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agdaproofs/Groups/Subgroups/Normal/Examples.agda
2020-01-05 15:06:35 +00:00

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Agda

{-# OPTIONS --safe --warning=error --without-K #-}
open import Groups.Definition
open import Setoids.Setoids
open import Sets.EquivalenceRelations
open import Groups.Subgroups.Normal.Definition
module Groups.Subgroups.Normal.Examples {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A A A} (G : Group S _+_) where
open import Groups.Subgroups.Examples G
open Setoid S
open Equivalence eq
open Group G
trivialSubgroupIsNormal : normalSubgroup G trivialSubgroup
trivialSubgroupIsNormal {g} k=0 = transitive (+WellDefined reflexive (transitive (+WellDefined k=0 reflexive) identLeft)) (invRight {g})
improperSubgroupIsNormal : normalSubgroup G improperSubgroup
improperSubgroupIsNormal _ = record {}