J. Terlouw, Een nadere bewijstheoretische analyse van GSTT's (incl. appendix), Manuscript, Faculty of Mathematics and Computer Science, University of Nijmegen, Netherlands, March, April 1989. (In Dutch)  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in 1910 in the famous Principia Mathematica [28] and simpli ed by Ramsey and Hilbert and Ackermann. In 1940, Church combined his theory of functions, the calculus, with the simpli ed type theory resulting in the in uential simple theory of types [9] In 1988 1989, Berardi [4] and Terlouw [27] gave as an extension of Barendregt s work [1] a general framework for type systems, which is at the basis of the so The rst practical use of the propositions as types principle is found in Automath. called Pure Type Systems (PTSs [1] PTSs include many of the type systems that play an ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Technical report, Department of Computer Science, University of Nijmegen, 1989.

....according to the system considered. In this section we systematize this observation by reviewing the general notion of domain free pure type systems from [12] The development here is completely analogous to the classical development of pure type systems by Barendregt [4] Berardi [13] and Terlouw [72]. In the rst subsection we introduce the notion of a domain free pure type system. In the second subsection we show that the systems of the domain free cube can be viewed as domain free pure type systems. The third subsection introduces the notion of logical specication, due to Coquand and ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Manuscript (in Dutch), 1989.

....which di erent type systems can be described may turn out to be useful for providing general criteria and results concerning the various systems. A general framework also helps comparing the di erent systems. In 1988 and 1989, a general framework was given independently by Terlouw and Berardi in [5, 21] which classi es di erent known type theories. This framework is known as the Pure Type Systems (PTSs for short) framework. In [4] a description of PTSs and of a cube of eight di erent systems that are all PTSs can be found. Important type systems that are PTSs include Church s simply typed ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Technical report, Department of Computer Science, University of Nijmegen, 1989.

....In this paper, we consider typing a la Church. We present what is known as Pure Type Systems or PTSs. Important type systems that are PTSs include Church s simply typed calculus [8] and the calculus of constructions [9] which are also systems of the Barendregt cube [4] Berardi [5] and Terlouw [47] have independently generalised the method of generating type systems into the pure type systems framework. This generalisation has many advantages. First, it enables one to introduce eight logical systems that are in close correspondence with the systems of the Barendregt cube. Those eight ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Technical report, Department of Computer Science, University of Nijmegen, 1989.

.... Barthe, Hatclioe and S#rensen [6] which have been respectively used to dene j long normal forms and CPS translations for the systems of Barendregt s cube [2, 3] 1 Introduction Pure Type Systems (PTSs) provide a description of typed calculi that is parametric in the notion of type discipline [2, 3, 9, 13, 14, 26]. The parametricity of PTSs allows many logics and type systems that have been studied in the literature to arise as specic instances of PTSs. Indeed, many well known typed calculi are embodied in Barendregt s cube [2, 3] which provides a ne grain analysis of the Calculus of Constructions ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Manuscript (in Dutch), 1989.

....logic. To motivate this we give some examples of derivable judgements in PRED . For more information on Pure Type Systems and typed calculus in general, we refer to [Barendregt 1992] and [Geuvers 1993] Pure Type Systems or PTSs were first introduced by Berardi [Berardi 1990] and Terlouw [Terlouw 1989a] with slightly different definitions. The advantage of the class of PTSs is that many known systems can be seen as PTSs. So, many specific results for specific systems are immediate instances of general properties of PTSs. In the following we will mention a number of these properties. ....

J. Terlouw, Een nadere bewijstheoretische analyse van GSTT's (incl.

....using Russell s notion of order. There are however many important bonuses that result from our study: 1. We give the rst presentation of a subsystem of the proof checker Nuprl as a PTS. In Section 2 we give a formal description of a part of the type system of Nuprl as a Pure Type System (PTS) [38]. The systems of the Barendregt cube are examples of PTSs. Nuprl in PTS style enables us to formalize the concept of order in Nuprl and to show its correctness. This order classi es types and terms of Nuprl into their relevant hierarchy. 2. We give a formal presentation of rtt. Such a formal ....

....system on which Nuprl is based (see [17, 6] We do not give a full presentation of all of Nuprl s type constructors, as we will only need parts of it. The description of the typing rules is given in a natural deduction style similar to that used in the Barendregt Cube [1] and Pure Type Systems [38]. Below we assume V to be a set of variables, Z to be the set of integers, and S= f 1 ; 2 ; g a set of sorts. The intuition behind the sort a is that it represents the propositions (and, more general, the types) of order a. a corresponds to the Universe of Types U a in [17, 27] ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Technical report, Department of Computer Science, University of Nijmegen, 1989.

....(M) A; tc ok ) Ok( true : All cases in the proof of this Proposition are by an easy induction. 3.3. Pure Type Systems The system HOL is just an instance of a general class of typed calculi, the so called Pure Type Systems or PTSs. These were rst introduced by Berardi [1988] and Terlouw [1989], under di erent names and with slightly di erent de nitions, as a generalisation of the cube, see Barendregt [1992] The reason for de ning the class of PTSs is that many known systems are (or better: can be seen as) PTSs. This makes it fruitful to study the general properties of PTSs in order ....

