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* use improved ringsolver * delete one more line * code from old branch * port to CommRings * our lemma at the right place * give up for today * give up for today * find a new strategy * give up * simplify * prepare for presheaf * Add boiler plate code needed * done * clean up
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| {-# OPTIONS --safe #-} | ||
| module Cubical.Algebra.Algebra.Properties where | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.Function | ||
| open import Cubical.Foundations.Equiv | ||
| open import Cubical.Foundations.Equiv.HalfAdjoint | ||
| open import Cubical.Foundations.HLevels | ||
| open import Cubical.Foundations.Isomorphism | ||
| open import Cubical.Foundations.Univalence | ||
| open import Cubical.Foundations.Transport | ||
| open import Cubical.Foundations.SIP | ||
| open import Cubical.Foundations.GroupoidLaws | ||
| open import Cubical.Foundations.Path | ||
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| open import Cubical.Data.Sigma | ||
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| open import Cubical.Structures.Axioms | ||
| open import Cubical.Structures.Auto | ||
| open import Cubical.Structures.Macro | ||
| open import Cubical.Algebra.Module | ||
| open import Cubical.Algebra.Ring | ||
| open import Cubical.Algebra.AbGroup | ||
| open import Cubical.Algebra.Group | ||
| open import Cubical.Algebra.Monoid | ||
| open import Cubical.Algebra.Algebra.Base | ||
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| open Iso | ||
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| private | ||
| variable | ||
| ℓ ℓ' ℓ'' ℓ''' : Level | ||
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| module AlgebraTheory (R : Ring ℓ) (A : Algebra R ℓ') where | ||
| open RingStr (snd R) renaming (_+_ to _+r_ ; _·_ to _·r_) | ||
| open AlgebraStr (A .snd) | ||
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| 0-actsNullifying : (x : ⟨ A ⟩) → 0r ⋆ x ≡ 0a | ||
| 0-actsNullifying x = | ||
| let idempotent-+ = 0r ⋆ x ≡⟨ cong (λ u → u ⋆ x) (sym (RingTheory.0Idempotent R)) ⟩ | ||
| (0r +r 0r) ⋆ x ≡⟨ ⋆-ldist 0r 0r x ⟩ | ||
| (0r ⋆ x) + (0r ⋆ x) ∎ | ||
| in RingTheory.+Idempotency→0 (Algebra→Ring A) (0r ⋆ x) idempotent-+ | ||
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| ⋆Dist· : (x y : ⟨ R ⟩) (a b : ⟨ A ⟩) → (x ·r y) ⋆ (a · b) ≡ (x ⋆ a) · (y ⋆ b) | ||
| ⋆Dist· x y a b = (x ·r y) ⋆ (a · b) ≡⟨ ⋆-rassoc _ _ _ ⟩ | ||
| a · ((x ·r y) ⋆ b) ≡⟨ cong (a ·_) (⋆-assoc _ _ _) ⟩ | ||
| a · (x ⋆ (y ⋆ b)) ≡⟨ sym (⋆-rassoc _ _ _) ⟩ | ||
| x ⋆ (a · (y ⋆ b)) ≡⟨ sym (⋆-lassoc _ _ _) ⟩ | ||
| (x ⋆ a) · (y ⋆ b) ∎ | ||
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| module AlgebraHoms {R : Ring ℓ} where | ||
| open IsAlgebraHom | ||
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| idAlgebraHom : (A : Algebra R ℓ') → AlgebraHom A A | ||
| fst (idAlgebraHom A) x = x | ||
| pres0 (snd (idAlgebraHom A)) = refl | ||
| pres1 (snd (idAlgebraHom A)) = refl | ||
| pres+ (snd (idAlgebraHom A)) x y = refl | ||
