module Data.Vec.Properties where
open import Algebra
open import Category.Applicative.Indexed
open import Category.Monad
open import Category.Monad.Identity
open import Data.Vec
open import Data.List.Any
using (here; there) renaming (module Membership-≡ to List)
open import Data.Nat
open import Data.Empty using (⊥-elim)
import Data.Nat.Properties as Nat
open import Data.Fin as Fin using (Fin; zero; suc; toℕ; fromℕ)
open import Data.Fin.Properties using (_+′_)
open import Data.Empty using (⊥-elim)
open import Data.Product as Prod using (_×_; _,_; proj₁; proj₂; <_,_>)
open import Function
open import Function.Inverse using (_↔_)
open import Relation.Binary
module UsingVectorEquality {s₁ s₂} (S : Setoid s₁ s₂) where
private module SS = Setoid S
open SS using () renaming (Carrier to A)
import Data.Vec.Equality as VecEq
open VecEq.Equality S
replicate-lemma : ∀ {m} n x (xs : Vec A m) →
replicate {n = n} x ++ (x ∷ xs) ≈
replicate {n = 1 + n} x ++ xs
replicate-lemma zero x xs = refl (x ∷ xs)
replicate-lemma (suc n) x xs = SS.refl ∷-cong replicate-lemma n x xs
xs++[]=xs : ∀ {n} (xs : Vec A n) → xs ++ [] ≈ xs
xs++[]=xs [] = []-cong
xs++[]=xs (x ∷ xs) = SS.refl ∷-cong xs++[]=xs xs
map-++-commute : ∀ {b m n} {B : Set b}
(f : B → A) (xs : Vec B m) {ys : Vec B n} →
map f (xs ++ ys) ≈ map f xs ++ map f ys
map-++-commute f [] = refl _
map-++-commute f (x ∷ xs) = SS.refl ∷-cong map-++-commute f xs
open import Relation.Binary.PropositionalEquality as P
using (_≡_; _≢_; refl; _≗_)
open import Relation.Binary.HeterogeneousEquality using (_≅_; refl)
∷-injective : ∀ {a n} {A : Set a} {x y : A} {xs ys : Vec A n} →
(x ∷ xs) ≡ (y ∷ ys) → x ≡ y × xs ≡ ys
∷-injective refl = refl , refl
lookup-morphism :
∀ {a n} (i : Fin n) →
Morphism (applicative {a = a})
(RawMonad.rawIApplicative IdentityMonad)
lookup-morphism i = record
{ op = lookup i
; op-pure = lookup-replicate i
; op-⊛ = lookup-⊛ i
}
where
lookup-replicate : ∀ {a n} {A : Set a} (i : Fin n) →
lookup i ∘ replicate {A = A} ≗ id {A = A}
lookup-replicate zero = λ _ → refl
lookup-replicate (suc i) = lookup-replicate i
lookup-⊛ : ∀ {a b n} {A : Set a} {B : Set b}
i (fs : Vec (A → B) n) (xs : Vec A n) →
lookup i (fs ⊛ xs) ≡ (lookup i fs $ lookup i xs)
lookup-⊛ zero (f ∷ fs) (x ∷ xs) = refl
lookup-⊛ (suc i) (f ∷ fs) (x ∷ xs) = lookup-⊛ i fs xs
lookup∘tabulate : ∀ {a n} {A : Set a} (f : Fin n → A) (i : Fin n) →
lookup i (tabulate f) ≡ f i
