{-# OPTIONS --universe-polymorphism #-}
module Data.List.Properties where
open import Algebra
open import Category.Monad
open import Data.List as List
open import Data.Nat
open import Data.Nat.Properties
open import Data.Bool
open import Function
open import Data.Product as Prod hiding (map)
open import Data.Maybe
open import Relation.Binary.PropositionalEquality as P
using (_≡_; _≢_; _≗_; refl)
import Relation.Binary.EqReasoning as EqR
open RawMonadPlus List.monadPlus
private
module LM {a} {A : Set a} = Monoid (List.monoid A)
∷-injective : ∀ {a} {A : Set a} {x y xs ys} →
(x ∷ xs ∶ List A) ≡ y ∷ ys → x ≡ y × xs ≡ ys
∷-injective refl = (refl , refl)
∷ʳ-injective : ∀ {a} {A : Set a} {x y} xs ys →
(xs ∷ʳ x ∶ List A) ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y
∷ʳ-injective [] [] refl = (refl , refl)
∷ʳ-injective (x ∷ xs) (y ∷ ys) eq with ∷-injective eq
∷ʳ-injective (x ∷ xs) (.x ∷ ys) eq | (refl , eq′) =
Prod.map (P.cong (_∷_ x)) id $ ∷ʳ-injective xs ys eq′
∷ʳ-injective [] (_ ∷ []) ()
∷ʳ-injective [] (_ ∷ _ ∷ _) ()
∷ʳ-injective (_ ∷ []) [] ()
∷ʳ-injective (_ ∷ _ ∷ _) [] ()
right-identity-unique : ∀ {a} {A : Set a} (xs : List A) {ys} →
xs ≡ xs ++ ys → ys ≡ []
right-identity-unique [] refl = refl
right-identity-unique (x ∷ xs) eq =
right-identity-unique xs (proj₂ (∷-injective eq))
left-identity-unique : ∀ {A : Set} {xs} (ys : List A) →
xs ≡ ys ++ xs → ys ≡ []
left-identity-unique [] _ = refl
left-identity-unique {xs = []} (y ∷ ys) ()
left-identity-unique {xs = x ∷ xs} (y ∷ ys) eq
with left-identity-unique (ys ++ [ x ]) (begin
xs ≡⟨ proj₂ (∷-injective eq) ⟩
ys ++ x ∷ xs ≡⟨ P.sym (LM.assoc ys [ x ] xs) ⟩
(ys ++ [ x ]) ++ xs ∎)
where open P.≡-Reasoning
left-identity-unique {xs = x ∷ xs} (y ∷ [] ) eq | ()
left-identity-unique {xs = x ∷ xs} (y ∷ _ ∷ _) eq | ()
map-++-commute : ∀ {a b} {A : Set a} {B : Set b} (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)
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 CommutativeSemiring commutativeSemiring hiding (_+_)
open P.≡-Reasoning
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-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))
idIsFold : ∀ {a} {A : Set a} → id {A = List A} ≗ foldr _∷_ []
idIsFold = foldr-universal id _∷_ [] refl (λ _ _ → refl)
++IsFold : ∀ {a} {A : Set a} (xs ys : List A) →
xs ++ ys ≡ foldr _∷_ ys xs
++IsFold xs ys =
begin
xs ++ ys
≡⟨ P.cong (λ xs → xs ++ ys) (idIsFold xs) ⟩
foldr _∷_ [] xs ++ ys
≡⟨ foldr-fusion (λ xs → xs ++ ys) [] (λ _ _ → refl) xs ⟩
foldr _∷_ ([] ++ ys) xs
≡⟨ refl ⟩
foldr _∷_ ys xs
∎
where open P.≡-Reasoning
mapIsFold : ∀ {a b} {A : Set a} {B : Set b} {f : A → B} →
map f ≗ foldr (λ x ys → f x ∷ ys) []
mapIsFold {f = f} =
begin
map f
≈⟨ P.cong (map f) ∘ idIsFold ⟩
map f ∘ foldr _∷_ []
≈⟨ foldr-fusion (map f) [] (λ _ _ → refl) ⟩
foldr (λ x ys → f x ∷ ys) []
∎
where open EqR (P._→-setoid_ _ _)
