Module Fin2Restrict.
Notation n2f := Fin.of_nat_lt.
Definition f2n {n} (x:Fin.t n) := proj1_sig (Fin.to_nat x).
Definition f2n_ok n (x:Fin.t n) : f2n x < n := proj2_sig (Fin.to_nat x).
Definition n2f_f2n : forall n x, n2f (f2n_ok x) = x := @Fin.of_nat_to_nat_inv.
Definition f2n_n2f x n h : f2n (n2f h) = x := f_equal (@proj1_sig _ _) (@Fin.to_nat_of_nat x n h).
Definition n2f_ext : forall x n h h', n2f h = n2f h' := @Fin.of_nat_ext.
Definition f2n_inj : forall n x y, f2n x = f2n y -> x = y := @Fin.to_nat_inj.
Definition extend n (f:Fin.t n -> Fin.t n) : (nat->nat) :=
fun x =>
match le_lt_dec n x with
| left _ => 0
| right h => f2n (f (n2f h))
end.
Definition restrict n (f:nat->nat)(hf : bFun n f) : (Fin.t n -> Fin.t n) :=
fun x => let (x',h) := Fin.to_nat x in n2f (hf _ h).
Ltac break_dec H :=
let H' := fresh "H" in
destruct le_lt_dec as [H'|H'];
[elim (Lt.le_not_lt _ _ H' H)
|try rewrite (n2f_ext H' H) in *; try clear H'].
End Fin2Restrict.
(** We can now use Proof via the equivalence ... *)

Lemma bSurjective_bBijective n (f:nat->nat) :
bFun n f -> bSurjective n f ->
exists g, bFun n g /\ forall x, x < n -> g (f x) = x /\ f (g x) = x.