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Separated category of elgot algebras to own file
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2 changed files with 246 additions and 165 deletions
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@ -14,7 +14,7 @@ private
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o ℓ e : Level
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module _ {C𝒞 : CocartesianCategory o ℓ e} where
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module _ (C𝒞 : CocartesianCategory o ℓ e) where
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open CocartesianCategory C𝒞 renaming (U to 𝒞; id to idC)
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--*
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@ -103,170 +103,9 @@ module _ {C𝒞 : CocartesianCategory o ℓ e} where
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[ (idC +₁ i₁) ∘ f , i₂ ∘ h ] ∘ [ i₁ , h ] ∎))) ⟩
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([ (idC +₁ i₁) ∘ f , i₂ ∘ i₂ ] ∘ [ i₁ , h ])# ∘ i₂ ∎
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-- divergence constant
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⊥ₑ : ⊥ ⇒ A
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⊥ₑ = i₂ #
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--*
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-- let's define the category of elgot-algebras
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--*
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-- iteration preversing morphism between two elgot-algebras
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module _ (E₁ E₂ : Elgot-Algebra) where
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open Elgot-Algebra E₁ renaming (_# to _#₁)
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open Elgot-Algebra E₂ renaming (_# to _#₂; A to B)
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record Elgot-Algebra-Morphism : Set (o ⊔ ℓ ⊔ e) where
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field
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h : A ⇒ B
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preserves : ∀ {X} {f : X ⇒ A + X} → h ∘ (f #₁) ≈ ((h +₁ idC) ∘ f)#₂
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-- the category of elgot algebras for a given category
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Elgot-Algebras : Category (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e) e
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Elgot-Algebras = record
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{ Obj = Elgot-Algebra
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; _⇒_ = Elgot-Algebra-Morphism
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; _≈_ = λ f g → Elgot-Algebra-Morphism.h f ≈ Elgot-Algebra-Morphism.h g
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; id = λ {EB} → let open Elgot-Algebra EB in
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record { h = idC; preserves = λ {X : Obj} {f : X ⇒ A + X} → begin
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idC ∘ f # ≈⟨ identityˡ ⟩
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(f #) ≈⟨ sym $ #-resp-≈ identityˡ ⟩
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((idC ∘ f) #) ≈⟨ sym (#-resp-≈ (∘-resp-≈ˡ +-η)) ⟩
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(([ i₁ , i₂ ] ∘ f)#) ≈⟨ sym $ #-resp-≈ (∘-resp-≈ˡ ([]-cong₂ identityʳ identityʳ)) ⟩
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(([ i₁ ∘ idC , i₂ ∘ idC ] ∘ f)#) ≈⟨ sym $ #-resp-≈ (∘-resp-≈ˡ []∘+₁) ⟩
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((([ i₁ , i₂ ] ∘ (idC +₁ idC)) ∘ f)#) ≈⟨ #-resp-≈ assoc ⟩
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(([ i₁ , i₂ ] ∘ (idC +₁ idC) ∘ f)#) ≈⟨ #-resp-≈ (∘-resp-≈ˡ +-η) ⟩
