Variables (A : {set aT}) (S T : {set rT}).
Hypotheses (AactS : [acts A, on S | to]) (AactT : [acts A, on T | to]).

(* Specifications: *)
Lemma astabU : 'C(S :|: T | to) = 'C(S | to) :&: 'C(T | to).
Lemma astabsD : 'N(S | to) :&: 'N(T | to) \subset 'N(S :\: T| to).
Lemma subset_faithful A B S :
B \subset A -> [faithful A, on S | to] -> [faithful B, on S | to].
Lemma reindex_acts A a F : [acts A, on S | to] -> a \in A ->
\big[op/idx]_(i in S) F i = \big[op/idx]_(i in S) F (to i a).
Lemma actXin x a i : a \in D -> to x (a ^+ i) = iter i (to^~ a) x.
Lemma afix1 : 'Fix_to(1) = setT.
Lemma orbit_in_trans G : G \subset D -> transitive (orbit_rel G).
Lemma orbit_in_eqP G x y :
G \subset D -> reflect (orbit to G x = orbit to G y) (x \in orbit to G y).
Lemma orbit_in_transl G x y z :
G \subset D -> y \in orbit to G x ->
(y \in orbit to G z) = (x \in orbit to G z).
Lemma orbit_act_in x a G :
G \subset D -> a \in G -> orbit to G (to x a) = orbit to G x.
Lemma orbit_lcoset_in A a x :
A \subset D -> a \in D ->
orbit to (a *: A) x = orbit to A (to x a).
Lemma orbit_rcoset_in A a x y :
A \subset D -> a \in D ->
(to y a \in orbit to (A :* a) x) = (y \in orbit to A x).
Lemma orbit_conjsg_in A a x y :
A \subset D -> a \in D ->
(to y a \in orbit to (A :^ a) (to x a)) = (y \in orbit to A x).
Lemma orbit1P G x : reflect (orbit to G x = [set x]) (x \in 'Fix_to(G)).
Lemma card_orbit1 G x : #|orbit to G x| = 1%N -> orbit to G x = [set x].
Lemma orbit_partition G S :
[acts G, on S | to] -> partition (orbit to G @: S) S.