lookup∘tabulate f zero = refl
lookup∘tabulate f (suc i) = lookup∘tabulate (f ∘ suc) i
tabulate∘lookup : ∀ {a n} {A : Set a} (xs : Vec A n) →
tabulate (flip lookup xs) ≡ xs
tabulate∘lookup [] = refl
tabulate∘lookup (x ∷ xs) = P.cong (_∷_ x) $ tabulate∘lookup xs
lookup-allFin : ∀ {n} (i : Fin n) → lookup i (allFin n) ≡ i
lookup-allFin = lookup∘tabulate id
lookup-++-< : ∀ {a} {A : Set a} {m n}
(xs : Vec A m) (ys : Vec A n) i (i<m : toℕ i < m) →
lookup i (xs ++ ys) ≡ lookup (Fin.fromℕ≤ i<m) xs
lookup-++-< [] ys i ()
lookup-++-< (x ∷ xs) ys zero (s≤s z≤n) = refl
lookup-++-< (x ∷ xs) ys (suc i) (s≤s (s≤s i<m)) =
lookup-++-< xs ys i (s≤s i<m)
lookup-++-≥ : ∀ {a} {A : Set a} {m n}
(xs : Vec A m) (ys : Vec A n) i (i≥m : toℕ i ≥ m) →
lookup i (xs ++ ys) ≡ lookup (Fin.reduce≥ i i≥m) ys
lookup-++-≥ [] ys i i≥m = refl
lookup-++-≥ (x ∷ xs) ys zero ()
lookup-++-≥ (x ∷ xs) ys (suc i) (s≤s i≥m) = lookup-++-≥ xs ys i i≥m
lookup-++-inject+ : ∀ {a} {A : Set a} {m n}
(xs : Vec A m) (ys : Vec A n) i →
lookup (Fin.inject+ n i) (xs ++ ys) ≡ lookup i xs
lookup-++-inject+ [] ys ()
lookup-++-inject+ (x ∷ xs) ys zero = refl
lookup-++-inject+ (x ∷ xs) ys (suc i) = lookup-++-inject+ xs ys i
lookup-++-+′ : ∀ {a} {A : Set a} {m n}
(xs : Vec A m) (ys : Vec A n) i →
lookup (fromℕ m +′ i) (xs ++ ys) ≡ lookup i ys
lookup-++-+′ [] ys zero = refl
lookup-++-+′ [] (y ∷ xs) (suc i) = lookup-++-+′ [] xs i
lookup-++-+′ (x ∷ xs) ys i = lookup-++-+′ xs ys i
lookup∘update : ∀ {a} {A : Set a} {n}
(i : Fin n) (xs : Vec A n) x →
lookup i (xs [ i ]≔ x) ≡ x
lookup∘update zero (_ ∷ xs) x = refl
lookup∘update (suc i) (_ ∷ xs) x = lookup∘update i xs x
lookup∘update′ : ∀ {a} {A : Set a} {n} {i j : Fin n} →
i ≢ j → ∀ (xs : Vec A n) y →
lookup i (xs [ j ]≔ y) ≡ lookup i xs
lookup∘update′ {i = zero} {zero} i≢j xs y = ⊥-elim (i≢j refl)
lookup∘update′ {i = zero} {suc j} i≢j (x ∷ xs) y = refl
lookup∘update′ {i = suc i} {zero} i≢j (x ∷ xs) y = refl
lookup∘update′ {i = suc i} {suc j} i≢j (x ∷ xs) y =
lookup∘update′ (i≢j ∘ P.cong suc) xs y
map-cong : ∀ {a b n} {A : Set a} {B : Set b} {f g : A → B} →
f ≗ g → _≗_ {A = Vec A n} (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-id : ∀ {a n} {A : Set a} → _≗_ {A = Vec A n} (map id) id
map-id [] = refl
map-id (x ∷ xs) = P.cong (_∷_ x) (map-id xs)
map-∘ : ∀ {a b c n} {A : Set a} {B : Set b} {C : Set c}
(f : B → C) (g : A → B) →
_≗_ {A = Vec A n} (map (f ∘ g)) (map f ∘ map g)
