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 Qmax_nondecreasing: forall q1 q2 p1 p2, q1 <= p1 -> q2 <= p2 -> Qmax q1 q2 <= Qmax p1 p2.