Hal, I think you really might want to read some introductory textbook on
logic and formal systems, to check the standard definitions of `proof'
and `theorem.'

BTW, the following remarkable method heavily depends on what's provable.
I believe it will find its way into general computer science textbooks.

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The fastest and shortest algorithm for all well-defined problems
Marcus Hutter, IDSIA

An algorithm M is described that solves any well-defined problem p as
quickly as the fastest algorithm computing a solution to p, save for
a factor of 5 and low-order additive terms. M optimally distributes
resources between the execution of provably correct p-solving programs
and an enumeration of all proofs, including relevant proofs of program
correctness and of time bounds on program runtimes. M avoids Blum's
speed-up theorem by ignoring programs without correctness proof. M has
broader applicability and can be faster than Levin's universal search, the
fastest method for inverting functions save for a large multiplicative
constant. An extension of Kolmogorov complexity and two novel natural
measures of function complexity are used to show that the most efficient
program computing some function f is also among the shortest programs
provably computing f.

ftp://ftp.idsia.ch/pub/techrep/IDSIA-16-00.ps.gz

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Juergen Schmidhuber                                      www.idsia.ch

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