Ltac qif_monotonic_tac :=
  repeat (apply forall2_le_suml || apply forall2_le_maxl ||
          apply Plus.plus_le_compat_r || apply Plus.plus_le_compat_l || apply Plus.plus_le_compat ||
          apply Nat.max_le_compat_r || apply Nat.max_le_compat_l || apply Nat.max_le_compat ||
          apply Mult.mult_le_compat_l || apply Mult.mult_le_compat_r
         ); trivial.

Lemma qi_is_compatible : compatible_QI qic qif.