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stuck on final proof for stability

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Leon Vatthauer 2024-02-02 17:59:59 +01:00
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43
agda/src/Algebra/Elgot/Properties.lagda.md
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@ -26,10 +26,11 @@ module Algebra.Elgot.Properties {o ℓ e} {C : Category o ℓ e} (distributive :
open Category C open Category C
open HomReasoning open HomReasoning
open Equiv open Equiv
open MR C
open Distributive distributive open Distributive distributive
open import Categories.Category.Distributive.Properties distributive open import Categories.Category.Distributive.Properties distributive
open Cartesian cartesian open Cartesian cartesian
open BinaryProducts products open BinaryProducts products hiding (η) renaming (unique to ⟨⟩-unique)
open Cocartesian cocartesian open Cocartesian cocartesian
ccc : CartesianClosed C ccc : CartesianClosed C
ccc = record { cartesian = cartesian ; exp = expos } ccc = record { cartesian = cartesian ; exp = expos }
@ -39,19 +40,43 @@ module Algebra.Elgot.Properties {o ℓ e} {C : Category o ℓ e} (distributive :
open import Category.Construction.ElgotAlgebras {C = C} open import Category.Construction.ElgotAlgebras {C = C}
open Cat cocartesian open Cat cocartesian
open Exponentials distributive expos open Exponentials distributive expos
open import Algebra.Elgot cocartesian
open import Algebra.Elgot.Free cocartesian open import Algebra.Elgot.Free cocartesian
open import Algebra.Elgot.Stable distributive open import Algebra.Elgot.Stable distributive
open CartesianClosed Elgot-CCC using () renaming (λg to λg#; _^_ to _^#_) open CartesianClosed Elgot-CCC using () renaming (λg to λg#; _^_ to _^#_)
open Elgot-Algebra-Morphism renaming (h to <_>)
stable : ∀ {Y} (fe : FreeElgotAlgebra Y) → IsStableFreeElgotAlgebraˡ fe stable : ∀ {Y} (fe : FreeElgotAlgebra Y) → IsStableFreeElgotAlgebraˡ fe
IsStableFreeElgotAlgebraˡ.[_,_]♯ˡ (stable {Y} fe) {X} A f = (eval′ ∘ (Elgot-Algebra-Morphism.h ((fe FreeObject.*) {A = B^A-Helper {A} {X}} (λg f)) ⁂ id)) IsStableFreeElgotAlgebraˡ.[_,_]♯ˡ (stable {Y} fe) {X} A f = (eval′ ∘ (< FreeObject._* fe {B^A-Helper {A} {X}} (λg f) > ⁂ id))
where open FreeObject fe
IsStableFreeElgotAlgebraˡ.♯ˡ-law (stable {Y} fe) {X} {A} f = sym (begin IsStableFreeElgotAlgebraˡ.♯ˡ-law (stable {Y} fe) {X} {A} f = sym (begin
{! [ stable fe , A ]♯ˡ f ∘ (η fe ⁂ id) !} ≈⟨ {! !} ⟩ (eval′ ∘ (< (λg f)* > ⁂ id)) ∘ (η ⁂ id) ≈⟨ pullʳ ⁂∘⁂ ⟩
{! !} ≈⟨ {! !} ⟩ eval′ ∘ ((< (λg f)* > ∘ η) ⁂ id ∘ id) ≈⟨ refl⟩∘⟨ (⁂-cong₂ (*-lift (λg f)) identity²) ⟩
{! !} ≈⟨ {! !} ⟩ eval′ ∘ (λg f ⁂ id) ≈⟨ β′ ⟩
{! !} ≈⟨ {! !} ⟩ f ∎)
f ∎) where open FreeObject fe
IsStableFreeElgotAlgebraˡ.♯ˡ-preserving (stable {Y} fe) {X} {B} f {Z} h = {! !} IsStableFreeElgotAlgebraˡ.♯ˡ-preserving (stable {Y} fe) {X} {B} f {Z} h = begin
