------------------------------------------------------------------------
-- The Agda standard library
--
-- Propositional equality
--
-- This file contains some core definitions which are re-exported by
-- Relation.Binary.PropositionalEquality.
------------------------------------------------------------------------
 
{-# 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
 
------------------------------------------------------------------------
-- Propositional equality
 
open import Agda.Builtin.Equality public
 
infix  _≢_
_≢_ : {A : Set a} → Rel A a
x ≢ y = ¬ x ≡ y
 
------------------------------------------------------------------------
-- Pointwise equality
 
infix  _≗_
 
_≗_ : (f g : A → B) → Set _
_≗_ {A = A} {B = B} f g = ∀ x → f x ≡ g x
 
 
------------------------------------------------------------------------
-- A variant of `refl` where the argument is explicit
 
pattern erefl x = refl {x = x}
 
------------------------------------------------------------------------
-- Congruence lemmas
 
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
 
------------------------------------------------------------------------
-- Properties of _≡_
 
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 _∼_
 
------------------------------------------------------------------------
-- Properties of _≢_
 
≢-sym : Symmetric {A = A} _≢_
≢-sym x≢y =  x≢y ∘ sym