133 | | @Hl |
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| 134 | ] |
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| 135 | qed. |
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| 136 | |
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[726] | 137 | let rec lookup_opt (A: Type[0]) (n: nat) |
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| 138 | (b: BitVector n) (t: BitVectorTrie A n) on t |
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| 139 | : option A ≝ |
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| 140 | (match t return λx.λ_. BitVector x → option A with |
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| 141 | [ Leaf l ⇒ λ_.Some ? l |
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| 142 | | Node h l r ⇒ λb. lookup_opt A ? (tail … b) (if head' … b then r else l) |
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| 143 | | Stub _ ⇒ λ_.None ? |
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| 144 | ]) b. |
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[1052] | 145 | |
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| 146 | definition member ≝ |
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| 147 | λA. |
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| 148 | λn. |
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| 149 | λb: BitVector n. |
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| 150 | λt: BitVectorTrie A n. |
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| 151 | match lookup_opt A n b t with |
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| 152 | [ None ⇒ false |
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| 153 | | _ ⇒ true |
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| 154 | ]. |
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| 155 | |
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| 156 | definition member_p ≝ |
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| 157 | λA. |
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| 158 | λn. |
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| 159 | λb: BitVector n. |
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| 160 | λt: BitVectorTrie A n. |
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| 161 | match lookup_opt A n b t with |
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| 162 | [ None ⇒ False |
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| 163 | | _ ⇒ True |
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| 164 | ]. |
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[1006] | 165 | |
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| 166 | lemma forall_lookup: |
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| 167 | ∀A. |
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| 168 | ∀n. |
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| 169 | ∀t:BitVectorTrie A n. |
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| 170 | ∀P:BitVector n → A → Prop. |
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| 171 | forall A n t P → ∀a:A.∀b.lookup_opt A n b t = Some ? a → P b a. |
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[1516] | 172 | #A #n #t #P generalize in match (refl ? n); elim t in P ⊢ (???% → ??%%? → ? → ? → ??(??%%%)? → ?); |
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[1006] | 173 | [ #x #f #_ #Hf #a #b whd in Hf; #Hb normalize in Hb; destruct >(BitVector_O b) @(proj1 ? ? Hf) |
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| 174 | | #h #l #r #Hl #Hr #f #_ #Hf #a #b #Hb cases (BitVector_Sn h b) |
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| 175 | #hd #bla elim bla -bla #tl #Htl >Htl in Hb; #Hb cases hd in Hb; |
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| 176 | [ #Hb normalize in Hb; @(Hr (λx.λa.f (true:::x) a) (refl ? h)) |
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| 177 | [ @(forall_noder A h l r f Hf) |
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| 178 | | @Hb |
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| 179 | ] |
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| 180 | | #Hb normalize in Hb; @(Hl (λx.λa.f (false:::x) a) (refl ? h)) |
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| 181 | [ @(forall_nodel A h l r f Hf) |
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| 182 | | @Hb |
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| 183 | ] |
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| 184 | ] |
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| 185 | | #n #f #_ #Hf #a #b #Hb normalize in Hb; destruct |
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| 186 | qed. |
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[726] | 187 | |
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[1044] | 188 | lemma lookup_forall: |
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| 189 | ∀A:Type[0].∀n.∀t:BitVectorTrie A n.∀P:BitVector n → A → Prop. |
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| 190 | (∀a:A.∀b:BitVector n.lookup_opt A n b t = Some ? a → P b a) → forall A n t P. |
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| 191 | #A #n #t elim t |
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| 192 | [ #x #P #HP normalize %1 [ @HP normalize @refl | // ] |
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| 193 | | #h #l #r #Hl #Hr #P #HP @forall_node |
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| 194 | [ @Hl #a #b #Hlookup @HP normalize @Hlookup |
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| 195 | | @Hr #a #b #Hlookup @HP normalize @Hlookup |
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| 196 | ] |
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| 197 | | #n #P #HP normalize // |
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| 198 | ] |
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| 199 | qed. |
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| 200 | |
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[475] | 201 | let rec lookup (A: Type[0]) (n: nat) |
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| 202 | (b: BitVector n) (t: BitVectorTrie A n) (a: A) on b |
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| 203 | : A ≝ |