IsStableFreeElgotAlgebraˡ.♯ˡ-unique (stable {Y} fe) {X} {B} f g eq₁ eq₂ = {! !} (eval′ ∘ (< (λg f)* > ⁂ id)) ∘ (h #ʸ ⁂ id) ≈⟨ pullʳ ⁂∘⁂ ⟩
eval′ ∘ (< (λg f)* > ∘ h #ʸ ⁂ id ∘ id) ≈⟨ refl⟩∘⟨ (⁂-cong₂ (preserves ((λg f)*)) identity²) ⟩
eval′ ∘ (λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((< (λg f)* > +₁ id) ∘ h ⁂ id)) #ᵇ) ⁂ id) ≈⟨ β′ ⟩
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((< (λg f)* > +₁ id) ∘ h ⁂ id)) #ᵇ ≈˘⟨ #ᵇ-resp-≈ (refl⟩∘⟨ refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl identity²)) ⟩
((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((< (λg f)* > +₁ id) ⁂ id) ∘ (h ⁂ id)) #ᵇ ≈⟨ #ᵇ-resp-≈ (refl⟩∘⟨ pullˡ (sym (distributeʳ⁻¹-natural id < (λg f) * > id))) ⟩
((eval′ +₁ id) ∘ ((< (λg f)* > ⁂ id +₁ id ⁂ id) ∘ distributeʳ⁻¹) ∘ (h ⁂ id)) #ᵇ ≈⟨ #ᵇ-resp-≈ (refl⟩∘⟨ assoc ○ pullˡ (+₁∘+₁ ○ +₁-cong₂ refl (elimʳ (⟨⟩-unique id-comm id-comm)))) ⟩
((eval′ ∘ (< (λg f)* > ⁂ id) +₁ id) ∘ distributeʳ⁻¹ ∘ (h ⁂ id)) #ᵇ ∎
where
open FreeObject fe renaming (FX to KY)
open Elgot-Algebra B using () renaming (_# to _#ᵇ; #-resp-≈ to #ᵇ-resp-≈)
open Elgot-Algebra KY using () renaming (_# to _#ʸ)
IsStableFreeElgotAlgebraˡ.♯ˡ-unique (stable {Y} fe) {X} {B} f g eq₁ eq₂ = sym (begin
eval′ ∘ (< (λg f)* > ⁂ id) ≈⟨ refl⟩∘⟨ (⁂-cong₂ (*-cong (λ-cong eq₁)) refl) ⟩
eval′ ∘ (< (λg (g ∘ (η ⁂ id)))* > ⁂ id) ≈˘⟨ refl⟩∘⟨ (⁂-cong₂ (*-cong subst) refl) ⟩
eval′ ∘ (< (λg g ∘ η)* > ⁂ id) ≈˘⟨ refl⟩∘⟨ (⁂-cong₂ (*-uniq (λg g ∘ η) (record { h = λg g ; preserves = {! !} }) refl) refl) ⟩
eval′ ∘ (λg g ⁂ id) ≈⟨ β′ ⟩
g ∎)
where
open FreeObject fe renaming (FX to KY)
open Elgot-Algebra B using () renaming (_# to _#ᵇ; #-resp-≈ to #ᵇ-resp-≈)
open Elgot-Algebra KY using () renaming (_# to _#ʸ)
``` ```

69
agda/src/Category/Construction/ElgotAlgebras.lagda.md
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@ -314,7 +314,7 @@ module Category.Construction.ElgotAlgebras {o ℓ e} {C : Category o ℓ e} wher
[ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ≈˘⟨ cancelʳ (IsIso.isoˡ isIsoʳ) ⟩ [ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ≈˘⟨ cancelʳ (IsIso.isoˡ isIsoʳ) ⟩
([ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ∘ distributeʳ⁻¹) ∘ distributeʳ ∎) ([ (id +₁ i₁) ∘ (eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (f ⁂ id) , i₂ ∘ distributeʳ⁻¹ ∘ (h ⁂ id) ] ∘ distributeʳ⁻¹) ∘ distributeʳ ∎)
<_> = Elgot-Algebra-Morphism.h open Elgot-Algebra-Morphism renaming (h to <_>)
cccc : CanonicalCartesianClosed Elgot-Algebras cccc : CanonicalCartesianClosed Elgot-Algebras
CanonicalCartesianClosed.⊤ cccc = Terminal.⊤ Terminal-Elgot-Algebras CanonicalCartesianClosed.⊤ cccc = Terminal.⊤ Terminal-Elgot-Algebras
@ -366,21 +366,60 @@ module Category.Construction.ElgotAlgebras {o ℓ e} {C : Category o ℓ e} wher
(eval′ +₁ id) ∘ (i₂ ∘ (id ⁂ id)) ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ pullˡ (pullˡ inject₂) ⟩ (eval′ +₁ id) ∘ (i₂ ∘ (id ⁂ id)) ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ pullˡ (pullˡ inject₂) ⟩
