{-# OPTIONS --without-K --safe #-}
module Data.List.Properties where
open import Algebra
import Algebra.Structures as Structures
open import Algebra.FunctionProperties
open import Data.Bool.Base using (Bool; false; true; not; if_then_else_)
open import Data.Fin using (Fin; zero; suc; cast; toℕ)
open import Data.List 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
open import Data.Nat.Properties
open import Data.Product as Prod hiding (map; zip)
open import Data.These as These using (These; this; that; these)
open import Function
import Relation.Binary as B
import Relation.Binary.Reasoning.Setoid as EqR
open import Relation.Binary.PropositionalEquality as P
using (_≡_; _≢_; _≗_; refl ; sym ; cong)
open import Relation.Nullary using (¬_; yes; no)
open import Relation.Nullary.Negation using (contradiction)
open import Relation.Nullary.Decidable using (⌊_⌋)
open import Relation.Unary using (Pred; Decidable; ∁)
open import Relation.Unary.Properties using (∁?)
module _ {a} {A : Set a} {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
module _ {a} {A : Set a} where
≡-dec : B.Decidable _≡_ → B.Decidable {A = List A} _≡_
≡-dec _≟_ [] [] = yes refl
≡-dec _≟_ (x ∷ xs) [] = no λ()
≡-dec _≟_ [] (y ∷ ys) = no λ()
≡-dec _≟_ (x ∷ xs) (y ∷ ys) with x ≟ y | ≡-dec _≟_ xs ys
... | no x≢y | _ = no (x≢y ∘ ∷-injectiveˡ)
... | yes _ | no xs≢ys = no (xs≢ys ∘ ∷-injectiveʳ)
... | yes refl | yes refl = yes refl
module _ {a} {A : Set a} where
map-id : map id ≗ id {A = List A}
map-id [] = refl
map-id (x ∷ xs) = P.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) = P.cong₂ _∷_ fx≡x (map-id₂ pxs)
module _ {a b} {A : Set a} {B : Set b} where
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 = P.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) = P.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) = P.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) = P.cong suc (length-map f xs)
module _ {a b c} {A : Set a} {B : Set b} {C : Set c} where
map-compose : {g : B → C} {f : A → B} → map (g ∘ f) ≗ map g ∘ map f
map-compose [] = refl
map-compose (x ∷ xs) = P.cong (_ ∷_) (map-compose xs)
module _ {a} {A : Set a} where
mapMaybe-just : (xs : List A) → mapMaybe just xs ≡ xs
mapMaybe-just [] = refl
mapMaybe-just (x ∷ xs) = P.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 _ {a b} {A : Set a} {B : Set b} (f : A → Maybe B) where
mapMaybe-concatMap : mapMaybe f ≗ concatMap (fromMaybe ∘ f)
mapMaybe-concatMap [] = refl
mapMaybe-concatMap (x ∷ xs) with f x
... | just y = P.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)
module _ {a} {A : Set a} where
++-assoc : Associative {A = List A} _≡_ _++_
++-assoc [] ys zs = refl
++-assoc (x ∷ xs) ys zs = P.cong (x ∷_) (++-assoc xs ys zs)
++-identityˡ : LeftIdentity {A = List A} _≡_ [] _++_
++-identityˡ xs = refl
++-identityʳ : RightIdentity {A = List A} _≡_ [] _++_
++-identityʳ [] = refl
++-identityʳ (x ∷ xs) = P.cong (x ∷_) (++-identityʳ xs)
++-identity : Identity {A = List A} _≡_ [] _++_
++-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 = []} (y ∷ ys) ()
++-identityˡ-unique {xs = x ∷ xs} (y ∷ ys) eq
