proposition: Lemma sub_in_le_big [I : eqType] (s : seq I) (P P' : {pred I}) (F : I -> R) : {in s, forall i, P i -> P' i} -> le (\big[op/x]_(i <- s | P i) F i) (\big[op/x]_(i <- s | P' i) F i).
proof: Proof. move=> PP'; apply: sub_le_big_seq_cond => i; rewrite leq_count_subseq//. rewrite subseq_filter filter_subseq andbT; apply/allP => j. by rewrite !mem_filter => /andP[/PP'/[apply]->]. Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq path) mathcomp(div fintype tuple finfun)