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Summary file/missing cohomology lemmas (#1092)
* fix * stuff * stuff * more * .. * done? * wups * wups * wups * fixes * ugh...
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| {-# OPTIONS --safe #-} | ||
| module Cubical.Algebra.AbGroup.Instances.DirectProduct where | ||
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| open import Cubical.Data.Sigma | ||
| open import Cubical.Algebra.AbGroup.Base | ||
| open import Cubical.Algebra.Group.DirProd | ||
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| AbDirProd : ∀ {ℓ ℓ'} → AbGroup ℓ → AbGroup ℓ' → AbGroup (ℓ-max ℓ ℓ') | ||
| AbDirProd G H = | ||
| Group→AbGroup (DirProd (AbGroup→Group G) (AbGroup→Group H)) comm | ||
| where | ||
| comm : (x y : fst G × fst H) → _ ≡ _ | ||
| comm (g1 , h1) (g2 , h2) = | ||
| ΣPathP (AbGroupStr.+Comm (snd G) g1 g2 | ||
| , AbGroupStr.+Comm (snd H) h1 h2) |
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| {-# OPTIONS --safe #-} | ||
| module Cubical.Algebra.AbGroup.Instances.Int where | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Data.Int | ||
| open import Cubical.Algebra.AbGroup.Base | ||
| open import Cubical.Algebra.Group.Base | ||
| open import Cubical.Algebra.Group.Instances.Int | ||
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| ℤAbGroup : AbGroup ℓ-zero | ||
| ℤAbGroup = Group→AbGroup ℤGroup +Comm |
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| {-# OPTIONS --safe #-} | ||
| module Cubical.Algebra.AbGroup.Instances.IntMod where | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.Isomorphism | ||
| open import Cubical.Foundations.HLevels | ||
| open import Cubical.Foundations.Function | ||
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| open import Cubical.Algebra.AbGroup | ||
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| open import Cubical.Algebra.Group.Instances.Int | ||
| open import Cubical.Algebra.AbGroup.Instances.Int | ||
| open import Cubical.Algebra.Group.Instances.IntMod | ||
| open import Cubical.Algebra.Group.Base | ||
| open import Cubical.Algebra.Group.MorphismProperties | ||
| open import Cubical.Algebra.Group.ZAction | ||
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| open import Cubical.Data.Empty as ⊥ | ||
| open import Cubical.Data.Nat renaming (_+_ to _+ℕ_ ; _·_ to _·ℕ_) | ||
| open import Cubical.Data.Nat.Order | ||
| open import Cubical.Data.Int | ||
| renaming (_+_ to _+ℤ_ ; _·_ to _·ℤ_ ; -_ to -ℤ_) | ||
| open import Cubical.Data.Fin | ||
| open import Cubical.Data.Fin.Arithmetic | ||
| open import Cubical.Data.Sigma | ||
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| open import Cubical.HITs.SetQuotients as SQ | ||
| open import Cubical.HITs.PropositionalTruncation as PT | ||
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| ℤAbGroup/_ : ℕ → AbGroup ℓ-zero | ||
| ℤAbGroup/ n = Group→AbGroup (ℤGroup/ n) (comm n) | ||
| where | ||
| comm : (n : ℕ) (x y : fst (ℤGroup/ n)) | ||
| → GroupStr._·_ (snd (ℤGroup/ n)) x y | ||
| ≡ GroupStr._·_ (snd (ℤGroup/ n)) y x | ||
| comm zero = +Comm | ||
| comm (suc n) = +ₘ-comm | ||
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| ℤ/2 : AbGroup ℓ-zero | ||
| ℤ/2 = ℤAbGroup/ 2 | ||
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| ℤ/2[2]≅ℤ/2 : AbGroupIso (ℤ/2 [ 2 ]ₜ) ℤ/2 | ||
| Iso.fun (fst ℤ/2[2]≅ℤ/2) = fst | ||
| Iso.inv (fst ℤ/2[2]≅ℤ/2) x = x , cong (x +ₘ_) (+ₘ-rUnit x) ∙ x+x x | ||
| where | ||
| x+x : (x : ℤ/2 .fst) → x +ₘ x ≡ fzero | ||
| x+x = ℤ/2-elim refl refl | ||
| Iso.rightInv (fst ℤ/2[2]≅ℤ/2) x = Σ≡Prop (λ _ → isProp≤) refl | ||
| Iso.leftInv (fst ℤ/2[2]≅ℤ/2) x = Σ≡Prop (λ _ → isSetFin _ _) refl | ||
| snd ℤ/2[2]≅ℤ/2 = makeIsGroupHom λ _ _ → refl | ||
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| ℤ/2/2≅ℤ/2 : AbGroupIso (ℤ/2 /^ 2) ℤ/2 | ||
| Iso.fun (fst ℤ/2/2≅ℤ/2) = | ||
