{-# OPTIONS --without-K --safe #-}
open import Relation.Binary
module Relation.Binary.Reasoning.Base.Double {a ℓ₁ ℓ₂} {A : Set a}
{_≈_ : Rel A ℓ₁} {_∼_ : Rel A ℓ₂} (isPreorder : IsPreorder _≈_ _∼_)
where
open import Data.Product using (proj₁; proj₂)
open import Level using (Level; _⊔_; Lift; lift)
open import Function.Base using (case_of_; id)
open import Relation.Binary.PropositionalEquality.Core
using (_≡_; refl; sym)
open import Relation.Nullary using (Dec; yes; no)
open import Relation.Nullary.Decidable using (True; toWitness)
open IsPreorder isPreorder
infix 4 _IsRelatedTo_
data _IsRelatedTo_ (x y : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
nonstrict : (x∼y : x ∼ y) → x IsRelatedTo y
equals : (x≈y : x ≈ y) → x IsRelatedTo y
data IsEquality {x y} : x IsRelatedTo y → Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
isEquality : ∀ x≈y → IsEquality (equals x≈y)
IsEquality? : ∀ {x y} (x≲y : x IsRelatedTo y) → Dec (IsEquality x≲y)
IsEquality? (nonstrict _) = no λ()
IsEquality? (equals x≈y) = yes (isEquality x≈y)
extractEquality : ∀ {x y} {x≲y : x IsRelatedTo y} → IsEquality x≲y → x ≈ y
extractEquality (isEquality x≈y) = x≈y
infix 1 begin_ begin-equality_
infixr 2 step-∼ step-≈ step-≈˘ step-≡ step-≡˘ _≡⟨⟩_
infix 3 _∎
begin_ : ∀ {x y} (r : x IsRelatedTo y) → x ∼ y
begin (nonstrict x∼y) = x∼y
begin (equals x≈y) = reflexive x≈y
begin-equality_ : ∀ {x y} (r : x IsRelatedTo y) → {s : True (IsEquality? r)} → x ≈ y
begin-equality_ r {s} = extractEquality (toWitness s)
step-∼ : ∀ (x : A) {y z} → y IsRelatedTo z → x ∼ y → x IsRelatedTo z
step-∼ x (nonstrict y∼z) x∼y = nonstrict (trans x∼y y∼z)
step-∼ x (equals y≈z) x∼y = nonstrict (∼-respʳ-≈ y≈z x∼y)
step-≈ : ∀ (x : A) {y z} → y IsRelatedTo z → x ≈ y → x IsRelatedTo z
step-≈ x (nonstrict y∼z) x≈y = nonstrict (∼-respˡ-≈ (Eq.sym x≈y) y∼z)
step-≈ x (equals y≈z) x≈y = equals (Eq.trans x≈y y≈z)
step-≈˘ : ∀ x {y z} → y IsRelatedTo z → y ≈ x → x IsRelatedTo z
step-≈˘ x y∼z x≈y = step-≈ x y∼z (Eq.sym x≈y)
step-≡ : ∀ (x : A) {y z} → y IsRelatedTo z → x ≡ y → x IsRelatedTo z
step-≡ x (nonstrict y∼z) x≡y = nonstrict (case x≡y of λ where refl → y∼z)
step-≡ x (equals y≈z) x≡y = equals (case x≡y of λ where refl → y≈z)
step-≡˘ : ∀ x {y z} → y IsRelatedTo z → y ≡ x → x IsRelatedTo z
step-≡˘ x y∼z x≡y = step-≡ x y∼z (sym x≡y)
_≡⟨⟩_ : ∀ (x : A) {y} → x IsRelatedTo y → x IsRelatedTo y
x ≡⟨⟩ x≲y = x≲y
_∎ : ∀ x → x IsRelatedTo x
x ∎ = equals Eq.refl
syntax step-∼ x y∼z x∼y = x ∼⟨ x∼y ⟩ y∼z
syntax step-≈ x y∼z x≈y = x ≈⟨ x≈y ⟩ y∼z
syntax step-≈˘ x y∼z y≈x = x ≈˘⟨ y≈x ⟩ y∼z
syntax step-≡ x y∼z x≡y = x ≡⟨ x≡y ⟩ y∼z
syntax step-≡˘ x y∼z y≡x = x ≡˘⟨ y≡x ⟩ y∼z