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Worked on commutativity

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src/Monad/Instance/K/Commutative.lagda.md
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@ -29,15 +29,15 @@ module Monad.Instance.K.Commutative {o ℓ e} (ambient : Ambient o ℓ e) (MK :
# K is a commutative monad # K is a commutative monad
The proof is analogous to the ones for strength, this is the relevant diagram is: The proof is analogous to the ones for strength, this is the relevant diagram is:
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```agda ```agda
KCommutative : CommutativeMonad {C = C} {V = monoidal} symmetric KStrong KCommutative : CommutativeMonad {C = C} {V = monoidal} symmetric KStrong
KCommutative = record { commutes = commutes' } KCommutative = record { commutes = commutes' }
where where
open monadK using (μ) open monadK using (μ)
open StrongMonad KStrong open StrongMonad KStrong using (strengthen)
open IsStableFreeUniformIterationAlgebra using (♯-law; ♯-preserving; ♯-unique) open IsStableFreeUniformIterationAlgebra using (♯-law; ♯-preserving; ♯-unique)
open Uniform-Iteration-Algebra using (#-Uniformity; #-Fixpoint; #-resp-≈) open Uniform-Iteration-Algebra using (#-Uniformity; #-Fixpoint; #-resp-≈)
@ -48,10 +48,19 @@ The proof is analogous to the ones for strength, this is the relevant diagram is
σ : ∀ ((X , Y) : Obj ×f Obj) → K.₀ X × Y ⇒ K.₀ (X × Y) σ : ∀ ((X , Y) : Obj ×f Obj) → K.₀ X × Y ⇒ K.₀ (X × Y)
σ _ = K.₁ swap ∘ (τ _) ∘ swap σ _ = K.₁ swap ∘ (τ _) ∘ swap
σ-preserve : ∀ {X Y Z : Obj} (h : Z ⇒ K.₀ Y + Z) → σ (Y , X) ∘ (h # ⁂ idC) ≈ ((σ _ +₁ idC) ∘ distributeʳ⁻¹ ∘ (h ⁂ idC))#
{-
K.₁ swap ∘ τ ∘ swap ∘ (h # ⁂ idC)
≈ K.₁ swap ∘ τ ∘ (idC ⁂ h #) ∘ swap
≈ K.₁ swap ∘ ()# ∘ swap
≈ ((K.₁ swap +₁ τ) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ∘ swap
-}
σ-preserve {Z} h = {! !}
commutes' : ∀ {X Y : Obj} → μ.η _ ∘ K.₁ (σ _) ∘ τ (K.₀ X , Y) ≈ μ.η _ ∘ K.₁ (τ _) ∘ σ _ commutes' : ∀ {X Y : Obj} → μ.η _ ∘ K.₁ (σ _) ∘ τ (K.₀ X , Y) ≈ μ.η _ ∘ K.₁ (τ _) ∘ σ _
commutes' {X} {Y} = begin commutes' {X} {Y} = begin
μ.η _ ∘ K.₁ (σ _) ∘ τ _ ≈⟨ ♯-unique (stable _) (σ _) (μ.η (X × Y) ∘ K.₁ (σ _) ∘ τ _) comm₁ comm₂ ⟩ μ.η _ ∘ K.₁ (σ _) ∘ τ _ ≈⟨ ♯-unique (stable _) (σ _) (μ.η (X × Y) ∘ K.₁ (σ _) ∘ τ _) comm₁ comm₂ ⟩
