------------------------------------------------------------------------ -- 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) : ℕ → 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 → 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 : ℕ → 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 : ℕ → 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 (λ 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 n → ℕ sum = foldr _ _+_ 0 count : ∀ {P : Pred A p} → Decidable P → ∀ {n} → Vec A n → 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) (λ 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