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Fix an effective enumeration $f_{e,n}$ of the $n$-ary partial recursive functions.

The S-m-n theorem says that program inputs can be compiled into an index: for every $m,n\ge 1$ there is a total recursive $s_n^m$ such thatHere $\simeq$ means that either both sides are undefined or both are defined and equal.

The recursion theorem says that every total recursive transformation $h$ of program indices has a semantic fixed point: some $e$ satisfies

The Rice theorem says that every nontrivial extensional property of programs computed by partial recursive functions is undecidable.

By the S-m-n theorem, there is a total recursive function $h$ such thatfor all $x,y$. Apply the recursion theorem to $h$. It supplies an index $e$ for whichConsequently

Fix an acceptable effective enumeration $f_{e,n}$ of the $n$-ary partial computable functions. The S-m-n theorem states that for every $m,n\ge 1$ there is a total computable function $s_n^m$ such thatwhere $\simeq$ means that both sides have the same value when defined and are otherwise both undefined.

The recursion theorem states that every total computable transformation $h$ of program indices has a semantic fixed point: there is an $e$ such that

Apply the S-m-n theorem to $g$. It gives a total computable function $h$ such thatThe recursion theorem supplies an index $e$ satisfying $f_{e,1}\simeq f_{h(e),1}$. Consequently

For each $x$, let $h(x)$ be an effectively obtained index of the partial functionThe existence of the total computable index transformation $h$ follows from the S-m-n theorem. By the recursion theorem, some $m$ satisfies $f_{m,1}\simeq f_{h(m),1}$. Therefore its domain isand hence

Kleene's recursion theorem says that for every total recursive map $f$ on program indices there is an index $e$ with the same partial function as program $f(e)$.