Definition is_groupAction (to : actT) :=Definition is_groupAction (to : actT) :=Definition is_groupAction (to : actT) :=: {set aT}) (R : {set rT}).
Implicit Type A : {set aT}.
Implicit Type S : {set rT}.Definition gacent to A := 'Fix_(R | to)(D :&: A).

Definition acLemma actmM a : {in R &, {morph actm to a : x y(R : {group rT}) (to : groupAction D R).
Implicit Type A B : {set aT}./setIP=> Rx /afixP cAx /setIP[Ry /afixP cAy].
rewrite inE groupM //; apply/afixP=> a Aa.
by rewrite gactM ?cAx ?cAy //; case/setIP: Aa.Lemma ract_is_groupAction : is_groupAction R (to \ sAD).rewrite actpermE /= /actby (negbTE Gx).
apply/morphicP=> x y Gx Gy; rewrite !actpermE /= /actby Aa groupM ?Gx ?Gy //=.
by case nGAg; move/acts_dom; do 2!move/subsetP=> ?; rewrite gact

(* Specifications: *)
by rewrite -defHy; apply: mem_imset; apply: rcoset_refl.