------------------------------------------------------------------------ -- The Agda standard library -- -- Lists, basic types and operations ------------------------------------------------------------------------ -- See README.Data.List for examples of how to use and reason about -- lists. {-# OPTIONS --cubical-compatible --safe #-} module Data.List.Base where open import Algebra.Bundles.Raw using (RawMagma; RawMonoid) open import Data.Bool.Base as Bool using (Bool; false; true; not; _∧_; _∨_; if_then_else_) open import Data.Fin.Base using (Fin; zero; suc) open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe′) open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _+_; _*_ ; _≤_ ; s≤s) open import Data.Product.Base as Product using (_×_; _,_; map₁; map₂′) open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂) open import Data.These.Base as These using (These; this; that; these) open import Function.Base using (id; _∘_ ; _∘′_; _∘₂_; _$_; const; flip) open import Level using (Level) open import Relation.Unary using (Pred; Decidable) open import Relation.Binary.Core using (Rel) import Relation.Binary.Definitions as B open import Relation.Binary.PropositionalEquality.Core using (_≡_) open import Relation.Nullary.Decidable.Core using (T?; does; ¬?) private variable a b c p ℓ : Level A : Set a B : Set b C : Set c ------------------------------------------------------------------------ -- Types open import Agda.Builtin.List public using (List; []; _∷_) ------------------------------------------------------------------------ -- Operations for transforming lists map : (A → B) → List A → List B map f [] = [] map f (x ∷ xs) = f x ∷ map f xs infixr 5 _++_ _++_ : List A → List A → List A [] ++ ys = ys (x ∷ xs) ++ ys = x ∷ (xs ++ ys) intersperse : A → List A → List A intersperse x [] = [] intersperse x (y ∷ []) = y ∷ [] intersperse x (y ∷ ys) = y ∷ x ∷ intersperse x ys intercalate : List A → List (List A) → List A intercalate xs [] = [] intercalate xs (ys ∷ []) = ys intercalate xs (ys ∷ yss) = ys ++ xs ++ intercalate xs yss cartesianProductWith : (A → B → C) → List A → List B → List C cartesianProductWith f [] _ = [] cartesianProductWith f (x ∷ xs) ys = map (f x) ys ++ cartesianProductWith f xs ys cartesianProduct : List A → List B → List (A × B) cartesianProduct = cartesianProductWith _,_ ------------------------------------------------------------------------ -- Aligning and zipping alignWith : (These A B → C) → List A → List B → List C 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 zipWith : (A → B → C) → List A → List B → List C zipWith f (x ∷ xs) (y ∷ ys) = f x y ∷ zipWith f xs ys zipWith f _ _ = [] unalignWith : (A → These B C) → List A → List B × List C unalignWith f [] = [] , [] unalignWith f (a ∷ as) with f a ... | this b = Product.map₁ (b ∷_) (unalignWith f as) ... | that c = Product.map₂ (c ∷_) (unalignWith f as) ... | these b c = Product.map (b ∷_) (c ∷_) (unalignWith f as) unzipWith : (A → B × C) → List A → List B × List C unzipWith f [] = [] , [] unzipWith f (xy ∷ xys) = Product.zip _∷_ _∷_ (f xy) (unzipWith f xys) partitionSumsWith : (A → B ⊎ C) → List A → List B × List C partitionSumsWith f = unalignWith (These.fromSum ∘′ f) align : List A → List B → List (These A B) align = alignWith id zip : List A → List B → List (A × B) zip = zipWith (_,_) unalign : List (These A B) → List A × List B unalign = unalignWith id unzip : List (A × B) → List A × List B unzip = unzipWith id partitionSums : List (A ⊎ B) → List A × List B partitionSums = partitionSumsWith id merge : {R : Rel A ℓ} → B.Decidable R → List A → List A → List A merge R? [] ys = ys merge R? xs [] = xs merge R? x∷xs@(x ∷ xs) y∷ys@(y ∷ ys) = if does (R? x y) then x ∷ merge R? xs y∷ys else y ∷ merge R? x∷xs ys ------------------------------------------------------------------------ -- Operations for reducing lists foldr : (A → B → B) → B → List A → B foldr c n [] = n foldr c n (x ∷ xs) = c x (foldr c n xs) foldl : (A → B → A) → A → List B → A foldl