Revision history [back]

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions or show that there are no solution for such a system using Sage.

Here an example of a linear system of inequalities that came up in my research and was generated by using GAP:

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The list of inequalitis are then the inequalities W[i][1]>=W[i][2][k] for all i (the number of lists in W) and for all k (the length of the list W[i][2]).

So in the above examples the inequalities are 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions or show that there are no solution for such a system using Sage.

Here an example of a linear system of inequalities that came up in my research and was generated by using GAP:

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The list of inequalitis are then the inequalities W[i][1]>=W[i][2][k] for all i (the number of lists in W) and for all k (the length of the list W[i][2]).

So in the above examples the inequalities are 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions or show that there are no solution for such a system using Sage.

Here an example of a linear system of inequalities that came up in my research and was generated by using GAP:

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The list of inequalitis are then the inequalities W[i][1]>=W[i][2][k] for all i (the (from 1 to the the number of lists in W) and for all k (the (from 1 to the the length of the list W[i][2]).

So in the above examples the inequalities are 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions or show that there are no solution for such a system using Sage.

Here an example of a linear system of inequalities that came up in my research and was generated by using GAP:

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The list of inequalitis are then the inequalities W[i][1]>=W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions or show that there are no solution for such a system using Sage.Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of inequalities that came up in my research and was generated by using GAP:

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The list of inequalitis are then the inequalities W[i][1]>=W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions (it is not necessary to find all, just one solution is enough!) or show that there are no solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of inequalities that came up in my research and was generated by using GAP:

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The list of inequalitis are then the inequalities W[i][1]>=W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

Thanks for any help!

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions (it is not necessary to find all, just one solution is enough!) or show that there are no solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of inequalities that came up in my research and was generated by using GAP:GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way. Is there some command like the latex command \verbatim in this forum?)

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The list of inequalitis are then the inequalities W[i][1]>=W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

Thanks for any help!

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions (it is not necessary to find all, just one solution is enough!) or show that there are no solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of inequalities that came up in my research and was generated by using GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way. way in this forum. Is there some command like the latex command \verbatim in this forum?)

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The list of inequalitis are then the inequalities W[i][1]>=W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

Thanks for any help!

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions (it is not necessary to find all, just one solution is enough!) or show that there are no solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of inequalities that came up in my research and was generated by using GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way in this forum. Is there some command like the latex command \verbatim in this forum?)

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The complete list of inequalitis we are interested in are then the inequalities W[i][1]>=W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

Thanks for any help!

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions (it is not necessary to find all, just one solution is enough!) or show that there are no solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of inequalities inequalities, where the variable names are xi, that came up in my research and was generated by using GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way in this forum. Is there some command like the latex command \verbatim in this forum?)

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The complete list of inequalitis we are interested in are then the inequalities W[i][1]>=W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

Thanks for any help!

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find solutions (it is not necessary to find all, just one solution is enough!) or show that there are no solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of inequalities, where the variable names are xi, that came up in my research and was generated by using GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way in this forum. Is there some command like the latex command \verbatim in this forum?)

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The complete list of inequalitis we are interested in are then the inequalities W[i][1]>=W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are for W[1]: 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

Thanks for any help!

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find integer solutions (it is not necessary to find all, just one solution is enough!) or show that there are no integer solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of inequalities, where the variable names are xi, that came up in my research and was generated by using GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way in this forum. Is there some command like the latex command \verbatim in this forum?)

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The complete list of inequalitis we are interested in are then the inequalities W[i][1]>=W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are for W[1]: 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

Thanks for any help!

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find integer solutions (it is not necessary to find all, just one solution is enough!) or show that there are no integer solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of (strict) inequalities, where the variable names are xi, that came up in my research and was generated by using GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way in this forum. Is there some command like the latex command \verbatim in this forum?)

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The complete list of inequalitis we are interested in are then the inequalities W[i][1]>=W[i][2][k] W[i][1]> W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are for W[1]: 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7>= x7> 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >= > 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >=1/8>1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

Note that all inequalities are meant to be strict. If it helps, we can also assume that we work over the field of rational numbers instead of the integers.

Thanks for any help!

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find integer solutions (it is not necessary to find all, just one solution is enough!) or show that there are no integer solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of (strict) inequalities, where the variable names are xi, that came up in my research and was generated by using GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way in this forum. Is there some command like the latex command \verbatim in this forum?)

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The complete list of inequalitis we are interested in are then the inequalities W[i][1]> W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are for W[1]: 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7> 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 > 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

Note that all inequalities are meant to be strict. If it helps, we can also assume that we work over the field of rational numbers instead of the integers.

Thanks for any help!

