{-# 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 Numbers.Integers.Integers
open import Numbers.Integers.Addition
open import Sets.FinSet
open import Groups.Homomorphisms.Definition
open import Groups.Groups
open import Groups.Lemmas
open import Groups.Subgroups.Definition
open import Groups.Abelian.Definition
open import Groups.Definition
open import Groups.Cyclic.Definition
open import Groups.Cyclic.DefinitionLemmas
open import Semirings.Definition

module Groups.Cyclic.Lemmas where

elementPowerHom : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_·_ : A → A → A} (G : Group S _·_) (x : A) → GroupHom ℤGroup G (λ i → elementPower G x i)
GroupHom.groupHom (elementPowerHom {S = S} G x) {a} {b} = symmetric (elementPowerCollapse G x a b)
  where
    open Equivalence (Setoid.eq S)
GroupHom.wellDefined (elementPowerHom {S = S} G x) {.y} {y} refl = reflexive
  where
    open Equivalence (Setoid.eq S)

subgroupOfCyclicIsCyclic : {a b c d : _} {A : Set a} {B : Set b} {S : Setoid {a} {c} A} {T : Setoid {b} {d} B} {_+A_ : A → A → A} {_+B_ : B → B → B} {G : Group S _+A_} {H : Group T _+B_} {f : B → A} {fHom : GroupHom H G f} → Subgroup G H fHom → CyclicGroup G → CyclicGroup H
CyclicGroup.generator (subgroupOfCyclicIsCyclic {f = f} subgrp record { generator = generator ; cyclic = cyclic }) = {!f generator!}
CyclicGroup.cyclic (subgroupOfCyclicIsCyclic subgrp gCyclic) = {!!}

-- Prefer to prove that subgroup of cyclic is cyclic, then deduce this like with abelian groups
{-
cyclicIsGroupProperty : {m n o p : _} {A : Set m} {B : Set o} {S : Setoid {m} {n} A} {T : Setoid {o} {p} B} {_+_ : A → A → A} {_*_ : B → B → B} {G : Group S _+_} {H : Group T _*_} → GroupsIsomorphic G H → CyclicGroup G → CyclicGroup H
CyclicGroup.generator (cyclicIsGroupProperty {H = H} iso G) = GroupsIsomorphic.isomorphism iso (CyclicGroup.generator G)
CyclicGroup.cyclic (cyclicIsGroupProperty {H = H} iso G) {a} with GroupIso.surj (GroupsIsomorphic.proof iso) {a}
CyclicGroup.cyclic (cyclicIsGroupProperty {H = H} iso G) {a} | a' , b with CyclicGroup.cyclic G {a'}
... | pow , prPow = pow , {!!}
-}

-- Proof of abelianness of cyclic groups: a cyclic group is the image of elementPowerHom into Z, so is isomorphic to a subgroup of Z. All subgroups of an abelian group are abelian.
cyclicGroupIsAbelian : {m n : _} {A : Set m} {S : Setoid {m} {n} A} {_+_ : A → A → A} {G : Group S _+_} (cyclic : CyclicGroup G) → AbelianGroup G
cyclicGroupIsAbelian cyclic = {!!}