Z is a Euclidean domain (#86)

Groups/Subgroups/Examples.agda

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+{-# OPTIONS --safe --warning=error --without-K #-}
+
+open import LogicalFormulae
+open import Setoids.Setoids
+open import Sets.EquivalenceRelations
+open import Functions
+open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
+open import Numbers.Naturals.Naturals
+open import Sets.FinSet
+open import Groups.Definition
+
+module Groups.Subgroups.Examples {a b : _} {A : Set a} {S : Setoid {a} {b} A} {_+_ : A → A → A} (G : Group S _+_) where
+
+open import Groups.Subgroups.Definition G
+open import Groups.Lemmas G
+
+open Group G
+open Setoid S
+open Equivalence eq
+
+trivialSubgroupPred : A → Set b
+trivialSubgroupPred a = (a ∼ 0G)
+
+trivialSubgroup : Subgroup trivialSubgroupPred
+Subgroup.isSubset trivialSubgroup x=y x=0 = transitive (symmetric x=y) x=0
+Subgroup.closedUnderPlus trivialSubgroup x=0 y=0 = transitive (+WellDefined x=0 y=0) identLeft
+Subgroup.containsIdentity trivialSubgroup = reflexive
+Subgroup.closedUnderInverse trivialSubgroup x=0 = transitive (inverseWellDefined x=0) invIdent
+
+improperSubgroupPred : A → Set
+improperSubgroupPred a = True
+
+improperSubgroup : Subgroup improperSubgroupPred
+Subgroup.isSubset improperSubgroup _ _ = record {}
+Subgroup.closedUnderPlus improperSubgroup _ _ = record {}
+Subgroup.containsIdentity improperSubgroup = record {}
+Subgroup.closedUnderInverse improperSubgroup _ = record {}