S-M-N Theorem

The S-M-N theorem essentially states that we can do Closure Conversion for Partial Recursive Functions.

More specifically, the s-m-n theorem states that for every Partial Recursive Function f 𝑓 f with m + n 𝑚 𝑛 m+n arguments and n 𝑛 n -ary vector y ¯ ¯ 𝑦 \overline{y} of Natural Numbers, there exists a partial recursive function f ′ superscript 𝑓 ′ f^{\prime} with m 𝑚 m arguments such that f ′ ⁢ ( x ¯ ) ≃ f ⁢ ( x ¯ , y ¯ ) similar-to-or-equals superscript 𝑓 ′ ¯ 𝑥 𝑓 ¯ 𝑥 ¯ 𝑦 f^{\prime}(\overline{x})\simeq f(\overline{x},\overline{y}) for every x ¯ ¯ 𝑥 \overline{x} . Moreover, this process itself is Primitive Recursive.