Definition Qmax p q := if Q_le_lt_dec p q then q else p.Definition Qmin p q := if Q_le_lt_dec p q then p else q.

Lemma Qmin_nondecreasing: forall q1 q2 p1 p2, q1 <= p1 -> q2 <= p2 -> Qmin q1 q2 <= Qmin p1 p2.