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((idC ∘ (idC +₁ idC) ∘ f)#) ≈⟨ #-resp-≈ identityˡ ⟩
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((idC +₁ idC) ∘ f) # ∎ }
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; _∘_ = λ {EA} {EB} {EC} f g → let
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open Elgot-Algebra-Morphism f renaming (h to hᶠ; preserves to preservesᶠ)
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open Elgot-Algebra-Morphism g renaming (h to hᵍ; preserves to preservesᵍ)
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open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ)
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open Elgot-Algebra EB using () renaming (_# to _#ᵇ; A to B)
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open Elgot-Algebra EC using () renaming (_# to _#ᶜ; A to C; #-resp-≈ to #ᶜ-resp-≈)
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in record { h = hᶠ ∘ hᵍ; preserves = λ {X} {f : X ⇒ A + X} → begin
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(hᶠ ∘ hᵍ) ∘ (f #ᵃ) ≈⟨ assoc ⟩
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(hᶠ ∘ hᵍ ∘ (f #ᵃ)) ≈⟨ ∘-resp-≈ʳ preservesᵍ ⟩
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(hᶠ ∘ (((hᵍ +₁ idC) ∘ f) #ᵇ)) ≈⟨ preservesᶠ ⟩
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(((hᶠ +₁ idC) ∘ (hᵍ +₁ idC) ∘ f) #ᶜ) ≈⟨ #ᶜ-resp-≈ sym-assoc ⟩
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((((hᶠ +₁ idC) ∘ (hᵍ +₁ idC)) ∘ f) #ᶜ) ≈⟨ #ᶜ-resp-≈ (∘-resp-≈ˡ +₁∘+₁) ⟩
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((((hᶠ ∘ hᵍ) +₁ (idC ∘ idC)) ∘ f) #ᶜ) ≈⟨ #ᶜ-resp-≈ (∘-resp-≈ˡ (+₁-cong₂ refl (identity²))) ⟩
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((hᶠ ∘ hᵍ +₁ idC) ∘ f) #ᶜ ∎ }
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; identityˡ = identityˡ
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; identityʳ = identityʳ
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; identity² = identity²
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; assoc = assoc
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; sym-assoc = sym-assoc
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; equiv = record
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{ refl = refl
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; sym = sym
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; trans = trans}
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; ∘-resp-≈ = ∘-resp-≈
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}
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where
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open Elgot-Algebra-Morphism
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open HomReasoning
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open Equiv
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--*
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-- products and exponentials of elgot-algebras
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--*
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-- if the carriers of the algebra form a product, so do the algebras
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Product-Elgot-Algebra : ∀ {EA EB : Elgot-Algebra} → Product 𝒞 (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Elgot-Algebra
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Product-Elgot-Algebra {EA} {EB} p = record
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{ A = A×B
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; _# = λ {X : Obj} (h : X ⇒ A×B + X) → ⟨ ((π₁ +₁ idC) ∘ h)#ᵃ , ((π₂ +₁ idC) ∘ h)#ᵇ ⟩
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; #-Fixpoint = λ {X} {f} → begin
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⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ≈⟨ ⟨⟩-cong₂ #ᵃ-Fixpoint #ᵇ-Fixpoint ⟩
