intros.
apply constructive_definite_description.
destruct H.
exists x.
red; split.
assumption.
intros.
apply f_inj.
transitivity y; trivial.
symmetry; trivial.
Defined.
Definition ginv: forall x:X, (exists y:Y, g y = x) ->
{ y:Y | g y = x }.
intros.
apply constructive_definite_description.
destruct H.
exists x0.
red; split.
assumption.
intros.
apply g_inj.
transitivity x; trivial; symmetry; trivial.
Defined.
Definition ginv_odd: forall x:X, X_odd x ->
{ y:Y | g y = x }.
intros.
apply ginv.
destruct H.
exists y.
reflexivity.
Defined.
Definition finv_noteven: forall y:Y, ~ Y_even y ->
{ x:X | f x = y }.
intros.
apply finv.
apply NNPP.
red; intro.
contradict H.
constructor 1.
intro; red; intro.
apply H0.
exists x.
assumption.
Defined.
Definition CSB_bijection (x:X) : Y :=
match (classic_dec (X_odd x)) with
| left o => proj1_sig (ginv_odd x o)
| right _ => f x
end.
Definition CSB_bijection2 (y:Y) : X :=
match (classic_dec (Y_even y)) with
| left _ => g y
| right ne => proj1_sig (finv_noteven y ne)
end.

Lemma CSB_comp1: forall x:X, CSB_bijection2 (CSB_bijection x) = x.