------------------------------------------------------------------------
-- The Agda standard library
--
-- The basic code for equational reasoning with a partial relation
------------------------------------------------------------------------

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

open import Relation.Binary

module Relation.Binary.Reasoning.Base.Partial
  {a ℓ} {A : Set a} (_∼_ : Rel A ℓ) (trans : Transitive _∼_)
  where

open import Level using (_⊔_)
open import Relation.Binary.PropositionalEquality.Core as P
  using (_≡_)

infix   _IsRelatedTo_
infix   _∎⟨_⟩
infixr  _∼⟨_⟩_ _≡⟨_⟩_ _≡˘⟨_⟩_ _≡⟨⟩_
infix   begin_

-- This seemingly unnecessary type is used to make it possible to
-- infer arguments even if the underlying equality evaluates.

data _IsRelatedTo_ (x y : A) : Set (a ⊔ ℓ) where
  relTo : (x∼y : x ∼ y) → x IsRelatedTo y

begin_ : ∀ {x y} → x IsRelatedTo y → x ∼ y
begin relTo x∼y = x∼y

_∼⟨_⟩_ : ∀ x {y z} → x ∼ y → y IsRelatedTo z → x IsRelatedTo z
_ ∼⟨ x∼y ⟩ relTo y∼z = relTo (trans x∼y y∼z)

_≡⟨_⟩_ : ∀ x {y z} → x ≡ y → y IsRelatedTo z → x IsRelatedTo z
_ ≡⟨ P.refl ⟩ x∼z = x∼z

_≡˘⟨_⟩_ : ∀ x {y z} → y ≡ x → y IsRelatedTo z → x IsRelatedTo z
_ ≡˘⟨ P.refl ⟩ x∼z = x∼z

_≡⟨⟩_ : ∀ x {y} → x IsRelatedTo y → x IsRelatedTo y
_ ≡⟨⟩ x∼y = _ ≡⟨ P.refl ⟩ x∼y

_∎⟨_⟩ : ∀ x → x ∼ x → x IsRelatedTo x
_ ∎⟨ x∼x ⟩  = relTo x∼x