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find integer solutions (it is not necessary to find all, just one solution is enough!) or show that there are no integer solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of (strict) inequalities, where the variable names are xi, that came up in my research and was generated by using GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way in this forum. Is there some command like the latex command \verbatim in this forum?)

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/3x4 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

This is a list of lists. The first entry of this list W is for example [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ]

The complete list of inequalitis we are interested in are then the inequalities W[i][1]> W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the list W[i][2]).

So in the above two examples the inequalities we obtain are for W[1]: 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7> 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 > 1/4x2+1/4x3+1/4x4+1/4x5 and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

Note that all inequalities are meant to be strict. If it helps, we can also assume that we work over the field of rational numbers instead of the integers.integers. The optimal output would be to say whether there is a solution or there is no solution and output a solution if there is one. The main problem here is the large number of inequalities which seemingly makes it impossible to use the standard command solve_ineq.

Thanks for any help!

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find integer solutions (it is not necessary to find all, just one solution is enough!) or show that there are no integer solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of (strict) inequalities, where the variable names are xi, that came up in my research and was generated by using GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way in this forum. Is there some command like the latex command \verbatim in this forum?)

W=[ [ [ 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7 ], [ 1/4x3+1/4x4+1/4x5+1/4x7 ] ], [ [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7 ], [ 1/3x2+1/3x3+1/31/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7 ], [ 1/4*x3+1/4*x4+1/4*x5+1/4*x7 ] ], [ [ 1/7*x2+2/7*x3+2/7*x4+1/7*x5+1/7*x7 ], [ 1/3*x2+1/3*x3+1/3*x4 ] ], 
  [ [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+1/11*x6+1/11*x7 ], [ 1/8*x1+1/8*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x6+1/8*x7 ] ], 
  [ [ 1/13*x1+2/13*x2+3/13*x3+2/13*x4+2/13*x5+1/13*x6+2/13*x7 ], [ 1/5*x2+1/5*x3+1/5*x4+1/5*x5+1/5*x7 ] ], [ [ 1/9*x1+2/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x7 ], [ 1/4*x1+1/4*x2+1/4*x3+1/4*x4 ] ], 
  [ [ 1/6*x1+1/6*x2+1/3*x3+1/6*x4+1/6*x7 ], [ 1/2*x3+1/2*x7 ] ], [ [ 1/3*x2+1/3*x3+1/3*x7 ], [ x2 ] ], [ [ 1/2*x1+1/2*x2 ], [ x1 ] ], [ [ 1/4*x3+1/4*x4+1/4*x5+1/4*x6 ], [ 1/3*x3+1/3*x4+1/3*x5 ] ], 
  [ [ 1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x5, 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ] ], 
  [ [ 1/13*x1+2/13*x2+3/13*x3+3/13*x4+2/13*x5+1/13*x6+1/13*x7 ], [ 1/7*x2+2/7*x3+2/7*x4+1/7*x5+1/7*x7, 1/9*x1+1/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x6+1/9*x7, 
          1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/6*x5+1/12*x6+1/12*x7 ] ], 
  [ [ 1/15*x1+2/15*x2+4/15*x3+1/5*x4+2/15*x5+1/15*x6+2/15*x7 ], [ 1/6*x2+1/3*x3+1/6*x4+1/6*x5+1/6*x7, 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+1/11*x6+1/11*x7, 
          1/14*x1+1/7*x2+3/14*x3+3/14*x4+1/7*x5+1/14*x6+1/7*x7 ] ], 
  [ [ 1/16*x1+3/16*x2+1/4*x3+3/16*x4+1/8*x5+1/16*x6+1/8*x7 ], [ 1/13*x1+2/13*x2+3/13*x3+2/13*x4+2/13*x5+1/13*x6+2/13*x7, 1/10*x1+1/5*x2+1/5*x3+1/5*x4+1/10*x5+1/10*x6+1/10*x7, 
          1/10*x1+1/5*x2+3/10*x3+1/5*x4+1/10*x5+1/10*x7 ] ], 
  [ [ 2/17*x1+3/17*x2+4/17*x3+3/17*x4+2/17*x5+1/17*x6+2/17*x7 ], [ 1/9*x1+2/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x7, 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/12*x5+1/12*x6+1/6*x7, 
          1/7*x1+1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x7 ] ], 
  [ [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+2/11*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x7, 1/6*x1+1/6*x2+1/3*x3+1/6*x4+1/6*x7, 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ] ], 
  [ [ 1/7*x1+2/7*x2+2/7*x3+1/7*x4+1/7*x7 ], [ 1/3*x2+1/3*x3+1/3*x7, 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7, 1/3*x1+1/3*x2+1/3*x3 ] ], [ [ 1/4*x1+1/4*x2+1/4*x3+1/4*x7 ], [ x7, 1/2*x1+1/2*x2 ] ], 
