{-# OPTIONS --without-K --safe #-}
module Data.Vec.Base where
open import Data.Bool.Base using (Bool; true; false; if_then_else_)
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
m n : ℕ
infixr 5 _∷_
data Vec (A : Set a) : ℕ → Set a where
[] : Vec A zero
_∷_ : ∀ (x : A) (xs : Vec A n) → Vec A (suc n)
infix 4 _[_]=_
data _[_]=_ {A : Set a} : Vec A n → Fin n → A → Set a where
here : ∀ {x} {xs : Vec A n} → x ∷ xs [ zero ]= x
there : ∀ {i} {x y} {xs : Vec A n}
(xs[i]=x : xs [ i ]= x) → y ∷ xs [ suc i ]= x
length : Vec A n → ℕ
length {n = n} _ = n
head : Vec A (1 + n) → A
head (x ∷ xs) = x
tail : Vec A (1 + n) → Vec A n
tail (x ∷ xs) = xs
lookup : Vec A n → Fin n → A
lookup (x ∷ xs) zero = x
lookup (x ∷ xs) (suc i) = lookup xs i
iterate : (A → A) → A → ∀ {n} → Vec A n
iterate s z {zero} = []
iterate s z {suc n} = z ∷ iterate s (s z)
insert : 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 : 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 : 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
infixl 6 _[_]%=_
_[_]%=_ : Vec A n → Fin n → (A → A) → Vec A n
xs [ i ]%= f = updateAt i f xs
infixl 6 _[_]≔_
_[_]≔_ : Vec A n → Fin n → A → Vec A n
xs [ i ]≔ y = xs [ i ]%= const y
map : (A → B) → Vec A n → Vec B n
map f [] = []
map f (x ∷ xs) = f x ∷ map f xs
infixr 5 _++_
_++_ : Vec A m → Vec A n → Vec A (m + n)
[] ++ ys = ys
(x ∷ xs) ++ ys = x ∷ (xs ++ ys)
concat : Vec (Vec A m) n → Vec A (n * m)
concat [] = []
concat (xs ∷ xss) = xs ++ concat xss
alignWith : (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 : (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 : (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 : (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 : Vec A m → Vec B n → Vec (These A B) (m ⊔ n)
align = alignWith id
restrict : Vec A m → Vec B n → Vec (A × B) (m ⊓ n)
restrict = restrictWith _,_
zip : Vec A n → Vec B n → Vec (A × B) n
zip = zipWith _,_
unzip : Vec (A × B) n → Vec A n × Vec B n
unzip = unzipWith id
infixr 5 _⋎_
_⋎_ : Vec A m → Vec A n → Vec A (m +⋎ n)
[] ⋎ ys = ys
(x ∷ xs) ⋎ ys = x ∷ (ys ⋎ xs)
infixl 4 _⊛_
_⊛_ : Vec (A → B) n → Vec A n → Vec B n
[] ⊛ [] = []
(f ∷ fs) ⊛ (x ∷ xs) = f x ∷ (fs ⊛ xs)
module CartesianBind where
infixl 1 _>>=_
_>>=_ : Vec A m → (A → Vec B n) → Vec B (m * n)
xs >>= f = concat (map f xs)
infixl 4 _⊛*_
_⊛*_ : Vec (A → B) m → Vec A n → Vec B (m * n)
fs ⊛* xs = fs CartesianBind.>>= λ f → map f xs
allPairs : Vec A m → Vec B n → Vec (A × B) (m * n)
allPairs xs ys = map _,_ xs ⊛* ys
diagonal : Vec (Vec A n) n → Vec A n
diagonal [] = []
diagonal (xs ∷ xss) = head xs ∷ diagonal (map tail xss)
module DiagonalBind where
infixl 1 _>>=_
_>>=_ : Vec A n → (A → Vec B n) → Vec B n
xs >>= f = diagonal (map f xs)
module _ (A : Set a) (B : ℕ → Set b) where
FoldrOp = ∀ {n} → A → B n → B (suc n)
FoldlOp = ∀ {n} → B n → A → B (suc n)
foldr : ∀ (B : ℕ → Set b) → FoldrOp A B → B zero → Vec A n → B n
foldr B _⊕_ e [] = e
foldr B _⊕_ e (x ∷ xs) = x ⊕ foldr B _⊕_ e xs
foldl : ∀ (B : ℕ → Set b) → FoldlOp A B → B zero → Vec A n → B n
foldl B _⊕_ e [] = e
foldl B _⊕_ e (x ∷ xs) = foldl (B ∘ suc) _⊕_ (e ⊕ x) xs
foldr′ : (A → B → B) → B → Vec A n → B
foldr′ _⊕_ = foldr _ _⊕_
foldl′ : (B → A → B) → B → Vec A n → B
foldl′ _⊕_ = foldl _ _⊕_
foldr₁ : (A → A → A) → Vec A (suc n) → A
foldr₁ _⊕_ (x ∷ []) = x
foldr₁ _⊕_ (x ∷ y ∷ ys) = x ⊕ foldr₁ _⊕_ (y ∷ ys)
foldl₁ : (A → A → A) → Vec A (suc n) → A
foldl₁ _⊕_ (x ∷ xs) = foldl _ _⊕_ x xs
sum : Vec ℕ n → ℕ
sum = foldr _ _+_ 0
countᵇ : (A → Bool) → Vec A n → ℕ
countᵇ p [] = zero
countᵇ p (x ∷ xs) = if p x then suc (countᵇ p xs) else countᵇ p xs
count : ∀ {P : Pred A p} → Decidable P → Vec A n → ℕ
count P? = countᵇ (does ∘ P?)
[_] : A → Vec A 1
[ x ] = x ∷ []
replicate : A → Vec A n
replicate {n = zero} x = []
replicate {n = suc n} x = x ∷ replicate x
tabulate : (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
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 : 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 : Vec A (suc n) → A × Vec A n
uncons (x ∷ xs) = x , xs
truncate : ∀ {m n} → m ≤ n → Vec A n → Vec A m
truncate z≤n _ = []
truncate (s≤s le) (x ∷ xs) = x ∷ (truncate le xs)
padRight : ∀ {m n} → m ≤ n → A → Vec A m → Vec A n
padRight z≤n a xs = replicate a
padRight (s≤s le) a (x ∷ xs) = x ∷ padRight le a xs
toList : 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
infixl 5 _∷ʳ_
_∷ʳ_ : Vec A n → A → Vec A (suc n)
[] ∷ʳ y = [ y ]
(x ∷ xs) ∷ʳ y = x ∷ (xs ∷ʳ y)
reverse : Vec A n → Vec A n
reverse = foldl (Vec _) (λ rev x → x ∷ rev) []
infix 5 _ʳ++_
_ʳ++_ : Vec A m → Vec A n → Vec A (m + n)
xs ʳ++ ys = foldl (Vec _ ∘ (_+ _)) (λ rev x → x ∷ rev) ys xs
initLast : ∀ (xs : Vec A (1 + n)) → ∃₂ λ ys y → xs ≡ ys ∷ʳ y
initLast {n = zero} (x ∷ []) = ([] , x , refl)
initLast {n = suc n} (x ∷ xs) with initLast xs
... | (ys , y , refl) = (x ∷ ys , y , refl)
init : Vec A (1 + n) → Vec A n
init xs with initLast xs
... | (ys , y , refl) = ys
last : Vec A (1 + n) → A
last xs with initLast xs
... | (ys , y , refl) = y
transpose : Vec (Vec A n) m → Vec (Vec A m) n
transpose [] = replicate []
transpose (as ∷ ass) = replicate _∷_ ⊛ as ⊛ transpose ass