{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Relation.Binary.Reasoning.Base.Single
{a ℓ} {A : Set a} (_∼_ : Rel A ℓ)
(refl : Reflexive _∼_) (trans : Transitive _∼_)
where
open import Level using (_⊔_)
open import Relation.Binary.PropositionalEquality as P using (_≡_)
infix 4 _IsRelatedTo_
infix 3 _∎
infixr 2 _∼⟨_⟩_ _≡⟨_⟩_ _≡˘⟨_⟩_ _≡⟨⟩_
infix 1 begin_
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 IsRelatedTo x
_∎ _ = relTo refl