{-# OPTIONS --cubical-compatible --safe #-}
module Relation.Binary.PropositionalEquality.Core where
open import Data.Product.Base using (_,_)
open import Function.Base using (_∘_)
open import Level
open import Relation.Binary.Core
open import Relation.Binary.Definitions
open import Relation.Nullary.Negation.Core using (¬_)
private
variable
a b ℓ : Level
A B C : Set a
open import Agda.Builtin.Equality public
infix 4 _≢_
_≢_ : {A : Set a} → Rel A a
x ≢ y = ¬ x ≡ y
infix 4 _≗_
_≗_ : (f g : A → B) → Set _
_≗_ {A = A} {B = B} f g = ∀ x → f x ≡ g x
pattern erefl x = refl {x = x}
cong : ∀ (f : A → B) {x y} → x ≡ y → f x ≡ f y
cong f refl = refl
cong′ : ∀ {f : A → B} x → f x ≡ f x
cong′ _ = refl
icong : ∀ {f : A → B} {x y} → x ≡ y → f x ≡ f y
icong = cong _
icong′ : ∀ {f : A → B} x → f x ≡ f x
icong′ _ = refl
cong₂ : ∀ (f : A → B → C) {x y u v} → x ≡ y → u ≡ v → f x u ≡ f y v
cong₂ f refl refl = refl
cong-app : ∀ {A : Set a} {B : A → Set b} {f g : (x : A) → B x} →
f ≡ g → (x : A) → f x ≡ g x
cong-app refl x = refl
sym : Symmetric {A = A} _≡_
sym refl = refl
trans : Transitive {A = A} _≡_
trans refl eq = eq
subst : Substitutive {A = A} _≡_ ℓ
subst P refl p = p
subst₂ : ∀ (_∼_ : REL A B ℓ) {x y u v} → x ≡ y → u ≡ v → x ∼ u → y ∼ v
subst₂ _ refl refl p = p
resp : ∀ (P : A → Set ℓ) → P Respects _≡_
resp P refl p = p
respˡ : ∀ (∼ : Rel A ℓ) → ∼ Respectsˡ _≡_
respˡ _∼_ refl x∼y = x∼y
respʳ : ∀ (∼ : Rel A ℓ) → ∼ Respectsʳ _≡_
respʳ _∼_ refl x∼y = x∼y
resp₂ : ∀ (∼ : Rel A ℓ) → ∼ Respects₂ _≡_
resp₂ _∼_ = respʳ _∼_ , respˡ _∼_
≢-sym : Symmetric {A = A} _≢_
≢-sym x≢y = x≢y ∘ sym