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Analytic proof
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<h2 id="structural-proof-theory">Structural proof theory</h2> <p>In proof theory, the notion of analytic proof provides the fundamental concept that brings out the similarities between a number of essentially distinct <a href="/facts/Proof_calculus/oXTtCnhC">proof calculi</a>, so defining the subfield of <a href="/facts/Structural_proof_theory/zF4GemsX">structural proof theory</a>. There is no uncontroversial general definition of analytic proof, but for several proof calculi there is an accepted notion. For example: </p> <ul><li>In <a href="/facts/Gerhard_Gentzen/9ujN0ssI">Gerhard Gentzen</a>'s <a href="/facts/Natural_deduction_calculus/PPtgwoaJ">natural deduction calculus</a> the analytic proofs are those in normal form; that is, no formula occurrence is both the principal premise of an elimination rule and the conclusion of an introduction rule;</li> <li>In Gentzen's <a href="/facts/Sequent_calculus/62NMylhL">sequent calculus</a> the analytic proofs are those that do not use the <a href="/facts/Cut-elimination/o9nltnr0">cut rule</a>.</li></ul> <p>However, it is possible to extend the <a href="/facts/Inference_rule/QV5NklQ9">inference rules</a> of both calculi so that there are proofs that satisfy the condition but are not analytic. For example, a particularly tricky example of this is the <i>analytic cut rule</i>, used widely in the <a href="/facts/Tableau_method/Tyrp5src">tableau method</a>, which is a special case of the cut rule where the cut formula is a <a href="/facts/Subformula/YpTVlzuj">subformula</a> of side formulae of the cut rule: a proof that contains an analytic cut is by virtue of that rule not analytic. </p><p>Furthermore, proof calculi that are not analogous to Gentzen's calculi have other notions of analytic proof. For example, the <a href="/facts/Calculus_of_structures/9wgtV7qm">calculus of structures</a> organises its inference rules into pairs, called the up fragment and the down fragment, and an analytic proof is one that only contains the down fragment. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Proof-theoretic_semantics/qo68D5Py">Proof-theoretic semantics</a></li></ul> <ul><li><a href="/facts/Bernard_Bolzano/BUbRVY4p">Bernard Bolzano</a> (1817). Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation. In <i>Abhandlungen der koniglichen bohmischen Gesellschaft der Wissenschaften</i> Vol. V, pp.225-48.</li> <li><a href="/facts/Frank_Pfenning/TI1350yj">Frank Pfenning</a> (1984). Analytic and Non-analytic Proofs. In <i>Proc. 7th International <a href="/facts/Conference_on_Automated_Deduction/tJCaU8T7">Conference on Automated Deduction</a></i>.</li> <li>Jan Šebestik (2007). <a href="http://plato.stanford.edu/entries/bolzano-logic">Bolzano's Logic</a>. Entry in the <i><a href="/facts/Stanford_Encyclopedia_of_Philosophy/LDuL5HW4">Stanford Encyclopedia of Philosophy</a></i>.</li></ul>