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Changed to 2 space indentation, still needs refactor

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Leon Vatthauer 2023-08-07 20:17:20 +02:00
parent f7dfe31f3d
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42
ElgotAlgebras.agda
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@ -45,7 +45,7 @@ module _ (D : ExtensiveDistributiveCategory o ℓ e) where
h : A ⇒ B
preserves : ∀ {X} {f : X ⇒ A + X} → h ∘ (f #₁) ≈ ((h +₁ idC) ∘ f)#₂
-- the category of elgot algebras for a given (cocartesian-)category
-- the category of elgot algebras for a given category
Elgot-Algebras : Category (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e) e
Elgot-Algebras = record
{ Obj = Elgot-Algebra D
@ -84,7 +84,8 @@ module _ (D : ExtensiveDistributiveCategory o ℓ e) where
; equiv = record
{ refl = refl
; sym = sym
; trans = trans}
; trans = trans
}
; ∘-resp-≈ = ∘-resp-≈
}
where
@ -98,18 +99,20 @@ module _ (D : ExtensiveDistributiveCategory o ℓ e) where
-- if the carrier contains a terminal, so does elgot-algebras
Terminal-Elgot-Algebras : Terminal C → Terminal Elgot-Algebras
Terminal-Elgot-Algebras T = record {
⊤ = record
Terminal-Elgot-Algebras T = record
{ ⊤ = record
{ A = ⊤
; _# = λ x → !
; #-Fixpoint = λ {_ f} → !-unique ([ idC , ! ] ∘ f)
; #-Uniformity = λ {_ _ _ _ h} _ → !-unique (! ∘ h)
; #-Folding = refl
; #-resp-≈ = λ _ → refl
} ;
⊤-is-terminal = record
}
; ⊤-is-terminal = record
{ ! = λ {A} → record { h = ! ; preserves = λ {X} {f} → sym (!-unique (! ∘ (A Elgot-Algebra.#) f)) }
; !-unique = λ {A} f → !-unique (Elgot-Algebra-Morphism.h f) } }
; !-unique = λ {A} f → !-unique (Elgot-Algebra-Morphism.h f)
}
}
where
open Terminal T
open Equiv
@ -124,14 +127,12 @@ module _ (D : ExtensiveDistributiveCategory o ℓ e) where
⟨ [ idC , ((π₁ +₁ idC) ∘ f)#ᵃ ] ∘ ((π₁ +₁ idC) ∘ f) , [ idC , ((π₂ +₁ idC) ∘ f)#ᵇ ] ∘ ((π₂ +₁ idC) ∘ f) ⟩ ≈⟨ ⟨⟩-cong₂ sym-assoc sym-assoc ⟩
⟨ ([ idC , ((π₁ +₁ idC) ∘ f)#ᵃ ] ∘ (π₁ +₁ idC)) ∘ f , ([ idC , ((π₂ +₁ idC) ∘ f)#ᵇ ] ∘ (π₂ +₁ idC)) ∘ f ⟩ ≈⟨ ⟨⟩-cong₂ (∘-resp-≈ˡ []∘+₁) (∘-resp-≈ˡ []∘+₁) ⟩
⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] ∘ f , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ∘ f ⟩ ≈⟨ sym ⟨⟩∘ ⟩
(⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ⟩ ∘ f) ≈⟨ ∘-resp-≈ˡ (unique′
(begin
(⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ⟩ ∘ f) ≈⟨ ∘-resp-≈ˡ (unique′ (begin
π₁ ∘ ⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ⟩ ≈⟨ project₁ ⟩
[ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] ≈⟨ []-cong₂ identityˡ identityʳ ⟩
[ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ] ≈⟨ sym ([]-cong₂ identityʳ project₁) ⟩
[ π₁ ∘ idC , π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ≈⟨ sym ∘[] ⟩
π₁ ∘ [ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ∎)
(begin
