{-# OPTIONS --without-K --safe #-}
open import Relation.Binary.Indexed.Heterogeneous.Core
module Relation.Binary.Indexed.Heterogeneous.Structures
{i a ℓ} {I : Set i} (A : I → Set a) (_≈_ : IRel A ℓ)
where
open import Function.Base
open import Level using (suc; _⊔_)
open import Relation.Binary using (_⇒_)
open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
open import Relation.Binary.Indexed.Heterogeneous.Definitions
record IsIndexedEquivalence : Set (i ⊔ a ⊔ ℓ) where
field
refl : Reflexive A _≈_
sym : Symmetric A _≈_
trans : Transitive A _≈_
reflexive : ∀ {i} → _≡_ ⟨ _⇒_ ⟩ _≈_ {i}
reflexive P.refl = refl
record IsIndexedPreorder {ℓ₂} (_∼_ : IRel A ℓ₂) : Set (i ⊔ a ⊔ ℓ ⊔ ℓ₂) where
field
isEquivalence : IsIndexedEquivalence
reflexive : ∀ {i j} → (_≈_ {i} {j}) ⟨ _⇒_ ⟩ _∼_
trans : Transitive A _∼_
module Eq = IsIndexedEquivalence isEquivalence
refl : Reflexive A _∼_
refl = reflexive Eq.refl