(σ _) ♯ ≈⟨ sym (♯-unique (stable _) (σ _) (μ.η _ ∘ K.₁ (τ _) ∘ σ _) comm₃ {! !}) ⟩ (σ _) ♯ ≈⟨ sym (♯-unique (stable _) (σ _) (μ.η _ ∘ K.₁ (τ _) ∘ σ _) comm₃ comm₄) ⟩
μ.η _ ∘ K.₁ (τ _) ∘ σ _ ∎ μ.η _ ∘ K.₁ (τ _) ∘ σ _ ∎
where where
comm₁ : σ _ ≈ (μ.η _ ∘ K.₁ (σ _) ∘ τ _) ∘ (idC ⁂ η _) comm₁ : σ _ ≈ (μ.η _ ∘ K.₁ (σ _) ∘ τ _) ∘ (idC ⁂ η _)
@ -71,14 +80,54 @@ The proof is analogous to the ones for strength, this is the relevant diagram is
((μ.η _ ∘ K.₁ (σ _) ∘ τ _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ∎ ((μ.η _ ∘ K.₁ (σ _) ∘ τ _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ∎
comm₃ : σ _ ≈ (μ.η _ ∘ K.₁ (τ _) ∘ σ _) ∘ (idC ⁂ η _) comm₃ : σ _ ≈ (μ.η _ ∘ K.₁ (τ _) ∘ σ _) ∘ (idC ⁂ η _)
comm₃ = sym (begin comm₃ = sym (begin
-- idea use swap epi and K.₁ swap mono: (μ.η _ ∘ K.₁ (τ _) ∘ σ _) ∘ (idC ⁂ η _) ≈⟨ pullʳ (pullʳ (pullʳ (pullʳ swap∘⁂))) ⟩
{- μ.η _ ∘ K.₁ (τ _) ∘ K.₁ swap ∘ τ _ ∘ (η _ ⁂ idC) ∘ swap ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ ⁂-cong₂ refl (sym K.identity) ⟩∘⟨refl ⟩
K.₁ swap ∘ (μ.η _ ∘ K.₁ (K.₁ swap ∘ τ _) ∘ σ _) ∘ (idC ⁂ η _) ∘ swap μ.η _ ∘ K.₁ (τ _) ∘ K.₁ swap ∘ τ _ ∘ (η _ ⁂ K.₁ idC) ∘ swap ≈⟨ refl⟩∘⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (strengthen.commute (η _ , idC)) ⟩
≈ (μ.η _ ∘ K.₁ (σ _) ∘ (τ _)) ∘ (η _ ⁂ idC) μ.η _ ∘ K.₁ (τ _) ∘ K.₁ swap ∘ (K.₁ (η _ ⁂ idC) ∘ τ _) ∘ swap ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (pullˡ (sym K.homomorphism)) ⟩
-} μ.η _ ∘ K.₁ (τ _) ∘ (K.₁ (swap ∘ (η _ ⁂ idC)) ∘ τ _) ∘ swap ≈⟨ refl⟩∘⟨ (pullˡ (pullˡ (sym K.homomorphism))) ⟩
(μ.η _ ∘ K.₁ (τ _) ∘ σ _) ∘ (idC ⁂ η _) ≈⟨ {! !} ⟩ μ.η _ ∘ (K.₁ (τ _ ∘ swap ∘ (η _ ⁂ idC)) ∘ τ _) ∘ swap ≈⟨ refl⟩∘⟨ (((K.F-resp-≈ (refl⟩∘⟨ swap∘⁂)) ⟩∘⟨refl) ⟩∘⟨refl) ⟩
{! !} ≈⟨ {! !} ⟩ μ.η _ ∘ (K.₁ (τ _ ∘ (idC ⁂ η _) ∘ swap) ∘ τ _) ∘ swap ≈⟨ refl⟩∘⟨ (K.F-resp-≈ (pullˡ (τ-η _))) ⟩∘⟨refl ⟩∘⟨refl ⟩
{! !} ≈⟨ {! !} ⟩ μ.η _ ∘ (K.₁ (η _ ∘ swap) ∘ τ _) ∘ swap ≈⟨ refl⟩∘⟨ ((K.homomorphism ⟩∘⟨refl) ⟩∘⟨refl) ⟩
{! !} ≈⟨ {! !} ⟩ μ.η _ ∘ ((K.₁ (η _) ∘ K.₁ swap) ∘ τ _) ∘ swap ≈⟨ pullˡ (pullˡ (cancelˡ monadK.identityˡ)) ⟩
(K.₁ swap ∘ τ _) ∘ swap ≈⟨ assoc ⟩
σ _ ∎) σ _ ∎)
comm₄ : ∀ {Z : Obj} (h : Z ⇒ K.₀ Y + Z) → (μ.η _ ∘ K.₁ (τ _) ∘ σ _) ∘ (idC ⁂ h #) ≈ ((μ.η _ ∘ K.₁ (τ _) ∘ σ _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))#
comm₄ {Z} h = begin
(μ.η _ ∘ K.₁ (τ _) ∘ σ _) ∘ (idC ⁂ h #) ≈⟨ {! !} ⟩
μ.η (X × Y) ∘ K.₁ (τ _) ∘ K.₁ swap ∘ τ _ ∘ (h # ⁂ idC) ∘ swap ≈⟨ {! !} ⟩