map-∘ f g [] = refl
map-∘ f g (x ∷ xs) = P.cong (_∷_ (f (g x))) (map-∘ f g xs)
tabulate-∘ : ∀ {n a b} {A : Set a} {B : Set b}
(f : A → B) (g : Fin n → A) →
tabulate (f ∘ g) ≡ map f (tabulate g)
tabulate-∘ {zero} f g = refl
tabulate-∘ {suc n} f g =
P.cong (_∷_ (f (g zero))) (tabulate-∘ f (g ∘ suc))
tabulate-allFin : ∀ {n a} {A : Set a} (f : Fin n → A) →
tabulate f ≡ map f (allFin n)
tabulate-allFin f = tabulate-∘ f id
allFin-map : ∀ n → allFin (suc n) ≡ zero ∷ map suc (allFin n)
allFin-map n = P.cong (_∷_ zero) $ tabulate-∘ suc id
map-lookup-allFin : ∀ {a} {A : Set a} {n} (xs : Vec A n) →
map (λ x → lookup x xs) (allFin n) ≡ xs
map-lookup-allFin {n = n} xs = begin
map (λ x → lookup x xs) (allFin n) ≡⟨ P.sym $ tabulate-∘ (λ x → lookup x xs) id ⟩
tabulate (λ x → lookup x xs) ≡⟨ tabulate∘lookup xs ⟩
xs ∎
where open P.≡-Reasoning
∈-tabulate : ∀ {n a} {A : Set a} (f : Fin n → A) i → f i ∈ tabulate f
∈-tabulate f zero = here
∈-tabulate f (suc i) = there (∈-tabulate (f ∘ suc) i)
∈-allFin : ∀ {n} (i : Fin n) → i ∈ allFin n
∈-allFin = ∈-tabulate id
∈-map : ∀ {a b m} {A : Set a} {B : Set b} {x : A} {xs : Vec A m}
(f : A → B) → x ∈ xs → f x ∈ map f xs
∈-map f here = here
∈-map f (there x∈xs) = there (∈-map f x∈xs)
∈-++ₗ : ∀ {a m n} {A : Set a} {x : A} {xs : Vec A m} {ys : Vec A n}
→ x ∈ xs → x ∈ xs ++ ys
∈-++ₗ here = here
∈-++ₗ (there x∈xs) = there (∈-++ₗ x∈xs)
∈-++ᵣ : ∀ {a m n} {A : Set a} {x : A} (xs : Vec A m) {ys : Vec A n}
→ x ∈ ys → x ∈ xs ++ ys
∈-++ᵣ [] x∈ys = x∈ys
∈-++ᵣ (x ∷ xs) x∈ys = there (∈-++ᵣ xs x∈ys)
∈-allPairs : ∀ {a b} {A : Set a} {B : Set b} {m n : ℕ}
{xs : Vec A m} {ys : Vec B n}
{x y} → x ∈ xs → y ∈ ys → (x , y) ∈ allPairs xs ys
∈-allPairs {xs = x ∷ xs} {ys} here y∈ys =
∈-++ₗ (∈-map (x ,_) y∈ys)
∈-allPairs {xs = x ∷ xs} {ys} (there x∈xs) y∈ys =
∈-++ᵣ (map (x ,_) ys) (∈-allPairs x∈xs y∈ys)
sum-++-commute : ∀ {m n} (xs : Vec ℕ m) {ys : Vec ℕ n} →
sum (xs ++ ys) ≡ sum xs + sum ys
sum-++-commute [] = refl
sum-++-commute (x ∷ xs) {ys} = begin
x + sum (xs ++ ys)
≡⟨ P.cong (λ p → x + p) (sum-++-commute xs) ⟩
x + (sum xs + sum ys)
≡⟨ P.sym (+-assoc x (sum xs) (sum ys)) ⟩
sum (x ∷ xs) + sum ys
∎
where
open P.≡-Reasoning
open CommutativeSemiring Nat.commutativeSemiring hiding (_+_; sym)
foldr-cong : ∀ {a b} {A : Set a}
{B₁ : ℕ → Set b}
{f₁ : ∀ {n} → A → B₁ n → B₁ (suc n)} {e₁}
{B₂ : ℕ → Set b}
{f₂ : ∀ {n} → A → B₂ n → B₂ (suc n)} {e₂} →
