H. Geuvers. Logics and Type Systems. PhD thesis, Department of Mathematics and Computer Science, University of Nijmegen, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[PP=NN] PPeq=nateq] PPeqiffQ=nateqcharacter] PPinf=NNinf] VV=NN] VVeq=nateq] VVeqiffQ=nateqcharacter] VVinf=NNinf] The Cut command is described in section 2.1.4. Since this changes the context destructively, we should now re load from the definition of PP to regain abstractness. 2 [GN91,Geu93] do put structure on VV, partitioning it into disjoint countably infinite sets, one for each sort. This can be formalized as VV Theta SS, rather than partitioning VV, so Trm depends on SS but VV does not. Chapter 3. Lambda Calculus: How to Handle Free Names 38 3.3 Reduction and Conversion In ....

....syntax directed presentation of a class of type theories including CC and P . That paper, in improved form, is published as a section in [vBJMP94] Here I present a similar development, extended to a class of type systems including ECC as well. The concept is also used, for different purposes, in [Geu93] where it is credited to [vBJMP94] Chapter 5. Semi Full and Cumulative PTS : Typechecking ECC 99 Inductive [sfts:Cxt Trm Trm Prop] NoReductions Constructors [sfAx: s1,s2 SS sc:ax s1 s2 sfts nilCxt (sort s1) sort s2) sfStart: G Cxt A Trm s SS p PP noccG:isff (Poccur ....

Herman Geuvers. Logics and Type Systems. PhD thesis, Department of Mathematics and Computer Science, University of Nijmegen, 1993.

....of context morphisms. We study this result in the context of Pure Type Systems ( PTS ) introduced as a general syntactic characterisation of type theories [Bar92] PTS gives a framework for showing syntactic results, such as strengthening and subject reduction, for a broad class of systems [vBJ93, Geu93]. The second author gave an informal proof of closure under substitution and thinning for Geuvers systems (which extend the rules Submitted to Journal of Functional Programming 1 for PTS with both j conversion, and strengthening as a rule of inference, complicating their meta theory a good ....

Herman Geuvers. Logics and Type Systems. PhD thesis, Department of Mathematics and Computer Science, University of Nijmegen, 1993.

....P , is only semi full. To the best of my knowledge, this definition first appeared in [Pol92] where I used it to give a syntax directed presentation of a class of type theories including CC and P . That paper, in improved form, is published as a section in [vBJMP94] The notion is also used in [Geu93] where it is credited to [vBJMP94] For our purposes, the interesting fact about semifull PTS is that, because of the possibility to simplify the Lda rule mentioned above, we can give an alternative presentation of such systems: relation sdsf (syntax directed semifull) is defined inductively by ....

Herman Geuvers. Logics and Type Systems. PhD thesis, Department of Mathematics and Computer Science, University of Nijmegen, 1993.

....systems with j rule, some properties of Section 3.2 can be complicated. This was so, for example, the Church Rosser property for the system LF with j reduction, that was first proved by A. Salvesen in [Sal90] or the same theorem in normalizing P.T.S. proved in the thesis of H. Geuvers [Geu93] As is well known in the literature, when the j rule is added to system F2, the subject reduction property for typable terms is lost, as is shown in the next example: Example 4.4.1 Consider the following j reduction between well typed terms in F2: x:yx j y. It is easy to verify that x:yx ....

Herman Geuvers. Logics and Type Systems. PhD thesis, Department of Mathematics and Computer Science, University of Nijmegen, 1993.

....include program specification and data refinement [Luo91] strong normalization of System F [Alt93] synthetic domain theory [Reu95, Reu96] and operational semantics for imperative programs [Sch97] 1. 1 Why PTS have a beautiful meta theory, developed informally in [Bar92, Ber90, GN91, vBJ93, Geu93] These papers are unusually clear and mathematical, and there is little doubt about the correctness of their results, so why write a machine checked development The informal presentations leave many decisions unspecified and many facts unproved. They are far from the detail of representation ....

....with an application as its subject, there is no confusion whether it is derived by App or Weak. Thus, with atomic weakening, any judgement may only be derived by tConv or by exactly one of the remaining rules. 29 The Lambda Rule For the rule for typing a lambda in informal presentations [Bar92, Geu93] is Gamma[x:A] M : B Gamma fx:AgB : s The conventional understanding is that the bound variable, x , doesn t really occur in the conclusion of rule , as the notations [x:A]M and fx:AgB refer to alpha equivalence classes . Thus, in concrete notation, the subject and ....

Herman Geuvers. Logics and Type Systems. PhD thesis, Department of Mathematics and Computer Science, University of Nijmegen, 1993.

....that keep acquiring more and more types by reduction As an aside, notice that this PTS is strongly normalizing. It can be mapped into three levels of a strongly normalizing predicative type hierarchy (for definiteness we mention ECC [Luo90] but much weaker systems will do) by the PTS morphism ([Geu, Geu93]) 7 Type(0) 4 7 Type(1) 5 7 Type(1) 7 Type(2) Since this mapping preserves sorts, axioms and rules, any well typed term in the system above has a well typed image, and therefore strongly normalizes. Further, since S is finite, this system has decidable typechecking [vBJ93] ut It is ....

Herman Geuvers. Logics and Type Systems. PhD thesis, Department of Mathematics and Computer Science, University of Nijmegen, 1993.

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H. Geuvers. Logics and Type Systems. PhD thesis, Department of Mathematics and Computer Science, University of Nijmegen, 1993.

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