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 big_nat_widen m n1 n2 (P : pred nat) F : n1 <= n2 -> \big[op/idx]_(m <= i < n1 | P i) F i = \big[op/idx]_(m <= i < n2 | P i && (i < n1)) F i.