(∀ {n x} {y₁ : B₁ n} {y₂ : B₂ n} →
y₁ ≅ y₂ → f₁ x y₁ ≅ f₂ x y₂) →
e₁ ≅ e₂ →
∀ {n} (xs : Vec A n) →
foldr B₁ f₁ e₁ xs ≅ foldr B₂ f₂ e₂ xs
foldr-cong _ e₁=e₂ [] = e₁=e₂
foldr-cong {B₁ = B₁} f₁=f₂ e₁=e₂ (x ∷ xs) =
f₁=f₂ (foldr-cong {B₁ = B₁} f₁=f₂ e₁=e₂ xs)
foldl-cong : ∀ {a b} {A : Set a}
{B₁ : ℕ → Set b}
{f₁ : ∀ {n} → B₁ n → A → B₁ (suc n)} {e₁}
{B₂ : ℕ → Set b}
{f₂ : ∀ {n} → B₂ n → A → B₂ (suc n)} {e₂} →
(∀ {n x} {y₁ : B₁ n} {y₂ : B₂ n} →
y₁ ≅ y₂ → f₁ y₁ x ≅ f₂ y₂ x) →
e₁ ≅ e₂ →
∀ {n} (xs : Vec A n) →
foldl B₁ f₁ e₁ xs ≅ foldl B₂ f₂ e₂ xs
foldl-cong _ e₁=e₂ [] = e₁=e₂
foldl-cong {B₁ = B₁} f₁=f₂ e₁=e₂ (x ∷ xs) =
foldl-cong {B₁ = B₁ ∘ suc} f₁=f₂ (f₁=f₂ e₁=e₂) xs
foldr-universal : ∀ {a b} {A : Set a} (B : ℕ → Set b)
(f : ∀ {n} → A → B n → B (suc n)) {e}
(h : ∀ {n} → Vec A n → B n) →
h [] ≡ e →
(∀ {n} x → h ∘ _∷_ x ≗ f {n} x ∘ h) →
∀ {n} → h ≗ foldr B {n} f e
foldr-universal B f h base step [] = base
foldr-universal B f {e} h base step (x ∷ xs) = begin
h (x ∷ xs)
≡⟨ step x xs ⟩
f x (h xs)
≡⟨ P.cong (f x) (foldr-universal B f h base step xs) ⟩
f x (foldr B f e xs)
∎
where open P.≡-Reasoning
foldr-fusion : ∀ {a b c} {A : Set a}
{B : ℕ → Set b} {f : ∀ {n} → A → B n → B (suc n)} e
{C : ℕ → Set c} {g : ∀ {n} → A → C n → C (suc n)}
(h : ∀ {n} → B n → C n) →
(∀ {n} x → h ∘ f {n} x ≗ g x ∘ h) →
∀ {n} → h ∘ foldr B {n} f e ≗ foldr C g (h e)
foldr-fusion {B = B} {f} e {C} h fuse =
foldr-universal C _ _ refl (λ x xs → fuse x (foldr B f e xs))
idIsFold : ∀ {a n} {A : Set a} → id ≗ foldr (Vec A) {n} _∷_ []
idIsFold = foldr-universal _ _ id refl (λ _ _ → refl)
∈⇒List-∈ : ∀ {a} {A : Set a} {n x} {xs : Vec A n} →
x ∈ xs → x List.∈ toList xs
∈⇒List-∈ here = here P.refl
∈⇒List-∈ (there x∈) = there (∈⇒List-∈ x∈)
List-∈⇒∈ : ∀ {a} {A : Set a} {x : A} {xs} →
x List.∈ xs → x ∈ fromList xs
List-∈⇒∈ (here P.refl) = here
List-∈⇒∈ (there x∈) = there (List-∈⇒∈ x∈)
proof-irrelevance-[]= : ∀ {a} {A : Set a} {n} {xs : Vec A n} {i x} →
(p q : xs [ i ]= x) → p ≡ q
proof-irrelevance-[]= here here = refl
proof-irrelevance-[]= (there xs[i]=x) (there xs[i]=x') =
P.cong there (proof-irrelevance-[]= xs[i]=x xs[i]=x')
[]=↔lookup : ∀ {a n i} {A : Set a} {x} {xs : Vec A n} →
xs [ i ]= x ↔ lookup i xs ≡ x
[]=↔lookup {i = i} {x = x} {xs} = record
{ to = P.→-to-⟶ to
; from = P.→-to-⟶ (from i xs)
; inverse-of = record
{ left-inverse-of = λ _ → proof-irrelevance-[]= _ _
