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https://github.com/Smaug123/agdaproofs
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21 lines
1.0 KiB
Agda
21 lines
1.0 KiB
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import Setoids.Setoids
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import Groups.Definition
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module Groups.Homomorphisms.Definition where
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record GroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} (G : Group S _·A_) (H : Group T _·B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
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open Group H
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open Setoid T
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field
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groupHom : {x y : A} → f (x ·A y) ∼ (f x) ·B (f y)
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wellDefined : {x y : A} → Setoid._∼_ S x y → f x ∼ f y
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record InjectiveGroupHom {m n o p : _} {A : Set m} {S : Setoid {m} {o} A} {_·A_ : A → A → A} {B : Set n} {T : Setoid {n} {p} B} {_·B_ : B → B → B} {G : Group S _·A_} {H : Group T _·B_} {underf : A → B} (f : GroupHom G H underf) : Set (m ⊔ n ⊔ o ⊔ p) where
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open Setoid S renaming (_∼_ to _∼A_)
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open Setoid T renaming (_∼_ to _∼B_)
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field
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injective : SetoidInjection S T underf
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