------------------------------------------------------------------------
-- 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.Base using (_∘_)
open import Level
open import Relation.Binary.Core
open import Relation.Binary.Definitions
open import Relation.Nullary using (¬_)

private
  variable
    a b  : Level
    A : Set a
    B : Set b

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

open import Agda.Builtin.Equality public

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

------------------------------------------------------------------------
-- 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

cong :  (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 _∼_

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

trans-reflʳ :  {x y : A} (p : x  y)  trans p refl  p
trans-reflʳ refl = refl

trans-assoc :  {x y z u : A} (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 :  {x y : A} (p : x  y)  trans (sym p) p  refl
trans-symˡ refl = refl

trans-symʳ :  {x y : A} (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 : 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