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Irreducible and maximal (#87)

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Patrick Stevens
2019-12-07 18:53:08 +00:00
committed by GitHub
parent e192f0e1f1
commit 99c38495ce
9 changed files with 123 additions and 19 deletions
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7
Rings/Associates/Lemmas.agda
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@@ -22,6 +22,7 @@ open import Rings.Divisible.Definition R
open import Rings.IntegralDomains.Lemmas intDom
open import Rings.Associates.Definition intDom
open import Rings.Units.Definition R
open import Rings.Ideals.Definition R
open Setoid S
open Ring R
open Equivalence eq
@@ -42,3 +43,9 @@ mutualDivisionImpliesAssociate' {a} {b} (r , ar=b) (s , bs=a) a=0 = 1R , ((1R ,
where
b=0 : b ∼ 0R
b=0 = transitive (symmetric ar=b) (transitive (transitive *Commutative (*WellDefined reflexive a=0)) (timesZero {r}))
associateImpliesGeneratedIdealsEqual : {a b : A} → Associates a b → {x : A} → generatedIdealPred a x → generatedIdealPred b x
associateImpliesGeneratedIdealsEqual {a} {b} (r , ((s , rs=1) ,, a=br)) {x} (c , ac=x) = (r * c) , transitive *Associative (transitive (*WellDefined (symmetric a=br) reflexive) ac=x)
associateImpliesGeneratedIdealsEqual' : {a b : A} → Associates a b → {x : A} → generatedIdealPred b x → generatedIdealPred a x
associateImpliesGeneratedIdealsEqual' assoc = associateImpliesGeneratedIdealsEqual (Equivalence.symmetric associatesEquiv assoc)