A codimension 5 example with details -- Constructing a minimal resolution for a codimension 5 cyclic polytope detailed.

We give all the details of the unprojection construction of a codim 5 example.

Δ(2,7):

i1 : R=QQ[x_1..x_7];

i2 : cc=cycRes(2,R,verbose=>2);

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

delta(2,{x_1, x_2, x_3, x_4, x_5})   +   delta(0,{z, x_2, x_3, x_4})   ->   delta(2,{x_1, x_2, x_3, x_4, x_5, x_6})

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res(I): 

a_1 = | x_2x_4 x_3x_5 x_1x_4 x_2x_5 x_1x_3 |

a_2 = {2} | 0    x_1  0    0    -x_5 |
      {2} | -x_1 0    x_2  0    0    |
      {2} | 0    -x_2 0    x_3  0    |
      {2} | 0    0    -x_3 0    x_4  |
      {2} | x_5  0    0    -x_4 0    |

a_3 = {3} | x_2x_4 |
      {3} | x_3x_5 |
      {3} | x_1x_4 |
      {3} | x_2x_5 |
      {3} | x_1x_3 |


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res(J): 

b_1 = | z x_3 x_4 x_2 |

b_2 = {1} | x_3 x_4 x_2 0    0    0    |
      {1} | -z  0   0   0    x_2  -x_4 |
      {1} | 0   -z  0   -x_2 0    x_3  |
      {1} | 0   0   -z  x_4  -x_3 0    |

b_3 = {2} | 0    x_2  -x_4 0   |
      {2} | -x_2 0    x_3  0   |
      {2} | x_4  -x_3 0    0   |
      {2} | z    0    0    x_3 |
      {2} | 0    z    0    x_4 |
      {2} | 0    0    z    x_2 |

b_4 = {3} | x_3 |
      {3} | x_4 |
      {3} | x_2 |
      {3} | -z  |

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phi: {z, x_3, x_4, x_2} -> {x_1*x_5, 0, 0, 0}

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alpha_1 = {1} | 0   0   0   0   0   |
          {1} | 0   x_5 0   0   x_1 |
          {1} | 0   0   x_1 0   0   |
          {1} | x_4 0   0   x_5 0   |

alpha_2 = {2} | 0 0   0   0   0 |
          {2} | 0 0   0   0   0 |
          {2} | 0 0   0   0   0 |
          {2} | 0 x_1 0   0   0 |
          {2} | 0 0   x_5 0   0 |
          {2} | 0 0   0   x_1 0 |

alpha_3 = {3} | 0      |
          {3} | 0      |
          {3} | 0      |
          {3} | x_1x_5 |


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beta_1 = | -x_1x_5 0 0 0 |

beta_2 = {2} | 0    0    0    0 0 0 |
         {2} | -x_1 0    0    0 0 0 |
         {2} | 0    -x_5 0    0 0 0 |
         {2} | 0    0    -x_1 0 0 0 |
         {2} | 0    0    0    0 0 0 |

beta_3 = {3} | 0    0    -x_4 0 |
         {3} | -x_5 0    0    0 |
         {3} | 0    -x_1 0    0 |
         {3} | 0    0    -x_5 0 |
         {3} | -x_1 0    0    0 |


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u = -1

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h_1 = 0

h_2 = 0

h_3 = 0


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f_1 = | x_2x_4 x_3x_5 x_1x_4 x_2x_5 x_1x_3 x_6z-x_1x_5 x_6x_3 x_6x_4 x_6x_2 |

f_2 = | 0    x_1  0    0    -x_5 0    0    0    0    0    0    x_6  0    0    0    0    |
      | -x_1 0    x_2  0    0    -x_1 0    0    0    0    0    0    x_6  0    0    0    |
      | 0    -x_2 0    x_3  0    0    -x_5 0    0    0    0    0    0    x_6  0    0    |
      | 0    0    -x_3 0    x_4  0    0    -x_1 0    0    0    0    0    0    x_6  0    |
      | x_5  0    0    -x_4 0    0    0    0    0    0    0    0    0    0    0    x_6  |
      | 0    0    0    0    0    -x_3 -x_4 -x_2 0    0    0    0    0    0    0    0    |
      | 0    0    0    0    0    z    0    0    0    -x_2 x_4  0    -x_5 0    0    -x_1 |
      | 0    0    0    0    0    0    z    0    x_2  0    -x_3 0    0    -x_1 0    0    |
      | 0    0    0    0    0    0    0    z    -x_4 x_3  0    -x_4 0    0    -x_5 0    |

