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A web interface for matita
 In Proceedings of Intelligent Computer Mathematics (CICM 2012
"... This article describes a prototype implementation of a web interface for the Matita proof assistant [2]. The motivations behind our work are similar to those of several recent, related efforts [7, 9, 1, 8] (see also [6]). In particular: 1. creation of a web collaborative working environment for inte ..."
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This article describes a prototype implementation of a web interface for the Matita proof assistant [2]. The motivations behind our work are similar to those of several recent, related efforts [7, 9, 1, 8] (see also [6]). In particular: 1. creation of a web collaborative working environment for interactive theorem proving, aimed at fostering knowledgeintensive cooperation, content creation and management; 2. exploitation of the markup in order to enrich the document with several kinds of annotations or active elements; annotations may have both a presentational/hypertextual nature, aimed to improve the quality of the proof script as a human readable document, or a more semantic nature, aimed to help the system in its processing (or reprocessing) of the script; 3. platform independence with respect to operating systems, and wider accessibility also for users using devices with limited resources; 4. overcoming the installation issues typical of interactive provers, also in view of attracting a wider audience, especially in the mathematical community.
Recycling Proof Patterns in Coq: Case Studies
 Journal Mathematics in Computer Science, accepted
, 2014
"... Abstract. Development of Interactive Theorem Provers has led to the creation of big libraries and varied infrastructures for formal proofs. However, despite (or perhaps due to) their sophistication, the reuse of libraries by nonexperts or across domains is a challenge. In this paper, we provide de ..."
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Abstract. Development of Interactive Theorem Provers has led to the creation of big libraries and varied infrastructures for formal proofs. However, despite (or perhaps due to) their sophistication, the reuse of libraries by nonexperts or across domains is a challenge. In this paper, we provide detailed case studies and evaluate the machinelearning tool ML4PG built to interactively datamine the electronic libraries of proofs, and to provide user guidance on the basis of proof patterns found in the existing libraries.
Consistency of the minimalist foundation with Church thesis and Bar Induction. submitted
, 2010
"... We consider a version of the minimalist foundation previously introduced to formalize predicative constructive mathematics. This foundation is equipped with two levels to meet the usual informal practice of developing mathematics in an extensional set theory (its extensional level) with the possibil ..."
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We consider a version of the minimalist foundation previously introduced to formalize predicative constructive mathematics. This foundation is equipped with two levels to meet the usual informal practice of developing mathematics in an extensional set theory (its extensional level) with the possibility of formalizing it in an intensional theory enjoying a proofs as programs semantics (its intensional level). For the intensional level we show a realizability interpretation validating Bar Induction and formal Church thesis for typetheoretic functions. This is possible because in our foundation the wellknown result by Kleene that Brouwer’s principle of Bar Induction is inconsistent with the formal Church thesis for choice sequences can be decomposed as follows: Brouwer’s Bar Induction, where choice sequences are functional relations, is inconsistent with the formal Church thesis for typetheoretic functions (from natural numbers to natural numbers) and the axiom of unique choice transforming a functional relation between natural numbers into a typetheoretic function. As a consequence this model disproves the validity of the axiom of unique choice in our foundation. This model can serve to interpret the whole foundation in a classical predicative set theory by keeping the computational interpretation of predicative sets as data types and their typetheoretic functions as programs. Moreover it shows that choice sequences of Cantor space, those of Baire space, and real numbers both as Dedekind cuts or Cauchy sequences, do not form a set in the minimalist foundation.
Formalizing Turing Machines
"... Abstract. We discuss the formalization, in the Matita Theorem Prover, of a few, basic results on Turing Machines, up to the existence of a (certified) Universal Machine. The work is meant to be a preliminary step towards the creation of a formal repository in Complexity Theory, and is a small piece ..."
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Abstract. We discuss the formalization, in the Matita Theorem Prover, of a few, basic results on Turing Machines, up to the existence of a (certified) Universal Machine. The work is meant to be a preliminary step towards the creation of a formal repository in Complexity Theory, and is a small piece in our Reverse Complexity program, aiming to a comfortable, machine independent axiomatization of the field. 1
Why topology in the minimalist foundation must be pointfree
, 2013
"... We give arguments explaining why, when adopting a minimalist approach to constructive mathematics as that formalized in our twolevel minimalist foundation, the choice for a pointfree approach to topology is not just a matter of convenience or mathematical elegance, but becomes compulsory. The main ..."
