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@ -1,6 +1,5 @@
<!-- <!--
```agda ```agda
{-# OPTIONS --allow-unsolved-metas #-}
open import Level open import Level
open import Categories.Functor renaming (id to idF) open import Categories.Functor renaming (id to idF)
open import Categories.Functor.Algebra open import Categories.Functor.Algebra
@ -134,9 +133,10 @@ Here we give a different Characterization and show that it is equal.
by-uni = begin by-uni = begin
([ h +₁ i₁ ∘ i₁ , i₂ ∘ i₂ ] ∘ [ i₁ , f ]) ∘ (i₁ +₁ idC) ≈⟨ ((∘[] ○ []-cong₂ inject₁ refl) ⟩∘⟨refl) ⟩ ([ h +₁ i₁ ∘ i₁ , i₂ ∘ i₂ ] ∘ [ i₁ , f ]) ∘ (i₁ +₁ idC) ≈⟨ ((∘[] ○ []-cong₂ inject₁ refl) ⟩∘⟨refl) ⟩
[ h +₁ i₁ ∘ i₁ , [ h +₁ i₁ ∘ i₁ , i₂ ∘ i₂ ] ∘ f ] ∘ (i₁ +₁ idC) ≈⟨ ([]∘+₁ ○ []-cong₂ +₁∘i₁ identityʳ) ⟩ [ h +₁ i₁ ∘ i₁ , [ h +₁ i₁ ∘ i₁ , i₂ ∘ i₂ ] ∘ f ] ∘ (i₁ +₁ idC) ≈⟨ ([]∘+₁ ○ []-cong₂ +₁∘i₁ identityʳ) ⟩
[ i₁ ∘ h , [ h +₁ i₁ ∘ i₁ , i₂ ∘ i₂ ] ∘ f ] ≈˘⟨ []-cong₂ (pullˡ (+₁∘i₁ ○ identityʳ)) (([]-cong₂ (+₁∘+₁ ○ +₁-cong₂ identityˡ +₁∘i₁) (pullˡ +₁∘i₂ ○ pullʳ (+₁∘i₂ ○ identityʳ))) ⟩∘⟨refl) ⟩ -- TODO all these steps work
[ (idC +₁ (i₁ +₁ idC)) ∘ i₁ ∘ h , [ (idC +₁ (i₁ +₁ idC)) ∘ (h +₁ i₁) , (idC +₁ (i₁ +₁ idC)) ∘ i₂ ∘ i₂ ] ∘ f ] ≈˘⟨ []-cong₂ refl (pullˡ ∘[]) ⟩ [ i₁ ∘ h , [ h +₁ i₁ ∘ i₁ , i₂ ∘ i₂ ] ∘ f ] ≈⟨ {! !} ⟩
[ (idC +₁ (i₁ +₁ idC)) ∘ i₁ ∘ h , (idC +₁ (i₁ +₁ idC)) ∘ [ (h +₁ i₁) , i₂ ∘ i₂ ] ∘ f ] ≈˘⟨ ∘[] ⟩ [ (idC +₁ (i₁ +₁ idC)) ∘ i₁ ∘ h , [ (idC +₁ (i₁ +₁ idC)) ∘ (h +₁ i₁) , (idC +₁ (i₁ +₁ idC)) ∘ i₂ ∘ i₂ ] ∘ f ] ≈⟨ {! !} ⟩
[ (idC +₁ (i₁ +₁ idC)) ∘ i₁ ∘ h , (idC +₁ (i₁ +₁ idC)) ∘ [ (h +₁ i₁) , i₂ ∘ i₂ ] ∘ f ] ≈⟨ {! !} ⟩
(idC +₁ (i₁ +₁ idC)) ∘ [ i₁ ∘ h , [ (h +₁ i₁) , i₂ ∘ i₂ ] ∘ f ] ∎ (idC +₁ (i₁ +₁ idC)) ∘ [ i₁ ∘ h , [ (h +₁ i₁) , i₂ ∘ i₂ ] ∘ f ] ∎
-- TODO Proposition 41 -- TODO Proposition 41
@ -147,24 +147,22 @@ Here we give a different Characterization and show that it is equal.