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⟨ [ 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 ⟩
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⟨ ([ 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) ⟩
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⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] ∘ f , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ∘ f ⟩ ≈⟨ sym ∘-distribʳ-⟨⟩ ⟩
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(⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ⟩ ∘ f) ≈⟨ ∘-resp-≈ˡ (unique′
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(begin
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π₁ ∘ ⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ⟩ ≈⟨ project₁ ⟩
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[ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] ≈⟨ []-cong₂ identityˡ identityʳ ⟩
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[ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ] ≈⟨ sym ([]-cong₂ identityʳ project₁) ⟩
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[ π₁ ∘ idC , π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ≈⟨ sym ∘[] ⟩
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π₁ ∘ [ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ∎)
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(begin
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π₂ ∘ ⟨ [ idC ∘ π₁ , ((π₁ +₁ idC) ∘ f)#ᵃ ∘ idC ] , [ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ⟩ ≈⟨ project₂ ⟩
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[ idC ∘ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ∘ idC ] ≈⟨ []-cong₂ identityˡ identityʳ ⟩
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[ π₂ , ((π₂ +₁ idC) ∘ f)#ᵇ ] ≈⟨ sym ([]-cong₂ identityʳ project₂) ⟩
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[ π₂ ∘ idC , π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ≈⟨ sym ∘[] ⟩
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π₂ ∘ [ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ∎) )⟩
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([ idC , ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ∘ f) ∎
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; #-Uniformity = λ {X Y f g h} uni → unique′ (
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begin
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π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ≈⟨ project₁ ⟩
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(((π₁ +₁ idC) ∘ f)#ᵃ) ≈⟨ #ᵃ-Uniformity (
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begin
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(idC +₁ h) ∘ (π₁ +₁ idC) ∘ f ≈⟨ sym-assoc ⟩
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((idC +₁ h) ∘ (π₁ +₁ idC)) ∘ f ≈⟨ ∘-resp-≈ˡ +₁∘+₁ ⟩
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(idC ∘ π₁ +₁ h ∘ idC) ∘ f ≈⟨ ∘-resp-≈ˡ (+₁-cong₂ identityˡ identityʳ) ⟩
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((π₁ +₁ h) ∘ f) ≈⟨ sym (∘-resp-≈ˡ (+₁-cong₂ identityʳ identityˡ)) ⟩
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(((π₁ ∘ idC +₁ idC ∘ h)) ∘ f) ≈⟨ sym (∘-resp-≈ˡ +₁∘+₁) ⟩
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((π₁ +₁ idC) ∘ (idC +₁ h)) ∘ f ≈⟨ assoc ⟩
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(π₁ +₁ idC) ∘ ((idC +₁ h) ∘ f) ≈⟨ ∘-resp-≈ʳ uni ⟩
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(π₁ +₁ idC) ∘ g ∘ h ≈⟨ sym-assoc ⟩
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((π₁ +₁ idC) ∘ g) ∘ h ∎
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)⟩
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(((π₁ +₁ idC) ∘ g)#ᵃ ∘ h) ≈⟨ sym (∘-resp-≈ˡ project₁) ⟩
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((π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ g)#ᵃ , ((π₂ +₁ idC) ∘ g)#ᵇ ⟩) ∘ h) ≈⟨ assoc ⟩
