------------------------------------------------------------------------
-- The Agda standard library
--
-- Indexed binary relations
------------------------------------------------------------------------

-- The contents of this module should be accessed via
-- `Relation.Binary.Indexed.Heterogeneous`.

{-# OPTIONS --cubical-compatible --safe #-}

module Relation.Binary.Indexed.Heterogeneous.Bundles where

open import Function.Base
open import Level using (suc; _⊔_)
open import Relation.Binary.Core using (_⇒_)
open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
open import Relation.Binary.Indexed.Heterogeneous.Core
open import Relation.Binary.Indexed.Heterogeneous.Structures

------------------------------------------------------------------------
-- Definitions

record IndexedSetoid {i} (I : Set i) c ℓ : Set (suc (i ⊔ c ⊔ ℓ)) where
  infix  _≈_
  field
    Carrier       : I → Set c
    _≈_           : IRel Carrier ℓ
    isEquivalence : IsIndexedEquivalence Carrier _≈_

  open IsIndexedEquivalence isEquivalence public


record IndexedPreorder {i} (I : Set i) c ℓ₁ ℓ₂ :
                       Set (suc (i ⊔ c ⊔ ℓ₁ ⊔ ℓ₂)) where
  infix  _≈_ _∼_
  field
    Carrier    : I → Set c
    _≈_        : IRel Carrier ℓ₁  -- The underlying equality.
    _∼_        : IRel Carrier ℓ₂  -- The relation.
    isPreorder : IsIndexedPreorder Carrier _≈_ _∼_

  open IsIndexedPreorder isPreorder public