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We give arguments explaining why, when adopting a minimalist approach to constructive mathematics as that formalized in our twolevel minimalist foundation, the choice for a pointfree approach to topology is not just a matter of convenience or mathematical elegance, but becomes compulsory. The main reason is that in our foundation real numbers, either as Dedekind cuts or as Cauchy sequences, do not form a set.
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.
The Strategy Challenge in SMT Solving
"... Abstract. Highperformance SMT solvers contain many tightly integrated, handcrafted heuristic combinations of algorithmic proof methods. While these heuristic combinations tend to be highly tuned for known classes of problems, they may easily perform badly on classes of problems not anticipated by ..."
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Abstract. Highperformance SMT solvers contain many tightly integrated, handcrafted heuristic combinations of algorithmic proof methods. While these heuristic combinations tend to be highly tuned for known classes of problems, they may easily perform badly on classes of problems not anticipated by solver developers. This issue is becoming increasingly pressing as SMT solvers begin to gain the attention of practitioners in diverse areas of science and engineering. We present a challenge to the SMT community: to develop methods through which users can exert strategic control over core heuristic aspects of SMT solvers. We present evidence that the adaptation of ideas of strategy prevalent both within the Argonne and LCF theorem proving paradigms can go a long way towards realizing this goal. Prologue. Bill McCune, Kindness and Strategy, by Grant Passmore I would like to tell a short story about Bill, of how I met him, and one way his work and kindness impacted my life.
A formal proof of borodintrakhtenbrot’s gap theorem
 In Certified Programs and Proofs  Third International Conference, CPP 2013
"... Abstract. In this paper, we discuss the formalization of the well known Gap Theorem of Complexity Theory, asserting the existence of arbitrarily large gaps between complexity classes. The proof is done at an abstract, machine independent level, and is particularly aimed to identify the minimal set ..."
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Abstract. In this paper, we discuss the formalization of the well known Gap Theorem of Complexity Theory, asserting the existence of arbitrarily large gaps between complexity classes. The proof is done at an abstract, machine independent level, and is particularly aimed to identify the minimal set of assumptions required to prove the result (smaller than expected, actually). The work is part of a long term reverse complexity program, whose goal is to obtain, via a reverse methodological approach, a formal treatment of Complexity Theory at a comfortable level of abstraction and logical rigor. 1
Elaboration in dependent type theory
"... Abstract. We describe the elaboration algorithm that is used in Lean, a new interactive theorem prover based on dependent type theory. To be practical, interactive theorem provers must provide mechanisms to resolve ambiguities and infer implicit information, thereby supporting convenient input of ex ..."
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Abstract. We describe the elaboration algorithm that is used in Lean, a new interactive theorem prover based on dependent type theory. To be practical, interactive theorem provers must provide mechanisms to resolve ambiguities and infer implicit information, thereby supporting convenient input of expressions and proofs. Lean’s elaborator supports higherorder unification, adhoc overloading, insertion of coercions, type class inference, the use of tactics, and the computational reduction of terms. The interactions between these components are subtle and complex, and Lean’s elaborator has been carefully designed to balance efficiency and usability. 1
Foundational Extensible Corecursion
, 2014
"... This paper presents a theoretical framework for defining corecursive functions safely in a total setting, based on corecursion upto and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under wellbehaved operations, including con ..."
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This paper presents a theoretical framework for defining corecursive functions safely in a total setting, based on corecursion upto and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under wellbehaved operations, including constructors. Corecursive functions that are well behaved can be registered as such, thereby increasing the corecursor’s expressiveness. To the extensible corecursor corresponds an equally flexible coinduction principle. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The approach is foundational: The corecursor is derived from first principles, without requiring new axioms or extensions of the logic. This ensures that no inconsistencies can be introduced by omissions in a termination or productivity check.