```------------------------------------------------------------------------
-- Lists
------------------------------------------------------------------------

module Data.List where

open import Data.Nat
open import Data.Sum as Sum using (_⊎_; inj₁; inj₂)
open import Data.Bool
open import Data.Maybe using (Maybe; nothing; just)
open import Data.Product as Prod using (_×_; _,_)
open import Data.Function
open import Algebra
import Relation.Binary.PropositionalEquality as PropEq
import Algebra.FunctionProperties as FunProp

infixr 5 _∷_ _++_

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

data List (A : Set) : Set where
[]  : List A
_∷_ : (x : A) (xs : List A) → List A

{-# BUILTIN LIST List #-}
{-# BUILTIN NIL  []   #-}
{-# BUILTIN CONS _∷_  #-}

------------------------------------------------------------------------
-- Some operations

-- * Basic functions

[_] : ∀ {a} → a → List a
[ x ] = x ∷ []

_++_ : ∀ {a} → List a → List a → List a
[]       ++ ys = ys
(x ∷ xs) ++ ys = x ∷ (xs ++ ys)

null : ∀ {a} → List a → Bool
null []       = true
null (x ∷ xs) = false

-- * List transformations

map : ∀ {a b} → (a → b) → List a → List b
map f []       = []
map f (x ∷ xs) = f x ∷ map f xs

reverse : ∀ {a} → List a → List a
reverse xs = rev xs []
where
rev : ∀ {a} → List a → List a → List a
rev []       ys = ys
rev (x ∷ xs) ys = rev xs (x ∷ ys)

replicate : ∀ {a} → (n : ℕ) → a → List a
replicate zero    x = []
replicate (suc n) x = x ∷ replicate n x

zipWith : ∀ {A B C} → (A → B → C) → List A → List B → List C
zipWith f (x ∷ xs) (y ∷ ys) = f x y ∷ zipWith f xs ys
zipWith f _        _        = []

zip : ∀ {A B} → List A → List B → List (A × B)
zip = zipWith (_,_)

intersperse : ∀ {A} → A → List A → List A
intersperse x []           = []
intersperse x (y ∷ [])     = [ y ]
intersperse x (y ∷ z ∷ zs) = y ∷ x ∷ intersperse x (z ∷ zs)

-- * Reducing lists (folds)

foldr : {a b : Set} → (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 : Set} → (a → b → a) → a → List b → a
foldl c n []       = n
foldl c n (x ∷ xs) = foldl c (c n x) xs

-- ** Special folds

concat : ∀ {a} → List (List a) → List a
concat = foldr _++_ []

concatMap : ∀ {a b} → (a → List b) → List a → List b
concatMap f = concat ∘ map f

and : List Bool → Bool
and = foldr _∧_ true

or : List Bool → Bool
or = foldr _∨_ false

any : ∀ {a} → (a → Bool) → List a → Bool
any p = or ∘ map p

all : ∀ {a} → (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 : ∀ {a} → List a → ℕ
length = foldr (λ _ → suc) 0

-- * Building lists

-- ** Scans

scanr : ∀ {a b} → (a → b → b) → b → List a → List b
scanr f e []       = e ∷ []
scanr f e (x ∷ xs) with scanr f e xs
... | []     = []                -- dead branch
... | y ∷ ys = f x y ∷ y ∷ ys

scanl : ∀ {a b} → (a → b → a) → a → List b → List a
scanl f e []       = e ∷ []
scanl f e (x ∷ xs) = e ∷ scanl f (f e x) xs

-- ** Unfolding

-- Unfold. Uses a measure (a natural number) to ensure termination.

unfold : {A : Set} (B : ℕ → Set)
(f : ∀ {n} → B (suc n) → Maybe (A × B n)) →
∀ {n} → B n → List A
unfold B f {n = zero}  s = []
unfold B f {n = suc n} s with f s
... | nothing       = []
... | just (x , s') = x ∷ unfold B f s'

-- downFrom 3 = 2 ∷ 1 ∷ 0 ∷ [].

