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But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice).
(In order to do that, we'll need to add lines the lines $-e_i$ for $1\le i\le\dim(x)$ to $A$, in order to get $-x_i\le 0$)
Yes, lower and upper bounds on variables can be enforced via explicit constraints. In practice, however, bounds are handled implicitly because the explicit constraints determine the size of the basis.
