Work on commutativity

.gitignore

@@ -4,3 +4,4 @@
 Everything.agda
 public/
 .direnv
+.DS_Store

src/Category/Instance/AmbientCategory.lagda.md

@@ -96,6 +96,7 @@ module Category.Instance.AmbientCategory where
     dstr-law₃ = (refl⟩∘⟨ (sym inject₁)) ○ (cancelˡ (IsIso.isoˡ isIsoʳ))
     dstr-law₄ : ∀ {A B C} → distributeʳ⁻¹ {A} {B} {C} ∘ (i₂ ⁂ idC) ≈ i₂
     dstr-law₄ = (refl⟩∘⟨ (sym inject₂)) ○ (cancelˡ (IsIso.isoˡ isIsoʳ))
+    -- TODO this is double...
     dstr-law₅ : ∀ {A B C} → (π₂ +₁ π₂) ∘ distributeˡ⁻¹ {A} {B} {C} ≈ π₂
     dstr-law₅ = Iso⇒Epi C (IsIso.iso isIsoˡ) ((π₂ +₁ π₂) ∘ distributeˡ⁻¹) π₂ (begin 
       (((π₂ +₁ π₂) ∘ distributeˡ⁻¹) ∘ distributeˡ) ≈⟨ cancelʳ (IsIso.isoˡ isIsoˡ) ⟩ 

src/Monad/Instance/K/Commutative.lagda.md

@@ -27,7 +27,7 @@ module Monad.Instance.K.Commutative {o ℓ e} (ambient : Ambient o ℓ e) (MK :
 ```
 
 # 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, the relevant diagram is:
 
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@@ -57,10 +57,13 @@ The proof is analogous to the ones for strength, this is the relevant diagram is
       ≈ ((K.₁ swap +₁ τ) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ∘ swap
       -}
       σ-preserve {Z} h = {!   !}
+      σ-preserve' : ∀ {X Y Z : Obj} (h : Z ⇒ K.₀ Y + Z) → σ (X , K.₀ Y) ∘ (idC ⁂ h #) ≈ ((σ _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))#
+      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} = begin 
         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 comm₄) ⟩
+        {!   !} ≈⟨ {!   !} ⟩
         μ.η _ ∘ K.₁ (τ _) ∘ σ _ ∎
         where
           comm₁ : σ _ ≈ (μ.η _ ∘ K.₁ (σ _) ∘ τ _) ∘ (idC ⁂ η _)
@@ -94,25 +97,22 @@ The proof is analogous to the ones for strength, this is the relevant diagram is
           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 _) ∘ σ _) ∘ ((i₁ # ∘ idC) ⁂ h #) ≈˘⟨ {!   !} ⟩
-            μ.η _ ∘ K.₁ (τ _) ∘ K.₁ swap ∘ τ _ ∘ (h # ⁂ K.₁ idC) ∘ swap ≈⟨ {!   !} ⟩ 
+            (μ.η _ ∘ K.₁ (τ _) ∘ σ _) ∘ (((i₁ #) ⁂ h #)) ≈˘⟨ refl⟩∘⟨ ⟨⟩-cong₂ (#-Uniformity (algebras _) helper₁) {!   !} ⟩
-            μ.η _ ∘ K.₁ (τ _) ∘ K.₁ swap ∘ K.₁ (h # ⁂ idC) ∘ τ _ ∘ swap ≈⟨ {!   !} ⟩ 
+            (μ.η _ ∘ K.₁ (τ _) ∘ σ _) ∘ ⟨ ((π₁ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) # , ((π₂ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) # ⟩ ≈⟨ {!   !} ⟩
-            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
+              -- this leads nowhere
+              helper₁ : (idC +₁ π₁) ∘ (π₁ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h) ≈ i₁ ∘ π₁
+              helper₁ = begin 
+                (idC +₁ π₁) ∘ (π₁ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h) ≈⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ identityˡ identityʳ) ⟩ 
+                (π₁ +₁ π₁) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h) ≈⟨ {!   !} ⟩ 
+                i₁ ∘ π₁ ∘ (idC ⁂ h) ≈⟨ refl⟩∘⟨ π₁∘⁂ ⟩ 
+                i₁ ∘ idC ∘ π₁ ≈⟨ refl⟩∘⟨ identityˡ ⟩ 
+                i₁ ∘ π₁ ∎
               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∘⁂)) ⟩