Base.agda

------------------------------------------------------------------------
-- 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 []       = [] ∷ []
inits (x ∷ xs) = [] ∷ map (x ∷_) (inits xs)

tails : List A → List (List A)
tails []       = [] ∷ []
tails (x ∷ xs) = (x ∷ xs) ∷ tails 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

-- Scans

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
... | y ∷ ys = f x y ∷ y ∷ ys

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

-- 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 with f s
... | nothing       = []
... | just (x , s′) = x ∷ unfold P 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) with does (P? x)
... | true  = x ∷ takeWhile P? xs
... | false = []

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) with does (P? x)
... | true  = dropWhile P? xs
... | false = 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) with does (P? x)
... | false = filter P? xs
... | true  = x ∷ filter P? 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) with does (P? x) | partition P? xs
... | true  | (ys , zs) = (x ∷ ys , zs)
... | false | (ys , zs) = (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) with does (P? x)
... | true  = Product.map (x ∷_) id (span P? xs)
... | false = ([] , 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) with does (P? c)
  ... | true  = reverse (Maybe.fromMaybe [] acc) ∷ go nothing cs
  ... | false = go (just (c ∷ Maybe.fromMaybe [] acc)) cs

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) with does (P? c)
  ... | true  = cons acc (go [] cs)
  ... | false = 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 ∷ _)) with does (R? x y) | derun R? xs
... | true  | ys = ys
... | false | ys = 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."
#-}