------------------------------------------------------------------------
-- The Agda standard library
--
-- Lists, basic types and operations
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

module Data.List.Base where

open import Data.Bool.Base as Bool
  using (Bool; false; true; not; _∧_; _∨_; if_then_else_)
open import Data.Fin using (Fin; zero; suc)
open import Data.Maybe.Base as Maybe using (Maybe; nothing; just)
open import Data.Nat.Base as  using (; zero; suc; _+_; _*_ ; _≤_ ; s≤s)
open import Data.Product as Prod using (_×_; _,_)
open import Data.Sum as Sum using (_⊎_; inj₁; inj₂)
open import Data.These as These using (These; this; that; these)
open import Function using (id; _∘_ ; _∘′_; const; flip)
open import Level using (Level)
open import Relation.Nullary using (yes; no)
open import Relation.Unary using (Pred; Decidable)
open import Relation.Unary.Properties using (∁?)

private
  variable
    a b c p : 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

mapMaybe : (A  Maybe B)  List A  List B
mapMaybe p []       = []
mapMaybe p (x  xs) with p x
... | just y  = y  mapMaybe p xs
... | nothing =     mapMaybe p 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

------------------------------------------------------------------------
-- 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    = Prod.map₁ (b ∷_) (unalignWith f as)
... | that c    = Prod.map₂ (c ∷_) (unalignWith f as)
... | these b c = Prod.map (b ∷_) (c ∷_) (unalignWith f as)

unzipWith : (A  B × C)  List A  List B × List C
unzipWith f []         = [] , []
unzipWith f (xy  xys) = Prod.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

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

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   
sum = foldr _+_ 0

product : List   
product = foldr _*_ 1

length : List A  
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 :   A  List A
replicate zero    x = []
replicate (suc n) x = x  replicate n x

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

-- 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 : (  A)    List A
applyUpTo f zero    = []
applyUpTo f (suc n) = f zero  applyUpTo (f  suc) n

applyDownFrom : (  A)    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 :   List 
upTo = applyUpTo id

downFrom :   List 
downFrom = applyDownFrom id

allFin :  n  List (Fin n)
allFin n = tabulate id

-- Other

unfold :  (P :   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 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

take :   List A  List A
take zero    xs       = []
take (suc n) []       = []
take (suc n) (x  xs) = x  take n xs

drop :   List A  List A
drop zero    xs       = xs
drop (suc n) []       = []
drop (suc n) (x  xs) = drop n xs

splitAt :   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 :  {P : Pred A p}  Decidable P  List A  List A
takeWhile P? []       = []
takeWhile P? (x  xs) with P? x
... | yes _ = x  takeWhile P? xs
... | no  _ = []

dropWhile :  {P : Pred A p}  Decidable P  List A  List A
dropWhile P? []       = []
dropWhile P? (x  xs) with P? x
... | yes _ = dropWhile P? xs
... | no  _ = x  xs

filter :  {P : Pred A p}  Decidable P  List A  List A
filter P? [] = []
filter P? (x  xs) with P? x
... | no  _ = filter P? xs
... | yes _ = x  filter P? xs

partition :  {P : Pred A p}  Decidable P  List A  (List A × List A)
partition P? []       = ([] , [])
partition P? (x  xs) with P? x | partition P? xs
... | yes _ | (ys , zs) = (x  ys , zs)
... | no  _ | (ys , zs) = (ys , x  zs)

span :  {P : Pred A p}  Decidable P  List A  (List A × List A)
span P? []       = ([] , [])
span P? (x  xs) with P? x
... | yes _ = Prod.map (x ∷_) id (span P? xs)
... | no  _ = ([] , x  xs)

break :  {P : Pred A p}  Decidable P  List A  (List A × List A)
break P? = span (∁? P?)

------------------------------------------------------------------------
-- Actions on single elements

infixl 5 _[_]%=_ _[_]∷=_ _─_

_[_]%=_ : (xs : List A)  Fin (length xs)  (A  A)  List A
(x  xs) [ zero  ]%= f = f x  xs
(x  xs) [ suc k ]%= f = x  (xs [ k ]%= f)

_[_]∷=_ : (xs : List A)  Fin (length xs)  A  List A
xs [ k ]∷= v = xs [ k ]%= const v

_─_ : (xs : List A)  Fin (length xs)  List A
(x  xs)  zero  = xs
(x  xs)  suc k = x  (xs  k)

------------------------------------------------------------------------
-- Operations for reversing lists

reverseAcc : List A  List A  List A
reverseAcc = foldl (flip _∷_)

reverse : List A  List A
reverse = reverseAcc []

-- Snoc.

infixl 5 _∷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

------------------------------------------------------------------------
-- DEPRECATED
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.
--
-- Note that the `boolX` functions are not given warnings as they are
-- used by other deprecated proofs throughout the library.

-- Version 0.15

gfilter = mapMaybe
{-# WARNING_ON_USAGE gfilter
"Warning: gfilter was deprecated in v0.15.
Please use mapMaybe instead."
#-}

boolFilter : (A  Bool)  List A  List A
boolFilter p = mapMaybe  x  if p x then just x else nothing)

boolPartition : (A  Bool)  List A  (List A × List A)
boolPartition p []       = ([] , [])
boolPartition p (x  xs) with p x | boolPartition p xs
... | true  | (ys , zs) = (x  ys , zs)
... | false | (ys , zs) = (ys , x  zs)

-- Version 0.16

boolTakeWhile : (A  Bool)  List A  List A
boolTakeWhile p []       = []
boolTakeWhile p (x  xs) with p x
... | true  = x  boolTakeWhile p xs
... | false = []

boolDropWhile : (A  Bool)  List A  List A
boolDropWhile p []       = []
boolDropWhile p (x  xs) with p x
... | true  = boolDropWhile p xs
... | false = x  xs

boolSpan : (A  Bool)  List A  (List A × List A)
boolSpan p []       = ([] , [])
boolSpan p (x  xs) with p x
... | true  = Prod.map (x ∷_) id (boolSpan p xs)
... | false = ([] , x  xs)

boolBreak : (A  Bool)  List A  (List A × List A)
boolBreak p = boolSpan (not  p)