Require Import Basics.Overture Basics.Tactics WildCat.Core AbelianGroup
  AbGroups.Z Spaces.Int Groups.QuotientGroup.
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(** * Cyclic groups *)

(** The [n]-th cyclic group is the cokernel of [ab_mul n]. *)
Definition cyclic (n : nat) : AbGroup
  := ab_cokernel (ab_mul (A:=abgroup_Z) n).

Definition cyclic_in (n : nat) : abgroup_Z $-> cyclic n
  := grp_quotient_map. 

Definition ab_mul_cyclic_in (n : nat) (x y : abgroup_Z)
  : ab_mul y (cyclic_in n x) = cyclic_in n (y * x)%int.
n: nat
x, y: abgroup_Z
ab_mul y (cyclic_in n x) = cyclic_in n (y * x)%int
Proof.
n: nat
x, y: abgroup_Z
ab_mul y (cyclic_in n x) = cyclic_in n (y * x)%int
  lhs_V napply ab_mul_natural.
n: nat
x, y: abgroup_Z
cyclic_in n (ab_mul y x) = cyclic_in n (y * x)%int
  apply ap, abgroup_Z_ab_mul.
Defined.