------------------------------------------------------------------------
-- The Agda standard library
--
-- List-related properties
------------------------------------------------------------------------

-- Note that the lemmas below could be generalised to work with other
-- equalities than _≡_.

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

module Data.List.Properties where

open import Algebra.Bundles
open import Algebra.Definitions as AlgebraicDefinitions using (Involutive)
import Algebra.Structures as AlgebraicStructures
open import Data.Bool.Base using (Bool; false; true; not; if_then_else_)
open import Data.Fin.Base using (Fin; zero; suc; cast; toℕ; inject₁)
open import Data.List.Base as List
open import Data.List.Relation.Unary.All using (All; []; _∷_)
open import Data.List.Relation.Unary.Any using (Any; here; there)
open import Data.Maybe.Base using (Maybe; just; nothing)
open import Data.Nat.Base
open import Data.Nat.Properties
open import Data.Product as Prod hiding (map; zip)
import Data.Product.Relation.Unary.All as Prod using (All)
open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
open import Data.These.Base as These using (These; this; that; these)
open import Function
open import Level using (Level)
open import Relation.Binary as B using (DecidableEquality)
import Relation.Binary.Reasoning.Setoid as EqR
open import Relation.Binary.PropositionalEquality as P hiding ([_])
open import Relation.Binary as B using (Rel)
open import Relation.Nullary.Reflects using (invert)
open import Relation.Nullary using (¬_; Dec; does; _because_; yes; no)
open import Relation.Nullary.Negation using (contradiction; ¬?)
open import Relation.Nullary.Decidable as Decidable using (isYes; map′; ⌊_⌋)
open import Relation.Nullary.Product using (_×-dec_)
open import Relation.Unary using (Pred; Decidable; )
open import Relation.Unary.Properties using (∁?)

open ≡-Reasoning

private
  variable
    a b c d e p : Level
    A : Set a
    B : Set b
    C : Set c
    D : Set d
    E : Set e

-----------------------------------------------------------------------
-- _∷_

module _ {x y : A} {xs ys : List A} where

  ∷-injective : x  xs  y List.∷ ys  x  y × xs  ys
  ∷-injective refl = (refl , refl)

  ∷-injectiveˡ : x  xs  y List.∷ ys  x  y
  ∷-injectiveˡ refl = refl

  ∷-injectiveʳ : x  xs  y List.∷ ys  xs  ys
  ∷-injectiveʳ refl = refl

  ∷-dec : Dec (x  y)  Dec (xs  ys)  Dec (x List.∷ xs  y  ys)
  ∷-dec x≟y xs≟ys = Decidable.map′ (uncurry (cong₂ _∷_)) ∷-injective (x≟y ×-dec xs≟ys)

≡-dec : DecidableEquality A  DecidableEquality (List A)
≡-dec _≟_ []       []       = yes refl
≡-dec _≟_ (x  xs) []       = no λ()
≡-dec _≟_ []       (y  ys) = no λ()
≡-dec _≟_ (x  xs) (y  ys) = ∷-dec (x  y) (≡-dec _≟_ xs ys)

------------------------------------------------------------------------
-- map

map-id : map id  id {A = List A}
map-id []       = refl
map-id (x  xs) = cong (x ∷_) (map-id xs)

map-id₂ :  {f : A  A} {xs}  All  x  f x  x) xs  map f xs  xs
map-id₂ []           = refl
map-id₂ (fx≡x  pxs) = cong₂ _∷_ fx≡x (map-id₂ pxs)

map-++-commute :  (f : A  B) xs ys 
                 map f (xs ++ ys)  map f xs ++ map f ys
map-++-commute f []       ys = refl
map-++-commute f (x  xs) ys = cong (f x ∷_) (map-++-commute f xs ys)

map-cong :  {f g : A  B}  f  g  map f  map g
map-cong f≗g []       = refl
map-cong f≗g (x  xs) = cong₂ _∷_ (f≗g x) (map-cong f≗g xs)

map-cong₂ :  {f g : A  B} {xs} 
            All  x  f x  g x) xs  map f xs  map g xs
map-cong₂ []                = refl
map-cong₂ (fx≡gx  fxs≡gxs) = cong₂ _∷_ fx≡gx (map-cong₂ fxs≡gxs)

length-map :  (f : A  B) xs  length (map f xs)  length xs
length-map f []       = refl
length-map f (x  xs) = cong suc (length-map f xs)

map-compose : {g : B  C} {f : A  B}  map (g  f)  map g  map f
map-compose []       = refl
map-compose (x  xs) = cong (_ ∷_) (map-compose xs)

map-injective :  {f : A  B}  Injective _≡_ _≡_ f  Injective _≡_ _≡_ (map f)
map-injective finj {[]} {[]} eq = refl
map-injective finj {x  xs} {y  ys} eq =
  let fx≡fy , fxs≡fys = ∷-injective eq in
  cong₂ _∷_ (finj fx≡fy) (map-injective finj fxs≡fys)

------------------------------------------------------------------------
-- mapMaybe

mapMaybe-just : (xs : List A)  mapMaybe just xs  xs
mapMaybe-just []       = refl
mapMaybe-just (x  xs) = cong (x ∷_) (mapMaybe-just xs)

mapMaybe-nothing : (xs : List A) 
                   mapMaybe {B = A}  _  nothing) xs  []
mapMaybe-nothing []       = refl
mapMaybe-nothing (x  xs) = mapMaybe-nothing xs

module _ (f : A  Maybe B) where

  mapMaybe-concatMap : mapMaybe f  concatMap (fromMaybe  f)
  mapMaybe-concatMap [] = refl
  mapMaybe-concatMap (x  xs) with f x
  ... | just y  = cong (y ∷_) (mapMaybe-concatMap xs)
  ... | nothing = mapMaybe-concatMap xs

  length-mapMaybe :  xs  length (mapMaybe f xs)  length xs
  length-mapMaybe []       = z≤n
  length-mapMaybe (x  xs) with f x
  ... | just y  = s≤s (length-mapMaybe xs)
  ... | nothing = ≤-step (length-mapMaybe xs)

