Note. The title was modified. Previous title was "Every theorem t has a proof no more complex than~|t|. Is this right?"

The question ("Is Kolmogorov complexity (KC) relevant for proof theory?") arises because every theorem has a proof with low Kolmogorov complexity". To be more specific,

*Every theorem $t$ has a proof $p$ with $K(p)\leq K(t)+c\leq|t|+c'$, where $c$ and $c'$ are constants.*

*Proof:* It is assumed that there is a proof checking algorithm $C(x)$ that
outputs TRUE if $x$ is a correct proof, FALSE otherwise.
Then there exists a fixed (depending on the formal system) Turing
machine $M$ with
Input: $t$ a string (that may be a theorem)

- $M$ enumerates all the proofs until (and if) $t$ is proved.

- When $t$ is proved, print the proof $p$ and halt.

Thus, if $t$ is a theorem, $M(t)$ prints a proof of $t$.

Using the universal TM, we get
$ K(p) \leq K(t) + c \leq |t| + c' $
where $c$ and $c'$ are constants.

Thus, and apart from a constant, no theorem needs a proof with Kolmogorov complexity greater than $|t|$. To me this seems a bit strange.

In other words, the number of bits of "inspiration" (or "non-deterministic bits", or "oracle bits") needed to prove any theorem $t$ is at most $|t|+c$.

If this is true, the very lengthy proofs that sometimes are needed to prove some simple to state theorems are necessarily very regular / structured / compressible (synonymous).