From Coq Require Export Ensembles.
From ZornsLemma Require Export Image.
From Coq Require Export Finite_sets.
From ZornsLemma Require Export FunctionProperties.
Set Asymmetric Patterns.
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 inj_finite: forall (X Y:Type) (f:X->Y),
FiniteT Y -> injective f ->
(forall y:Y, (exists x:X, f x = y) \/
(~ exists x:X, f x = y)) ->
FiniteT X.