module Data.Vec.Properties where
open import Data.Empty using (⊥-elim)
open import Data.Fin as Fin using (Fin; zero; suc; toℕ; fromℕ)
open import Data.Fin.Properties using (_+′_)
open import Data.List.Any using (here; there)
import Data.List.Any.Membership.Propositional as List
open import Data.Nat
open import Data.Nat.Properties using (+-assoc)
open import Data.Product as Prod using (_×_; _,_; proj₁; proj₂; <_,_>)
open import Data.Vec
open import Function
open import Function.Inverse using (_↔_)
open import Relation.Binary
open import Relation.Binary.PropositionalEquality as P
using (_≡_; _≢_; refl; _≗_)
open import Relation.Binary.HeterogeneousEquality using (_≅_; refl)
module _ {a} {A : Set a} {n} {x y : A} {xs ys : Vec A n} where
∷-injectiveˡ : x ∷ xs ≡ y ∷ ys → x ≡ y
∷-injectiveˡ refl = refl
∷-injectiveʳ : x ∷ xs ≡ y ∷ ys → xs ≡ ys
∷-injectiveʳ refl = refl
∷-injective : (x ∷ xs) ≡ (y ∷ ys) → x ≡ y × xs ≡ ys
∷-injective refl = refl , refl
module _ {a} {A : Set a} where
[]=-irrelevance : ∀ {n} {xs : Vec A n} {i x} →
(p q : xs [ i ]= x) → p ≡ q
[]=-irrelevance here here = refl
[]=-irrelevance (there xs[i]=x) (there xs[i]=x') =
P.cong there ([]=-irrelevance xs[i]=x xs[i]=x')
module _ {a} {A : Set a} where
[]=⇒lookup : ∀ {n} {x : A} {xs} {i : Fin n} →
xs [ i ]= x → lookup i xs ≡ x
[]=⇒lookup here = refl
[]=⇒lookup (there xs[i]=x) = []=⇒lookup xs[i]=x
lookup⇒[]= : ∀ {n} (i : Fin n) {x : A} xs →
lookup i xs ≡ x → xs [ i ]= x
lookup⇒[]= zero (_ ∷ _) refl = here
lookup⇒[]= (suc i) (_ ∷ xs) p = there (lookup⇒[]= i xs p)
[]=↔lookup : ∀ {n i} {x} {xs : Vec A n} →
xs [ i ]= x ↔ lookup i xs ≡ x
[]=↔lookup = record
{ to = P.→-to-⟶ []=⇒lookup
; from = P.→-to-⟶ (lookup⇒[]= _ _)
; inverse-of = record
{ left-inverse-of = λ _ → []=-irrelevance _ _
; right-inverse-of = λ _ → P.≡-irrelevance _ _
}
}
module _ {a} {A : Set a} where
[]≔-idempotent : ∀ {n} (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} (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} (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} (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)
[]≔-lookup : ∀ {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
lookup∘update : ∀ {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′ : ∀ {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-id : ∀ {a n} {A : Set a} → map {n = n} {A} id ≗ id
map-id [] = refl
map-id (x ∷ xs) = P.cong (x ∷_) (map-id xs)
map-cong : ∀ {a b n} {A : Set a} {B : Set b} {f g : A → B} →
f ≗ g → map {n = n} 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-∘ : ∀ {a b c n} {A : Set a} {B : Set b} {C : Set c}
(f : B → C) (g : A → B) →
map {n = n} (f ∘ g) ≗ map f ∘ map g
map-∘ f g [] = refl
map-∘ f g (x ∷ xs) = P.cong (f (g x) ∷_) (map-∘ f g xs)
lookup-map : ∀ {a b n} {A : Set a} {B : Set b}
(i : Fin n) (f : A → B) (xs : Vec A n) →
lookup i (map f xs) ≡ f (lookup i xs)
lookup-map zero f (x ∷ xs) = refl
lookup-map (suc i) f (x ∷ xs) = lookup-map i f xs
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
module _ {a} {A : Set a} where
lookup-++-< : ∀ {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-++-≥ : ∀ {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+ : ∀ {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-++-+′ : ∀ {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
[]≔-++-inject+ : ∀ {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
module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} where
zipWith-map₁ : ∀ {n} (_⊕_ : B → C → D) (f : A → B)
(xs : Vec A n) (ys : Vec C n) →
zipWith _⊕_ (map f xs) ys ≡ zipWith (λ x y → f x ⊕ y) xs ys
zipWith-map₁ _⊕_ f [] [] = refl
zipWith-map₁ _⊕_ f (x ∷ xs) (y ∷ ys) =
P.cong (f x ⊕ y ∷_) (zipWith-map₁ _⊕_ f xs ys)
zipWith-map₂ : ∀ {n} (_⊕_ : A → C → D) (f : B → C)
(xs : Vec A n) (ys : Vec B n) →
zipWith _⊕_ xs (map f ys) ≡ zipWith (λ x y → x ⊕ f y) xs ys
zipWith-map₂ _⊕_ f [] [] = refl
zipWith-map₂ _⊕_ f (x ∷ xs) (y ∷ ys) =
P.cong (x ⊕ f y ∷_) (zipWith-map₂ _⊕_ f xs ys)
module _ {a b} {A : Set a} {B : Set b} where
lookup-zip : ∀ {n} (i : Fin n) (xs : Vec A n) (ys : Vec B n) →
lookup i (zip xs ys) ≡ (lookup i xs , lookup i ys)
lookup-zip zero (x ∷ xs) (y ∷ ys) = refl
lookup-zip (suc i) (x ∷ xs) (y ∷ ys) = lookup-zip i xs ys
map-proj₁-zip : ∀ {n} (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 : ∀ {n} (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 b c n} {A : Set a} {B : Set b} {C : Set c}
