------------------------------------------------------------------------
-- The Agda standard library
--
-- Propositional equality
--
-- This file contains some core definitions which are re-exported by
-- Relation.Binary.PropositionalEquality.
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

module Relation.Binary.PropositionalEquality.Core where

open import Data.Product using (_,_)
open import Function     using (_∘_)
open import Level
open import Relation.Binary.Core
open import Relation.Nullary using (¬_)

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

open import Agda.Builtin.Equality public

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

------------------------------------------------------------------------
-- Properties of _≡_

module _ {a} {A : Set a} where

  sym : Symmetric {A = A} _≡_
  sym refl = refl

  trans : Transitive {A = A} _≡_
  trans refl eq = eq

  subst :  {p}  Substitutive {A = A} _≡_ p
  subst P refl p = p

  cong :  {b} {B : Set b} (f : A  B) {x y}  x  y  f x  f y
  cong f refl = refl

  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 _∼_

  isEquivalence : IsEquivalence {A = A} _≡_
  isEquivalence = record
    { refl  = refl
    ; sym   = sym
    ; trans = trans
    }

------------------------------------------------------------------------
-- Various equality rearrangement lemmas

module _ {a} {A : Set a} {x y : A} where

  trans-reflʳ : (p : x  y)  trans p refl  p
  trans-reflʳ refl = refl

  trans-assoc :  {z u} (p : x  y) {q : y  z} {r : z  u} 
    trans (trans p q) r  trans p (trans q r)
  trans-assoc refl = refl

  trans-symˡ : (p : x  y)  trans (sym p) p  refl
  trans-symˡ refl = refl

  trans-symʳ : (p : x  y)  trans p (sym p)  refl
  trans-symʳ refl = refl

------------------------------------------------------------------------
-- 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
-- readably.

module ≡-Reasoning {a} {A : Set a} where

  infix  3 _∎
  infixr 2 _≡⟨⟩_ _≡⟨_⟩_ _≡˘⟨_⟩_
  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

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

  _≡˘⟨_⟩_ :  (x {y z} : A)  y  x  y  z  x  z
  _ ≡˘⟨ y≡x  y≡z = trans (sym y≡x) y≡z

  _∎ :  (x : A)  x  x
  _∎ _ = refl