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{-# OPTIONS --without-K --safe #-}
open import Categories.Category using (Category)
open import Category.Dagger.Semiadditive using (IdempotentSemiadditiveDagger)
open import Level using (Level)
module Data.WiringDiagram.Directed
{o ℓ e : Level}
{𝒞 : Category o ℓ e}
(S : IdempotentSemiadditiveDagger 𝒞)
where
import Categories.Category.Monoidal.Properties as ⊗-Properties
import Categories.Category.Monoidal.Reasoning as ⊗-Reasoning
import Categories.Morphism.Reasoning as ⇒-Reasoning
open import Categories.Category.Helper using (categoryHelper)
open import Categories.Category.Monoidal using (Monoidal)
open import Categories.Category.Monoidal.Utilities using (module Shorthands)
open import Data.WiringDiagram.Core S using (Box; WiringDiagram; _≈-⧈_; _□_; _⧈_; _⌸_; id-⧈; _⌻_; ≈-isEquiv)
open Category 𝒞
open IdempotentSemiadditiveDagger S
open Monoidal +-monoidal
open Shorthands +-monoidal using (α⇒; α⇐; λ⇒; λ⇐; ρ⇒; ρ⇐)
open ⊗-Properties +-monoidal using (coherence₁)
private
⌻-resp-≈ : {A B C : Box} {f h : WiringDiagram B C} {g i : WiringDiagram A B} → f ≈-⧈ h → g ≈-⧈ i → f ⌻ g ≈-⧈ h ⌻ i
⌻-resp-≈ {A} {B} {C} {fᵢ ⧈ fₒ} {hᵢ ⧈ hₒ} {gᵢ ⧈ gₒ} {iᵢ ⧈ iₒ} (fᵢ≈hᵢ ⌸ fₒ≈hₒ) (gᵢ≈iᵢ ⌸ gₒ≈iₒ) = ≈ᵢ ⌸ ∘-resp-≈ fₒ≈hₒ gₒ≈iₒ
where
open ⊗-Reasoning +-monoidal
≈ᵢ : gᵢ ∘ id ⊕₁ (fᵢ ∘ gₒ ⊕₁ id) ∘ α⇒ ∘ △ {Box.ₒ A} ⊕₁ id
≈ iᵢ ∘ id ⊕₁ (hᵢ ∘ iₒ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈ᵢ = gᵢ≈iᵢ ⟩∘⟨ refl⟩⊗⟨ (fᵢ≈hᵢ ⟩∘⟨ gₒ≈iₒ ⟩⊗⟨refl) ⟩∘⟨refl
⌻-assoc : {A B C D : Box} {f : WiringDiagram A B} {g : WiringDiagram B C} {h : WiringDiagram C D} → (h ⌻ g) ⌻ f ≈-⧈ h ⌻ (g ⌻ f)
⌻-assoc {Aᵢ □ Aₒ} {Bᵢ □ Bₒ} {Cᵢ □ Cₒ} {Dᵢ □ Dₒ} {fᵢ ⧈ fₒ} {gᵢ ⧈ gₒ} {hᵢ ⧈ hₒ} = ≈ᵢ ⌸ assoc
where
open ⊗-Reasoning +-monoidal
term₁ : Aₒ ⊕₀ Dᵢ ⇒ Cᵢ
term₁ = hᵢ ∘ (gₒ ∘ fₒ) ⊕₁ id
term₂ : Bₒ ⊕₀ Dᵢ ⇒ Cᵢ
term₂ = hᵢ ∘ gₒ ⊕₁ id
term₃ : Aₒ ⊕₀ Cᵢ ⇒ Bᵢ
term₃ = gᵢ ∘ fₒ ⊕₁ id
open ⇒-Reasoning 𝒞
lemma₁ : α⇒ {Aₒ ⊕₀ Aₒ} {Aₒ} {Dᵢ} ∘ (α⇐ ∘ id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id ≈ △ ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id
lemma₁ = begin
α⇒ ∘ (α⇐ ∘ id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ pushˡ split₁ˡ ⟩
α⇒ ∘ α⇐ ⊕₁ id ∘ (id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ merge₁ˡ ⟩
α⇒ ∘ α⇐ ⊕₁ id ∘ (id ⊕₁ △ ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ △-assoc ⟩⊗⟨refl ⟩
α⇒ ∘ α⇐ ⊕₁ id ∘ (α⇒ ∘ △ ⊕₁ id ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ merge₁ʳ ⟩
α⇒ ∘ (α⇐ ∘ α⇒ ∘ △ ⊕₁ id ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ cancelˡ associator.isoˡ ⟩⊗⟨refl ⟩