[ (idC +₁ i₁) ∘ g , f ] # ∘ i₂ ≈˘⟨ ((#-resp-≈ ([]-cong₂ refl (elimˡ ([]-unique id-comm-sym id-comm-sym)))) ⟩∘⟨refl) ⟩ [ (idC +₁ i₁) ∘ g , f ] # ∘ i₂ ≈˘⟨ ((#-resp-≈ ([]-cong₂ refl (elimˡ ([]-unique id-comm-sym id-comm-sym)))) ⟩∘⟨refl) ⟩
[ (idC +₁ i₁) ∘ g , (idC +₁ idC) ∘ f ] # ∘ i₂ ≈˘⟨ ((#-resp-≈ ([]-cong₂ (pullˡ (+₁∘+₁ ○ +₁-cong₂ inject₁ identityˡ)) (pullˡ (+₁∘+₁ ○ +₁-cong₂ inject₂ identityˡ)))) ⟩∘⟨refl) ⟩ [ (idC +₁ i₁) ∘ g , (idC +₁ idC) ∘ f ] # ∘ i₂ ≈˘⟨ ((#-resp-≈ ([]-cong₂ (pullˡ (+₁∘+₁ ○ +₁-cong₂ inject₁ identityˡ)) (pullˡ (+₁∘+₁ ○ +₁-cong₂ inject₂ identityˡ)))) ⟩∘⟨refl) ⟩
[ ([ idC , idC ] +₁ idC) ∘ ((i₁ +₁ i₁) ∘ g) , ([ idC , idC ] +₁ idC) ∘ ((i₂ +₁ idC) ∘ f) ] # ∘ i₂ ≈˘⟨ ((#-resp-≈ ∘[]) ⟩∘⟨refl) ⟩ [ ([ idC , idC ] +₁ idC) ∘ ((i₁ +₁ i₁) ∘ g) , ([ idC , idC ] +₁ idC) ∘ ((i₂ +₁ idC) ∘ f) ] # ∘ i₂ ≈˘⟨ ((#-resp-≈ ∘[]) ⟩∘⟨refl) ⟩
(([ idC , idC ] +₁ idC) ∘ [ ((i₁ +₁ i₁) ∘ g) , ((i₂ +₁ idC) ∘ f) ]) # ∘ i₂ ≈⟨ ((#-Stutter [ (i₁ +₁ i₁) ∘ g , (i₂ +₁ idC) ∘ f ] idC) ⟩∘⟨refl) ⟩ (([ idC , idC ] +₁ idC) ∘ [ ((i₁ +₁ i₁) ∘ g) , ((i₂ +₁ idC) ∘ f) ]) # ∘ i₂ ≈⟨ {! !} ⟩ -- lemma 40
([ i₁ ∘ idC , [ idC +₁ i₁ , i₂ ∘ i₂ ] ∘ [ (i₁ +₁ i₁) ∘ g , (i₂ +₁ idC) ∘ f ] ] # ∘ i₂) ∘ i₂ ≈⟨ (assoc ○ ((#-resp-≈ ([]-cong₂ identityʳ refl)) ⟩∘⟨refl)) ⟩
[ i₁ , ([ idC +₁ i₁ , i₂ ∘ i₂ ] ∘ [ (i₁ +₁ i₁) ∘ g , (i₂ +₁ idC) ∘ f ]) ] # ∘ i₂ ∘ i₂ ≈⟨ ((#-resp-≈ ([]-cong₂ refl (∘[] ○ []-cong₂ (pullˡ []∘+₁) (pullˡ []∘+₁)))) ⟩∘⟨refl) ⟩ [ i₁ , ([ idC +₁ i₁ , i₂ ∘ i₂ ] ∘ [ (i₁ +₁ i₁) ∘ g , (i₂ +₁ idC) ∘ f ]) ] # ∘ i₂ ∘ i₂ ≈⟨ ((#-resp-≈ ([]-cong₂ refl (∘[] ○ []-cong₂ (pullˡ []∘+₁) (pullˡ []∘+₁)))) ⟩∘⟨refl) ⟩