; right-inverse-of = λ _ → P.proof-irrelevance _ _
}
}
where
to : ∀ {n xs} {i : Fin n} → xs [ i ]= x → lookup i xs ≡ x
to here = refl
to (there xs[i]=x) = to xs[i]=x
from : ∀ {n} (i : Fin n) xs → lookup i xs ≡ x → xs [ i ]= x
from zero (.x ∷ _) refl = here
from (suc i) (_ ∷ xs) p = there (from i xs p)
[]≔-idempotent :
∀ {n a} {A : Set a} (xs : Vec A n) (i : Fin n) {x₁ x₂ : A} →
(xs [ i ]≔ x₁) [ i ]≔ x₂ ≡ xs [ i ]≔ x₂
[]≔-idempotent [] ()
[]≔-idempotent (x ∷ xs) zero = refl
[]≔-idempotent (x ∷ xs) (suc i) = P.cong (_∷_ x) $ []≔-idempotent xs i
[]≔-commutes :
∀ {n a} {A : Set a} (xs : Vec A n) (i j : Fin n) {x y : A} →
i ≢ j → (xs [ i ]≔ x) [ j ]≔ y ≡ (xs [ j ]≔ y) [ i ]≔ x
[]≔-commutes [] () () _
[]≔-commutes (x ∷ xs) zero zero 0≢0 = ⊥-elim $ 0≢0 refl
[]≔-commutes (x ∷ xs) zero (suc i) _ = refl
[]≔-commutes (x ∷ xs) (suc i) zero _ = refl
[]≔-commutes (x ∷ xs) (suc i) (suc j) i≢j =
P.cong (_∷_ x) $ []≔-commutes xs i j (i≢j ∘ P.cong suc)
[]≔-updates : ∀ {n a} {A : Set a} (xs : Vec A n) (i : Fin n) {x : A} →
(xs [ i ]≔ x) [ i ]= x
[]≔-updates [] ()
[]≔-updates (x ∷ xs) zero = here
[]≔-updates (x ∷ xs) (suc i) = there ([]≔-updates xs i)
[]≔-minimal :
∀ {n a} {A : Set a} (xs : Vec A n) (i j : Fin n) {x y : A} →
i ≢ j → xs [ i ]= x → (xs [ j ]≔ y) [ i ]= x
[]≔-minimal [] () () _ _
[]≔-minimal (x ∷ xs) .zero zero 0≢0 here = ⊥-elim $ 0≢0 refl
[]≔-minimal (x ∷ xs) .zero (suc j) _ here = here
[]≔-minimal (x ∷ xs) (suc i) zero _ (there loc) = there loc
[]≔-minimal (x ∷ xs) (suc i) (suc j) i≢j (there loc) =
there ([]≔-minimal xs i j (i≢j ∘ P.cong suc) loc)
map-[]≔ : ∀ {n a b} {A : Set a} {B : Set b}
(f : A → B) (xs : Vec A n) (i : Fin n) {x : A} →
map f (xs [ i ]≔ x) ≡ map f xs [ i ]≔ f x
map-[]≔ f [] ()
map-[]≔ f (x ∷ xs) zero = refl
map-[]≔ f (x ∷ xs) (suc i) = P.cong (_∷_ _) $ map-[]≔ f xs i
[]≔-lookup : ∀ {a} {A : Set a} {n} (xs : Vec A n) (i : Fin n) →
xs [ i ]≔ lookup i xs ≡ xs
[]≔-lookup [] ()
[]≔-lookup (x ∷ xs) zero = refl
[]≔-lookup (x ∷ xs) (suc i) = P.cong (_∷_ x) $ []≔-lookup xs i
[]≔-++-inject+ : ∀ {a} {A : Set a} {m n x}
(xs : Vec A m) (ys : Vec A n) i →
(xs ++ ys) [ Fin.inject+ n i ]≔ x ≡ xs [ i ]≔ x ++ ys
[]≔-++-inject+ [] ys ()
[]≔-++-inject+ (x ∷ xs) ys zero = refl
[]≔-++-inject+ (x ∷ xs) ys (suc i) =
P.cong (_∷_ x) $ []≔-++-inject+ xs ys i
unzip∘zip : ∀ {a n} {A B : Set a} (xs : Vec A n) (ys : Vec B n) →
unzip (zip xs ys) ≡ (xs , ys)
unzip∘zip [] [] = refl