(f : A → B) (g : A → C) (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 (_ ∷_) (map-<,>-zip f g xs)
map-zip : ∀ {a b c d n} {A : Set a} {B : Set b} {C : Set c} {D : Set d}
(f : A → B) (g : C → D) (xs : Vec A n) (ys : Vec C 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 (_ ∷_) (map-zip f g xs ys)
module _ {a b} {A : Set a} {B : Set b} where
lookup-unzip : ∀ {n} (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
map-unzip : ∀ {c d n} {C : Set c} {D : Set d}
(f : A → B) (g : C → D) (xys : Vec (A × C) 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)
unzip∘zip : ∀ {n} (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 : ∀ {n} (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× : ∀ {n} → (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
}
}
module _ {a b} {A : Set a} {B : Set b} where
lookup-⊛ : ∀ {n} 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
map-is-⊛ : ∀ {n} (f : A → B) (xs : Vec A n) →
map f xs ≡ (replicate f ⊛ xs)
map-is-⊛ f [] = refl
map-is-⊛ f (x ∷ xs) = P.cong (_ ∷_) (map-is-⊛ f xs)
⊛-is-zipWith : ∀ {n} (fs : Vec (A → B) n) (xs : Vec A n) →
(fs ⊛ xs) ≡ zipWith _$_ fs xs
⊛-is-zipWith [] [] = refl
⊛-is-zipWith (f ∷ fs) (x ∷ xs) = P.cong (f x ∷_) (⊛-is-zipWith fs xs)
zipWith-is-⊛ : ∀ {c} {C : Set c} {n} (f : A → B → C) →
(xs : Vec A n) (ys : Vec B n) →
zipWith f xs ys ≡ (replicate f ⊛ xs ⊛ ys)
zipWith-is-⊛ f [] [] = refl
zipWith-is-⊛ f (x ∷ xs) (y ∷ ys) = P.cong (_ ∷_) (zipWith-is-⊛ f xs ys)
foldr-cong : ∀ {a b} {A : Set a}
{B : ℕ → Set b} {f : ∀ {n} → A → B n → B (suc n)} {d}
{C : ℕ → Set b} {g : ∀ {n} → A → C n → C (suc n)} {e} →
(∀ {n x} {y : B n} {z : C n} → y ≅ z → f x y ≅ g x z) →
d ≅ e → ∀ {n} (xs : Vec A n) →
foldr B f d xs ≅ foldr C g e xs
foldr-cong _ d≅e [] = d≅e
foldr-cong f≅g d≅e (x ∷ xs) = f≅g (foldr-cong f≅g d≅e 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)
foldl-cong : ∀ {a b} {A : Set a}
{B : ℕ → Set b} {f : ∀ {n} → B n → A → B (suc n)} {d}
{C : ℕ → Set b} {g : ∀ {n} → C n → A → C (suc n)} {e} →
(∀ {n x} {y : B n} {z : C n} → y ≅ z → f y x ≅ g z x) →
d ≅ e → ∀ {n} (xs : Vec A n) →
foldl B f d xs ≅ foldl C g e xs
foldl-cong _ d≅e [] = d≅e
foldl-cong f≅g d≅e (x ∷ xs) = foldl-cong f≅g (f≅g d≅e) xs
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 (x +_) (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
lookup-replicate : ∀ {a n} {A : Set a} (i : Fin n) (x : A) →
lookup i (replicate x) ≡ x
lookup-replicate zero = λ _ → refl
lookup-replicate (suc i) = lookup-replicate i
map-replicate : ∀ {a b} {A : Set a} {B : Set b} (f : A → B) (x : A) →
∀ n → map f (replicate x) ≡ replicate {n = n} (f x)
map-replicate f x zero = refl
map-replicate f x (suc n) = P.cong (f x ∷_) (map-replicate f x n)
module _ {a b c} {A : Set a} {B : Set b} {C : Set c} where
zipWith-replicate₁ : ∀ {n} (_⊕_ : A → B → C) (x : A) (ys : Vec B n) →
zipWith _⊕_ (replicate x) ys ≡ map (x ⊕_) ys
zipWith-replicate₁ _⊕_ x [] = refl
zipWith-replicate₁ _⊕_ x (y ∷ ys) =
P.cong (x ⊕ y ∷_) (zipWith-replicate₁ _⊕_ x ys)
zipWith-replicate₂ : ∀ {n} (_⊕_ : A → B → C) (xs : Vec A n) (y : B) →
zipWith _⊕_ xs (replicate y) ≡ map (_⊕ y) xs
zipWith-replicate₂ _⊕_ [] y = refl
zipWith-replicate₂ _⊕_ (x ∷ xs) y =
P.cong (x ⊕ y ∷_) (zipWith-replicate₂ _⊕_ xs y)
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)
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-cong : ∀ {n a} {A : Set a} {f g : Fin n → A} → f ≗ g → tabulate f ≡ tabulate g
tabulate-cong {zero} p = refl
tabulate-cong {suc n} p = P.cong₂ _∷_ (p zero) (tabulate-cong (p ∘ suc))
lookup-allFin : ∀ {n} (i : Fin n) → lookup i (allFin n) ≡ i
lookup-allFin = lookup∘tabulate id
allFin-map : ∀ n → allFin (suc n) ≡ zero ∷ map suc (allFin n)
allFin-map n = P.cong (zero ∷_) $ tabulate-∘ suc id
tabulate-allFin : ∀ {n a} {A : Set a} (f : Fin n → A) →
tabulate f ≡ map f (allFin n)
tabulate-allFin f = tabulate-∘ f 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
∈-++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)
∈-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)
∈-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
∈-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)
∈⇒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-[]= = []=-irrelevance