(* Concatenating two matrices, in either direction. *)

Fact row_mx_key : unit.Definition row_mx (A1 : 'M_(m, n1)) (A2 : 'M_(m, n2)) : 'M[R]_(m, n1 + n2) :=
  \matrix[row_mx_key]_(i, j)
     match split j with inl j1 => A1 i j1 | inr j2 => A2 i j2 end.Fact col_mx_key : unit.Definition col_mx (A1 : 'M_(m1, n)) (A2 : 'M_(m2, n)) : 'M[R]_(m1 + m2, n) :=
  \matrix[col_mx_key]_(i, j)
     match split i with inl i1 => A1 i1 j | inr i2 => A2 i2 j end.(* Left/Right | Up/Down submatrices of a rows | columns matrix.   *)
(* The shape of the (dependent) width parameters of the type of A *)
(* determines which submatrix is selected.                        *)

Fact lsubmx_key : unit.Definition lsubmx (A : 'M[R]_(m, n1 + n2)) :=
  \matrix[lsubmx_key]_(i, j) A i (lshift n2 j).Fact rsubmx_key : unit.Definition rsubmx (A : 'M[R]_(m, n1 + n2)) :=
  \matrix[rsubmx_key]_(i, j) A i (rshift n1 j).Fact usubmx_key : unit.Definition usubmx (A : 'M[R]_(m1 + m2, n)) :=
  \matrix[usubmx_key]_(i, j) A (lshift m2 i) j.Fact dsubmx_key : unit.Definition dsubmx (A : 'M[R]_(m1 + m2, n)) :=
  \matrix[dsubmx_key]_(i, j) A (rshift m1 i) j.

Lemma trmx_usub m1 m2 n (A : 'M_(m1 + m2, n)) : (usubmx A)^T = lsubmx A^T.