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| {{distinguish|Herbrand–Ribet theorem|Ramification_group#Ramification_groups_in_upper_numbering{{!}}Herbrand's theorem on ramification groups}}
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| '''Herbrand's theorem''' is a fundamental result of [[mathematical logic]] obtained by [[Jacques Herbrand]] (1930).<ref>J. Herbrand: Recherches sur la theorie de la demonstration. Travaux de la Societe des Sciences et des Lettres de Varsovie, Class III, Sciences Mathematiques et Physiques, 33, 1930.</ref> It essentially allows a certain kind of reduction of [[first-order logic]] to [[propositional logic]]. Although Herbrand originally proved his theorem for arbitrary formulas of first-order logic,<ref>Samuel R. Buss: "Handbook of Proof Theory". Chapter 1, "An Introduction to Proof Theory". Elsevier, 1998.</ref> the simpler version shown here, restricted to formulas in [[prenex form]] containing only existential quantifiers became more popular.
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| Let
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| :<math>(\exists y_1,\ldots,y_n)F(y_1,\ldots,y_n)</math>
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| be a formula of first-order logic with
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| :<math>F(y_1,\ldots,y_n)</math> quantifier-free.
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| Then
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| :<math>(\exists y_1,\ldots,y_n)F(y_1,\ldots,y_n)</math>
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| is valid if and only if there exists a finite sequence of terms: <math>t_{ij}</math>, with
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| :<math>1 \le i \le k</math> and <math>1 \le j \le n</math>,
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| such that
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| :<math>F(t_{11},\ldots,t_{1n}) \vee \ldots \vee F(t_{k1},\ldots,t_{kn})</math>
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| is valid. If it is valid,
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| :<math>F(t_{11},\ldots,t_{1n}) \vee \ldots \vee F(t_{k1},\ldots,t_{kn})</math>
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| is called a ''Herbrand disjunction'' for
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| :<math>(\exists y_1,\ldots,y_n)F(y_1,\ldots,y_n).</math>
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| Informally: a formula <math>A</math> in [[prenex form]] containing existential quantifiers only is provable (valid) in first-order logic if and only if a disjunction composed of [[substitution instance]]s of the quantifier-free subformula of <math>A</math> is a [[tautology (logic)|tautology]] (propositionally derivable).
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| The restriction to formulas in prenex form containing only existential quantifiers does not limit the generality of the theorem, because formulas can be converted to prenex form and their universal quantifiers can be removed by [[Herbrandization]]. Conversion to prenex form can be avoided, if ''structural'' Herbrandization is performed. Herbrandization can be avoided by imposing additional restrictions on the variable dependencies allowed in the Herbrand disjunction.
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| ==Proof Sketch==
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| A proof of the non-trivial direction of the theorem can be constructed according to the following steps:
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| # If the formula <math>(\exists y_1,\ldots,y_n)F(y_1,\ldots,y_n)</math> is valid, then by completeness of cut-free [[sequent calculus]], which follows from [[Gentzen]]'s [[cut-elimination]] theorem, there is a cut-free proof of <math>\vdash (\exists y_1,\ldots,y_n)F(y_1,\ldots,y_n)</math>.
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| # Starting from above downwards, remove the inferences that introduce existential quantifiers.
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| # Remove contraction-inferences on previously existentially quantified formulas, since the formulas (now with terms substituted for previously quantified variables) might not be identical anymore after the removal of the quantifier inferences.
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| # The removal of contractions accumulates all the relevant substitution instances of <math>F(y_1,\ldots,y_n)</math> in the right side of the sequent, thus resulting in a proof of <math>\vdash F(t_{11},\ldots,t_{1n}), \ldots, F(t_{k1},\ldots,t_{kn})</math>, from which the Herbrand disjunction can be obtained.
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| However, [[sequent calculus]] and [[cut-elimination]] were not known at the time of Herbrand's theorem, and Herbrand had to prove his theorem in a more complicated way.
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| ==Generalizations of Herbrand's Theorem==
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| * Herbrand's theorem has been extended to arbitrary [[higher-order logics]] by using [[expansion-tree proof]]s.<ref>Dale Miller: A Compact Representation of Proofs. Studia Logica, 46(4), pp. 347--370, 1987.</ref> The deep representation of [[expansion-tree proofs]] correspond to Herbrand disjunctions, when restricted to first-order logic.
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| * Herbrand disjunctions and expansion-tree proofs have been extended with a notion of cut. Due to the complexity of cut-elimination, herbrand disjunctions with cuts can be non-elementarily smaller than a standard herbrand disjunction.
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| * Herbrand disjunctions have been generalized to Herbrand sequents, allowing Herbrand's theorem to be stated for sequents: "a skolemized sequent is derivable iff it has a Herbrand sequent".
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| ==See also==
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| * [[Herbrand structure]]
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| * [[Herbrand interpretation]]
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| * [[Herbrand universe]]
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| * [[Compactness theorem]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{Citation | last1=Buss | first1=Samuel R. | editor1-last=Maurice | editor1-first=Daniel | editor2-last=Leivant | editor2-first=Raphaël | title=Logic and Computational Complexity | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Computer Science | isbn=978-3-540-60178-4 | year=1995 | chapter=On Herbrand's Theorem | chapterurl=http://math.ucsd.edu/~sbuss/ResearchWeb/herbrandtheorem/ | pages=195–209}}.
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| [[Category:Proof theory]]
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| [[Category:Theorems in the foundations of mathematics]]
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| [[Category:Metatheorems]]
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Stonemason Truman from Prince Albert, likes to spend time wall art, property developers in singapore and psychology. Finds the charm in going to destinations around the entire world, of late just coming back from Historic Centre of Ceský Krumlov.
Look into my blog post; http://www.thewrightview.com/