[ i₁ , [ [ (idC +₁ i₁) ∘ i₁ , (i₂ ∘ i₂) ∘ i₁ ] ∘ g , [ (idC +₁ i₁) ∘ i₂ , (i₂ ∘ i₂) ∘ idC ] ∘ f ] ] # ∘ i₂ ∘ i₂ ≈⟨ ((#-resp-≈ ([]-cong₂ refl ([]-cong₂ (([]-cong₂ (+₁∘i₁ ○ identityʳ) assoc) ⟩∘⟨refl) (([]-cong₂ +₁∘i₂ identityʳ) ⟩∘⟨refl)))) ⟩∘⟨refl) ⟩ [ i₁ , [ [ (idC +₁ i₁) ∘ i₁ , (i₂ ∘ i₂) ∘ i₁ ] ∘ g , [ (idC +₁ i₁) ∘ i₂ , (i₂ ∘ i₂) ∘ idC ] ∘ f ] ] # ∘ i₂ ∘ i₂ ≈⟨ ((#-resp-≈ ([]-cong₂ refl ([]-cong₂ (([]-cong₂ (+₁∘i₁ ○ identityʳ) assoc) ⟩∘⟨refl) (([]-cong₂ +₁∘i₂ identityʳ) ⟩∘⟨refl)))) ⟩∘⟨refl) ⟩
[ i₁ , [ [ i₁ , i₂ ∘ i₂ ∘ i₁ ] ∘ g , [ i₂ ∘ i₁ , i₂ ∘ i₂ ] ∘ f ] ] # ∘ i₂ ∘ i₂ ≈˘⟨ ((#-resp-≈ ([]-cong₂ refl ([]-cong₂ (pullˡ ([]∘+₁ ○ []-cong₂ identityʳ refl)) (∘[] ⟩∘⟨refl)))) ⟩∘⟨refl) ⟩ [ i₁ , [ [ i₁ , i₂ ∘ i₂ ∘ i₁ ] ∘ g , [ i₂ ∘ i₁ , i₂ ∘ i₂ ] ∘ f ] ] # ∘ i₂ ∘ i₂ ≈˘⟨ ((#-resp-≈ ([]-cong₂ refl ([]-cong₂ (pullˡ ([]∘+₁ ○ []-cong₂ identityʳ refl)) (∘[] ⟩∘⟨refl)))) ⟩∘⟨refl) ⟩
[ i₁ , [ [ i₁ , i₂ ] ∘ (idC +₁ i₂ ∘ i₁) ∘ g , (i₂ ∘ [ i₁ , i₂ ]) ∘ f ] ] # ∘ i₂ ∘ i₂ ≈⟨ ((#-resp-≈ ([]-cong₂ refl ([]-cong₂ (elimˡ +-η) ((elimʳ +-η) ⟩∘⟨refl)))) ⟩∘⟨refl) ⟩ [ i₁ , [ [ i₁ , i₂ ] ∘ (idC +₁ i₂ ∘ i₁) ∘ g , (i₂ ∘ [ i₁ , i₂ ]) ∘ f ] ] # ∘ i₂ ∘ i₂ ≈⟨ ((#-resp-≈ ([]-cong₂ refl ([]-cong₂ (elimˡ +-η) ((elimʳ +-η) ⟩∘⟨refl)))) ⟩∘⟨refl) ⟩
[ i₁ , [ (idC +₁ i₂ ∘ i₁) ∘ g , i₂ ∘ f ] ] # ∘ i₂ ∘ i₂ {A = X} {B = X} ≈⟨ sym (#-Uniformity (sym by-uni₂)) ⟩ [ i₁ , [ (idC +₁ i₂ ∘ i₁) ∘ g , i₂ ∘ f ] ] # ∘ i₂ ∘ i₂ ≈⟨ {! !} ⟩
-- [ i₁ , [ (idC +₁ i₂ ∘ i₁) ∘ g , i₂ ∘ f ] ] # ∘ i₂ ∘ i₂ ≈⟨ pullˡ (sym (#-Uniformity (sym by-uni₂))) ⟩
-- {! !} ≈⟨ {! !} ⟩
-- [ {! !} , [ [ i₁ , (idC +₁ i₂ ∘ i₁ ∘ i₂) ∘ g ] , i₂ ∘ i₂ ∘ h ] ] # ∘ i₂ ∘ i₂ ≈˘⟨ (#-Uniformity by-uni₃ ⟩∘⟨refl) ○ assoc ⟩
[ [ i₁ , (idC +₁ i₁ ∘ i₂) ∘ g ] , i₂ ∘ h ] # ∘ i₂ ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩ {! !} ≈⟨ {! !} ⟩
(([ {! !} , {! !} ] ∘ f) #) ≈⟨ #-Uniformity (sym by-uni₃) ⟩ {! !} ≈⟨ {! !} ⟩
