(** Instead of working on a finite subset of nat, another
    solution is to use restricted [nat->nat] functions, and
    to consider them only below a certain bound [n]. *)

Definition bFun n (f:nat->nat) := forall x, x < n -> f x < n.Definition bInjective n (f:nat->nat) :=
 forall x y, x < n -> y < n -> f x = f y -> x = y.Definition bSurjective n (f:nat->nat) :=
 forall y, y < n -> exists x, x < n /\ f x = y.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 (proj1 (Nat.le_ngt _ _) H' H)
  |try rewrite (n2f_ext H' H) in *; try clear H'].

(* Specifications: *)
Lemma extend_ok n f : bFun n (@extend n f).
Lemma extend_f2n n f (x:Fin.t n) : extend f (f2n x) = f2n (f x).
Lemma extend_n2f n f x (h:x<n) : n2f (extend_ok f h) = f (n2f h).
Lemma restrict_f2n n f hf (x:Fin.t n) : f2n (@restrict n f hf x) = f (f2n x).
Lemma restrict_n2f n f hf x (h:x<n) : @restrict n f hf (n2f h) = n2f (hf _ h).
Lemma extend_surjective n f : bSurjective n (@extend n f) <-> Surjective f.
Lemma extend_injective n f : bInjective n (@extend n f) <-> Injective f.
Lemma restrict_surjective n f h : Surjective (@restrict n f h) <-> bSurjective n f.
Lemma restrict_injective n f h : Injective (@restrict n f h) <-> bInjective n f.
Lemma bInjective_bSurjective n (f:nat->nat) : bFun n f -> (bInjective n f <-> bSurjective n f).
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.