So, if you look at the “big picture”, the upper bound is exponential. But if you look at it locally, the introduction of new rules and premises makes it combinatorial.

]]>I believe my statement is correct. It grows exponentially w.r.t. the proof length, that is what Marcus Hutter’s famous paper The Fastest and Shortest Algorithm for All Well-Defined Problems says. He means the length of the binary representation of the proof, but I think it applies here too. There is a finite number of rules and axioms, ultimately a inference tree of size S can be turned into a binary string of length O(S).

Maybe you mean w.r.t. to something else. For instance the complexity w.r.t. the length of the theorem is unbounded.

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