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 finite_dec_exists: forall (X:Type) (P:X->Prop), FiniteT X -> (forall x:X, {P x} + {~ P x}) -> { exists x:X, P x } + { forall x:X, ~ P x }.