  [ [ 1/3*x4+1/3*x5+1/3*x6 ], [ 1/2*x4+1/2*x5 ] ], [ [ 1/3*x3+1/3*x4+1/3*x5 ], [ 1/2*x3+1/2*x4 ] ], [ [ 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ], [ 1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ] ], 
  [ [ 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/6*x5+1/12*x6+1/12*x7 ], [ 1/11*x1+2/11*x2+2/11*x3+2/11*x4+2/11*x5+1/11*x6+1/11*x7 ] ], 
  [ [ 1/14*x1+1/7*x2+3/14*x3+3/14*x4+1/7*x5+1/14*x6+1/7*x7 ], [ 1/8*x1+1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x7 ] ], [ [ 1/10*x1+1/5*x2+3/10*x3+1/5*x4+1/10*x5+1/10*x7 ], [ 1/5*x2+2/5*x3+1/5*x4+1/5*x7 ] ], 
  [ [ 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/12*x5+1/12*x6+1/6*x7 ], [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+2/11*x7, 1/9*x1+2/9*x2+2/9*x3+1/9*x4+1/9*x5+1/9*x6+1/9*x7 ] ], 
  [ [ 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ], [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7, 1/7*x1+2/7*x2+2/7*x3+1/7*x4+1/7*x7 ] ], 
  [ [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7 ], [ 1/4*x1+1/4*x2+1/4*x3+1/4*x7 ] ], [ [ 1/2*x5+1/2*x6 ], [ x5 ] ], [ [ 1/2*x4+1/2*x5 ], [ x4 ] ], 
  [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7 ], [ 1/8x1+1/8x2+1/4x3+1/8x4+1/8x5+1/8x6+1/8x7 ] ], 
  [ [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7 ], [ 1/5x2+1/5x3+1/5x4+1/5x5+1/5x7 ] ], [ [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x4 ] ], 
  [ [ 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7 ], [ 1/2x3+1/2x7 ] ], [ [ 1/3x2+1/3x3+1/3x7 ], [ x2 ] ], [ [ 1/2x1+1/2x2 ], [ x1 ] ], [ [ 1/4x3+1/4x4+1/4x5+1/4x6 ], [ 1/3x3+1/3x4+1/3x5 ] ], 
  [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ] ], 
  [ [ 1/13x1+2/13x2+3/13x3+3/13x4+2/13x5+1/13x6+1/13x7 ], [ 1/7x2+2/7x3+2/7x4+1/7x5+1/7x7, 1/9x1+1/9x2+2/9x3+2/9x4+1/9x5+1/9x6+1/9x7, 
          1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ] ], 
  [ [ 1/15x1+2/15x2+4/15x3+1/5x4+2/15x5+1/15x6+2/15x7 ], [ 1/6x2+1/3x3+1/6x4+1/6x5+1/6x7, 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+1/11x6+1/11x7, 
          1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ] ], 
  [ [ 1/16x1+3/16x2+1/4x3+3/16x4+1/8x5+1/16x6+1/8x7 ], [ 1/13x1+2/13x2+3/13x3+2/13x4+2/13x5+1/13x6+2/13x7, 1/10x1+1/5x2+1/5x3+1/5x4+1/10x5+1/10x6+1/10x7, 
          1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ] ], 
  [ [ 2/17x1+3/17x2+4/17x3+3/17x4+2/17x5+1/17x6+2/17x7 ], [ 1/9x1+2/9x2+2/9x3+2/9x4+1/9x5+1/9x7, 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7, 
          1/7x1+1/7x2+2/7x3+1/7x4+1/7x5+1/7x7 ] ], 
  [ [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x7, 1/6x1+1/6x2+1/3x3+1/6x4+1/6x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], 
  [ [ 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ], [ 1/3x2+1/3x3+1/3x7, 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7, 1/3x1+1/3x2+1/3[ [ 1/2*x3+1/2*x4 ], [ x3 ] ], [ [ 1/4x1+1/4x2+1/4x3+1/4x7 ], [ x7, 1/2x1+1/2x2 ] ], 
  [ [ 1/3x4+1/3x5+1/3x6 ], [ 1/2x4+1/2x5 ] ], [ [ 1/3x3+1/3x4+1/3x5 ], [ 1/2x3+1/2x4 ] ], [ [ 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 ], [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ] ], 
  [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/6x5+1/12x6+1/12x7 ], [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ] ], 
  [ [ 1/14x1+1/7x2+3/14x3+3/14x4+1/7x5+1/14x6+1/7x7 ], [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ] ], [ [ 1/10x1+1/5x2+3/10x3+1/5x4+1/10x5+1/10x7 ], [ 1/5x2+2/5x3+1/5x4+1/5x7 ] ], 
  [ [ 1/12x1+1/6x2+1/4x3+1/6x4+1/12x5+1/12x6+1/6x7 ], [ 1/11x1+2/11x2+3/11x3+2/11x4+1/11x5+2/11x7, 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ] ], 
  [ [ 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7, 1/7x1+2/7x2+2/7x3+1/7x4+1/7x7 ] ], 
  [ [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ], [ 1/4x1+1/4x2+1/4x3+1/4x7 ] ], [ [ 1/2x5+1/2x6 ], [ x5 ] ], [ [ 1/2x4+1/2x5 ], [ x4 ] ], [ [ 1/2x3+1/2x4 ], [ x3 ] ], 
  [ [ 1/7x2+2/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x2+1/6x3+1/6x4+1/6x5+1/6x6+1/6x7 ] ], 
  [ [ 1/11x1+2/11x2+2/11x3+2/11x4+2/11x5+1/11x6+1/11x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x5 ] ], [ [ 1/8x1+1/8x2+1/4x3+1/4x4+1/8x5+1/8x7 ], [ 1/3x3+1/3x4+1/3x7 ] ], 
  [ [ 1/5x2+2/5x3+1/5x4+1/5x7 ], [ 1/2x2+1/2x3 ] ], 
  [ [ 1/9x1+2/9x2+2/9x3+1/9x4+1/9x5+1/9x6+1/9x7 ], [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7, 1/8x1+1/4x2+1/4x3+1/8x4+1/8x5+1/8x7 ] ], 
  [ [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ], [ 1/5x1+1/5x2+1/5x3+1/5x4+1/5x7 ] ], [ [ 1/7x1+1/7x2+1/7x3+1/7x4+1/7x5+1/7x6+1/7x7 ], [ 1/6x1+1/6x2+1/6x3+1/6x4+1/6x5+1/6x7 ] ] ]