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π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ g)#ᵃ , ((π₂ +₁ idC) ∘ g)#ᵇ ⟩ ∘ h ∎
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) (
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begin
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π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ≈⟨ project₂ ⟩
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((π₂ +₁ idC) ∘ f)#ᵇ ≈⟨ #ᵇ-Uniformity (
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begin
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(idC +₁ h) ∘ (π₂ +₁ idC) ∘ f ≈⟨ sym-assoc ⟩
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(((idC +₁ h) ∘ (π₂ +₁ idC)) ∘ f) ≈⟨ ∘-resp-≈ˡ +₁∘+₁ ⟩
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((idC ∘ π₂ +₁ h ∘ idC) ∘ f) ≈⟨ ∘-resp-≈ˡ (+₁-cong₂ identityˡ identityʳ) ⟩
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((π₂ +₁ h) ∘ f) ≈⟨ sym (∘-resp-≈ˡ (+₁-cong₂ identityʳ identityˡ)) ⟩
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((((π₂ ∘ idC +₁ idC ∘ h)) ∘ f)) ≈⟨ sym (∘-resp-≈ˡ +₁∘+₁) ⟩
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((π₂ +₁ idC) ∘ ((idC +₁ h))) ∘ f ≈⟨ assoc ⟩
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(π₂ +₁ idC) ∘ ((idC +₁ h)) ∘ f ≈⟨ ∘-resp-≈ʳ uni ⟩
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(π₂ +₁ idC) ∘ g ∘ h ≈⟨ sym-assoc ⟩
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((π₂ +₁ idC) ∘ g) ∘ h ∎
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)⟩
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((π₂ +₁ idC) ∘ g)#ᵇ ∘ h ≈⟨ sym (∘-resp-≈ˡ project₂) ⟩
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((π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ g)#ᵃ , ((π₂ +₁ idC) ∘ g)#ᵇ ⟩) ∘ h) ≈⟨ assoc ⟩
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π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ g)#ᵃ , ((π₂ +₁ idC) ∘ g)#ᵇ ⟩ ∘ h ∎
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)
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; #-Folding = λ {X} {Y} {f} {h} → ⟨⟩-cong₂ (foldingˡ {X} {Y}) (foldingʳ {X} {Y})
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; #-resp-≈ = λ fg → ⟨⟩-cong₂ (#ᵃ-resp-≈ (∘-resp-≈ʳ fg)) (#ᵇ-resp-≈ (∘-resp-≈ʳ fg))
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}
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where
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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-≈)
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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-≈)
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open Product 𝒞 p
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open HomReasoning
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open Equiv
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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
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foldingˡ {X} {Y} {f} {h} = begin
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((π₁ +₁ idC) ∘ (⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ h))#ᵃ ≈⟨ #ᵃ-resp-≈ +₁∘+₁ ⟩
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((π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ idC ∘ h)#ᵃ) ≈⟨ #ᵃ-resp-≈ (+₁-cong₂ project₁ identityˡ) ⟩
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((((π₁ +₁ idC) ∘ f)#ᵃ +₁ h)#ᵃ) ≈⟨ #ᵃ-Folding ⟩
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([ (idC +₁ i₁) ∘ ((π₁ +₁ idC) ∘ f) , i₂ ∘ h ] #ᵃ) ≈⟨ #ᵃ-resp-≈ ([]-congʳ sym-assoc) ⟩
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([ ((idC +₁ i₁) ∘ (π₁ +₁ idC)) ∘ f , i₂ ∘ h ] #ᵃ) ≈⟨ #ᵃ-resp-≈ ([]-congʳ (∘-resp-≈ˡ +₁∘+₁)) ⟩
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([ ((idC ∘ π₁ +₁ i₁ ∘ idC)) ∘ f , i₂ ∘ h ] #ᵃ) ≈⟨ #ᵃ-resp-≈ ([]-congʳ (∘-resp-≈ˡ (+₁-cong₂ identityˡ identityʳ))) ⟩