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| 204 | (match b return λx.λ_. x = n → A with |
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| 205 | [ VEmpty ⇒ |
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| 206 | (match t return λx.λ_. O = x → A with |
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| 207 | [ Leaf l ⇒ λ_.l |
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| 208 | | Node h l r ⇒ λK.⊥ |
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| 209 | | Stub s ⇒ λ_.a |
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| 210 | ]) |
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| 211 | | VCons o hd tl ⇒ |
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| 212 | match t return λx.λ_. (S o) = x → A with |
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| 213 | [ Leaf l ⇒ λK.⊥ |
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| 214 | | Node h l r ⇒ |
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| 215 | match hd with |
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| 216 | [ true ⇒ λK. lookup A h (tl⌈o ↦ h⌉) r a |
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| 217 | | false ⇒ λK. lookup A h (tl⌈o ↦ h⌉) l a |
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| 218 | ] |
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| 219 | | Stub s ⇒ λ_. a] |
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| 220 | ]) (refl ? n). |
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| 221 | [1,2: |
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| 222 | destruct |
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| 223 | |*: |
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| 224 | @ injective_S |
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| 225 | // |
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| 226 | ] |
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| 227 | qed. |
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| 228 | |
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[1474] | 229 | alias id "bvt_lookup" = "cic:/matita/cerco/ASM/BitVectorTrie/lookup.fix(0,2,5)". |
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| 230 | |
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[475] | 231 | let rec prepare_trie_for_insertion (A: Type[0]) (n: nat) (b: BitVector n) (a:A) on b : BitVectorTrie A n ≝ |
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| 232 | match b with |
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| 233 | [ VEmpty ⇒ Leaf A a |
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| 234 | | VCons o hd tl ⇒ |
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| 235 | match hd with |
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| 236 | [ true ⇒ Node A o (Stub A o) (prepare_trie_for_insertion A o tl a) |
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| 237 | | false ⇒ Node A o (prepare_trie_for_insertion A o tl a) (Stub A o) |
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| 238 | ] |
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| 239 | ]. |
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| 240 | |
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| 241 | let rec insert (A: Type[0]) (n: nat) (b: BitVector n) (a: A) on b: BitVectorTrie A n → BitVectorTrie A n ≝ |
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| 242 | (match b with |
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| 243 | [ VEmpty ⇒ λ_. Leaf A a |
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| 244 | | VCons o hd tl ⇒ λt. |
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| 245 | match t return λy.λ_. S o = y → BitVectorTrie A (S o) with |
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| 246 | [ Leaf l ⇒ λprf.⊥ |
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| 247 | | Node p l r ⇒ λprf. |
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| 248 | match hd with |
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| 249 | [ true ⇒ Node A o (l⌈p ↦ o⌉) (insert A o tl a (r⌈p ↦ o⌉)) |
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| 250 | | false ⇒ Node A o (insert A o tl a (l⌈p ↦ o⌉)) (r⌈p ↦ o⌉) |
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| 251 | ] |
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| 252 | | Stub p ⇒ λprf. (prepare_trie_for_insertion A ? (hd:::tl) a) |
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| 253 | ] (refl ? (S o)) |
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| 254 | ]). |
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| 255 | [ destruct |
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| 256 | |*: |
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| 257 | @ injective_S |
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| 258 | // |
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| 259 | ] |
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[761] | 260 | qed. |
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[1034] | 261 | |
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[761] | 262 | let rec update (A: Type[0]) (n: nat) (b: BitVector n) (a: A) on b: BitVectorTrie A n → option (BitVectorTrie A n) ≝ |
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| 263 | (match b with |
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| 264 | [ VEmpty ⇒ λt. match t return λy.λ_. O = y → option (BitVectorTrie A O) with |
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| 265 | [ Leaf _ ⇒ λ_. Some ? (Leaf A a) |
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| 266 | | Stub _ ⇒ λ_. None ? |
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| 267 | | Node _ _ _ ⇒ λprf. ⊥ |
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| 268 | ] (refl ? O) |
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| 269 | | VCons o hd tl ⇒ λt. |
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| 270 | match t return λy.λ_. S o = y → option (BitVectorTrie A (S o)) with |
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| 271 | [ Leaf l ⇒ λprf.⊥ |
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| 272 | | Node p l r ⇒ λprf. |
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| 273 | match hd with |
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| 274 | [ true ⇒ option_map ?? (λv. Node A o (l⌈p ↦ o⌉) v) (update A o tl a (r⌈p ↦ o⌉)) |