unzip∘zip (x ∷ xs) (y ∷ ys) =
P.cong (Prod.map (_∷_ x) (_∷_ y)) (unzip∘zip xs ys)
zip∘unzip : ∀ {a n} {A B : Set a} (xys : Vec (A × B) n) →
(Prod.uncurry zip) (unzip xys) ≡ xys
zip∘unzip [] = refl
zip∘unzip ((x , y) ∷ xys) = P.cong (_∷_ (x , y)) (zip∘unzip xys)
×v↔v× : ∀ {a n} {A B : Set a} → (Vec A n × Vec B n) ↔ Vec (A × B) n
×v↔v× = record
{ to = P.→-to-⟶ (Prod.uncurry zip)
; from = P.→-to-⟶ unzip
; inverse-of = record
{ left-inverse-of = Prod.uncurry unzip∘zip
; right-inverse-of = zip∘unzip
}
}
map-proj₁-zip : ∀ {a n} {A B : Set a} (xs : Vec A n) (ys : Vec B n) →
map proj₁ (zip xs ys) ≡ xs
map-proj₁-zip [] [] = refl
map-proj₁-zip (x ∷ xs) (y ∷ ys) = P.cong (_∷_ x) (map-proj₁-zip xs ys)
map-proj₂-zip : ∀ {a n} {A B : Set a} (xs : Vec A n) (ys : Vec B n) →
map proj₂ (zip xs ys) ≡ ys
map-proj₂-zip [] [] = refl
map-proj₂-zip (x ∷ xs) (y ∷ ys) = P.cong (_∷_ y) (map-proj₂-zip xs ys)
map-<,>-zip : ∀ {a n} {A B₁ B₂ : Set a}
(f : A → B₁) (g : A → B₂) (xs : Vec A n) →
map < f , g > xs ≡ zip (map f xs) (map g xs)
map-<,>-zip f g [] = P.refl
map-<,>-zip f g (x ∷ xs) = P.cong (_∷_ (f x , g x)) (map-<,>-zip f g xs)
map-zip : ∀ {a n} {A₁ A₂ B₁ B₂ : Set a}
(f : A₁ → A₂) (g : B₁ → B₂) (xs : Vec A₁ n) (ys : Vec B₁ n) →
map (Prod.map f g) (zip xs ys) ≡ zip (map f xs) (map g ys)
map-zip f g [] [] = refl
map-zip f g (x ∷ xs) (y ∷ ys) = P.cong (_∷_ (f x , g y)) (map-zip f g xs ys)
map-unzip : ∀ {a n} {A₁ A₂ B₁ B₂ : Set a}
(f : A₁ → A₂) (g : B₁ → B₂) (xys : Vec (A₁ × B₁) n) →
let xs , ys = unzip xys
in (map f xs , map g ys) ≡ unzip (map (Prod.map f g) xys)
map-unzip f g [] = refl
map-unzip f g ((x , y) ∷ xys) =
P.cong (Prod.map (_∷_ (f x)) (_∷_ (g y))) (map-unzip f g xys)
lookup-unzip : ∀ {a n} {A B : Set a} (i : Fin n) (xys : Vec (A × B) n) →
let xs , ys = unzip xys
in (lookup i xs , lookup i ys) ≡ lookup i xys
lookup-unzip () []
lookup-unzip zero ((x , y) ∷ xys) = refl
lookup-unzip (suc i) ((x , y) ∷ xys) = lookup-unzip i xys
lookup-zip : ∀ {a n} {A B : Set a}
(i : Fin n) (xs : Vec A n) (ys : Vec B n) →
lookup i (zip xs ys) ≡ (lookup i xs , lookup i ys)
lookup-zip i xs ys = begin
lookup i (zip xs ys)
≡⟨ P.sym (lookup-unzip i (zip xs ys)) ⟩
lookup i (proj₁ (unzip (zip xs ys))) , lookup i (proj₂ (unzip (zip xs ys)))
≡⟨ P.cong₂ _,_ (P.cong (lookup i ∘ proj₁) (unzip∘zip xs ys))
(P.cong (lookup i ∘ proj₂) (unzip∘zip xs ys)) ⟩
lookup i xs , lookup i ys
∎
where open P.≡-Reasoning