TITLE: What does "simplification of proofs as evaluation of programs" mean?
QUESTION [12 upvotes]: I am currently going through Philip Walder's "Proposition as Types" and a passage of the introduction has struck me:

for each way to simplify a proof
  there is a corresponding way to evaluate a program

What does simplify a proof means here? I have searched in my logic textbook and on the Internet but surprisingly enough I could not find any direct, informal explanation.

REPLY [9 votes]: Andrej's answer gives the technical explanation, but it might be useful to note, in addition, that "proof simplification" is a technical term that roughly corresponds to "lemma elimination", and is analogous to cut-elimination in the sequent calculus.
Informally, this operation is the opposite of a "proof simplification"! The things humans usually do to simplify proofs, e.g. introduce new notations, prove intermediate lemmas, generalize conclusions are the things that are eliminated by the operation you are referring to.