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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 Even if Chebyshev’s result has been later superseded by the stronger prime numb ..."
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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 Even if Chebyshev’s result has been later superseded by the stronger prime number theorem, his machinery, and in particular the two functions ψ and θ still play a central role in the modern development of number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
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
Formalizing Arrow’s theorem
"... Abstract. We present a small project in which we encoded a proof of Arrow’s theorem – probably the most famous results in the economics field of social choice theory – in the computer using the Mizar system. We both discuss the details of this specific project, as well as describe the process of for ..."
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Abstract. We present a small project in which we encoded a proof of Arrow’s theorem – probably the most famous results in the economics field of social choice theory – in the computer using the Mizar system. We both discuss the details of this specific project, as well as describe the process of formalization (encoding proofs in the computer) in general. Keywords: formalization of mathematics, Mizar, social choice theory, Arrow’s theorem, GibbardSatterthwaite theorem, proof errors.
Stateless HOL Dedicated to Roel de Vrijer, in the tradition of Automath
"... Abstract. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far has been moved out of the kernel. This means that ..."
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Abstract. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far has been moved out of the kernel. This means that the kernel now is purely functional. The changes to the system are small. All existing HOL Light developments can be run by the stateless system with only minor changes. The basic principle behind the system is not to name constants by strings, but by pairs consisting of a string and a definition. This means that the data structures for the terms are all merged into one big graph. OCaml – the implementation language of the system – can use pointer equality to establish equality of data structures fast. This allows the system to run at acceptable speeds. Our system is about 1 6 version of HOL Light. th slower than the stateful
A proof of Bertrand’s postulate
"... We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand’s postulate. Even if Chebyshev’s result has been later superseded by the stronger prime number theorem, his machinery ..."
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We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, 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. The proof makes use of most part of the machinery of elementary arithmetics, and in particular of properties of prime numbers, gcd, products and summations, providing a natural benchmark for assessing the actual development of the arithmetical knowledge base. 1.
Stateless HOL
"... Dedicated to Roel de Vrijer, in the tradition of Automath. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far h ..."
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Dedicated to Roel de Vrijer, in the tradition of Automath. We present a version of the HOL Light system that supports undoing definitions in such a way that this does not compromise the soundness of the logic. In our system the code that keeps track of the constants that have been defined thus far has been moved out of the kernel. This means that the kernel now is purely functional. The changes to the system are small. All existing HOL Light developments can be run by the stateless system with only minor changes. The basic principle behind the system is not to name constants by strings, but by pairs consisting of a string and a definition. This means that the data structures for the terms are all merged into one big graph. OCaml – the implementation language of the system – can use pointer equality to establish equality of data structures fast. This allows the system to run at acceptable speeds. Our system runs at about 85 % of the speed of the stateful version of HOL Light.
UITP 2010 Pollackinconsistency
"... For interactive theorem provers a very desirable property is consistency: it should not be possible to prove false theorems. However, this is not enough: it also should not be possible to think that a theorem that actually is false has been proved. More precisely: the user should be able to know wha ..."
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For interactive theorem provers a very desirable property is consistency: it should not be possible to prove false theorems. However, this is not enough: it also should not be possible to think that a theorem that actually is false has been proved. More precisely: the user should be able to know what it is that the interactive theorem prover is proving. To make these issues concrete we introduce the notion of Pollackconsistency. This property is related to a system being able to correctly parse formulas that it printed itself. In current systems it happens regularly that this fails. We argue that a good interactive theorem prover should be Pollackconsistent. We show with examples that many interactive theorem provers currently are not Pollackconsistent. Finally we describe a simple approach for making a system Pollackconsistent, which only consists of a small modification to the printing code of the system. The most intelligent creature in the universe is a rock. None would know it because they have lousy I/O. — quote from the Internet
Nonconstructive complex analysis in Coq
"... Winding numbers are fundamental objects arising in algebraic topology, with many applications in nonconstructive complex analysis. We present a formalization in Coq of the winding numbers and their main properties. As an application of this development, we also give nonconstructive proofs of the f ..."
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Winding numbers are fundamental objects arising in algebraic topology, with many applications in nonconstructive complex analysis. We present a formalization in Coq of the winding numbers and their main properties. As an application of this development, we also give nonconstructive proofs of the following theorems: the Fundamental Theorem of Algebra, the
A SYNTHESIS OF THE PROCEDURAL AND DECLARATIVE STYLES OF INTERACTIVE THEOREM PROVING
"... Abstract. We propose a synthesis of the two proof styles of interactive theorem proving: the procedural style (where proofs are scripts of commands, like in Coq) and the declarative style (where proofs are texts in a controlled natural language, like in Isabelle/Isar). Our approach combines the adva ..."
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Abstract. We propose a synthesis of the two proof styles of interactive theorem proving: the procedural style (where proofs are scripts of commands, like in Coq) and the declarative style (where proofs are texts in a controlled natural language, like in Isabelle/Isar). Our approach combines the advantages of the declarative style – the possibility to write formal proofs like normal mathematical text – and the procedural style – strong automation and help with shaping the proofs, including determining the statements of intermediate steps. Our approach is new, and differs significantly from the ways in which the procedural and declarative proof styles have been combined before in the Isabelle, Ssreflect and Matita systems. Our approach is generic and can be implemented on top of any procedural interactive theorem prover, regardless of its architecture and logical foundations. To show the viability of our proposed approach, we fully implemented it as a proof interface called miz3, on top of the HOL Light interactive theorem prover. The declarative language that this interface uses is a slight variant of the language of the Mizar system, and can be used for any interactive theorem prover regardless of its logical foundations. The miz3 interface allows easy access to the full set of tactics and formal libraries of HOL Light, and as such has ‘industrial strength’. Our approach gives a way to automatically convert any procedural proof to a declarative counterpart, where the converted proof is similar in size to the original. As all declarative systems have essentially the same proof language, this gives a straightforward way to port proofs between interactive theorem provers. 1.