  [ [ 1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ], [ 1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x6+1/6*x7 ] ], 
  [ [ 1/11*x1+2/11*x2+2/11*x3+2/11*x4+2/11*x5+1/11*x6+1/11*x7 ], [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x5 ] ], [ [ 1/8*x1+1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x7 ], [ 1/3*x3+1/3*x4+1/3*x7 ] ], 
  [ [ 1/5*x2+2/5*x3+1/5*x4+1/5*x7 ], [ 1/2*x2+1/2*x3 ] ], 
  [ [ 1/9*x1+2/9*x2+2/9*x3+1/9*x4+1/9*x5+1/9*x6+1/9*x7 ], [ 1/7*x1+1/7*x2+1/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7, 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ] ], 
  [ [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7 ], [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7 ] ], [ [ 1/7*x1+1/7*x2+1/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ], [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7 ] ] ]

[ [ 1/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7 ], [ 1/4*x3+1/4*x4+1/4*x5+1/4*x7 ] ]

and the tenth entry is: [ [ 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 ], [ 1/4x2+1/4x3+1/4x4+1/4x5, 1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7 [ [ 1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x5, 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ] ]

The complete list of inequalitis we are interested in are then the inequalities W[i][1]> W[i][2][k] W[i][2][k] for all i i (from 1 to the the number of lists in W) W) and for all k k (from 1 to the the length of the list W[i][2]).listW[i][2]).

So in the above two examples the inequalities we obtain are for W[1]: 1/10x1+1/10x2+1/5x3+1/5x4+1/5x5+1/10x6+1/10x7> 1/4x3+1/4x4+1/4x5+1/4*x7 for W[1]

1/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7> 1/4*x3+1/4*x4+1/4*x5+1/4*x7

and for W[10] we get the two inequalities: 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7

1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 > 1/4x2+1/4x3+1/4x4+1/4x5 1/4*x2+1/4*x3+1/4*x4+1/4*x5

and 1/9x2+2/9x3+2/9x4+2/9x5+1/9x6+1/9x7 >1/8x2+1/4x3+1/4x4+1/8x5+1/8x6+1/8x7.

1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 >1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7.

Note that all inequalities are meant to be strict. If it helps, we can also assume that we work over the field of rational numbers instead of the integers. The optimal output would be to say whether there is a solution or there is no solution and output a solution if there is one. The main problem here is the large number of inequalities which seemingly makes it impossible to use the standard command solve_ineq.

Thanks for any help!

I have no experience with linear inequality systems but I wonder whethere there is an easy way to find integer solutions (it is not necessary to find all, just one solution is enough!) or show that there are no integer solution for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of (strict) inequalities, where the variable names are xi, that came up in my research and was generated by using GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way in this forum. Is there some command like the latex command \verbatim in this forum?)

