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  "body": "\n## Summary\n\nProof complexity is the part of [[proof theory|N4EDJNMTEF]] and of computational complexity that studies the size of formal proofs.\nIts main objects are proof systems for [[propositional logic|09K9MWD2QQ]], and the question is how short a proof of a given [[tautology|05006AJ7DZ]] can be.\nCook and Reckhow showed in 1979 that a proof system with short proofs of all tautologies exists if and only if NP equals coNP.\nSo lower bounds on proof size for ever stronger systems are a route toward separating complexity classes.\n\n## Explanations\n\n### L1 human\n\nA tautology is a formula that is true under every truth assignment.\nOne can check this with a truth table, but the table for $n$ variables has $2^n$ rows.\nA proof can be much shorter, but it is not clear that a short proof always exists.\nProof complexity asks how long the shortest proof must be, in a given proof system.\nFor example, in the resolution system, every proof of the pigeonhole principle, \"$n + 1$ pigeons do not fit in $n$ holes one per hole\", has size exponential in $n$.\n",
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