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21
A modular formalisation of finite group theory
 In TPHOLs
, 2007
"... Abstract. In this paper, we present a formalisation of elementary group theory done in Coq. This work is the first milestone of a longterm effort to formalise FeitThompson theorem. As our further developments will heavily rely on this initial base, we took special care to articulate it in the most ..."
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Cited by 18 (6 self)
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Abstract. In this paper, we present a formalisation of elementary group theory done in Coq. This work is the first milestone of a longterm effort to formalise FeitThompson theorem. As our further developments will heavily rely on this initial base, we took special care to articulate it in the most compositional way. 1
SourceLevel Proof Reconstruction for Interactive Theorem Proving
"... Abstract. Interactive proof assistants should verify the proofs they receive from automatic theorem provers. Normally this proof reconstruction takes place internally, forming part of the integration between the two tools. We have implemented sourcelevel proof reconstruction: resolution proofs are ..."
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Cited by 16 (2 self)
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Abstract. Interactive proof assistants should verify the proofs they receive from automatic theorem provers. Normally this proof reconstruction takes place internally, forming part of the integration between the two tools. We have implemented sourcelevel proof reconstruction: resolution proofs are automatically translated to Isabelle proof scripts. Users can insert this text into their proof development or (if they wish) examine it manually. Each step of a proof is justified by calling Hurd’s Metis prover, which we have ported to Isabelle. A recurrent issue in this project is the treatment of Isabelle’s axiomatic type classes. 1
Towards Automatic Proofs of Inequalities Involving Elementary Functions
 In Pragmatics of Decision Procedures in Automated Reasoning (PDPAR
, 2006
"... Inequalities involving functions such as sines, exponentials and logarithms lie outside the scope of decision procedures, and can only be solved using heuristic methods. Preliminary investigations suggest that many such problems can be solved by reduction to algebraic inequalities, which can then be ..."
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Cited by 11 (5 self)
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Inequalities involving functions such as sines, exponentials and logarithms lie outside the scope of decision procedures, and can only be solved using heuristic methods. Preliminary investigations suggest that many such problems can be solved by reduction to algebraic inequalities, which can then be decided by a decision procedure for the theory of real closed fields (RCF). The reduction involves replacing each occurrence of a function by a lower or upper bound (as appropriate) typically derived from a power series expansion. Typically this requires splitting the domain of the function being replaced, since most bounds are only valid for specific intervals. 1
Combining decision procedures for the reals
 In preparation
"... Vol. 2 (4:4) 2006, pp. 1–42 www.lmcsonline.org ..."
About the formalization of some results by Chebyshev in number theory
 Proceedings of TYPES’08, Vol. 5497 of LNCS
, 2009
"... Abstract. We discuss the formalization, in the Matita Interactive Theorem Prover, of a famous result by Chebyshev concerning the distribution of prime numbers, essentially subsuming, as a corollary, Bertrand’s postulate. Even if Chebyshev’s result has been later superseded by the stronger prime numb ..."
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Cited by 8 (4 self)
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Abstract. We discuss the formalization, in the Matita Interactive Theorem Prover, of a famous result by Chebyshev concerning the distribution of prime numbers, essentially subsuming, as a corollary, Bertrand’s postulate. Even if Chebyshev’s result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions ψ and θ still play a central role in the modern development of number theory. Differently from other recent formalizations of other results in number theory, our proof is entirely arithmetical. It makes use of most part of the machinery of elementary arithmetics, and in particular of properties of prime numbers, factorization, products and summations, providing a natural benchmark for assessing the actual development of the arithmetical knowledge base. 1
Extending a Resolution Prover for Inequalities on Elementary Functions
 In Logic for Programming, Artificial Intelligence, and Reasoning (LPAR), LNCS 4790
, 2007
"... Abstract. Experiments show that many inequalities involving exponentials and logarithms can be proved automatically by combining a resolution theorem prover with a decision procedure for the theory of real closed fields (RCF). The method should be applicable to any functions for which polynomial upp ..."
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Cited by 6 (4 self)
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Abstract. Experiments show that many inequalities involving exponentials and logarithms can be proved automatically by combining a resolution theorem prover with a decision procedure for the theory of real closed fields (RCF). The method should be applicable to any functions for which polynomial upper and lower bounds are known. Most bounds only hold for specific argument ranges, but resolution can automatically perform the necessary case analyses. The system consists of a superposition prover (Metis) combined with John Harrison’s RCF solver and a small amount of code to simplify literals with respect to the RCF theory. 1
Proof Assistants: history, ideas and future
"... In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assista ..."
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Cited by 3 (0 self)
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In this paper we will discuss the fundamental ideas behind proof assistants: What are they and what is a proof anyway? We give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics formalized. We will also briefly discuss the places where proof assistants are used and how we envision their extended use in the future. While being an introduction into the world of proof assistants and the main issues behind them, this paper is also a position paper that pushes the further use of proof assistants. We believe that these systems will become the future of mathematics, where definitions, statements, computations and proofs are all available in a computerized form. An important application is and will be in computer supported modelling and verification of systems. But their is still along road ahead and we will indicate what we believe is needed for the further proliferation of proof assistants.
Some considerations on the usability of Interactive Provers
"... Abstract. In spite of the remarkable achievements recently obtained in the field of mechanization of formal reasoning, the overall usability of interactive provers does not seem to be sensibly improved since the advent of the “second generation ” of systems, in the mid of the eighties. We try to ana ..."
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Cited by 3 (1 self)
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Abstract. In spite of the remarkable achievements recently obtained in the field of mechanization of formal reasoning, the overall usability of interactive provers does not seem to be sensibly improved since the advent of the “second generation ” of systems, in the mid of the eighties. We try to analyze the reasons of such a slow progress, pointing out the main problems and suggesting some possible research directions. 1
A Decision Procedure for Linear ”Big O” Equations
 J. Autom. Reason
"... Let F be the set of functions from an infinite set, S, to an ordered ring, R. For f, g, and h in F, the assertion f = g + O(h) means that for some constant C, f(x) − g(x)  ≤ Ch(x)  for every x in S. Let L be the firstorder language with variables ranging over such functions, symbols for 0,+, ..."
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Cited by 3 (0 self)
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Let F be the set of functions from an infinite set, S, to an ordered ring, R. For f, g, and h in F, the assertion f = g + O(h) means that for some constant C, f(x) − g(x)  ≤ Ch(x)  for every x in S. Let L be the firstorder language with variables ranging over such functions, symbols for 0,+, −,min,max, and absolute value, and a ternary relation f = g + O(h). We show that the set of quantifierfree formulas in this language that are valid in the intended class of interpretations is decidable, and does not depend on the underlying set, S, or the ordered ring, R. If R is a subfield of the real numbers, we can add a constant 1 function, as well as multiplication by constants from any computable subfield. We obtain further decidability results for certain situations in which one adds symbols denoting the elements of a fixed sequence of functions of strictly increasing rates of growth. 1
Understanding proofs
 The Philosophy of Mathematical Practice
, 2008
"... “Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a springcarriage ashore. But the awful lonesomeness is intolerable. The intense concentration of self in the middle of such a heartless immensity, my God! who can tell it? Mark, how when sailors in a ..."
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Cited by 3 (0 self)
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“Now, in calm weather, to swim in the open ocean is as easy to the practised swimmer as to ride in a springcarriage ashore. But the awful lonesomeness is intolerable. The intense concentration of self in the middle of such a heartless immensity, my God! who can tell it? Mark, how when sailors in a dead calm bathe in the open sea—mark how closely they hug their ship and only coast along her