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_cat_nat n m p (P : pred nat) F : m <= n -> n <= p -> \big[*%M/1]_(m <= i < p | P i) F i = (\big[*%M/1]_(m <= i < n | P i) F i) * (\big[*%M/1]_(n <= i < p | P i) F i).