W=[ [ [ 1/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7 ], [ 1/4*x3+1/4*x4+1/4*x5+1/4*x7 ] ], [ [ 1/7*x2+2/7*x3+2/7*x4+1/7*x5+1/7*x7 ], [ 1/3*x2+1/3*x3+1/3*x4 ] ], 
  [ [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+1/11*x6+1/11*x7 ], [ 1/8*x1+1/8*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x6+1/8*x7 ] ], 
  [ [ 1/13*x1+2/13*x2+3/13*x3+2/13*x4+2/13*x5+1/13*x6+2/13*x7 ], [ 1/5*x2+1/5*x3+1/5*x4+1/5*x5+1/5*x7 ] ], [ [ 1/9*x1+2/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x7 ], [ 1/4*x1+1/4*x2+1/4*x3+1/4*x4 ] ], 
  [ [ 1/6*x1+1/6*x2+1/3*x3+1/6*x4+1/6*x7 ], [ 1/2*x3+1/2*x7 ] ], [ [ 1/3*x2+1/3*x3+1/3*x7 ], [ x2 ] ], [ [ 1/2*x1+1/2*x2 ], [ x1 ] ], [ [ 1/4*x3+1/4*x4+1/4*x5+1/4*x6 ], [ 1/3*x3+1/3*x4+1/3*x5 ] ], 
  [ [ 1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x5, 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ] ], 
  [ [ 1/13*x1+2/13*x2+3/13*x3+3/13*x4+2/13*x5+1/13*x6+1/13*x7 ], [ 1/7*x2+2/7*x3+2/7*x4+1/7*x5+1/7*x7, 1/9*x1+1/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x6+1/9*x7, 
          1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/6*x5+1/12*x6+1/12*x7 ] ], 
  [ [ 1/15*x1+2/15*x2+4/15*x3+1/5*x4+2/15*x5+1/15*x6+2/15*x7 ], [ 1/6*x2+1/3*x3+1/6*x4+1/6*x5+1/6*x7, 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+1/11*x6+1/11*x7, 
          1/14*x1+1/7*x2+3/14*x3+3/14*x4+1/7*x5+1/14*x6+1/7*x7 ] ], 
  [ [ 1/16*x1+3/16*x2+1/4*x3+3/16*x4+1/8*x5+1/16*x6+1/8*x7 ], [ 1/13*x1+2/13*x2+3/13*x3+2/13*x4+2/13*x5+1/13*x6+2/13*x7, 1/10*x1+1/5*x2+1/5*x3+1/5*x4+1/10*x5+1/10*x6+1/10*x7, 
          1/10*x1+1/5*x2+3/10*x3+1/5*x4+1/10*x5+1/10*x7 ] ], 
  [ [ 2/17*x1+3/17*x2+4/17*x3+3/17*x4+2/17*x5+1/17*x6+2/17*x7 ], [ 1/9*x1+2/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x7, 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/12*x5+1/12*x6+1/6*x7, 
          1/7*x1+1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x7 ] ], 
  [ [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+2/11*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x7, 1/6*x1+1/6*x2+1/3*x3+1/6*x4+1/6*x7, 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ] ], 
  [ [ 1/7*x1+2/7*x2+2/7*x3+1/7*x4+1/7*x7 ], [ 1/3*x2+1/3*x3+1/3*x7, 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7, 1/3*x1+1/3*x2+1/3*x3 ] ], [ [ 1/4*x1+1/4*x2+1/4*x3+1/4*x7 ], [ x7, 1/2*x1+1/2*x2 ] ], 
  [ [ 1/3*x4+1/3*x5+1/3*x6 ], [ 1/2*x4+1/2*x5 ] ], [ [ 1/3*x3+1/3*x4+1/3*x5 ], [ 1/2*x3+1/2*x4 ] ], [ [ 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ], [ 1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ] ], 
  [ [ 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/6*x5+1/12*x6+1/12*x7 ], [ 1/11*x1+2/11*x2+2/11*x3+2/11*x4+2/11*x5+1/11*x6+1/11*x7 ] ], 
  [ [ 1/14*x1+1/7*x2+3/14*x3+3/14*x4+1/7*x5+1/14*x6+1/7*x7 ], [ 1/8*x1+1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x7 ] ], [ [ 1/10*x1+1/5*x2+3/10*x3+1/5*x4+1/10*x5+1/10*x7 ], [ 1/5*x2+2/5*x3+1/5*x4+1/5*x7 ] ], 
  [ [ 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/12*x5+1/12*x6+1/6*x7 ], [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+2/11*x7, 1/9*x1+2/9*x2+2/9*x3+1/9*x4+1/9*x5+1/9*x6+1/9*x7 ] ], 
  [ [ 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ], [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7, 1/7*x1+2/7*x2+2/7*x3+1/7*x4+1/7*x7 ] ], 
  [ [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7 ], [ 1/4*x1+1/4*x2+1/4*x3+1/4*x7 ] ], [ [ 1/2*x5+1/2*x6 ], [ x5 ] ], [ [ 1/2*x4+1/2*x5 ], [ x4 ] ], [ [ 1/2*x3+1/2*x4 ], [ x3 ] ], 
  [ [ 1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ], [ 1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x6+1/6*x7 ] ], 
  [ [ 1/11*x1+2/11*x2+2/11*x3+2/11*x4+2/11*x5+1/11*x6+1/11*x7 ], [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x5 ] ], [ [ 1/8*x1+1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x7 ], [ 1/3*x3+1/3*x4+1/3*x7 ] ], 
  [ [ 1/5*x2+2/5*x3+1/5*x4+1/5*x7 ], [ 1/2*x2+1/2*x3 ] ], 
  [ [ 1/9*x1+2/9*x2+2/9*x3+1/9*x4+1/9*x5+1/9*x6+1/9*x7 ], [ 1/7*x1+1/7*x2+1/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7, 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ] ], 
  [ [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7 ], [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7 ] ], [ [ 1/7*x1+1/7*x2+1/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ], [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7 ] ] ]

