I am trying to understand the following definition:
Def: A cutting plane proof from the system $Ax\leq b$ for an inequality $c^Tx\leq d$ is a sequence of inequalities $c_i^T\leq d_i$, $(i = 1,\ldots, k)$ with the following properties
i) every $c_i$ is integral,
ii) $c_k = c$ and $d_k = d$,
iii) for every $i$ there is a number $d_i'$ satisfying $\lfloor d_i'\rfloor \leq d_i$, such that $c_i^T x\leq d_i'$ is a nonnegative combination of the inequalities $Ax\leq b$ and $c_1^Tx\leq d_1,\ldots, c_{i-1}^Tx\leq d_{i-1}$.
Question: Could someone explain to me how the number $k$ is specified, and give an example of a cutting plane proof? I have tried to find examples that match this definition, but I haven't found any.
Thanks in advance!