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https://github.com/Smaug123/agdaproofs
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26 lines
984 B
Agda
26 lines
984 B
Agda
{-# OPTIONS --safe --warning=error --without-K #-}
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open import LogicalFormulae
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open import Groups.Groups
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open import Groups.Homomorphisms.Definition
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open import Groups.Definition
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open import Numbers.Naturals.Naturals
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open import Setoids.Orders
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open import Setoids.Setoids
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open import Functions
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open import Sets.EquivalenceRelations
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open import Rings.Definition
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Rings.Homomorphisms.Definition where
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record RingHom {m n o p : _} {A : Set m} {B : Set n} {SA : Setoid {m} {o} A} {SB : Setoid {n} {p} B} {_+A_ : A → A → A} {_*A_ : A → A → A} (R : Ring SA _+A_ _*A_) {_+B_ : B → B → B} {_*B_ : B → B → B} (S : Ring SB _+B_ _*B_) (f : A → B) : Set (m ⊔ n ⊔ o ⊔ p) where
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open Ring S
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open Group additiveGroup
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open Setoid SB
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field
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preserves1 : f (Ring.1R R) ∼ 1R
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ringHom : {r s : A} → f (r *A s) ∼ (f r) *B (f s)
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groupHom : GroupHom (Ring.additiveGroup R) additiveGroup f
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