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Align proofs

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Leon Vatthauer 2023-10-22 15:07:39 +02:00
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src/Monad/Instance/Delay/Commutative.lagda.md
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@ -44,7 +44,6 @@ module Monad.Instance.Delay.Commutative {o ℓ e} (ambient : Ambient o ℓ e) (D
open Functor functor using () renaming (F₁ to D₁; identity to D-identity; homomorphism to D-homomorphism; F-resp-≈ to D-resp-≈)
open RMonad kleisli using (extend; extend-≈) renaming (assoc to k-assoc; identityʳ to k-identityʳ; identityˡ to k-identityˡ)
open Monad monad using (η; μ)
-- open StrongMonad strongMonad using (strengthen)
open NaturalTransformation (StrongMonad.strengthen strongMonad) using () renaming (commute to τ-commute)
```
@ -145,8 +144,11 @@ module Monad.Instance.Delay.Commutative {o ℓ e} (ambient : Ambient o ℓ e) (D
helper₃ = begin
[ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ (out {X} ⁂ out {Y}) ≈⟨ refl⟩∘⟨ helper ⟩
[ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ [ [ i₁ ∘ i₁ , i₂ ∘ i₁ ] , [ (i₁ ∘ i₂) , (i₂ ∘ i₂) ] ] ∘ w ≈⟨ pullˡ ∘[] ⟩
[ [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ [ i₁ ∘ i₁ , i₂ ∘ i₁ ] , [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ [ (i₁ ∘ i₂) , (i₂ ∘ i₂) ] ] ∘ w ≈⟨ ([]-cong₂ ∘[] ∘[]) ⟩∘⟨refl ⟩
[ [ [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ i₁ ∘ i₁ , [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ i₂ ∘ i₁ ] , [ [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ (i₁ ∘ i₂) , [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ (i₂ ∘ i₂) ] ] ∘ w ≈⟨ ([]-cong₂ ([]-cong₂ (pullˡ inject₁) (pullˡ inject₂ ○ assoc)) ([]-cong₂ (pullˡ inject₁) (pullˡ inject₂ ○ assoc))) ⟩∘⟨refl ⟩
[ [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ [ i₁ ∘ i₁ , i₂ ∘ i₁ ]
, [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ [ (i₁ ∘ i₂) , (i₂ ∘ i₂) ] ] ∘ w ≈⟨ ([]-cong₂ ∘[] ∘[]) ⟩∘⟨refl ⟩
[ [ [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ i₁ ∘ i₁ , [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ i₂ ∘ i₁ ]
, [ [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ (i₁ ∘ i₂)
, [ idC +₁ τ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ] ∘ (i₂ ∘ i₂) ] ] ∘ w ≈⟨ ([]-cong₂ ([]-cong₂ (pullˡ inject₁) (pullˡ inject₂ ○ assoc)) ([]-cong₂ (pullˡ inject₁) (pullˡ inject₂ ○ assoc))) ⟩∘⟨refl ⟩
[ [ (idC +₁ τ) ∘ i₁ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ∘ i₁ ] , [ (idC +₁ τ) ∘ i₂ , i₂ ∘ [ D₁ swap ∘ τ ∘ swap , ▷ ∘ extend τ ∘ σ ] ∘ i₂ ] ] ∘ w ≈⟨ ([]-cong₂ ([]-cong₂ (+₁∘i₁ ○ identityʳ) (refl⟩∘⟨ inject₁)) ([]-cong₂ +₁∘i₂ (refl⟩∘⟨ inject₂))) ⟩∘⟨refl ⟩
[ [ i₁ , i₂ ∘ σ ] , [ i₂ ∘ τ , i₂ ∘ ▷ ∘ extend τ ∘ σ ] ] ∘ w ≈˘⟨ ([]-cong₂ ([]-cong₂ identityʳ refl) ∘[]) ⟩∘⟨refl ⟩
[ idC +₁ σ , i₂ ∘ [ τ , ▷ ∘ extend τ ∘ σ ] ] ∘ w ∎
