Variables (A : {set aT}) (sAD : A \subset D).

(* Specifications: *)
Lemma ractpermE : actperm (to \ sAD) =1 actperm to.
Canonical morph_action := Action mact_is_action.
Lemma mactE x a : morph_action x a = phi a x. Proof. by []. Qed.
Lemma injm_faithful : 'injm phi -> [faithful D, on setT | morph_action].
Lemma perm_mact a : actperm morph_action a = phi a.
Canonical comp_action := Action comp_is_action.
Lemma comp_actE x e : comp_action x e = to x (f e). Proof. by []. Qed.
Lemma afix_comp (A : {set gT}) :
A \subset B -> 'Fix_comp_action(A) = 'Fix_to(f @* A).
Lemma astab_comp S : 'C(S | comp_action) = f @*^-1 'C(S | to).
Canonical perm_action := Action aperm_is_action.
Lemma pcycleE a : pcycle a = orbit perm_action <[a]>%g.
Lemma perm_faithful A : [faithful A, on setT | perm_action].
Lemma actperm_id p : actperm perm_action p = p.
Lemma pcycle_actperm (a : aT) :
a \in D -> pcycle (actperm to a) =1 orbit to <[a]>.