concat-map : ∀ {a b} {A : Set a} {B : Set b} {f : A → B} →
concat ∘ map (map f) ≗ map f ∘ concat
concat-map {b = b} {f = f} =
begin
concat ∘ map (map f)
≈⟨ P.cong concat ∘ mapIsFold {b = b} ⟩
concat ∘ foldr (λ xs ys → map f xs ∷ ys) []
≈⟨ foldr-fusion {b = b} concat [] (λ _ _ → refl) ⟩
foldr (λ ys zs → map f ys ++ zs) []
≈⟨ P.sym ∘
foldr-fusion (map f) [] (λ ys zs → map-++-commute f ys zs) ⟩
map f ∘ concat
∎
where open EqR (P._→-setoid_ _ _)
map-id : ∀ {a} {A : Set a} → map id ≗ id {A = List A}
map-id {A = A} = begin
map id ≈⟨ mapIsFold ⟩
foldr _∷_ [] ≈⟨ P.sym ∘ idIsFold {A = A} ⟩
id ∎
where open EqR (P._→-setoid_ _ _)
map-compose : ∀ {a b c} {A : Set a} {B : Set b} {C : Set c}
{g : B → C} {f : A → B} →
map (g ∘ f) ≗ map g ∘ map f
map-compose {A = A} {B} {g = g} {f} =
begin
map (g ∘ f)
≈⟨ P.cong (map (g ∘ f)) ∘ idIsFold ⟩
map (g ∘ f) ∘ foldr _∷_ []
≈⟨ foldr-fusion (map (g ∘ f)) [] (λ _ _ → refl) ⟩
foldr (λ a y → g (f a) ∷ y) []
≈⟨ P.sym ∘ foldr-fusion (map g) [] (λ _ _ → refl) ⟩
map g ∘ foldr (λ a y → f a ∷ y) []
≈⟨ P.cong (map g) ∘ P.sym ∘ mapIsFold {A = A} {B = B} ⟩
map g ∘ map f
∎
where open EqR (P._→-setoid_ _ _)
foldr-cong : ∀ {a b} {A : Set a} {B : Set b}
{f₁ f₂ : A → B → B} {e₁ e₂ : B} →
(∀ x y → f₁ x y ≡ f₂ x y) → e₁ ≡ e₂ →
foldr f₁ e₁ ≗ foldr f₂ e₂
foldr-cong {f₁ = f₁} {f₂} {e} f₁≗₂f₂ refl =
begin
foldr f₁ e
≈⟨ P.cong (foldr f₁ e) ∘ idIsFold ⟩
foldr f₁ e ∘ foldr _∷_ []
≈⟨ foldr-fusion (foldr f₁ e) [] (λ x xs → f₁≗₂f₂ x (foldr f₁ e xs)) ⟩
foldr f₂ e
∎
where open EqR (P._→-setoid_ _ _)
map-cong : ∀ {a b} {A : Set a} {B : Set b} {f g : A → B} →
f ≗ g → map f ≗ map g
map-cong {A = A} {B} {f} {g} f≗g =
begin
map f
≈⟨ mapIsFold ⟩
foldr (λ x ys → f x ∷ ys) []
≈⟨ foldr-cong (λ x ys → P.cong₂ _∷_ (f≗g x) refl) refl ⟩
foldr (λ x ys → g x ∷ ys) []
≈⟨ P.sym ∘ mapIsFold {A = A} {B = B} ⟩
map g
∎
where open EqR (P._→-setoid_ _ _)
take++drop : ∀ {a} {A : Set a}
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 (λ xs → x ∷ xs) (take++drop n xs)
splitAt-defn : ∀ {a} {A : Set a} 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
takeWhile++dropWhile : ∀ {a} {A : Set a} (p : A → Bool) (xs : List A) →
takeWhile p xs ++ dropWhile p xs ≡ xs
takeWhile++dropWhile p [] = refl
takeWhile++dropWhile p (x ∷ xs) with p x
... | true = P.cong (_∷_ x) (takeWhile++dropWhile p xs)
... | false = refl
span-defn : ∀ {a} {A : Set a} (p : A → Bool) →
span p ≗ < takeWhile p , dropWhile p >
span-defn p [] = refl
span-defn p (x ∷ xs) with p x
... | true = P.cong (Prod.map (_∷_ x) id) (span-defn p xs)
... | false = refl
partition-defn : ∀ {a} {A : Set a} (p : A → Bool) →
partition p ≗ < filter p , filter (not ∘ p) >
partition-defn p [] = refl
partition-defn p (x ∷ xs)
with p x | partition p xs | partition-defn p xs
... | true | (ys , zs) | eq = P.cong (Prod.map (_∷_ x) id) eq
... | false | (ys , zs) | eq = P.cong (Prod.map id (_∷_ x)) eq