[ [ i₁ , (idC +₁ i₁ ∘ i₂) ∘ g ] , i₂ ∘ h ] # ∘ i₂ ≈˘⟨ ((#-resp-≈ ([]-cong₂ (∘[] ○ []-cong₂ (+₁∘i₁ ○ identityʳ) (pullˡ (+₁∘+₁ ○ +₁-cong₂ identity² refl))) refl)) ⟩∘⟨refl) ⟩ [ [ i₁ , (idC +₁ i₁ ∘ i₂) ∘ g ] , [ i₂ ∘ i₁ ∘ i₁ , i₂ ∘ (i₂ +₁ idC) ] ∘ f ] # ∘ i₂ ≈˘⟨ ((#-resp-≈ ([]-cong₂ (∘[] ○ []-cong₂ (+₁∘i₁ ○ identityʳ) (pullˡ (+₁∘+₁ ○ +₁-cong₂ identity² refl))) (pullˡ ∘[]))) ⟩∘⟨refl) ⟩
[ (idC +₁ i₁) ∘ [ i₁ , (idC +₁ i₂) ∘ g ] , i₂ ∘ h ] # ∘ i₂ ≈⟨ (sym #-Folding) ⟩∘⟨refl ⟩ [ (idC +₁ i₁) ∘ [ i₁ , (idC +₁ i₂) ∘ g ] , i₂ ∘ h ] # ∘ i₂ ≈⟨ (sym #-Folding) ⟩∘⟨refl ⟩
([ i₁ , (idC +₁ i₂) ∘ g ] # +₁ h)# ∘ i₂ ≈⟨ ((#-resp-≈ (+₁-cong₂ by-fix refl)) ⟩∘⟨refl) ⟩ ([ i₁ , (idC +₁ i₂) ∘ g ] # +₁ h)# ∘ i₂ ≈⟨ {! !} ⟩
([ idC , g # ] +₁ h ) # ∘ i₂ ≈˘⟨ ((#-resp-≈ ([]-cong₂ refl ((sym ∘[] ○ elimʳ +-η) ⟩∘⟨refl))) ⟩∘⟨refl) ⟩ {! !} ≈⟨ {! !} ⟩
[ i₁ ∘ [ idC , g # ] , [ i₂ ∘ i₁ , i₂ ∘ i₂ ] ∘ h ] # ∘ i₂ ≈˘⟨ ((#-resp-≈ ([]-cong₂ refl (pullˡ ([]∘+₁ ○ []-cong₂ inject₂ identityʳ)))) ⟩∘⟨refl) ⟩
[ i₁ ∘ [ idC , g # ] , [ [ idC , g # ] +₁ i₁ , i₂ ∘ i₂ ] ∘ (i₂ +₁ idC) ∘ h ] # ∘ i₂ ≈⟨ sym (#-Stutter ((i₂ +₁ idC) ∘ h) [ idC , g # ]) ⟩
(([ [ idC , g # ] , [ idC , g # ] ] +₁ idC) ∘ (i₂ +₁ idC) ∘ h) # ≈⟨ #-resp-≈ (pullˡ (+₁∘+₁ ○ +₁-cong₂ inject₂ identity²)) ⟩
((([ idC , g # ] +₁ idC)) ∘ h) # ≈⟨ #-resp-≈ (pullˡ (∘[] ○ []-cong₂ (pullˡ +₁∘i₁) +₁∘+₁)) ⟩
([ (i₁ ∘ [ idC , g # ]) ∘ i₁ , [ idC , g # ] ∘ i₂ +₁ idC ∘ idC ] ∘ f) # ≈⟨ #-resp-≈ (([]-cong₂ (cancelʳ inject₁) (+₁-cong₂ inject₂ identity²)) ⟩∘⟨refl) ⟩
([ i₁ , g # +₁ idC ] ∘ f) # ∎ ([ i₁ , g # +₁ idC ] ∘ f) # ∎
where where
g = (idC +₁ [ idC , idC ]) ∘ f g = (idC +₁ [ idC , idC ]) ∘ f
@ -176,27 +174,22 @@ Here we give a different Characterization and show that it is equal.
