------------------------------------------------------------------------
-- The Agda standard library
--
-- Lemmas relating algebraic definitions (such as associativity and
-- commutativity) that don't require the equality relation to be a setoid.
------------------------------------------------------------------------
 
{-# OPTIONS --cubical-compatible --safe #-}
 
module Algebra.Consequences.Base
  {a} {A : Set a} where
 
open import Algebra.Core
open import Algebra.Definitions
open import Data.Sum.Base
open import Relation.Binary.Core
open import Relation.Binary.Definitions using (Reflexive)
 
module _ {ℓ} {_•_ : Op₂ A} (_≈_ : Rel A ℓ) where
 
  sel⇒idem : Selective _≈_ _•_ → Idempotent _≈_ _•_
  sel⇒idem sel x = reduce (sel x x)
 
module _ {ℓ} {f : Op₁ A} (_≈_ : Rel A ℓ) where
 
  reflexive∧selfInverse⇒involutive : Reflexive _≈_ →
                                     SelfInverse _≈_ f →
                                     Involutive _≈_ f
  reflexive∧selfInverse⇒involutive refl inv _ = inv refl
 
------------------------------------------------------------------------
-- DEPRECATED NAMES
------------------------------------------------------------------------
-- Please use the new names as continuing support for the old names is
-- not guaranteed.
 
-- Version 2.0
 
reflexive+selfInverse⇒involutive = reflexive∧selfInverse⇒involutive
{-# WARNING_ON_USAGE reflexive+selfInverse⇒involutive
"Warning: reflexive+selfInverse⇒involutive was deprecated in v2.0.
Please use reflexive∧selfInverse⇒involutive instead."
#-}