Section SubDsumsmx.
Variables (P : pred I) (m n : nat) (A : 'M[F]_(m, n)) (B : I -> 'M[F]_n).
Variant sub_dsumsmx_spec : Prop :=
SubDsumsmxSpec A_ of forall i, P i -> (A_ i <= B i)%MS
& A = \sum_(i | P i) A_ i
& forall C, (forall i, P i -> C i <= B i)%MS ->
A = \sum_(i | P i) C i -> {in SimplPred P, C =1 A_}.

Lemma sub_dsumsmx :
mxdirect (\sum_(i | P i) B i) -> (A <= \sum_(i | P i) B i)%MS ->
sub_dsumsmx_spec.