{-# 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)
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-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 (λ 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 (λ 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-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} (λ _ → 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
}
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)
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-identityˡ : ∀ xs → zipWith f [] xs ≡ []
zipWith-identityˡ [] = refl
zipWith-identityˡ (x ∷ xs) = refl
zipWith-identityʳ : ∀ xs → zipWith f xs [] ≡ []
zipWith-identityʳ [] = refl
zipWith-identityʳ (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 (λ 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 (λ 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-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)
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-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
(λ x xs → fuse x (foldr f e xs))
id-is-foldr : id {A = List A} ≗ foldr _∷_ []
id-is-foldr = foldr-universal id _∷_ [] refl (λ _ _ → 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) [] (λ _ _ → 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 (λ 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) [] (λ _ _ → refl) xs ⟩
foldr (λ 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)
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-++ : ∀ (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-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 (λ xs → map f xs ∷_) [] xss) ≡⟨ foldr-fusion concat [] (λ _ _ → refl) xss ⟩
foldr (λ ys → map f ys ++_) [] xss ≡⟨ sym (foldr-fusion (map f) [] (map-++-commute f) xss) ⟩
map f (concat xss) ∎
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 ∎
length-replicate : ∀ n {x : A} → length (replicate n x) ≡ n
length-replicate zero = refl
length-replicate (suc n) = cong suc (length-replicate n)
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 (λ z → f x z ∷_) z≡fy⦇f⦈xs eq
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))
∎)
length-applyUpTo : ∀ (f : ℕ → 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 : ℕ → 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
module _ (f : ℕ → 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
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
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-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 (λ 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)
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)
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-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
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
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
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
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)
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 ∎
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 ∎
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-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-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 ∎
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) ∎
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 ∎
unfold-reverse : ∀ (x : A) xs → reverse (x ∷ xs) ≡ reverse xs ∷ʳ x
unfold-reverse x xs = ʳ++-defn xs
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-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-injective : ∀ {xs ys : List A} → reverse xs ≡ reverse ys → xs ≡ ys
reverse-injective = subst₂ _≡_ (reverse-involutive _) (reverse-involutive _) ∘ cong reverse
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
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)
gfilter-just = mapMaybe-just
{-# WARNING_ON_USAGE gfilter-just
"Warning: gfilter-just was deprecated in v0.15.
Please use mapMaybe-just instead."
#-}
gfilter-nothing = mapMaybe-nothing
{-# WARNING_ON_USAGE gfilter-nothing
"Warning: gfilter-nothing was deprecated in v0.15.
Please use mapMaybe-nothing instead."
#-}
gfilter-concatMap = mapMaybe-concatMap
{-# WARNING_ON_USAGE gfilter-concatMap
"Warning: gfilter-concatMap was deprecated in v0.15.
Please use mapMaybe-concatMap instead."
#-}
length-gfilter = length-mapMaybe
{-# WARNING_ON_USAGE length-gfilter
"Warning: length-gfilter was deprecated in v0.15.
Please use length-mapMaybe instead."
#-}
right-identity-unique = ++-identityʳ-unique
{-# WARNING_ON_USAGE right-identity-unique
"Warning: right-identity-unique was deprecated in v0.15.
Please use ++-identityʳ-unique instead."
#-}
left-identity-unique = ++-identityˡ-unique
{-# WARNING_ON_USAGE left-identity-unique
"Warning: left-identity-unique was deprecated in v0.15.
Please use ++-identityˡ-unique instead."
#-}
module _ (p : A → Bool) where
boolTakeWhile++boolDropWhile : ∀ xs → boolTakeWhile p xs ++ boolDropWhile p xs ≡ xs
boolTakeWhile++boolDropWhile [] = refl
boolTakeWhile++boolDropWhile (x ∷ xs) with p x
... | true = cong (x ∷_) (boolTakeWhile++boolDropWhile xs)
... | false = refl
{-# WARNING_ON_USAGE boolTakeWhile++boolDropWhile
"Warning: boolTakeWhile and boolDropWhile were deprecated in v0.16.
Please use takeWhile and dropWhile instead."
#-}
boolSpan-defn : boolSpan p ≗ < boolTakeWhile p , boolDropWhile p >
boolSpan-defn [] = refl
boolSpan-defn (x ∷ xs) with p x
... | true = cong (Prod.map (x ∷_) id) (boolSpan-defn xs)
... | false = refl
{-# WARNING_ON_USAGE boolSpan-defn
"Warning: boolSpan, boolTakeWhile and boolDropWhile were deprecated in v0.16.
Please use span, takeWhile and dropWhile instead."
#-}
length-boolFilter : ∀ xs → length (boolFilter p xs) ≤ length xs
length-boolFilter xs =
length-mapMaybe (λ x → if p x then just x else nothing) xs
{-# WARNING_ON_USAGE length-boolFilter
"Warning: boolFilter was deprecated in v0.16.
Please use filter instead."
#-}
boolPartition-defn : boolPartition p ≗ < boolFilter p , boolFilter (not ∘ p) >
boolPartition-defn [] = refl
boolPartition-defn (x ∷ xs) with p x
... | true = cong (Prod.map (x ∷_) id) (boolPartition-defn xs)
... | false = cong (Prod.map id (x ∷_)) (boolPartition-defn xs)
{-# WARNING_ON_USAGE boolPartition-defn
"Warning: boolPartition and boolFilter were deprecated in v0.16.
Please use partition and filter instead."
#-}
module _ (P : A → Set p) (P? : Decidable P) where
boolFilter-filters : ∀ xs → All P (boolFilter (isYes ∘ P?) xs)
boolFilter-filters [] = []
boolFilter-filters (x ∷ xs) with P? x
... | true because [px] = invert [px] ∷ boolFilter-filters xs
... | false because _ = boolFilter-filters xs
{-# WARNING_ON_USAGE boolFilter-filters
"Warning: boolFilter was deprecated in v0.16.
Please use filter instead."
#-}
idIsFold = id-is-foldr
{-# WARNING_ON_USAGE idIsFold
"Warning: idIsFold was deprecated in v0.17.
Please use id-is-foldr instead."
#-}
++IsFold = ++-is-foldr
{-# WARNING_ON_USAGE ++IsFold
"Warning: ++IsFold was deprecated in v0.17.
Please use ++-is-foldr instead."
#-}
mapIsFold = map-is-foldr
{-# WARNING_ON_USAGE mapIsFold
"Warning: mapIsFold was deprecated in v0.17.
Please use map-is-foldr instead."
#-}