scanr-defn : ∀ {a b} {A : Set a} {B : Set b}
(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 (x₁ ∷ x₂ ∷ xs)
with scanr f e (x₂ ∷ xs) | scanr-defn f e (x₂ ∷ xs)
... | [] | ()
... | y ∷ ys | eq with ∷-injective eq
... | y≡fx₂⦇f⦈xs , _ = P.cong₂ (λ z zs → f x₁ z ∷ zs) y≡fx₂⦇f⦈xs eq
scanl-defn : ∀ {a b} {A : Set a} {B : Set b}
(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
length-map : ∀ {a b} {A : Set a} {B : Set b} (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)
length-++ : ∀ {a} {A : Set a} (xs : List A) {ys} →
length (xs ++ ys) ≡ length xs + length ys
length-++ [] = refl
length-++ (x ∷ xs) = P.cong suc (length-++ xs)
length-gfilter : ∀ {a b} {A : Set a} {B : Set b} (p : A → Maybe B) xs →
length (gfilter p xs) ≤ length xs
length-gfilter p [] = z≤n
length-gfilter p (x ∷ xs) with p x
length-gfilter p (x ∷ xs) | just y = s≤s (length-gfilter p xs)
length-gfilter p (x ∷ xs) | nothing = ≤-step (length-gfilter p xs)
module Monad where
left-zero : {A B : Set} (f : A → List B) → (∅ >>= f) ≡ ∅
left-zero f = refl
right-zero : {A B : Set} (xs : List A) →
(xs >>= const ∅) ≡ (∅ ∶ List B)
right-zero [] = refl
right-zero (x ∷ xs) = right-zero xs
private
not-left-distributive :
let xs = true ∷ false ∷ []; f = return; g = return in
(xs >>= λ x → f x ∣ g x) ≢ ((xs >>= f) ∣ (xs >>= g))
not-left-distributive ()
right-distributive : {A B : Set} (xs ys : List A) (f : A → List B) →
(xs ∣ ys >>= f) ≡ ((xs >>= f) ∣ (ys >>= f))
right-distributive [] ys f = refl
right-distributive (x ∷ xs) ys f = begin
f x ∣ (xs ∣ ys >>= f) ≡⟨ P.cong (_∣_ (f x)) $ right-distributive xs ys f ⟩
f x ∣ ((xs >>= f) ∣ (ys >>= f)) ≡⟨ P.sym $ LM.assoc (f x) _ _ ⟩
(f x ∣ (xs >>= f)) ∣ (ys >>= f) ∎
where open P.≡-Reasoning
left-identity : {A B : Set} (x : A) (f : A → List B) →
(return x >>= f) ≡ f x
left-identity x f = proj₂ LM.identity (f x)
right-identity : {A : Set} (xs : List A) →
(xs >>= return) ≡ xs
right-identity [] = refl
right-identity (x ∷ xs) = P.cong (_∷_ x) (right-identity xs)
associative : {A B C : Set}
(xs : List A) (f : A → List B) (g : B → List C) →
(xs >>= λ x → f x >>= g) ≡ (xs >>= f >>= g)
associative [] f g = refl
associative (x ∷ xs) f g = begin
(f x >>= g) ∣ (xs >>= λ x → f x >>= g) ≡⟨ P.cong (_∣_ (f x >>= g)) $ associative xs f g ⟩
(f x >>= g) ∣ (xs >>= f >>= g) ≡⟨ P.sym $ right-distributive (f x) (xs >>= f) g ⟩
(f x ∣ (xs >>= f) >>= g) ∎
where open P.≡-Reasoning
cong : ∀ {A B : Set} {xs₁ xs₂} {f₁ f₂ : A → List B} →
xs₁ ≡ xs₂ → f₁ ≗ f₂ → (xs₁ >>= f₁) ≡ (xs₂ >>= f₂)
cong {xs₁ = xs} refl f₁≗f₂ = P.cong concat (map-cong f₁≗f₂ xs)
module Applicative where
open P.≡-Reasoning
private
pam : {A B : Set} → List A → (A → B) → List B
pam xs f = xs >>= return ∘ f
left-zero : ∀ {A B} xs → (∅ ∶ List (A → B)) ⊛ xs ≡ ∅
left-zero xs = begin
∅ ⊛ xs ≡⟨ refl ⟩
(∅ >>= pam xs) ≡⟨ Monad.left-zero (pam xs) ⟩
∅ ∎
right-zero : ∀ {A B} (fs : List (A → B)) → fs ⊛ ∅ ≡ ∅
