Z is a Euclidean domain (#86)

Groups/Subgroups/Normal/Examples.agda

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+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import Groups.Groups
+open import Groups.Definition
+open import Orders
+open import Numbers.Integers.Integers
+open import Setoids.Setoids
+open import LogicalFormulae
+open import Sets.FinSet
+open import Functions
+open import Sets.EquivalenceRelations
+open import Numbers.Naturals.Naturals
+open import Groups.Homomorphisms.Definition
+open import Groups.Homomorphisms.Lemmas
+open import Groups.Isomorphisms.Definition
+open import Groups.Subgroups.Definition
+open import Groups.Lemmas
+open import Groups.Abelian.Definition
+open import Groups.QuotientGroup.Definition
+open import Groups.Subgroups.Normal.Definition
+
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+
+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 {}