Introduction to combinators and &lgr;-calculus
Introduction to combinators and &lgr;-calculus
Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
Programming with Proofs: A Second Order Type Theory
ESOP '88 Proceedings of the 2nd European Symposium on Programming
TYPES '95 Selected papers from the International Workshop on Types for Proofs and Programs
A Model for Impredicative Type Systems, Universes, Intersection Types and Subtyping
LICS '00 Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science
Mathematical Structures in Computer Science
Constructing a small category of setoids
Mathematical Structures in Computer Science
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We extend the proof-irrelevant model defined in Smith (1988) to the whole of Martin-Löf's logical framework. The main difference here is the existence of a type whose objects themselves represent types rather than proof-objects. This means that the model must now be able to distinguish between objects with different degree of relevance: those that denote proofs are irrelevant whereas those that denote types are not. In fact a whole hierarchy of relevance exists.Another difference is the higher level of detail in the formulation of the formal theory, such as the explicit manipulation of contexts and substitutions. This demands an equally detailed definition of the model, including interpreting contexts and substitutions.We are thus led to a whole reformulation of the proof-irrelevant model. We present a model that is built up from an arbitrary model of the untyped lambda calculus. We also show how to extend it when the logical framework itself is enlarged with inductive definitions. In doing so, a variant of Church numerals is introduced.As in Smith (1988), the model can only be defined in the absence of universes, and it is useful to obtain an elementary proof of consistency and to prove the independence of Peano's fourth axiom.