------------------------------------------------------------------------
-- The Agda standard library
--
-- Vectors defined as functions from a finite set to a type.
------------------------------------------------------------------------

-- This implementation is designed for reasoning about fixed-size
-- vectors where ease of retrieval of elements is prioritised.

-- See `Data.Vec` for an alternative implementation using inductive
-- data-types, which is more suitable for reasoning about vectors that
-- will grow or shrink in size.

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

module Data.Vec.Functional where

open import Data.Fin.Base
open import Data.List.Base as L using (List)
open import Data.Nat.Base as  using (; zero; suc)
open import Data.Product using (Σ; ; _×_; _,_; proj₁; proj₂; uncurry)
open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
open import Data.Vec.Base as V using (Vec)
open import Function.Base
open import Level using (Level)

infixr 5 _∷_ _++_
infixl 4 _⊛_
infixl 1 _>>=_

private
  variable
    a b c : Level
    A : Set a
    B : Set b
    C : Set c

------------------------------------------------------------------------
-- Definition

Vector : Set a    Set a
Vector A n = Fin n  A

------------------------------------------------------------------------
-- Conversion

toVec :  {n}  Vector A n  Vec A n
toVec = V.tabulate

fromVec :  {n}  Vec A n  Vector A n
fromVec = V.lookup

toList :  {n}  Vector A n  List A
toList = L.tabulate

fromList :  (xs : List A)  Vector A (L.length xs)
fromList = L.lookup

------------------------------------------------------------------------
-- Basic operations

[] : Vector A zero
[] ()

_∷_ :  {n}  A  Vector A n  Vector A (suc n)
(x  xs) zero    = x
(x  xs) (suc i) = xs i

length :  {n}  Vector A n  
length {n = n} _ = n

head :  {n}  Vector A (suc n)  A
head xs = xs zero

tail :  {n}  Vector A (suc n)  Vector A n
tail xs = xs  suc

uncons :  {n}  Vector A (suc n)  A × Vector A n
uncons xs = head xs , tail xs

replicate :  {n}  A  Vector A n
replicate = const

insert :  {n}  Vector A n  Fin (suc n)  A  Vector A (suc n)
insert {n = n}     xs zero    v zero    = v
insert {n = n}     xs zero    v (suc j) = xs j
insert {n = suc n} xs (suc i) v zero    = head xs
insert {n = suc n} xs (suc i) v (suc j) = insert (tail xs) i v j

remove :  {n}  Fin (suc n)  Vector A (suc n)  Vector A n
remove i t = t  punchIn i

updateAt :  {n}  Fin n  (A  A)  Vector A n  Vector A n
updateAt {n = suc n} zero    f xs zero    = f (head xs)
updateAt {n = suc n} zero    f xs (suc j) = xs (suc j)
updateAt {n = suc n} (suc i) f xs zero    = head xs
updateAt {n = suc n} (suc i) f xs (suc j) = updateAt i f (tail xs) j

------------------------------------------------------------------------
-- Transformations

map : (A  B)   {n}  Vector A n  Vector B n
map f xs = f  xs

_++_ :  {m n}  Vector A m  Vector A n  Vector A (m ℕ.+ n)
_++_ {m = m} xs ys i = [ xs , ys ] (splitAt m i)

concat :  {m n}  Vector (Vector A m) n  Vector A (n ℕ.* m)
concat {m = m} xss i = uncurry (flip xss) (quotRem m i)

foldr : (A  B  B)  B   {n}  Vector A n  B
foldr f z {n = zero}  xs = z
foldr f z {n = suc n} xs = f (head xs) (foldr f z (tail xs))

foldl : (B  A  B)  B   {n}  Vector A n  B
foldl f z {n = zero}  xs = z
foldl f z {n = suc n} xs = foldl f (f z (head xs)) (tail xs)

rearrange :  {m n}  (Fin m  Fin n)  Vector A n  Vector A m
rearrange r xs = xs  r

_⊛_ :  {n}  Vector (A  B) n  Vector A n  Vector B n
_⊛_ = _ˢ_

_>>=_ :  {m n}  Vector A m  (A  Vector B n)  Vector B (m ℕ.* n)
xs >>= f = concat (map f xs)

zipWith : (A  B  C)   {n}  Vector A n  Vector B n  Vector C n
zipWith f xs ys i = f (xs i) (ys i)

unzipWith :  {n}  (A  B × C)  Vector A n  Vector B n × Vector C n
unzipWith f xs = proj₁  f  xs , proj₂  f  xs

zip :  {n}  Vector A n  Vector B n  Vector (A × B) n
zip = zipWith _,_

unzip :  {n}  Vector (A × B) n  Vector A n × Vector B n
unzip = unzipWith id

take :  m {n}  Vector A (m ℕ.+ n)  Vector A m
take _ {n = n} xs = xs  inject+ n

drop :  m {n}  Vector A (m ℕ.+ n)  Vector A n
drop m xs = xs  raise m

reverse :  {n}  Vector A n  Vector A n
reverse xs = xs  opposite

init :  {n}  Vector A (suc n)  Vector A n
init xs = xs  inject₁

last :  {n}  Vector A (suc n)  A
last {n = n} xs = xs (fromℕ n)

transpose :  {m n}  Vector (Vector A n) m  Vector (Vector A m) n
transpose = flip