((i₂ ∘ id) ∘ (id ⁂ id)) ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ (cancelʳ (identityˡ ○ ⟨⟩-unique id-comm id-comm)) ⟩∘⟨refl ⟩ ((i₂ ∘ id) ∘ (id ⁂ id)) ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ≈⟨ (cancelʳ (identityˡ ○ ⟨⟩-unique id-comm id-comm)) ⟩∘⟨refl ⟩
i₂ ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ∎ i₂ ∘ ⟨ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ f) ⟩ ∎
Elgot-Algebra-Morphism.h (CanonicalCartesianClosed.curry cccc {A} {B} {C} f) = λg < f > < CanonicalCartesianClosed.curry cccc {A} {B} {C} f > = λg < f >
Elgot-Algebra-Morphism.preserves (CanonicalCartesianClosed.curry cccc {A} {B} {C} f) {X} {g} = λ-unique′ (begin preserves (CanonicalCartesianClosed.curry cccc {A} {B} {C} f) {X} {g} = begin
eval′ ∘ ((λg < f > ∘ (Elgot-Algebra._# A) g) ⁂ id) ≈⟨ refl⟩∘⟨ ⁂-cong₂ subst refl ⟩ λg < f > ∘ g #ᵃ ≈⟨ subst ⟩
eval′ ∘ ((λg (< f > ∘ (((Elgot-Algebra._# A) g) ⁂ id))) ⁂ id) ≈⟨ β′ ⟩ λg (< f > ∘ (g #ᵃ ⁂ id)) ≈⟨ {! !} ⟩
< f > ∘ (((Elgot-Algebra._# A) g) ⁂ id) ≈⟨ {! !} ⟩
{! !} ≈⟨ {! !} ⟩ {! !} ≈⟨ {! !} ⟩
{! !} ≈˘⟨ {! !} ⟩ {! !} ≈⟨ {! !} ⟩
{! !} ≈˘⟨ {! !} ⟩ λg (< f > ∘ ⟨ ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵃ , ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ⟩) ≈⟨ λ-cong (preserves f) ⟩
{! !} ≈˘⟨ {! !} ⟩ λg (((< f > +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᶜ) ≈⟨ {! !} ⟩
< f > ∘ ⟨ [ id , (Elgot-Algebra._# A) ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id))) ] ∘ ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id))) , [ id , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id))) ] ∘ ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id))) ⟩ ≈˘⟨ refl⟩∘⟨ (⟨⟩-cong₂ (Elgot-Algebra.#-Fixpoint A) (Elgot-Algebra.#-Fixpoint B)) ⟩ {! !} ≈⟨ {! !} ⟩
< f > ∘ ⟨ (Elgot-Algebra._# A) ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id))) , (Elgot-Algebra._# B) ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id))) ⟩ ≈⟨ Elgot-Algebra-Morphism.preserves f ⟩ {! !} ≈⟨ {! !} ⟩
(Elgot-Algebra._# C) ((< f > +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) ≈˘⟨ {! !} ⟩ λg (((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (((λg < f > +₁ id) ∘ g) ⁂ id)) #ᶜ) ∎
(Elgot-Algebra._# C) ((eval′ +₁ id) ∘ (((λg < f > ⁂ id) +₁ (id ⁂ id)) ∘ distributeʳ⁻¹) ∘ (g ⁂ id)) ≈˘⟨ Elgot-Algebra.#-resp-≈ C (refl⟩∘⟨ (pullˡ (sym (distributeʳ⁻¹-natural id (λg < f >) id)))) ⟩ where
(Elgot-Algebra._# C) ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((λg < f > +₁ id) ⁂ id) ∘ (g ⁂ id)) ≈⟨ Elgot-Algebra.#-resp-≈ C (refl⟩∘⟨ (refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl identity²))) ⟩ open Elgot-Algebra A using () renaming (_# to _#ᵃ; #-Uniformity to #ᵃ-Uniformity)
(Elgot-Algebra._# C) ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (((λg < f > +₁ id) ∘ g) ⁂ id)) ∎) open Elgot-Algebra B using () renaming (_# to _#ᵇ; #-Uniformity to #ᵇ-Uniformity)
