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48
A Pointfree approach to Constructive Analysis in Type Theory
, 1997
"... The first paper in this thesis presents a machine checked formalisation, in MartinLöf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is defined as a formal space from ..."
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The first paper in this thesis presents a machine checked formalisation, in MartinLöf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is defined as a formal space from a base of rational intervals. Then the closed rational interval [a, b] is defined as a formal space, in terms of the continuum, and the HeineBorel covering theorem is proved constructively. The basic definitions for a pointfree approach to functional analysis are given in such a way that the linear functionals from a seminormed linear space to the reals are points of a particular formal space, and in this setting the Alaoglu and the HahnBanach theorems are proved in an entirely constructive way. The proofs have been carried out in intensional MartinLöf type theory with one universe and finitary inductive definitions, and the proofs have also been mechanically checked in an implementation of that system. ...
A modular typechecking algorithm for type theory with singleton types and proof irrelevance
 IN TLCA’09, VOLUME 5608 OF LNCS
, 2009
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Greedy Bidirectional Polymorphism
, 2009
"... Bidirectional typechecking has become popular in advanced type systems because it works in many situations where inference is undecidable. In this paper, I show how to cleanly handle parametric polymorphism in a bidirectional setting. The key contribution is a bidirectional type system for a subset ..."
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Bidirectional typechecking has become popular in advanced type systems because it works in many situations where inference is undecidable. In this paper, I show how to cleanly handle parametric polymorphism in a bidirectional setting. The key contribution is a bidirectional type system for a subset of ML that supports firstclass (higherrank and even impredicative) polymorphism, and is complete for predicative polymorphism (including MLstyle polymorphism and higherrank polymorphism). The system’s power comes from bidirectionality combined with a “greedy ” method of finding polymorphic instances inspired by Cardelli’s early work on System F<:. This work demonstrates that bidirectionality is a good foundation for traditionally vexing features like firstclass polymorphism.
Constructor subtyping in the Calculus of Inductive Constructions
 Proceedings of FOSSACS'00, LNCS 1784
, 2000
"... The Calculus of Inductive Constructions (CIC) is a powerful type system, featuring dependent types and inductive definitions, that forms the basis of proofassistant systems such as Coq and Lego. We extend CIC with constructor subtyping, a basic form of subtyping in which an inductive type &sigm ..."
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The Calculus of Inductive Constructions (CIC) is a powerful type system, featuring dependent types and inductive definitions, that forms the basis of proofassistant systems such as Coq and Lego. We extend CIC with constructor subtyping, a basic form of subtyping in which an inductive type &sigma; is viewed as a subtype of another inductive type &tau; if &tau; has more elements than &sigma;. It is shown that the calculus is wellbehaved and provides a suitable basis for formalizing natural semantics in proofdevelopment systems.
Typed Applicative Structures and Normalization by Evaluation for System F ω
"... Abstract. We present a normalizationbyevaluation (NbE) algorithm for System F ω with βηequality, the simplest impredicative type theory with computation on the type level. Values are kept abstract and requirements on values are kept to a minimum, allowing many different implementations of the alg ..."
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Abstract. We present a normalizationbyevaluation (NbE) algorithm for System F ω with βηequality, the simplest impredicative type theory with computation on the type level. Values are kept abstract and requirements on values are kept to a minimum, allowing many different implementations of the algorithm. The algorithm is verified through a general model construction using typed applicative structures, called type and object structures. Both soundness and completeness of NbE are conceived as an instance of a single fundamental theorem.
A general method to prove the normalization theorem for first and second order typed λcalculi
 Mathematical Structures in Computer Science
, 1999
"... and second order typed λcalculi ..."
Towards Normalization by Evaluation for the βηCalculus of Constructions
"... Abstract. We consider the Calculus of Constructions with typed betaeta equality and an algorithm which computes long normal forms. The normalization algorithm evaluates terms into a semantic domain, and reifies the values back to terms in normal form. To show termination, we interpret types as part ..."
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Abstract. We consider the Calculus of Constructions with typed betaeta equality and an algorithm which computes long normal forms. The normalization algorithm evaluates terms into a semantic domain, and reifies the values back to terms in normal form. To show termination, we interpret types as partial equivalence relations between values and type constructors as operators on PERs. This models also yields consistency of the betaetaCalculus of Constructions. The model construction can be carried out directly in impredicative type theory, enabling a formalization in Coq. 1
On the Algebraic Foundation of Proof Assistants for Intuitionistic Type Theory
, 2008
"... An algebraic presentation of MartinLöf’s intuitionistic type theory is given which is based on the notion of a category with families with extra structure. We then present a typechecking algorithm for the normal forms of this theory, and sketch how it gives rise to an initial category with familie ..."
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An algebraic presentation of MartinLöf’s intuitionistic type theory is given which is based on the notion of a category with families with extra structure. We then present a typechecking algorithm for the normal forms of this theory, and sketch how it gives rise to an initial category with families with extra structure. In this way we obtain a purely algebraic formulation of the correctness of the typechecking algorithm which provides the core of proof assistants for intuitionistic type theory.
Dependent Record Types, Subtyping and Proof Reutilization
"... . We present an example of formalization of systems of algebras using an extension of MartinLof's theory of types with record types and subtyping. This extension has been presented in [5]. In this paper we intend to illustrate all the features of the extended theory that we consider relevant f ..."
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. We present an example of formalization of systems of algebras using an extension of MartinLof's theory of types with record types and subtyping. This extension has been presented in [5]. In this paper we intend to illustrate all the features of the extended theory that we consider relevant for the task of formalizing algebraic constructions. We also provide code of the formalization as accepted by a type checker that has been implemented. 1. Introduction We shall use an extension of MartinLof's theory of logical types [14] with dependent record types and subtyping as the formal language in which constructions concerning systems of algebras are going to be represented. The original formulation of MartinLof's theory of types, from now on referred to as the logical framework, has been presented in [15, 7]. The system of types that this calculus embodies are the type Set (the type of inductively defined sets), dependent function types and for each set A, the type of the elements of A...
ON IRRELEVANCE AND ALGORITHMIC EQUALITY IN PREDICATIVE TYPE THEORY
"... Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To obtain reasonable performance of not only the compiled program but also the type checker such static terms need to be erased as early as ..."
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Abstract. Dependently typed programs contain an excessive amount of static terms which are necessary to please the type checker but irrelevant for computation. To obtain reasonable performance of not only the compiled program but also the type checker such static terms need to be erased as early as possible, preferably immediately after type checking. To this end, Pfenning’s type theory with irrelevant quantification, that models a distinction between static and dynamic code, is extended to universes and large eliminations. Normalization, consistency, and decidability are obtained via a universal Kripke model based on algorithmic equality. 1. Introduction and Related