------------------------------------------------------------------------
-- _++_

length-++ :  (xs : List A) {ys} 
            length (xs ++ ys)  length xs + length ys
length-++ []       = refl
length-++ (x  xs) = cong suc (length-++ xs)

module _ {A : Set a} where

  open AlgebraicDefinitions {A = List A} _≡_
  open AlgebraicStructures  {A = List A} _≡_

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

  ++-identityˡ : LeftIdentity [] _++_
  ++-identityˡ xs = refl

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

  ++-identity : Identity [] _++_
  ++-identity = ++-identityˡ , ++-identityʳ

  ++-identityʳ-unique :  (xs : List A) {ys}  xs  xs ++ ys  ys  []
  ++-identityʳ-unique []       refl = refl
  ++-identityʳ-unique (x  xs) eq   =
    ++-identityʳ-unique xs (proj₂ (∷-injective eq))

  ++-identityˡ-unique :  {xs} (ys : List A)  xs  ys ++ xs  ys  []
  ++-identityˡ-unique               []       _  = refl
  ++-identityˡ-unique {xs = x  xs} (y  ys) eq
    with ++-identityˡ-unique (ys ++ [ x ]) (begin
         xs                  ≡⟨ proj₂ (∷-injective eq) 
         ys ++ x  xs        ≡⟨ sym (++-assoc ys [ x ] xs) 
         (ys ++ [ x ]) ++ xs )
  ++-identityˡ-unique {xs = x  xs} (y  []   ) eq | ()
  ++-identityˡ-unique {xs = x  xs} (y  _  _) eq | ()

  ++-cancelˡ :  xs {ys zs : List A}  xs ++ ys  xs ++ zs  ys  zs
  ++-cancelˡ []       ys≡zs             = ys≡zs
  ++-cancelˡ (x  xs) x∷xs++ys≡x∷xs++zs = ++-cancelˡ xs (∷-injectiveʳ x∷xs++ys≡x∷xs++zs)

  ++-cancelʳ :  {xs : List A} ys zs  ys ++ xs  zs ++ xs  ys  zs
  ++-cancelʳ {_}  []       []       _             = refl
  ++-cancelʳ {xs} []       (z  zs) eq =
    contradiction (trans (cong length eq) (length-++ (z  zs))) (m≢1+n+m (length xs))
  ++-cancelʳ {xs} (y  ys) []       eq =
    contradiction (trans (sym (length-++ (y  ys))) (cong length eq)) (m≢1+n+m (length xs)  sym)
  ++-cancelʳ {_}  (y  ys) (z  zs) eq =
    cong₂ _∷_ (∷-injectiveˡ eq) (++-cancelʳ ys zs (∷-injectiveʳ eq))

  ++-cancel : Cancellative _++_
  ++-cancel = ++-cancelˡ , ++-cancelʳ

  ++-conicalˡ :  (xs ys : List A)  xs ++ ys  []  xs  []
  ++-conicalˡ []       _ refl = refl

  ++-conicalʳ :  (xs ys : List A)  xs ++ ys  []  ys  []
  ++-conicalʳ []       _ refl = refl

  ++-conical : Conical [] _++_
  ++-conical = ++-conicalˡ , ++-conicalʳ

  ++-isMagma : IsMagma _++_
  ++-isMagma = record
    { isEquivalence = isEquivalence
    ; ∙-cong        = cong₂ _++_
    }

  ++-isSemigroup : IsSemigroup _++_
  ++-isSemigroup = record
    { isMagma = ++-isMagma
    ; assoc   = ++-assoc
    }

  ++-isMonoid : IsMonoid _++_ []
  ++-isMonoid = record
    { isSemigroup = ++-isSemigroup
    ; identity    = ++-identity
    }

module _ (A : Set a) where

  ++-semigroup : Semigroup a a
  ++-semigroup = record
    { Carrier     = List A
    ; isSemigroup = ++-isSemigroup
    }

  ++-monoid : Monoid a a
  ++-monoid = record
    { Carrier  = List A
    ; isMonoid = ++-isMonoid
    }

------------------------------------------------------------------------
-- alignWith

module _ {f g : These A B  C} where

  alignWith-cong : f  g   as  alignWith f as  alignWith g as
  alignWith-cong f≗g []         bs       = map-cong (f≗g  that) bs
  alignWith-cong f≗g as@(_  _) []       = map-cong (f≗g  this) as
  alignWith-cong f≗g (a  as)   (b  bs) =
    cong₂ _∷_ (f≗g (these a b)) (alignWith-cong f≗g as bs)

  length-alignWith :  xs ys 
                   length (alignWith f xs ys)  length xs  length ys
  length-alignWith []         ys       = length-map (f ∘′ that) ys
  length-alignWith xs@(_  _) []       = length-map (f ∘′ this) xs
  length-alignWith (x  xs)   (y  ys) = cong suc (length-alignWith xs ys)

  alignWith-map : (g : D  A) (h : E  B) 
                   xs ys  alignWith f (map g xs) (map h ys) 
                            alignWith (f ∘′ These.map g h) xs ys
  alignWith-map g h []         ys     = sym (map-compose ys)
  alignWith-map g h xs@(_  _) []     = sym (map-compose xs)
  alignWith-map g h (x  xs) (y  ys) =
    cong₂ _∷_ refl (alignWith-map g h xs ys)

  map-alignWith :  (g : C  D)   xs ys 
                  map g (alignWith f xs ys) 
                  alignWith (g ∘′ f) xs ys
  map-alignWith g []         ys     = sym (map-compose ys)
  map-alignWith g xs@(_  _) []     = sym (map-compose xs)
  map-alignWith g (x  xs) (y  ys) =
    cong₂ _∷_ refl (map-alignWith g xs ys)