[ (idC +₁ idC) ∘ g , (idC +₁ [ idC , idC ]) ∘ f ] ≈⟨ []-cong₂ (elimˡ ([]-unique id-comm-sym id-comm-sym)) refl ⟩ [ (idC +₁ idC) ∘ g , (idC +₁ [ idC , idC ]) ∘ f ] ≈⟨ []-cong₂ (elimˡ ([]-unique id-comm-sym id-comm-sym)) refl ⟩
[ g , g ] ≈⟨ sym (∘[] ○ []-cong₂ identityʳ identityʳ) ⟩ [ g , g ] ≈⟨ sym (∘[] ○ []-cong₂ identityʳ identityʳ) ⟩
g ∘ [ idC , idC ] ∎ g ∘ [ idC , idC ] ∎
by-uni₂ : [ i₁ , [ (idC +₁ i₂ ∘ i₁) ∘ g , i₂ ∘ f ] ] ∘ i₂ ∘ i₂ ≈ (idC +₁ i₂ ∘ i₂) ∘ {! !} by-uni₂ : [ i₁ , [ (idC +₁ i₂ ∘ i₁) ∘ g , i₂ ∘ f ] ] ∘ i₂ ≈ (idC +₁ i₂) ∘ {! !}
by-uni₂ = begin by-uni₂ = begin
[ i₁ , [ (idC +₁ i₂ ∘ i₁) ∘ g , i₂ ∘ f ] ] ∘ i₂ ∘ i₂ ≈⟨ (pullˡ inject₂) ○ inject₂ ⟩ [ i₁ , [ (idC +₁ i₂ ∘ i₁) ∘ g , i₂ ∘ f ] ] ∘ i₂ ≈⟨ inject₂ ⟩
i₂ ∘ f ≈⟨ {! !} ⟩ [ (idC +₁ i₂ ∘ i₁) ∘ g , i₂ ∘ f ] ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
(idC +₁ i₂) ∘ [ ((idC +₁ i₁) ∘ g) , {! h !} ] ∎
by-uni₃ : (idC +₁ i₂) ∘ [ [ i₁ , (idC +₁ i₁ ∘ i₂) ∘ g ] , i₂ ∘ h ] ≈ {! !} ∘ i₂
by-uni₃ = begin
(idC +₁ i₂) ∘ [ [ i₁ , (idC +₁ i₁ ∘ i₂) ∘ g ] , i₂ ∘ h ] ≈⟨ ∘[] ⟩
[ (idC +₁ i₂) ∘ [ i₁ , (idC +₁ i₁ ∘ i₂) ∘ g ] , (idC +₁ i₂) ∘ i₂ ∘ h ] ≈⟨ []-cong₂ ∘[] (pullˡ +₁∘i₂) ⟩
[ [ (idC +₁ i₂) ∘ i₁ , (idC +₁ i₂) ∘ (idC +₁ i₁ ∘ i₂) ∘ g ] , (i₂ ∘ i₂) ∘ h ] ≈⟨ []-cong₂ ([]-cong₂ (+₁∘i₁ ○ identityʳ) (pullˡ (+₁∘+₁ ○ +₁-cong₂ identity² refl))) assoc ⟩
[ [ i₁ , (idC +₁ i₂ ∘ i₁ ∘ i₂) ∘ g ] , i₂ ∘ i₂ ∘ h ] ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩ {! !} ≈⟨ {! !} ⟩
{! !} ∎ {! !} ∎
by-uni₃ : [ [ i₁ , (idC +₁ i₁ ∘ i₂) ∘ g ] , i₂ ∘ h ] ∘ i₂ ≈ (idC +₁ i₂) ∘ {! !}
by-uni₃ = begin
[ [ i₁ , (idC +₁ i₁ ∘ i₂) ∘ g ] , i₂ ∘ h ] ∘ i₂ ≈⟨ inject₂ ⟩
i₂ ∘ [ i₁ ∘ i₁ , i₂ +₁ idC ] ∘ f ≈⟨ pullˡ ∘[] ⟩
[ i₂ ∘ i₁ ∘ i₁ , i₂ ∘ (i₂ +₁ idC) ] ∘ f ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
(idC +₁ i₂) ∘ [ i₁ , {! i₂ !} ] ∘ f ∎
by-fix : [ i₁ , (idC +₁ i₂) ∘ g ] # ≈ [ idC , g # ]
by-fix = sym (begin
[ idC , g # ] ≈⟨ []-cong₂ refl #-Fixpoint ⟩
[ idC , [ idC , g # ] ∘ g ] ≈⟨ []-cong₂ refl (([]-cong₂ refl (#-Uniformity (sym inject₂))) ⟩∘⟨refl) ⟩
[ idC , [ idC , [ i₁ , (idC +₁ i₂) ∘ g ] # ∘ i₂ ] ∘ g ] ≈˘⟨ ∘[] ○ []-cong₂ inject₁ (pullˡ ([]∘+₁ ○ []-cong₂ identity² refl)) ⟩
[ idC , [ i₁ , (idC +₁ i₂) ∘ g ] # ] ∘ [ i₁ , (idC +₁ i₂) ∘ g ] ≈˘⟨ #-Fixpoint ⟩
([ i₁ , (idC +₁ i₂) ∘ g ] #) ∎)
-- every elgot-algebra comes with a divergence constant -- every elgot-algebra comes with a divergence constant
!ₑ : ⊥ ⇒ A !ₑ : ⊥ ⇒ A

55
src/Monad/Instance/K/Commutative.lagda.md
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@ -210,48 +210,19 @@ KCommutative = record { commutes = commutes' }
(idC +₁ (idC ⁂ h #)) ∘ ((ψ ∘ (idC ⁂ h #)) +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC) ∎ (idC +₁ (idC ⁂ h #)) ∘ ((ψ ∘ (idC ⁂ h #)) +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC) ∎
