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  "body": "\n## Statement\n\nFix a standard [[Gödel numbering|MFJXPAF0R0]] of programs, and let $\\varphi^{(k)}_e$ be the partial [[computable function|KZSB5E471K]] of $k$ arguments computed by program $e$.\nLet $m, n \\ge 1$.\nThen there is a [[primitive recursive|5NVA2JRHPV]] function $s^m_n \\colon \\N^{m+1} \\to \\N$ such that for all $e \\in \\N$, $\\bar x \\in \\N^m$ and $\\bar y \\in \\N^n$:\n$$\\varphi^{(n)}_{s^m_n(e, \\bar x)}(\\bar y) \\simeq \\varphi^{(m+n)}_e(\\bar x, \\bar y).$$\nHere $\\simeq$ means that both sides are undefined, or both are defined and equal.\n\n## Remarks\n\nThe program $s^m_n(e, \\bar x)$ first writes the fixed numbers $\\bar x$ in front of its input $\\bar y$, then runs program $e$.\nBuilding this program from $e$ and $\\bar x$ is a simple manipulation of codes, so $s^m_n$ is primitive recursive.\n\nExample: let $e$ be a program for $(x, y) \\mapsto x + y$.\nThen $s^1_1(e, 5)$ is the number of a program for $y \\mapsto 5 + y$, and $x \\mapsto s^1_1(e, x)$ computes an index for \"add $x$\" from $x$.\n\nThe theorem is also called the parameter theorem, and it is due to Kleene.\n\n## Explanations\n\n### L1 human\n\nIf a program takes two inputs and you fix the first one, you get a program with one input.\nThe smn theorem says that the new program can be produced from the old one and the fixed input by a simple mechanical rule.\nIn programming terms, it is partial application, or currying, done on source code.\n\n### L2 human\n\nThe smn theorem is used to turn a construction \"uniform in a parameter\" into a computable function on indices.\nIt is the main tool in the proof of [[Kleene's recursion theorem|F3Z30M606H]].\nIt is also used to build many-one reductions, for example in the proof of [[Rice's theorem|AKDR5YBBE2]].\nNumberings that satisfy the smn theorem and have a universal function are called acceptable, and any two of them are computably isomorphic by a theorem of Rogers.\n",
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