patrick/agdaproofs
1
0
Fork 0
mirror of https://github.com/Smaug123/agdaproofs synced 2025-10-19 10:08:42 +00:00
Code Issues Packages Projects Releases Wiki Activity

Quotient ring (#81)

Browse Source
This commit is contained in:
Patrick Stevens
2019-11-21 07:28:25 +00:00
committed by GitHub
parent d4f51a3efe
commit b33baa5fb7
12 changed files with 125 additions and 64 deletions
Download Patch File Download Diff File Expand all files Collapse all files

10
Groups/Homomorphisms/Kernel.agda
View File

@@ -34,11 +34,11 @@ groupKernelPred a = Setoid._∼_ T (f a) (Group.0G H)
groupKernelPredWd : {x y : A} → (Setoid._∼_ S x y) → groupKernelPred x → groupKernelPred y
groupKernelPredWd x=y fx=0 = transitive (GroupHom.wellDefined fHom (Equivalence.symmetric (Setoid.eq S) x=y)) fx=0
groupKernelIsSubgroup : subgroup G groupKernelPred
_&_&_.one (_&&_.snd groupKernelIsSubgroup) fg=0 fh=0 = transitive (transitive (GroupHom.groupHom fHom) (Group.+WellDefined H fg=0 fh=0)) (Group.identLeft H)
_&_&_.two (_&&_.snd groupKernelIsSubgroup) = imageOfIdentityIsIdentity fHom
_&_&_.three (_&&_.snd groupKernelIsSubgroup) fg=0 = transitive (homRespectsInverse fHom) (transitive (inverseWellDefined H fg=0) (invIdent H))
_&&_.fst groupKernelIsSubgroup = groupKernelPredWd
groupKernelIsSubgroup : Subgroup G groupKernelPred
Subgroup.closedUnderPlus groupKernelIsSubgroup fg=0 fh=0 = transitive (transitive (GroupHom.groupHom fHom) (Group.+WellDefined H fg=0 fh=0)) (Group.identLeft H)
Subgroup.containsIdentity groupKernelIsSubgroup = imageOfIdentityIsIdentity fHom
Subgroup.closedUnderInverse groupKernelIsSubgroup fg=0 = transitive (homRespectsInverse fHom) (transitive (inverseWellDefined H fg=0) (invIdent H))
Subgroup.isSubset groupKernelIsSubgroup = groupKernelPredWd
groupKernelIsNormalSubgroup : normalSubgroup G groupKernelIsSubgroup
groupKernelIsNormalSubgroup {g} fk=0 = transitive (transitive (transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H reflexive (transitive (GroupHom.groupHom fHom) (transitive (Group.+WellDefined H fk=0 reflexive) (Group.identLeft H)))) (symmetric (GroupHom.groupHom fHom)))) (GroupHom.wellDefined fHom (Group.invRight G {g}))) (imageOfIdentityIsIdentity fHom)