| pres· (snd (idAlgebraHom A)) x y = refl | ||
| pres- (snd (idAlgebraHom A)) x = refl | ||
| pres⋆ (snd (idAlgebraHom A)) r x = refl | ||
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| compIsAlgebraHom : {A : Algebra R ℓ'} {B : Algebra R ℓ''} {C : Algebra R ℓ'''} | ||
| {g : ⟨ B ⟩ → ⟨ C ⟩} {f : ⟨ A ⟩ → ⟨ B ⟩} | ||
| → IsAlgebraHom (B .snd) g (C .snd) | ||
| → IsAlgebraHom (A .snd) f (B .snd) | ||
| → IsAlgebraHom (A .snd) (g ∘ f) (C .snd) | ||
| compIsAlgebraHom {g = g} {f} gh fh .pres0 = cong g (fh .pres0) ∙ gh .pres0 | ||
| compIsAlgebraHom {g = g} {f} gh fh .pres1 = cong g (fh .pres1) ∙ gh .pres1 | ||
| compIsAlgebraHom {g = g} {f} gh fh .pres+ x y = cong g (fh .pres+ x y) ∙ gh .pres+ (f x) (f y) | ||
| compIsAlgebraHom {g = g} {f} gh fh .pres· x y = cong g (fh .pres· x y) ∙ gh .pres· (f x) (f y) | ||
| compIsAlgebraHom {g = g} {f} gh fh .pres- x = cong g (fh .pres- x) ∙ gh .pres- (f x) | ||
| compIsAlgebraHom {g = g} {f} gh fh .pres⋆ r x = cong g (fh .pres⋆ r x) ∙ gh .pres⋆ r (f x) | ||
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| compAlgebraHom : {A : Algebra R ℓ'} {B : Algebra R ℓ''} {C : Algebra R ℓ'''} | ||
| → AlgebraHom A B → AlgebraHom B C → AlgebraHom A C | ||
| compAlgebraHom f g .fst = g .fst ∘ f .fst | ||
| compAlgebraHom f g .snd = compIsAlgebraHom (g .snd) (f .snd) | ||
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| syntax compAlgebraHom f g = g ∘a f | ||
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| compIdAlgebraHom : {A B : Algebra R ℓ'} (φ : AlgebraHom A B) → compAlgebraHom (idAlgebraHom A) φ ≡ φ | ||
| compIdAlgebraHom φ = AlgebraHom≡ refl | ||
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| idCompAlgebraHom : {A B : Algebra R ℓ'} (φ : AlgebraHom A B) → compAlgebraHom φ (idAlgebraHom B) ≡ φ | ||
| idCompAlgebraHom φ = AlgebraHom≡ refl | ||
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| compAssocAlgebraHom : {A B C D : Algebra R ℓ'} | ||
| (φ : AlgebraHom A B) (ψ : AlgebraHom B C) (χ : AlgebraHom C D) | ||
| → compAlgebraHom (compAlgebraHom φ ψ) χ ≡ compAlgebraHom φ (compAlgebraHom ψ χ) | ||
| compAssocAlgebraHom _ _ _ = AlgebraHom≡ refl | ||
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| module AlgebraEquivs {R : Ring ℓ} where | ||
| open IsAlgebraHom | ||
| open AlgebraHoms | ||
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| compIsAlgebraEquiv : {A : Algebra R ℓ'} {B : Algebra R ℓ''} {C : Algebra R ℓ'''} | ||
| {g : ⟨ B ⟩ ≃ ⟨ C ⟩} {f : ⟨ A ⟩ ≃ ⟨ B ⟩} | ||
| → IsAlgebraEquiv (B .snd) g (C .snd) | ||
| → IsAlgebraEquiv (A .snd) f (B .snd) | ||
| → IsAlgebraEquiv (A .snd) (compEquiv f g) (C .snd) | ||
| compIsAlgebraEquiv {g = g} {f} gh fh = compIsAlgebraHom {g = g .fst} {f .fst} gh fh | ||
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| compAlgebraEquiv : {A : Algebra R ℓ'} {B : Algebra R ℓ''} {C : Algebra R ℓ'''} | ||
| → AlgebraEquiv A B → AlgebraEquiv B C → AlgebraEquiv A C | ||
| fst (compAlgebraEquiv f g) = compEquiv (f .fst) (g .fst) | ||