with ++-identityˡ-unique (ys ++ [ x ]) (begin
xs ≡⟨ proj₂ (∷-injective eq) ⟩
ys ++ x ∷ xs ≡⟨ P.sym (++-assoc ys [ x ] xs) ⟩
(ys ++ [ x ]) ++ xs ∎)
where open P.≡-Reasoning
++-identityˡ-unique {xs = x ∷ xs} (y ∷ [] ) eq | ()
++-identityˡ-unique {xs = x ∷ xs} (y ∷ _ ∷ _) eq | ()
length-++ : ∀ (xs : List A) {ys} → length (xs ++ ys) ≡ length xs + length ys
length-++ [] = refl
length-++ (x ∷ xs) = P.cong suc (length-++ xs)
module _ {a} {A : Set a} where
open Structures {A = List A} _≡_
++-isMagma : IsMagma _++_
++-isMagma = record
{ isEquivalence = P.isEquivalence
; ∙-cong = P.cong₂ _++_
}
++-isSemigroup : IsSemigroup _++_
++-isSemigroup = record
{ isMagma = ++-isMagma
; assoc = ++-assoc
}
++-isMonoid : IsMonoid _++_ []
++-isMonoid = record
{ isSemigroup = ++-isSemigroup
; identity = ++-identity
}
module _ {a} (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 _ {a b c} {A : Set a} {B : Set b} {C : Set c}
{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) =
P.cong₂ _∷_ (f≗g (these a b)) (alignWith-cong f≗g as bs)
module _ {a b c} {A : Set a} {B : Set b} {C : Set c}
(f : These A B → C) where
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) = P.cong suc (length-alignWith xs ys)
alignWith-map : ∀ {d e} {D : Set d} {E : Set e} (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) =
P.cong₂ _∷_ refl (alignWith-map g h xs ys)
map-alignWith : ∀ {d} {D : Set d} (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) =
P.cong₂ _∷_ refl (map-alignWith g xs ys)
module _ {a b} {A : Set a} {B : Set b} (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) =
P.cong₂ _∷_ (f-comm x y) (zipWith-comm f-comm xs ys)
module _ {a b c} {A : Set a} {B : Set b} {C : Set c}
(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) = P.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) =
P.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) =
P.cong₂ _∷_ refl (map-zipWith g xs ys)
module _ {a b} {A : Set a} {B : Set b} where
unalignWith-this : unalignWith ((A → These A B) ∋ this) ≗ (_, [])
unalignWith-this [] = refl
unalignWith-this (a ∷ as) = P.cong (Prod.map₁ (a ∷_)) (unalignWith-this as)
unalignWith-that : unalignWith ((B → These A B) ∋ that) ≗ ([] ,_)
unalignWith-that [] = refl
unalignWith-that (b ∷ bs) = P.cong (Prod.map₂ (b ∷_)) (unalignWith-that bs)
module _ {a b c} {A : Set a} {B : Set b} {C : Set c}
{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 = P.cong (Prod.map₁ (a ∷_)) (unalignWith-cong f≗g cs)
... | that b | ._ | refl = P.cong (Prod.map₂ (b ∷_)) (unalignWith-cong f≗g cs)
... | these a b | ._ | refl = P.cong (Prod.map (a ∷_) (b ∷_)) (unalignWith-cong f≗g cs)
module _ {a b c} {A : Set a} {B : Set b} {C : Set c}
(f : C → These A B) where
unalignWith-map : ∀ {d} {D : Set d} (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 = P.cong (Prod.map₁ (a ∷_)) (unalignWith-map g ds)
... | that b = P.cong (Prod.map₂ (b ∷_)) (unalignWith-map g ds)
... | these a b = P.cong (Prod.map (a ∷_) (b ∷_)) (unalignWith-map g ds)
map-unalignWith : ∀ {d e} {D : Set d} {E : Set e} (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 = P.cong (Prod.map₁ (g a ∷_)) (map-unalignWith g h cs)
