allGradings -- add Grading to a chainComplex

i1 : a=3,b=3

o1 = (3, 3)

o1 : Sequence

i2 : I=carpet(a,b);

                ZZ
o2 : Ideal of -----[x ..x , y ..y ]
              32003  0   3   0   3

i3 : F = res(I,FastNonminimal=>true,LengthLimit=>2);

i4 : betti F

            0  1  2
o4 = total: 1 10 25
         0: 1  .  .
         1: . 10 16
         2: .  .  9

o4 : BettiTally

i5 : degs=apply(a+1,i->{1,0,i})|apply(b+1,j->{0,1,j})

o5 = {{1, 0, 0}, {1, 0, 1}, {1, 0, 2}, {1, 0, 3}, {0, 1, 0}, {0, 1, 1}, {0,
     ------------------------------------------------------------------------
     1, 2}, {0, 1, 3}}

o5 : List

i6 : S=coefficientRing ring I[gens ring I,Degrees=>degs]

o6 = S

o6 : PolynomialRing

i7 : Fall = allGradings(F,S)

      1      10      25
o7 = S  <-- S   <-- S
                     
     0      1       2

o7 : ChainComplex

i8 : netList apply(length Fall+1,i->tally degrees Fall_i)

     +---------------------+
o8 = |Tally{{0, 0, 0} => 1}|
     +---------------------+
     |Tally{{0, 2, 2} => 1}|
     |      {0, 2, 3} => 1 |
     |      {0, 2, 4} => 1 |
     |      {1, 1, 2} => 1 |
     |      {1, 1, 3} => 2 |
     |      {1, 1, 4} => 1 |
     |      {2, 0, 2} => 1 |
     |      {2, 0, 3} => 1 |
     |      {2, 0, 4} => 1 |
     +---------------------+
     |Tally{{0, 3, 4} => 1}|
     |      {0, 3, 5} => 1 |
     |      {1, 2, 3} => 1 |
     |      {1, 2, 4} => 2 |
     |      {1, 2, 5} => 2 |
     |      {1, 2, 6} => 1 |
     |      {1, 3, 6} => 1 |
     |      {1, 3, 7} => 1 |
     |      {2, 1, 3} => 1 |
     |      {2, 1, 4} => 2 |
     |      {2, 1, 5} => 2 |
     |      {2, 1, 6} => 1 |
     |      {2, 2, 4} => 1 |
     |      {2, 2, 5} => 1 |
     |      {2, 2, 6} => 1 |
     |      {2, 2, 7} => 1 |
     |      {2, 2, 8} => 1 |
     |      {3, 0, 4} => 1 |
     |      {3, 0, 5} => 1 |
     |      {3, 1, 5} => 1 |
     |      {3, 1, 6} => 1 |
     +---------------------+