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([ ((π₁ +₁ i₁)) ∘ f , i₂ ∘ h ] #ᵃ) ≈⟨ sym (#ᵃ-resp-≈ ([]-cong₂ (∘-resp-≈ˡ (+₁-cong₂ identityʳ identityˡ)) (∘-resp-≈ˡ identityʳ))) ⟩
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(([ (π₁ ∘ idC +₁ idC ∘ i₁) ∘ f , (i₂ ∘ idC) ∘ h ])#ᵃ) ≈⟨ sym (#ᵃ-resp-≈ ([]-cong₂ (∘-resp-≈ˡ +₁∘+₁) (∘-resp-≈ˡ +₁∘i₂))) ⟩
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(([ ((π₁ +₁ idC) ∘ (idC +₁ i₁)) ∘ f , ((π₁ +₁ idC) ∘ i₂) ∘ h ])#ᵃ) ≈⟨ #ᵃ-resp-≈ ([]-cong₂ assoc assoc) ⟩
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(([ (π₁ +₁ idC) ∘ (idC +₁ i₁) ∘ f , (π₁ +₁ idC) ∘ i₂ ∘ h ])#ᵃ) ≈⟨ sym (#ᵃ-resp-≈ ∘[]) ⟩
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((π₁ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵃ ∎
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foldingʳ : ∀ {X} {Y} {f} {h} → ((π₂ +₁ idC) ∘ (⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ h))#ᵇ ≈ ((π₂ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ
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foldingʳ {X} {Y} {f} {h} = begin
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((π₂ +₁ idC) ∘ (⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ h))#ᵇ ≈⟨ #ᵇ-resp-≈ +₁∘+₁ ⟩
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((π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ idC ∘ h)#ᵇ) ≈⟨ #ᵇ-resp-≈ (+₁-cong₂ project₂ identityˡ) ⟩
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((((π₂ +₁ idC) ∘ f)#ᵇ +₁ h)#ᵇ) ≈⟨ #ᵇ-Folding ⟩
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[ (idC +₁ i₁) ∘ ((π₂ +₁ idC) ∘ f) , i₂ ∘ h ] #ᵇ ≈⟨ #ᵇ-resp-≈ ([]-congʳ sym-assoc) ⟩
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([ ((idC +₁ i₁) ∘ (π₂ +₁ idC)) ∘ f , i₂ ∘ h ] #ᵇ) ≈⟨ #ᵇ-resp-≈ ([]-congʳ (∘-resp-≈ˡ +₁∘+₁)) ⟩
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([ ((idC ∘ π₂ +₁ i₁ ∘ idC)) ∘ f , i₂ ∘ h ] #ᵇ) ≈⟨ #ᵇ-resp-≈ ([]-congʳ (∘-resp-≈ˡ (+₁-cong₂ identityˡ identityʳ))) ⟩
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([ ((π₂ +₁ i₁)) ∘ f , i₂ ∘ h ] #ᵇ) ≈⟨ sym (#ᵇ-resp-≈ ([]-cong₂ (∘-resp-≈ˡ (+₁-cong₂ identityʳ identityˡ)) (∘-resp-≈ˡ identityʳ))) ⟩
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(([ (π₂ ∘ idC +₁ idC ∘ i₁) ∘ f , (i₂ ∘ idC) ∘ h ])#ᵇ) ≈⟨ sym (#ᵇ-resp-≈ ([]-cong₂ (∘-resp-≈ˡ +₁∘+₁) (∘-resp-≈ˡ +₁∘i₂))) ⟩
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(([ ((π₂ +₁ idC) ∘ (idC +₁ i₁)) ∘ f , ((π₂ +₁ idC) ∘ i₂) ∘ h ])#ᵇ) ≈⟨ #ᵇ-resp-≈ ([]-cong₂ assoc assoc) ⟩
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(([ (π₂ +₁ idC) ∘ (idC +₁ i₁) ∘ f , (π₂ +₁ idC) ∘ i₂ ∘ h ])#ᵇ) ≈⟨ sym (#ᵇ-resp-≈ ∘[]) ⟩
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((π₂ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ ∎
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-- every elgot-algebra comes with a divergence constant
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!ₑ : ⊥ ⇒ A
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!ₑ = i₂ #
|
||||
|
||||
--*
|
||||
-- Here follows the proof of equivalence for unguarded and Id-guarded Elgot-Algebras
|
||||
|
|
|
|||
242
ElgotAlgebras.agda
Normal file
242
ElgotAlgebras.agda
Normal file
|
|
@ -0,0 +1,242 @@
|
|||
open import Level renaming (suc to ℓ-suc)
|
||||
open import Function using (_$_) renaming (id to idf; _∘_ to _∘ᶠ_)
|
||||
open import Data.Product using (_,_) renaming (_×_ to _∧_)
|
||||
|
||||
open import Categories.Category.Cocartesian.Bundle using (CocartesianCategory)
|
||||
open import Categories.Category.Cartesian
|
||||
open import Categories.Functor renaming (id to idF)
|
||||
open import Categories.Functor.Algebra
|
||||
open import Categories.Object.Terminal
|
||||
open import Categories.Object.Product
|
||||
open import Categories.Object.Coproduct
|
||||
open import Categories.Category.BinaryProducts
|
||||