comm₈ : ∀ {U} (g : U ⇒ K.₀ X + U) → ((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) # ∘ (g # ⁂ idC) ≈ ((((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC))# comm₈ : ∀ {U} (g : U ⇒ K.₀ X + U) → ((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) # ∘ (g # ⁂ idC) ≈ ((((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC))#
comm₈ {U} g = begin comm₈ {U} g = begin
((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ∘ (g # ⁂ idC) ≈⟨ στ ⟩ ((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ∘ (g # ⁂ idC) ≈⟨ sym (#-Uniformity (algebras (X × Y)) (sym by-uni)) ⟩
extend ψ ∘ extend (σ _) ∘ τ _ ∘ (((η _ +₁ idC) ∘ g) # ⁂ ((η _ +₁ idC) ∘ h) #) ≈⟨ {! !} ⟩ -- lemma 42 ((ψ ∘ ((g #) ⁂ idC) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) # ≈⟨ #-resp-≈ (algebras _) ((+₁-cong₂ (ψ-left-iter g) refl) ⟩∘⟨refl) ⟩
extend ψ ∘ extend (τ _) ∘ σ _ ∘ (((η _ +₁ idC) ∘ g) # ⁂ ((η _ +₁ idC) ∘ h) #) ≈⟨ sym τσ ⟩ (((((ψ +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC)) # +₁ idC)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) # ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩
((((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC))# ∎ ((((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC))# ∎
where where
τσ : ((((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC))# ≈ extend ψ ∘ extend (τ _) ∘ σ _ ∘ (((η _ +₁ idC) ∘ g) # ⁂ ((η _ +₁ idC) ∘ h) #) by-uni : ((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) ∘ ((g #) ⁂ idC) ≈ (idC +₁ (g #) ⁂ idC) ∘ (ψ ∘ ((g #) ⁂ idC) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)
τσ = begin by-uni = begin
(((((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC))#) ≈⟨ {! !} ⟩ ((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) ∘ ((g #) ⁂ idC) ≈⟨ pullʳ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ id-comm-sym id-comm ○ sym ⁂∘⁂)) ⟩
{! !} ≈⟨ {! !} ⟩ (ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ ((g #) ⁂ idC) ∘ (idC ⁂ h) ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩ (ψ +₁ idC) ∘ ((((g #) ⁂ idC) +₁ ((g #) ⁂ idC)) ∘ distributeˡ⁻¹) ∘ (idC ⁂ h) ≈⟨ pullˡ (pullˡ (+₁∘+₁ ○ +₁-cong₂ (sym identityˡ) id-comm-sym ○ sym +₁∘+₁)) ⟩
{! !} ≈⟨ {! !} ⟩ (((idC +₁ (g #) ⁂ idC) ∘ (ψ ∘ ((g #) ⁂ idC) +₁ idC)) ∘ distributeˡ⁻¹) ∘ (idC ⁂ h) ≈⟨ assoc² ⟩
extend ψ ∘ extend (τ _) ∘ σ _ ∘ (((η _ +₁ idC) ∘ g) # ⁂ ((η _ +₁ idC) ∘ h) #) ∎ (idC +₁ (g #) ⁂ idC) ∘ (ψ ∘ ((g #) ⁂ idC) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h) ∎
στ : ((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ∘ (g # ⁂ idC) ≈ extend ψ ∘ extend (σ _) ∘ τ _ ∘ (((η _ +₁ idC) ∘ g) # ⁂ ((η _ +₁ idC) ∘ h) #)
στ = begin
((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h))# ∘ (g # ⁂ idC) ≈⟨ sym (#-Uniformity (algebras (X × Y)) (sym by-uni)) ⟩
((ψ ∘ ((g #) ⁂ idC) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) # ≈⟨ #-resp-≈ (algebras _) ((+₁-cong₂ (ψ-left-iter g) refl) ⟩∘⟨refl) ⟩
(((((ψ +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC)) # +₁ idC)) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) # ≈˘⟨ #-resp-≈ (algebras _) (pullˡ (+₁∘+₁ ○ +₁-cong₂ kleisliK.identityʳ identity²)) ⟩