------------------------------------------------------------------------
-- zipWith

module _ (f : A  A  B) where

  zipWith-comm : (∀ x y  f x y  f y x) 
                  xs ys  zipWith f xs ys  zipWith f ys xs
  zipWith-comm f-comm []       []       = refl
  zipWith-comm f-comm []       (x  ys) = refl
  zipWith-comm f-comm (x  xs) []       = refl
  zipWith-comm f-comm (x  xs) (y  ys) =
    cong₂ _∷_ (f-comm x y) (zipWith-comm f-comm xs ys)

module _ (f : A  B  C) where

  zipWith-zeroˡ :  xs  zipWith f [] xs  []
  zipWith-zeroˡ []       = refl
  zipWith-zeroˡ (x  xs) = refl

  zipWith-zeroʳ :  xs  zipWith f xs []  []
  zipWith-zeroʳ []       = refl
  zipWith-zeroʳ (x  xs) = refl

  length-zipWith :  xs ys 
                   length (zipWith f xs ys)  length xs  length ys
  length-zipWith []       []       = refl
  length-zipWith []       (y  ys) = refl
  length-zipWith (x  xs) []       = refl
  length-zipWith (x  xs) (y  ys) = cong suc (length-zipWith xs ys)

  zipWith-map :  {d e} {D : Set d} {E : Set e} (g : D  A) (h : E  B) 
                 xs ys  zipWith f (map g xs) (map h ys) 
                          zipWith  x y  f (g x) (h y)) xs ys
  zipWith-map g h []       []       = refl
  zipWith-map g h []       (y  ys) = refl
  zipWith-map g h (x  xs) []       = refl
  zipWith-map g h (x  xs) (y  ys) =
    cong₂ _∷_ refl (zipWith-map g h xs ys)

  map-zipWith :  {d} {D : Set d} (g : C  D)   xs ys 
                map g (zipWith f xs ys) 
                zipWith  x y  g (f x y)) xs ys
  map-zipWith g []       []       = refl
  map-zipWith g []       (y  ys) = refl
  map-zipWith g (x  xs) []       = refl
  map-zipWith g (x  xs) (y  ys) =
    cong₂ _∷_ refl (map-zipWith g xs ys)

------------------------------------------------------------------------
-- unalignWith

unalignWith-this : unalignWith ((A  These A B)  this)  (_, [])
unalignWith-this []       = refl
unalignWith-this (a  as) = cong (Prod.map₁ (a ∷_)) (unalignWith-this as)

unalignWith-that : unalignWith ((B  These A B)  that)  ([] ,_)
unalignWith-that []       = refl
unalignWith-that (b  bs) = cong (Prod.map₂ (b ∷_)) (unalignWith-that bs)

module _ {f g : C  These A B} where

  unalignWith-cong : f  g  unalignWith f  unalignWith g
  unalignWith-cong f≗g []       = refl
  unalignWith-cong f≗g (c  cs) with f c | g c | f≗g c
  ... | this a    | ._ | refl = cong (Prod.map₁ (a ∷_)) (unalignWith-cong f≗g cs)
  ... | that b    | ._ | refl = cong (Prod.map₂ (b ∷_)) (unalignWith-cong f≗g cs)
  ... | these a b | ._ | refl = cong (Prod.map (a ∷_) (b ∷_)) (unalignWith-cong f≗g cs)

module _ (f : C  These A B) where

  unalignWith-map : (g : D  C)   ds 
                    unalignWith f (map g ds)  unalignWith (f ∘′ g) ds
  unalignWith-map g []       = refl
  unalignWith-map g (d  ds) with f (g d)
  ... | this a    = cong (Prod.map₁ (a ∷_)) (unalignWith-map g ds)
  ... | that b    = cong (Prod.map₂ (b ∷_)) (unalignWith-map g ds)
  ... | these a b = cong (Prod.map (a ∷_) (b ∷_)) (unalignWith-map g ds)

  map-unalignWith : (g : A  D) (h : B  E) 
    Prod.map (map g) (map h) ∘′ unalignWith f  unalignWith (These.map g h ∘′ f)
  map-unalignWith g h []       = refl
  map-unalignWith g h (c  cs) with f c
  ... | this a    = cong (Prod.map₁ (g a ∷_)) (map-unalignWith g h cs)
  ... | that b    = cong (Prod.map₂ (h b ∷_)) (map-unalignWith g h cs)
  ... | these a b = cong (Prod.map (g a ∷_) (h b ∷_)) (map-unalignWith g h cs)

  unalignWith-alignWith : (g : These A B  C)  f ∘′ g  id   as bs 
                          unalignWith f (alignWith g as bs)  (as , bs)
  unalignWith-alignWith g g∘f≗id []         bs = begin
    unalignWith f (map (g ∘′ that) bs) ≡⟨ unalignWith-map (g ∘′ that) bs 
    unalignWith (f ∘′ g ∘′ that) bs    ≡⟨ unalignWith-cong (g∘f≗id  that) bs 
    unalignWith that bs                ≡⟨ unalignWith-that bs 
    [] , bs                            
  unalignWith-alignWith g g∘f≗id as@(_  _) [] = begin
    unalignWith f (map (g ∘′ this) as) ≡⟨ unalignWith-map (g ∘′ this) as 
    unalignWith (f ∘′ g ∘′ this) as    ≡⟨ unalignWith-cong (g∘f≗id  this) as 
    unalignWith this as                ≡⟨ unalignWith-this as 
    as , []                            
  unalignWith-alignWith g g∘f≗id (a  as)   (b  bs)
    rewrite g∘f≗id (these a b) =
    cong (Prod.map (a ∷_) (b ∷_)) (unalignWith-alignWith g g∘f≗id as bs)