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| 275 | | false ⇒ option_map ?? (λv. Node A o v (r⌈p ↦ o⌉)) (update A o tl a (l⌈p ↦ o⌉)) |
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| 276 | ] |
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| 277 | | Stub p ⇒ λprf. None ? |
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| 278 | ] (refl ? (S o)) |
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| 279 | ]). |
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| 280 | [ 1,2: destruct |
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| 281 | |*: |
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| 282 | @ injective_S @sym_eq @prf |
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| 283 | ] |
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| 284 | qed. |
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[779] | 285 | |
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| 286 | let rec merge (A: Type[0]) (n: nat) (b: BitVectorTrie A n) on b: BitVectorTrie A n → BitVectorTrie A n ≝ |
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| 287 | match b return λx. λ_. BitVectorTrie A x → BitVectorTrie A x with |
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| 288 | [ Stub _ ⇒ λc. c |
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| 289 | | Leaf l ⇒ λc. match c with [ Leaf a ⇒ Leaf ? a | _ ⇒ Leaf ? l ] |
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| 290 | | Node p l r ⇒ |
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| 291 | λc. |
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| 292 | (match c return λx. λ_. x = (S p) → BitVectorTrie A (S p) with |
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| 293 | [ Node p' l' r' ⇒ λprf. Node ? ? (merge ?? l (l'⌈p' ↦ p⌉)) (merge ?? r (r'⌈p' ↦ p⌉)) |
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| 294 | | Stub _ ⇒ λprf. Node ? p l r |
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| 295 | | Leaf _ ⇒ λabsd. ? |
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| 296 | ] (refl ? (S p))) |
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| 297 | ]. |
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| 298 | [1: |
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| 299 | destruct(absd) |
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| 300 | |2,3: |
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| 301 | @ injective_S |
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| 302 | assumption |
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| 303 | ] |
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| 304 | qed. |
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[985] | 305 | |
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| 306 | lemma BitVectorTrie_O: |
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| 307 | ∀A:Type[0].∀v:BitVectorTrie A 0.(∃w. v ≃ Leaf A w) ∨ v ≃ Stub A 0. |
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[1516] | 308 | #A #v generalize in match (refl … O); cases v in ⊢ (??%? → (?(??(λ_.?%%??)))(?%%??)); |
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[985] | 309 | [ #w #_ %1 %[@w] % |
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[990] | 310 | | #n #l #r #abs @⊥ destruct(abs) |
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[985] | 311 | | #n #EQ %2 >EQ %] |
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| 312 | qed. |
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| 313 | |
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| 314 | lemma BitVectorTrie_Sn: |
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| 315 | ∀A:Type[0].∀n.∀v:BitVectorTrie A (S n).(∃l,r. v ≃ Node A n l r) ∨ v ≃ Stub A (S n). |
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[1516] | 316 | #A #n #v generalize in match (refl … (S n)); cases v in ⊢ (??%? → (?(??(λ_.??(λ_.?%%??))))%); |
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[990] | 317 | [ #m #abs @⊥ destruct(abs) |
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[985] | 318 | | #m #l #r #EQ %1 <(injective_S … EQ) %[@l] %[@r] // |
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| 319 | | #m #EQ %2 // ] |
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| 320 | qed. |
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| 321 | |
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| 322 | lemma lookup_prepare_trie_for_insertion_hit: |
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| 323 | ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n. |
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| 324 | lookup … b (prepare_trie_for_insertion … b v) a = v. |
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| 325 | #A #a #v #n #b elim b // #m #hd #tl #IH cases hd normalize // |
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| 326 | qed. |
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| 327 | |
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| 328 | lemma lookup_insert_hit: |
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| 329 | ∀A:Type[0].∀a,v:A.∀n.∀b:BitVector n.∀t:BitVectorTrie A n. |
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| 330 | lookup … b (insert … b v t) a = v. |
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| 331 | #A #a #v #n #b elim b -b -n // |
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| 332 | #n #hd #tl #IH #t cases(BitVectorTrie_Sn … t) |
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| 333 | [ * #l * #r #JMEQ >JMEQ cases hd normalize // |
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| 334 | | #JMEQ >JMEQ cases hd normalize @lookup_prepare_trie_for_insertion_hit ] |
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| 335 | qed. |
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| 336 | |
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| 337 | lemma lookup_prepare_trie_for_insertion_miss: |
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| 338 | ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n. |
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| 339 | (notb (eq_bv ? b c)) → lookup … b (prepare_trie_for_insertion … c v) a = a. |
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| 340 | #A #a #v #n #c elim c |