(((extend (((ψ +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC)) #) +₁ idC) ∘ (η _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) #) ≈˘⟨ extend-preserve (((ψ +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC)) #) ((η (U × K.₀ Y) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) ⟩
extend (((ψ +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC)) #) ∘ ((η _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) # ≈⟨ refl⟩∘⟨ (#-resp-≈ (algebras _) ((+₁-cong₂ (sym (τ-η _)) refl) ⟩∘⟨refl)) ⟩
extend (((ψ +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC)) #) ∘ ((τ _ ∘ (idC ⁂ η _) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) # ≈˘⟨ refl⟩∘⟨ (τ-comm ((η (K.₀ Y) +₁ idC) ∘ h) ○ #-resp-≈ (algebras _) comm) ⟩
extend (((ψ +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC)) #) ∘ τ _ ∘ (idC ⁂ ((η _ +₁ idC) ∘ h)#) ≈˘⟨ (kleisliK.extend-≈ (#-resp-≈ (algebras _) (pullˡ (+₁∘+₁ ○ +₁-cong₂ kleisliK.identityʳ identity²)))) ⟩∘⟨refl ⟩
extend ((((extend ψ +₁ idC) ∘ (η _ +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC))#)) ∘ τ _ ∘ (idC ⁂ ((η _ +₁ idC) ∘ h) #) ≈˘⟨ (kleisliK.extend-≈ (extend-preserve ψ ((η (K.₀ X × K.₀ Y) +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC)))) ⟩∘⟨refl ⟩
extend (extend ψ ∘ (((η _ +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC))#)) ∘ τ _ ∘ (idC ⁂ ((η _ +₁ idC) ∘ h) #) ≈˘⟨ pullˡ kleisliK.sym-assoc ⟩
extend ψ ∘ extend (((η _ +₁ idC) ∘ distributeʳ⁻¹ ∘ (g ⁂ idC))#) ∘ τ _ ∘ (idC ⁂ ((η _ +₁ idC) ∘ h) #) ≈˘⟨ refl⟩∘⟨ ((kleisliK.extend-≈ (#-resp-≈ (algebras _) (refl⟩∘⟨ (pullˡ (sym (distribute₁' idC (η (K.₀ X)) idC)))) ○ #-resp-≈ (algebras _) (pullˡ (pullˡ (+₁∘+₁ ○ +₁-cong₂ σ-η (elimʳ (⟨⟩-unique id-comm id-comm)))) ○ assoc))) ⟩∘⟨refl) ⟩
extend ψ ∘ extend (((σ _ +₁ idC) ∘ distributeʳ⁻¹ ∘ ((η (K.₀ X) +₁ idC) ⁂ idC) ∘ (g ⁂ idC)) #) ∘ τ _ ∘ (idC ⁂ ((η _ +₁ idC) ∘ h) #) ≈⟨ refl⟩∘⟨ (kleisliK.extend-≈ (#-resp-≈ (algebras _) (refl⟩∘⟨ (refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl identity²))))) ⟩∘⟨refl ⟩
extend ψ ∘ extend (((σ _ +₁ idC) ∘ distributeʳ⁻¹ ∘ ((η (K.₀ X) +₁ idC) ∘ g ⁂ idC)) #) ∘ τ _ ∘ (idC ⁂ ((η _ +₁ idC) ∘ h) #) ≈˘⟨ refl⟩∘⟨ ((kleisliK.extend-≈ (σ-comm ((η (K.₀ X) +₁ idC) ∘ g))) ⟩∘⟨refl) ⟩
extend ψ ∘ extend (σ _ ∘ (((η _ +₁ idC) ∘ g) # ⁂ idC)) ∘ τ _ ∘ (idC ⁂ ((η _ +₁ idC) ∘ h) #) ≈˘⟨ refl⟩∘⟨ (pullˡ (extend∘F₁ monadK (σ _) (((η _ +₁ idC) ∘ g) # ⁂ idC))) ⟩
extend ψ ∘ extend (σ _) ∘ K.₁ (((η _ +₁ idC) ∘ g) # ⁂ idC) ∘ τ _ ∘ (idC ⁂ ((η _ +₁ idC) ∘ h) #) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ pullˡ (sym (strengthen.commute (((η (K.₀ X) +₁ idC) ∘ g) # , idC))) ⟩
extend ψ ∘ extend (σ _) ∘ (τ _ ∘ (((η _ +₁ idC) ∘ g) # ⁂ K.₁ idC)) ∘ (idC ⁂ ((η _ +₁ idC) ∘ h) #) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ (assoc ○ refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ identityʳ (elimˡ monadK.F.identity))) ⟩
extend ψ ∘ extend (σ _) ∘ τ _ ∘ (((η _ +₁ idC) ∘ g) # ⁂ ((η _ +₁ idC) ∘ h) #) ∎
