Thomas Streicher. Semantics of Type Theory. Springer-Verlag, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from the classifying type. Interestingly, recent research on dependently typed rewriting [Vir99] has also isolated equivalence classes of terms modulo conversion of the type labels as a critical concept. In some of the original work on Martin Lof type theory [NPS90] and some subsequent studies [Str91], type theories without type labels have been studied, but to our knowledge they have not been considered with respect to bi directional type checking or adequacy proofs in logical framework representations. There is now significant evidence that our construction is robust with respect to ....

Thomas Streicher. Semantics of Type Theory. Birkhauser, 1991.

.... programming syntax and semantics have enjoyed widespread use for over a decade [56] Categorical models have been used to give clean, implementation independent approaches to side effects and state [47, 62, 68, 52] non determinism [53] type disciplines [15, 33] and other logics for computation [7, 63]. The mathematical treatment of some features, such as parametricity and polymorphism, have required categorical tools [57, 21] Logic programming, however, has developed within a different semantic tradition than that of functional or imperative programming. The divide has narrowed in the last ....

T. Streicher. Semantics of Type Theory. Birkhuser, 1992.

....it more difficult to show that the type formers are injective; an auxiliary property required to establish the subject reduction property for an untyped rewrite system derived from definitional equality. Also, in many models rule U Pi Ty is not valid under the canonical interpretation of Pi, see (Streicher 1991) and the example following Def. 3.20. Closure under natural numbers is described by Gamma ctxt Gamma N : U U N and further rules introducing term formers 0, Suc, and R N witnessing that El( N) behaves like N. Again we could instead impose the equality Gamma El( N) N type ....

....which can safely be omitted without violating this property. Discuss the properties a type theory must have so that the type annotation in application can be omitted. In other words when can we replace App [x:oe] M; N) by App(M;N) in the official syntax without violating uniqueness of types. See (Streicher 1991) for a thorough discussion of this point. 2.4 Context morphisms Definition 2.11 Let Gamma and Delta def = x 1 : oe 1 ; x n : oe n be valid contexts. If f def = M 1 ; M n ) is a sequence of n pre terms we write Gamma f ) Delta and say that f is a context morphism from ....

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Streicher, T. (1991). Semantics of Type Theory. Birkhauser.

....algorithm is sound and complete with respect to the semantics above. This shows furthermore that the usual notation with terms is not ambiguous; there may be several derivations of the same typing judgement, but they are all convertible with each other. The idea of this proof comes from Streicher [19] (chapter IV) 5.1 Definition of terms We mutually define the set of terms, T 2 Set, and substitutions, S 2 Set as x 2 Name v(x) 2 T x 2 Name t 2 T lam(x; t) 2 T t 1 2 T t 2 2 T app(t 1 ; t 2 ) 2 T s 2 S t 2 T sub(t; s) 2 T id 2 S s 2 S t 2 T upd(s; x; t) 2 S s 1 ; s 2 2 S com(s 1 ; ....

T. Streicher. Semantics of Type Theory. Birkhuser, 1991.

....functor has a right adjoint. In our presentation we interpret Pi types as the subset of the set theoretic dependent functions which have a realizer. Although equivalent to the categorical construction this presentation seems more natural. 12 Streicher s thesis has been published as a monograph [Str91] In future we will refer to the monograph only. 13 e.g. see [ACCL90] Chapter 1. Introduction 17 Given that our main motivation for semantics is to show admissible properties of the system and to do this we have to give sound interpretations, I decided not to use categories when constructing ....

....use a particular notion of reduction tight reduction which is essential to our approach. This presentation should be compared to [Rit92] where essentially the same goals are achieved using categorical combinators. 2.1. The judgement presentation of CC In our presentation we largely follow [Str91] i.e. ffl We use equality as a judgement. ffl We do not confuse types and terms, i.e. we avoid chains of colons. Therefore we have to introduce an explicit reflection operator El and differentiate between Pi for types and 8 for Set. Chapter 2. The Calculus of Constructions 22 ffl We consider ....

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Thomas Streicher. Semantics of Type Theory. Birkhauser, 1991.

