State_Code_Function_is_Primitive_Recursive

Let $k \in \N^*$. Let $e = \gamma \left({P}\right)$ be the code number of a URM program $P$. Let $\left({n_1, n_2, \ldots, n_k}\right)$ be the input of $P$. Let $S_k: \N^{k+2} \to \N$ be the function defined as $S_k \left({e, n_1, n_2, \ldots, n_k, t}\right)$ is the state code for $P$ at stage $t$ of computation of $P$. If $e$ does not code a URM Program then $S_k = 0$. Also, if $P$ terminates at stage $t_0$, then we put $\forall t > t_0: S_k \left({e, n_1, n_2, \ldots, n_k, t}\right) = S_k \left({e, n_1, n_2, \ldots, n_k, t_0}\right)$. Then for all $k \ge 1$, the function $S_k$ is primitive recursive.