Definition phi (a : U) (n : nat) : U := exp (pos n) a.

(* Specifications: *)
Lemma phi_unfold : forall (a : U) (n : nat), In U G a -> phi a (S n) = star a (phi a n).
Lemma positive_powers : forall (a : U) (n : nat), In U H a -> In U H (phi a n).
Lemma tech_exp : forall (a : U) (n : nat), In U G a -> star (phi a n) a = phi a (S n).
Lemma tech_exp' : forall n : nat, phi e n = e.
Lemma phi_exp : forall (a : U) (n m : nat), In U G a -> star (phi a n) (phi a m) = phi a (S (n + m)).
Lemma powers_repeat : forall (a : U) (n m : nat), In U G a -> phi a n = phi a (S (S (n + m))) -> phi a m = inv a.