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49
Analyzing Proofs in Analysis
 LOGIC: FROM FOUNDATIONS TO APPLICATIONS. EUROPEAN LOGIC COLLOQUIUM (KEELE
, 1993
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General logical metatheorems for functional analysis
, 2008
"... In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds ..."
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Cited by 45 (26 self)
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In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, HölderLipschitz, uniformly continuous, bounded and weakly quasinonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
Effective bounds from ineffective proofs in analysis: an application of functional interpretation and majorization
 J. Symbolic Logic
, 1992
"... We show how to extract effective bounds Φ for ∀u1∀v ≤γ tu∃wηG0–sentences which depend on u only (i.e. ∀u∀v ≤γ tu∃w ≤η ΦuG0) from arithmetical proofs which use analytical assumptions of the form (∗)∀xδ∃y ≤ρ sx∀zτF0 (δ, ρ, τ are arbitrary finite types, η ≤ 2, G0, F0 are quantifier–free and s, t close ..."
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Cited by 43 (22 self)
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We show how to extract effective bounds Φ for ∀u1∀v ≤γ tu∃wηG0–sentences which depend on u only (i.e. ∀u∀v ≤γ tu∃w ≤η ΦuG0) from arithmetical proofs which use analytical assumptions of the form (∗)∀xδ∃y ≤ρ sx∀zτF0 (δ, ρ, τ are arbitrary finite types, η ≤ 2, G0, F0 are quantifier–free and s, t closed terms). If τ ≤ 2, (∗) can be weakened to ∀xδ, zτ∃y ≤ρ sx∀z ̃ ≤τ zF0. This is used to establish new conservation results about weak Knig’s lemma WKL. Applications to proofs in classical analysis, especially uniqueness proofs in approximation theory, will be given in subsequent papers. 1 Introduction and basic notions Various theorems in classical analysis have the form A ≡ ∀x ∈ X∃y ∈ Kx ⊆ Y A1(x, y), where X,Y are complete separable metric spaces, Kx is compact in Y and A1 is purely universal. If an analytical sentence B is proved in using besides lemmata A only arithmetical constructions
Mathematically Strong Subsystems of Analysis With Low Rate of Growth of Provably Recursive Functionals
, 1995
"... This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of ..."
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Cited by 40 (24 self)
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This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. We establish the following extraction rule for an extension of GnA # by quantifierfree choice ACqf and analytical axioms # having the form #x # #y ## sx#z # F0 (including also a `non standard' axiom F  which does not hold in the full settheoretic model but in the strongly majorizable functionals): From a proof GnA # +ACqf + # # #u 1 , k 0 #v ## tuk#w 0 A0(u, k, v, w) one can extract a uniform bound # such that #u 1 , k 0 #v ## tuk#w # #ukA0 (u, k, v, w) holds in the full settheoretic type structure. In case n = 2 (resp. n = 3) #uk is a polynomial (resp. an elementary recursive function) in k, u M := #x. max(u0, . . . , ux). In the present paper we show that for n # 2, GnA # +ACqf+F  proves a generalization of the binary Knig's lemma yielding new conservation results since the conclusion of the above rule can be verified in G max(3,n) A # in this case. In a subsequent paper we will show that many important ine#ective analytical principles and theorems can be proved already in G2A # +ACqf+# for suitable #. 1
Modified Bar Recursion and Classical Dependent Choice
 In Logic Colloquium 2001
"... We introduce a variant of Spector's bar recursion in nite types to give a realizability interpretation of the classical axiom of dependent choice allowing for the extraction of witnesses from proofs of 1 formulas in classical analysis. We also give a bar recursive denition of the fan functional ..."
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Cited by 35 (17 self)
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We introduce a variant of Spector's bar recursion in nite types to give a realizability interpretation of the classical axiom of dependent choice allowing for the extraction of witnesses from proofs of 1 formulas in classical analysis. We also give a bar recursive denition of the fan functional and study the relationship of our variant of bar recursion with others. x1.
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 19 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Selection Functions, Bar Recursion and Backward Induction
 MATHEMATICAL STRUCTURES IN COMPUTER SCIENCE, 20, PP 127168
, 2010
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Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
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Cited by 16 (2 self)
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This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question