Section pos_def.
Variable r : G.
Fixpoint pos_to_group (p : positive) : G :=
match p with
| xH => r
| xO p' => group_square G (pos_to_group p')
| xI p' => sgroup_law G (group_square G (pos_to_group p')) r
end.

Lemma pos_nat_group :
forall p : positive, Equal (pos_to_group p) (nat_to_group r (nat_of_P p)).