Definition Z_to_group_fun : ZZ -> G.

(* Specifications: *)
Lemma nat_to_group_inverse : forall (n : nat) (r : G), Equal (group_inverse G (nat_to_group r n)) (nat_to_group (group_inverse G r) n).
Lemma Z_to_group_fun_eq : forall z : ZZ, Equal (Z_to_group_fun z) (Z_to_group_nat r z).