------------------------------------------------------------------------
-- The Agda standard library
--
-- Vectors, basic types and operations
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

module Data.Vec.Base where

open import Data.Bool.Base
open import Data.Nat.Base
open import Data.Fin.Base using (Fin; zero; suc)
open import Data.List.Base as List using (List)
open import Data.Product as Prod using (; ∃₂; _×_; _,_)
open import Data.These.Base as These using (These; this; that; these)
open import Function.Base using (const; _∘′_; id; _∘_)
open import Level using (Level)
open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
open import Relation.Nullary using (does)
open import Relation.Unary using (Pred; Decidable)

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

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

infixr 5 _∷_

data Vec (A : Set a) :   Set a where
  []  : Vec A zero
  _∷_ :  {n} (x : A) (xs : Vec A n)  Vec A (suc n)

infix 4 _[_]=_

data _[_]=_ {A : Set a} :  {n}  Vec A n  Fin n  A  Set a where
  here  :  {n}     {x}   {xs : Vec A n}  x  xs [ zero ]= x
  there :  {n} {i} {x y} {xs : Vec A n}
          (xs[i]=x : xs [ i ]= x)  y  xs [ suc i ]= x

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

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

head :  {n}  Vec A (1 + n)  A
head (x  xs) = x

tail :  {n}  Vec A (1 + n)  Vec A n
tail (x  xs) = xs

lookup :  {n}  Vec A n  Fin n  A
lookup (x  xs) zero    = x
lookup (x  xs) (suc i) = lookup xs i

insert :  {n}  Vec A n  Fin (suc n)  A  Vec A (suc n)
insert xs       zero     v = v  xs
insert (x  xs) (suc i)  v = x  insert xs i v

remove :  {n}  Vec A (suc n)  Fin (suc n)  Vec A n
remove (_  xs)     zero     = xs
remove (x  y  xs) (suc i)  = x  remove (y  xs) i

updateAt :  {n}  Fin n  (A  A)  Vec A n  Vec A n
updateAt zero    f (x  xs) = f x  xs
updateAt (suc i) f (x  xs) = x    updateAt i f xs

-- xs [ i ]%= f  modifies the i-th element of xs according to f

infixl 6 _[_]%=_

_[_]%=_ :  {n}  Vec A n  Fin n  (A  A)  Vec A n
xs [ i ]%= f = updateAt i f xs

-- xs [ i ]≔ y  overwrites the i-th element of xs with y

infixl 6 _[_]≔_

_[_]≔_ :  {n}  Vec A n  Fin n  A  Vec A n
xs [ i ]≔ y = xs [ i ]%= const y

------------------------------------------------------------------------
-- Operations for transforming vectors

map :  {n}  (A  B)  Vec A n  Vec B n
map f []       = []
map f (x  xs) = f x  map f xs

-- Concatenation.

infixr 5 _++_

_++_ :  {m n}  Vec A m  Vec A n  Vec A (m + n)
[]       ++ ys = ys
(x  xs) ++ ys = x  (xs ++ ys)

concat :  {m n}  Vec (Vec A m) n  Vec A (n * m)
concat []         = []
concat (xs  xss) = xs ++ concat xss

-- Align, Restrict, and Zip.

alignWith :  {m n}  (These A B  C)  Vec A m  Vec B n  Vec C (m  n)
alignWith f []         bs       = map (f ∘′ that) bs
alignWith f as@(_  _) []       = map (f ∘′ this) as
alignWith f (a  as)   (b  bs) = f (these a b)  alignWith f as bs

restrictWith :  {m n}  (A  B  C)  Vec A m  Vec B n  Vec C (m  n)
restrictWith f []       bs       = []
restrictWith f (_  _)  []       = []
restrictWith f (a  as) (b  bs) = f a b  restrictWith f as bs

zipWith :  {n}  (A  B  C)  Vec A n  Vec B n  Vec C n
zipWith f []       []       = []
zipWith f (x  xs) (y  ys) = f x y  zipWith f xs ys

unzipWith :  {n}  (A  B × C)  Vec A n  Vec B n × Vec C n
unzipWith f []       = [] , []
unzipWith f (a  as) = Prod.zip _∷_ _∷_ (f a) (unzipWith f as)

align :  {m n}  Vec A m  Vec B n  Vec (These A B) (m  n)
align = alignWith id

restrict :  {m n}  Vec A m  Vec B n  Vec (A × B) (m  n)
restrict = restrictWith _,_

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

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

-- Interleaving.

infixr 5 _⋎_

_⋎_ :  {m n}  Vec A m  Vec A n  Vec A (m +⋎ n)
[]        ys = ys
(x  xs)  ys = x  (ys  xs)

-- Pointwise application

infixl 4 _⊛_

_⊛_ :  {n}  Vec (A  B) n  Vec A n  Vec B n
[]        []       = []
(f  fs)  (x  xs) = f x  (fs  xs)

-- Multiplication

module CartesianBind where
  infixl 1 _>>=_

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

infixl 4 _⊛*_

_⊛*_ :  {m n}  Vec (A  B) m  Vec A n  Vec B (m * n)
fs ⊛* xs = fs CartesianBind.>>= λ f  map f xs

allPairs :  {m n}  Vec A m  Vec B n  Vec (A × B) (m * n)
allPairs xs ys = map _,_ xs ⊛* ys

