Notation "\big [ op / idx ]_ ( i < n ) F" :=
  (\big[op/idx]_(i : ordinal n) F) : big_scope.Local Notation "x + y" := (plus x y).Notation "\max_ ( i 'in' A ) F" :=
 (\big[maxn/0%N]_(i in A) F%N) : nat_scope.

Lemma leq_sum I r (P : pred I) (E1 E2 : I -> nat) : (forall i, P i -> E1 i <= E2 i) -> \sum_(i <- r | P i) E1 i <= \sum_(i <- r | P i) E2 i.