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| 341 | [ #b >(BitVector_O … b) normalize #abs @⊥ // |
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| 342 | | #m #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ |
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| 343 | cases hd cases hd' normalize |
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| 344 | [2,3: #_ cases tl' // |
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| 345 | |*: change with (bool_to_Prop (notb (eq_bv ???)) → ?) /2/ ]] |
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| 346 | qed. |
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| 347 | |
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| 348 | lemma lookup_insert_miss: |
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| 349 | ∀A:Type[0].∀a,v:A.∀n.∀c,b:BitVector n.∀t:BitVectorTrie A n. |
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| 350 | (notb (eq_bv ? b c)) → lookup … b (insert … c v t) a = lookup … b t a. |
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| 351 | #A #a #v #n #c elim c -c -n |
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[1516] | 352 | [ #b #t #DIFF @⊥ whd in DIFF; >(BitVector_O … b) in DIFF; // |
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[985] | 353 | | #n #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ |
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| 354 | #t cases(BitVectorTrie_Sn … t) |
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| 355 | [ * #l * #r #JMEQ >JMEQ cases hd cases hd' #H normalize in H; |
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[1521] | 356 | [1,4: change with (bool_to_Prop (notb (eq_bv ???))) in H; ] normalize // @IH // |
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[985] | 357 | | #JMEQ >JMEQ cases hd cases hd' #H normalize in H; |
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[1521] | 358 | [1,4: change with (bool_to_Prop (notb (eq_bv ???))) in H; ] normalize |
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[985] | 359 | [3,4: cases tl' // | *: @lookup_prepare_trie_for_insertion_miss //]]] |
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| 360 | qed. |
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[1034] | 361 | |
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| 362 | lemma lookup_stub: |
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| 363 | ∀A.∀n.∀b.∀a. |
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| 364 | lookup A n b (Stub A ?) a = a. |
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[1516] | 365 | #A #n #b #a cases n in b ⊢ (??(??%%%?)?); |
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[1034] | 366 | [ #b >(BitVector_O b) normalize @refl |
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| 367 | | #h #b cases (BitVector_Sn h b) #hd #X elim X -X; #tl #Hb >Hb cases hd |
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| 368 | [ normalize @refl |
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| 369 | | normalize @refl |
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| 370 | ] |
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| 371 | ] |
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| 372 | qed. |
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| 373 | |
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[1393] | 374 | lemma lookup_opt_lookup_miss: |
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[1034] | 375 | ∀A:Type[0].∀n:nat.∀b:BitVector n.∀t:BitVectorTrie A n.∀a:A. |
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[1393] | 376 | lookup_opt A n b t = None A → ∀x.lookup A n b t x = x. |
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[1516] | 377 | #A #n #b #t #a generalize in match (refl ? n); elim t in b ⊢ (???% → ??(??%%%)? → ? → ?); |
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[1393] | 378 | [ #a #B #_ #H #x normalize in H; >(BitVector_O B) normalize destruct |
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| 379 | | #h #l #r #Hl #Hr #b #_ #H #x cases (BitVector_Sn h b) #hd #X elim X -X; #tl #Hb |
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| 380 | >Hb >Hb in H; cases hd |
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| 381 | [ normalize #Hlookup @(Hr ? (refl ? h)) @Hlookup |
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| 382 | | normalize #Hlookup @(Hl ? (refl ? h)) @Hlookup |
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| 383 | ] |
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| 384 | | #n #B #_ #H #x @lookup_stub |
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| 385 | ] |
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| 386 | qed. |
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| 387 | |
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| 388 | lemma lookup_opt_lookup_hit: |
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| 389 | ∀A:Type[0].∀n:nat.∀b:BitVector n.∀t:BitVectorTrie A n.∀a:A. |
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[1034] | 390 | lookup_opt A n b t = Some A a → ∀x.lookup A n b t x = a. |
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[1516] | 391 | #A #n #b #t #a generalize in match (refl ? n); elim t in b ⊢ (???% → ??(??%%%)? → ? → ?); |
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[1034] | 392 | [ #a #B #_ #H #x normalize in H; >(BitVector_O B) normalize destruct @refl |
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[1393] | 393 | | #h #l #r #Hl #Hr #b #_ #H #x cases (BitVector_Sn h b) #hd #X elim X -X; #tl #Hb |
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[1034] | 394 | >Hb >Hb in H; cases hd |
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| 395 | [ normalize #Hlookup @(Hr ? (refl ? h)) @Hlookup |
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| 396 | | normalize #Hlookup @(Hl ? (refl ? h)) @Hlookup |
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| 397 | ] |
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| 398 | | #n #B #_ #H #x normalize in H; destruct |
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| 399 | ] |
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| 400 | qed. |
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| 401 | |
