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A computational approach to pocklington certificates in type theory
 In Proc. of the 8th Int. Symp. on Functional and Logic Programming, volume 3945 of LNCS
, 2006
"... Abstract. Pocklington certificates are known to provide short proofs of primality. We show how to perform this in the framework of formal, mechanically checked, proofs. We present an encoding of certificates for the proof system Coq which yields radically improved performances by relying heavily on ..."
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Cited by 14 (4 self)
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Abstract. Pocklington certificates are known to provide short proofs of primality. We show how to perform this in the framework of formal, mechanically checked, proofs. We present an encoding of certificates for the proof system Coq which yields radically improved performances by relying heavily on computations inside and outside of the system (twolevel approach). 1 Formal Computational Proofs 1.1 Machines and the Quest for Correctness It is generally considered that modern mathematical logic was born towards the end of 19 th century, with the work of logicians like Frege, Peano, Russell or Zermelo, which lead to the precise definition of the notion of logical deduction and to formalisms like arithmetic, set theory or early type theory. From then on, a mathematical proof could be understood as a mathematical object itself, whose correction obeys some welldefined syntactical rules. In most formalisms, a formal proof is viewed as some treestructure; in natural deduction for instance, given to formal proofs σA and σB respectively of propositions A and B, these can be combined in order to build a proof of A ∧ B: σA σB ⊢ A ⊢ B ⊢ A ∧ B To sum things up, the logical point of view is that a mathematical statement holds in a given formalism if there exists a formal proof of this statement which follows the syntactical rules of the formalism. A traditional mathematical text can then be understood as an informal description of the formal proof. Things changed in the 1960ties, when N.G. de Bruijn’s team started to use computers to actually build formal proofs and verify their correctness. Using the fact that datastructures like formal proofs are very naturally represented in a computer’s memory, they delegated the proofverification work to the machine; their software Automath is considered as the first proofsystem and is the common
Formal proof—theory and practice
 Notices AMS
, 2008
"... Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are nume ..."
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Cited by 12 (1 self)
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Aformal proof is a proof written in a precise artificial language that admits only a fixed repertoire of stylized steps. This formal language is usually designed so that there is a purely mechanical process by which the correctness of a proof in the language can be verified. Nowadays, there are numerous computer programs known as proof assistants that can check, or even partially construct, formal proofs written in their preferred proof language. These can be considered as practical, computerbased realizations of the traditional systems of formal symbolic logic and set theory proposed as foundations for mathematics. Why should we wish to create formal proofs?
Integration of Deduction and Computation
 Applications of Computer Algebra
, 2000
"... We outline some of our approaches to the integration of Computer Algebra Systems and Automated Theorem Provers. Experimental couplings led to the development of the OMSCS framework, an architecture to specify the coupling of computational and reasoning systems. A model defining the context of a ..."
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Cited by 4 (1 self)
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We outline some of our approaches to the integration of Computer Algebra Systems and Automated Theorem Provers. Experimental couplings led to the development of the OMSCS framework, an architecture to specify the coupling of computational and reasoning systems. A model defining the context of a computation is proposed next. Finally, a multiagent approach, built upon our KOMET project, is then outlined through the integration of Mathematica.
System Description: Interface between Theorema And External Automated Deduction Systems
 In Linton and Sebastiani [175
, 2001
"... The interface between the Theorema system and external automated deduction systems is described. It provides a tool to access external provers within a Theorema session in the same way as \internal" Theorema provers. Currently 11 external systems are supported. The design of the interface allows ..."
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Cited by 3 (0 self)
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The interface between the Theorema system and external automated deduction systems is described. It provides a tool to access external provers within a Theorema session in the same way as \internal" Theorema provers. Currently 11 external systems are supported. The design of the interface allows combining external systems with each other as well as with \internal" Theorema provers.
Under consideration for publication in Formal Aspects of Computing The Mechanical Generation of Fault Trees for Reactive Systems via Retrenchment I: Combinational Circuits 1
"... Abstract. The manual construction of fault trees for complex systems is an errorprone and timeconsuming activity, encouraging automated techniques. In this paper we show how the retrenchment approach to formal system model evolution can be developed into a versatile structured approach for the mec ..."
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Abstract. The manual construction of fault trees for complex systems is an errorprone and timeconsuming activity, encouraging automated techniques. In this paper we show how the retrenchment approach to formal system model evolution can be developed into a versatile structured approach for the mechanical construction of fault trees. The system structure and the structure of retrenchment concessions interact to generate fault trees with appropriately deep nesting. We show how this approach can be extended to deal with minimisation, thereby diminishing the post hoc subsumption workload and potentially rendering some infeasible cases feasible.