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diff --git a/Data/WiringDiagram/Equalities.agda b/Data/WiringDiagram/Equalities.agda new file mode 100644 index 0000000..37b8f38 --- /dev/null +++ b/Data/WiringDiagram/Equalities.agda @@ -0,0 +1,153 @@ +{-# 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.Equalities {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.Monoidal using (module Monoidal) +open import Categories.Category.Monoidal.Utilities using (module Shorthands) +open import Data.WiringDiagram.Core S using (_⌸_; _⌻_; _≈-⧈_; ≈-trans; loop; push; pull; merge; split) + +open Category 𝒞 +open IdempotentSemiadditiveDagger S +open Monoidal +-monoidal using (module unitorˡ; module unitorʳ; triangle; assoc-commute-from; unitorˡ-commute-from) +open Shorthands +-monoidal using (α⇒; α⇐; λ⇒; λ⇐; ρ⇒; ρ⇐) +open ⊗-Properties +-monoidal using (coherence₁) + +loop∘loop : {A : Obj} → loop ⌻ loop ≈-⧈ loop {A} +loop∘loop {A} = ≈ᵢ ⌸ identity² + where + open ⇒-Reasoning 𝒞 + open ⊗-Reasoning +-monoidal + ≈ᵢ : ▽ ∘ id ⊕₁ (▽ ∘ id ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈ ▽ + ≈ᵢ = begin + ▽ ∘ id ⊕₁ (▽ ∘ id ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ elimʳ ⊕.identity ⟩∘⟨refl ⟩ + ▽ ∘ id ⊕₁ ▽ ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ assoc ⟨ + ▽ ∘ (id ⊕₁ ▽ ∘ α⇒) ∘ △ ⊕₁ id ≈⟨ extendʳ ▽-assoc ⟨ + ▽ ∘ ▽ ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ merge₁ˡ ⟩ + ▽ ∘ (▽ ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ ▽∘△ ⟩⊗⟨refl ⟩ + ▽ ∘ id ⊕₁ id ≈⟨ elimʳ ⊕.identity ⟩ + ▽ ∎ + +loop∘push∘loop≈merge : {A B : Obj} (f : A ⇒ B) → id ≤ ((f †) ∘ f) → loop ⌻ push f ⌻ loop ≈-⧈ merge f +loop∘push∘loop≈merge f id≤f†∘f = ≈ᵢ ⌸ (identityˡ ○ identityʳ) + where + open ⇒-Reasoning 𝒞 + open ⊗-Reasoning +-monoidal + ≈ᵢ : (▽ ∘ (id ⊕₁ ((f † ∘ p₂) ∘ id ⊕₁ id)) ∘ α⇒ ∘ △ ⊕₁ id) ∘ id ⊕₁ (▽ ∘ (f ∘ id) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id + ≈ f † ∘ ▽ ∘ f ⊕₁ id + ≈ᵢ = begin + (▽ ∘ id ⊕₁ ((f † ∘ p₂) ∘ id ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id) ∘ id ⊕₁ (▽ ∘ (f ∘ id) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ (refl⟩∘⟨ refl⟩⊗⟨ elimʳ ⊕.identity ⟩∘⟨refl) ⟩∘⟨refl ⟩ + (▽ ∘ id ⊕₁ (f † ∘ p₂) ∘ α⇒ ∘ △ ⊕₁ id) ∘ id ⊕₁ (▽ ∘ (f ∘ id) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ (identityʳ ⟩⊗⟨refl)) ⟩∘⟨refl ⟩ + (▽ ∘ id ⊕₁ (f † ∘ p₂) ∘ α⇒ ∘ △ ⊕₁ id) ∘ id ⊕₁ (▽ ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ pushˡ (refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ p₂-⊕) ⟩∘⟨refl) ⟩ + ▽ ∘ (id ⊕₁ (f † ∘ λ⇒ ∘ ! ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id) ∘ id ⊕₁ (▽ ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ extendʳ (pullʳ (pullʳ (Equiv.sym serialize₁₂))) ⟩ + ▽ ∘ id ⊕₁ (f † ∘ λ⇒ ∘ ! ⊕₁ id) ∘ (α⇒ ∘ △ ⊕₁ (▽ ∘ f ⊕₁ id)) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ pushˡ split₂ˡ ⟩ + ▽ ∘ id ⊕₁ (f †) ∘ id ⊕₁ (λ⇒ ∘ ! ⊕₁ id) ∘ (α⇒ ∘ △ ⊕₁ (▽ ∘ f ⊕₁ id)) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ split₂ˡ ⟩ + ▽ ∘ id ⊕₁ (f †) ∘ id ⊕₁ λ⇒ ∘ id ⊕₁ ! ⊕₁ id ∘ (α⇒ ∘ △ ⊕₁ (▽ ∘ f ⊕₁ id)) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ (pullˡ (Equiv.sym assoc-commute-from)) ⟩ + ▽ ∘ id ⊕₁ (f †) ∘ id ⊕₁ λ⇒ ∘ (α⇒ ∘ (id ⊕₁ !) ⊕₁ id) ∘ △ ⊕₁ (▽ ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ (pullˡ triangle) ⟩ + ▽ ∘ id ⊕₁ (f †) ∘ ρ⇒ ⊕₁ id ∘ (id ⊕₁ !) ⊕₁ id ∘ △ ⊕₁ (▽ ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ merge₁ˡ ⟩ + ▽ ∘ id ⊕₁ (f †) ∘ (ρ⇒ ∘ id ⊕₁ !) ⊕₁ id ∘ △ ⊕₁ (▽ ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ merge₁ˡ ⟩ + ▽ ∘ id ⊕₁ (f †) ∘ ((ρ⇒ ∘ id ⊕₁ !) ∘ △) ⊕₁ (▽ ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullʳ △-identityʳ ⟩⊗⟨refl ⟩∘⟨refl ⟩ + ▽ ∘ id ⊕₁ (f †) ∘ (ρ⇒ ∘ ρ⇐) ⊕₁ (▽ ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ unitorʳ.isoʳ ⟩⊗⟨refl ⟩∘⟨refl ⟩ + ▽ ∘ id ⊕₁ (f †) ∘ id ⊕₁ (▽ ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩ + ▽ ∘ id ⊕₁ (f † ∘ ▽ ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ extendʳ ⇒▽ ⟩∘⟨refl ⟩ + ▽ ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ (f †) ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ merge₁ʳ) ⟩∘⟨refl ⟩ + ▽ ∘ id ⊕₁ (▽ ∘ (f † ∘ f) ⊕₁ (f †)) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ pushˡ split₂ˡ ⟩ + ▽ ∘ id ⊕₁ ▽ ∘ id ⊕₁ (f † ∘ f) ⊕₁ (f †) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ assoc-commute-from ⟨ + ▽ ∘ id ⊕₁ ▽ ∘ α⇒ ∘ (id ⊕₁ (f † ∘ f)) ⊕₁ (f †) ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ merge₁ʳ ⟩ + ▽ ∘ id ⊕₁ ▽ ∘ α⇒ ∘ (id ⊕₁ (f † ∘ f) ∘ △) ⊕₁ (f †) ≈⟨ refl⟩∘⟨ sym-assoc ⟩ + ▽ ∘ (id ⊕₁ ▽ ∘ α⇒) ∘ (id ⊕₁ (f † ∘ f) ∘ △) ⊕₁ (f †) ≈⟨ extendʳ ▽-assoc ⟨ + ▽ ∘ ▽ ⊕₁ id ∘ (id ⊕₁ (f † ∘ f) ∘ △) ⊕₁ (f †) ≈⟨ refl⟩∘⟨ merge₁ˡ ⟩ + ▽ ∘ (id + (f † ∘ f)) ⊕₁ (f †) ≈⟨ refl⟩∘⟨ id≤f†∘f ⟩⊗⟨refl ⟩ + ▽ ∘ (f † ∘ f) ⊕₁ (f †) ≈⟨ refl⟩∘⟨ split₁ʳ ⟩ + ▽ ∘ (f †) ⊕₁ (f †) ∘ f ⊕₁ id ≈⟨ extendʳ ⇒▽ ⟨ + f † ∘ ▽ ∘ f ⊕₁ id ∎ + +merge≈loop∘push : {A B : Obj} (f : A ⇒ B) → merge f ≈-⧈ loop ⌻ push f +merge≈loop∘push f = ≈ᵢ ⌸ Equiv.sym identityˡ + where + open ⇒-Reasoning 𝒞 + open ⊗-Reasoning +-monoidal + ≈ᵢ : f † ∘ ▽ ∘ f ⊕₁ id + ≈ (f † ∘ p₂) ∘ id ⊕₁ (▽ ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id + ≈ᵢ = begin + f † ∘ ▽ ∘ f ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ introʳ unitorˡ.isoʳ ⟩⊗⟨refl ⟩ + f † ∘ ▽ ∘ (f ∘ λ⇒ ∘ λ⇐) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ (refl⟩∘⟨ refl⟩∘⟨ △-identityˡ ) ⟩⊗⟨refl ⟨ + f † ∘ ▽ ∘ (f ∘ λ⇒ ∘ ! ⊕₁ id ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ unitorˡ-commute-from ⟩⊗⟨refl ⟨ + f † ∘ ▽ ∘ (λ⇒ ∘ id ⊕₁ f ∘ ! ⊕₁ id ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ (refl⟩∘⟨ pullˡ (Equiv.sym serialize₂₁)) ⟩⊗⟨refl ⟩ + f † ∘ ▽ ∘ (λ⇒ ∘ ! ⊕₁ f ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ split₁ˡ ⟩ + f † ∘ ▽ ∘ λ⇒ ⊕₁ id ∘ (! ⊕₁ f ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ split₁ˡ ⟩ + f † ∘ ▽ ∘ λ⇒ ⊕₁ id ∘ (! ⊕₁ f) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ (Equiv.sym coherence₁) ⟩ + f † ∘ ▽ ∘ λ⇒ ∘ α⇒ ∘ (! ⊕₁ f) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ extendʳ unitorˡ-commute-from ⟨ + f † ∘ λ⇒ ∘ id ⊕₁ ▽ ∘ α⇒ ∘ (! ⊕₁ f) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ assoc-commute-from ⟩ + f † ∘ λ⇒ ∘ id ⊕₁ ▽ ∘ ! ⊕₁ f ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩ + f † ∘ λ⇒ ∘ ! ⊕₁ (▽ ∘ (f ⊕₁ id)) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ split₂ʳ ⟩ + f † ∘ λ⇒ ∘ ! ⊕₁ ▽ ∘ id ⊕₁ f ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ serialize₁₂ ⟩ + f † ∘ λ⇒ ∘ ! ⊕₁ id ∘ id ⊕₁ ▽ ∘ id ⊕₁ f ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ pushʳ (pullˡ (Equiv.sym p₂-⊕)) ⟩ + (f † ∘ p₂) ∘ id ⊕₁ ▽ ∘ id ⊕₁ f ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩ + (f † ∘ p₂) ∘ id ⊕₁ (▽ ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ∎ + +loop∘pull∘loop≈split : {A B : Obj} (f : A ⇒ B) → (f ∘ (f †)) ≤ id → loop ⌻ pull f ⌻ loop ≈-⧈ split f +loop∘pull∘loop≈split f f∘f†≤id = ≈ᵢ ⌸ (identityˡ ○ identityʳ) + where + open ⇒-Reasoning 𝒞 + open ⊗-Reasoning +-monoidal + ≈ᵢ : (▽ ∘ (id ⊕₁ ((f ∘ p₂) ∘ id ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id)) ∘ id ⊕₁ (▽ ∘ (f † ∘ id) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id + ≈ ▽ ∘ id ⊕₁ f + ≈ᵢ = begin + (▽ ∘ (id ⊕₁ ((f ∘ p₂) ∘ id ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id)) ∘ id ⊕₁ (▽ ∘ (f † ∘ id) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ (refl⟩∘⟨ refl⟩⊗⟨ elimʳ ⊕.identity ⟩∘⟨refl) ⟩∘⟨refl ⟩ + (▽ ∘ (id ⊕₁ (f ∘ p₂) ∘ α⇒ ∘ △ ⊕₁ id)) ∘ id ⊕₁ (▽ ∘ (f † ∘ id) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ pullʳ (pullʳ (pullʳ (refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ identityʳ ⟩⊗⟨refl) ⟩∘⟨refl)))⟩ + ▽ ∘ id ⊕₁ (f ∘ p₂) ∘ α⇒ ∘ △ ⊕₁ id ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ pushˡ split₂ˡ ⟩ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ p₂ ∘ α⇒ ∘ △ ⊕₁ id ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ p₂-⊕ ⟩∘⟨refl ⟩ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ (λ⇒ ∘ ! ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ split₂ˡ ⟩ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ λ⇒ ∘ id ⊕₁ ! ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ assoc-commute-from ⟨ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ λ⇒ ∘ α⇒ ∘ (id ⊕₁ !) ⊕₁ id ∘ △ ⊕₁ id ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ triangle ⟩ + ▽ ∘ id ⊕₁ f ∘ ρ⇒ ⊕₁ id ∘ (id ⊕₁ !) ⊕₁ id ∘ △ ⊕₁ id ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ merge₁ˡ ⟩ + ▽ ∘ id ⊕₁ f ∘ ρ⇒ ⊕₁ id ∘ (id ⊕₁ ! ∘ △) ⊕₁ id ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ △-identityʳ ⟩⊗⟨refl ⟩∘⟨refl ⟩ + ▽ ∘ id ⊕₁ f ∘ ρ⇒ ⊕₁ id ∘ ρ⇐ ⊕₁ id ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ merge₁ˡ ⟩ + ▽ ∘ id ⊕₁ f ∘ (ρ⇒ ∘ ρ⇐) ⊕₁ id ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ unitorʳ.isoʳ ⟩⊗⟨refl ⟩∘⟨refl ⟩ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ id ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ elimˡ ⊕.identity ⟩ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ (▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩ + ▽ ∘ id ⊕₁ (f ∘ ▽ ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ extendʳ ⇒▽ ⟩∘⟨refl ⟩ + ▽ ∘ id ⊕₁ (▽ ∘ f ⊕₁ f ∘ (f †) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ merge₁ʳ) ⟩∘⟨refl ⟩ + ▽ ∘ id ⊕₁ (▽ ∘ (f ∘ f †) ⊕₁ f) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ pushˡ split₂ˡ ⟩ + ▽ ∘ id ⊕₁ ▽ ∘ id ⊕₁ (f ∘ f †) ⊕₁ f ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ pushʳ (extendʳ (Equiv.sym assoc-commute-from)) ⟩ + ▽ ∘ (id ⊕₁ ▽ ∘ α⇒) ∘ (id ⊕₁ (f ∘ f †)) ⊕₁ f ∘ △ ⊕₁ id ≈⟨ extendʳ ▽-assoc ⟨ + ▽ ∘ ▽ ⊕₁ id ∘ (id ⊕₁ (f ∘ f †)) ⊕₁ f ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ merge₁ʳ ⟩ + ▽ ∘ ▽ ⊕₁ id ∘ (id ⊕₁ (f ∘ f †) ∘ △) ⊕₁ f ≈⟨ refl⟩∘⟨ merge₁ˡ ⟩ + ▽ ∘ id + (f ∘ f †) ⊕₁ f ≈⟨ refl⟩∘⟨ +-commutative ⟩⊗⟨refl ⟩ + ▽ ∘ (f ∘ f †) + id ⊕₁ f ≈⟨ refl⟩∘⟨ f∘f†≤id ⟩⊗⟨refl ⟩ + ▽ ∘ id ⊕₁ f ∎ + +split≈pull∘loop : {A B : Obj} (f : A ⇒ B) → split f ≈-⧈ pull f ⌻ loop +split≈pull∘loop f = ≈ᵢ ⌸ Equiv.sym identityʳ + where + open ⇒-Reasoning 𝒞 + open ⊗-Reasoning +-monoidal + ≈ᵢ : ▽ ∘ id ⊕₁ f + ≈ ▽ ∘ id ⊕₁ ((f ∘ p₂) ∘ id ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id + ≈ᵢ = begin + ▽ ∘ id ⊕₁ f ≈⟨ refl⟩∘⟨ introʳ ⊕.identity ⟩ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ unitorʳ.isoʳ ⟩⊗⟨refl ⟨ + ▽ ∘ id ⊕₁ f ∘ (ρ⇒ ∘ ρ⇐) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ split₁ˡ ⟩ + ▽ ∘ id ⊕₁ f ∘ ρ⇒ ⊕₁ id ∘ ρ⇐ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ △-identityʳ ⟩⊗⟨refl ⟨ + ▽ ∘ id ⊕₁ f ∘ ρ⇒ ⊕₁ id ∘ (id ⊕₁ ! ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ (Equiv.sym triangle) ⟩ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ λ⇒ ∘ α⇒ ∘ (id ⊕₁ ! ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ split₁ˡ ⟩ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ λ⇒ ∘ α⇒ ∘ (id ⊕₁ !) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ assoc-commute-from ⟩ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ λ⇒ ∘ id ⊕₁ ! ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ (λ⇒ ∘ ! ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ p₂-⊕ ⟩∘⟨refl ⟨ + ▽ ∘ id ⊕₁ f ∘ id ⊕₁ p₂ ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩ + ▽ ∘ id ⊕₁ (f ∘ p₂) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩⊗⟨ (pushʳ (introʳ ⊕.identity)) ⟩∘⟨refl ⟩ + ▽ ∘ id ⊕₁ ((f ∘ p₂) ∘ id ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ∎ + +loop∘push∘loop : {A B : Obj} (f : A ⇒ B) → id ≤ ((f †) ∘ f) → loop ⌻ push f ⌻ loop ≈-⧈ loop ⌻ push f +loop∘push∘loop f id≤f†∘f = ≈-trans (loop∘push∘loop≈merge f id≤f†∘f) (merge≈loop∘push f) + +loop∘pull∘loop : {A B : Obj} → (f : A ⇒ B) → (f ∘ (f †)) ≤ id → loop ⌻ pull f ⌻ loop ≈-⧈ pull f ⌻ loop +loop∘pull∘loop f f∘f†≤id = ≈-trans (loop∘pull∘loop≈split f f∘f†≤id) (split≈pull∘loop f) |