Polynomial ring (#76)

Rings/Polynomial/Ring.agda

@@ -0,0 +1,45 @@
+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Groups.Groups
+open import Groups.Abelian.Definition
+open import Groups.Homomorphisms.Definition
+open import Groups.Definition
+open import Numbers.Naturals.Definition
+open import Setoids.Orders
+open import Setoids.Setoids
+open import Functions
+open import Sets.EquivalenceRelations
+open import Rings.Definition
+open import Vectors
+open import Lists.Lists
+open import Maybe
+open import Rings.Homomorphisms.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+module Rings.Polynomial.Ring {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ _*_ : A → A → A} (R : Ring S _+_ _*_) where
+
+open Ring R
+open Setoid S
+open Equivalence eq
+open import Groups.Polynomials.Group additiveGroup
+open import Groups.Polynomials.Definition additiveGroup
+open import Rings.Polynomial.Definition R
+open import Rings.Polynomial.Multiplication R
+
+polyRing : Ring naivePolySetoid _+P_ _*P_
+Ring.additiveGroup polyRing = polyGroup
+Ring.*WellDefined polyRing {a} {b} {c} {d} = *PwellDefined {a} {b} {c} {d}
+Ring.1R polyRing = 1R :: []
+Ring.groupIsAbelian polyRing {x} {y} = AbelianGroup.commutative (abelian (record { commutative = Ring.groupIsAbelian R })) {x} {y}
+Ring.*Associative polyRing {a} {b} {c} = *Passoc {a} {b} {c}
+Ring.*Commutative polyRing {a} {b} = p*Commutative {a} {b}
+Ring.*DistributesOver+ polyRing {a} {b} {c} = *Pdistrib {a} {b} {c}
+Ring.identIsIdent polyRing {a} = *Pident {a}
+
+polyInjectionIsHom : RingHom R polyRing polyInjection
+RingHom.preserves1 polyInjectionIsHom = reflexive ,, record {}
+RingHom.ringHom polyInjectionIsHom = reflexive ,, record {}
+GroupHom.groupHom (RingHom.groupHom polyInjectionIsHom) = reflexive ,, record {}
+GroupHom.wellDefined (RingHom.groupHom polyInjectionIsHom) = SetoidInjection.wellDefined polyInjectionIsInj