open import Categories.Category
|
||||
open import ElgotAlgebra
|
||||
|
||||
module ElgotAlgebras where
|
||||
|
||||
private
|
||||
variable
|
||||
o ℓ e : Level
|
||||
|
||||
module _ (CC : CocartesianCategory o ℓ e) where
|
||||
open CocartesianCategory CC renaming (U to C; id to idC)
|
||||
|
||||
--*
|
||||
-- let's define the category of elgot-algebras
|
||||
--*
|
||||
|
||||
-- iteration preversing morphism between two elgot-algebras
|
||||
module _ (E₁ E₂ : Elgot-Algebra CC) where
|
||||
open Elgot-Algebra E₁ renaming (_# to _#₁)
|
||||
open Elgot-Algebra E₂ renaming (_# to _#₂; A to B)
|
||||
record Elgot-Algebra-Morphism : Set (o ⊔ ℓ ⊔ e) where
|
||||
field
|
||||
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
|
||||
Elgot-Algebras : Category (o ⊔ ℓ ⊔ e) (o ⊔ ℓ ⊔ e) e
|
||||
Elgot-Algebras = record
|
||||
{ Obj = Elgot-Algebra CC
|
||||
; _⇒_ = Elgot-Algebra-Morphism
|
||||
; _≈_ = λ f g → Elgot-Algebra-Morphism.h f ≈ Elgot-Algebra-Morphism.h g
|
||||
; id = λ {EB} → let open Elgot-Algebra EB in
|
||||
record { h = idC; preserves = λ {X : Obj} {f : X ⇒ A + X} → begin
|
||||
idC ∘ f # ≈⟨ identityˡ ⟩
|
||||
(f #) ≈⟨ sym $ #-resp-≈ identityˡ ⟩
|
||||
((idC ∘ f) #) ≈⟨ sym (#-resp-≈ (∘-resp-≈ˡ +-η)) ⟩
|
||||
(([ i₁ , i₂ ] ∘ f)#) ≈⟨ sym $ #-resp-≈ (∘-resp-≈ˡ ([]-cong₂ identityʳ identityʳ)) ⟩
|
||||
(([ i₁ ∘ idC , i₂ ∘ idC ] ∘ f)#) ≈⟨ sym $ #-resp-≈ (∘-resp-≈ˡ []∘+₁) ⟩
|
||||
((([ i₁ , i₂ ] ∘ (idC +₁ idC)) ∘ f)#) ≈⟨ #-resp-≈ assoc ⟩
|
||||
(([ i₁ , i₂ ] ∘ (idC +₁ idC) ∘ f)#) ≈⟨ #-resp-≈ (∘-resp-≈ˡ +-η) ⟩
|
||||
((idC ∘ (idC +₁ idC) ∘ f)#) ≈⟨ #-resp-≈ identityˡ ⟩
|
||||
((idC +₁ idC) ∘ f) # ∎ }
|
||||
; _∘_ = λ {EA} {EB} {EC} f g → let
|
||||
open Elgot-Algebra-Morphism f renaming (h to hᶠ; preserves to preservesᶠ)
|
||||
open Elgot-Algebra-Morphism g renaming (h to hᵍ; preserves to preservesᵍ)
|
||||
open Elgot-Algebra EA using (A) renaming (_# to _#ᵃ)
|
||||
open Elgot-Algebra EB using () renaming (_# to _#ᵇ; A to B)
|
||||
open Elgot-Algebra EC using () renaming (_# to _#ᶜ; A to C; #-resp-≈ to #ᶜ-resp-≈)
|
||||
in record { h = hᶠ ∘ hᵍ; preserves = λ {X} {f : X ⇒ A + X} → begin
|
||||
(hᶠ ∘ hᵍ) ∘ (f #ᵃ) ≈⟨ assoc ⟩
|
||||
(hᶠ ∘ hᵍ ∘ (f #ᵃ)) ≈⟨ ∘-resp-≈ʳ preservesᵍ ⟩
|
||||
(hᶠ ∘ (((hᵍ +₁ idC) ∘ f) #ᵇ)) ≈⟨ preservesᶠ ⟩
|
||||
(((hᶠ +₁ idC) ∘ (hᵍ +₁ idC) ∘ f) #ᶜ) ≈⟨ #ᶜ-resp-≈ sym-assoc ⟩
|
||||
((((hᶠ +₁ idC) ∘ (hᵍ +₁ idC)) ∘ f) #ᶜ) ≈⟨ #ᶜ-resp-≈ (∘-resp-≈ˡ +₁∘+₁) ⟩
|
||||
((((hᶠ ∘ hᵍ) +₁ (idC ∘ idC)) ∘ f) #ᶜ) ≈⟨ #ᶜ-resp-≈ (∘-resp-≈ˡ (+₁-cong₂ refl (identity²))) ⟩
|
||||
((hᶠ ∘ hᵍ +₁ idC) ∘ f) #ᶜ ∎ }
|
||||
; identityˡ = identityˡ
|
||||
; identityʳ = identityʳ
|
||||
; identity² = identity²
|
||||
; assoc = assoc
|
||||
; sym-assoc = sym-assoc
|
||||
; equiv = record
|
||||
{ refl = refl
|
||||
; sym = sym
|
||||
; trans = trans}
|
||||
; ∘-resp-≈ = ∘-resp-≈
|
||||
}
|
||||
where
|
||||
open Elgot-Algebra-Morphism
|
||||
open HomReasoning
|
||||
open Equiv
|
||||
|
||||
--*
|
||||
-- products and exponentials of elgot-algebras
|
||||
--*
|
||||
|
||||
-- if the carrier contains a terminal, so does elgot-algebras
|
||||
Terminal-Elgot-Algebras : Terminal C → Terminal Elgot-Algebras
|
||||
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
|
||||
{ ! = λ {A} → record { h = ! ; preserves = λ {X} {f} → sym (!-unique (! ∘ (A Elgot-Algebra.#) f)) }
|
||||
; !-unique = λ {A} f → !-unique (Elgot-Algebra-Morphism.h f) } }
|
||||
where
|
||||
open Terminal T
|
||||
open Equiv
|
||||
|
||||
-- if the carriers of the algebra form a product, so do the algebras
|
||||
A×B-Helper : ∀ {EA EB : Elgot-Algebra CC} → Product C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Elgot-Algebra CC
|
||||
A×B-Helper {EA} {EB} p = record
|
||||
{ A = A×B
|
||||
; _# = λ {X : Obj} (h : X ⇒ A×B + X) → ⟨ ((π₁ +₁ idC) ∘ h)#ᵃ , ((π₂ +₁ idC) ∘ h)#ᵇ ⟩
|
||||
; #-Fixpoint = λ {X} {f} → begin
