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Hierarchical Reflection
"... Abstract. The technique of reflection is a way to automate proof construction in type theoretical proof assistants. Reflection is based on the definition of a type of syntactic expressions that gets interpreted in the domain of discourse. By allowing the interpretation function to be partial or even ..."
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Abstract. The technique of reflection is a way to automate proof construction in type theoretical proof assistants. Reflection is based on the definition of a type of syntactic expressions that gets interpreted in the domain of discourse. By allowing the interpretation function to be partial or even a relation one gets a more general method known as ``partial reflection''. In this paper we show how one can take advantage of the partiality of the interpretation to uniformly define a family of tactics for equational reasoning that will work in different algebraic structures. The tactics then follow the hierarchy of those algebraic structures in a natural way.
Extensional normalization in the logical framework with proof irrelevant equality
 In Workshop on Normalization by Evaluation, affiliated to LiCS 2009, Los Angeles
, 2009
"... We extend the Logical Framework by proof irrelevant equality types and present an algorithm that computes unique long normal forms. The algorithm is inspired by normalizationbyevaluation. Equality proofs which are not reflexivity are erased to a single object ∗. The algorithm decides judgmental eq ..."
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We extend the Logical Framework by proof irrelevant equality types and present an algorithm that computes unique long normal forms. The algorithm is inspired by normalizationbyevaluation. Equality proofs which are not reflexivity are erased to a single object ∗. The algorithm decides judgmental equality, its completeness is established by a PER model. 1.
A Framework for Internalizing Relations into Type Theory
"... Abstract. This paper introduces the concept of internalization structure, which can be used to incorporate certain relations into F Π, a variant of system F, while maintaining termination of the new system. We will call this process of incorporation internalization, F Π the base system and the new s ..."
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Abstract. This paper introduces the concept of internalization structure, which can be used to incorporate certain relations into F Π, a variant of system F, while maintaining termination of the new system. We will call this process of incorporation internalization, F Π the base system and the new system after the incorporation the internalized system. We first specify the syntax, and then the semantics of F Π via the TaitGirard reducibility method. We then define internalization structure. We show that we can obtain a terminating internalized system from an internalization structure. Finally, as motivating examples, we demonstrate how our framework can be applied to internalize subtyping, fullbeta term equality and termtype inhabitation relations. 1
Implementing Certified Programming Language Tools in Dependent Type Theory
, 2007
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Fractional Types
"... Abstract. In previous work, we developed a firstorder, informationpreserving, and reversible programming language Π founded on type isomorphisms. Being restricted to firstorder types limits the expressiveness of the language: it is not possible, for example, to abstract common program fragments in ..."
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Abstract. In previous work, we developed a firstorder, informationpreserving, and reversible programming language Π founded on type isomorphisms. Being restricted to firstorder types limits the expressiveness of the language: it is not possible, for example, to abstract common program fragments into a higherlevel combinator. In this paper, we introduce a higherorder extension of Π based on the novel concept of fractional types 1/b. Intuitively, a value of a fractional type 1/v represents negative information. A function is modeled by a pair (1/v1, v2) with 1/v1 representing the needed argument and v2 representing the result. Fractional values are firstclass: they can be freely propagated and transformed but must ultimately — in a complete program — be offset by the corresponding amount of positive information. 1
MartinLöf Complexes
, 2009
"... In this paper we define MartinLöf complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional MartinLöf type theory. We then study the resulting categories of algebras for several theories. Our principal resu ..."
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In this paper we define MartinLöf complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional MartinLöf type theory. We then study the resulting categories of algebras for several theories. Our principal result is that there exists a cofibrantly generated Quillen model structure on
LogiCal Project
, 2004
"... This document is the Reference Manual of version 8.0 of the COQ proof assistant. A companion volume, the COQ Tutorial, is provided for the beginners. It is advised to read the Tutorial first. A new book [13] on practical uses of the COQ system will be published in 2004 and is a good support for both ..."
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This document is the Reference Manual of version 8.0 of the COQ proof assistant. A companion volume, the COQ Tutorial, is provided for the beginners. It is advised to read the Tutorial first. A new book [13] on practical uses of the COQ system will be published in 2004 and is a good support for both the beginner and the advanced user.
And was defended before a jury composed of
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