Section Binary.
Variable (m1 m2 n : nat) (S1 : mxsum_expr m1 n) (S2 : mxsum_expr m2 n).
Fact binary_mxsum_proof :
mxsum_spec (unwrap S1 + unwrap S2)
(unwrap (mxsum_rank S1) + unwrap (mxsum_rank S2)).
Canonical nary_mxsum_expr := ProperMxsumExpr nary_mxsum_proof.
End Nary.
Definition mxdirect_def m n T of phantom 'M_(m, n) (unwrap (mxsum_val T)) :=
\rank (unwrap T) == unwrap (mxsum_rank T).
End SumExpr.
Notation mxdirect A := (mxdirect_def (Phantom 'M_(_,_) A%MS)).

Lemma mxdirect_sumsP (A_ : I -> 'M_n) :
reflect (forall i, P i -> A_ i :&: (\sum_(j | P j && (j != i)) A_ j) = 0)%MS
(mxdirect (\sum_(i | P i) A_ i)).