Nederpelt, R.P., Strong normalisation in a typed lambda calculus with lambda structured types, Ph.D. thesis, Eindhoven University of Technology, Department of Mathematics and Computer Science, 1973. ALso in Nederpelt, R.P., Geuvers, J.H. and de Vrijer, R.C., eds., Selected Papaers on Automath, North Holland, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and was carried out at Boston University to whom and especially to Assaf Kfoury, I am grateful. Department of Computing Science, 17 Lilybank Gardens, University of Glasgow, Glasgow G12 8QQ, Scotland, email: fairouz dcs.glasgow.ac.uk, fax 44 41 3304913. I was told by Rob Nederpelt that in [Nederpelt 73] he assumed that the preservation of Strong Normalisation for fi e is easy to establish yet when editing [NGV 94] he retracted his assumption. All these (related) rules attempt to make more redexes visible. fl C for example, makes sure that y and Q form a redex even before the redex based on ....

....when the term is Strongly Normalising (SN) Vid 89] also introduces reductions similar to those of [Reg 94] Furthermore, KTU 94] uses (and other reductions) to show that typability in ML is equivalent to acyclic semi unification. SF 92] uses a reduction which has some common themes to . Nederpelt 73] and [dG 93] use whereas [KW 95a] uses fl to reduce the problem of strong normalisation for fi reduction to the problem of weak normalisation for related reductions. KW 94] uses amongst other things, and fl to reduce typability in the rank 2 restriction of system F to the problem of acyclic ....

Nederpelt, R.P., Strong normalisation in a typed lambda calculus with lambda structured types, Ph.D. thesis, Eindhoven University of Technology, Department of Mathematics and Computer Science, 1973. Also appears in [NGV 94].

....(R) z] P ) x]fN [y : Q]g Surely this is clearer than writing ( z : x : y :N)P )RQ g ( z : x :N [y : Q]P )R. 2 An overview of generalised reduction in the literature Generalized reduction was rst introduced by Nederpelt in 1973 to aid in proving the strong normalization of AUTOMATH [19]. Kamareddine and Nederpelt have shown how generalised reduction makes more redexes visible, allowing exibility in reducing a term [7] Bloo, Kamareddine, and Nederpelt show that with generalised reduction one may indeed avoid size explosion without the cost of a longer reduction path and that ....

R.P. Nederpelt. Strong normalisation in a typed lambda calculus with lambda structured types. Ph.D. thesis, Eindhoven University of Technology, Department of Mathematics and Computer Science, 1973. Also appears in [20].

....was carried out at Boston University to whom and especially to Assaf Kfoury, I am grateful. y Department of Computing Science, 17 Lilybank Gardens, University of Glasgow, Glasgow G12 8QQ, Scotland, email: fairouz dcs.glasgow.ac.uk, fax 44 41 3304913. 1 I was told by Rob Nederpelt that in [Nederpelt 73] he assumed that the preservation of Strong Normalisation for fi e is easy to establish yet when editing [NGV 94] he retracted his assumption. All these (related) rules attempt to make more redexes visible. fl C for example, makes sure that y and Q form a redex even before the redex based ....

....when the term is Strongly Normalising (SN) Vid 89] also introduces reductions similar to those of [Reg 94] Furthermore, KTU 94] uses (and other reductions) to show that typability in ML is equivalent to acyclic semi unification. SF 92] uses a reduction which has some common themes to . Nederpelt 73] and [dG 93] use whereas [KW 95a] uses fl to reduce the problem of strong normalisation for fi reduction to the problem of weak normalisation for related reductions. KW 94] uses amongst other things, and fl to reduce typability in the rank 2 restriction of system F to the problem of acyclic ....

Nederpelt, R.P., Strong normalisation in a typed lambda calculus with lambda structured types, Ph.D. thesis, Eindhoven University of Technology, Department of Mathematics and Computer Science, 1973. Also appears in [NGV 94].

....(PTS fij s) 9, 10, 17] form a variant of PTS fi s in which definitional equality is understood as fij conversion. However, the meta theory of PTS fij s is significantly more complex than the one of PTS fi s, see loc. cit. because fij reduction is not confluent on the set of untyped terms [16]. In fact, Geuvers and Werner [12] have shown that such proofs ought to rely on normalization and cannot be achieved by combinatorial means. j expansion j is an alternative computational interpretation of j conversion, with numerous applications in categorical rewriting, unification and ....

....Lemma 4 (Context conversion) Assume that Gamma 2 G is legal, Delta M : A and Gamma = fij Delta. Then Gamma M : A. Proof. By induction on the length of Gamma . A non result: confluence It is folklore that fij reduction is not confluent on pseudo terms. Nederpelt s counter example [16] provides an easy proof of this fact: for every pairwise disjoint A; B; x; y 2 V , we have x : A: x fi x : A: y : B: y) x j y : B: y (Note that the above example also shows the failure of Uniqueness of Normal Forms and of local confluence. The failure of confluence makes it difficult to ....

R. Nederpelt. Strong normalisation in a typed lambda calculus with lambda structured types. PhD thesis, Technical University of Eindhoven, 1973.

....which the conversion rule uses = fi . In order to avoid confusion, we refer to the latter presentation as the usual cube, the usual Calculus of Constructions. j reduction The study of fi reduction j reduction in dependent type theories dates back to the early 70 s, with Nederpelt s thesis [21]. Nederpelt showed that fi reduction j reduction is not confluent on the pseudoterms of a dependently typed language a la Church, i.e. with typed abstractions. Later, van Daalen [12] proved confluence of fi reduction j reduction for (the typed terms of) a language of the Automath family. More ....