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[1393] | 402 | lemma lookup_lookup_opt_hit: |
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[1074] | 403 | ∀A.∀n.∀b.∀t.∀x,a. |
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| 404 | lookup A n b t x = a → x ≠ a → lookup_opt A n b t = Some A a. |
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[1516] | 405 | #A #n #b #t #x #a generalize in match (refl ? n); elim t in b ⊢ (???% → ? → ?); |
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[1074] | 406 | [ #z #B #_ #H #Hx >(BitVector_O B) in H; normalize #H >H @refl |
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| 407 | | #h #l #r #Hl #Hr #B #_ #H #Hx cases (BitVector_Sn h B) #hd #X elim X; -X #tl #HB |
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| 408 | >HB >HB in H; cases hd |
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| 409 | [ normalize #H >(Hr tl (refl ? h) H Hx) @refl |
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| 410 | | normalize #H >(Hl tl (refl ? h) H Hx) @refl |
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| 411 | ] |
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| 412 | | #n #B #_ #H #Hx cases B in H; |
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| 413 | [ normalize #Hx' | #n' #b #v normalize #Hx' ] |
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| 414 | @⊥ @(absurd (eq ? x a)) [1,3: @Hx' |2,4: @Hx ] |
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| 415 | ] |
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| 416 | qed. |
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| 417 | |
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[1034] | 418 | lemma lookup_opt_prepare_trie_for_insertion_hit: |
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| 419 | ∀A:Type[0].∀v:A.∀n.∀b:BitVector n. |
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| 420 | lookup_opt … b (prepare_trie_for_insertion … b v) = Some A v. |
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| 421 | #A #v #n #b elim b // #m #hd #tl #IH cases hd normalize // |
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| 422 | qed. |
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| 423 | |
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[1070] | 424 | lemma lookup_opt_prepare_trie_for_insertion_miss: |
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| 425 | ∀A:Type[0].∀v:A.∀n.∀c,b:BitVector n. |
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| 426 | (notb (eq_bv ? b c)) → lookup_opt … b (prepare_trie_for_insertion … c v) = None ?. |
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| 427 | #A #v #n #c elim c |
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| 428 | [ #b >(BitVector_O … b) normalize #abs @⊥ // |
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| 429 | | #m #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ |
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| 430 | cases hd cases hd' normalize |
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| 431 | [2,3: #_ cases tl' // |
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| 432 | |*: change with (bool_to_Prop (notb (eq_bv ???)) → ?) @IH ]] |
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| 433 | qed. |
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| 434 | |
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[1034] | 435 | lemma lookup_opt_insert_hit: |
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| 436 | ∀A:Type[0].∀v:A.∀n.∀b:BitVector n.∀t:BitVectorTrie A n. |
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| 437 | lookup_opt … b (insert … b v t) = Some A v. |
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[1516] | 438 | #A #v #n #b #t elim t in b ⊢ (??(??%%%)?); |
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[1034] | 439 | [ #x #b >(BitVector_O b) normalize @refl |
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| 440 | | #h #l #r #Hl #Hr #b cases (BitVector_Sn h b) #hd #X elim X -X; #tl #Hb >Hb cases hd |
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| 441 | [ normalize @Hr |
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| 442 | | normalize @Hl |
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| 443 | ] |
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[1516] | 444 | | #n' #b cases n' in b ⊢ ?; |
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[1034] | 445 | [ #b >(BitVector_O b) normalize @refl |
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| 446 | | #m #b cases (BitVector_Sn m b) #hd #X elim X -X; #tl #Hb >Hb cases hd |
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| 447 | normalize @lookup_opt_prepare_trie_for_insertion_hit |
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| 448 | ] |
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| 449 | ] |
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| 450 | qed. |
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[1070] | 451 | |
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| 452 | lemma lookup_opt_insert_miss: |
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| 453 | ∀A:Type[0].∀v:A.∀n.∀c,b:BitVector n.∀t:BitVectorTrie A n. |
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| 454 | (notb (eq_bv ? b c)) → lookup_opt … b (insert … c v t) = lookup_opt … b t. |
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| 455 | #A #v #n #c elim c -c -n |
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[1516] | 456 | [ #b #t #DIFF @⊥ whd in DIFF; >(BitVector_O … b) in DIFF; // |
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[1070] | 457 | | #n #hd #tl #IH #b cases(BitVector_Sn … b) #hd' * #tl' #JMEQ >JMEQ |
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| 458 | #t cases(BitVectorTrie_Sn … t) |
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| 459 | [ * #l * #r #JMEQ >JMEQ cases hd cases hd' #H normalize in H; |
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[1521] | 460 | [1,4: change with (bool_to_Prop (notb (eq_bv ???))) in H; ] normalize // @IH // |
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[1070] | 461 | | #JMEQ >JMEQ cases hd cases hd' #H normalize in H; |
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[1521] | 462 | [1,4: change with (bool_to_Prop (notb (eq_bv ???))) in H; ] normalize |
