------------------------------------------------------------------------
-- The Agda standard library
--
-- The binary relation defined by a constant
------------------------------------------------------------------------
{-# OPTIONS --cubical-compatible --safe #-}
module Relation.Binary.Construct.Constant where
open import Level using (Level)
open import Relation.Binary.Bundles using (Setoid)
open import Relation.Binary.Structures using (IsEquivalence)
open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)
private
variable
a b c : Level
A : Set a
B : Set b
------------------------------------------------------------------------
-- Re-export definition
open import Relation.Binary.Construct.Constant.Core public
------------------------------------------------------------------------
-- Properties
module _ {a c} (A : Set a) {C : Set c} where
refl : C → Reflexive {A = A} (Const C)
refl c = c
sym : Symmetric {A = A} (Const C)
sym c = c
trans : Transitive {A = A} (Const C)
trans c d = c
isEquivalence : C → IsEquivalence {A = A} (Const C)
isEquivalence c = record
{ refl = λ {x} → refl c {x}
; sym = λ {x} {y} → sym {x} {y}
; trans = λ {x} {y} {z} → trans {x} {y} {z}
}
setoid : C → Setoid a c
setoid x = record { isEquivalence = isEquivalence x }
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