Singular

LIB "gradedModules.lib";
ring r=32003,(x,y,z),dp;
module A = grobj( module([x+y, x, 0, 3], [0, x+y, y, 2], [y, y, z, 1]), intvec(0,0,0,1) );
grview(A);
==> Graded homomorphism: r^3 + r(-1) <- r(-1)^3, given by a matrix, with degr\
   ees: 
==>      ..1 ..2 ..3 ....
==>      --- --- --- +...
==>   0 :  1   -   1 |..1
==>   0 :  1   1   1 |..2
==>   0 :  -   1   1 |..3
==>   1 :  0   0   0 |..4
==>      === === ===     
==>        1   1   1     
A = grgroebner(A);
grview(A);
==> Graded homomorphism: r^3 + r(-1) <- r(-1)^3 + r(-2) + r(-3), given by a m\
   atrix, with degrees: 
==>      ..1 ..2 ..3 ..4 ..5 ....
==>      --- --- --- --- --- +...
==>   0 :  1   1   1   2   - |..1
==>   0 :  1   -   1   -   - |..2
==>   0 :  1   1   1   2   3 |..3
==>   1 :  0   0   0   1   2 |..4
==>      === === === === ===     
==>        1   1   1   2   3     
module B = grsyz(A);
grview(B);
==> Graded homomorphism: r(-1)^3 + r(-2) + r(-3) <- r(-2) + r(-3), given by a\
    matrix, with degrees: 
==>      ..1 ..2 ....
==>      --- --- +...
==>   1 :  1   - |..1
==>   1 :  1   2 |..2
==>   1 :  1   - |..3
==>   2 :  0   1 |..4
==>   3 :  -   0 |..5
==>      === ===     
==>        2   3     
print(B);
==> x, 0,    
==> -y,y2,   
==> -y,0,    
==> 1, -x-2y,
==> 0, 1     
module D = grprod( A, B );
grview(D);
==> Graded homomorphism: r^3 + r(-1) <- r(-2) + r(-3), given by zero (4 x 2) \
   matrix.
print(D); // must be all zeroes due to syzygy property!
==> 0,0,
==> 0,0,
==> 0,0,
==> 0,0 
ASSUME(0, size(D) == 0);