c n [] = n foldl c n (x ∷ xs) = foldl c (c n x) xs concat : List (List A) → List A concat = foldr _++_ [] concatMap : (A → List B) → List A → List B concatMap f = concat ∘ map f ap : List (A → B) → List A → List B ap fs as = concatMap (flip map as) fs catMaybes : List (Maybe A) → List A catMaybes = foldr (maybe′ _∷_ id) [] mapMaybe : (A → Maybe B) → List A → List B mapMaybe p = catMaybes ∘ map p null : List A → Bool null [] = true null (x ∷ xs) = false and : List Bool → Bool and = foldr _∧_ true or : List Bool → Bool or = foldr _∨_ false any : (A → Bool) → List A → Bool any p = or ∘ map p all : (A → Bool) → List A → Bool all p = and ∘ map p sum : List ℕ → ℕ sum = foldr _+_ 0 product : List ℕ → ℕ product = foldr _*_ 1 length : List A → ℕ length = foldr (const suc) 0 ------------------------------------------------------------------------ -- Operations for constructing lists [_] : A → List A [ x ] = x ∷ [] fromMaybe : Maybe A → List A fromMaybe (just x) = [ x ] fromMaybe nothing = [] replicate : ℕ → A → List A replicate zero x = [] replicate (suc n) x = x ∷ replicate n x iterate : (A → A) → A → ℕ → List A iterate f e zero = [] iterate f e (suc n) = e ∷ iterate f (f e) n inits : List A → List (List A) inits {A = A} = λ xs → [] ∷ tail xs module Inits where tail : List A → List (List A) tail [] = [] tail (x ∷ xs) = [ x ] ∷ map (x ∷_) (tail xs) tails : List A → List (List A) tails {A = A} = λ xs → xs ∷ tail xs module Tails where tail : List A → List (List A) tail [] = [] tail (_ ∷ xs) = xs ∷ tail xs insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A insertAt xs zero v = v ∷ xs insertAt (x ∷ xs) (suc i) v = x ∷ insertAt xs i v updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A updateAt (x ∷ xs) zero f = f x ∷ xs updateAt (x ∷ xs) (suc i) f = x ∷ updateAt xs i f -- Tabulation applyUpTo : (ℕ → A) → ℕ → List A applyUpTo f zero = [] applyUpTo f (suc n) = f zero ∷ applyUpTo (f ∘ suc) n applyDownFrom : (ℕ → A) → ℕ → List A applyDownFrom f zero = [] applyDownFrom f (suc n) = f n ∷ applyDownFrom f n tabulate : ∀ {n} (f : Fin n → A) → List A tabulate {n = zero} f = [] tabulate {n = suc n} f = f zero ∷ tabulate (f ∘ suc) lookup : ∀ (xs : List A) → Fin (length xs) → A lookup (x ∷ xs) zero = x lookup (x ∷ xs) (suc i) = lookup xs i -- Numerical upTo : ℕ → List ℕ upTo = applyUpTo id downFrom : ℕ → List ℕ downFrom = applyDownFrom id allFin : ∀ n → List (Fin n) allFin n = tabulate id unfold : ∀ (P : ℕ → Set b) (f : ∀ {n} → P (suc n) → Maybe (A × P n)) → ∀ {n} → P n → List A unfold P f {n = zero} s = [] unfold P f {n = suc n} s = maybe′ (λ (x , s′) → x ∷ unfold P f s′) [] (f s) ------------------------------------------------------------------------ -- Operations for reversing lists reverseAcc : List A → List A → List A reverseAcc = foldl (flip _∷_) reverse : List A → List A reverse = reverseAcc [] -- "Reverse append" xs ʳ++ ys = reverse xs ++ ys infixr 5 _ʳ++_ _ʳ++_ : List A → List A → List A _ʳ++_ = flip reverseAcc -- Snoc: Cons, but from the right. infixl 6 _∷ʳ_ _∷ʳ_ : List A → A → List A xs ∷ʳ x = xs ++ [ x ] -- Backwards initialisation infixl 5 _∷ʳ′_ data InitLast {A : Set a} : List A → Set a where [] : InitLast [] _∷ʳ′_ : (xs : List A) (x : A) → InitLast (xs ∷ʳ x) initLast : (xs : List A) → InitLast xs initLast [] = [] initLast (x ∷ xs) with initLast xs ... | [] = [] ∷ʳ′ x ... | ys ∷ʳ′ y = (x ∷ ys) ∷ʳ′ y -- uncons, but from the right unsnoc : List A → Maybe (List A × A) unsnoc as with initLast as ... | [] = nothing ... | xs ∷ʳ′ x = just (xs , x) ------------------------------------------------------------------------ -- Operations for deconstructing lists -- Note that although the following three combinators can be useful for -- programming, when proving it is often a better idea to manually -- destruct a list argument as each branch of the pattern-matching will -- have a refined type. uncons : List A → Maybe (A × List A) uncons [] = nothing uncons (x ∷ xs) = just (x , xs) head : List A → Maybe A head [] = nothing head (x ∷ _) = just x tail : List A → Maybe (List A) tail [] = nothing tail (_ ∷ xs) = just xs last : List A → Maybe A last [] = nothing last (x ∷ []) = just x last (_ ∷ xs) = last xs take : ℕ → List A → List A take zero xs = [] take (suc n) [] = [] take (suc n) (x ∷ xs) = x ∷ take n xs drop : ℕ → List A → List A drop zero xs = xs drop (suc n) [] = [] drop (suc n) (x ∷ xs) = drop n xs splitAt : ℕ → List A → List A × List A splitAt zero xs = ([] , xs) splitAt (suc n) [] = ([] , []) splitAt (suc n) (x ∷ xs) = Product.map₁ (x ∷_) (splitAt n xs) removeAt : (xs : List A) → Fin (length xs) → List A removeAt (x ∷ xs) zero = xs removeAt (x ∷ xs) (suc i) = x ∷ removeAt xs i ------------------------------------------------------------------------ -- Operations for filtering lists -- The following are a variety of functions that can be used to -- construct sublists using a predicate. -- -- Each function has two forms. The first main variant uses a -- proof-relevant decidable predicate, while the second variant uses -- a irrelevant boolean predicate and are suffixed with a `ᵇ` character, -- typed as \^b. -- -- The decidable versions have several advantages: 1) easier to prove -- properties, 2) better meta-variable inference and 3) most of the rest -- of the library is set-up to work with decidable predicates. However, -- in rare cases the boolean versions can be useful, mainly when one -- wants to minimise dependencies. -- -- In summary, in most cases you probably want to use the decidable -- versions over the boolean versions, e.g. use `takeWhile (_≤? 10) xs` -- rather than `takeWhileᵇ (_≤ᵇ 10) xs`. takeWhile : ∀ {P : Pred A p} → Decidable P → List A → List A takeWhile P? [] = [] takeWhile P? (x ∷ xs) = if does (P? x) then x ∷ takeWhile P? xs else [] takeWhileᵇ : (A → Bool) → List A → List A takeWhileᵇ p = takeWhile (T? ∘ p) dropWhile : ∀ {P : Pred A p} → Decidable P → List A → List A dropWhile P? [] = [] dropWhile P? (x ∷ xs) = if does (P? x) then dropWhile P? xs else x ∷ xs dropWhileᵇ : (A → Bool) → List A → List A dropWhileᵇ p = dropWhile (T? ∘ p) filter : ∀ {P : Pred A p} → Decidable P → List A → List A filter P? [] = [] filter P? (x ∷ xs) = let xs′ = filter P? xs in if does (P? x) then x ∷ xs′ else xs′ filterᵇ : (A → Bool) → List A → List A filterᵇ p = filter (T? ∘ p) partition : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A) partition P? [] = [] , [] partition P? (x ∷ xs) = let ys , zs = partition P? xs in if does (P? x) then (x ∷ ys , zs) else (ys , x ∷ zs) partitionᵇ : (A → Bool) → List A → List A × List A partitionᵇ p = partition (T? ∘ p) span : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A) span P? [] = [] , [] span P? ys@(x ∷ xs) = if does (P? x) then Product.map (x ∷_) id (span P? xs) else ([] , ys) spanᵇ : (A → Bool) → List A → List A × List A spanᵇ p = span (T? ∘ p) break : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A) break P? = span (¬? ∘ P?) breakᵇ : (A → Bool) → List A → List A × List A breakᵇ p = break (T? ∘ p) -- The predicate `P` represents the notion of newline character for the -- type `A`. It is used to split the input list into a list of lines. -- Some lines may be empty if the input contains at least two -- consecutive newline characters. linesBy : ∀ {P : Pred A p} → Decidable P → List A → List (List A) linesBy {A = A} P? = go nothing where go : Maybe (List A) → List A → List (List A) go acc [] = maybe′ ([_] ∘′ reverse) [] acc go acc (c ∷ cs) = if does (P? c) then reverse acc′ ∷ go nothing cs else go (just (c ∷ acc′)) cs where acc′ = Maybe.fromMaybe [] acc linesByᵇ : (A → Bool) → List A → List (List A) linesByᵇ p = linesBy (T? ∘ p) -- The predicate `P` represents the notion of space character for the -- type `A`. It is used to split the input list into a list of words. -- All the words are non empty and the output does not contain any space -- characters. wordsBy : ∀ {P : Pred A p} → Decidable P → List A → List (List A) wordsBy {A = A} P? = go [] where cons : List A → List (List A) → List (List A) cons [] ass = ass cons as ass = reverse as ∷ ass go : List A → List A → List (List A) go acc [] = cons acc [] go acc (c ∷ cs) = if does (P? c) then cons acc (go [] cs) else go (c ∷ acc) cs wordsByᵇ : (A → Bool) → List A → List (List A) wordsByᵇ p = wordsBy (T? ∘ p) derun : ∀ {R : Rel A p} → B.Decidable R → List A → List A derun R? [] = [] derun R? (x ∷ []) = x ∷ [] derun R? (x ∷ xs@(y ∷ _)) = let ys = derun R? xs in if does (R? x y) then ys else x ∷ ys derunᵇ : (A → A → Bool) → List A → List A derunᵇ r = derun (T? ∘₂ r) deduplicate : ∀ {R : Rel A p} → B.Decidable R → List A → List A deduplicate R? [] = [] deduplicate R? (x ∷ xs) = x ∷ filter (¬? ∘ R? x) (deduplicate R? xs) deduplicateᵇ : (A → A → Bool) → List A → List A deduplicateᵇ r = deduplicate (T? ∘₂ r) -- Finds the first element satisfying the boolean predicate find : ∀ {P : Pred A p} → Decidable P → List A → Maybe A find P? [] = nothing find P? (x ∷ xs) = if does (P? x) then just x else find P? xs findᵇ : (A → Bool) → List A → Maybe A findᵇ p = find (T? ∘ p) -- Finds the index of the first element satisfying the boolean predicate findIndex : ∀ {P : Pred A p} → Decidable P → (xs : List A) → Maybe $ Fin (length xs) findIndex P? [] = nothing findIndex P? (x ∷ xs) = if does (P? x) then just zero else Maybe.map suc (findIndex P? xs) findIndexᵇ : (A → Bool) → (xs : List A) → Maybe $ Fin (length xs) findIndexᵇ p = findIndex (T? ∘ p) -- Finds indices of all the elements satisfying the boolean predicate findIndices : ∀ {P : Pred A p} → Decidable P → (xs : List A) → List $ Fin (length xs) findIndices P? [] = [] findIndices P? (x ∷ xs) = if does (P? x) then zero ∷ indices else indices where indices = map suc (findIndices P? xs) findIndicesᵇ : (A → Bool) → (xs : List A) → List $ Fin (length xs) findIndicesᵇ p = findIndices (T? ∘ p) ------------------------------------------------------------------------ -- Actions on single elements infixl 5 _[_]%=_ _[_]∷=_ -- xs [ i ]%= f modifies the i-th element of xs according to f _[_]%=_ : (xs : List A) → Fin (length xs) → (A → A) → List A xs [ i ]%= f = updateAt xs i f -- xs [ i ]≔ y overwrites the i-th element of xs with y _[_]∷=_ : (xs : List A) → Fin (length xs) → A → List A xs [ k ]∷= v = xs [ k ]%= const v ------------------------------------------------------------------------ -- Conditional versions of cons and snoc infixr 5 _?∷_ _?∷_ : Maybe A → List A → List A _?∷_ = maybe′ _∷_ id infixl 6 _∷ʳ?_ _∷ʳ?_ : List A → Maybe A → List A xs ∷ʳ? x = maybe′ (xs ∷ʳ_) xs x ------------------------------------------------------------------------ -- Raw algebraic bundles module _ (A : Set a) where ++-rawMagma : RawMagma a _ ++-rawMagma = record { Carrier = List A ; _≈_ = _≡_ ; _∙_ = _++_ } ++-[]-rawMonoid : RawMonoid a _ ++-[]-rawMonoid = record { Carrier = List A ; _≈_ = _≡_ ; _∙_ = _++_ ; ε = [] } ------------------------------------------------------------------------ -- DEPRECATED ------------------------------------------------------------------------ -- Please use the new names as continuing support for the old names is -- not guaranteed. -- Version 1.4 infixl 5 _∷ʳ'_ _∷ʳ'_ : (xs : List A) (x : A) → InitLast (xs ∷ʳ x) _∷ʳ'_ = InitLast._∷ʳ′_ {-# WARNING_ON_USAGE _∷ʳ'_ "Warning: _∷ʳ'_ (ending in an apostrophe) was deprecated in v1.4. Please use _∷ʳ′_ (ending in a prime) instead." #-} -- Version 2.0 infixl 5 _─_ _─_ = removeAt {-# WARNING_ON_USAGE _─_ "Warning: _─_ was deprecated in v2.0. Please use removeAt instead." #-} -- Version 2.1 scanr : (A → B → B) → B → List A → List B scanr f e [] = e ∷ [] scanr f e (x ∷ xs) with scanr f e xs ... | [] = [] -- dead branch ... | ys@(y ∷ _) = f x y ∷ ys {-# WARNING_ON_USAGE scanr "Warning: scanr was deprecated in v2.1. Please use Data.List.Scans.Base.scanr instead." #-} scanl : (A → B → A) → A → List B → List A scanl f e [] = e ∷ [] scanl f e (x ∷ xs) = e ∷ scanl f (f e x) xs {-# WARNING_ON_USAGE scanl "Warning: scanl was deprecated in v2.1. Please use Data.List.Scans.Base.scanl instead." #-}