Section RawAction.
(* Lemmas that do not require the group structure on the action domain. *)
(* Some lemmas like actMin would be actually be valid for arbitrary rT, *)
(* e.g., for actions on a function type, but would be difficult to use  *)
(* as a view due to the confusion between parameters and assumptions.   *)
Variables (aT : finGroupType) (D : {set aT}) (rT : finType) (to : action D rT).
Implicit Types (a : aT) (x y : rT) (A B : {set aT}) (S T : {set rT}).
Arguments act_inj : clear implicits.
Canonical set_action := Action setact_is_action.

Lemma orbitP A x y :
reflect (exists2 a, a \in A & to x a = y) (y \in orbit to A x).