This is a list of lists. The first entry of this list W is for example

[ [ 1/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7 ], [ 1/4*x3+1/4*x4+1/4*x5+1/4*x7 ] ]

and the tenth entry is: is:

[ [ 1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x5, 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ] ]
]

The complete list of inequalitis we are interested in are then the inequalities W[i][1]> W[i][2][k] for all i (from 1 to the the number of lists in W) and for all k (from 1 to the the length of the listW[i][2]).

So in the above two examples the inequalities we obtain are for W[1]:

1/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7> 1/4*x3+1/4*x4+1/4*x5+1/4*x7

and for W[10] we get the two inequalities:

1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 > 1/4*x2+1/4*x3+1/4*x4+1/4*x5

and

1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 >1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7.

Note that all inequalities are meant to be strict. If it helps, we can also assume that we work over the field of rational numbers instead of the integers. The optimal output would be to say whether there is a solution or there is no solution and output a solution if there is one. The main problem here is the large number of inequalities which seemingly makes it impossible to use the standard command solve_ineq.

Thanks for any help!

I have no experience Given a matrix A with linear inequality systems but I wonder whethere there is an easy way to find integer solutions (it is not necessary to find all, just one solution is enough!) or show that there are no integer solution rational entries, for such a system using Sage (or if it is not possible with Sage maybe someone can recommend another software).

Here an example of a linear system of (strict) inequalities, where the variable names are xi, that came up in my research and was generated by using GAP: (it is not displayed very nicely, maybe someone can edit my post to show how to post such a GAP list in a nice way in this forum. Is there some command like the latex command \verbatim in this forum?)the 36x7 matrix:

