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16
Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
MetiTarski: An Automatic Prover for the Elementary Functions
"... Abstract. Many inequalities involving the functions ln, exp, sin, cos, etc., can be proved automatically by MetiTarski: a resolution theorem prover (Metis) modified to call a decision procedure (QEPCAD) for the theory of real closed fields. The decision procedure simplifies clauses by deleting liter ..."
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Abstract. Many inequalities involving the functions ln, exp, sin, cos, etc., can be proved automatically by MetiTarski: a resolution theorem prover (Metis) modified to call a decision procedure (QEPCAD) for the theory of real closed fields. The decision procedure simplifies clauses by deleting literals that are inconsistent with other algebraic facts, while deleting as redundant clauses that follow algebraically from other clauses. MetiTarski includes special code to simplify arithmetic expressions.
Formalizing integration theory with an application to probabilistic algorithms
 Proceedings of TPHOLs 2004. Number 3223 in LNCS, Pack City
, 2004
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Automating Proofs in Category Theory
"... Abstract. We introduce a semiautomated proof system for basic categorytheoretic reasoning. It is based on a firstorder sequent calculus that captures the basic properties of categories, functors and natural transformations as well as a small set of proof tactics that automate proof search in this ..."
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Abstract. We introduce a semiautomated proof system for basic categorytheoretic reasoning. It is based on a firstorder sequent calculus that captures the basic properties of categories, functors and natural transformations as well as a small set of proof tactics that automate proof search in this calculus. We demonstrate our approach by automating the proof that the functor categories Fun[C × D,E] and Fun[C,Fun[D,E]] are naturally isomorphic. 1
A theoretical analysis of hierarchical proofs
 In Asperti et al
, 2003
"... www.uclic.ucl.ac.uk/imp Abstract. Hierarchical proof presentations are ubiquitous within logic and computer science, but have made little impact on mathematics in general. The reasons for this are not currently known, and need to be understood if mathematical knowledge management systems are to gain ..."
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www.uclic.ucl.ac.uk/imp Abstract. Hierarchical proof presentations are ubiquitous within logic and computer science, but have made little impact on mathematics in general. The reasons for this are not currently known, and need to be understood if mathematical knowledge management systems are to gain acceptance in the mathematical community. We report on some initial experiments with three users of a set of webbased hierarchical proofs, which suggest that usability problems could be a factor. In order to better understand these problems we present a theoretical analysis of hierarchical proofs using Cognitive Dimensions [6]. The analysis allows us to formulate some concrete hypotheses about the usability of hierarchical proof presentations. 1
The Ergo 5 Generic Proof Engine
, 1997
"... s and compressed postscript files are available via http://svrc.it.uq.edu.au The Ergo 5 Generic Proof Engine Mark Utting Abstract This paper describes the design principles and the architecture of the latest version of the Ergo proof engine, Ergo 5. Ergo 5 is a generic interactive theorem pro ..."
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s and compressed postscript files are available via http://svrc.it.uq.edu.au The Ergo 5 Generic Proof Engine Mark Utting Abstract This paper describes the design principles and the architecture of the latest version of the Ergo proof engine, Ergo 5. Ergo 5 is a generic interactive theorem prover, similar to Isabelle, but based on sequent calculus rather than natural deduction and with a quite different approach to handling variable scoping. An efficient implementation of Ergo 5, based on QuProlog, is also described, together with some benchmark results. 1 Motivation The Software Verification Research Centre, a special research centre of the Australian Research Council, is developing a suite of tools for reasoning about Z specifications and verifying refinement of specifications to code. There are several different projects investigating various aspects and approaches. To gain synergy, we want a common proof tool for all the projects, even though they have differing requi...
Some Mathematical Case Studies in ProofPowerHOL
, 2004
"... TPHOLs conference covers all aspects of theorem proving in higher order logics as well as related topics in theorem proving and verification. There were 42 papers submitted to TPHOLs 2004 in the full research category, each of which was refereed by at least 3 reviewers selected by the program commit ..."
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TPHOLs conference covers all aspects of theorem proving in higher order logics as well as related topics in theorem proving and verification. There were 42 papers submitted to TPHOLs 2004 in the full research category, each of which was refereed by at least 3 reviewers selected by the program committee. Of these submissions, 21 were accepted for presentation at the conference and publication in volume 3223 of Springer’s Lecture Notes in Computer Science series. In keeping with longstanding tradition, TPHOLs 2004 also offered a venue for the presentation of work in progress, where researchers invite discussion by means of a brief introductory talk and then discuss their work at a poster
Computer Theorem Proving in Math
"... We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context. ..."
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We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context.
An Inductive Approach to Formalizing Notions of Number Theory Proofs
, 2001
"... Introduction During the past 2025 years many parts of mathematics have been formalized and mechanized in various settings and using various systems. Mechanizations as such are thus (in general) no longer seen as achievements by themselves. But what does it mean that a result a has been mechaniz ..."
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Introduction During the past 2025 years many parts of mathematics have been formalized and mechanized in various settings and using various systems. Mechanizations as such are thus (in general) no longer seen as achievements by themselves. But what does it mean that a result a has been mechanized? At least it means that there is some computer system in which the result can be formulated and that the system can check (more or less automatically) that the proof is correct. But does it also mean that it is formulated in a language which is similar to that in, say, a textbook and/or that the proof follows the same lines of reasoning and uses the same concepts as the proof in the textbook? This is not always the case. One of the arguments in favor of formalized mathematics b is that it helps clarify subtle arguments and this in turn can be helpful for developing new theory. But this use of a formalization gets dicult if the mechanization is too