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Cleanup proofs in Kleisli category construction #378

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Jul 15, 2023
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Cleanup proofs in Kleisli category construction #378

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c3fc767
Add variant of glue
Taneb Jul 15, 2023
781d16b
Clean up proofs in Kleisli category construction
Taneb Jul 15, 2023
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33 changes: 11 additions & 22 deletions src/Categories/Category/Construction/Kleisli.agda
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Expand Up @@ -7,7 +7,7 @@ open import Categories.Category
open import Categories.Functor using (Functor; module Functor)
open import Categories.NaturalTransformation hiding (id)
open import Categories.Monad
import Categories.Morphism.Reasoning as MR
import Categories.Morphism.Reasoning.Core as MR

private
variable
Expand Down Expand Up @@ -36,34 +36,23 @@ Kleisli {𝒞 = 𝒞} M = record
open HomReasoning
open MR 𝒞

-- shorthands to make the proofs nicer
F≈ = F-resp-≈

assoc′ : ∀ {A B C D} {f : A ⇒ F₀ B} {g : B ⇒ F₀ C} {h : C ⇒ F₀ D}
→ (μ.η D ∘ (F₁ ((μ.η D ∘ F₁ h) ∘ g))) ∘ f ≈ (μ.η D ∘ F₁ h) ∘ ((μ.η C ∘ F₁ g) ∘ f)
assoc′ {A} {B} {C} {D} {f} {g} {h} =
begin
(μ.η D ∘ F₁ ((μ.η D ∘ F₁ h) ∘ g)) ∘ f ≈⟨ pullʳ (F≈ assoc ⟩∘⟨refl) ⟩
μ.η D ∘ (F₁ (μ.η D ∘ (F₁ h ∘ g)) ∘ f) ≈⟨ refl⟩∘⟨ (homomorphism ⟩∘⟨refl) ⟩
μ.η D ∘ ((F₁ (μ.η D) ∘ F₁ (F₁ h ∘ g)) ∘ f) ≈⟨ pushʳ assoc ⟩
(μ.η D ∘ F₁ (μ.η D)) ∘ (F₁ (F₁ h ∘ g) ∘ f) ≈⟨ pushˡ M.assoc ⟩
μ.η D ∘ (μ.η (F₀ D) ∘ F₁ (F₁ h ∘ g) ∘ f) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ homomorphism ⟩∘⟨refl ⟩
μ.η D ∘ μ.η (F₀ D) ∘ (F₁ (F₁ h) ∘ F₁ g) ∘ f ≈⟨ refl⟩∘⟨ center⁻¹ (μ.commute h) Equiv.refl ⟩
μ.η D ∘ ((F₁ h ∘ μ.η C) ∘ F₁ g ∘ f) ≈⟨ pushʳ (center Equiv.refl) ⟩
(μ.η D ∘ F₁ h) ∘ ((μ.η C ∘ F₁ g) ∘ f) ∎
assoc′ {A} {B} {C} {D} {f} {g} {h} = begin
(μ.η D ∘ F₁ ((μ.η D ∘ F₁ h) ∘ g)) ∘ f ≈⟨ pushʳ homomorphism ⟩∘⟨refl ⟩
((μ.η D ∘ F₁ (μ.η D ∘ F₁ h)) ∘ F₁ g) ∘ f ≈⟨ pushˡ (∘-resp-≈ˡ (∘-resp-≈ʳ homomorphism)) ⟩
(μ.η D ∘ (F₁ (μ.η D) ∘ F₁ (F₁ h))) ∘ (F₁ g ∘ f) ≈⟨ pushˡ (glue′ M.assoc (μ.commute h)) ⟩
(μ.η D ∘ F₁ h) ∘ (μ.η C ∘ (F₁ g ∘ f)) ≈⟨ refl⟩∘⟨ sym-assoc ⟩
(μ.η D ∘ F₁ h) ∘ ((μ.η C ∘ F₁ g) ∘ f) ∎

identityˡ′ : ∀ {A B} {f : A ⇒ F₀ B} → (μ.η B ∘ F₁ (η.η B)) ∘ f ≈ f
identityˡ′ {A} {B} {f} = elimˡ M.identityˡ

identityʳ′ : ∀ {A B} {f : A ⇒ F₀ B} → (μ.η B ∘ F₁ f) ∘ η.η A ≈ f
identityʳ′ {A} {B} {f} =
begin
(μ.η B ∘ F₁ f) ∘ η.η A ≈⟨ assoc ⟩
μ.η B ∘ (F₁ f ∘ η.η A) ≈˘⟨ refl⟩∘⟨ η.commute f ⟩
μ.η B ∘ (η.η (F₀ B) ∘ f) ≈⟨ sym-assoc ⟩
(μ.η B ∘ η.η (F₀ B)) ∘ f ≈⟨ elimˡ M.identityʳ ⟩
f
∎
identityʳ′ {A} {B} {f} = begin
(μ.η B ∘ F₁ f) ∘ η.η A ≈˘⟨ extendˡ (η.commute f) ⟩
(μ.η B ∘ η.η (F₀ B)) ∘ f ≈⟨ elimˡ M.identityʳ ⟩
f ∎

identity²′ : {A : Obj} → (μ.η A ∘ F₁ (η.η A)) ∘ η.η A ≈ η.η A
identity²′ = elimˡ M.identityˡ
10 changes: 10 additions & 0 deletions src/Categories/Morphism/Reasoning/Core.agda
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Expand Up @@ -157,6 +157,16 @@ glue {c′ = c′} {a′ = a′} {a = a} {c″ = c″} {c = c} {b′ = b′} {b
(c″ ∘ a′) ∘ b′ ≈⟨ assoc ⟩
c″ ∘ (a′ ∘ b′) ∎

-- A "rotated" version of glue′. Equivalent to 'Equiv.sym (glue (Equiv.sym sq-a) (Equiv.sym sq-b))'
glue′ : CommutativeSquare a′ c′ c″ a →
CommutativeSquare b′ c c′ b →
CommutativeSquare (a′ ∘ b′) c c″ (a ∘ b)
glue′ {a′ = a′} {c′ = c′} {c″ = c″} {a = a} {b′ = b′} {c = c} {b = b} sq-a sq-b = begin
c″ ∘ (a′ ∘ b′) ≈⟨ pullˡ sq-a ⟩
(a ∘ c′) ∘ b′ ≈⟨ pullʳ sq-b ⟩
a ∘ (b ∘ c) ≈⟨ sym-assoc ⟩
(a ∘ b) ∘ c ∎

glue◃◽ : a ∘ c′ ≈ c″ → CommutativeSquare c b′ b c′ → CommutativeSquare c b′ (a ∘ b) c″
glue◃◽ {a = a} {c′ = c′} {c″ = c″} {c = c} {b′ = b′} {b = b} tri-a sq-b = begin
(a ∘ b) ∘ c ≈⟨ pullʳ sq-b ⟩
Expand Down