@ -155,8 +157,10 @@ module Monad.Instance.Delay.Commutative {o ℓ e} (ambient : Ambient o ℓ e) (D
helper = sym (begin
[ [ i₁ ∘ i₁ , i₂ ∘ i₁ ] , [ (i₁ ∘ i₂) , (i₂ ∘ i₂) ] ] ∘ w ≈⟨ pullˡ []∘+₁ ⟩
[ (i₁ +₁ i₁) ∘ distributeʳ⁻¹ , (i₂ +₁ i₂) ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ ([]-cong₂ ((+₁-cong₂ (sym dstr-law₁) (sym dstr-law₁)) ⟩∘⟨refl) ((+₁-cong₂ (sym dstr-law₂) (sym dstr-law₂)) ⟩∘⟨refl)) ⟩∘⟨refl ⟩
[ (distributeˡ⁻¹ ∘ (idC ⁂ i₁) +₁ distributeˡ⁻¹ ∘ (idC ⁂ i₁)) ∘ distributeʳ⁻¹ , (distributeˡ⁻¹ ∘ (idC ⁂ i₂) +₁ distributeˡ⁻¹ ∘ (idC ⁂ i₂)) ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ sym (([]-cong₂ (pullˡ +₁∘+₁) (pullˡ +₁∘+₁)) ⟩∘⟨refl) ⟩
[ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ ((idC ⁂ i₁) +₁ (idC ⁂ i₁)) ∘ distributeʳ⁻¹ , (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ ((idC ⁂ i₂) +₁ (idC ⁂ i₂)) ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ sym (pullˡ ∘[]) ⟩
[ (distributeˡ⁻¹ ∘ (idC ⁂ i₁) +₁ distributeˡ⁻¹ ∘ (idC ⁂ i₁)) ∘ distributeʳ⁻¹
, (distributeˡ⁻¹ ∘ (idC ⁂ i₂) +₁ distributeˡ⁻¹ ∘ (idC ⁂ i₂)) ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ sym (([]-cong₂ (pullˡ +₁∘+₁) (pullˡ +₁∘+₁)) ⟩∘⟨refl) ⟩
[ (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ ((idC ⁂ i₁) +₁ (idC ⁂ i₁)) ∘ distributeʳ⁻¹
, (distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ ((idC ⁂ i₂) +₁ (idC ⁂ i₂)) ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ sym (pullˡ ∘[]) ⟩
(distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ [ ((idC ⁂ i₁) +₁ (idC ⁂ i₁)) ∘ distributeʳ⁻¹ , ((idC ⁂ i₂) +₁ (idC ⁂ i₂)) ∘ distributeʳ⁻¹ ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ refl⟩∘⟨ []-cong₂ (distribute₁' i₁ idC idC) (distribute₁' i₂ idC idC) ⟩∘⟨refl ⟩
(distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ [ distributeʳ⁻¹ ∘ ((idC +₁ idC) ⁂ i₁) , distributeʳ⁻¹ ∘ ((idC +₁ idC) ⁂ i₂) ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ refl⟩∘⟨ (sym (pullˡ ∘[])) ⟩
(distributeˡ⁻¹ +₁ distributeˡ⁻¹) ∘ distributeʳ⁻¹ ∘ [ ((idC +₁ idC) ⁂ i₁) , ((idC +₁ idC) ⁂ i₂) ] ∘ distributeˡ⁻¹ ∘ (out ⁂ out) ≈⟨ refl⟩∘⟨ refl⟩∘⟨ ([]-cong₂ (⁂-cong₂ ([]-unique id-comm-sym id-comm-sym) refl) (⁂-cong₂ ([]-unique id-comm-sym id-comm-sym) refl)) ⟩∘⟨refl ⟩
@ -204,9 +208,4 @@ module Monad.Instance.Delay.Commutative {o ℓ e} (ambient : Ambient o ℓ e) (D
[ out ∘ [ now , f ] , i₂ ∘ extend [ now , f ] ] ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ pullʳ (cancelˡ (_≅_.isoʳ out-≅)) ⟩
([ out ∘ [ now , f ] , i₂ ∘ extend [ now , f ] ] ∘ out) ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ≈˘⟨ pullˡ (extendlaw [ now , f ]) ⟩
out ∘ extend [ now , f ] ∘ out⁻¹ ∘ [ i₁ +₁ D₁ i₁ ∘ σ , i₂ ∘ [ D₁ i₁ ∘ τ , ▷ ∘ now ∘ i₂ ] ] ∘ w ∎
{-
TODO there's an error in the paper, at the end of the proof of proposition two:
the last line of the 3 line calulation 'f = ....'
should be ▷ ∘ η ∘ inr, but is η ∘ inr!!
-}
```