Proof assistant

this article we will not attempt to describe all the dierent possible choicesof type theories. Instead we want to discuss the main underlying ideas, with a specialfocus on the use of type theory as the formalism for the description of theoriesincluding proofs

Wouldn’t it be nice to be able to conveniently use ordinary real number expressions within proof assistants? In this paper we outline how this can be done within a theorem proving framework. First, we formally establish upper and lower bounds for trigonometric and transcendental functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations can be performed in an algebraic setting. This pragmatic approach has been implemented as a strategy in PVS. The strategy provides a safe way to perform explicit calculations over real numbers in formal proofs.

La Mathematique n'est pas une science naturelle, susceptible de verification experimentale. La verite mathematique repose sur une notion de coherence logique, mais le phenomene d'incompletude fait qu'on ne peut esperer un fondement mathematique rigoureux intrinseque. De plus, le developpement de tres grandes preuves ne permet plus une verification manuelle. Cette fragilisation de l'edifice mathematique est preoccupante a l'heure ou notre tissu industriel, economique, social et medical repose sur l'utilisation de methodes mathematiques a grande echelle. Le developpement de l'Informatique permet neanmoins d'y remedier, notamment par la mise au point de nouvelles notations de preuves (theorie des types) ayant de bonnes proprietes calculatoires et linguistiques. L'augmentation des puissances de calcul disponibles, couplee a l'utilisation d'algorithmes sophistiques, permet une verification mecanique des demonstrations les plus touffues, comme e...

We address the problem of representing mathematical struc- tures in a proof assistant which: 1) is based on a type theory with de- pendent types, telescopes and a computational version of Leibniz equal- ity; 2) implements coercive subtyping, accepting multiple coherent paths between type families; 3) implements a restricted form of higher order unication and type reconstruction. We show how

We show how to write generic programs and proofs in MartinL of type theory. To this end we consider several extensions of MartinL of's logical framework for dependent types. Each extension has a universes of codes (signatures) for inductively defined sets with generic formation, introduction, elimination, and equality rules. These extensions are modeled on Dybjer and Setzer's finitely axiomatized theories of inductive-recursive definitions, which also have a universe of codes for sets, and generic formation, introduction, elimination, and equality rules.

Abstract: This article describes uses of dependencies in tools to maintainbig proofs in interactive proof assistants. The aim of this article isto describe the notion of dependency in proof assistants, how it can beused to design tools to maintain large proofs and how those tools couldbe integrated in proof environment.

This paper presents some experiments about the notion of generalization in proof assistants based on proof theory. We propose a mechanism which, starting from a proved theorem, allows to get a less specific result that can be instantiated and reused in other contexts.

We have formalized material from an introductory real analysis textbook in the proof assistant Scunak. Scunak is a system based on set theory encoded in a dependent type theory. We use the formalized material to illustrate some interesting aspects of the relationship between informal presentations of mathematics and their formal representation. We focus especially on a representative example proved using

We investigate how projective plane geometry can be formalized in a proof assistant such as Coq. Such a formalization increases the reliability of textbook proofs whose details and particular cases are often overlooked and left to the reader as exercises. Projective plane geometry is described through two different axiom systems which are formally proved equivalent. Usual properties such as decidability of equality of points (and lines) are then proved in a constructive way. The duality principle as well as formal models of projective plane geometry are then studied and implemented in Coq. Finally, we formally prove in Coq that Desargues’ property is independent of the axioms of projective plane geometry.