Terlouw, J. [1989]. Een nadere bewijstheoretische analyse van GSTT's, Technical report, Department of Computer Science, University of Nijmegen.

....Maude [7] of a proof assistant for OCC, the open calculus of constructions, an equational extension of the calculus of constructions. 1 Introduction This paper is a detailed case study on the ease and naturalness with which a family of higher order formal systems, namely pure type systems (PTS) [4, 32], can be represented in the rst order logical framework of rewriting logic [25] PTS systems generalize the cube [1] which already contains important calculi like [6] F [12, 29] F [12] a system P close to the logical framework LF [13] and their combination, the calculus of ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTTs. Manuscript, University of Nijmegen, The Netherlands, 1989.

....logics using Russell s notion of order. There are however many important bonuses that result from our study: 1. We give the first presentation of the proof checker Nuprl as a PTS. In Section 2 we give a formal description of a part of the type system of Nuprl as a Pure Type System (PTS) [22]. The systems of the Barendregt cube are examples of PTSs. Nuprl in PTS style enables us to formalize the concept of order in Nuprl and to show its correctness. This order classifies types and terms of Nuprl into their relevant hierarchy. 1 This Theorem states that any non empty set of real ....

....system on which Nuprl is based (see [9, 3] We don t give a full presentation of all of Nuprl s type constructors, as we will only need parts of it. The description of the typing rules is given in a natural deduction style similar to that used in the Barendregt Cube [1] and Pure Type Systems [22]. Below we assume V to be a set of variables, Z to be the set of integers, and S= f 1 ; 2 ; g a set of sorts. The intuition behind the sort a is that it represents the propositions (and, more general, the types) of order a. a corresponds to the Universe of Types U a in [9, 16] ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Technical report, Department of Computer Science, University of Nijmegen, 1989.

....unified manner. Such a treatment provides a step towards the generalisation of type systems which is an important topic of research at the present time. For example, Barendregt s taxonomy of type systems in [Barendregt 92] but also Pure Type Systems (PTSs) introduced by Terlouw and Berardi (see [Ter 89] and our generalised system in [NK 94] are attempts at combining all the important results of type systems in a compact and elegant way. As a step towards this goal, we believe that conversion should apply to both types and terms. In fact, Pi is indeed a kind of , hence eligible for an ....

Terlouw, J., Een nadere bewijstheoretische analyse van GSTT's. Technical report, Department of Computer Science, University of Nijmegen, 1989.

.... for a variant of injective Pure Type Systems where the problematic clause in the (abstraction) rule is replaced in favour of constraints over elmt( j: and sort( j: 1 Introduction Pure Type Systems provide an elegant and general framework for the definition and study of typed calculi [2, 7, 9, 10, 19]. One central issue in the theory of Pure Type Systems is the problem of type checking. Given a Pure Type System S, type checking consists in deciding whether a judgment Gamma M : A is derivable according to the rules of Pure Type Systems. Although type checking is undecidable in general [8] ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Manuscript (in Dutch), 1989.

.... Barthe, Hatcliff and S rensen [5] which have been respectively used to define j long normal forms and CPS translations for the systems of Barendregt s cube [2, 3] 1 Introduction Pure Type Systems (PTSs) provide a description of typed calculi that is parametric in the notion of type discipline [2, 3, 9, 13, 14, 25]. The parametricity of PTSs allows many logics and type systems that have been studied in the literature to arise as specific instances of PTSs. Indeed, many well known typed calculi are embodied in Barendregt s cube [2, 3] which provides a fine grain analysis of the Calculus of Constructions ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Manuscript (in Dutch), 1989.

....and Ackermann [19] In 1940, Church combined his theory of functions, the calculus ( 9, 10] with the simplified type theory, resulting in the so called Simple Theory of Types [11] This system has served as a basis for the many systems that have been developed since then. In 1989, Terlouw [30] presented, as an extension of Barendregt s work [3] a general framework for type systems, which is at the basis of the so called Pure Type Systems (PTSs; see [16] 3] 15] The theory of PTSs nowadays plays a central role in type theory and typed calculus. This paper will focus on the ....

....j reduction, Pi application and Pi reduction. In Section 6 we compare the definition system of Aut 68 with several other, more modern, type systems with definitions. 2 Pure Type Systems Pure Type Systems (PTSs) were introduced (in a somewhat different way than presented below) by Terlouw [30] in 1989 and were also implicitly present in the work of Berardi [5] Many type systems can be described as a PTS and this makes PTSs a central notion in type theory. Below we repeat the definition of PTS as presented in [3] In [3] one can also find the basic properties of PTSs, and some ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Technical report, Department of Computer Science, University of Nijmegen, 1989.

