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On the strength of proofirrelevant type theories
 of Lecture Notes in Computer Science
, 2006
"... Vol. 4 (3:13) 2008, pp. 1–20 ..."
Type Theory via Exact Categories (Extended Abstract)
 In Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science LICS '98
, 1998
"... Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why ..."
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Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R ` N\ThetaN on th...
A Simple Model Construction for the Calculus of Constructions
 Types for Proofs and Programs, International Workshop TYPES'95
, 1996
"... . We present a model construction for the Calculus of Constructions (CC) where all dependencies are carried out in a settheoretical setting. The Soundness Theorem is proved and as a consequence of it Strong Normalization for CC is obtained. Some other applications of our model constructions are: sh ..."
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. We present a model construction for the Calculus of Constructions (CC) where all dependencies are carried out in a settheoretical setting. The Soundness Theorem is proved and as a consequence of it Strong Normalization for CC is obtained. Some other applications of our model constructions are: showing that CC + Classical logic is consistent (by constructing a model for it) and showing that the Axiom of Choice is not derivable in CC (by constructing a model in which the type that represents the Axiom of Choice is empty). 1 Introduction In the literature there are many investigations on the semantics of polymorphic calculus with dependent types (see for example [12, 11, 10, 1, 5, 13]). Most of the existing models present a semantics for systems in which the inhabitants of the impredicative universe (types) are "lifted" to inhabitants of the predicative universe (kinds) (see [16]). Such systems are convenient to be modeled by locally Cartesianclosed categories having small Cartesia...