{-# OPTIONS --without-K --safe #-}
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)
module _ {a} {A : Set a} where
private
Aᵢ : I → Set a
Aᵢ i = A
isIndexedEquivalence : ∀ {ℓ} {_≈_ : Rel A ℓ} → 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
indexedSetoid : ∀ {a ℓ} → Setoid a ℓ → IndexedSetoid I a ℓ
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