f_3 = | 0    0    -x_4 0    -x_6 0    0    0    0    |
      | -x_5 0    0    0    0    -x_6 0    0    0    |
      | 0    -x_1 0    0    0    0    -x_6 0    0    |
      | 0    0    -x_5 0    0    0    0    -x_6 0    |
      | -x_1 0    0    0    0    0    0    0    -x_6 |
      | 0    -x_2 x_4  0    0    0    0    0    0    |
      | x_2  0    -x_3 0    0    0    0    0    0    |
      | -x_4 x_3  0    0    0    0    0    0    0    |
      | -z   0    0    -x_3 0    -x_1 0    0    0    |
      | 0    -z   0    -x_4 0    0    -x_5 0    0    |
      | 0    0    -z   -x_2 0    0    0    -x_1 0    |
      | 0    0    0    0    0    x_1  0    0    -x_5 |
      | 0    0    0    0    -x_1 0    x_2  0    0    |
      | 0    0    0    0    0    -x_2 0    x_3  0    |
      | 0    0    0    0    0    0    -x_3 0    x_4  |
      | 0    0    0    0    x_5  0    0    -x_4 0    |

f_4 = | -x_6x_3     |
      | -x_6x_4     |
      | -x_6x_2     |
      | x_6z-x_1x_5 |
      | x_2x_4      |
      | x_3x_5      |
      | x_1x_4      |
      | x_2x_5      |
      | x_1x_3      |


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++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

delta(2,{x_1, x_2, x_3, x_4, x_5, x_6})   +   delta(0,{z, x_2, x_3, x_4, x_5})   ->   delta(2,{x_1, x_2, x_3, x_4, x_5, x_6, x_7})

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res(I): 

a_1 = | x_2x_4 x_3x_5 x_1x_4 x_2x_5 x_1x_3 -x_1x_5 x_3x_6 x_4x_6 x_2x_6 |

a_2 = {2} | 0    x_1  0    0    -x_5 0    0    0    0    0    0    x_6  0    0    0    0    |
      {2} | -x_1 0    x_2  0    0    -x_1 0    0    0    0    0    0    x_6  0    0    0    |
      {2} | 0    -x_2 0    x_3  0    0    -x_5 0    0    0    0    0    0    x_6  0    0    |
      {2} | 0    0    -x_3 0    x_4  0    0    -x_1 0    0    0    0    0    0    x_6  0    |
      {2} | x_5  0    0    -x_4 0    0    0    0    0    0    0    0    0    0    0    x_6  |
      {2} | 0    0    0    0    0    -x_3 -x_4 -x_2 0    0    0    0    0    0    0    0    |
      {2} | 0    0    0    0    0    0    0    0    0    -x_2 x_4  0    -x_5 0    0    -x_1 |
      {2} | 0    0    0    0    0    0    0    0    x_2  0    -x_3 0    0    -x_1 0    0    |
      {2} | 0    0    0    0    0    0    0    0    -x_4 x_3  0    -x_4 0    0    -x_5 0    |

a_3 = {3} | 0    0    -x_4 0    -x_6 0    0    0    0    |
      {3} | -x_5 0    0    0    0    -x_6 0    0    0    |
      {3} | 0    -x_1 0    0    0    0    -x_6 0    0    |
      {3} | 0    0    -x_5 0    0    0    0    -x_6 0    |
      {3} | -x_1 0    0    0    0    0    0    0    -x_6 |
      {3} | 0    -x_2 x_4  0    0    0    0    0    0    |
      {3} | x_2  0    -x_3 0    0    0    0    0    0    |
      {3} | -x_4 x_3  0    0    0    0    0    0    0    |
      {3} | 0    0    0    -x_3 0    -x_1 0    0    0    |
      {3} | 0    0    0    -x_4 0    0    -x_5 0    0    |
      {3} | 0    0    0    -x_2 0    0    0    -x_1 0    |
      {3} | 0    0    0    0    0    x_1  0    0    -x_5 |
      {3} | 0    0    0    0    -x_1 0    x_2  0    0    |
      {3} | 0    0    0    0    0    -x_2 0    x_3  0    |
      {3} | 0    0    0    0    0    0    -x_3 0    x_4  |
      {3} | 0    0    0    0    x_5  0    0    -x_4 0    |

a_4 = {4} | -x_3x_6 |
      {4} | -x_4x_6 |
      {4} | -x_2x_6 |
      {4} | -x_1x_5 |
      {4} | x_2x_4  |
      {4} | x_3x_5  |
      {4} | x_1x_4  |
      {4} | x_2x_5  |
      {4} | x_1x_3  |


