Section CSB.
Variable X Y:Type.
Variable f:X->Y.
Variable g:Y->X.
Hypothesis f_inj: injective f.
Hypothesis g_inj: injective g.
Inductive X_even: X->Prop :=
| not_g_img: forall x:X, (forall y:Y, g y <> x) -> X_even x
| g_Y_odd: forall y:Y, Y_odd y -> X_even (g y)
with Y_odd: Y->Prop :=
| f_X_even: forall x:X, X_even x -> Y_odd (f x).
Inductive X_odd: X->Prop :=
| g_Y_even: forall y:Y, Y_even y -> X_odd (g y)
with Y_even: Y->Prop :=
| not_f_img: forall y:Y, (forall x:X, f x <> y) -> Y_even y
| f_X_odd: forall x:X, X_odd x -> Y_even (f x).
Scheme X_even_coind := Minimality for X_even Sort Prop
with Y_odd_coind := Minimality for Y_odd Sort Prop.
Scheme X_odd_coind := Minimality for X_odd Sort Prop
with Y_even_coind := Minimality for Y_even Sort Prop.
Definition finv: forall y:Y, (exists x:X, f x = y) ->
{ x:X | f x = y }.
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.