------------------------------------------------------------------------
-- The Agda standard library
--
-- Creates trivially indexed records from their non-indexed counterpart.
------------------------------------------------------------------------

{-# OPTIONS --without-K #-}

module Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
  {i} {I : Set i} where

open import Relation.Binary
open import Relation.Binary.Indexed.Heterogeneous hiding (Rel)
  hiding (IsEquivalence; Setoid)

------------------------------------------------------------------------
-- Structures

module _ {a} {A : Set a} where

  private
    Aᵢ : I  Set a
    Aᵢ i = A

  isIndexedEquivalence :  {} {_≈_ : Rel A }  IsEquivalence _≈_ 
                         IsIndexedEquivalence Aᵢ _≈_
  isIndexedEquivalence isEq = record
    { refl  = refl
    ; sym   = sym
    ; trans = trans
    }
    where open IsEquivalence isEq

  isIndexedPreorder :  {ℓ₁ ℓ₂} {_≈_ : Rel A ℓ₁} {_∼_ : Rel A ℓ₂} 
                      IsPreorder _≈_ _∼_ 
                      IsIndexedPreorder Aᵢ _≈_ _∼_
  isIndexedPreorder isPreorder = record
    { isEquivalence = isIndexedEquivalence isEquivalence
    ; reflexive     = reflexive
    ; trans         = trans
    }
    where open IsPreorder isPreorder

------------------------------------------------------------------------
-- Packages

indexedSetoid :  {a }  Setoid a   IndexedSetoid I a 
indexedSetoid S = record
  { isEquivalence = isIndexedEquivalence isEquivalence
  }
  where open Setoid S

indexedPreorder :  {a ℓ₁ ℓ₂}  Preorder a ℓ₁ ℓ₂ 
                  IndexedPreorder I a ℓ₁ ℓ₂
indexedPreorder O = record
  { isPreorder = isIndexedPreorder isPreorder
  }
  where open Preorder O