Definition possible_image {X Y} (STEP:X-> Y-> X (*-> Y*) -> Prop) (valid: X -> Y -> Prop)  st st' U:= valid st U /\ (* exists U',  *) STEP st U st' (* U' *).Definition finite_on_x {X Y} (A:X->Y->Prop):=
    exists n (f: nat -> X), forall x y, A x y -> exists i, (i < n) /\ f i = x.

Lemma power_set_finite {X}: (forall P : Prop, P \/ ~ P) -> forall P: X-> Prop, finite P -> finite (fun A => subset A P).