mirror of
https://github.com/Smaug123/agdaproofs
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45 lines
1.8 KiB
Agda
45 lines
1.8 KiB
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.Abelian.Definition
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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.Definition
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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 Vectors
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open import Lists.Lists
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open import Maybe
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open import Rings.Homomorphisms.Definition
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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module Rings.Polynomial.Ring {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
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open Ring R
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open Setoid S
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open Equivalence eq
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open import Groups.Polynomials.Group additiveGroup
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open import Groups.Polynomials.Definition additiveGroup
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open import Rings.Polynomial.Multiplication R
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polyRing : Ring naivePolySetoid _+P_ _*P_
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Ring.additiveGroup polyRing = polyGroup
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Ring.*WellDefined polyRing {a} {b} {c} {d} = *PwellDefined {a} {b} {c} {d}
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Ring.1R polyRing = 1R :: []
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Ring.groupIsAbelian polyRing {x} {y} = AbelianGroup.commutative (abelian (record { commutative = Ring.groupIsAbelian R })) {x} {y}
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Ring.*Associative polyRing {a} {b} {c} = *Passoc {a} {b} {c}
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Ring.*Commutative polyRing {a} {b} = p*Commutative {a} {b}
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Ring.*DistributesOver+ polyRing {a} {b} {c} = *Pdistrib {a} {b} {c}
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Ring.identIsIdent polyRing {a} = *Pident {a}
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polyInjectionIsHom : RingHom R polyRing polyInjection
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RingHom.preserves1 polyInjectionIsHom = reflexive ,, record {}
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RingHom.ringHom polyInjectionIsHom = reflexive ,, (reflexive ,, record {})
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GroupHom.groupHom (RingHom.groupHom polyInjectionIsHom) = reflexive ,, record {}
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GroupHom.wellDefined (RingHom.groupHom polyInjectionIsHom) = SetoidInjection.wellDefined polyInjectionIsInj
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