| snd (compAlgebraEquiv f g) = compIsAlgebraEquiv {g = g .fst} {f = f .fst} (g .snd) (f .snd) | ||
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| -- the Algebra version of uaCompEquiv | ||
| module AlgebraUAFunctoriality {R : Ring ℓ} where | ||
| open AlgebraStr | ||
| open AlgebraEquivs | ||
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| Algebra≡ : (A B : Algebra R ℓ') → ( | ||
| Σ[ p ∈ ⟨ A ⟩ ≡ ⟨ B ⟩ ] | ||
| Σ[ q0 ∈ PathP (λ i → p i) (0a (snd A)) (0a (snd B)) ] | ||
| Σ[ q1 ∈ PathP (λ i → p i) (1a (snd A)) (1a (snd B)) ] | ||
| Σ[ r+ ∈ PathP (λ i → p i → p i → p i) (_+_ (snd A)) (_+_ (snd B)) ] | ||
| Σ[ r· ∈ PathP (λ i → p i → p i → p i) (_·_ (snd A)) (_·_ (snd B)) ] | ||
| Σ[ s- ∈ PathP (λ i → p i → p i) (-_ (snd A)) (-_ (snd B)) ] | ||
| Σ[ s⋆ ∈ PathP (λ i → ⟨ R ⟩ → p i → p i) (_⋆_ (snd A)) (_⋆_ (snd B)) ] | ||
| PathP (λ i → IsAlgebra R (q0 i) (q1 i) (r+ i) (r· i) (s- i) (s⋆ i)) (isAlgebra (snd A)) | ||
| (isAlgebra (snd B))) | ||
| ≃ (A ≡ B) | ||
| Algebra≡ A B = isoToEquiv theIso | ||
| where | ||
| open Iso | ||
| theIso : Iso _ _ | ||
| fun theIso (p , q0 , q1 , r+ , r· , s- , s⋆ , t) i = p i | ||
| , algebrastr (q0 i) (q1 i) (r+ i) (r· i) (s- i) (s⋆ i) (t i) | ||
| inv theIso x = cong ⟨_⟩ x , cong (0a ∘ snd) x , cong (1a ∘ snd) x | ||
| , cong (_+_ ∘ snd) x , cong (_·_ ∘ snd) x , cong (-_ ∘ snd) x , cong (_⋆_ ∘ snd) x | ||
| , cong (isAlgebra ∘ snd) x | ||
| rightInv theIso _ = refl | ||
| leftInv theIso _ = refl | ||
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| caracAlgebra≡ : {A B : Algebra R ℓ'} (p q : A ≡ B) → cong ⟨_⟩ p ≡ cong ⟨_⟩ q → p ≡ q | ||
| caracAlgebra≡ {A = A} {B = B} p q P = | ||
| sym (transportTransport⁻ (ua (Algebra≡ A B)) p) | ||
| ∙∙ cong (transport (ua (Algebra≡ A B))) helper | ||
| ∙∙ transportTransport⁻ (ua (Algebra≡ A B)) q | ||
| where | ||
| helper : transport (sym (ua (Algebra≡ A B))) p ≡ transport (sym (ua (Algebra≡ A B))) q | ||
| helper = Σ≡Prop | ||
| (λ _ → isPropΣ | ||
| (isOfHLevelPathP' 1 (is-set (snd B)) _ _) | ||
| λ _ → isPropΣ (isOfHLevelPathP' 1 (is-set (snd B)) _ _) | ||
| λ _ → isPropΣ (isOfHLevelPathP' 1 (isSetΠ2 λ _ _ → is-set (snd B)) _ _) | ||
| λ _ → isPropΣ (isOfHLevelPathP' 1 (isSetΠ2 λ _ _ → is-set (snd B)) _ _) | ||
| λ _ → isPropΣ (isOfHLevelPathP' 1 (isSetΠ λ _ → is-set (snd B)) _ _) | ||
| λ _ → isPropΣ (isOfHLevelPathP' 1 (isSetΠ2 λ _ _ → is-set (snd B)) _ _) | ||
| λ _ → isOfHLevelPathP 1 (isPropIsAlgebra _ _ _ _ _ _ _) _ _) | ||
| (transportRefl (cong ⟨_⟩ p) ∙ P ∙ sym (transportRefl (cong ⟨_⟩ q))) | ||
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| uaCompAlgebraEquiv : {A B C : Algebra R ℓ'} (f : AlgebraEquiv A B) (g : AlgebraEquiv B C) | ||
| → uaAlgebra (compAlgebraEquiv f g) ≡ uaAlgebra f ∙ uaAlgebra g | ||
| uaCompAlgebraEquiv f g = caracAlgebra≡ _ _ ( | ||
| cong ⟨_⟩ (uaAlgebra (compAlgebraEquiv f g)) | ||
| ≡⟨ uaCompEquiv _ _ ⟩ | ||
| cong ⟨_⟩ (uaAlgebra f) ∙ cong ⟨_⟩ (uaAlgebra g) | ||
| ≡⟨ sym (cong-∙ ⟨_⟩ (uaAlgebra f) (uaAlgebra g)) ⟩ | ||
| cong ⟨_⟩ (uaAlgebra f ∙ uaAlgebra g) ∎) |
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