Section TotalActions.
(* These lemmas are only established for total actions (domain = [set: rT]) *)
Variable (aT : finGroupType) (rT : finType).
Variable to : {action aT &-> rT}.
Implicit Types (a b : aT) (x y z : rT) (A B : {set aT}) (G H : {group aT}).
Implicit Type S : {set rT}.
Arguments actsP {A S}.

Lemma atrans_acts_card G S :
[transitive G, on S | to] =
[acts G, on S | to] && (#|orbit to G @: S| == 1%N).