Unique Code for URM Instruction

Theorem
Each basic instruction $$I$$ in a URM Program can be identified with a unique code number $$\beta \left({I}\right)$$.

We also define the following sets:
 * $$\operatorname{Zinstr}$$ is the set of codes of all the Zero instructions;
 * $$\operatorname{Sinstr}$$ is the set of codes of all the Successor instructions;
 * $$\operatorname{Cinstr}$$ is the set of codes of all the Copy instructions;
 * $$\operatorname{Jinstr}$$ is the set of codes of all the Jump instructions.

Then we define $$\operatorname{Instr}$$ to be the set of codes of all basic URM instructions.

That is:
 * $$\operatorname{Instr} = \operatorname{Zinstr} \cup \operatorname{Sinstr} \cup \operatorname{Cinstr} \cup \operatorname{Jinstr}$$.

Proof
Each basic URM instruction is of one of the following forms:

Let $$\mathbb{I}$$ be the set of all basic URM instructions.

We define the mapping $$\beta: \mathbb{I} \to \N$$ as follows:
 * $$\beta \left({Z \left({n}\right)}\right) = 6 n - 3$$;
 * $$\beta \left({S \left({n}\right)}\right) = 6 n$$;
 * $$\beta \left({C \left({m, n}\right)}\right) = 2^m 3^n + 1$$;
 * $$\beta \left({J \left({m, n, q}\right)}\right) = 2^m 3^n 5^q + 2$$.

We note that:
 * $$\beta \left({Z \left({n}\right)}\right) \equiv 3 \pmod 6$$;
 * $$\beta \left({S \left({n}\right)}\right) \equiv 0 \pmod 6$$;
 * $$\beta \left({C \left({m, n}\right)}\right) \equiv 1 \pmod 3$$;
 * $$\beta \left({J \left({m, n, q}\right)}\right) \equiv 2 \pmod 3$$.

So, if $$\beta \left({I_1}\right) = \beta \left({I_2}\right)$$, then both instructions must be of the same type.

Hence it follows from the Fundamental Theorem of Arithmetic that $$\beta$$ is uniquely specified for any given basic URM instruction.

Thus $$\beta$$ is an injection.

Note
The coding scheme for $$\mathbb{I}$$ is not unique.

This particular scheme lends itself especially to number-theoretical analysis techniques.