------------------------------------------------------------------------
-- unzipWith

module _ (f : A  B × C) where

  length-unzipWith₁ :  xys 
                     length (proj₁ (unzipWith f xys))  length xys
  length-unzipWith₁ []        = refl
  length-unzipWith₁ (x  xys) = cong suc (length-unzipWith₁ xys)

  length-unzipWith₂ :  xys 
                     length (proj₂ (unzipWith f xys))  length xys
  length-unzipWith₂ []        = refl
  length-unzipWith₂ (x  xys) = cong suc (length-unzipWith₂ xys)

  zipWith-unzipWith : (g : B  C  A)  uncurry′ g  f  id 
                      uncurry′ (zipWith g)  (unzipWith f)   id
  zipWith-unzipWith g f∘g≗id []       = refl
  zipWith-unzipWith g f∘g≗id (x  xs) =
    cong₂ _∷_ (f∘g≗id x) (zipWith-unzipWith g f∘g≗id xs)

------------------------------------------------------------------------
-- foldr

foldr-universal :  (h : List A  B) f e  (h []  e) 
                  (∀ x xs  h (x  xs)  f x (h xs)) 
                  h  foldr f e
foldr-universal h f e base step []       = base
foldr-universal h f e base step (x  xs) = begin
  h (x  xs)          ≡⟨ step x xs 
  f x (h xs)          ≡⟨ cong (f x) (foldr-universal h f e base step xs) 
  f x (foldr f e xs)  

foldr-cong :  {f g : A  B  B} {d e : B} 
             (∀ x y  f x y  g x y)  d  e 
             foldr f d  foldr g e
foldr-cong f≗g refl []      = refl
foldr-cong f≗g d≡e (x  xs) rewrite foldr-cong f≗g d≡e xs = f≗g x _

foldr-fusion :  (h : B  C) {f : A  B  B} {g : A  C  C} (e : B) 
               (∀ x y  h (f x y)  g x (h y)) 
               h  foldr f e  foldr g (h e)
foldr-fusion h {f} {g} e fuse =
  foldr-universal (h  foldr f e) g (h e) refl
                   x xs  fuse x (foldr f e xs))

id-is-foldr : id {A = List A}  foldr _∷_ []
id-is-foldr = foldr-universal id _∷_ [] refl  _ _  refl)

++-is-foldr : (xs ys : List A)  xs ++ ys  foldr _∷_ ys xs
++-is-foldr xs ys = begin
  xs ++ ys                ≡⟨ cong (_++ ys) (id-is-foldr xs) 
  foldr _∷_ [] xs ++ ys   ≡⟨ foldr-fusion (_++ ys) []  _ _  refl) xs 
  foldr _∷_ ([] ++ ys) xs ≡⟨⟩
  foldr _∷_ ys xs         

foldr-++ :  (f : A  B  B) x ys zs 
           foldr f x (ys ++ zs)  foldr f (foldr f x zs) ys
foldr-++ f x []       zs = refl
foldr-++ f x (y  ys) zs = cong (f y) (foldr-++ f x ys zs)

map-is-foldr : {f : A  B}  map f  foldr  x ys  f x  ys) []
map-is-foldr {f = f} xs = begin
  map f xs                        ≡⟨ cong (map f) (id-is-foldr xs) 
  map f (foldr _∷_ [] xs)         ≡⟨ foldr-fusion (map f) []  _ _  refl) xs 
  foldr  x ys  f x  ys) [] xs 

foldr-∷ʳ :  (f : A  B  B) x y ys 
           foldr f x (ys ∷ʳ y)  foldr f (f y x) ys
foldr-∷ʳ f x y []       = refl
foldr-∷ʳ f x y (z  ys) = cong (f z) (foldr-∷ʳ f x y ys)

-- Interaction with predicates

module _ {P : Pred A p} {f : A  A  A} where

  foldr-forcesᵇ : (∀ x y  P (f x y)  P x × P y) 
                   e xs  P (foldr f e xs)  All P xs
  foldr-forcesᵇ _      _ []       _     = []
  foldr-forcesᵇ forces _ (x  xs) Pfold with forces _ _ Pfold
  ... | (px , pfxs) = px  foldr-forcesᵇ forces _ xs pfxs

  foldr-preservesᵇ : (∀ {x y}  P x  P y  P (f x y)) 
                      {e xs}  P e  All P xs  P (foldr f e xs)
  foldr-preservesᵇ _    Pe []         = Pe
  foldr-preservesᵇ pres Pe (px  pxs) = pres px (foldr-preservesᵇ pres Pe pxs)

  foldr-preservesʳ : (∀ x {y}  P y  P (f x y)) 
                      {e}  P e   xs  P (foldr f e xs)
  foldr-preservesʳ pres Pe []       = Pe
  foldr-preservesʳ pres Pe (_  xs) = pres _ (foldr-preservesʳ pres Pe xs)

  foldr-preservesᵒ : (∀ x y  P x  P y  P (f x y)) 
                      e xs  P e  Any P xs  P (foldr f e xs)
  foldr-preservesᵒ pres e []       (inj₁ Pe)          = Pe
  foldr-preservesᵒ pres e (x  xs) (inj₁ Pe)          =
    pres _ _ (inj₂ (foldr-preservesᵒ pres e xs (inj₁ Pe)))
  foldr-preservesᵒ pres e (x  xs) (inj₂ (here px))   = pres _ _ (inj₁ px)
  foldr-preservesᵒ pres e (x  xs) (inj₂ (there pxs)) =
    pres _ _ (inj₂ (foldr-preservesᵒ pres e xs (inj₂ pxs)))

------------------------------------------------------------------------
-- foldl

foldl-++ :  (f : A  B  A) x ys zs 
           foldl f x (ys ++ zs)  foldl f (foldl f x ys) zs
foldl-++ f x []       zs = refl
foldl-++ f x (y  ys) zs = foldl-++ f (f x y) ys zs

foldl-∷ʳ :  (f : A  B  A) x y ys 
           foldl f x (ys ∷ʳ y)  f (foldl f x ys) y
foldl-∷ʳ f x y []       = refl
foldl-∷ʳ f x y (z  ys) = foldl-∷ʳ f (f x z) y ys

