------------------------------------------------------------------------
-- 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  : Level
A B C : Set a

------------------------------------------------------------------------
-- Propositional equality

open import Agda.Builtin.Equality public

infix 4 _≢_
_≢_ : {A : Set a}  Rel A a
x  y = ¬ x  y

------------------------------------------------------------------------
-- 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} _≡_
subst P refl p = p

subst₂ :  (_∼_ : REL A B ) {x y u v}  x  y  u  v  x  u  y  v
subst₂ _ refl refl p = p

resp :  (P : A  Set )  P Respects _≡_
resp P refl p = p

respˡ :  ( : Rel A )   Respectsˡ _≡_
respˡ _∼_ refl x∼y = x∼y

respʳ :  ( : Rel A )   Respectsʳ _≡_
respʳ _∼_ refl x∼y = x∼y

resp₂ :  ( : Rel A )   Respects₂ _≡_
resp₂ _∼_ = respʳ _∼_ , respˡ _∼_

------------------------------------------------------------------------
-- Properties of _≢_

≢-sym : Symmetric {A = A} _≢_
≢-sym x≢y =  x≢y  sym

------------------------------------------------------------------------
-- Convenient syntax for equational reasoning

-- This is a special instance of `Relation.Binary.Reasoning.Setoid`.
-- Rather than instantiating the latter with (setoid A), we reimplement
-- equation chains from scratch since then goals are printed much more

module ≡-Reasoning {A : Set a} where

infix  3 _∎
infixr 2 _≡⟨⟩_ step-≡ step-≡˘
infix  1 begin_

begin_ : ∀{x y : A}  x  y  x  y
begin_ x≡y = x≡y

_≡⟨⟩_ :  (x {y} : A)  x  y  x  y
_ ≡⟨⟩ x≡y = x≡y

step-≡ :  (x {y z} : A)  y  z  x  y  x  z
step-≡ _ y≡z x≡y = trans x≡y y≡z

step-≡˘ :  (x {y z} : A)  y  z  y  x  x  z
step-≡˘ _ y≡z y≡x = trans (sym y≡x) y≡z

_∎ :  (x : A)  x  x
_∎ _ = refl

syntax step-≡  x y≡z x≡y = x ≡⟨  x≡y  y≡z
syntax step-≡˘ x y≡z y≡x = x ≡˘⟨ y≡x  y≡z