open Elgot-Algebra C using () renaming (_# to _#ᶜ; #-resp-≈ to #ᶜ-resp-≈)
-- Elgot-Algebra-Morphism.preserves (CanonicalCartesianClosed.curry cccc {A} {B} {C} f) {X} {g} = λ-unique′ (begin
-- eval′ ∘ ((λg < f > ∘ g #ᵃ) ⁂ id) ≈⟨ refl⟩∘⟨ ⁂-cong₂ subst refl ⟩
-- eval′ ∘ ((λg (< f > ∘ (g #ᵃ ⁂ id))) ⁂ id) ≈⟨ β′ ⟩
-- < f > ∘ (g #ᵃ ⁂ id) ≈⟨ refl⟩∘⟨ (⟨⟩-unique by-π₁ by-π₂) ⟩
-- < f > ∘ ⟨ ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵃ , ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ⟩ ≈⟨ Elgot-Algebra-Morphism.preserves f ⟩
-- ((< f > +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id)) #ᶜ ≈˘⟨ #ᶜ-resp-≈ (refl⟩∘⟨ assoc ○ pullˡ (+₁∘+₁ ○ +₁-cong₂ β′ (elimʳ (⟨⟩-unique id-comm id-comm)))) ⟩
-- ((eval′ +₁ id) ∘ (((λg < f > ⁂ id) +₁ (id ⁂ id)) ∘ distributeʳ⁻¹) ∘ (g ⁂ id)) #ᶜ ≈˘⟨ Elgot-Algebra.#-resp-≈ C (refl⟩∘⟨ (pullˡ (sym (distributeʳ⁻¹-natural id (λg < f >) id)))) ⟩
-- ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ ((λg < f > +₁ id) ⁂ id) ∘ (g ⁂ id)) #ᶜ ≈⟨ Elgot-Algebra.#-resp-≈ C (refl⟩∘⟨ (refl⟩∘⟨ (⁂∘⁂ ○ ⁂-cong₂ refl identity²))) ⟩
-- ((eval′ +₁ id) ∘ distributeʳ⁻¹ ∘ (((λg < f > +₁ id) ∘ g) ⁂ id)) #ᶜ ∎)
-- where
-- open Elgot-Algebra A using () renaming (_# to _#ᵃ; #-Uniformity to #ᵃ-Uniformity)
-- open Elgot-Algebra B using () renaming (_# to _#ᵇ; #-Uniformity to #ᵇ-Uniformity)
-- open Elgot-Algebra C using () renaming (_# to _#ᶜ; #-resp-≈ to #ᶜ-resp-≈)
-- by-π₁ : π₁ ∘ ⟨ ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵃ , ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ⟩ ≈ g #ᵃ ∘ π₁
-- by-π₁ = project₁ ○ #ᵃ-Uniformity by-uni
-- where
-- by-uni : (id +₁ π₁) ∘ (π₁ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id) ≈ g ∘ π₁
-- by-uni = begin
-- (id +₁ π₁) ∘ (π₁ +₁ id) ∘ distributeʳ⁻¹ ∘ (g ⁂ id) ≈⟨ pullˡ (+₁∘+₁ ○ +₁-cong₂ identityˡ identityʳ) ⟩
-- (π₁ +₁ π₁) ∘ distributeʳ⁻¹ ∘ (g ⁂ id) ≈⟨ pullˡ distributeʳ⁻¹-π₁ ⟩
-- π₁ ∘ (g ⁂ id) ≈⟨ project₁ ⟩
-- g ∘ π₁ ∎
-- by-π₂ : π₂ ∘ ⟨ ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵃ , ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ⟩ ≈ id ∘ π₂
-- by-π₂ = begin
-- π₂ ∘ ⟨ ((π₁ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵃ , ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ⟩ ≈⟨ project₂ ⟩
-- ((π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)))#ᵇ ≈⟨ #ᵇ-Uniformity by-uni ⟩
-- {! !} #ᵇ ∘ π₂ ≈⟨ {! !} ⟩
-- {! !} ≈⟨ {! !} ⟩
-- {! !} ≈⟨ {! !} ⟩
-- {! !} ≈⟨ {! !} ⟩
-- id ∘ π₂ ∎
-- where
-- by-uni : (id +₁ π₂) ∘ (π₂ +₁ id) ∘ (distributeʳ⁻¹ ∘ (g ⁂ id)) ≈ {! !} ∘ π₂
-- by-uni = begin
-- {! !} ≈⟨ {! !} ⟩
-- {! !} ≈⟨ {! !} ⟩
-- {! !} ≈⟨ {! !} ⟩
-- {! !} ≈⟨ {! !} ⟩
-- {! !} ∎
CanonicalCartesianClosed.eval-comp cccc {A} {B} {C} {f} = β′ CanonicalCartesianClosed.eval-comp cccc {A} {B} {C} {f} = β′
CanonicalCartesianClosed.curry-unique cccc {A} {B} {C} {f} {g} eq = λ-unique′ eq CanonicalCartesianClosed.curry-unique cccc {A} {B} {C} {f} {g} eq = λ-unique′ eq