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