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[1070] | 463 | [3,4: cases tl' // | *: @lookup_opt_prepare_trie_for_insertion_miss //]]] |
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| 464 | qed. |
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| 465 | |
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[1479] | 466 | lemma insert_lookup_opt: |
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| 467 | ∀A:Type[0].∀v,a:A.∀n.∀c,b:BitVector n.∀t:BitVectorTrie A n. |
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| 468 | lookup_opt … b (insert … c v t) = Some A a → lookup_opt … b t = Some A a ∨ a = v. |
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| 469 | #A #v #a #n #c elim c -c; -n; |
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| 470 | [ #b #t #Hl normalize in Hl; %2 destruct (Hl) @refl |
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| 471 | | #n #hd #tl #Hind #b cases (BitVector_Sn … b) #hd' * #tl' #Heq >Heq |
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| 472 | #t cases (BitVectorTrie_Sn … t) |
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| 473 | [ * #l * #r #Heq2 >Heq2 cases hd cases hd' #H normalize in H; normalize |
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| 474 | [1,4: @(Hind tl' ? H) |
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| 475 | |2,3: %1 @H |
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| 476 | ] |
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| 477 | | #Heq2 >Heq2 cases hd cases hd' #H normalize in H; normalize |
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[1516] | 478 | [1,4: lapply (refl ? (eq_bv ? tl' tl)) cases (eq_bv ? tl' tl) in ⊢ (???% → %); #Heq3 |
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[1479] | 479 | [1,3: >(eq_bv_eq … Heq3) in H; >lookup_opt_prepare_trie_for_insertion_hit #X destruct (X) %2 // |
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[1516] | 480 | |2,4: >(lookup_opt_prepare_trie_for_insertion_miss) in H; |
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[1479] | 481 | [1,3: #X %1 // |
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| 482 | |2,4: >Heq3 // |
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| 483 | ] |
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| 484 | ] |
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| 485 | |2,3: destruct (H) |
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| 486 | ] |
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| 487 | qed. |
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| 488 | |
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[1034] | 489 | lemma forall_insert_inv1: |
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| 490 | ∀A.∀n.∀b.∀a.∀t.∀P. |
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| 491 | forall A n (insert A n b a t) P → P b a. |
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| 492 | #A #n #b #a #t #P #H @(forall_lookup ? ? (insert A n b a t)) |
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| 493 | [ @H |
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| 494 | | >(lookup_opt_insert_hit A ? n b) @(refl ? (Some A a)) |
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| 495 | ] |
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| 496 | qed. |
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| 497 | |
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| 498 | lemma forall_insert_inv2a: |
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| 499 | ∀A:Type[0].∀n:nat.∀b.∀a.∀t.∀P. |
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| 500 | lookup_opt A n b t = (None A) → forall A n (insert A n b a t) P → forall A n t P. |
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[1516] | 501 | #A #n #b #a #t generalize in match (refl ? n); elim t in b ⊢ (???% → ? → ??(??%%%)? → ??%%% → ??%%%); |
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[1034] | 502 | [ #x #b #_ #P >(BitVector_O b) normalize #H destruct |
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| 503 | | #h #l #r #Hl #Hr #b #_ #P cases (BitVector_Sn h b) #hd #X elim X -X; #tl #Hb >Hb cases hd #Hlookup #H |
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| 504 | [ normalize in H; normalize |
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| 505 | @(fold_eq … (fold A ? ? (λx.λa0.λacc.P (true:::x) a0∧acc) (insert … tl a r) True) … H) |
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| 506 | [ #Hfold @(Hr tl (refl ? h) ? Hlookup Hfold) |
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| 507 | | #x #t' #X #Y #HXY #HP %1 [ @(proj1 ? ? HP) | @(HXY (proj2 ? ?HP)) ] |
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| 508 | ] |
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| 509 | | normalize in H; normalize |
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| 510 | @(fold_eq … True) |
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| 511 | [ #_ @(fold_init A h (λx.λa0.λacc.P (false:::x) a0 ∧ acc) (insert A h tl a l)) |
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| 512 | [ #z #t' #X #HX @(proj2 ? ? HX) |
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| 513 | | @H ] |
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| 514 | | #z #t' #X #Y #HXY #HP %1 [ @(proj1 ? ? HP) | @(HXY (proj2 ? ? HP)) ] |
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| 515 | | @(Hl tl (refl ? h) ? Hlookup) normalize |
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| 516 | @(fold_eq … (fold A ? ? (λx.λa0.λacc.P (true:::x) a0∧acc) r True)) |
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| 517 | [ // |
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| 518 | | #z #t' #X #Y #HXY #HP %1 [ @(proj1 ? ? HP) | @(HXY (proj2 ? ? HP)) ] |
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| 519 | | @H |
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| 520 | ] |
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| 521 | ] |
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| 522 | ] |
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| 523 | | #n #b #_ #P #Hlookup #Hf normalize // ] |
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[1393] | 524 | qed. |
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[1034] | 525 | |