... | that b = P.cong (Prod.map₂ (h b ∷_)) (map-unalignWith g h cs)
... | these a b = P.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 ∎ where open P.≡-Reasoning
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 , [] ∎ where open P.≡-Reasoning
unalignWith-alignWith g g∘f≗id (a ∷ as) (b ∷ bs)
rewrite g∘f≗id (these a b) = let ih = unalignWith-alignWith g g∘f≗id as bs in
P.cong (Prod.map (a ∷_) (b ∷_)) ih
module _ {a b c} {A : Set a} {B : Set b} {C : Set c}
(f : A → B × C) where
length-unzipWith₁ : ∀ xys →
length (proj₁ (unzipWith f xys)) ≡ length xys
length-unzipWith₁ [] = refl
length-unzipWith₁ (x ∷ xys) = P.cong suc (length-unzipWith₁ xys)
length-unzipWith₂ : ∀ xys →
length (proj₂ (unzipWith f xys)) ≡ length xys
length-unzipWith₂ [] = refl
length-unzipWith₂ (x ∷ xys) = P.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) =
P.cong₂ _∷_ (f∘g≗id x) (zipWith-unzipWith g f∘g≗id xs)
foldr-universal : ∀ {a b} {A : Set a} {B : Set b}
(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)
≡⟨ P.cong (f x) (foldr-universal h f e base step xs) ⟩
f x (foldr f e xs)
∎
where open P.≡-Reasoning
foldr-cong : ∀ {a b} {A : Set a} {B : Set b}
{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 : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c}
(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 : ∀ {a} {A : Set a} → id {A = List A} ≗ foldr _∷_ []
id-is-foldr = foldr-universal id _∷_ [] refl (λ _ _ → refl)
++-is-foldr : ∀ {a} {A : Set a} (xs ys : List A) →
xs ++ ys ≡ foldr _∷_ ys xs
++-is-foldr xs ys =
begin
xs ++ ys
≡⟨ P.cong (_++ ys) (id-is-foldr xs) ⟩
foldr _∷_ [] xs ++ ys
≡⟨ foldr-fusion (_++ ys) [] (λ _ _ → refl) xs ⟩
foldr _∷_ ([] ++ ys) xs
≡⟨⟩
foldr _∷_ ys xs
∎
where open P.≡-Reasoning
module _ {a b} {A : Set a} {B : Set b} where
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 = P.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} =
begin
map f
≈⟨ P.cong (map f) ∘ id-is-foldr ⟩
map f ∘ foldr _∷_ []
≈⟨ foldr-fusion (map f) [] (λ _ _ → refl) ⟩
foldr (λ x ys → f x ∷ ys) []
∎ where open EqR (P._→-setoid_ _ _)
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) = P.cong (f z) (foldr-∷ʳ f x y ys)
module _ {a b} {A : Set a} {B : Set b} where
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
module _ {a b} {A : Set a} {B : Set b} where
concat-map : ∀ {f : A → B} → concat ∘ map (map f) ≗ map f ∘ concat
concat-map {f = f} =
begin
concat ∘ map (map f)
≈⟨ P.cong concat ∘ map-is-foldr ⟩
concat ∘ foldr (λ xs → map f xs ∷_) []
≈⟨ foldr-fusion concat [] (λ _ _ → refl) ⟩
foldr (λ ys → map f ys ++_) []
≈⟨ P.sym ∘ foldr-fusion (map f) [] (map-++-commute f) ⟩
map f ∘ concat
∎
where open EqR (P._→-setoid_ _ _)
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) ≡⟨ P.cong (x +_) (sum-++-commute xs ys) ⟩
x + (sum xs + sum ys) ≡⟨ P.sym (+-assoc x _ _) ⟩
(x + sum xs) + sum ys ∎
where open P.≡-Reasoning
module _ {a} {A : Set a} where
length-replicate : ∀ n {x : A} → length (replicate n x) ≡ n
length-replicate zero = refl
length-replicate (suc n) = P.cong suc (length-replicate n)