------------------------------------------------------------------------
-- concat

concat-map :  {f : A  B}  concat  map (map f)  map f  concat
concat-map {f = f} xss = begin
  concat (map (map f) xss)                   ≡⟨ cong concat (map-is-foldr xss) 
  concat (foldr  xs  map f xs ∷_) [] xss) ≡⟨ foldr-fusion concat []  _ _  refl) xss 
  foldr  ys  map f ys ++_) [] xss         ≡⟨ sym (foldr-fusion (map f) [] (map-++-commute f) xss) 
  map f (concat xss)                         

concat-++ : (xss yss : List (List A))  concat xss ++ concat yss  concat (xss ++ yss)
concat-++ [] yss = refl
concat-++ ([]  xss) yss = concat-++ xss yss
concat-++ ((x  xs)  xss) yss = cong (x ∷_) (concat-++ (xs  xss) yss)

concat-concat : concat {A = A}  map concat  concat  concat
concat-concat [] = refl
concat-concat (xss  xsss) = begin
  concat (map concat (xss  xsss))   ≡⟨ cong (concat xss ++_) (concat-concat xsss) 
  concat xss ++ concat (concat xsss) ≡⟨ concat-++ xss (concat xsss) 
  concat (concat (xss  xsss))       

concat-[-] : concat {A = A}  map [_]  id
concat-[-] [] = refl
concat-[-] (x  xs) = cong (x ∷_) (concat-[-] xs)

------------------------------------------------------------------------
-- sum

sum-++-commute :  xs ys  sum (xs ++ ys)  sum xs + sum ys
sum-++-commute []       ys = refl
sum-++-commute (x  xs) ys = begin
  x + sum (xs ++ ys)     ≡⟨ cong (x +_) (sum-++-commute xs ys) 
  x + (sum xs + sum ys)  ≡⟨ sym (+-assoc x _ _) 
  (x + sum xs) + sum ys  

------------------------------------------------------------------------
-- replicate

length-replicate :  n {x : A}  length (replicate n x)  n
length-replicate zero    = refl
length-replicate (suc n) = cong suc (length-replicate n)

------------------------------------------------------------------------
-- scanr

scanr-defn :  (f : A  B  B) (e : B) 
             scanr f e  map (foldr f e)  tails
scanr-defn f e []             = refl
scanr-defn f e (x  [])       = refl
scanr-defn f e (x  y  xs)
  with scanr f e (y  xs) | scanr-defn f e (y  xs)
... | []     | ()
... | z  zs | eq with ∷-injective eq
...   | z≡fy⦇f⦈xs , _ = cong₂  z  f x z ∷_) z≡fy⦇f⦈xs eq

------------------------------------------------------------------------
-- scanl

scanl-defn :  (f : A  B  A) (e : A) 
             scanl f e  map (foldl f e)  inits
scanl-defn f e []       = refl
scanl-defn f e (x  xs) = cong (e ∷_) (begin
   scanl f (f e x) xs
 ≡⟨ scanl-defn f (f e x) xs 
   map (foldl f (f e x)) (inits xs)
 ≡⟨ refl 
   map (foldl f e  (x ∷_)) (inits xs)
 ≡⟨ map-compose (inits xs) 
   map (foldl f e) (map (x ∷_) (inits xs))
 )

------------------------------------------------------------------------
-- applyUpTo

length-applyUpTo :  (f :   A) n  length (applyUpTo f n)  n
length-applyUpTo f zero    = refl
length-applyUpTo f (suc n) = cong suc (length-applyUpTo (f  suc) n)

lookup-applyUpTo :  (f :   A) n i  lookup (applyUpTo f n) i  f (toℕ i)
lookup-applyUpTo f (suc n) zero    = refl
lookup-applyUpTo f (suc n) (suc i) = lookup-applyUpTo (f  suc) n i

------------------------------------------------------------------------
-- applyUpTo

module _ (f :   A) where

  length-applyDownFrom :  n  length (applyDownFrom f n)  n
  length-applyDownFrom zero    = refl
  length-applyDownFrom (suc n) = cong suc (length-applyDownFrom n)

  lookup-applyDownFrom :  n i  lookup (applyDownFrom f n) i  f (n  (suc (toℕ i)))
  lookup-applyDownFrom (suc n) zero    = refl
  lookup-applyDownFrom (suc n) (suc i) = lookup-applyDownFrom n i

------------------------------------------------------------------------
-- upTo

length-upTo :  n  length (upTo n)  n
length-upTo = length-applyUpTo id

lookup-upTo :  n i  lookup (upTo n) i  toℕ i
lookup-upTo = lookup-applyUpTo id

------------------------------------------------------------------------
-- downFrom

length-downFrom :  n  length (downFrom n)  n
length-downFrom = length-applyDownFrom id

lookup-downFrom :  n i  lookup (downFrom n) i  n  (suc (toℕ i))
lookup-downFrom = lookup-applyDownFrom id

------------------------------------------------------------------------
-- tabulate

tabulate-cong :  {n} {f g : Fin n  A} 
                f  g  tabulate f  tabulate g
tabulate-cong {n = zero}  p = refl
tabulate-cong {n = suc n} p = cong₂ _∷_ (p zero) (tabulate-cong (p  suc))

tabulate-lookup :  (xs : List A)  tabulate (lookup xs)  xs
tabulate-lookup []       = refl
tabulate-lookup (x  xs) = cong (_ ∷_) (tabulate-lookup xs)

length-tabulate :  {n}  (f : Fin n  A) 
                  length (tabulate f)  n
length-tabulate {n = zero} f = refl
length-tabulate {n = suc n} f = cong suc (length-tabulate  z  f (suc z)))

lookup-tabulate :  {n}  (f : Fin n  A) 
                   i  let i′ = cast (sym (length-tabulate f)) i
                        in lookup (tabulate f) i′  f i
lookup-tabulate f zero    = refl
lookup-tabulate f (suc i) = lookup-tabulate (f  suc) i

map-tabulate :  {n} (g : Fin n  A) (f : A  B) 
               map f (tabulate g)  tabulate (f  g)
map-tabulate {n = zero}  g f = refl
map-tabulate {n = suc n} g f = cong (_ ∷_) (map-tabulate (g  suc) f)

