Inductive FiniteT : Type -> Prop :=
  | empty_finite: FiniteT False
  | add_finite: forall T:Type, FiniteT T -> FiniteT (option T)
  | bij_finite: forall (X Y:Type) (f:X->Y), FiniteT X ->
    invertible f -> FiniteT Y.Inductive FiniteT : Type -> Prop :=
  | empty_finite: FiniteT False
  | add_finite: forall T:Type, FiniteT T -> FiniteT (option T)
  | bij_finite: forall (X Y:Type) (f:X->Y), FiniteT X ->
    invertible f -> FiniteT Y.Inductive FiniteT : Type -> Prop :=
  | empty_finite: FiniteT False
  | add_finite: forall T:Type, FiniteT T -> FiniteT (option T)
  | bij_finite: forall (X Y:Type) (f:X->Y), FiniteT X ->
    invertible f -> FiniteT Y.Inductive FiniteT : Type -> Prop :=
  | empty_finite: FiniteT False
  | add_finite: forall T:Type, FiniteT T -> FiniteT (option T)
  | bij_finite: forall (X Y:Type) (f:X->Y), FiniteT X ->
    invertible f -> FiniteT Y.

Lemma surj_finite: forall (X Y:Type) (f:X->Y), FiniteT X -> surjective f -> (forall y1 y2:Y, y1=y2 \/ y1<>y2) -> FiniteT Y.