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| 526 | lemma forall_insert_inv2b: |
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| 527 | ∀A:Type[0].∀n:nat.∀b:BitVector n.∀a:A.∀t.∀P:(BitVector n → A → Prop). |
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| 528 | (∀x.(lookup_opt A n b t = Some A x) → P b x) → forall A n (insert A n b a t) P → forall A n t P. |
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[1516] | 529 | #A #n #b #a #t generalize in match (refl ? n); elim t in b ⊢ (???% → % → ? → ??%%% → ?); |
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[1034] | 530 | [ #x #b #_ #P >(BitVector_O b) normalize #HP #Hf %1 [ @HP @refl | @(proj2 ? ? Hf) ] |
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| 531 | | #h #l #r #Hl #Hr #b #_ cases (BitVector_Sn h b) #hd #X elim X -X; #tl #Hb >Hb cases hd #P #HP #Hf |
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| 532 | [ normalize in Hf; normalize |
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| 533 | @(fold_eq … (fold A ? ? (λx.λa0.λacc.P (true:::x) a0∧acc) (insert … tl a r) True) … Hf) |
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| 534 | [ #Hfold @(Hr tl (refl ? h) ? HP Hfold) |
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| 535 | | #x #t' #X #Y #HXY #HP %1 [ @(proj1 ? ? HP) | @(HXY (proj2 ? ? HP)) ] |
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| 536 | ] |
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| 537 | | normalize in H; normalize |
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| 538 | @(fold_eq … True) |
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| 539 | [ #_ @(fold_init A h (λx.λa0.λacc.P (false:::x) a0 ∧ acc) (insert A h tl a l)) |
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| 540 | [ #z #t' #X #HX @(proj2 ? ? HX) |
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| 541 | | @Hf ] |
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| 542 | | #z #t' #X #Y #HXY #HP %1 [ @(proj1 ? ? HP) | @(HXY (proj2 ? ? HP)) ] |
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| 543 | | @(Hl tl (refl ? h) ? HP) normalize |
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| 544 | @(fold_eq … (fold A ? ? (λx.λa0.λacc.P (true:::x) a0∧acc) r True)) |
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| 545 | [ // |
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| 546 | | #z #t' #X #Y #HXY #HP %1 [ @(proj1 ? ? HP) | @(HXY (proj2 ? ? HP)) ] |
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| 547 | | @Hf |
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| 548 | ] |
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| 549 | ] |
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| 550 | ] |
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| 551 | | #n #b #_ #P #Hlookup #Hf normalize // ] |
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[1038] | 552 | qed. |
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| 553 | |
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| 554 | lemma forall_prepare_tree_for_insertion: |
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| 555 | ∀A:Type[0].∀h:nat.∀b:BitVector h.∀a:A.∀P. |
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| 556 | P b a → |
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| 557 | forall A h (prepare_trie_for_insertion A h b a) P. |
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| 558 | #A #h #b elim b |
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| 559 | [ #a #P #HP normalize %1 [ @HP | // ] |
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| 560 | | #h #x #tl #Ha #a #P cases x #HP normalize @Ha @HP |
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| 561 | ] |
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| 562 | qed. |
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| 563 | |
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| 564 | lemma forall_insert: |
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| 565 | ∀A:Type[0].∀n:nat.∀b:BitVector n.∀a:A.∀t.∀P. |
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| 566 | forall A n t P → P b a → forall A n (insert A n b a t) P. |
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[1516] | 567 | #A #n #b #a #t generalize in match (refl ? n); elim t in b ⊢ (???% → % → ??%%% → %%? → ??%%%); |
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[1038] | 568 | [ #x #b #_ #P >(BitVector_O b) normalize #H1 #H2 /2/ |
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| 569 | | #h #l #r #Hl #Hr #b #_ cases (BitVector_Sn h b) #hd #X elim X -X; #tl #Hb >Hb cases hd #P #Hf #HP |
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| 570 | [ normalize in Hf; normalize |
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| 571 | @(fold_eq A … (fold A … (λx.λa0.λacc.P (true:::x) a0∧acc) r True) … Hf) |
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| 572 | [ #Hp @(Hr tl (refl ? h) ? Hp HP) |
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| 573 | | #z #t' #X #Y #HXY #HP %1 [ @(proj1 ? ? HP) | @(HXY (proj2 ? ? HP)) ] |
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| 574 | ] |
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| 575 | | normalize in Hf; normalize |
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| 576 | @(fold_eq … True) |
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| 577 | [ #_ @(fold_init A h (λx.λa0.λacc.P (false:::x) a0∧acc) l) |
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| 578 | [ #z #t' #X #HX @(proj2 ? ? HX) |
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| 579 | | @Hf ] |
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| 580 | | #z #t' #X #Y #HXY #HP %1 [ @(proj1 ? ? HP) | @(HXY (proj2 ? ? HP)) ] |
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| 581 | | @(Hl tl (refl ? h) ? ? HP) |
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| 582 | normalize @(fold_eq … (fold A ? ? (λx.λa0.λacc.P (true:::x) a0∧acc) r True)) |
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| 583 | [ // |
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| 584 | | #z #t' #X #Y #HXY #HP %1 [ @(proj1 ? ? HP) | @(HXY (proj2 ? ? HP)) ] |
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| 585 | | @Hf |
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| 586 | ] |