α⇒ ∘ (△ ⊕₁ id ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ split₁ˡ ⟩
α⇒ ∘ (△ ⊕₁ id) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ extendʳ assoc-commute-from ⟩
△ ⊕₁ id ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩⊗⟨ ⊕.identity ⟩∘⟨refl ⟩
△ ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ∎
lemma₂ : △ ⊕₁ id {Dᵢ} ∘ fₒ ⊕₁ id ≈ (fₒ ⊕₁ fₒ) ⊕₁ id ∘ △ ⊕₁ id
lemma₂ = begin
△ ⊕₁ id ∘ fₒ ⊕₁ id ≈⟨ merge₁ʳ ⟩
(△ ∘ fₒ) ⊕₁ id ≈⟨ ⇒△ ⟩⊗⟨refl ⟩
(fₒ ⊕₁ fₒ ∘ △) ⊕₁ id ≈⟨ split₁ʳ ⟩
(fₒ ⊕₁ fₒ) ⊕₁ id ∘ △ ⊕₁ id ∎
≈ᵢ : fᵢ ∘ id ⊕₁ ((gᵢ ∘ id ⊕₁ (hᵢ ∘ gₒ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id) ∘ fₒ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈ (fᵢ ∘ id ⊕₁ (gᵢ ∘ fₒ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id) ∘ id ⊕₁ (hᵢ ∘ (gₒ ∘ fₒ) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈ᵢ = begin
fᵢ ∘ id ⊕₁ ((gᵢ ∘ id ⊕₁ term₂ ∘ α⇒ ∘ △ ⊕₁ id) ∘ fₒ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩⊗⟨ extendˡ (extendˡ assoc) ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ ((gᵢ ∘ id ⊕₁ term₂ ∘ α⇒) ∘ △ ⊕₁ id ∘ fₒ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ lemma₂) ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ ((gᵢ ∘ id ⊕₁ term₂ ∘ α⇒) ∘ (fₒ ⊕₁ fₒ) ⊕₁ id ∘ △ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩⊗⟨ extendˡ assoc ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ ((gᵢ ∘ id ⊕₁ term₂) ∘ α⇒ ∘ (fₒ ⊕₁ fₒ) ⊕₁ id ∘ △ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ extendʳ assoc-commute-from ) ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ ((gᵢ ∘ id ⊕₁ term₂) ∘ fₒ ⊕₁ fₒ ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩⊗⟨ assoc ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ (gᵢ ∘ id ⊕₁ term₂ ∘ fₒ ⊕₁ fₒ ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ pullˡ merge₂ˡ ) ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ (gᵢ ∘ fₒ ⊕₁ (term₂ ∘ fₒ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ refl⟩⊗⟨ pullʳ merge₁ʳ ⟩∘⟨refl) ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ (gᵢ ∘ fₒ ⊕₁ term₁ ∘ α⇒ ∘ △ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩⊗⟨ assoc ⟩∘⟨refl ⟨
fᵢ ∘ id ⊕₁ ((gᵢ ∘ fₒ ⊕₁ term₁) ∘ α⇒ ∘ △ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩⊗⟨ extendˡ assoc ⟩∘⟨refl ⟨
fᵢ ∘ id ⊕₁ ((gᵢ ∘ fₒ ⊕₁ term₁ ∘ α⇒) ∘ △ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ pushˡ split₂ʳ ⟩
fᵢ ∘ id ⊕₁ (gᵢ ∘ fₒ ⊕₁ term₁ ∘ α⇒) ∘ id ⊕₁ △ ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ assoc-commute-from ⟨
