proposition: Theorem lt_div_O : forall p q : nat, p < q -> div p q = 0.
proof: Proof. intros p q H. generalize (pdiv_def p q); unfold div in |- *; case (pdiv p q); simpl in |- *. intros q1 r1; case q1; simpl in |- *; auto with arith. intros n H0; (absurd (q <= p); auto with arith). case H0; auto with arith. apply Nat.le_lt_trans with (2 := H); auto with arith. intros H1 H2; rewrite H1; auto with arith. Qed.
imports: Coq(ArithRing) Bertrand(Misc)