W=[ [ [ 1/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7 ], [ 1/4*x3+1/4*x4+1/4*x5+1/4*x7 ] ], [ [ 1/7*x2+2/7*x3+2/7*x4+1/7*x5+1/7*x7 ], [ 1/3*x2+1/3*x3+1/3*x4 ] ], 
A=[ [ 1/10, 1/10, -1/20, -1/20, -1/20, 1/10, -3/20 ], [ 0, -4/21, -1/21, -1/21, 1/7, 0, 1/7 ], [ -3/88, 5/88, 1/44, 5/88, -3/88, -3/88, -3/88 ], [ 1/13, -3/65, 2/65, -3/65, -3/65, 1/13, -3/65 ],
[ -5/36, -1/36, -1/36, -1/36, 1/9, 0, 1/9 ], [ 1/6, 1/6, -1/6, 1/6, 0, 0, -1/3 ], [ 0, -2/3, 1/3, 0, 0, 0, 1/3 ], [ -1/2, 1/2, 0, 0, 0, 0, 0 ],
[ 0, 0, -1/12, -1/12, -1/12, 1/4, 0 ], [ 0, -5/36, -1/36, -1/36, -1/36, 1/9, 1/9 ], [ 1/13, 1/91, -5/91, -5/91, 1/91, 1/13, -6/91 ], [ 1/15, -1/30, -1/15, 1/30, -1/30, 1/15, -1/30 ],
[ -3/208, 7/208, 1/52, 7/208, -3/104, -3/208, -3/104 ], [ 1/153, -7/153, 2/153, -7/153, 1/153, 1/17, 1/153 ], [ 1/11, -3/44, 1/44, -3/44, 1/11, 0, -3/44 ], [ 1/7, -1/21, -1/21, 1/7, 0, 0, -4/21 ],
[ 1/4, 1/4, 1/4, 0, 0, 0, -3/4 ], [ 0, 0, 0, -1/6, -1/6, 1/3, 0 ], [ 0, 0, -1/6, -1/6, 1/3, 0, 0 ], [ 0, -1/56, -1/28, 3/28, -1/56, -1/56, -1/56 ],
 [ -1/132, -1/66, 3/44, -1/66, -1/66, -1/132, -1/132 ], [ -3/56, 1/56, -1/28, -1/28, 1/56, 1/14, 1/56 ], [ 1/10, 0, -1/10, 0, 1/10, 0, -1/10 ], [ -1/132, -1/66, -1/44, -1/66, -1/132, 1/12, -1/66 ],
 [ [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+1/11*x6+1/11*x7 ], [ 1/8*x1+1/8*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x6+1/8*x7 ] ], 
-1/24, 1/12, 1/12, -1/24, -1/24, 0, -1/24 ], [ -1/20, -1/20, -1/20, 1/5, 0, 0, -1/20 ], [ 0, 0, 0, 0, -1/2, 1/2, 0 ], [ 0, 0, 0, -1/2, 1/2, 0, 0 ],
 [ 0, 0, -1/2, 1/2, 0, 0, 0 ], [ 0, -1/42, 5/42, -1/42, -1/42, -1/42, -1/42 ], [ -6/55, -1/55, -1/55, -1/55, -1/55, 1/11, 1/11 ], [ 1/8, 1/8, -1/12, -1/12, 1/8, 0, -5/24 ],
 [ [ 1/13*x1+2/13*x2+3/13*x3+2/13*x4+2/13*x5+1/13*x6+2/13*x7 ], [ 1/5*x2+1/5*x3+1/5*x4+1/5*x5+1/5*x7 ] ], [ [ 1/9*x1+2/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x7 ], [ 1/4*x1+1/4*x2+1/4*x3+1/4*x4 ] ], 
  [ [ 1/6*x1+1/6*x2+1/3*x3+1/6*x4+1/6*x7 ], [ 1/2*x3+1/2*x7 ] ], [ [ 1/3*x2+1/3*x3+1/3*x7 ], [ x2 ] ], [ [ 1/2*x1+1/2*x2 ], [ x1 ] ], [ [ 1/4*x3+1/4*x4+1/4*x5+1/4*x6 ], [ 1/3*x3+1/3*x4+1/3*x5 ] ], 
  [ [ 1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x5, 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ] ], 
  [ [ 1/13*x1+2/13*x2+3/13*x3+3/13*x4+2/13*x5+1/13*x6+1/13*x7 ], [ 1/7*x2+2/7*x3+2/7*x4+1/7*x5+1/7*x7, 1/9*x1+1/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x6+1/9*x7, 
          1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/6*x5+1/12*x6+1/12*x7 ] ], 
  [ [ 1/15*x1+2/15*x2+4/15*x3+1/5*x4+2/15*x5+1/15*x6+2/15*x7 ], [ 1/6*x2+1/3*x3+1/6*x4+1/6*x5+1/6*x7, 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+1/11*x6+1/11*x7, 
          1/14*x1+1/7*x2+3/14*x3+3/14*x4+1/7*x5+1/14*x6+1/7*x7 ] ], 
  [ [ 1/16*x1+3/16*x2+1/4*x3+3/16*x4+1/8*x5+1/16*x6+1/8*x7 ], [ 1/13*x1+2/13*x2+3/13*x3+2/13*x4+2/13*x5+1/13*x6+2/13*x7, 1/10*x1+1/5*x2+1/5*x3+1/5*x4+1/10*x5+1/10*x6+1/10*x7, 
          1/10*x1+1/5*x2+3/10*x3+1/5*x4+1/10*x5+1/10*x7 ] ], 
  [ [ 2/17*x1+3/17*x2+4/17*x3+3/17*x4+2/17*x5+1/17*x6+2/17*x7 ], [ 1/9*x1+2/9*x2+2/9*x3+2/9*x4+1/9*x5+1/9*x7, 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/12*x5+1/12*x6+1/6*x7, 
          1/7*x1+1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x7 ] ], 
  [ [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+2/11*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x7, 1/6*x1+1/6*x2+1/3*x3+1/6*x4+1/6*x7, 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ] ], 
  [ [ 1/7*x1+2/7*x2+2/7*x3+1/7*x4+1/7*x7 ], [ 1/3*x2+1/3*x3+1/3*x7, 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7, 1/3*x1+1/3*x2+1/3*x3 ] ], [ [ 1/4*x1+1/4*x2+1/4*x3+1/4*x7 ], [ x7, 1/2*x1+1/2*x2 ] ], 
  [ [ 1/3*x4+1/3*x5+1/3*x6 ], [ 1/2*x4+1/2*x5 ] ], [ [ 1/3*x3+1/3*x4+1/3*x5 ], [ 1/2*x3+1/2*x4 ] ], [ [ 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ], [ 1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ] ], 
  [ [ 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/6*x5+1/12*x6+1/12*x7 ], [ 1/11*x1+2/11*x2+2/11*x3+2/11*x4+2/11*x5+1/11*x6+1/11*x7 ] ], 
  [ [ 1/14*x1+1/7*x2+3/14*x3+3/14*x4+1/7*x5+1/14*x6+1/7*x7 ], [ 1/8*x1+1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x7 ] ], [ [ 1/10*x1+1/5*x2+3/10*x3+1/5*x4+1/10*x5+1/10*x7 ], [ 1/5*x2+2/5*x3+1/5*x4+1/5*x7 ] ], 
  [ [ 1/12*x1+1/6*x2+1/4*x3+1/6*x4+1/12*x5+1/12*x6+1/6*x7 ], [ 1/11*x1+2/11*x2+3/11*x3+2/11*x4+1/11*x5+2/11*x7, 1/9*x1+2/9*x2+2/9*x3+1/9*x4+1/9*x5+1/9*x6+1/9*x7 ] ], 
  [ [ 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ], [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7, 1/7*x1+2/7*x2+2/7*x3+1/7*x4+1/7*x7 ] ], 
  [ [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7 ], [ 1/4*x1+1/4*x2+1/4*x3+1/4*x7 ] ], [ [ 1/2*x5+1/2*x6 ], [ x5 ] ], [ [ 1/2*x4+1/2*x5 ], [ x4 ] ], [ [ 1/2*x3+1/2*x4 ], [ x3 ] ], 
  [ [ 1/7*x2+2/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ], [ 1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x6+1/6*x7 ] ], 
  [ [ 1/11*x1+2/11*x2+2/11*x3+2/11*x4+2/11*x5+1/11*x6+1/11*x7 ], [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x5 ] ], [ [ 1/8*x1+1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x7 ], [ 1/3*x3+1/3*x4+1/3*x7 ] ], 
  [ [ 1/5*x2+2/5*x3+1/5*x4+1/5*x7 ], [ 1/2*x2+1/2*x3 ] ], 
  [ [ 1/9*x1+2/9*x2+2/9*x3+1/9*x4+1/9*x5+1/9*x6+1/9*x7 ], [ 1/7*x1+1/7*x2+1/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7, 1/8*x1+1/4*x2+1/4*x3+1/8*x4+1/8*x5+1/8*x7 ] ], 
  [ [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7 ], [ 1/5*x1+1/5*x2+1/5*x3+1/5*x4+1/5*x7 ] ], [ [ 1/7*x1+1/7*x2+1/7*x3+1/7*x4+1/7*x5+1/7*x6+1/7*x7 ], [ 1/6*x1+1/6*x2+1/6*x3+1/6*x4+1/6*x5+1/6*x7 ] 0, -3/10, -1/10, 1/5, 0, 0, 1/5 ], [ -2/63, 5/63, 5/63, -2/63, -2/63, -2/63, -2/63 ], [ -1/30, -1/30, -1/30, -1/30, 1/6, 0, -1/30 ], [ -1/42, -1/42, -1/42, -1/42, -1/42, 1/7, -1/42 ] ]

