Require Export Ensembles.
Require Export Image.
Require Export Finite_sets.
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.
Definition FiniteT_nat_cardinal (X:Type) (H:FiniteT X) : nat :=
proj1_sig (constructive_definite_description _
(FiniteT_has_nat_cardinal X H)).

Lemma injection_preserves_cardinal: forall (X Y:Type)
(f:X->Y) (n:nat) (S:Ensemble X), cardinal _ S n ->
injective f -> cardinal _ (Im S f) n.