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.
Definition FiniteT_nat_cardinal (X:Type) (H:FiniteT X) : nat :=
proj1_sig (constructive_definite_description _
(FiniteT_has_nat_cardinal X H)).

Lemma FiniteT_nat_cardinal_option:
forall (X:Type) (H:FiniteT X),
FiniteT_nat_cardinal (option X) (add_finite X H) =
S (FiniteT_nat_cardinal X H).