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(* 02110-1301 USA                                                     *)
Unset Standard Proposition Elimination Names.
Fixpoint rest (p r s n : nat) {struct n} : nat :=
match n with
| O => r
| S n' =>
match s with
| O => 0
| S O => rest p 0 p n'
| S s' => rest p (S r) s' n'
end
end.
Definition mod_ (m n : nat) : nat := rest n 0 n m.

Lemma mod_eq_0 : forall m : nat, mod_ 0 m = 0.
Proof with trivial.