fᵢ ∘ id ⊕₁ (gᵢ ∘ fₒ ⊕₁ term₁ ∘ α⇒) ∘ α⇒ ∘ (id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ extendʳ (refl⟩⊗⟨ assoc ⟩∘⟨refl) ⟨
fᵢ ∘ id ⊕₁ ((gᵢ ∘ fₒ ⊕₁ term₁) ∘ α⇒) ∘ α⇒ ∘ (id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ pushˡ split₂ʳ ⟩
fᵢ ∘ id ⊕₁ (gᵢ ∘ fₒ ⊕₁ term₁) ∘ id ⊕₁ α⇒ ∘ α⇒ ∘ (id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ insertˡ associator.isoʳ ⟩⊗⟨refl ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ (gᵢ ∘ fₒ ⊕₁ term₁) ∘ id ⊕₁ α⇒ ∘ α⇒ ∘ (α⇒ ∘ α⇐ ∘ id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ split₁ʳ ⟩
fᵢ ∘ id ⊕₁ (gᵢ ∘ fₒ ⊕₁ term₁) ∘ id ⊕₁ α⇒ ∘ α⇒ ∘ α⇒ ⊕₁ id ∘ (α⇐ ∘ id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ assoc ⟨
fᵢ ∘ id ⊕₁ (gᵢ ∘ fₒ ⊕₁ term₁) ∘ id ⊕₁ α⇒ ∘ (α⇒ ∘ α⇒ ⊕₁ id) ∘ (α⇐ ∘ id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ pentagon ⟩
fᵢ ∘ id ⊕₁ (gᵢ ∘ fₒ ⊕₁ term₁) ∘ α⇒ ∘ α⇒ ∘ (α⇐ ∘ id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩⊗⟨ pushʳ serialize₁₂ ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ (term₃ ∘ id ⊕₁ term₁) ∘ α⇒ ∘ α⇒ ∘ (α⇐ ∘ id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ pushˡ split₂ʳ ⟩
fᵢ ∘ id ⊕₁ term₃ ∘ id ⊕₁ id ⊕₁ term₁ ∘ α⇒ ∘ α⇒ ∘ (α⇐ ∘ id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ assoc-commute-from ⟨
fᵢ ∘ id ⊕₁ term₃ ∘ α⇒ ∘ (id ⊕₁ id) ⊕₁ term₁ ∘ α⇒ ∘ (α⇐ ∘ id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ ⊕.identity ⟩⊗⟨refl ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ term₃ ∘ α⇒ ∘ id ⊕₁ term₁ ∘ α⇒ ∘ (α⇐ ∘ id ⊕₁ △) ⊕₁ id ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ lemma₁ ⟩
fᵢ ∘ id ⊕₁ term₃ ∘ α⇒ ∘ id ⊕₁ term₁ ∘ △ ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ (Equiv.sym serialize₂₁ ○ serialize₁₂) ⟩
fᵢ ∘ id ⊕₁ term₃ ∘ α⇒ ∘ △ ⊕₁ id ∘ id ⊕₁ term₁ ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ refl⟩∘⟨ assoc ⟨
fᵢ ∘ id ⊕₁ term₃ ∘ (α⇒ ∘ △ ⊕₁ id) ∘ id ⊕₁ term₁ ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ refl⟩∘⟨ assoc ⟨
fᵢ ∘ (id ⊕₁ term₃ ∘ α⇒ ∘ △ ⊕₁ id) ∘ id ⊕₁ term₁ ∘ α⇒ ∘ △ ⊕₁ id
≈⟨ assoc ⟨
(fᵢ ∘ id ⊕₁ term₃ ∘ α⇒ ∘ △ ⊕₁ id) ∘ id ⊕₁ term₁ ∘ α⇒ ∘ △ ⊕₁ id ∎
⌻-identityˡ : {A B : Box} {f : WiringDiagram A B} → id-⧈ ⌻ f ≈-⧈ f
⌻-identityˡ {Aᵢ □ Aₒ} {Bᵢ □ Bₒ} {fᵢ ⧈ fₒ} = ≈ᵢ ⌸ identityˡ
where
open ⇒-Reasoning 𝒞
open ⊗-Reasoning +-monoidal
≈ᵢ : fᵢ ∘ id ⊕₁ (p₂ ∘ fₒ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈ fᵢ
≈ᵢ = begin
fᵢ ∘ id ⊕₁ (p₂ ∘ fₒ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (p₂-⊕ ⟩∘⟨refl) ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ ((λ⇒ ∘ ! ⊕₁ id) ∘ fₒ ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ pullʳ merge₁ʳ ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ (λ⇒ ∘ (! ∘ fₒ) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ ⇒! ⟩⊗⟨refl) ⟩∘⟨refl ⟩