μ.η _ ∘ K.₁ (τ _) ∘ K.₁ swap ∘ τ _ ∘ (h # ⁂ K.₁ idC) ∘ swap ≈⟨ {! !} ⟩
μ.η _ ∘ K.₁ (τ _) ∘ K.₁ swap ∘ K.₁ (h # ⁂ idC) ∘ τ _ ∘ swap ≈⟨ {! !} ⟩
μ.η _ ∘ K.₁ (τ _) ∘ K.₁ (swap ∘ (h # ⁂ idC)) ∘ τ _ ∘ swap ≈⟨ {! !} ⟩
μ.η _ ∘ K.₁ (τ _) ∘ K.₁ ((idC ⁂ h #) ∘ swap) ∘ τ _ ∘ swap ≈⟨ {! !} ⟩
μ.η _ ∘ K.₁ (τ _) ∘ (K.₁ (idC ⁂ h #) ∘ K.₁ swap) ∘ τ _ ∘ swap ≈⟨ {! !} ⟩
μ.η _ ∘ (K.₁ (τ _ ∘ (idC ⁂ h #)) ∘ K.₁ swap) ∘ τ _ ∘ swap ≈⟨ {! !} ⟩
μ.η _ ∘ (K.₁ (((τ _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #) ∘ K.₁ swap) ∘ τ _ ∘ swap ≈⟨ {! !} ⟩
μ.η _ ∘ K.₁ (((τ _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #) ∘ σ _ ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
μ.η (X × Y) ∘ K.₁ (τ _) ∘ K.₁ swap ∘ ((τ _ +₁ idC) ∘ distributeʳ⁻¹ ∘ (h ⁂ idC)) # ∘ swap ≈⟨ {! !} ⟩
μ.η _ ∘ K.₁ (τ _) ∘ K.₁ swap ∘ ((τ _ ∘ swap +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ≈⟨ {! !} ⟩
μ.η _ ∘ K.₁ (τ _) ∘ ((σ _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ≈⟨ {! !} ⟩
μ.η _ ∘ ((K.₁ (τ _) ∘ σ _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ≈⟨ {! !} ⟩
((μ.η _ ∘ K.₁ (τ _) ∘ σ _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ∎
where
test : ((τ (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ∘ swap ≈ ((τ (X , Y) ∘ swap +₁ idC) ∘ distributeʳ⁻¹ ∘ (h ⁂ idC)) #
test = sym (#-Uniformity (algebras _) (sym (begin
((τ (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) ∘ swap ≈⟨ pullʳ (pullʳ (sym swap∘⁂)) ⟩
(τ (X , Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ swap ∘ (h ⁂ idC) ≈⟨ refl⟩∘⟨ (pullˡ distributeˡ⁻¹∘swap) ⟩
(τ (X , Y) +₁ idC) ∘ ((swap +₁ swap) ∘ distributeʳ⁻¹) ∘ (h ⁂ idC) ≈⟨ pullˡ (pullˡ (+₁∘+₁ ○ +₁-cong₂ (sym identityˡ) id-comm-sym)) ⟩
((idC ∘ τ (X , Y) ∘ swap +₁ swap ∘ idC) ∘ distributeʳ⁻¹) ∘ (h ⁂ idC) ≈⟨ assoc ○ (sym +₁∘+₁) ⟩∘⟨refl ⟩
((idC +₁ swap) ∘ (τ (X , Y) ∘ swap +₁ idC)) ∘ distributeʳ⁻¹ ∘ (h ⁂ idC) ≈⟨ assoc ⟩
(idC +₁ swap) ∘ (τ (X , Y) ∘ swap +₁ idC) ∘ distributeʳ⁻¹ ∘ (h ⁂ idC) ∎)))
helper : τ _ ∘ (h # ⁂ idC) ∘ swap ≈ ((τ _ +₁ idC) ∘ distributeʳ⁻¹ ∘ (h ⁂ idC)) # ∘ swap
helper = {! !}
τ∘swap-preserving : τ (K.₀ Y , X) ∘ (h # ⁂ idC) ≈ ((τ _ +₁ idC) ∘ distributeʳ⁻¹ ∘ (h ⁂ idC)) #
τ∘swap-preserving = begin
τ (K.₀ Y , X) ∘ (h # ⁂ idC) ≈⟨ {! !} ⟩
τ (K.₀ Y , X) ∘ (h # ⁂ K.₁ idC) ≈⟨ {! !} ⟩
K.₁ (h # ⁂ idC) ∘ τ _ ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
((τ _ +₁ idC) ∘ distributeʳ⁻¹ ∘ (h ⁂ idC)) # ∎
``` ```