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| 587 | ] |
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| 588 | ] |
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[1516] | 589 | | #n #b #_ elim b in t ⊢ (% → ? → ? → ??%%%); |
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[1038] | 590 | [ #b #P #Hf #HP normalize %1 [ @HP | // ] |
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| 591 | | #h #hd #tl #H #b #P #Hf cases hd #HP normalize @(forall_prepare_tree_for_insertion A h tl a ? HP) |
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| 592 | ] |
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| 593 | ] |
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[1393] | 594 | qed. |
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[1316] | 595 | |
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| 596 | lemma update_fail : ∀A,n,b,a,t. |
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| 597 | update A n b a t = None ? → |
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| 598 | lookup_opt A n b t = None ?. |
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| 599 | #A #n elim n |
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| 600 | [ #b @(vector_inv_n … b) #a #t cases (BitVectorTrie_O … t) |
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| 601 | [ * #x #E >E normalize #NE destruct |
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| 602 | | #E >E normalize // |
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| 603 | ] |
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| 604 | | #m #IH #b @(vector_inv_n … b) #hd #tl #a #t cases (BitVectorTrie_Sn … t) |
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[1516] | 605 | [ * #t1 * #t2 #E >E cases hd whd in ⊢ (??%? → ??%?); |
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[1316] | 606 | #X lapply (option_map_none … X) @IH |
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| 607 | | #E >E normalize // |
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| 608 | ] |
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| 609 | ] qed. |
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| 610 | |
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| 611 | lemma update_lookup_opt_same : ∀A,n,b,a,t,t'. |
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| 612 | update A n b a t = Some ? t' → |
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| 613 | lookup_opt A n b t' = Some ? a. |
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| 614 | #A #n elim n |
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| 615 | [ #b #a #t #t' @(vector_inv_n … b) |
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| 616 | cases (BitVectorTrie_O … t) |
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| 617 | [ * #x #E >E normalize #E' destruct @refl |
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| 618 | | #E >E normalize #E' destruct |
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| 619 | ] |
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| 620 | | #m #IH #b #a #t #t' |
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| 621 | @(vector_inv_n … b) #bhd #btl |
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| 622 | cases (BitVectorTrie_Sn … t) |
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| 623 | [ * #t1 * #t2 #E' >E' |
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[1516] | 624 | whd in ⊢ (??%? → ??%?); cases bhd #U |
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[1316] | 625 | cases (option_map_some ????? U) |
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| 626 | #tn' * #U' #E'' <E'' |
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[1516] | 627 | whd in ⊢ (??%?); whd in ⊢ (??(???%%)?); |
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[1316] | 628 | @(IH … U') |
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| 629 | | #E >E normalize #E' destruct |
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| 630 | ] |
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| 631 | ] qed. |
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| 632 | |
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| 633 | lemma update_lookup_opt_other : ∀A,n,b,a,t,t'. |
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| 634 | update A n b a t = Some ? t' → |
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| 635 | ∀b'. b ≠ b' → |
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| 636 | lookup_opt A n b' t = lookup_opt A n b' t'. |
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| 637 | #A #n elim n |
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| 638 | [ #b #a #t #t' #E #b' |
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| 639 | @(vector_inv_n … b) @(vector_inv_n … b') |
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| 640 | * #NE cases (NE (refl ??)) |
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| 641 | | #m #IH #b #a #t #t' |
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| 642 | @(vector_inv_n … b) #bhd #btl |
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| 643 | cases (BitVectorTrie_Sn … t) |
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[1516] | 644 | [ * #t1 * #t2 #E >E whd in ⊢ (??%? → ?); cases bhd |
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[1316] | 645 | #U cases (option_map_some ????? U) #tn' * #U' #E' <E' |
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| 646 | #b' @(vector_inv_n … b') #bhd' #btl' |
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| 647 | cases bhd' |
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| 648 | [ 2,3: #_ @refl |
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[1516] | 649 | | *: #NE whd in ⊢ (??%%); whd in ⊢ (??(???%%)(???%%)); |
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| 650 | @(IH … U') % #E'' >E'' in NE; * #H @H @refl |
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[1316] | 651 | ] |
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[1516] | 652 | | #E >E whd in ⊢ (??%? → ?); #NE destruct |
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[1316] | 653 | ] |
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[1424] | 654 | ] qed. |
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