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 bigmax_sup (I : finType) i0 (P : pred I) m F : P i0 -> m <= F i0 -> m <= \max_(i | P i) F i.