-- Diagonal

diagonal :  {n}  Vec (Vec A n) n  Vec A n
diagonal [] = []
diagonal (xs  xss) = head xs  diagonal (map tail xss)

module DiagonalBind where
  infixl 1 _>>=_

  _>>=_ :  {n}  Vec A n  (A  Vec B n)  Vec B n
  xs >>= f = diagonal (map f xs)

------------------------------------------------------------------------
-- Operations for reducing vectors

foldr :  {a b} {A : Set a} (B :   Set b) {m} 
        (∀ {n}  A  B n  B (suc n)) 
        B zero 
        Vec A m  B m
foldr b _⊕_ n []       = n
foldr b _⊕_ n (x  xs) = x  foldr b _⊕_ n xs

foldr₁ :  {n}  (A  A  A)  Vec A (suc n)  A
foldr₁ _⊕_ (x  [])     = x
foldr₁ _⊕_ (x  y  ys) = x  foldr₁ _⊕_ (y  ys)

foldl :  {a b} {A : Set a} (B :   Set b) {m} 
        (∀ {n}  B n  A  B (suc n)) 
        B zero 
        Vec A m  B m
foldl b _⊕_ n []       = n
foldl b _⊕_ n (x  xs) = foldl  n  b (suc n)) _⊕_ (n  x) xs

foldl₁ :  {n}  (A  A  A)  Vec A (suc n)  A
foldl₁ _⊕_ (x  xs) = foldl _ _⊕_ x xs

-- Special folds

sum :  {n}  Vec  n  
sum = foldr _ _+_ 0

count :  {P : Pred A p}  Decidable P   {n}  Vec A n  
count P? []       = zero
count P? (x  xs) with does (P? x)
... | true  = suc (count P? xs)
... | false = count P? xs

------------------------------------------------------------------------
-- Operations for building vectors

[_] : A  Vec A 1
[ x ] = x  []

replicate :  {n}  A  Vec A n
replicate {n = zero}  x = []
replicate {n = suc n} x = x  replicate x

tabulate :  {n}  (Fin n  A)  Vec A n
tabulate {n = zero}  f = []
tabulate {n = suc n} f = f zero  tabulate (f  suc)

allFin :  n  Vec (Fin n) n
allFin _ = tabulate id

------------------------------------------------------------------------
-- Operations for dividing vectors

splitAt :  m {n} (xs : Vec A (m + n)) 
          ∃₂ λ (ys : Vec A m) (zs : Vec A n)  xs  ys ++ zs
splitAt zero    xs                = ([] , xs , refl)
splitAt (suc m) (x  xs)          with splitAt m xs
splitAt (suc m) (x  .(ys ++ zs)) | (ys , zs , refl) =
  ((x  ys) , zs , refl)

take :  m {n}  Vec A (m + n)  Vec A m
take m xs          with splitAt m xs
take m .(ys ++ zs) | (ys , zs , refl) = ys

drop :  m {n}  Vec A (m + n)  Vec A n
drop m xs          with splitAt m xs
drop m .(ys ++ zs) | (ys , zs , refl) = zs

group :  n k (xs : Vec A (n * k)) 
         λ (xss : Vec (Vec A k) n)  xs  concat xss
group zero    k []                  = ([] , refl)
group (suc n) k xs                  with splitAt k xs
group (suc n) k .(ys ++ zs)         | (ys , zs , refl) with group n k zs
group (suc n) k .(ys ++ concat zss) | (ys , ._ , refl) | (zss , refl) =
  ((ys  zss) , refl)

split :  {n}  Vec A n  Vec A  n /2⌉ × Vec A  n /2⌋
split []           = ([]     , [])
split (x  [])     = (x  [] , [])
split (x  y  xs) = Prod.map (x ∷_) (y ∷_) (split xs)

uncons :  {n}  Vec A (suc n)  A × Vec A n
uncons (x  xs) = x , xs

------------------------------------------------------------------------
-- Operations for converting between lists

toList :  {n}  Vec A n  List A
toList []       = List.[]
toList (x  xs) = List._∷_ x (toList xs)

fromList : (xs : List A)  Vec A (List.length xs)
fromList List.[]         = []
fromList (List._∷_ x xs) = x  fromList xs

------------------------------------------------------------------------
-- Operations for reversing vectors

reverse :  {n}  Vec A n  Vec A n
reverse {A = A} = foldl (Vec A)  rev x  x  rev) []

infixl 5 _∷ʳ_

_∷ʳ_ :  {n}  Vec A n  A  Vec A (1 + n)
[]       ∷ʳ y = [ y ]
(x  xs) ∷ʳ y = x  (xs ∷ʳ y)

initLast :  {n} (xs : Vec A (1 + n)) 
           ∃₂ λ (ys : Vec A n) (y : A)  xs  ys ∷ʳ y
initLast {n = zero}  (x  [])         = ([] , x , refl)
initLast {n = suc n} (x  xs)         with initLast xs
initLast {n = suc n} (x  .(ys ∷ʳ y)) | (ys , y , refl) =
  ((x  ys) , y , refl)

init :  {n}  Vec A (1 + n)  Vec A n
init xs         with initLast xs
init .(ys ∷ʳ y) | (ys , y , refl) = ys

last :  {n}  Vec A (1 + n)  A
last xs         with initLast xs
last .(ys ∷ʳ y) | (ys , y , refl) = y

------------------------------------------------------------------------
-- Other operations

transpose :  {m n}  Vec (Vec A n) m  Vec (Vec A m) n
transpose []         = replicate []
transpose (as  ass) = replicate _∷_  as  transpose ass