Definition:Unlimited Register Machine/Program/Highest Register

Definition
Let $\Bbb U$ denote the set of all URM programs. Let $P \in \Bbb U$ be a URM program.

By definition, $P$ uses a finite number of registers.

We define the function $\rho: \Bbb U \to \N$ as follows:
 * $\forall P \in \Bbb U: \map \rho P =$ the highest register number used by $P$

That is, in any URM program $P$, no instruction refers to any register with index greater than $\map \rho P$.