....for my taste too much on pure type systems. If one is interested in non dependent typed calculi Girard et al. s book is certainly worthwhile reading [GLT89] Roy Crole s book [Cro93] approaches the subject from the viewpoint of Category Theory but is generally self contained. Thomas Streicher [Str91] investigates the Calculus of Constructions also using Category Theory but seems to be a bit heavy going for the uninitiated. A good general reference and introduction for categorical models are Martin Hofmann s lecture notes [Hof95b] On a more practical level it is certainly exciting to use one ....

Thomas Streicher. Semantics of Type Theory. Birkhauser, 1991.

....of this, namely the categorical semantics of dependent products in Section 6.5. Several different, but interconnected, categorical structures have been proposed for interpreting the basic framework of dependent types by Seely [ 1984 ] Cartmell [ 1986 ] Taylor [ 1986 ] Ehrhard [ 1988 ] Streicher [ 1989, 1991 ] Hyland and Pitts [ 1989 ] Obtu lowicz [ 1989 ] Curien [ 1989 ] and Jacobs [ 1991 ] This reflects the fact that the categorical interpretation of dependent types is undoubtedly more complicated than the other varieties of categorical logic explained in this chapter. This is due to the ....

Th. Streicher. Semantics of Type Theory. Birkhauser, 1991.

....let u 2 be a Level 2 environment, assigning meanings to pre defined identifiers. The superscript on the environments and the semantic functions distinguishes among the environments and semantic functions for each Level. We propose a model suggested by the term model defined by Thomas Streicher [50], and based on the semantics function of Level 2 terms. This model meets the criteria that only provably well defined contexts, types, and objects are given an interpretation. The advantage of this model is that it incorporates the context for all terms in the elements of the model, which is ....

Streicher, T. Semantics of Type Theory. Birkhauser, 1991.

....products and not only small ones which amounts to having Set as a full internal internally complete subcategory of the ambient category. This stronger assumption is reasonable as it is part of the assumption that the ambient category is a model of impredicative constructive type theory, cf. [9, 10]. Of course, employing the type theoretic universe Set with S 2 Set the property S replete can be expressed by a formula of the internal language of the ambient category of constructive sets. Hyland s abovementioned construction of repletion together with its verification could be formalised ....

.... of type theory (and 2 of set theory as well) the existence of such universes in Grothendieck toposes is not known (rather one knows that non trivial impredicative universes do not exist in Grothendieck toposes) 1 The Ambient Logic We work in impredicative constructive type theory, see e.g. [9, 10], where besides an impredicative universe P rop of proof irrelevant propositions (i.e. all proofs of a proposition are equal as objects) we assume a further universe Set that contains P rop as a subtype but not as an element and that is closed under small dependent sums and (small) dependent ....

T. Streicher. Semantics of Type Theory. Birkhauser, 1991.

....product of an arbitrary family of small sets is again small. Furthermore, we require subset types for equality predicates (i.e. equalisers) and quotients of classical (i.e. closed) equivalence relations. A model for such a metalanguage is furnished by sets and pers modest sets as described in (Streicher 1991). For a proof that the set model admits the required quotients see for instance Ch. II of loc. cit. We use an informal (impredicative) extensional Martin Lof type theory to denote constructions in the metalanguage. In particular, we write Pi and Sigma for dependent product and sum, and we use ....

Streicher, T. (1991). Semantics of Type Theory. Birkhauser.

....under inductive definitions. Impredicativity means that the product of an arbitrary family of small sets is again small. Furthermore, we require subset types for equality predicates (i.e. equalisers) A model for such a metalanguage is furnished by Sets and Pers modest sets as described in [11]. We use an informal (impredicative) extensional Martin Lof type theory to denote constructions in the metalanguage. In particular, we write Pi and Sigma for dependent product and sum, and we use and juxtaposition for abstraction and application and h Gamma; Gammai for pairing and :1, 2 for ....

T. Streicher. Semantics of Type Theory. Birkhauser, 1991.

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Thomas Streicher. Semantics of Type Theory. Springer-Verlag, 1991.

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