π₁ ∘ [ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ∎) (begin
π₂ ∘ ⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ⟩ ≈⟨ project₂ ⟩
[ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ≈⟨ []-cong₂ identityˡ identityʳ ⟩
[ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ] ≈⟨ sym ([]-cong₂ identityʳ project₂) ⟩
@ -139,11 +140,9 @@ module _ (D : ExtensiveDistributiveCategory o ℓ e) where
π₂ ∘ [ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ∎)
)⟩
([ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ∘ f) ∎
; #-Uniformity = λ {X Y f g h} uni → unique′ (
begin
; #-Uniformity = λ {X Y f g h} uni → unique′ (begin
π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ≈⟨ project₁ ⟩
(((π₁ +₁ idC) ∘ f)#ᵃ) ≈⟨ #ᵃ-Uniformity (
begin
(((π₁ +₁ idC) ∘ f)#ᵃ) ≈⟨ #ᵃ-Uniformity (begin
(idC +₁ h) ∘ (π₁ +₁ idC) ∘ f ≈⟨ sym-assoc ⟩
((idC +₁ h) ∘ (π₁ +₁ idC)) ∘ f ≈⟨ ∘-resp-≈ˡ +₁∘+₁ ⟩
(idC ∘ π₁ +₁ h ∘ idC) ∘ f ≈⟨ ∘-resp-≈ˡ (+₁-cong₂ identityˡ identityʳ) ⟩
@ -157,11 +156,9 @@ module _ (D : ExtensiveDistributiveCategory o ℓ e) where
(((π₁ +₁ idC) ∘ g)#ᵃ ∘ h) ≈⟨ sym (∘-resp-≈ˡ project₁) ⟩
((π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ g)#ᵃ , ((π₂ +₁ idC) ∘ g)#ᵇ ⟩) ∘ h) ≈⟨ assoc ⟩
π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ g)#ᵃ , ((π₂ +₁ idC) ∘ g)#ᵇ ⟩ ∘ h ∎
) (
begin
) (begin
π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ≈⟨ project₂ ⟩
((π₂ +₁ idC) ∘ f)#ᵇ ≈⟨ #ᵇ-Uniformity (
begin
((π₂ +₁ idC) ∘ f)#ᵇ ≈⟨ #ᵇ-Uniformity (begin
(idC +₁ h) ∘ (π₂ +₁ idC) ∘ f ≈⟨ sym-assoc ⟩
(((idC +₁ h) ∘ (π₂ +₁ idC)) ∘ f) ≈⟨ ∘-resp-≈ˡ +₁∘+₁ ⟩
((idC ∘ π₂ +₁ h ∘ idC) ∘ f) ≈⟨ ∘-resp-≈ˡ (+₁-cong₂ identityˡ identityʳ) ⟩
@ -183,7 +180,6 @@ module _ (D : ExtensiveDistributiveCategory o ℓ e) where
open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ; #-Fixpoint to #ᵃ-Fixpoint; #-Uniformity to #ᵃ-Uniformity; #-Folding to #ᵃ-Folding; #-resp-≈ to #ᵃ-resp-≈)
open Elgot-Algebra EB using () renaming (A to B; _# to _#ᵇ; #-Fixpoint to #ᵇ-Fixpoint; #-Uniformity to #ᵇ-Uniformity; #-Folding to #ᵇ-Folding; #-resp-≈ to #ᵇ-resp-≈)
open HomReasoning
-- open Product (product {A} {B})
open Equiv
foldingˡ : ∀ {X} {Y} {f} {h} → (((π₁ +₁ idC) ∘ (⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ h))#ᵃ) ≈ ((π₁ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵃ
foldingˡ {X} {Y} {f} {h} = begin
@ -242,9 +238,9 @@ module _ (D : ExtensiveDistributiveCategory o ℓ e) where
-- if the carrier is cartesian, so is the category of algebras
Cartesian-Elgot-Algebras : Cartesian Elgot-Algebras
Cartesian-Elgot-Algebras = record {
terminal = Terminal-Elgot-Algebras terminal;
products = record { product = λ {EA EB} → Product-Elgot-Algebras EA EB }
Cartesian-Elgot-Algebras = record
{ terminal = Terminal-Elgot-Algebras terminal
; products = record { product = λ {EA EB} → Product-Elgot-Algebras EA EB }
}
where
open Equiv