{-# OPTIONS --safe #-}
module Cubical.Algebra.AbGroup.Properties where

open import Cubical.Foundations.Prelude

open import Cubical.Algebra.AbGroup.Base
open import Cubical.Algebra.Group
open import Cubical.Algebra.Group.Morphisms
open import Cubical.Algebra.Group.MorphismProperties
open import Cubical.Algebra.Group.QuotientGroup
open import Cubical.Algebra.Group.Subgroup
open import Cubical.Algebra.Group.ZAction

open import Cubical.HITs.SetQuotients as SQ hiding (_/_)

open import Cubical.Data.Int using (ℤ ; pos ; negsuc)
open import Cubical.Data.Nat hiding (_+_)
open import Cubical.Data.Sigma

private variable
  ℓ : Level

module AbGroupTheory (A : AbGroup ℓ) where
  open GroupTheory (AbGroup→Group A)
  open AbGroupStr (snd A)

  comm-4 : ∀ a b c d → (a + b) + (c + d) ≡ (a + c) + (b + d)
  comm-4 a b c d =
                 (a + b) + (c + d)  ≡⟨ +Assoc (a + b) c d ⟩
                 ((a + b) + c) + d  ≡⟨ cong (λ X → X + d) (sym (+Assoc a b c)) ⟩
                 (a + (b + c)) + d  ≡⟨ cong (λ X → (a + X) + d) (+Comm b c) ⟩
                 (a + (c + b)) + d  ≡⟨ cong (λ X → X + d) (+Assoc a c b) ⟩
                 ((a + c) + b) + d  ≡⟨ sym (+Assoc (a + c) b d) ⟩
                 (a + c) + (b + d)  ∎

  implicitInverse : ∀ {a b} → a + b ≡ 0g → b ≡ - a
  implicitInverse b+a≡0 = invUniqueR b+a≡0

addGroupHom : (A B : AbGroup ℓ) (ϕ ψ : AbGroupHom A B) → AbGroupHom A B
fst (addGroupHom A B ϕ ψ) x = AbGroupStr._+_ (snd B) (ϕ .fst x) (ψ .fst x)
snd (addGroupHom A B ϕ ψ) = makeIsGroupHom
  λ x y → cong₂ (AbGroupStr._+_ (snd B))
                 (IsGroupHom.pres· (snd ϕ) x y)
                 (IsGroupHom.pres· (snd ψ) x y)
        ∙ AbGroupTheory.comm-4 B (fst ϕ x) (fst ϕ y) (fst ψ x) (fst ψ y)

negGroupHom : (A B : AbGroup ℓ) (ϕ : AbGroupHom A B) → AbGroupHom A B
fst (negGroupHom A B ϕ) x = AbGroupStr.-_ (snd B) (ϕ .fst x)
snd (negGroupHom A B ϕ) =
  makeIsGroupHom λ x y
    → sym (IsGroupHom.presinv (snd ϕ) (AbGroupStr._+_ (snd A) x y))
     ∙ cong (fst ϕ) (GroupTheory.invDistr (AbGroup→Group A) x y
                  ∙ AbGroupStr.+Comm (snd A) _ _)
     ∙ IsGroupHom.pres· (snd ϕ) _ _
     ∙ cong₂ (AbGroupStr._+_ (snd B))
             (IsGroupHom.presinv (snd ϕ) x)
             (IsGroupHom.presinv (snd ϕ) y)

subtrGroupHom : (A B : AbGroup ℓ) (ϕ ψ : AbGroupHom A B) → AbGroupHom A B
subtrGroupHom A B ϕ ψ = addGroupHom A B ϕ (negGroupHom A B ψ)



-- ℤ-multiplication defines a homomorphism for abelian groups
private module _ (G : AbGroup ℓ) where
  ℤ·isHom-pos : (n : ℕ) (x y : fst G)
    → (pos n ℤ[ (AbGroup→Group G) ]· (AbGroupStr._+_ (snd G) x y))
     ≡ AbGroupStr._+_ (snd G) ((pos n) ℤ[ (AbGroup→Group G) ]· x)
                              ((pos n) ℤ[ (AbGroup→Group G) ]· y)
  ℤ·isHom-pos zero x y =
    sym (AbGroupStr.+IdR (snd G) (AbGroupStr.0g (snd G)))
  ℤ·isHom-pos (suc n) x y =
    cong₂ (AbGroupStr._+_ (snd G))
          refl
          (ℤ·isHom-pos n x y)
    ∙ AbGroupTheory.comm-4 G _ _ _ _

  ℤ·isHom-neg : (n : ℕ) (x y : fst G)
    → (negsuc n ℤ[ (AbGroup→Group G) ]· (AbGroupStr._+_ (snd G) x y))
     ≡ AbGroupStr._+_ (snd G) ((negsuc n) ℤ[ (AbGroup→Group G) ]· x)
                              ((negsuc n) ℤ[ (AbGroup→Group G) ]· y)
  ℤ·isHom-neg zero x y =
    GroupTheory.invDistr (AbGroup→Group G) _ _
    ∙ AbGroupStr.+Comm (snd G) _ _
  ℤ·isHom-neg (suc n) x y =
    cong₂ (AbGroupStr._+_ (snd G))
          (GroupTheory.invDistr (AbGroup→Group G) _ _
            ∙ AbGroupStr.+Comm (snd G) _ _)
          (ℤ·isHom-neg n x y)
    ∙ AbGroupTheory.comm-4 G _ _ _ _

ℤ·isHom : (n : ℤ) (G : AbGroup ℓ) (x y : fst G)
  → (n ℤ[ (AbGroup→Group G) ]· (AbGroupStr._+_ (snd G) x y))
   ≡ AbGroupStr._+_ (snd G) (n ℤ[ (AbGroup→Group G) ]· x)
                            (n ℤ[ (AbGroup→Group G) ]· y)
ℤ·isHom (pos n) G = ℤ·isHom-pos G n
ℤ·isHom (negsuc n) G = ℤ·isHom-neg G n

-- left multiplication as a group homomorphism
multₗHom : (G : AbGroup ℓ) (n : ℤ) → AbGroupHom G G
fst (multₗHom G n) g = n ℤ[ (AbGroup→Group G) ]· g
snd (multₗHom G n) = makeIsGroupHom (ℤ·isHom n G)

-- Abelian groups quotiented by a natural number
_/^_ : (G : AbGroup ℓ) (n : ℕ) → AbGroup ℓ
G /^ n =
  Group→AbGroup
    ((AbGroup→Group G)
     / (imSubgroup (multₗHom G (pos n))
      , isNormalIm (multₗHom G (pos n)) (AbGroupStr.+Comm (snd G))))
   (SQ.elimProp2 (λ _ _ → squash/ _ _)
     λ a b → cong [_] (AbGroupStr.+Comm (snd G) a b))

-- Torsion subgrous
_[_]ₜ : (G : AbGroup ℓ) (n : ℕ) → AbGroup ℓ
G [ n ]ₜ =
  Group→AbGroup (Subgroup→Group (AbGroup→Group G)
    (kerSubgroup (multₗHom G (pos n))))
    λ {(x , p) (y , q) → Σ≡Prop (λ _ → isPropIsInKer (multₗHom G (pos n)) _)
      (AbGroupStr.+Comm (snd G) _ _)}