downFrom : ℕ → List ℕ
downFrom n = unfold Singleton f (wrap n)
where
data Singleton : ℕ → Set where
wrap : (n : ℕ) → Singleton n

f : ∀ {n} → Singleton (suc n) → Maybe (ℕ × Singleton n)
f {n} (wrap .(suc n)) = just (n , wrap n)

-- ** Conversions

fromMaybe : ∀ {A} → Maybe A → List A
fromMaybe (just x) = [ x ]
fromMaybe nothing  = []

-- * Sublists

-- ** Extracting sublists

take : ∀ {a} → ℕ → List a → List a
take zero    xs       = []
take (suc n) []       = []
take (suc n) (x ∷ xs) = x ∷ take n xs

drop : ∀ {a} → ℕ → List a → List a
drop zero    xs       = xs
drop (suc n) []       = []
drop (suc n) (x ∷ xs) = drop n xs

splitAt : ∀ {a} → ℕ → List a → (List a × List a)
splitAt zero    xs       = ([] , xs)
splitAt (suc n) []       = ([] , [])
splitAt (suc n) (x ∷ xs) with splitAt n xs
... | (ys , zs) = (x ∷ ys , zs)

takeWhile : ∀ {a} → (a → Bool) → List a → List a
takeWhile p []       = []
takeWhile p (x ∷ xs) with p x
... | true  = x ∷ takeWhile p xs
... | false = []

dropWhile : ∀ {a} → (a → Bool) → List a → List a
dropWhile p []       = []
dropWhile p (x ∷ xs) with p x
... | true  = dropWhile p xs
... | false = x ∷ xs

span : ∀ {a} → (a → Bool) → List a → (List a × List a)
span p []       = ([] , [])
span p (x ∷ xs) with p x
... | true  = Prod.map (_∷_ x) id (span p xs)
... | false = ([] , x ∷ xs)

break : ∀ {a} → (a → Bool) → List a → (List a × List a)
break p = span (not ∘ p)

inits : ∀ {a} →  List a → List (List a)
inits []       = [] ∷ []
inits (x ∷ xs) = [] ∷ map (_∷_ x) (inits xs)

tails : ∀ {a} → List a → List (List a)
tails []       = [] ∷ []
tails (x ∷ xs) = (x ∷ xs) ∷ tails xs

-- * Searching lists

-- ** Searching with a predicate

-- A generalised variant of filter.

gfilter : ∀ {a b} → (a → Maybe b) → List a → List b
gfilter p []       = []
gfilter p (x ∷ xs) with p x
... | just y  = y ∷ gfilter p xs
... | nothing =     gfilter p xs

filter : ∀ {a} → (a → Bool) → List a → List a
filter p = gfilter (λ x → if p x then just x else nothing)

partition : ∀ {a} → (a → Bool) → List a → (List a × List a)
partition p []       = ([] , [])
partition p (x ∷ xs) with p x | partition p xs
... | true  | (ys , zs) = (x ∷ ys , zs)
... | false | (ys , zs) = (ys , x ∷ zs)

------------------------------------------------------------------------
-- List monoid

monoid : Set → Monoid
monoid A = record
{ carrier  = List A
; _≈_      = _≡_
; _∙_      = _++_
; ε        = []
; isMonoid = record
{ isSemigroup = record
{ isEquivalence = PropEq.isEquivalence
; assoc         = assoc
; ∙-pres-≈      = cong₂ _++_
}
; identity = ((λ _ → refl) , identity)
}
}
where
open PropEq
open FunProp _≡_

identity : RightIdentity [] _++_
identity []       = refl
identity (x ∷ xs) = cong (_∷_ x) (identity xs)

assoc : Associative _++_
assoc []       ys zs = refl
assoc (x ∷ xs) ys zs = cong (_∷_ x) (assoc xs ys zs)

------------------------------------------------------------------------

{ return = λ x → x ∷ []
; _>>=_  = λ xs f → concat (map f xs)
}

; ∅     = []
}

; _∣_       = _++_
}

------------------------------------------------------------------------

private