------------------------------------------------------------------------
-- _[_]%=_

length-%= :  xs k (f : A  A)  length (xs [ k ]%= f)  length xs
length-%= (x  xs) zero    f = refl
length-%= (x  xs) (suc k) f = cong suc (length-%= xs k f)

------------------------------------------------------------------------
-- _[_]∷=_

length-∷= :  xs k (v : A)  length (xs [ k ]∷= v)  length xs
length-∷= xs k v = length-%= xs k (const v)

map-∷= :  xs k (v : A) (f : A  B) 
         let eq = sym (length-map f xs) in
         map f (xs [ k ]∷= v)  map f xs [ cast eq k ]∷= f v
map-∷= (x  xs) zero    v f = refl
map-∷= (x  xs) (suc k) v f = cong (f x ∷_) (map-∷= xs k v f)

------------------------------------------------------------------------
-- _─_

length-─ :  (xs : List A) k  length (xs  k)  pred (length xs)
length-─ (x  xs) zero        = refl
length-─ (x  y  xs) (suc k) = cong suc (length-─ (y  xs) k)

map-─ :  xs k (f : A  B) 
        let eq = sym (length-map f xs) in
        map f (xs  k)  map f xs  cast eq k
map-─ (x  xs) zero    f = refl
map-─ (x  xs) (suc k) f = cong (f x ∷_) (map-─ xs k f)

------------------------------------------------------------------------
-- take

length-take :  n (xs : List A)  length (take n xs)  n  (length xs)
length-take zero    xs       = refl
length-take (suc n) []       = refl
length-take (suc n) (x  xs) = cong suc (length-take n xs)

------------------------------------------------------------------------
-- drop

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

take++drop :  n (xs : List A)  take n xs ++ drop n xs  xs
take++drop zero    xs       = refl
take++drop (suc n) []       = refl
take++drop (suc n) (x  xs) = cong (x ∷_) (take++drop n xs)

------------------------------------------------------------------------
-- splitAt

splitAt-defn :  n  splitAt {A = A} n  < take n , drop n >
splitAt-defn zero    xs       = refl
splitAt-defn (suc n) []       = refl
splitAt-defn (suc n) (x  xs) with splitAt n xs | splitAt-defn n xs
... | (ys , zs) | ih = cong (Prod.map (x ∷_) id) ih

------------------------------------------------------------------------
-- takeWhile, dropWhile, and span

module _ {P : Pred A p} (P? : Decidable P) where

  takeWhile++dropWhile :  xs  takeWhile P? xs ++ dropWhile P? xs  xs
  takeWhile++dropWhile []       = refl
  takeWhile++dropWhile (x  xs) with does (P? x)
  ... | true  = cong (x ∷_) (takeWhile++dropWhile xs)
  ... | false = refl

  span-defn : span P?  < takeWhile P? , dropWhile P? >
  span-defn []       = refl
  span-defn (x  xs) with does (P? x)
  ... | true  = cong (Prod.map (x ∷_) id) (span-defn xs)
  ... | false = refl

------------------------------------------------------------------------
-- filter

module _ {P : Pred A p} (P? : Decidable P) where

  length-filter :  xs  length (filter P? xs)  length xs
  length-filter []       = z≤n
  length-filter (x  xs) with does (P? x)
  ... | false = ≤-step (length-filter xs)
  ... | true  = s≤s (length-filter xs)

  filter-all :  {xs}  All P xs  filter P? xs  xs
  filter-all {[]}     []         = refl
  filter-all {x  xs} (px  pxs) with P? x
  ... | no          ¬px = contradiction px ¬px
  ... | true  because _ = cong (x ∷_) (filter-all pxs)

  filter-notAll :  xs  Any ( P) xs  length (filter P? xs) < length xs
  filter-notAll (x  xs) (here ¬px) with P? x
  ... | false because _ = s≤s (length-filter xs)
  ... | yes          px = contradiction px ¬px
  filter-notAll (x  xs) (there any) with does (P? x)
  ... | false = ≤-step (filter-notAll xs any)
  ... | true  = s≤s (filter-notAll xs any)

  filter-some :  {xs}  Any P xs  0 < length (filter P? xs)
  filter-some {x  xs} (here px)   with P? x
  ... | true because _ = s≤s z≤n
  ... | no         ¬px = contradiction px ¬px
  filter-some {x  xs} (there pxs) with does (P? x)
  ... | true  = ≤-step (filter-some pxs)
  ... | false = filter-some pxs

  filter-none :  {xs}  All ( P) xs  filter P? xs  []
  filter-none {[]}     []           = refl
  filter-none {x  xs} (¬px  ¬pxs) with P? x
  ... | false because _ = filter-none ¬pxs
  ... | yes          px = contradiction px ¬px

  filter-complete :  {xs}  length (filter P? xs)  length xs 
                    filter P? xs  xs
  filter-complete {[]}     eq = refl
  filter-complete {x  xs} eq with does (P? x)
  ... | false = contradiction eq (<⇒≢ (s≤s (length-filter xs)))
  ... | true  = cong (x ∷_) (filter-complete (suc-injective eq))

  filter-accept :  {x xs}  P x  filter P? (x  xs)  x  (filter P? xs)
  filter-accept {x} Px with P? x
  ... | true because _ = refl
  ... | no         ¬Px = contradiction Px ¬Px

  filter-reject :  {x xs}  ¬ P x  filter P? (x  xs)  filter P? xs
  filter-reject {x} ¬Px with P? x
  ... | yes          Px = contradiction Px ¬Px
  ... | false because _ = refl

  filter-idem : filter P?  filter P?  filter P?
  filter-idem []       = refl
  filter-idem (x  xs) with does (P? x) | inspect does (P? x)
  ... | false | _                   = filter-idem xs
  ... | true  | P.[ eq ] rewrite eq = cong (x ∷_) (filter-idem xs)

  filter-++ :  xs ys  filter P? (xs ++ ys)  filter P? xs ++ filter P? ys
  filter-++ []       ys = refl
  filter-++ (x  xs) ys with does (P? x)
  ... | true  = cong (x ∷_) (filter-++ xs ys)
  ... | false = filter-++ xs ys

