Efficiently expressing feasibility problems in Linear Systems, as feasibility problems in Asymptotic-Linear-Programs

We present a polynomial-time algorithm that obtains a set of Asymptotic Linear Programs (ALPs) from a given linear system S, such that one of these ALPs admits a feasible solution if and only if S admits a feasible solution. We also show how to use the same algorithm to determine whether or not S admits a non-trivial solution for any desired subset of its variables. S is allowed to consist of linear constraints over real variables with integer coefficients, where each constraint has either a lesser-than-or-equal-to, or a lesser-than, or a not-equal-to relational operator. Each constraint of the obtained ALPs has a lesser-than-or-equal-to relational operator, and the coefficients of its variables vary linearly with respect to the time parameter that tends to positive infinity.