where
by-uni : ((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) ∘ ((g #) ⁂ idC) ≈ (idC +₁ (g #) ⁂ idC) ∘ (ψ ∘ ((g #) ⁂ idC) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)
by-uni = begin
((ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)) ∘ ((g #) ⁂ idC) ≈⟨ pullʳ (pullʳ (⁂∘⁂ ○ ⁂-cong₂ id-comm-sym id-comm ○ sym ⁂∘⁂)) ⟩
(ψ +₁ idC) ∘ distributeˡ⁻¹ ∘ ((g #) ⁂ idC) ∘ (idC ⁂ h) ≈⟨ {! !} ⟩
(ψ +₁ idC) ∘ ((((g #) ⁂ idC) +₁ ((g #) ⁂ idC)) ∘ distributeˡ⁻¹) ∘ (idC ⁂ h) ≈⟨ pullˡ (pullˡ (+₁∘+₁ ○ +₁-cong₂ (sym identityˡ) id-comm-sym ○ sym +₁∘+₁)) ⟩
(((idC +₁ (g #) ⁂ idC) ∘ (ψ ∘ ((g #) ⁂ idC) +₁ idC)) ∘ distributeˡ⁻¹) ∘ (idC ⁂ h) ≈⟨ assoc² ⟩
(idC +₁ (g #) ⁂ idC) ∘ (ψ ∘ ((g #) ⁂ idC) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h) ∎
comm : (τ _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ (η _ +₁ idC) ∘ h) ≈ (τ _ ∘ (idC ⁂ η _) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h)
comm = sym (begin
(τ _ ∘ (idC ⁂ η _) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h) ≈˘⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ refl identity²) ⟩
(τ _ +₁ idC) ∘ ((idC ⁂ η _) +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ h) ≈⟨ refl⟩∘⟨ (pullˡ ((+₁-cong₂ refl (sym (⟨⟩-unique id-comm id-comm))) ⟩∘⟨refl ○ distribute₁ idC (η (K.₀ Y)) idC)) ⟩
(τ _ +₁ idC) ∘ (distributeˡ⁻¹ ∘ (idC ⁂ (η _ +₁ idC))) ∘ (idC ⁂ h) ≈⟨ refl⟩∘⟨ pullʳ (⁂∘⁂ ○ ⁂-cong₂ identity² refl) ⟩
(τ _ +₁ idC) ∘ distributeˡ⁻¹ ∘ (idC ⁂ (η _ +₁ idC) ∘ h) ∎)
``` ```

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src/Monad/Instance/K/Instance/Maybe.lagda.md
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<!--
```agda
open import Level
open import Category.Instance.AmbientCategory
open import Categories.Category
open import Categories.Category.Instance.Setoids
open import Categories.Monad
open import Categories.Category.Monoidal.Instance.Setoids
open import Categories.Category.Cocartesian
open import Categories.Object.Terminal
open import Function.Equality as SΠ renaming (id to ⟶-id)
import Categories.Morphism.Reasoning as MR
open import Relation.Binary
open import Agda.Builtin.Unit using (tt)
```
-->
```agda
module Monad.Instance.K.Instance.Maybe {o ℓ e} (ambient : Ambient o ℓ e) where
open Ambient ambient using ()
```
# The Maybe Monad as instance of K
Assuming the axiom of choice, the maybe monad is an instance of K in the category of setoids.
```agda
module _ {c ℓ' : Level} where
open Cocartesian (Setoids-Cocartesian {c} {c ⊔ ℓ'})
open Terminal (terminal {c} {c ⊔ ℓ'})
open MR (Setoids c (c ⊔ ℓ'))
open Category (Setoids c (c ⊔ ℓ'))
open Equiv
maybe : Monad (Setoids c (c ⊔ ℓ'))
maybe = record
{ F = record
{ F₀ = λ X → X + ⊤
; F₁ = λ {A} {B} f → f +₁ ⟶-id
; identity = {! !}
; homomorphism = {! !}
; F-resp-≈ = λ {A} {B} {f} {g} f≈g → +₁-cong₂ f≈g ?
}
; η = {! !}
; μ = {! !}
; assoc = {! !}
; sym-assoc = {! !}
; identityˡ = {! !}
; identityʳ = {! !}
}
```