mirror of
https://github.com/Smaug123/agdaproofs
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23 lines
836 B
Agda
23 lines
836 B
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Setoids.Setoids
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open import Functions
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Numbers.Naturals.Naturals
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open import Sets.FinSet
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module Groups.Definition where
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record Group {lvl1 lvl2} {A : Set lvl1} (S : Setoid {lvl1} {lvl2} A) (_·_ : A → A → A) : Set (lsuc lvl1 ⊔ lvl2) where
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open Setoid S
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field
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+WellDefined : {m n x y : A} → (m ∼ x) → (n ∼ y) → (m · n) ∼ (x · y)
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0G : A
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inverse : A → A
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+Associative : {a b c : A} → (a · (b · c)) ∼ (a · b) · c
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identRight : {a : A} → (a · 0G) ∼ a
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identLeft : {a : A} → (0G · a) ∼ a
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invLeft : {a : A} → (inverse a) · a ∼ 0G
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invRight : {a : A} → a · (inverse a) ∼ 0G
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