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 exchange_big_nat m1 n1 m2 n2 (P Q : pred nat) F : \big[*%M/1]_(m1 <= i < n1 | P i) \big[*%M/1]_(m2 <= j < n2 | Q j) F i j = \big[*%M/1]_(m2 <= j < n2 | Q j) \big[*%M/1]_(m1 <= i < n1 | P i) F i j.