module _ {a b} {A : Set a} {B : Set b} where
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 , _ = P.cong₂ (λ z → f x z ∷_) z≡fy⦇f⦈xs eq
module _ {a b} {A : Set a} {B : Set b} where
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) = P.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))
∎)
where open P.≡-Reasoning
module _ {a} {A : Set a} where
length-applyUpTo : ∀ (f : ℕ → A) n → length (applyUpTo f n) ≡ n
length-applyUpTo f zero = refl
length-applyUpTo f (suc n) = P.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 zero ()
lookup-applyUpTo f (suc n) zero = refl
lookup-applyUpTo f (suc n) (suc i) = lookup-applyUpTo (f ∘ suc) n i
module _ {a} {A : Set a} (f : ℕ → A) where
length-applyDownFrom : ∀ n → length (applyDownFrom f n) ≡ n
length-applyDownFrom zero = refl
length-applyDownFrom (suc n) = P.cong suc (length-applyDownFrom n)
lookup-applyDownFrom : ∀ n i → lookup (applyDownFrom f n) i ≡ f (n ∸ (suc (toℕ i)))
lookup-applyDownFrom zero ()
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
module _ {a} {A : Set a} where
tabulate-cong : ∀ {n} {f g : Fin n → A} →
f ≗ g → tabulate f ≡ tabulate g
tabulate-cong {zero} p = P.refl
tabulate-cong {suc n} p = P.cong₂ _∷_ (p zero) (tabulate-cong (p ∘ suc))
tabulate-lookup : ∀ (xs : List A) → tabulate (lookup xs) ≡ xs
tabulate-lookup [] = refl
tabulate-lookup (x ∷ xs) = P.cong (_ ∷_) (tabulate-lookup xs)
length-tabulate : ∀ {n} → (f : Fin n → A) →
length (tabulate f) ≡ n
length-tabulate {zero} f = refl
length-tabulate {suc n} f = P.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
module _ {a b} {A : Set a} {B : Set b} where
map-tabulate : ∀ {n} (g : Fin n → A) (f : A → B) →
map f (tabulate g) ≡ tabulate (f ∘ g)
map-tabulate {zero} g f = refl
map-tabulate {suc n} g f = P.cong (_ ∷_) (map-tabulate (g ∘ suc) f)
module _ {a} {A : Set a} where
length-%= : ∀ xs k (f : A → A) → length (xs [ k ]%= f) ≡ length xs
length-%= [] () f
length-%= (x ∷ xs) zero f = refl
length-%= (x ∷ xs) (suc k) f = P.cong suc (length-%= xs k f)
module _ {a} {A : Set a} where
length-∷= : ∀ xs k (v : A) → length (xs [ k ]∷= v) ≡ length xs
length-∷= xs k v = length-%= xs k (const v)
map-∷= : ∀ {b} {B : Set b} xs k (v : A) (f : A → B) →
let eq = P.sym (length-map f xs) in
map f (xs [ k ]∷= v) ≡ map f xs [ cast eq k ]∷= f v
map-∷= [] () v f
map-∷= (x ∷ xs) zero v f = refl
map-∷= (x ∷ xs) (suc k) v f = P.cong (f x ∷_) (map-∷= xs k v f)
module _ {a} {A : Set a} where
length-─ : ∀ (xs : List A) k → length (xs ─ k) ≡ pred (length xs)
length-─ [] ()
length-─ (x ∷ xs) zero = refl
length-─ (x ∷ []) (suc ())
length-─ (x ∷ y ∷ xs) (suc k) = P.cong suc (length-─ (y ∷ xs) k)
map-─ : ∀ {b} {B : Set b} xs k (f : A → B) →
let eq = P.sym (length-map f xs) in
map f (xs ─ k) ≡ map f xs ─ cast eq k
map-─ [] () f
map-─ (x ∷ xs) zero f = refl
map-─ (x ∷ xs) (suc k) f = P.cong (f x ∷_) (map-─ xs k f)
module _ {a} {A : Set a} where
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) = P.cong suc (length-take n xs)
module _ {a} {A : Set a} where
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) = P.cong (x ∷_) (take++drop n xs)
