Editing Kleene's recursion theorem (section)

=== Application to quines ===
A classic example using the second recursion theorem is the function <math>Q(x,y)=x</math>. The corresponding  index <math>p</math> in this case yields a computable function that outputs its own index when applied to any value.{{r|Cutland1980_204}} When expressed as computer programs, such indices are known as '''[[Quine (computing)|quine]]s'''.

The following example in [[Lisp programming language|Lisp]] illustrates how the <math>p</math> in the corollary can be effectively produced from the function <math>Q</math>.   The function <code>s11</code> in the code is the function of that name produced by the [[S-m-n theorem]].

<code>Q</code> can be changed to any two-argument function.
<syntaxhighlight lang="lisp">
(setq Q '(lambda (x y) x))
(setq s11 '(lambda (f x) (list 'lambda '(y) (list f x 'y))))
(setq n (list 'lambda '(x y) (list Q (list s11 'x 'x) 'y)))
(setq p (eval (list s11 n n)))
</syntaxhighlight>

The results of the following expressions should be the same. <math>\varphi</math> <code>p(nil)</code>
<syntaxhighlight lang="lisp">
(eval (list p nil))
</syntaxhighlight>
<code>Q(p, nil)</code>
<syntaxhighlight lang="lisp">
(eval (list Q p nil))
</syntaxhighlight>