right-zero fs = begin
fs ⊛ ∅ ≡⟨ refl ⟩
(fs >>= pam ∅) ≡⟨ (Monad.cong (refl {x = fs}) λ f →
Monad.left-zero (return ∘ f)) ⟩
(fs >>= λ _ → ∅) ≡⟨ Monad.right-zero fs ⟩
∅ ∎
right-distributive :
∀ {A B} (fs₁ fs₂ : List (A → B)) xs →
(fs₁ ∣ fs₂) ⊛ xs ≡ (fs₁ ⊛ xs ∣ fs₂ ⊛ xs)
right-distributive fs₁ fs₂ xs = begin
(fs₁ ∣ fs₂) ⊛ xs ≡⟨ refl ⟩
(fs₁ ∣ fs₂ >>= pam xs) ≡⟨ Monad.right-distributive fs₁ fs₂ (pam xs) ⟩
(fs₁ >>= pam xs) ∣ (fs₂ >>= pam xs) ≡⟨ refl ⟩
fs₁ ⊛ xs ∣ fs₂ ⊛ xs ∎
private
not-left-distributive :
let fs = id ∷ id ∷ []; xs₁ = true ∷ []; xs₂ = true ∷ false ∷ [] in
fs ⊛ (xs₁ ∣ xs₂) ≢ (fs ⊛ xs₁ ∣ fs ⊛ xs₂)
not-left-distributive ()
identity : ∀ {A} (xs : List A) → return id ⊛ xs ≡ xs
identity xs = begin
return id ⊛ xs ≡⟨ refl ⟩
(return id >>= pam xs) ≡⟨ Monad.left-identity id (pam xs) ⟩
(xs >>= return) ≡⟨ Monad.right-identity xs ⟩
xs ∎
private
pam-lemma : {A B C : Set}
(xs : List A) (f : A → B) (fs : B → List C) →
(pam xs f >>= fs) ≡ (xs >>= λ x → fs (f x))
pam-lemma xs f fs = begin
(pam xs f >>= fs) ≡⟨ P.sym $ Monad.associative xs (return ∘ f) fs ⟩
(xs >>= λ x → return (f x) >>= fs) ≡⟨ Monad.cong (refl {x = xs}) (λ x → Monad.left-identity (f x) fs) ⟩
(xs >>= λ x → fs (f x)) ∎
composition :
∀ {A B C} (fs : List (B → C)) (gs : List (A → B)) xs →
return _∘′_ ⊛ fs ⊛ gs ⊛ xs ≡ fs ⊛ (gs ⊛ xs)
composition fs gs xs = begin
return _∘′_ ⊛ fs ⊛ gs ⊛ xs ≡⟨ refl ⟩
(return _∘′_ >>= pam fs >>= pam gs >>= pam xs) ≡⟨ Monad.cong (Monad.cong (Monad.left-identity _∘′_ (pam fs))
(λ _ → refl))
(λ _ → refl) ⟩
(pam fs _∘′_ >>= pam gs >>= pam xs) ≡⟨ Monad.cong (pam-lemma fs _∘′_ (pam gs)) (λ _ → refl) ⟩
((fs >>= λ f → pam gs (_∘′_ f)) >>= pam xs) ≡⟨ P.sym $ Monad.associative fs (λ f → pam gs (_∘′_ f)) (pam xs) ⟩
(fs >>= λ f → pam gs (_∘′_ f) >>= pam xs) ≡⟨ (Monad.cong (refl {x = fs}) λ f →
pam-lemma gs (_∘′_ f) (pam xs)) ⟩
(fs >>= λ f → gs >>= λ g → pam xs (f ∘′ g)) ≡⟨ (Monad.cong (refl {x = fs}) λ f →
Monad.cong (refl {x = gs}) λ g →
P.sym $ pam-lemma xs g (return ∘ f)) ⟩
(fs >>= λ f → gs >>= λ g → pam (pam xs g) f) ≡⟨ (Monad.cong (refl {x = fs}) λ f →
Monad.associative gs (pam xs) (return ∘ f)) ⟩
(fs >>= pam (gs >>= pam xs)) ≡⟨ refl ⟩
fs ⊛ (gs ⊛ xs) ∎
homomorphism : ∀ {A B : Set} (f : A → B) x →
return f ⊛ return x ≡ return (f x)
homomorphism f x = begin
return f ⊛ return x ≡⟨ refl ⟩
(return f >>= pam (return x)) ≡⟨ Monad.left-identity f (pam (return x)) ⟩
pam (return x) f ≡⟨ Monad.left-identity x (return ∘ f) ⟩
return (f x) ∎
interchange : ∀ {A B} (fs : List (A → B)) {x} →
fs ⊛ return x ≡ return (λ f → f x) ⊛ fs
interchange fs {x} = begin
fs ⊛ return x ≡⟨ refl ⟩
(fs >>= pam (return x)) ≡⟨ (Monad.cong (refl {x = fs}) λ f →
Monad.left-identity x (return ∘ f)) ⟩
(fs >>= λ f → return (f x)) ≡⟨ refl ⟩
(pam fs (λ f → f x)) ≡⟨ P.sym $ Monad.left-identity (λ f → f x) (pam fs) ⟩
(return (λ f → f x) >>= pam fs) ≡⟨ refl ⟩
return (λ f → f x) ⊛ fs ∎