module _ {a} {A : Set a} where
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 = P.cong (Prod.map (x ∷_) id) ih
module _ {a p} {A : Set a} {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 P? x
... | yes _ = P.cong (x ∷_) (takeWhile++dropWhile xs)
... | no _ = refl
span-defn : span P? ≗ < takeWhile P? , dropWhile P? >
span-defn [] = refl
span-defn (x ∷ xs) with P? x
... | yes _ = P.cong (Prod.map (x ∷_) id) (span-defn xs)
... | no _ = refl
module _ {a p} {A : Set a} {P : A → Set p} (P? : Decidable P) where
length-filter : ∀ xs → length (filter P? xs) ≤ length xs
length-filter [] = z≤n
length-filter (x ∷ xs) with P? x
... | no _ = ≤-step (length-filter xs)
... | yes _ = 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
... | yes _ = P.cong (x ∷_) (filter-all pxs)
filter-notAll : ∀ xs → Any (∁ P) xs → length (filter P? xs) < length xs
filter-notAll [] ()
filter-notAll (x ∷ xs) (here ¬px) with P? x
... | no _ = s≤s (length-filter xs)
... | yes px = contradiction px ¬px
filter-notAll (x ∷ xs) (there any) with P? x
... | no _ = ≤-step (filter-notAll xs any)
... | yes _ = 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
... | yes _ = s≤s z≤n
... | no ¬px = contradiction px ¬px
filter-some {x ∷ xs} (there pxs) with P? x
... | yes _ = ≤-step (filter-some pxs)
... | no _ = filter-some pxs
filter-none : ∀ {xs} → All (∁ P) xs → filter P? xs ≡ []
filter-none {[]} [] = refl
filter-none {x ∷ xs} (¬px ∷ ¬pxs) with P? x
... | no _ = 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 P? x
... | no ¬px = contradiction eq (<⇒≢ (s≤s (length-filter xs)))
... | yes px = P.cong (x ∷_) (filter-complete (suc-injective eq))
module _ {a p} {A : Set a} {P : A → Set p} (P? : Decidable P) where
partition-defn : partition P? ≗ < filter P? , filter (∁? P?) >
partition-defn [] = refl
partition-defn (x ∷ xs) with P? x
... | yes Px = P.cong (Prod.map (x ∷_) id) (partition-defn xs)
... | no ¬Px = P.cong (Prod.map id (x ∷_)) (partition-defn xs)
module _ {a} {A : Set a} where
unfold-reverse : ∀ (x : A) xs → reverse (x ∷ xs) ≡ reverse xs ∷ʳ x
unfold-reverse x xs = helper [ x ] xs
where
open P.≡-Reasoning
helper : (xs ys : List A) → foldl (flip _∷_) xs ys ≡ reverse ys ++ xs
helper xs [] = refl
helper xs (y ∷ ys) = begin
foldl (flip _∷_) (y ∷ xs) ys ≡⟨ helper (y ∷ xs) ys ⟩
reverse ys ++ y ∷ xs ≡⟨ P.sym (++-assoc (reverse ys) _ _) ⟩
(reverse ys ∷ʳ y) ++ xs ≡⟨ P.sym $ P.cong (_++ xs) (unfold-reverse y ys) ⟩
reverse (y ∷ ys) ++ xs ∎
reverse-++-commute : (xs ys : List A) →
reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
reverse-++-commute [] ys = P.sym (++-identityʳ _)
reverse-++-commute (x ∷ xs) ys = begin
reverse (x ∷ xs ++ ys) ≡⟨ unfold-reverse x (xs ++ ys) ⟩
reverse (xs ++ ys) ++ [ x ] ≡⟨ P.cong (_++ [ x ]) (reverse-++-commute xs ys) ⟩
(reverse ys ++ reverse xs) ++ [ x ] ≡⟨ ++-assoc (reverse ys) _ _ ⟩
reverse ys ++ (reverse xs ++ [ x ]) ≡⟨ P.sym $ P.cong (reverse ys ++_) (unfold-reverse x xs) ⟩
reverse ys ++ reverse (x ∷ xs) ∎
where open P.≡-Reasoning
reverse-involutive : Involutive {A = List A} _≡_ reverse