------------------------------------------------------------------------
-- derun and deduplicate

module _ {R : Rel A p} (R? : B.Decidable R) where

  length-derun :  xs  length (derun R? xs)  length xs
  length-derun [] = ≤-refl
  length-derun (x  []) = ≤-refl
  length-derun (x  y  xs) with does (R? x y) | length-derun (y  xs)
  ... | true  | r = ≤-step r
  ... | false | r = s≤s r

  length-deduplicate :  xs  length (deduplicate R? xs)  length xs
  length-deduplicate [] = z≤n
  length-deduplicate (x  xs) = ≤-begin
    1 + length (filter (¬?  R? x) r) ≤⟨ s≤s (length-filter (¬?  R? x) r) 
    1 + length r                      ≤⟨ s≤s (length-deduplicate xs) 
    1 + length xs                     ≤-∎
    where
    open ≤-Reasoning renaming (begin_ to ≤-begin_; _∎ to _≤-∎)
    r = deduplicate R? xs

  derun-reject :  {x y} xs  R x y  derun R? (x  y  xs)  derun R? (y  xs)
  derun-reject {x} {y} xs Rxy with R? x y
  ... | yes _    = refl
  ... | no  ¬Rxy = contradiction Rxy ¬Rxy

  derun-accept :  {x y} xs  ¬ R x y  derun R? (x  y  xs)  x  derun R? (y  xs)
  derun-accept {x} {y} xs ¬Rxy with R? x y
  ... | yes Rxy = contradiction Rxy ¬Rxy
  ... | no  _   = refl

------------------------------------------------------------------------
-- partition

module _ {P : Pred A p} (P? : Decidable P) where

  partition-defn : partition P?  < filter P? , filter (∁? P?) >
  partition-defn []       = refl
  partition-defn (x  xs) with does (P? x)
  ...  | true  = cong (Prod.map (x ∷_) id) (partition-defn xs)
  ...  | false = cong (Prod.map id (x ∷_)) (partition-defn xs)

  length-partition :  xs  (let (ys , zs) = partition P? xs) 
                     length ys  length xs × length zs  length xs
  length-partition []       = z≤n , z≤n
  length-partition (x  xs) with does (P? x) | length-partition xs
  ...  | true  | rec = Prod.map s≤s ≤-step rec
  ...  | false | rec = Prod.map ≤-step s≤s rec

------------------------------------------------------------------------
-- _ʳ++_

ʳ++-defn :  (xs : List A) {ys}  xs ʳ++ ys  reverse xs ++ ys
ʳ++-defn [] = refl
ʳ++-defn (x  xs) {ys} = begin
  (x  xs)             ʳ++ ys   ≡⟨⟩
  xs         ʳ++   x      ys   ≡⟨⟩
  xs         ʳ++ [ x ]  ++ ys   ≡⟨ ʳ++-defn xs  
  reverse xs  ++ [ x ]  ++ ys   ≡⟨ sym (++-assoc (reverse xs) _ _) 
  (reverse xs ++ [ x ]) ++ ys   ≡⟨ cong (_++ ys) (sym (ʳ++-defn xs)) 
  (xs ʳ++ [ x ])        ++ ys   ≡⟨⟩
  reverse (x  xs)      ++ ys   

-- Reverse-append of append is reverse-append after reverse-append.

ʳ++-++ :  (xs {ys zs} : List A)  (xs ++ ys) ʳ++ zs  ys ʳ++ xs ʳ++ zs
ʳ++-++ [] = refl
ʳ++-++ (x  xs) {ys} {zs} = begin
  (x  xs ++ ys) ʳ++ zs   ≡⟨⟩
  (xs ++ ys) ʳ++ x  zs   ≡⟨ ʳ++-++ xs 
  ys ʳ++ xs ʳ++ x  zs    ≡⟨⟩
  ys ʳ++ (x  xs) ʳ++ zs  

-- Reverse-append of reverse-append is commuted reverse-append after append.

ʳ++-ʳ++ :  (xs {ys zs} : List A)  (xs ʳ++ ys) ʳ++ zs  ys ʳ++ xs ++ zs
ʳ++-ʳ++ [] = refl
ʳ++-ʳ++ (x  xs) {ys} {zs} = begin
  ((x  xs) ʳ++ ys) ʳ++ zs   ≡⟨⟩
  (xs ʳ++ x  ys) ʳ++ zs     ≡⟨ ʳ++-ʳ++ xs 
  (x  ys) ʳ++ xs ++ zs      ≡⟨⟩
  ys ʳ++ (x  xs) ++ zs      

-- Length of reverse-append

length-ʳ++ :  (xs {ys} : List A) 
             length (xs ʳ++ ys)  length xs + length ys
length-ʳ++ [] = refl
length-ʳ++ (x  xs) {ys} = begin
  length ((x  xs) ʳ++ ys)      ≡⟨⟩
  length (xs ʳ++ x  ys)        ≡⟨ length-ʳ++ xs 
  length xs + length (x  ys)   ≡⟨ +-suc _ _ 
  length (x  xs) + length ys   

-- map distributes over reverse-append.

map-ʳ++ : (f : A  B) (xs {ys} : List A) 
          map f (xs ʳ++ ys)  map f xs ʳ++ map f ys
map-ʳ++ f []            = refl
map-ʳ++ f (x  xs) {ys} = begin
  map f ((x  xs) ʳ++ ys)         ≡⟨⟩
  map f (xs ʳ++ x  ys)           ≡⟨ map-ʳ++ f xs 
  map f xs ʳ++ map f (x  ys)     ≡⟨⟩
  map f xs ʳ++ f x  map f ys     ≡⟨⟩
  (f x  map f xs) ʳ++ map f ys   ≡⟨⟩
  map f (x  xs)   ʳ++ map f ys   