|
||||
⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ≈⟨ ⟨⟩-cong₂ #ᵃ-Fixpoint #ᵇ-Fixpoint ⟩
|
||||
⟨ [ 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 ∘-distribʳ-⟨⟩ ⟩
|
||||
(⟨ [ 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
|
||||
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) ⟩
|
||||
[ π₂ ∘ idC , π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ] ≈⟨ sym ∘[] ⟩
|
||||
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
|
||||
π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ≈⟨ project₁ ⟩
|
||||
(((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) ⟩
|
||||
((π₁ +₁ h) ∘ f) ≈⟨ sym (∘-resp-≈ˡ (+₁-cong₂ identityʳ identityˡ)) ⟩
|
||||
(((π₁ ∘ idC +₁ idC ∘ h)) ∘ f) ≈⟨ sym (∘-resp-≈ˡ +₁∘+₁) ⟩
|
||||
((π₁ +₁ idC) ∘ (idC +₁ h)) ∘ f ≈⟨ assoc ⟩
|
||||
(π₁ +₁ idC) ∘ ((idC +₁ h) ∘ f) ≈⟨ ∘-resp-≈ʳ uni ⟩
|
||||
(π₁ +₁ idC) ∘ g ∘ h ≈⟨ sym-assoc ⟩
|
||||
((π₁ +₁ idC) ∘ g) ∘ h ∎
|
||||
)⟩
|
||||
(((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
|
||||
π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ ≈⟨ project₂ ⟩
|
||||
((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) ⟩
|
||||
((π₂ +₁ h) ∘ f) ≈⟨ sym (∘-resp-≈ˡ (+₁-cong₂ identityʳ identityˡ)) ⟩
|
||||
((((π₂ ∘ idC +₁ idC ∘ h)) ∘ f)) ≈⟨ sym (∘-resp-≈ˡ +₁∘+₁) ⟩
|
||||
((π₂ +₁ idC) ∘ ((idC +₁ h))) ∘ f ≈⟨ assoc ⟩
|
||||
(π₂ +₁ idC) ∘ ((idC +₁ h)) ∘ f ≈⟨ ∘-resp-≈ʳ uni ⟩
|
||||
(π₂ +₁ idC) ∘ g ∘ h ≈⟨ sym-assoc ⟩
|
||||
((π₂ +₁ idC) ∘ g) ∘ h ∎
|
||||
)⟩
|
||||
((π₂ +₁ idC) ∘ g)#ᵇ ∘ h ≈⟨ sym (∘-resp-≈ˡ project₂) ⟩
|
||||
((π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ g)#ᵃ , ((π₂ +₁ idC) ∘ g)#ᵇ ⟩) ∘ h) ≈⟨ assoc ⟩
|
||||
π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ g)#ᵃ , ((π₂ +₁ idC) ∘ g)#ᵇ ⟩ ∘ h ∎
|
||||
)
|
||||
; #-Folding = λ {X} {Y} {f} {h} → ⟨⟩-cong₂ (foldingˡ {X} {Y}) (foldingʳ {X} {Y})
|
||||
; #-resp-≈ = λ fg → ⟨⟩-cong₂ (#ᵃ-resp-≈ (∘-resp-≈ʳ fg)) (#ᵇ-resp-≈ (∘-resp-≈ʳ fg))
|
||||
}
|
||||
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 Product C p
|
||||
open HomReasoning
|
||||
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
|
||||
((π₁ +₁ idC) ∘ (⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ h))#ᵃ ≈⟨ #ᵃ-resp-≈ +₁∘+₁ ⟩
|
||||
((π₁ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ idC ∘ h)#ᵃ) ≈⟨ #ᵃ-resp-≈ (+₁-cong₂ project₁ identityˡ) ⟩
|
||||
((((π₁ +₁ idC) ∘ f)#ᵃ +₁ h)#ᵃ) ≈⟨ #ᵃ-Folding ⟩
|
||||
([ (idC +₁ i₁) ∘ ((π₁ +₁ idC) ∘ f) , i₂ ∘ h ] #ᵃ) ≈⟨ #ᵃ-resp-≈ ([]-congʳ sym-assoc) ⟩
|
||||
([ ((idC +₁ i₁) ∘ (π₁ +₁ idC)) ∘ f , i₂ ∘ h ] #ᵃ) ≈⟨ #ᵃ-resp-≈ ([]-congʳ (∘-resp-≈ˡ +₁∘+₁)) ⟩
|
||||
([ ((idC ∘ π₁ +₁ i₁ ∘ idC)) ∘ f , i₂ ∘ h ] #ᵃ) ≈⟨ #ᵃ-resp-≈ ([]-congʳ (∘-resp-≈ˡ (+₁-cong₂ identityˡ identityʳ))) ⟩
|
||||
([ ((π₁ +₁ i₁)) ∘ f , i₂ ∘ h ] #ᵃ) ≈⟨ sym (#ᵃ-resp-≈ ([]-cong₂ (∘-resp-≈ˡ (+₁-cong₂ identityʳ identityˡ)) (∘-resp-≈ˡ identityʳ))) ⟩
|
||||
(([ (π₁ ∘ idC +₁ idC ∘ i₁) ∘ f , (i₂ ∘ idC) ∘ h ])#ᵃ) ≈⟨ sym (#ᵃ-resp-≈ ([]-cong₂ (∘-resp-≈ˡ +₁∘+₁) (∘-resp-≈ˡ +₁∘i₂))) ⟩
|
||||
(([ ((π₁ +₁ idC) ∘ (idC +₁ i₁)) ∘ f , ((π₁ +₁ idC) ∘ i₂) ∘ h ])#ᵃ) ≈⟨ #ᵃ-resp-≈ ([]-cong₂ assoc assoc) ⟩
|
||||
(([ (π₁ +₁ idC) ∘ (idC +₁ i₁) ∘ f , (π₁ +₁ idC) ∘ i₂ ∘ h ])#ᵃ) ≈⟨ sym (#ᵃ-resp-≈ ∘[]) ⟩
|
||||
((π₁ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵃ ∎
|
||||
foldingʳ : ∀ {X} {Y} {f} {h} → ((π₂ +₁ idC) ∘ (⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ h))#ᵇ ≈ ((π₂ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ
|
||||
foldingʳ {X} {Y} {f} {h} = begin
|
||||
((π₂ +₁ idC) ∘ (⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ h))#ᵇ ≈⟨ #ᵇ-resp-≈ +₁∘+₁ ⟩
|
||||
((π₂ ∘ ⟨ ((π₁ +₁ idC) ∘ f)#ᵃ , ((π₂ +₁ idC) ∘ f)#ᵇ ⟩ +₁ idC ∘ h)#ᵇ) ≈⟨ #ᵇ-resp-≈ (+₁-cong₂ project₂ identityˡ) ⟩
|
||||
((((π₂ +₁ idC) ∘ f)#ᵇ +₁ h)#ᵇ) ≈⟨ #ᵇ-Folding ⟩
|
||||
[ (idC +₁ i₁) ∘ ((π₂ +₁ idC) ∘ f) , i₂ ∘ h ] #ᵇ ≈⟨ #ᵇ-resp-≈ ([]-congʳ sym-assoc) ⟩
|
||||