reverse-involutive [] = refl
reverse-involutive (x ∷ xs) = begin
reverse (reverse (x ∷ xs)) ≡⟨ P.cong reverse $ unfold-reverse x xs ⟩
reverse (reverse xs ∷ʳ x) ≡⟨ reverse-++-commute (reverse xs) ([ x ]) ⟩
x ∷ reverse (reverse (xs)) ≡⟨ P.cong (x ∷_) $ reverse-involutive xs ⟩
x ∷ xs ∎
where open P.≡-Reasoning
length-reverse : (xs : List A) → length (reverse xs) ≡ length xs
length-reverse [] = refl
length-reverse (x ∷ xs) = begin
length (reverse (x ∷ xs)) ≡⟨ P.cong length $ unfold-reverse x xs ⟩
length (reverse xs ∷ʳ x) ≡⟨ length-++ (reverse xs) ⟩
length (reverse xs) + 1 ≡⟨ P.cong (_+ 1) (length-reverse xs) ⟩
length xs + 1 ≡⟨ +-comm _ 1 ⟩
suc (length xs) ∎
where open P.≡-Reasoning
module _ {a b} {A : Set a} {B : Set b} where
reverse-map-commute : (f : A → B) (xs : List A) →
map f (reverse xs) ≡ reverse (map f xs)
reverse-map-commute f [] = refl
reverse-map-commute f (x ∷ xs) = begin
map f (reverse (x ∷ xs)) ≡⟨ P.cong (map f) $ unfold-reverse x xs ⟩
map f (reverse xs ∷ʳ x) ≡⟨ map-++-commute f (reverse xs) ([ x ]) ⟩
map f (reverse xs) ∷ʳ f x ≡⟨ P.cong (_∷ʳ f x) $ reverse-map-commute f xs ⟩
reverse (map f xs) ∷ʳ f x ≡⟨ P.sym $ unfold-reverse (f x) (map f xs) ⟩
reverse (map f (x ∷ xs)) ∎
where open P.≡-Reasoning
reverse-foldr : ∀ (f : A → B → B) x ys →
foldr f x (reverse ys) ≡ foldl (flip f) x ys
reverse-foldr f x [] = refl
reverse-foldr f x (y ∷ ys) = begin
foldr f x (reverse (y ∷ ys)) ≡⟨ P.cong (foldr f x) (unfold-reverse y ys) ⟩
foldr f x ((reverse ys) ∷ʳ y) ≡⟨ foldr-∷ʳ f x y (reverse ys) ⟩
foldr f (f y x) (reverse ys) ≡⟨ reverse-foldr f (f y x) ys ⟩
foldl (flip f) (f y x) ys ∎
where open P.≡-Reasoning
reverse-foldl : ∀ (f : A → B → A) x ys →
foldl f x (reverse ys) ≡ foldr (flip f) x ys
reverse-foldl f x [] = refl
reverse-foldl f x (y ∷ ys) = begin
foldl f x (reverse (y ∷ ys)) ≡⟨ P.cong (foldl f x) (unfold-reverse y ys) ⟩
foldl f x ((reverse ys) ∷ʳ y) ≡⟨ foldl-∷ʳ f x y (reverse ys) ⟩
f (foldl f x (reverse ys)) y ≡⟨ P.cong (flip f y) (reverse-foldl f x ys) ⟩
f (foldr (flip f) x ys) y ∎
where open P.≡-Reasoning
module _ {a} {A : Set a} where
∷ʳ-injective : ∀ {x y : A} 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 (P.cong (x ∷_)) id (∷ʳ-injective xs ys eq′)
∷ʳ-injective [] (_ ∷ []) ()
∷ʳ-injective [] (_ ∷ _ ∷ _) ()
∷ʳ-injective (_ ∷ []) [] ()
∷ʳ-injective (_ ∷ _ ∷ _) [] ()
∷ʳ-injectiveˡ : ∀ {x y : A} (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 : ∀ {x y : A} (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 _ {a} {A : Set a} (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 = P.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 = P.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 = P.cong (Prod.map (x ∷_) id) (boolPartition-defn xs)
... | false = P.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 _ {a p} {A : Set a} (P : A → Set p) (P? : Decidable P) where
boolFilter-filters : ∀ xs → All P (boolFilter (⌊_⌋ ∘ P?) xs)
boolFilter-filters [] = []
boolFilter-filters (x ∷ xs) with P? x
... | yes px = px ∷ boolFilter-filters xs
... | no ¬px = 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."
#-}