R. Nederpelt. Strong normalisation in a typed lambda calculus with lambda structured types. PhD thesis, Technical University of Eindhoven, 1973.

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Nederpelt, R.P., Strong normalisation in a typed lambda calculus with lambda structured types, Ph.D. thesis, Eindhoven University of Technology, Department of Mathematics and Computer Science, 1973. ALso in Nederpelt, R.P., Geuvers, J.H. and de Vrijer, R.C., eds., Selected Papaers on Automath, North Holland, 1994.

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Nederpelt, R.P., Strong normalisation in a typed lambda calculus with lambda structured types, Ph.D. thesis, Eindhoven University of Technology, Department of Mathematics and Computer Science, 1973.

....:X3 :x 1 x 4 )x 3 )x 2 . This reduction is difficult to carry out in the classical calculus. The item notation enables a new and important sort of reduction which has not yet been studied in relation to the standard calculus up to date. We believe that this generalised reduction (introduced in [Nederpelt 73] can only be obtained tidily in a system formulated using some form of our item notation. In fact, one is to compare the bracketing structure of the classical term t of Example 1.4, with the bracketing structure of the corresponding term in item notation: Example 3.6 The bracketing structure ....

Nederpelt, R.P., Strong normalisation in a typed lambda calculus with lambda structured types, Ph.D. thesis, Eindhoven University of Technology, Department of Mathematics and Computer Science, 1973. To appear in Nederpelt, R.P., Geuvers, J.H. and de Vrijer, R.C., eds., Selected Papaers on Automath, North Holland, 1994.

....relation generated out of: general fi) B ffi)s(C x )A , fi s(A[x : B] if s is well balanced General , fi is the reflexive and transitive closure of , fi and fi is the least equivalence relation generated by , fi . General fi reduction has firstly been introduced by Nederpelt in [Ned 73] in order to prove strong normalisation for a typed lambda calculus inspired by de Bruijn s Authomath. Example 4.2 cf. Example 1.1. As (cffi) P f ) mffi) Q x ) is a well balanced segment, then: A j (nffi) cffi) P f ) mffi) Q x ) R y ) yffi) xffi)f , fi (cffi) P f ) mffi) Q x ) nffi) xffi)f ....

Nederpelt, R.P. (1973), Strong Normalisation in a typed lambda calculus with lambda structured types, Ph.D. thesis, Eindhoven University of Technology.

....(4ffi)2 is the scope of (1ffi) 2) and if in t we remove (1ffi) 2) and replace (4ffi)2 by ( 4ffi)2 we get (3ffi)1. Hence t reduces to (3ffi)1. Lemma 3.24 If t ; t then all occurrences of variables in t are bound by the same s that bind them in t. This reduction was introduced in [21], where it was called fi 2 reduction. De Bruijn defines a mini reduction as being either a one step local fi reduction or a void reduction; see [7] Now we can describe the usual one step fi reduction as a combination of oe steps and steps: Definition 3.25 (one step fi reduction) One step ....

R.P. Nederpelt, "Strong normalisation in a typed lambda calculus with lambda structured types", Ph.D. thesis, Eindhoven University of Technology, Eindhoven, 1973.

No context found.

Nederpelt, R.P., Strong normalisation in a typed lambda calculus with lambda structured types, Ph.D. thesis, Eindhoven University of Technology, Department of Mathematics and Computer Science, 1973.

....relation generated out of: general fi) B ffi)s(C x )A , fi s(A[x : B] if s is well balanced General , fi is the reflexive and transitive closure of , fi and fi is the least equivalence relation generated by , fi . General fi reduction has firstly been introduced by Nederpelt in [Ned 73] in order to prove strong normalisation for a typed lambda calculus inspired by de Bruijn s Authomath. Example 4.2 cf. Example 1.1. As (cffi) P f ) mffi) Q x ) is a well balanced segment, then: A j (nffi) cffi) P f ) mffi) Q x ) R y ) yffi) xffi)f , fi (cffi) P f ) mffi) Q x ) nffi) xffi)f ....

Nederpelt, R.P. (1973), Strong Normalisation in a typed lambda calculus with lambda structured types, Ph.D. thesis, Eindhoven University of Technology.

....in t we remove (1ffi) 2) and replace (4ffi)2 by ( Gamma1) 4ffi)2 we get (3ffi)1. Hence t reduces to (3ffi)1. Lemma 3.24 If t ; t 0 then all occurrences of variables in t 0 are bound by the same s that bind them in t. Proof: Left to the reader. 2 j This reduction was introduced in [21], where it was called fi 2 reduction. De Bruijn defines a mini reduction as being either a one step local fi reduction or a void reduction; see [7] Now we can describe the usual one step fi reduction as a combination of oe steps and steps: Definition 3.25 (one step fi reduction) One step ....

R.P. Nederpelt, "Strong normalisation in a typed lambda calculus with lambda structured types", Ph.D. thesis, Eindhoven University of Technology, Eindhoven, 1973.

No context found.

R. P. Nederpelt, "Strong Normalisation in a typed lambda calculus with lambda structured types" in R. P. Nederpelt, J. H. Geuvers, R. C. de Vrijer Editors, "Selected Papers on Automath ", Studies in Logic, North Holland, volume 133, 1994.

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