------------------------------------------------------------------------
-- The Agda standard library
--
-- Bounded vectors, basic types and operations
------------------------------------------------------------------------

{-# OPTIONS --cubical-compatible --safe #-}

module Data.Vec.Bounded.Base where

open import Data.Nat.Base
import Data.Nat.Properties as 
open import Data.List.Base as List using (List)
open import Data.List.Extrema ℕ.≤-totalOrder
open import Data.List.Relation.Unary.All as All using (All)
import Data.List.Relation.Unary.All.Properties as All
open import Data.List.Membership.Propositional using (mapWith∈)
open import Data.Product.Base using (; _×_; _,_; proj₁; proj₂)
open import Data.Vec.Base as Vec using (Vec)
open import Data.These.Base as These using (These)
open import Function.Base using (_∘_; id; _$_)
open import Level using (Level)
open import Relation.Nullary.Decidable.Core using (recompute)
open import Relation.Binary.PropositionalEquality.Core as  using (_≡_; refl)
open import Relation.Binary.PropositionalEquality.Properties
  using (module ≡-Reasoning)

private
  variable
    a b c p : Level
    A : Set a
    B : Set b
    C : Set c
    m n : 

------------------------------------------------------------------------
-- Types

infix 4 _,_
record Vec≤ (A : Set a) (n : ) : Set a where
  constructor _,_
  field {length} : 
        vec      : Vec A length
        .bound   : length  n

-- projection to recompute irrelevant field
isBounded : (as : Vec≤ A n)  Vec≤.length as  n
isBounded as@(_ , m≤n) = recompute (_ ℕ.≤? _) m≤n

------------------------------------------------------------------------
-- Conversion functions

toVec : (as : Vec≤ A n)  Vec A (Vec≤.length as)
toVec as@(vs , _) = vs

fromVec : Vec A n  Vec≤ A n
fromVec v = v , ℕ.≤-refl

padRight : A  Vec≤ A n  Vec A n
padRight a as@(vs , m≤n)
  with k , reflℕ.m≤n⇒∃[o]m+o≡n m≤n
  = vs Vec.++ Vec.replicate k a

padLeft : A  Vec≤ A n  Vec A n
padLeft a record { length = m ; vec = vs ; bound = m≤n }
  with k , reflℕ.m≤n⇒∃[o]m+o≡n m≤n
  rewrite ℕ.+-comm m k
  = Vec.replicate k a Vec.++ vs

private
  split :  m k  m + k   k /2⌋ + (m +  k /2⌉)
  split m k = begin
    m + k                   ≡⟨ ≡.cong (m +_) (ℕ.⌊n/2⌋+⌈n/2⌉≡n k) 
    m + ( k /2⌋ +  k /2⌉) ≡⟨ ≡.cong (m +_) (ℕ.+-comm  k /2⌋  k /2⌉) 
    m + ( k /2⌉ +  k /2⌋) ≡⟨ ℕ.+-assoc m  k /2⌉  k /2⌋ 
    m +  k /2⌉ +  k /2⌋   ≡⟨ ℕ.+-comm _  k /2⌋ 
     k /2⌋ + (m +  k /2⌉) 
    where open ≡-Reasoning

padBoth : A  A  Vec≤ A n  Vec A n
padBoth aₗ aᵣ record { length = m ; vec = vs ; bound = m≤n }
  with k , reflℕ.m≤n⇒∃[o]m+o≡n m≤n
  rewrite split m k
  = Vec.replicate  k /2⌋ aₗ
      Vec.++ vs
      Vec.++ Vec.replicate  k /2⌉ aᵣ


fromList : (as : List A)  Vec≤ A (List.length as)
fromList = fromVec  Vec.fromList

toList : Vec≤ A n  List A
toList = Vec.toList  toVec

------------------------------------------------------------------------
-- Creating new Vec≤ vectors

replicate : .(m≤n : m  n)  A  Vec≤ A n
replicate m≤n a = Vec.replicate _ a , m≤n

[] : Vec≤ A n
[] = Vec.[] , z≤n

infixr 5 _∷_
_∷_ : A  Vec≤ A n  Vec≤ A (suc n)
a  (as , p) = a Vec.∷ as , s≤s p

------------------------------------------------------------------------
-- Modifying Vec≤ vectors

≤-cast : .(m≤n : m  n)  Vec≤ A m  Vec≤ A n
≤-cast m≤n (v , p) = v , ℕ.≤-trans p m≤n

≡-cast : .(eq : m  n)  Vec≤ A m  Vec≤ A n
≡-cast m≡n = ≤-cast (ℕ.≤-reflexive m≡n)

map : (A  B)  Vec≤ A n  Vec≤ B n
map f (v , p) = Vec.map f v , p

reverse : Vec≤ A n  Vec≤ A n
reverse (v , p) = Vec.reverse v , p

-- Align and Zip.

alignWith : (These A B  C)  Vec≤ A n  Vec≤ B n  Vec≤ C n
alignWith f (as , p) (bs , q) = Vec.alignWith f as bs , ℕ.⊔-lub p q

zipWith : (A  B  C)  Vec≤ A n  Vec≤ B n  Vec≤ C n
zipWith f (as , p) (bs , q) = Vec.restrictWith f as bs , ℕ.m≤n⇒m⊓o≤n _ p

zip : Vec≤ A n  Vec≤ B n  Vec≤ (A × B) n
zip = zipWith _,_

align : Vec≤ A n  Vec≤ B n  Vec≤ (These A B) n
align = alignWith id

-- take and drop

take :  n  Vec≤ A m  Vec≤ A (n  m)
take             zero    _                = []
take             (suc n) (Vec.[] , p)     = []
take {m = suc m} (suc n) (a Vec.∷ as , p) = a  take n (as , s≤s⁻¹ p)

drop :  n  Vec≤ A m  Vec≤ A (m  n)
drop             zero    v                = v
drop             (suc n) (Vec.[] , p)     = []
drop {m = suc m} (suc n) (a Vec.∷ as , p) = drop n (as , s≤s⁻¹ p)

------------------------------------------------------------------------
-- Lifting a collection of bounded vectors to the same size

rectangle : List ( (Vec≤ A))   (List  Vec≤ A)
rectangle {A = A} rows = width , padded where

  sizes = List.map proj₁ rows
  width = max 0 sizes

  all≤ : All  v  proj₁ v  width) rows
  all≤ = All.map⁻ (xs≤max 0 sizes)

  padded : List (Vec≤ A width)
  padded = mapWith∈ rows $ λ {x} x∈rows 
    ≤-cast (All.lookup all≤ x∈rows) (proj₂ x)