([ ((idC +₁ i₁) ∘ (π₂ +₁ idC)) ∘ f , i₂ ∘ h ] #ᵇ) ≈⟨ #ᵇ-resp-≈ ([]-congʳ (∘-resp-≈ˡ +₁∘+₁)) ⟩
|
||||
([ ((idC ∘ π₂ +₁ i₁ ∘ idC)) ∘ f , i₂ ∘ h ] #ᵇ) ≈⟨ #ᵇ-resp-≈ ([]-congʳ (∘-resp-≈ˡ (+₁-cong₂ identityˡ identityʳ))) ⟩
|
||||
([ ((π₂ +₁ i₁)) ∘ f , i₂ ∘ h ] #ᵇ) ≈⟨ sym (#ᵇ-resp-≈ ([]-cong₂ (∘-resp-≈ˡ (+₁-cong₂ identityʳ identityˡ)) (∘-resp-≈ˡ identityʳ))) ⟩
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(([ (π₂ ∘ idC +₁ idC ∘ i₁) ∘ f , (i₂ ∘ idC) ∘ h ])#ᵇ) ≈⟨ sym (#ᵇ-resp-≈ ([]-cong₂ (∘-resp-≈ˡ +₁∘+₁) (∘-resp-≈ˡ +₁∘i₂))) ⟩
|
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(([ ((π₂ +₁ idC) ∘ (idC +₁ i₁)) ∘ f , ((π₂ +₁ idC) ∘ i₂) ∘ h ])#ᵇ) ≈⟨ #ᵇ-resp-≈ ([]-cong₂ assoc assoc) ⟩
|
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(([ (π₂ +₁ idC) ∘ (idC +₁ i₁) ∘ f , (π₂ +₁ idC) ∘ i₂ ∘ h ])#ᵇ) ≈⟨ sym (#ᵇ-resp-≈ ∘[]) ⟩
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((π₂ +₁ idC) ∘ [ (idC +₁ i₁) ∘ f , i₂ ∘ h ])#ᵇ ∎
|
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|
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Product-Elgot-Algebras : ∀ (EA EB : Elgot-Algebra CC) → Product C (Elgot-Algebra.A EA) (Elgot-Algebra.A EB) → Product Elgot-Algebras EA EB
|
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Product-Elgot-Algebras EA EB p = record
|
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{ A×B = A×B-Helper {EA} {EB} p
|
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; π₁ = record { h = π₁ ; preserves = λ {X} {f} → project₁ }
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; π₂ = record { h = π₂ ; preserves = λ {X} {f} → project₂ }
|
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; ⟨_,_⟩ = λ {E} f g → let
|
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open Elgot-Algebra-Morphism f renaming (h to f′; preserves to preservesᶠ)
|
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open Elgot-Algebra-Morphism g renaming (h to g′; preserves to preservesᵍ)
|
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open Elgot-Algebra E renaming (_# to _#ᵉ) in record { h = ⟨ f′ , g′ ⟩ ; preserves = λ {X} {h} →
|
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begin
|
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⟨ f′ , g′ ⟩ ∘ (h #ᵉ) ≈⟨ ∘-distribʳ-⟨⟩ ⟩
|
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⟨ f′ ∘ (h #ᵉ) , g′ ∘ (h #ᵉ) ⟩ ≈⟨ ⟨⟩-cong₂ preservesᶠ preservesᵍ ⟩
|
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⟨ ((f′ +₁ idC) ∘ h) #ᵃ , ((g′ +₁ idC) ∘ h) #ᵇ ⟩ ≈⟨ sym (⟨⟩-cong₂ (#ᵃ-resp-≈ (∘-resp-≈ˡ (+₁-cong₂ project₁ identity²))) (#ᵇ-resp-≈ (∘-resp-≈ˡ (+₁-cong₂ project₂ identity²)))) ⟩
|
||||
⟨ ((π₁ ∘ ⟨ f′ , g′ ⟩ +₁ idC ∘ idC) ∘ h) #ᵃ , ((π₂ ∘ ⟨ f′ , g′ ⟩ +₁ idC ∘ idC) ∘ h) #ᵇ ⟩ ≈⟨ sym (⟨⟩-cong₂ (#ᵃ-resp-≈ (∘-resp-≈ˡ +₁∘+₁)) (#ᵇ-resp-≈ (∘-resp-≈ˡ +₁∘+₁))) ⟩
|
||||
⟨ (((π₁ +₁ idC) ∘ (⟨ f′ , g′ ⟩ +₁ idC)) ∘ h) #ᵃ , (((π₂ +₁ idC) ∘ (⟨ f′ , g′ ⟩ +₁ idC)) ∘ h) #ᵇ ⟩ ≈⟨ (⟨⟩-cong₂ (#ᵃ-resp-≈ assoc) (#ᵇ-resp-≈ assoc)) ⟩
|
||||
⟨ ((π₁ +₁ idC) ∘ (⟨ f′ , g′ ⟩ +₁ idC) ∘ h) #ᵃ , ((π₂ +₁ idC) ∘ (⟨ f′ , g′ ⟩ +₁ idC) ∘ h) #ᵇ ⟩ ∎ }
|
||||
; project₁ = project₁
|
||||
; project₂ = project₂
|
||||
; unique = unique
|
||||
}
|
||||
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 Elgot-Algebra (A×B-Helper {EA} {EB} p) using () renaming (_# to _#ᵖ)
|
||||
open Product C p
|
||||
open HomReasoning
|
||||
open Equiv
|
||||
|
||||
|
||||
-- if the carrier is cartesian, so is the category of algebras
|
||||
Cartesian-Elgot-Algebras : Cartesian C → Cartesian Elgot-Algebras
|
||||
Cartesian-Elgot-Algebras CaC = record {
|
||||
terminal = Terminal-Elgot-Algebras terminal;
|
||||
products = record { product = λ {EA EB} → Product-Elgot-Algebras EA EB product }
|
||||
}
|
||||
where
|
||||
open Cartesian CaC using (terminal; products)
|
||||
open BinaryProducts products using (product)
|
||||
open Equiv
|
||||
Loading…
Reference in a new issue