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res(J): 

b_1 = | z x_2 x_4 x_5 x_3 |

b_2 = {1} | x_2 x_4 x_5 x_3 0    0    0    0    0    0    |
      {1} | -z  0   0   0   x_4  x_5  x_3  0    0    0    |
      {1} | 0   -z  0   0   -x_2 0    0    0    x_3  -x_5 |
      {1} | 0   0   -z  0   0    -x_2 0    -x_3 0    x_4  |
      {1} | 0   0   0   -z  0    0    -x_2 x_5  -x_4 0    |

b_3 = {2} | x_4  x_5  x_3  0    0    0    0    0    0    0   |
      {2} | -x_2 0    0    0    x_3  -x_5 0    0    0    0   |
      {2} | 0    -x_2 0    -x_3 0    x_4  0    0    0    0   |
      {2} | 0    0    -x_2 x_5  -x_4 0    0    0    0    0   |
      {2} | z    0    0    0    0    0    0    x_3  -x_5 0   |
      {2} | 0    z    0    0    0    0    -x_3 0    x_4  0   |
      {2} | 0    0    z    0    0    0    x_5  -x_4 0    0   |
      {2} | 0    0    0    z    0    0    x_2  0    0    x_4 |
      {2} | 0    0    0    0    z    0    0    x_2  0    x_5 |
      {2} | 0    0    0    0    0    z    0    0    x_2  x_3 |

b_4 = {3} | 0    x_3  -x_5 0   0    |
      {3} | -x_3 0    x_4  0   0    |
      {3} | x_5  -x_4 0    0   0    |
      {3} | x_2  0    0    x_4 0    |
      {3} | 0    x_2  0    x_5 0    |
      {3} | 0    0    x_2  x_3 0    |
      {3} | -z   0    0    0   x_4  |
      {3} | 0    -z   0    0   x_5  |
      {3} | 0    0    -z   0   x_3  |
      {3} | 0    0    0    -z  -x_2 |

b_5 = {4} | x_4  |
      {4} | x_5  |
      {4} | x_3  |
      {4} | -x_2 |
      {4} | z    |

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phi: {z, x_2, x_4, x_5, x_3} -> {x_1*x_6, 0, 0, 0, 0}

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alpha_1 = {1} | 0   0   0   0   0   0    0   0   0   |
          {1} | x_4 0   0   x_5 0   0    0   0   x_6 |
          {1} | 0   0   x_1 0   0   0    0   x_6 0   |
          {1} | 0   0   0   0   0   -x_1 0   0   0   |
          {1} | 0   x_5 0   0   x_1 0    x_6 0   0   |

alpha_2 = {2} | 0 0   0    0   0 0    0   0    0    0   0    0 0 0 0 0 |
          {2} | 0 0   0    0   0 0    0   0    0    0   0    0 0 0 0 0 |
          {2} | 0 0   0    0   0 0    0   0    0    0   0    0 0 0 0 0 |
          {2} | 0 0   0    0   0 0    0   0    0    0   0    0 0 0 0 0 |
          {2} | 0 x_1 0    0   0 0    0   0    -x_6 0   0    0 0 0 0 0 |
          {2} | 0 0   0    0   0 0    0   -x_1 0    0   0    0 0 0 0 0 |
          {2} | 0 0   -x_5 0   0 0    0   0    0    x_6 0    0 0 0 0 0 |
          {2} | 0 0   0    0   0 -x_1 0   0    0    0   0    0 0 0 0 0 |
          {2} | 0 0   0    x_1 0 0    0   0    0    0   -x_6 0 0 0 0 0 |
          {2} | 0 0   0    0   0 0    x_1 0    0    0   0    0 0 0 0 0 |

