open import Level
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)

module Algebra.Group where
  record IsGroup {ℓ} (A : Set ℓ) (_∙_ : A → A → A) (ε : A) (_⁻¹ : A → A) : Set (suc ℓ) where
    field
      assoc : ∀ {f g h : A} → (f ∙ g) ∙ h ≡ f ∙ (g ∙ h)
      idˡ : ∀ {g : A} → ε ∙ g ≡ g
      idʳ : ∀ {g : A} → g ∙ ε ≡ g
      invˡ : ∀ {g : A} → g ∙ (g ⁻¹) ≡ ε
      invʳ : ∀ {g : A} → (g ⁻¹) ∙ g ≡ ε

  record Group (ℓ : Level) : Set (suc ℓ) where
    infix 8 _⁻¹
    infixl 7 _∙_
    field
      Carrier : Set ℓ
      _∙_ : Carrier → Carrier → Carrier
      ε : Carrier
      _⁻¹ : Carrier → Carrier

    field
      isGroup : IsGroup Carrier _∙_ ε _⁻¹
    open IsGroup isGroup public

    ∙-resp : ∀ {f h g i : Carrier} → f ≡ h → g ≡ i → f ∙ g ≡ h ∙ i
    ∙-resp {f} {h} {g} {i} f≡h g≡i = ≡.trans (≡.cong (f ∙_) g≡i) (≡.cong (_∙ i) f≡h)