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_choice : forall (A B:Type) (R:A->B->Prop), FiniteT A -> (forall x:A, exists y:B, R x y) -> exists f:A->B, forall x:A, R x (f x).