This is a list of lists. The first entry of this list W is for example

[ [ 1/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7 ], [ 1/4*x3+1/4*x4+1/4*x5+1/4*x7 ] ]

and the tenth entry is:

[ [ 1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 ], [ 1/4*x2+1/4*x3+1/4*x4+1/4*x5, 1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7 ] ]

The complete list of inequalitis we are interested in are then the inequalities W[i][1]> W[i][2][k] for all i (from 1 Is there an easy way using Sage (or another computer algebra system) to the the number of lists in W) and for all k (from 1 to the the length of the listW[i][2]).

So in the above two examples the inequalities we obtain are for W[1]:

1/10*x1+1/10*x2+1/5*x3+1/5*x4+1/5*x5+1/10*x6+1/10*x7> 1/4*x3+1/4*x4+1/4*x5+1/4*x7

and for W[10] we get the two inequalities:

1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 > 1/4*x2+1/4*x3+1/4*x4+1/4*x5

and

1/9*x2+2/9*x3+2/9*x4+2/9*x5+1/9*x6+1/9*x7 >1/8*x2+1/4*x3+1/4*x4+1/8*x5+1/8*x6+1/8*x7.

Note that all inequalities are meant to be strict. If it helps, we can also assume that we work over the field of rational numbers instead of the integers. The optimal output would be to say check whether there is a solution or to to the inequality systems Ax>0 (meaning every entry of the vector of Ax is strictly positive) in rational (or even real) numbers?

So the input should be the matrix A and the output should be whether there is no a solution and output of Ax>0 or not in the rationals numbers (and if possible give a solution if there is one. The main problem here is the large number of inequalities which seemingly makes it impossible to use the standard command solve_ineq.one).