diff options
Diffstat (limited to 'Data')
| -rw-r--r-- | Data/WiringDiagram/Balanced.agda | 65 | ||||
| -rw-r--r-- | Data/WiringDiagram/Core.agda | 20 | ||||
| -rw-r--r-- | Data/WiringDiagram/Directed.agda | 38 | ||||
| -rw-r--r-- | Data/WiringDiagram/Equalities.agda | 153 |
4 files changed, 265 insertions, 11 deletions
diff --git a/Data/WiringDiagram/Balanced.agda b/Data/WiringDiagram/Balanced.agda index 09656fc..ada297b 100644 --- a/Data/WiringDiagram/Balanced.agda +++ b/Data/WiringDiagram/Balanced.agda @@ -10,29 +10,78 @@ module Data.WiringDiagram.Balanced (S : IdempotentSemiadditiveDagger 𝒞) where +import Categories.Category.Monoidal.Reasoning as ⊗-Reasoning +import Categories.Morphism.Reasoning as ⇒-Reasoning + open import Categories.Functor using (Functor) -open import Data.WiringDiagram.Core S using (WiringDiagram; _□_) +open import Data.WiringDiagram.Core S using (WiringDiagram; _□_; _⌸_; push) +open import Categories.Category.Monoidal.Utilities using (module Shorthands) +open import Categories.Category.Monoidal using (Monoidal) open import Data.WiringDiagram.Directed S using (DWD) -open import Function using (id) +open import Function using () renaming (id to idf) -open Category 𝒞 using (Obj) +module 𝒞 = Category 𝒞 +module DWD = Category DWD -- The category of balanced wiring diagrams BWD : Category o ℓ e BWD = record - { Obj = Obj + { Obj = 𝒞.Obj ; _⇒_ = λ A B → WiringDiagram (A □ A) (B □ B) - ; Category DWD + ; DWD } -- Inclusion functor from BWD to DWD Include : Functor BWD DWD Include = record { F₀ = λ A → A □ A - ; F₁ = id + ; F₁ = idf ; identity = Equiv.refl ; homomorphism = Equiv.refl - ; F-resp-≈ = id + ; F-resp-≈ = idf + } + where + open DWD using (module Equiv) + +-- Covariant functor from underlying category to BWD +Push : Functor 𝒞 BWD +Push = record + { F₀ = idf + ; F₁ = push + ; identity = elimˡ †-identity ⌸ Equiv.refl + ; homomorphism = λ {A B C f g} → homoᵢ f g ⌸ Equiv.refl + ; F-resp-≈ = λ f≈g → (†-resp-≈ f≈g ⟩∘⟨refl) ⌸ f≈g } where - open Category DWD using (module Equiv) + open Category 𝒞 + open IdempotentSemiadditiveDagger S + using (+-monoidal; _†; p₂; _⊕₁_; △; !; p₁; p₂-⊕; ⇒!; ⇒△; ρ⇒≈p₁; p₁∘△; †-homomorphism; †-identity; †-resp-≈) + open Monoidal +-monoidal using (assoc-commute-from; unitorˡ-commute-from; triangle) + open Shorthands +-monoidal using (α⇒; λ⇒; ρ⇒) + open ⇒-Reasoning 𝒞 + open ⊗-Reasoning +-monoidal + homoᵢ + : {A B C : Obj} + (f : A ⇒ B) + (g : B ⇒ C) + → (g ∘ f) † ∘ p₂ + ≈ (f † ∘ p₂) ∘ id ⊕₁ ((g † ∘ p₂) ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id + homoᵢ f g = begin + (g ∘ f) † ∘ p₂ ≈⟨ pushˡ †-homomorphism ⟩ + f † ∘ g † ∘ p₂ ≈⟨ refl⟩∘⟨ refl⟩∘⟨ p₂-⊕ ⟩ + f † ∘ g † ∘ λ⇒ ∘ ! ⊕₁ id ≈⟨ refl⟩∘⟨ extendʳ unitorˡ-commute-from ⟨ + f † ∘ λ⇒ ∘ id ⊕₁ (g †) ∘ ! ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ insertˡ p₁∘△ ⟩⊗⟨refl ⟩ + f † ∘ λ⇒ ∘ id ⊕₁ (g †) ∘ (p₁ ∘ △ ∘ !) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ (ρ⇒≈p₁ ⟩∘⟨refl) ⟩⊗⟨refl ⟨ + f † ∘ λ⇒ ∘ id ⊕₁ (g †) ∘ (ρ⇒ ∘ △ ∘ !) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ split₁ˡ ⟩ + f † ∘ λ⇒ ∘ id ⊕₁ (g †) ∘ ρ⇒ ⊕₁ id ∘ (△ ∘ !) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ ⇒△ ⟩⊗⟨refl ⟩ + f † ∘ λ⇒ ∘ id ⊕₁ (g †) ∘ ρ⇒ ⊕₁ id ∘ (! ⊕₁ ! ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ split₁ˡ ⟩ + f † ∘ λ⇒ ∘ id ⊕₁ (g †) ∘ ρ⇒ ⊕₁ id ∘ (! ⊕₁ !) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ (Equiv.sym triangle) ⟩ + f † ∘ λ⇒ ∘ id ⊕₁ (g †) ∘ id ⊕₁ λ⇒ ∘ α⇒ ∘ (! ⊕₁ !) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩ + f † ∘ λ⇒ ∘ id ⊕₁ (g † ∘ λ⇒) ∘ α⇒ ∘ (! ⊕₁ !) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ assoc-commute-from ⟩ + f † ∘ λ⇒ ∘ id ⊕₁ (g † ∘ λ⇒) ∘ ! ⊕₁ ! ⊕₁ id ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩ + f † ∘ λ⇒ ∘ ! ⊕₁ ((g † ∘ λ⇒) ∘ ! ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ (pullʳ (refl⟩∘⟨ (Equiv.sym ⇒! ⟩⊗⟨refl))) ⟩∘⟨refl ⟩ + f † ∘ λ⇒ ∘ ! ⊕₁ (g † ∘ λ⇒ ∘ (! ∘ f) ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ (pushʳ split₁ˡ)) ⟩∘⟨refl ⟩ + f † ∘ λ⇒ ∘ ! ⊕₁ (g † ∘ (λ⇒ ∘ ! ⊕₁ id) ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ (pullˡ (refl⟩∘⟨ Equiv.sym p₂-⊕)) ⟩∘⟨refl ⟩ + f † ∘ λ⇒ ∘ ! ⊕₁ ((g † ∘ p₂) ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ pushʳ (extendʳ (pushʳ serialize₁₂)) ⟩ + (f † ∘ (λ⇒ ∘ ! ⊕₁ id)) ∘ id ⊕₁ ((g † ∘ p₂) ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ≈⟨ (refl⟩∘⟨ p₂-⊕) ⟩∘⟨refl ⟨ + (f † ∘ p₂) ∘ id ⊕₁ ((g † ∘ p₂) ∘ f ⊕₁ id) ∘ α⇒ ∘ △ ⊕₁ id ∎ diff --git a/Data/WiringDiagram/Core.agda b/Data/WiringDiagram/Core.agda index d2e01cd..9903ce5 100644 --- a/Data/WiringDiagram/Core.agda +++ b/Data/WiringDiagram/Core.agda @@ -14,7 +14,7 @@ open import Categories.Category.Monoidal.Utilities using (module Shorthands) open import Relation.Binary using (IsEquivalence) open Category 𝒞 using (Obj; _∘_; _⇒_; id; _≈_; module Equiv) -open IdempotentSemiadditiveDagger S using (_⊕₀_; _⊕₁_; p₂; +-monoidal; △; ▽) +open IdempotentSemiadditiveDagger S using (_⊕₀_; _⊕₁_; p₂; +-monoidal; △; ▽; _†) open Shorthands +-monoidal using (α⇒) -- A "Box" is a pair of objects from the underlying category, @@ -125,6 +125,22 @@ _⌻_ {Aᵢ □ Aₒ} {Bᵢ □ Bₒ} {Cᵢ □ Cₒ} (f′ ⧈ g′) (f ⧈ g) infixr 9 _⌻_ --- The loop wiring diagram +-- Special wiring diagrams + loop : {A : Obj} → WiringDiagram (A □ A) (A □ A) loop = ▽ ⧈ id + +pulsh : {A B C D : Obj} → A ⇒ B → C ⇒ D → WiringDiagram (B □ C) (A □ D) +pulsh f g = f ∘ p₂ ⧈ g + +push : {A B : Obj} → A ⇒ B → WiringDiagram (A □ A) (B □ B) +push f = pulsh (f †) f + +pull : {A B : Obj} → A ⇒ B → WiringDiagram (B □ B) (A □ A) +pull f = pulsh f (f †) + +merge : {A B : Obj} → A ⇒ B → WiringDiagram (A □ A) (B □ B) +merge f = f † ∘ ▽ ∘ f ⊕₁ id ⧈ f + +split : {A B : Obj} → A ⇒ B → WiringDiagram (B □ B) (A □ A) +split f = ▽ ∘ id ⊕₁ f ⧈ f † diff --git a/Data/WiringDiagram/Directed.agda b/Data/WiringDiagram/Directed.agda index 88c6008..f1cbb93 100644 --- a/Data/WiringDiagram/Directed.agda +++ b/Data/WiringDiagram/Directed.agda @@ -17,7 +17,9 @@ 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 import Categories.Functor.Bifunctor using (Bifunctor) +open import Data.Product using (_,_) +open import Data.WiringDiagram.Core S using (Box; WiringDiagram; _≈-⧈_; _□_; _⧈_; _⌸_; id-⧈; _⌻_; ≈-isEquiv; pulsh) open Category 𝒞 open IdempotentSemiadditiveDagger S @@ -182,3 +184,37 @@ DWD = categoryHelper record ; equiv = ≈-isEquiv ; ∘-resp-≈ = ⌻-resp-≈ } + +-- Every pair of morphisms in 𝒞 gives a wiring diagram +Pulsh : Bifunctor op 𝒞 DWD +Pulsh = record + { F₀ = λ (A , B) → A □ B + ; F₁ = λ (f , g) → pulsh f g + ; identity = elimˡ Equiv.refl ⌸ Equiv.refl + ; homomorphism = λ { {A} {B} {C} {f , g} {f′ , g′} → homoᵢ g g′ f f′ ⌸ Equiv.refl } + ; F-resp-≈ = λ (f≈f′ , g≈g′) → (f≈f′ ⟩∘⟨refl) ⌸ g≈g′ + } + where + open ⇒-Reasoning 𝒞 + open ⊗-Reasoning +-monoidal + homoᵢ : {A B C D E F : Obj} (g : A ⇒ B) (g′ : B ⇒ C) (f : E ⇒ F) (f′ : D ⇒ E) + → (f ∘ f′) ∘ p₂ + ≈ (f ∘ p₂) ∘ (id ⊕₁ ((f′ ∘ p₂) ∘ g ⊕₁ id)) ∘ α⇒ ∘ (△ ⊕₁ id) + homoᵢ g g′ f f′ = begin + (f ∘ f′) ∘ p₂ ≈⟨ refl⟩∘⟨ p₂-⊕ ⟩ + (f ∘ f′) ∘ λ⇒ ∘ ! ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ insertˡ p₁∘△ ⟩⊗⟨refl ⟩ + (f ∘ f′) ∘ λ⇒ ∘ (p₁ ∘ △ ∘ !) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ (ρ⇒≈p₁ ⟩∘⟨refl) ⟩⊗⟨refl ⟨ + (f ∘ f′) ∘ λ⇒ ∘ (ρ⇒ ∘ △ ∘ !) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ (refl⟩∘⟨ ⇒△) ⟩⊗⟨refl ⟩ + (f ∘ f′) ∘ λ⇒ ∘ (ρ⇒ ∘ ! ⊕₁ ! ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ split₁ʳ ⟩ + (f ∘ f′) ∘ λ⇒ ∘ ρ⇒ ⊕₁ id ∘ (! ⊕₁ ! ∘ △) ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ split₁ʳ ⟩ + (f ∘ f′) ∘ λ⇒ ∘ ρ⇒ ⊕₁ id ∘ (! ⊕₁ !) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ (Equiv.sym triangle) ⟩ + (f ∘ f′) ∘ λ⇒ ∘ id ⊕₁ λ⇒ ∘ α⇒ ∘ (! ⊕₁ !) ⊕₁ id ∘ △ ⊕₁ id ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ extendʳ assoc-commute-from ⟩ + (f ∘ f′) ∘ λ⇒ ∘ id ⊕₁ λ⇒ ∘ ! ⊕₁ ! ⊕₁ id ∘ α⇒ ∘ (△ ⊕₁ id) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩ + (f ∘ f′) ∘ λ⇒ ∘ ! ⊕₁ (λ⇒ ∘ ! ⊕₁ id) ∘ α⇒ ∘ (△ ⊕₁ id) ≈⟨ pullʳ (extendʳ (Equiv.sym unitorˡ-commute-from)) ⟩ + f ∘ λ⇒ ∘ id ⊕₁ f′ ∘ ! ⊕₁ (λ⇒ ∘ ! ⊕₁ id) ∘ α⇒ ∘ (△ ⊕₁ id) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ merge₂ˡ ⟩ + f ∘ λ⇒ ∘ ! ⊕₁ (f′ ∘ λ⇒ ∘ ! ⊕₁ id) ∘ α⇒ ∘ (△ ⊕₁ id) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ refl⟩∘⟨ ⇒! ⟩⊗⟨refl) ⟩∘⟨refl ⟨ + f ∘ λ⇒ ∘ ! ⊕₁ (f′ ∘ λ⇒ ∘ (! ∘ g) ⊕₁ id) ∘ α⇒ ∘ (△ ⊕₁ id) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ (refl⟩∘⟨ pushʳ split₁ˡ) ⟩∘⟨refl ⟩ + f ∘ λ⇒ ∘ ! ⊕₁ (f′ ∘ (λ⇒ ∘ ! ⊕₁ id) ∘ g ⊕₁ id) ∘ α⇒ ∘ (△ ⊕₁ id) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩⊗⟨ pushˡ (refl⟩∘⟨ p₂-⊕) ⟩∘⟨refl ⟨ + f ∘ λ⇒ ∘ ! ⊕₁ ((f′ ∘ p₂) ∘ g ⊕₁ id) ∘ α⇒ ∘ (△ ⊕₁ id) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pushˡ serialize₁₂ ⟩ + f ∘ λ⇒ ∘ ! ⊕₁ id ∘ (id ⊕₁ ((f′ ∘ p₂) ∘ g ⊕₁ id)) ∘ α⇒ ∘ (△ ⊕₁ id) ≈⟨ pushʳ (pullˡ (Equiv.sym p₂-⊕)) ⟩ + (f ∘ p₂) ∘ (id ⊕₁ ((f′ ∘ p₂) ∘ g ⊕₁ id)) ∘ α⇒ ∘ (△ ⊕₁ id) ∎ 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) |