alpha_3 = {3} | 0   0   0    0   0 0 0 0 0 |
          {3} | 0   0   0    0   0 0 0 0 0 |
          {3} | 0   0   0    0   0 0 0 0 0 |
          {3} | 0   0   0    0   0 0 0 0 0 |
          {3} | 0   0   0    0   0 0 0 0 0 |
          {3} | 0   0   0    0   0 0 0 0 0 |
          {3} | 0   x_1 0    0   0 0 0 0 0 |
          {3} | 0   0   0    x_6 0 0 0 0 0 |
          {3} | x_1 0   0    0   0 0 0 0 0 |
          {3} | 0   0   -x_1 0   0 0 0 0 0 |

alpha_4 = {4} | 0       |
          {4} | 0       |
          {4} | 0       |
          {4} | 0       |
          {4} | -x_1x_6 |


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beta_1 = | -x_1x_6 0 0 0 0 |

beta_2 = {2} | 0    0    0   0    0 0 0 0 0 0 |
         {2} | 0    0    0   0    0 0 0 0 0 0 |
         {2} | 0    0    0   0    0 0 0 0 0 0 |
         {2} | 0    0    0   0    0 0 0 0 0 0 |
         {2} | 0    0    0   0    0 0 0 0 0 0 |
         {2} | 0    0    x_6 0    0 0 0 0 0 0 |
         {2} | 0    0    0   -x_1 0 0 0 0 0 0 |
         {2} | 0    -x_1 0   0    0 0 0 0 0 0 |
         {2} | -x_1 0    0   0    0 0 0 0 0 0 |

beta_3 = {3} | 0   0   0    0   0   0    0 0 0 0 |
         {3} | 0   0   0    0   0   0    0 0 0 0 |
         {3} | 0   0   0    0   0   0    0 0 0 0 |
         {3} | 0   0   0    0   0   0    0 0 0 0 |
         {3} | 0   0   0    0   0   0    0 0 0 0 |
         {3} | 0   0   0    x_6 0   0    0 0 0 0 |
         {3} | 0   0   0    0   0   -x_6 0 0 0 0 |
         {3} | 0   x_6 0    0   0   0    0 0 0 0 |
         {3} | x_1 0   0    0   0   0    0 0 0 0 |
         {3} | 0   0   -x_1 0   0   0    0 0 0 0 |
         {3} | 0   0   0    0   x_1 0    0 0 0 0 |
         {3} | 0   0   0    0   0   0    0 0 0 0 |
         {3} | 0   0   0    x_1 0   0    0 0 0 0 |
         {3} | 0   0   0    0   0   -x_5 0 0 0 0 |
         {3} | 0   x_1 0    0   0   0    0 0 0 0 |
         {3} | 0   0   0    0   0   0    0 0 0 0 |

beta_4 = {4} | 0    0    -x_6 0    0 |
         {4} | -x_6 0    0    0    0 |
         {4} | 0    0    0    x_6  0 |
         {4} | 0    -x_1 0    0    0 |
         {4} | 0    0    0    -x_4 0 |
         {4} | 0    0    x_5  0    0 |
         {4} | x_1  0    0    0    0 |
         {4} | 0    0    0    -x_5 0 |
         {4} | 0    0    x_1  0    0 |


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u = 1

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h_1 = 0

h_2 = 0

h_3 = 0

h_4 = 0


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f_1 = | x_2x_4 x_3x_5 x_1x_4 x_2x_5 x_1x_3 -x_1x_5 x_3x_6 x_4x_6 x_2x_6 x_7z-x_1x_6 x_7x_2 x_7x_4 x_7x_5 x_7x_3 |

