2023-09-02 20:36:12 -07:00
|
|
|
|
module Equivalence where
|
|
|
|
|
|
|
|
|
|
|
|
open import Data.Product using (_×_; Σ; _,_; proj₁; proj₂)
|
|
|
|
|
|
open import Relation.Binary.Definitions
|
|
|
|
|
|
open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym)
|
|
|
|
|
|
|
2024-02-18 21:46:42 -08:00
|
|
|
|
module _ {a} (A : Set a) (_≈_ : A → A → Set a) where
|
|
|
|
|
|
IsReflexive : Set a
|
|
|
|
|
|
IsReflexive = ∀ {a : A} → a ≈ a
|
|
|
|
|
|
|
|
|
|
|
|
IsSymmetric : Set a
|
|
|
|
|
|
IsSymmetric = ∀ {a b : A} → a ≈ b → b ≈ a
|
|
|
|
|
|
|
|
|
|
|
|
IsTransitive : Set a
|
|
|
|
|
|
IsTransitive = ∀ {a b c : A} → a ≈ b → b ≈ c → a ≈ c
|
|
|
|
|
|
|
|
|
|
|
|
record IsEquivalence : Set a where
|
|
|
|
|
|
field
|
|
|
|
|
|
≈-refl : IsReflexive
|
|
|
|
|
|
≈-sym : IsSymmetric
|
|
|
|
|
|
≈-trans : IsTransitive
|
