ite !inE groupM ?(@astab_dom _ _ _ to S) //; apply/subsetP=> x Sx.
by rewrite inE actMin ?(@astab_dom _ _ _ to S) ?(astab_act _ Sx).
Qed.ite !inE groupM ?(@astabs_dom _ _ _ to S) //; apply/subsetP=> x Sx.
by rewrite inE actMin ?(@astabs_dom _ _ _ to S) ?astabs_act.
Qed.

(* Specifications: *)
Implicit Types (a b : aT) (x y z : rT) (A B : {s
Lemma actM x a b : to x (a * b)
Lemma actK : right_loop invg to.
Lemma actX x a n : to x (a ^+ n) = iter n (to^~ a) x.
Lemma actCJ a b x : to (to x a) b = to (to x b) (a ^ b).