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Proof Transformations in HigherOrder Logic
, 1987
"... We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, ..."
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We investigate the problem of translating between different styles of proof systems in higherorder logic: analytic proofs which are well suited for automated theorem proving, and nonanalytic deductions which are well suited for the mathematician. Analytic proofs are represented as expansion proofs, H, a form of the sequent calculus we define, nonanalytic proofs are represented by natural deductions. A nondeterministic translation algorithm between expansion proofs and Hdeductions is presented and its correctness is proven. We also present an algorithm for translation in the other direction and prove its correctness. A cutelimination algorithm for expansion proofs is given and its partial correctness is proven. Strong termination of this algorithm remains a conjecture for the full higherorder system, but is proven for the firstorder fragment. We extend the translations to a nonanalytic proof system which contains a primitive notion of equality, while leaving the notion of expansion proof unaltered. This is possible, since a nonextensional equality is definable in our system of type theory. Next we extend analytic and nonanalytic proof systems and the translations between them to include extensionality. Finally, we show how the methods and notions used so far apply to the problem of translating expansion proofs into natural deductions. Much care is taken to specify this translation in a
A nonconstructive proof of Gentzen’s Hauptsatz for second order predicate logic
 Bulletin of the American Math. Society
, 1966
"... Takeuti [3] showed that the consistency of analysis (i.e. second order number theory) is finitistically implied by the Hauptsatz for second order logic » i.e. by the proposition that every theorem of this system is derivable without cut. 1 We will prove that, conversely, the Hauptsatz for this syste ..."
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Cited by 7 (0 self)
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Takeuti [3] showed that the consistency of analysis (i.e. second order number theory) is finitistically implied by the Hauptsatz for second order logic » i.e. by the proposition that every theorem of this system is derivable without cut. 1 We will prove that, conversely, the Hauptsatz for this system follows from a certain generalization of the consistency of analysis; namely from: I. Every countable set of relations among natural numbers is included in an o)model. An comodel is a collection of relations among natural numbers which is closed under the second order comprehension axiom. Henkin [l] has shown that a second order formula is derivable with the cut rule if and only if it is valid in all (countable) comodels. When the given set of relations consists only of the successor relation, I asserts the consistency of analysis. The formalism for second order predicate logic which we will use is obtained from the system of predicate logic of finite order given in Schutte [2] by dropping all expressions and bound variables of types other than 0 (individuals), 1 (propositions) and (0, 0, • • • , 0) (relations among individuals). Thus, expressions of type 0 are built up from constants and free variables of type 0 using function constants. The expressions of type (0, • • • , 0) are constants, free variables and expressions Xx? • • • x£4(x?, • • • , x£), where A (a°u • • • , a£) is a wff (expression of type 1). The logical symbols other than X are —*, v and V. The notation and terminology of [2] will be assumed. In particular, the notions of strict derivation and partial valuation will be the same as in [2], except that they refer to the second order logic and not the full system of [2], and that we require of a partial valuation that whenever Vx T A(x T) is true (t), then so is A{a T) for some free 1 Actually, Takeuti asserts this result, not for second order logic, but for G
Cumulative HigherOrder Logic as a Foundation for Set Theory
"... The systems K of transnite cumulative types up to are extended to systems K 1 that include a natural innitary inference rule, the socalled limit rule. For countable a semantic completeness theorem for K 1 is proved by the method of reduction trees, and it is shown that every model of K 1 ..."
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The systems K of transnite cumulative types up to are extended to systems K 1 that include a natural innitary inference rule, the socalled limit rule. For countable a semantic completeness theorem for K 1 is proved by the method of reduction trees, and it is shown that every model of K 1 is equivalent to a cumulative hierarchy of sets. This is used to show that several axiomatic rstorder set theories can be interpreted in K 1 , for suitable . Keywords: cumulative types, innitary inference rule, logical foundations of set theory. MSC: 03B15 03B30 03E30 03F25 1 Introduction The idea of founding mathematics on a theory of types was rst proposed by Russell [20] (foreshadowed already in [19]), and subsequently implemented by Whitehead and Russell [26]. The formal systems presented in these works were later simplied and cast into their modern shape by Ramsey [18]. Godel [9] and Tarski [25] were the rst to restrict the type structure to types of unary predi...
Complete CutFree Tableaux for Equational Simple Type Theory
, 2009
"... We present a cutfree tableau system for a version of Church’s simple type normalization operator that completely hides the details of lambda conversion. We prove completeness of the system relative to Henkin models. The proof constructs Henkin models using the novel notion of a value system. 1 ..."
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We present a cutfree tableau system for a version of Church’s simple type normalization operator that completely hides the details of lambda conversion. We prove completeness of the system relative to Henkin models. The proof constructs Henkin models using the novel notion of a value system. 1