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The Gentle Art of Levitation
"... We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description—a firstclass value in a datatype of descriptions. Moreover, the latter itself has a de ..."
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We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description—a firstclass value in a datatype of descriptions. Moreover, the latter itself has a description. Datatypegeneric programming thus becomes ordinary programming. We show some of the resulting generic operations and deploy them in particular, useful ways on the datatype of datatype descriptions itself. Surprisingly this apparently selfsupporting setup is achievable without paradox or infinite regress. 1.
Why dependent types matter
 In preparation, http://www.epig.org/downloads/ydtm.pdf
, 2005
"... We exhibit the rationale behind the design of Epigram, a dependently typed programming language and interactive program development system, using refinements of a well known program—merge sort—as a running example. We discuss its relationship with other proposals to introduce aspects of dependent ty ..."
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We exhibit the rationale behind the design of Epigram, a dependently typed programming language and interactive program development system, using refinements of a well known program—merge sort—as a running example. We discuss its relationship with other proposals to introduce aspects of dependent types into functional programming languages and sketch some topics for further work in this area. 1.
Universe polymorphism in Coq
 In Proceedings of the 5th International Conference on Interactive Theorem Proving (ITP 2014), volume 8558 of LNCS
, 2014
"... Abstract. Universes are used in Type Theory to ensure consistency by checking that definitions are wellstratified according to a certain hierarchy. In the case of the Coq proof assistant, based on the predicative Calculus of Inductive Constructions (pCIC), this hierachy is built from an impredicat ..."
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Abstract. Universes are used in Type Theory to ensure consistency by checking that definitions are wellstratified according to a certain hierarchy. In the case of the Coq proof assistant, based on the predicative Calculus of Inductive Constructions (pCIC), this hierachy is built from an impredicative sort Prop and an infinite number of predicative Typei universes. A cumulativity relation represents the inclusion order of universes in the core theory. Originally, universes were thought to be floating levels, and definitions to implicitely constrain these levels in a consistent manner. This works well for most theories, however the globality of levels and constraints precludes generic constructions on universes that could work at different levels. Universe polymorphism extends this setup by adding local bindings of universes and constraints, supporting generic definitions over universes, reusable at different levels. This provides the same kind of code reuse facilities as MLstyle parametric polymorphism. However, the structure and hierarchy of universes is more complex than bare polymorphic type variables. In this paper, we introduce a conservative extension of pCIC supporting universe polymorphism and treating its whole hierarchy. This new design supports typical ambiguity and implicit polymorphic generalization at the same time, keeping it mostly transparent to the user. Benchmarking the implementation as an extension of the Coq proof assistant on realworld examples gives encouraging results. 1
Formalizing Overlap Algebras in Matita
 UNDER CONSIDERATION FOR PUBLICATION IN MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... We describe some formal topological results, formalized in Matita 1/2, presented in predicative intuitionistic logic and in terms of Overlap Algebras. Overlap Algebras are new algebraic structures designed to ease reasoning about subsets in an algebraic way within intuitionistic logic. We find that ..."
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We describe some formal topological results, formalized in Matita 1/2, presented in predicative intuitionistic logic and in terms of Overlap Algebras. Overlap Algebras are new algebraic structures designed to ease reasoning about subsets in an algebraic way within intuitionistic logic. We find that they also ease the formalization of formal topological results in an interactive theorem prover. Our main result is the existence of a functor between two categories of ‘generalized topological spaces’, one with points (Basic Pairs) and the other pointfree (Basic Topologies). The reported formalization is part as a wider scientific collaboration with the inventor of the theory, Giovanni Sambin. His goal is to verify in what sense, and with what difficulties, his theory is ‘implementable’. We check that all intermediate constructions respect the stringent size requirements imposed by predicative logic. The formalization is quite unusual, since it has to make explicit size information that is often hidden. We found that the version of Matita used for the formalization was largely inappropriate. The formalization drove several major improvements of Matita that will be integrated in the next major release (Matita 1.0). We show some motivating examples for these improvements, taken directly from the formalization. We also describe a possibly suboptimal solution in Matita 1/2, exploitable in other similar systems. We briefly discuss a better solution available in Matita 1.0.
Computer theorem proving in math
, 2004
"... Abstract—We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context. This is based on my talk at the PQR conference (Brussels, June 2003). ..."
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Abstract—We give an overview of issues surrounding computerverified theorem proving in the standard puremathematical context. This is based on my talk at the PQR conference (Brussels, June 2003).
unknown title
"... Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tara ..."
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Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tarattachment to the source file the revised and expanded version of the proof development which had been attached to math.HO/0311260.
Settheoretical mathematics in Coq
"... Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tara ..."
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Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tarattachment to the source file the revised and expanded version of the proof development which had been attached to math.HO/0311260.
unknown title
, 2004
"... Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tara ..."
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Abstract: We give a brief discussion of some of the issues which have arisen in the course of formalizing some classical settheoretical mathematics in the Coq system. This sprouts from, expands and replaces a chapter of math.HO/0311260 which will be removed in revision, and also contains as a tarattachment to the source file the revised and expanded version of the proof development which had been attached to math.HO/0311260.