{-# OPTIONS --cubical-compatible --safe #-}
module Function.Nary.NonDependent where
open import Level using (Level; 0ℓ; _⊔_; Lift)
open import Data.Nat.Base using (ℕ; zero; suc)
open import Data.Fin.Base using (Fin; zero; suc)
open import Data.Product.Base using (_×_; _,_)
open import Data.Product.Nary.NonDependent
open import Function.Base using (_∘′_; _$′_; const; flip)
open import Relation.Unary using (IUniversal)
open import Relation.Binary.PropositionalEquality
private
variable
a b r : Level
A : Set a
B : Set b
open import Function.Nary.NonDependent.Base public
ltabulate : ∀ n → (Fin n → Level) → Levels n
ltabulate zero f = _
ltabulate (suc n) f = f zero , ltabulate n (f ∘′ suc)
lreplicate : ∀ n → Level → Levels n
lreplicate n ℓ = ltabulate n (const ℓ)
0ℓs : ∀[ Levels ]
0ℓs = lreplicate _ 0ℓ
module _ n {ls} {as : Sets n ls} {R : Set r} (f : as ⇉ R) where
private
g : Product n as → R
g = uncurryₙ n f
Congruentₙ : Set (r Level.⊔ ⨆ n ls)
Congruentₙ = ∀ {l r} → Equalₙ n l r ⇉ (g l ≡ g r)
congₙ : Congruentₙ
congₙ = curryₙ n (cong g ∘′ fromEqualₙ n)
module _ m n {ls ls′} {as : Sets m ls} {bs : Sets n ls′}
(f : as ⇉ (A → bs ⇉ B)) where
private
g : Product m as → A → Product n bs → B
g vs a ws = uncurryₙ n (uncurryₙ m f vs a) ws
congAt : ∀ {vs ws a₁ a₂} → a₁ ≡ a₂ → g vs a₁ ws ≡ g vs a₂ ws
congAt {vs} {ws} = cong (λ a → g vs a ws)
module _ n {ls} {as : Sets n ls} {R : Set r} (con : as ⇉ R) where
private
c : Product n as → R
c = uncurryₙ n con
Injectiveₙ : Set (r Level.⊔ ⨆ n ls)
Injectiveₙ = ∀ {l r} → c l ≡ c r → Product n (Equalₙ n l r)
injectiveₙ : (∀ {l r} → c l ≡ c r → l ≡ r) → Injectiveₙ
injectiveₙ con-inj eq = toEqualₙ n (con-inj eq)