135 lines
4.6 KiB
Text
135 lines
4.6 KiB
Text
/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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Tuples are lists of a fixed size.
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It is implemented as a subtype.
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-/
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import data.list
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universe variables u v w
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def tuple (α : Type u) (n : ℕ) := {l : list α // list.length l = n}
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namespace tuple
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variables {α : Type u} {β : Type v} {φ : Type w}
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variable {n : ℕ}
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instance [decidable_eq α] : decidable_eq (tuple α n) :=
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begin unfold tuple, apply_instance end
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definition nil : tuple α 0 := ⟨[], rfl⟩
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definition cons : α → tuple α n → tuple α (nat.succ n)
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| a ⟨ v, h ⟩ := ⟨ a::v, congr_arg nat.succ h ⟩
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@[reducible] def length (v : tuple α n) : ℕ := n
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notation a :: b := cons a b
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notation `[` l:(foldr `, ` (h t, cons h t) nil `]`) := l
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open nat
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definition head : tuple α (nat.succ n) → α
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| ⟨list.nil, h ⟩ := let q : 0 = succ n := h in by contradiction
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| ⟨list.cons a v, h ⟩ := a
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theorem head_cons (a : α) : Π (v : tuple α n), head (a :: v) = a
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| ⟨ l, h ⟩ := rfl
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definition tail : tuple α (succ n) → tuple α n
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| ⟨ list.nil, h ⟩ := let q : 0 = succ n := h in by contradiction
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| ⟨ list.cons a v, h ⟩ := ⟨ v, congr_arg pred h ⟩
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theorem tail_cons (a : α) : Π (v : tuple α n), tail (a :: v) = v
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| ⟨ l, h ⟩ := rfl
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definition to_list : tuple α n → list α | ⟨ l, h ⟩ := l
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definition has_decidable_eq [decidable_eq α] {n:ℕ}
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: ∀ (x y : tuple α n), decidable (x = y)
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| ⟨s,p⟩ ⟨t,q⟩ :=
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match list.has_decidable_eq s t with
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| (is_true h) := is_true (subtype.eq h)
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| (is_false h) := is_false (λr, subtype.no_confusion r (λleq (peq : p == q), h leq))
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end
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/- append -/
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definition append {n m : nat} : tuple α n → tuple α m → tuple α (n + m)
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| ⟨ l₁, h₁ ⟩ ⟨ l₂, h₂ ⟩ :=
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let p := calc
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list.length (l₁ ++ l₂)
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= list.length l₁ + list.length l₂ : list.length_append l₁ l₂
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... = n + list.length l₂ : congr_arg (λi, i + list.length l₂) h₁
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... = n + m : congr_arg (λi, n + i) h₂ in
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⟨ list.append l₁ l₂, p ⟩
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/- map -/
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definition map (f : α → β) : tuple α n → tuple β n
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| ⟨ l, h ⟩ :=
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let q := calc list.length (list.map f l) = list.length l : list.length_map f l
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... = n : h in
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⟨ list.map f l, q ⟩
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theorem map_nil (f : α → β) : map f nil = nil := rfl
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theorem map_cons (f : α → β) (a : α)
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: Π (v : tuple α n), map f (a::v) = f a :: map f v
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| ⟨ l, h ⟩ := rfl
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definition map₂ (f : α → β → φ) : tuple α n → tuple β n → tuple φ n
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| ⟨ x, px ⟩ ⟨ y, py ⟩ :=
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let z : list φ := list.map₂ f x y in
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let pxx : list.length x = n := px in
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let pyy : list.length y = n := py in
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let p : list.length z = n := calc
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list.length z = min (list.length x) (list.length y) : list.length_map₂ f x y
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... = min n n : by rewrite [pxx, pyy]
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... = n : min_self n in
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⟨ z, p ⟩
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definition repeat (a : α) (n : ℕ) : tuple α n :=
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⟨list.repeat a n, list.length_repeat a n⟩
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definition dropn (i : ℕ) : tuple α n → tuple α (n - i)
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| ⟨l, p⟩ := ⟨list.dropn i l, p ▸ list.length_dropn i l⟩
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definition firstn (i : ℕ) : tuple α n → tuple α (min i n)
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| ⟨l, p⟩ :=
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let q := calc list.length (list.firstn i l)
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= min i (list.length l) : list.length_firstn i l
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... = min i n : congr_arg (min i) p in
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⟨list.firstn i l, q⟩
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section accum
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open prod
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variable {σ : Type}
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definition map_accumr
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: (α → σ → σ × β) → tuple α n → σ → σ × tuple β n
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| f ⟨ x, px ⟩ c :=
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let z := list.map_accumr f x c in
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let p := eq.trans (list.length_map_accumr f x c) px in
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(prod.fst z, ⟨ prod.snd z, p ⟩)
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definition map_accumr₂ {α β σ φ : Type} (f : α → β → σ → σ × φ)
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: tuple α n → tuple β n → σ → σ × tuple φ n
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| ⟨ x, px ⟩ ⟨ y, py ⟩ c :=
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let z := list.map_accumr₂ f x y c in
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let pxx : list.length x = n := px in
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let pyy : list.length y = n := py in
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let p := calc
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list.length (prod.snd (list.map_accumr₂ f x y c))
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= min (list.length x) (list.length y) : list.length_map_accumr₂ f x y c
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... = n : by rewrite [ pxx, pyy, min_self ] in
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(prod.fst z, ⟨prod.snd z, p ⟩)
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end accum
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end tuple
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instance decide_tuple_eq {A:Type.{1}} [decidable_eq A] {n:ℕ}
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: ∀ {x y : tuple A n}, decidable (x = y)
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:= tuple.has_decidable_eq
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