| SQ.rec isSetFin (λ x → x) lem | ||
| where | ||
| lem : _ | ||
| lem = ℤ/2-elim (ℤ/2-elim (λ _ → refl) | ||
| (PT.rec (isSetFin _ _) | ||
| (uncurry (ℤ/2-elim | ||
| (λ p → ⊥.rec (snotz (sym (cong fst p)))) | ||
| λ p → ⊥.rec (snotz (sym (cong fst p))))))) | ||
| (ℤ/2-elim (PT.rec (isSetFin _ _) | ||
| (uncurry (ℤ/2-elim | ||
| (λ p → ⊥.rec (snotz (sym (cong fst p)))) | ||
| λ p → ⊥.rec (snotz (sym (cong fst p)))))) | ||
| λ _ → refl) | ||
| Iso.inv (fst ℤ/2/2≅ℤ/2) = [_] | ||
| Iso.rightInv (fst ℤ/2/2≅ℤ/2) _ = refl | ||
| Iso.leftInv (fst ℤ/2/2≅ℤ/2) = | ||
| SQ.elimProp (λ _ → squash/ _ _) λ _ → refl | ||
| snd ℤ/2/2≅ℤ/2 = makeIsGroupHom | ||
| (SQ.elimProp (λ _ → isPropΠ λ _ → isSetFin _ _) | ||
| λ a → SQ.elimProp (λ _ → isSetFin _ _) λ b → refl) | ||
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| ℤTorsion : (n : ℕ) → isContr (fst (ℤAbGroup [ (suc n) ]ₜ)) | ||
| fst (ℤTorsion n) = AbGroupStr.0g (snd (ℤAbGroup [ (suc n) ]ₜ)) | ||
| snd (ℤTorsion n) (a , p) = Σ≡Prop (λ _ → isSetℤ _ _) | ||
| (sym (help a (ℤ·≡· (pos (suc n)) a ∙ p))) | ||
| where | ||
| help : (a : ℤ) → a +ℤ pos n ·ℤ a ≡ 0 → a ≡ 0 | ||
| help (pos zero) p = refl | ||
| help (pos (suc a)) p = | ||
| ⊥.rec (snotz (injPos (pos+ (suc a) (n ·ℕ suc a) | ||
| ∙ cong (pos (suc a) +ℤ_) (pos·pos n (suc a)) ∙ p))) | ||
| help (negsuc a) p = ⊥.rec | ||
| (snotz (injPos (cong -ℤ_ (negsuc+ a (n ·ℕ suc a) | ||
| ∙ (cong (negsuc a +ℤ_) | ||
| (cong (-ℤ_) (pos·pos n (suc a))) | ||
| ∙ sym (cong (negsuc a +ℤ_) (pos·negsuc n a)) ∙ p))))) |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,12 @@ | ||
| {-# OPTIONS --safe #-} | ||
| module Cubical.Algebra.AbGroup.Instances.Pi where | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.HLevels | ||
| open import Cubical.Algebra.AbGroup | ||
| open import Cubical.Algebra.Group.Instances.Pi | ||
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| module _ {ℓ ℓ' : Level} {X : Type ℓ} (G : X → AbGroup ℓ') where | ||
| ΠAbGroup : AbGroup (ℓ-max ℓ ℓ') | ||
| ΠAbGroup = Group→AbGroup (ΠGroup (λ x → AbGroup→Group (G x))) | ||
| λ f g i x → IsAbGroup.+Comm (AbGroupStr.isAbGroup (G x .snd)) (f x) (g x) i |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,32 @@ | ||
| {-# OPTIONS --safe #-} | ||
| module Cubical.Algebra.Group.Instances.Pi where | ||
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| open import Cubical.Foundations.Prelude | ||
| open import Cubical.Foundations.HLevels | ||
| open import Cubical.Algebra.Group.Base | ||
| open import Cubical.Algebra.Semigroup | ||
| open import Cubical.Algebra.Monoid | ||
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| open IsGroup | ||
| open GroupStr | ||
| open IsMonoid | ||
| open IsSemigroup | ||
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| module _ {ℓ ℓ' : Level} {X : Type ℓ} (G : X → Group ℓ') where | ||
| ΠGroup : Group (ℓ-max ℓ ℓ') | ||
| fst ΠGroup = (x : X) → fst (G x) | ||
| 1g (snd ΠGroup) x = 1g (G x .snd) | ||
| GroupStr._·_ (snd ΠGroup) f g x = GroupStr._·_ (G x .snd) (f x) (g x) | ||
| inv (snd ΠGroup) f x = inv (G x .snd) (f x) | ||
| is-set (isSemigroup (isMonoid (isGroup (snd ΠGroup)))) = | ||
| isSetΠ λ x → is-set (G x .snd) | ||
| IsSemigroup.·Assoc (isSemigroup (isMonoid (isGroup (snd ΠGroup)))) f g h = | ||
| funExt λ x → IsSemigroup.·Assoc (isSemigroup (isMonoid (G x .snd))) (f x) (g x) (h x) | ||
| IsMonoid.·IdR (isMonoid (isGroup (snd ΠGroup))) f = | ||
| funExt λ x → IsMonoid.·IdR (isMonoid (isGroup (snd (G x)))) (f x) | ||
| IsMonoid.·IdL (isMonoid (isGroup (snd ΠGroup))) f = | ||
| funExt λ x → IsMonoid.·IdL (isMonoid (isGroup (snd (G x)))) (f x) | ||
| ·InvR (isGroup (snd ΠGroup)) f = | ||
| funExt λ x → ·InvR (isGroup (snd (G x))) (f x) | ||
| ·InvL (isGroup (snd ΠGroup)) f = | ||
| funExt λ x → ·InvL (isGroup (snd (G x))) (f x) |
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