------------------------------------------------------------------------
-- The Agda standard library
--
-- Properties of n-ary products
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
module Data.Vec.Recursive.Properties where
open import Level using (Level)
open import Data.Nat.Base hiding (_^_)
open import Data.Product.Base
open import Data.Vec.Recursive
open import Data.Vec.Base using (Vec; _∷_)
open import Function.Bundles using (_↔_; mk↔ₛ′)
open import Relation.Binary.PropositionalEquality.Core
using (_≡_; refl; cong; cong₂)
open import Relation.Binary.PropositionalEquality.Properties
using (module ≡-Reasoning)
open ≡-Reasoning
private
variable
a : Level
A : Set a
------------------------------------------------------------------------
-- Basic proofs
cons-head-tail-identity : ∀ n (as : A ^ suc n) → cons n (head n as) (tail n as) ≡ as
cons-head-tail-identity 0 as = refl
cons-head-tail-identity (suc n) as = refl
head-cons-identity : ∀ n a (as : A ^ n) → head n (cons n a as) ≡ a
head-cons-identity 0 a as = refl
head-cons-identity (suc n) a as = refl
tail-cons-identity : ∀ n a (as : A ^ n) → tail n (cons n a as) ≡ as
tail-cons-identity 0 a as = refl
tail-cons-identity (suc n) a as = refl
append-cons : ∀ m n a (xs : A ^ m) ys →
append (suc m) n (cons m a xs) ys ≡ cons (m + n) a (append m n xs ys)
append-cons 0 n a xs ys = refl
append-cons (suc m) n a xs ys = refl
append-splitAt-identity : ∀ m n (as : A ^ (m + n)) → uncurry (append m n) (splitAt m n as) ≡ as
append-splitAt-identity 0 n as = refl
append-splitAt-identity (suc m) n as = begin
let x = head (m + n) as in
let (xs , ys) = splitAt m n (tail (m + n) as) in
append (suc m) n (cons m (head (m + n) as) xs) ys
≡⟨ append-cons m n x xs ys ⟩
cons (m + n) x (append m n xs ys)
≡⟨ cong (cons (m + n) x) (append-splitAt-identity m n (tail (m + n) as)) ⟩
cons (m + n) x (tail (m + n) as)
≡⟨ cons-head-tail-identity (m + n) as ⟩
as
∎
------------------------------------------------------------------------
-- Conversion to and from Vec
fromVec∘toVec : ∀ n (xs : A ^ n) → fromVec (toVec n xs) ≡ xs
fromVec∘toVec 0 _ = refl
fromVec∘toVec (suc n) xs = begin
cons n (head n xs) (fromVec (toVec n (tail n xs)))
≡⟨ cong (cons n (head n xs)) (fromVec∘toVec n (tail n xs)) ⟩
cons n (head n xs) (tail n xs)
≡⟨ cons-head-tail-identity n xs ⟩
xs ∎
toVec∘fromVec : ∀ {n} (xs : Vec A n) → toVec n (fromVec xs) ≡ xs
toVec∘fromVec Vec.[] = refl
toVec∘fromVec {n = suc n} (x Vec.∷ xs) = begin
head n (cons n x (fromVec xs)) Vec.∷ toVec n (tail n (cons n x (fromVec xs)))
≡⟨ cong₂ (λ x xs → x Vec.∷ toVec n xs) hd-prf tl-prf ⟩
x Vec.∷ toVec n (fromVec xs)
≡⟨ cong (x Vec.∷_) (toVec∘fromVec xs) ⟩
x Vec.∷ xs
∎ where
hd-prf = head-cons-identity _ x (fromVec xs)
tl-prf = tail-cons-identity _ x (fromVec xs)
↔Vec : ∀ n → A ^ n ↔ Vec A n
↔Vec n = mk↔ₛ′ (toVec n) fromVec toVec∘fromVec (fromVec∘toVec n)
------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.
-- Version 2.0
append-cons-commute = append-cons
{-# WARNING_ON_USAGE append-cons-commute
"Warning: append-cons-commute was deprecated in v2.0.
Please use append-cons instead."
#-}
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