f_2 = | 0    x_1  0    0    -x_5 0    0    0    0    0    0    x_6  0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_7  0    0    0    0    0   0    0    0    |
      | -x_1 0    x_2  0    0    -x_1 0    0    0    0    0    0    x_6  0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_7  0    0    0    0   0    0    0    |
      | 0    -x_2 0    x_3  0    0    -x_5 0    0    0    0    0    0    x_6  0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_7  0    0    0   0    0    0    |
      | 0    0    -x_3 0    x_4  0    0    -x_1 0    0    0    0    0    0    x_6  0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_7  0    0   0    0    0    |
      | x_5  0    0    -x_4 0    0    0    0    0    0    0    0    0    0    0    x_6  0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_7  0   0    0    0    |
      | 0    0    0    0    0    -x_3 -x_4 -x_2 0    0    0    0    0    0    0    0    0    0    x_6  0    0    0    0    0    0    0    0    0    0    0    0    x_7 0    0    0    |
      | 0    0    0    0    0    0    0    0    0    -x_2 x_4  0    -x_5 0    0    -x_1 0    0    0    -x_1 0    0    0    0    0    0    0    0    0    0    0    0   x_7  0    0    |
      | 0    0    0    0    0    0    0    0    x_2  0    -x_3 0    0    -x_1 0    0    0    -x_1 0    0    0    0    0    0    0    0    0    0    0    0    0    0   0    x_7  0    |
      | 0    0    0    0    0    0    0    0    -x_4 x_3  0    -x_4 0    0    -x_5 0    -x_1 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0   0    0    x_7  |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_2 -x_4 -x_5 -x_3 0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    z    0    0    0    -x_4 -x_5 -x_3 0    0    0    -x_4 0    0    -x_5 0    0   0    0    -x_6 |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    z    0    0    x_2  0    0    0    -x_3 x_5  0    0    -x_1 0    0    0   0    -x_6 0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    z    0    0    x_2  0    x_3  0    -x_4 0    0    0    0    0    x_1 0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    z    0    0    x_2  -x_5 x_4  0    0    -x_5 0    0    -x_1 0   -x_6 0    0    |

f_3 = | 0    0    -x_4 0    -x_6 0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |
      | -x_5 0    0    0    0    -x_6 0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    0    0    0    0    0    0    0    0    0    0    |
      | 0    -x_1 0    0    0    0    -x_6 0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    0    0    0    0    0    0    0    0    0    |
      | 0    0    -x_5 0    0    0    0    -x_6 0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    0    0    0    0    0    0    0    0    |
      | -x_1 0    0    0    0    0    0    0    -x_6 0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    0    0    0    0    0    0    0    |
      | 0    -x_2 x_4  0    0    0    0    0    0    0    0    0    x_6  0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    0    0    0    0    0    0    |
      | x_2  0    -x_3 0    0    0    0    0    0    0    0    0    0    0    -x_6 0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    0    0    0    0    0    |
      | -x_4 x_3  0    0    0    0    0    0    0    0    x_6  0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    0    0    0    0    |
      | 0    0    0    -x_3 0    -x_1 0    0    0    x_1  0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    0    0    0    |
      | 0    0    0    -x_4 0    0    -x_5 0    0    0    0    -x_1 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    0    0    |
      | 0    0    0    -x_2 0    0    0    -x_1 0    0    0    0    0    x_1  0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    0    |
      | 0    0    0    0    0    x_1  0    0    -x_5 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    0    |
      | 0    0    0    0    -x_1 0    x_2  0    0    0    0    0    x_1  0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    0    |
      | 0    0    0    0    0    -x_2 0    x_3  0    0    0    0    0    0    -x_5 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    0    |
      | 0    0    0    0    0    0    -x_3 0    x_4  0    x_1  0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 0    |
      | 0    0    0    0    x_5  0    0    -x_4 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_7 |
      | 0    0    0    0    0    0    0    0    0    -x_4 -x_5 -x_3 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    x_2  0    0    0    -x_3 x_5  0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    x_2  0    x_3  0    -x_4 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    x_2  -x_5 x_4  0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    -z   0    0    0    0    0    0    -x_3 x_5  0    0    -x_1 0    0    0    0    0    0    x_6  0    0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    -z   0    0    0    0    x_3  0    -x_4 0    0    0    0    0    0    0    0    x_1  0    0    0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    -z   0    0    0    -x_5 x_4  0    0    0    0    x_5  0    0    0    0    0    0    -x_6 0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    -z   0    0    -x_2 0    0    -x_4 0    0    0    0    0    x_1  0    0    0    0    0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    -z   0    0    -x_2 0    -x_5 0    0    0    -x_1 0    0    0    0    0    0    x_6  0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    -z   0    0    -x_2 -x_3 0    0    0    0    0    0    -x_1 0    0    0    0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_1  0    0    -x_5 0    0    0    0    0    0    x_6  0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_1 0    x_2  0    0    -x_1 0    0    0    0    0    0    x_6  0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_2 0    x_3  0    0    -x_5 0    0    0    0    0    0    x_6  0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_3 0    x_4  0    0    -x_1 0    0    0    0    0    0    x_6  0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_5  0    0    -x_4 0    0    0    0    0    0    0    0    0    0    0    x_6  |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_3 -x_4 -x_2 0    0    0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_2 x_4  0    -x_5 0    0    -x_1 |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_2  0    -x_3 0    0    -x_1 0    0    |
      | 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    -x_4 x_3  0    -x_4 0    0    -x_5 0    |

