Prove Non-Homogeneous Farkas' Lemma
Let $A\in\mathbb{R}^{m \times n},c\in\mathbb{R}^{n},b\in\mathbb{R}^{m},d\in\mathbb{R},$
Suppose that there exist $y\geq0$ such that $A^Ty=c$ Prove that excatly one of the following is feasible:
A.$Ax\leq b,c^Tx>d$
B. $A^Ty=c,b^Ty<d,y\geq0$
My attempt is the following.
Assume that B holds and we'll show that A doesn't.
$Ax\leq b\iff y^tAx\leq y^tb\iff (A^Ty)^Tx\leq d\iff c^Tx\leq d$ and A doesn't hold.
I'm not sure about this part of my proof
Assume that B doesn't holds and we'll show that A is feasible.
If B doesn't holds then $A^Ty=c,b^Ty=d$ doesn't holds aswell. Setting $\tilde{A}=\begin{pmatrix}
A^T\\b^T
\end{pmatrix},\tilde{c}=\begin{pmatrix}
c\\d
\end{pmatrix}$ Now from the homog. Farkas' Lemma we know that $\tilde{A}\tilde{x}\leq 0,\tilde{c}>0\iff Ax+x_{n+1}b\leq0,c^Tx+dx_{n+1}>0$ setting $x_{n+1}=-1$ we get $Ax\leq b,c^T>d$ as desired.\