fᵢ ∘ id ⊕₁ (λ⇒ ∘ ! ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ pushˡ split₂ʳ ⟩
fᵢ ∘ id ⊕₁ λ⇒ ∘ id ⊕₁ ! ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ assoc-commute-from ⟨
fᵢ ∘ id ⊕₁ λ⇒ ∘ α⇒ ∘ (id ⊕₁ !) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ merge₁ˡ ⟩
fᵢ ∘ id ⊕₁ λ⇒ ∘ α⇒ ∘ (id ⊕₁ ! ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ pullˡ triangle ⟩
fᵢ ∘ ρ⇒ ⊕₁ id ∘ (id ⊕₁ ! ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ merge₁ʳ ⟩
fᵢ ∘ (ρ⇒ ∘ id ⊕₁ ! ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ (refl⟩∘⟨ △-identityʳ ) ⟩⊗⟨refl ⟩
fᵢ ∘ (ρ⇒ ∘ ρ⇐) ⊕₁ id ≈⟨ refl⟩∘⟨ unitorʳ.isoʳ ⟩⊗⟨refl ⟩
fᵢ ∘ id ⊕₁ id ≈⟨ elimʳ ⊕.identity ⟩
fᵢ ∎
⌻-identityʳ : {A B : Box} {f : WiringDiagram A B} → f ⌻ id-⧈ ≈-⧈ f
⌻-identityʳ {Aᵢ □ Aₒ} {Bᵢ □ Bₒ} {fᵢ ⧈ fₒ} = ≈ᵢ ⌸ identityʳ
where
open ⇒-Reasoning 𝒞
open ⊗-Reasoning +-monoidal
≈ᵢ : p₂ ∘ id ⊕₁ (fᵢ ∘ id ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈ fᵢ
≈ᵢ = begin
p₂ ∘ id ⊕₁ (fᵢ ∘ id ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ p₂-⊕ ⟩∘⟨refl ⟩
(λ⇒ ∘ ! ⊕₁ id) ∘ id ⊕₁ (fᵢ ∘ id ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ elimʳ ⊕.identity ⟩∘⟨refl ⟩
(λ⇒ ∘ ! ⊕₁ id) ∘ id ⊕₁ fᵢ ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ pullˡ (pullʳ (Equiv.sym serialize₁₂)) ⟩
(λ⇒ ∘ ! ⊕₁ fᵢ) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ pullʳ (pushˡ serialize₂₁) ⟩
λ⇒ ∘ id ⊕₁ fᵢ ∘ ! ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ ⊕.identity ⟩∘⟨refl ⟨
λ⇒ ∘ id ⊕₁ fᵢ ∘ ! ⊕₁ id ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ assoc-commute-from ⟨
λ⇒ ∘ id ⊕₁ fᵢ ∘ α⇒ ∘ (! ⊕₁ id) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ merge₁ʳ ⟩
λ⇒ ∘ id ⊕₁ fᵢ ∘ α⇒ ∘ (! ⊕₁ id ∘ △) ⊕₁ id ≈⟨ extendʳ unitorˡ-commute-from ⟩
fᵢ ∘ λ⇒ ∘ α⇒ ∘ (! ⊕₁ id ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ △-identityˡ ⟩⊗⟨refl ⟩
fᵢ ∘ λ⇒ ∘ α⇒ ∘ λ⇐ ⊕₁ id ≈⟨ refl⟩∘⟨ pullˡ coherence₁ ⟩
fᵢ ∘ λ⇒ ⊕₁ id ∘ λ⇐ ⊕₁ id ≈⟨ refl⟩∘⟨ merge₁ʳ ⟩
fᵢ ∘ (λ⇒ ∘ λ⇐) ⊕₁ id ≈⟨ refl⟩∘⟨ unitorˡ.isoʳ ⟩⊗⟨refl ⟩
fᵢ ∘ id ⊕₁ id ≈⟨ elimʳ ⊕.identity ⟩
fᵢ ∎
-- The category of directed wiring diagrams
DWD : Category o ℓ e
DWD = categoryHelper record
{ Obj = Box
; _⇒_ = WiringDiagram
; _≈_ = _≈-⧈_
; id = id-⧈
; _∘_ = _⌻_
; assoc = ⌻-assoc
; identityˡ = ⌻-identityˡ
; identityʳ = ⌻-identityʳ
; equiv = ≈-isEquiv
; ∘-resp-≈ = ⌻-resp-≈
}
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