f_4 = | 0    0    -x_6 0    0    x_7  0    0    0    0    0    0    0    0    |
      | -x_6 0    0    0    0    0    x_7  0    0    0    0    0    0    0    |
      | 0    0    0    x_6  0    0    0    x_7  0    0    0    0    0    0    |
      | 0    -x_1 0    0    0    0    0    0    x_7  0    0    0    0    0    |
      | 0    0    0    -x_4 0    0    0    0    0    x_7  0    0    0    0    |
      | 0    0    x_5  0    0    0    0    0    0    0    x_7  0    0    0    |
      | x_1  0    0    0    0    0    0    0    0    0    0    x_7  0    0    |
      | 0    0    0    -x_5 0    0    0    0    0    0    0    0    x_7  0    |
      | 0    0    x_1  0    0    0    0    0    0    0    0    0    0    x_7  |
      | 0    -x_3 x_5  0    0    0    0    0    0    0    0    0    0    0    |
      | x_3  0    -x_4 0    0    0    0    0    0    0    0    0    0    0    |
      | -x_5 x_4  0    0    0    0    0    0    0    0    0    0    0    0    |
      | -x_2 0    0    -x_4 0    0    0    0    0    0    0    0    0    0    |
      | 0    -x_2 0    -x_5 0    0    0    0    0    0    0    0    0    0    |
      | 0    0    -x_2 -x_3 0    0    0    0    0    0    0    0    0    0    |
      | z    0    0    0    -x_4 0    -x_1 0    0    0    0    0    0    0    |
      | 0    z    0    0    -x_5 0    0    0    -x_6 0    0    0    0    0    |
      | 0    0    z    0    -x_3 -x_1 0    0    0    0    0    0    0    0    |
      | 0    0    0    z    x_2  0    0    x_1  0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    -x_4 0    -x_6 0    0    0    0    |
      | 0    0    0    0    0    -x_5 0    0    0    0    -x_6 0    0    0    |
      | 0    0    0    0    0    0    -x_1 0    0    0    0    -x_6 0    0    |
      | 0    0    0    0    0    0    0    -x_5 0    0    0    0    -x_6 0    |
      | 0    0    0    0    0    -x_1 0    0    0    0    0    0    0    -x_6 |
      | 0    0    0    0    0    0    -x_2 x_4  0    0    0    0    0    0    |
      | 0    0    0    0    0    x_2  0    -x_3 0    0    0    0    0    0    |
      | 0    0    0    0    0    -x_4 x_3  0    0    0    0    0    0    0    |
      | 0    0    0    0    0    0    0    0    -x_3 0    -x_1 0    0    0    |
      | 0    0    0    0    0    0    0    0    -x_4 0    0    -x_5 0    0    |
      | 0    0    0    0    0    0    0    0    -x_2 0    0    0    -x_1 0    |
      | 0    0    0    0    0    0    0    0    0    0    x_1  0    0    -x_5 |
      | 0    0    0    0    0    0    0    0    0    -x_1 0    x_2  0    0    |
      | 0    0    0    0    0    0    0    0    0    0    -x_2 0    x_3  0    |
      | 0    0    0    0    0    0    0    0    0    0    0    -x_3 0    x_4  |
      | 0    0    0    0    0    0    0    0    0    x_5  0    0    -x_4 0    |

f_5 = | -x_7x_4      |
      | -x_7x_5      |
      | -x_7x_3      |
      | x_7x_2       |
      | -x_7z+x_1x_6 |
      | -x_3x_6      |
      | -x_4x_6      |
      | -x_2x_6      |
      | -x_1x_5      |
      | x_2x_4       |
      | x_3x_5       |
      | x_1x_4       |
      | x_2x_5       |
      | x_1x_3       |


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