Single Instruction URM Programs/Identity Function

Theorem
The identity function $I_\N: \N \to \N$ defined as:
 * $\forall n \in \N: I_\N \left({n}\right) = n$

is URM computable by a single-instruction URM program.

Proof
Any of the following URM programs compute the identity function:

... where $m \ne 1$.

This sets the value $0$ into $R_m$ and then stops.

... where $m \ne 1$.

The input $n$ is in $R_1$ when the program starts.

The program adds $1$ to $r_m$ and then stops.

... where $m \ne 1$.

The input $n$ is in $R_1$ when the program starts.

The program copies $r_j$ to $r_m$ and then stops.

The input $n$ is in $R_1$ when the program starts.

The program copies $r_1$ to $r_1$, i.e. to itself, and then stops.

In none of these programs is $R_1$ affected, and so no change is effected to the input, which is returned as the output unchanged.

Hence they all compute the identity function.

Note that the latter program, consisting entirely of $C \left({1, 1}\right)$, is a direct implementation of the projection function program $\operatorname{pr}^1_1: \N \to \N$.

This latter function is the usual way of implementing the identity function as it is well-defined and obvious, and guaranteed to have no other side-effects when embedded in a larger program.