Offset URM Program is Primitive Recursive

== Theorem ==
There exists a primitive recursive function $\operatorname {Offset} : \N^2 \to \N$ such that, for $e, k \in \N$:
* If $e$ does not code a URM program, then $\map {\operatorname {Offset} } {e, k} = 0$.
* If $e$ codes a URM program $P$, then $\map {\operatorname {Offset} } {e, k}$ codes a program $P'$, with the following properties:
:$(1): \quad$ Every basic instruction of $P'$ with line number $i \in \closedint 1 k$ is $\map C {0,0}$.
:$(2): \quad$ Every Jump instruction in $P'$ is of the form $\map J {m,n,q}$, where $q > k$.
:$(3): \quad$ $P$ and $P'$ compute the same function.
== Proof ==
The basic instructions of $P'$ will be defined as:
:$P'_i = \begin{cases}
\map C {1,1} & : i \le k \\
P_{i - k} & : i > k \land P_{i - k} \notin \texttt {Jump} \\
\map J {m,n,q + k} & : i > k \land P_{i - k} = \map J {m,n,q}
\end{cases}$
Property $(1)$ holds by the first case.
Property $(2)$ holds as each Jump instruction has $q > 0$ by definition, so $q + k > k$ follows.
For Property $(3)$, compare the $j$-th stage of $P$ to the $j + k$-th of $P'$.
Fix the input values for the registers.
By induction, show that, if the $j$-th stage of $P$ has instruction pointer $i$, then the $j + k$-th stage of $P'$ has instruction pointer $i + k$ and exactly the same register values.
As each $\map C {1,1}$ instruction does not change the values of the registers, the $k$-th stage of $P'$ has the same register values as the $0$-th stage of $P$, when the inputs are the same.
Additionally, since the instruction pointers are $1$ and $1 + k$, respectively, the basis for the induction is satisfied.
Now, suppose the hypothesis holds.
Since the instruction pointer for $P'$ is $i + k > k$, the first case of the definition of $P'_{i + k}$ does not hold.
If $P_i \notin \texttt {Jump}$, then $P$ and $P'$ execute the same instruction on the same register values, resulting in the same register values once again.
The instruction pointers then proceed to $i + 1$ and $i + 1 + k$, respectively.
If $P_i = \map J {m,n,q} \in \texttt {Jump}$, then $P'_{i + k} = \map J {m,n,q + k}$.
If $R_m = R_n$ in $P$, then the same holds in $P'$ by assumption.
In that case, then, the next instruction pointers are $q$ and $q + k$, respectively.
If $R_m \ne R_n$, the same holds, and the instruction pointers become $i + 1$ and $i + 1 + k$.
In every case, it can be seen that the induction hypothesis is satisfied for $j + 1$.
Thus, by Principle of Mathematical Induction, the same holds for every state.
Therefore, either both $P$ and $P'$ output the same value, or they both fail to terminate.
Thus, Property $(3)$ holds.
The above definition for $P'_i$ can be regarded as a function $\map I {e,k,i}$ that produces the code for $P'_i$.
That function is primitive recursive by:
* Definition by Cases is Primitive Recursive
* Ordering Relations are Primitive Recursive
* Set of Codes for URM Instructions is Primitive Recursive, in particular Jinstr
* Set Operations on Primitive Recursive Relations
* Constant Function is Primitive Recursive, in particular $\map \beta {\map C {0,0} } = 2^1 \times 3^1 + 1 = 7$
* Cut-Off Subtraction is Primitive Recursive, as $i - k = i {\dot -} k$ whenever $i > k$
* As $\map \beta {\map J {m,n,q + k} } = 2^m \times 3^n \times 5^{q + k} + 2$:
:* Exponentiation is Primitive Recursive
:* Multiplication is Primitive Recursive
:* Addition is Primitive Recursive
:* Constant Function is Primitive Recursive
:* Cut-Off Subtraction is Primitive Recursive and Prime Exponent Function is Primitive Recursive, as $m = \paren {e_i - 2}_1$, and likewise for $n$ and $q$
Thus, the function $\operatorname {Offset}$ is given by:
:$\map {\operatorname {Offset} } {e,k} = \begin{cases}
\prod_{i = 1}^{\map {\operatorname {len} } e + k} {\map p i}^{\map I {e,k,i} } & : e \in \texttt{Prog} \\
0 & : \text {otherwise}
\end{cases}$
which is primitive recursive by:
* Definition by Cases is Primitive Recursive/Corollary
* Set of Codes for URM Programs is Primitive Recursive
* Bounded Product is Primitive Recursive
* Length Function is Primitive Recursive
* Prime Enumeration Function is Primitive Recursive
* Addition is Primitive Recursive
* Exponentiation is Primitive Recursive
* Constant Function is Primitive Recursive
Category:Primitive Recursive Functions
Category:URM Programs