(related to Theorem: Simulating LOOP Programs Using WHILE Programs)
The only LOOP command, which is not already included in the syntax and sematics of a WHILE program is the command
\[\mathtt{LOOP~r_i~DO~P~END;}\quad\quad ( * )\]
Therefore, in order to show that all LOOP-computable functions are included in all WHILE-computable functions, formally,
\[L O O P \subseteq W H I L E,\]
it is sufficient to find a \(W H I L E\) program simulating the LOOP command \( ( * )\). The following program does the trick:
\[\begin{array}{rll}\hline&\mathbf{\text{Algorithm:}}~\mathsf{\text{Simulation of }}\mathtt{LOOP~r_i~DO~P~END;}\mathsf{\text{ using W H I L E commands}}&\\ \hline \hline {\tiny 1}&{\hspace{0em}\mathtt{WHILE~r_i\neq 0~DO} }&//~\text{ simulate }r_n:=r_i\\ {\tiny 2}&{\hspace{2em}\mathtt{r_n:=r_n + 1;}}\\ {\tiny 3}&{\hspace{2em}\mathtt{r_{n+1}:=r_{n+1} + 1;}}\\ {\tiny 4}&{\hspace{2em}\mathtt{r_i:=r_i - 1;}}\\ {\tiny 5}&{\hspace{0em}\mathtt{END;}}\\ {\tiny 6}&{\hspace{0em}\mathtt{WHILE~r_{n+1}\neq 0~DO} }\\ {\tiny 7}&{\hspace{2em}\mathtt{r_i:=r_i + 1;}}\\ {\tiny 8}&{\hspace{2em}\mathtt{r_{n+1}:=r_{n+1} - 1;}}\\ {\tiny 9}&{\hspace{0em}\mathtt{END;}}\\ {\tiny 10}&{\hspace{0em}\mathtt{WHILE~r_n\neq 0~DO} }&//~\text{ simulate }LOOP\text{ command}\\ {\tiny 11}&{\hspace{2em}\mathtt{P;}}\\ {\tiny 12}&{\hspace{2em}\mathtt{r_n:=r_n - 1;}}\\ {\tiny 13}&{\hspace{0em}\mathtt{END;}}\\ \hline\end{array}\]