-- A foldr after a reverse is a foldl.

foldr-ʳ++ :  (f : A  B  B) b xs {ys} 
            foldr f b (xs ʳ++ ys)  foldl (flip f) (foldr f b ys) xs
foldr-ʳ++ f b []       {_}  = refl
foldr-ʳ++ f b (x  xs) {ys} = begin
  foldr f b ((x  xs) ʳ++ ys)              ≡⟨⟩
  foldr f b (xs ʳ++ x  ys)                ≡⟨ foldr-ʳ++ f b xs 
  foldl (flip f) (foldr f b (x  ys)) xs   ≡⟨⟩
  foldl (flip f) (f x (foldr f b ys)) xs   ≡⟨⟩
  foldl (flip f) (foldr f b ys) (x  xs)   

-- A foldl after a reverse is a foldr.

foldl-ʳ++ :  (f : B  A  B) b xs {ys} 
            foldl f b (xs ʳ++ ys)  foldl f (foldr (flip f) b xs) ys
foldl-ʳ++ f b []       {_}  = refl
foldl-ʳ++ f b (x  xs) {ys} = begin
  foldl f b ((x  xs) ʳ++ ys)              ≡⟨⟩
  foldl f b (xs ʳ++ x  ys)                ≡⟨ foldl-ʳ++ f b xs 
  foldl f (foldr (flip f) b xs) (x  ys)   ≡⟨⟩
  foldl f (f (foldr (flip f) b xs) x) ys   ≡⟨⟩
  foldl f (foldr (flip f) b (x  xs)) ys   

------------------------------------------------------------------------
-- reverse

-- reverse of cons is snoc of reverse.

unfold-reverse :  (x : A) xs  reverse (x  xs)  reverse xs ∷ʳ x
unfold-reverse x xs = ʳ++-defn xs

-- reverse is an involution with respect to append.

reverse-++-commute : (xs ys : List A) 
                     reverse (xs ++ ys)  reverse ys ++ reverse xs
reverse-++-commute xs ys = begin
  reverse (xs ++ ys)         ≡⟨⟩
  (xs ++ ys) ʳ++ []          ≡⟨ ʳ++-++ xs 
  ys ʳ++ xs ʳ++ []           ≡⟨⟩
  ys ʳ++ reverse xs          ≡⟨ ʳ++-defn ys 
  reverse ys ++ reverse xs   

-- reverse is involutive.

reverse-involutive : Involutive {A = List A} _≡_ reverse
reverse-involutive xs = begin
  reverse (reverse xs)  ≡⟨⟩
  (xs ʳ++ []) ʳ++ []    ≡⟨ ʳ++-ʳ++ xs 
  [] ʳ++  xs ++ []      ≡⟨⟩
  xs ++ []              ≡⟨ ++-identityʳ xs 
  xs                    

-- reverse is injective.

reverse-injective :  {xs ys : List A}  reverse xs  reverse ys  xs  ys
reverse-injective = subst₂ _≡_ (reverse-involutive _) (reverse-involutive _)  cong reverse

-- reverse preserves length.

length-reverse :  (xs : List A)  length (reverse xs)  length xs
length-reverse xs = begin
  length (reverse xs)   ≡⟨⟩
  length (xs ʳ++ [])    ≡⟨ length-ʳ++ xs 
  length xs + 0         ≡⟨ +-identityʳ _ 
  length xs             

reverse-map-commute : (f : A  B)  map f  reverse  reverse  map f
reverse-map-commute f xs = begin
  map f (reverse xs) ≡⟨⟩
  map f (xs ʳ++ [])  ≡⟨ map-ʳ++ f xs 
  map f xs ʳ++ []    ≡⟨⟩
  reverse (map f xs) 

reverse-foldr :  (f : A  B  B) b 
                foldr f b  reverse  foldl (flip f) b
reverse-foldr f b xs = foldr-ʳ++ f b xs

reverse-foldl :  (f : B  A  B) b xs 
                foldl f b (reverse xs)  foldr (flip f) b xs
reverse-foldl f b xs = foldl-ʳ++ f b xs

------------------------------------------------------------------------
-- _∷ʳ_

module _ {x y : A} where

  ∷ʳ-injective :  xs ys  xs ∷ʳ x  ys ∷ʳ y  xs  ys × x  y
  ∷ʳ-injective []          []          refl = (refl , refl)
  ∷ʳ-injective (x  xs)    (y   ys)   eq   with ∷-injective eq
  ... | refl , eq′ = Prod.map (cong (x ∷_)) id (∷ʳ-injective xs ys eq′)
  ∷ʳ-injective []          (_  _  _) ()
  ∷ʳ-injective (_  _  _) []          ()

  ∷ʳ-injectiveˡ :  (xs ys : List A)  xs ∷ʳ x  ys ∷ʳ y  xs  ys
  ∷ʳ-injectiveˡ xs ys eq = proj₁ (∷ʳ-injective xs ys eq)

  ∷ʳ-injectiveʳ :  (xs ys : List A)  xs ∷ʳ x  ys ∷ʳ y  x  y
  ∷ʳ-injectiveʳ xs ys eq = proj₂ (∷ʳ-injective xs ys eq)


------------------------------------------------------------------------
-- DEPRECATED
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.

-- Version 2.0

zipWith-identityˡ = zipWith-zeroˡ
{-# WARNING_ON_USAGE zipWith-identityˡ
"Warning: zipWith-identityˡ was deprecated in v2.0.
Please use zipWith-zeroˡ instead."
#-}

zipWith-identityʳ = zipWith-zeroʳ
{-# WARNING_ON_USAGE zipWith-identityʳ
"Warning: zipWith-identityʳ was deprecated in v2.0.
Please use zipWith-zeroʳ instead."
#-}