mirror of
https://github.com/Smaug123/agdaproofs
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40 lines
1.7 KiB
Agda
40 lines
1.7 KiB
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 Groups.Groups
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open import Groups.Definition
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open import Sets.EquivalenceRelations
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module Groups.Lemmas where
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invInv : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → {x : A} → Setoid._∼_ S (Group.inverse G (Group.inverse G x)) x
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invInv {S = S} G {x} = symmetric (transferToRight' G invRight)
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where
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open Setoid S
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open Group G
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open Equivalence eq
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invIdent : {a b : _} → {A : Set a} → {_·_ : A → A → A} → {S : Setoid {a} {b} A} → (G : Group S _·_) → Setoid._∼_ S (Group.inverse G (Group.0G G)) (Group.0G G)
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invIdent {S = S} G = symmetric (transferToRight' G (Group.identLeft G))
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where
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open Setoid S
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open Group G
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open Equivalence eq
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swapInv : {a b : _} {A : Set a} {_+_ : A → A → A} {S : Setoid {a} {b} A} (G : Group S _+_) → {x y : A} → Setoid._∼_ S (Group.inverse G x) y → Setoid._∼_ S x (Group.inverse G y)
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swapInv {S = S} G {x} {y} -x=y = transitive (symmetric (invInv G)) (inverseWellDefined G -x=y)
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where
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open Setoid S
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open Group G
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open Equivalence eq
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identityIsUnique : {a b : _} → {A : Set a} → {S : Setoid {a} {b} A} → {_·_ : A → A → A} → (G : Group S _·_) → (e : A) → ((b : A) → (Setoid._∼_ S (b · e) b)) → (Setoid._∼_ S e (Group.0G G))
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identityIsUnique {S = S} {_·_} g thing fb = transitive (symmetric identLeft) (fb 0G)
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where
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open Group g renaming (inverse to _^-1)
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open Setoid S
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open Equivalence eq
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