....[Rey74] 2 LAMBDA CALCULI WITH TYPES 5 2.1 Pure Type Systems In [Sev96] we work with pure type systems. They provide a framework to describe a large class of type systems a la Church in a uniform way. They were introduced independently by S. Berardi [Ber88] see also [Ber90] and J. Terlouw [Ter89]. Many systems can be described in this way, for instance the simply and the polymorphic typed lambda calculus, the systems of the AUTOMATH family [NGdV94] the Calculus of Constructions (and all the systems of the cube [Bar92] and the inconsistent system [Gir72] They are called pure because ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Manuscript, 1989.

.... 13: DS translation for the cube In this section we systematize this observation by reviewing the general notion of domain free pure type systems from [12] The development here is completely analogous to the classical development of pure type systems by Barendregt [4] Berardi [13] and Terlouw [72]. In the first subsection we introduce the notion of a domain free pure type system. In the second subsection we show that the systems of the domain free cube can be viewed as domain free pure type systems. The third subsection introduces the notion of logical specification, due to Coquand and ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Manuscript (in Dutch), 1989.

....for certain terms where some variables are bound more than once, such as x:x:x. Although one might believe that this system is closed under ff conversion, we show with a counterexample in section 5 that it is not. This system is usually given in its general formulation for pure type systems (see [Ber88, Ter89, Bar92]) and it has also been used as the type system for a small typed functional language in [Bov95] and [Tas97] The thinning rule for system Th is often seen as a generalisation of the weakening rule for Wk, where we have the possibility of adding several declarations at the same time instead of one ....

J. Terlouw. Een nadere bewijstheoretische analyse van gstt's. Technical report, Department of Computer Science, University of Nijmegen, 1989.

.... and application rules (to the right in the figure) In this section we systematize these observations by reviewing the general notion of domain free pure type systems from [8] The development here is completely analogous the classical development by Barendregt [3] Berardi [9] and Terlouw [59] of traditional pure type systems as a framework to define and study typed calculi with domain tags. Beyond their generality, the appeal of traditional and domain free pure type systems lies in their conceptual clarity; subtle differences between type systems can neatly be described by varying ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Manuscript (in Dutch), 1989.

....Bruijn isomorphism (see [8] It is constructive on two points: It is based on intuitionistic logic (as is the Curry Howard de Bruijn isomorphism) and it is based on predicative logic. In Section 2 we give a formal description of a part of the type system of Nuprl as a Pure Type System (PTS) [22]. The systems of the Barendregt cube are examples of PTSs. Nuprl in PTS style enables us to formalize the concept of order in Nuprl and to show its correctness. This order classifies types and terms of Nuprl into their relevant hierarchy. Before we can give any correspondence between rtt and ....

....system on which Nuprl is based (see [9, 3] We don t give a full presentation of all of Nuprl s type constructors, as we will only need parts of it. The description of the typing rules is given in a natural deduction style similar to that used in the Barendregt Cube [1] and Pure Type Systems [22]. Below we assume V to be a set of variables, Zto be the set of integers, and S= f 1 ; 2 ; g a set of sorts. The intuition behind the sort a is that it represents the propositions (and, more general, the types) of order a. a corresponds to the Universe of Types U a in [9, 16] ....

J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Technical report, Department of Computer Science, University of Nijmegen, 1989.

....: ff b then Delta c : fi d. A conversion morphism from A to B could be seen as a semantic of A in B. 3 Pure Type Systems Pure type systems provide a way to describe a large class of type systems a la Church in a uniform way. They were simultaneously introduced by Berardi [Ber88] and Terlouw [Ter89]. We will present the notion of pure type systems in a slightly different way than usual. We first define a functor from the category of specifications to the category of contextual abstract rewriting systems with typing. Then we define a pure type system as a value (S) of the functor for some ....

....in a slightly different way than usual. We first define a functor from the category of specifications to the category of contextual abstract rewriting systems with typing. Then we define a pure type system as a value (S) of the functor for some specification S. Definition 3.1. Ber88] [Ter89]) A specification is a triple S = S; A;R) such that 1. S is a set of symbols called sorts, 2. A S Theta S called set of axioms 3. R S Theta S Theta S called set of rules We denote the category whose objects are the specifications and morphisms are the obvious ones as Spec. Definition ....

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J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Manuscript, 1989.

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J. Terlouw, Een nadere bewijstheoretische analyse van GSTT's (incl. appendix), Manuscript, Faculty of Mathematics and Computer Science, University of Nijmegen, Netherlands, March, April 1989. (In Dutch)

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J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Technical report, Department of Computer Science, University of Nijmegen, 1989.

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J. Terlouw. Een nadere bewijstheoretische analyse van GSTT's. Technical report, Department of Computer Science, University of Nijmegen, 1989.

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Ter89. J. Terlouw. Een nadere bewijstheoretische analyse van GSTTs. Manuscript, University of Nijmegen, Toernooiveld I, 6525 ED Nijmegen, The Netherlands, 1989.

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