R.P. Nederpelt. Strong Normalization in a Typed Lambda Calculus with Lambda Structured Types. PhD thesis, Eindhoven University of Technology, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... the following critical pair is not joinable in the general case, e.g. assume A and B to be di#erent ground ##L normal forms: Shift0) IdS) Map) VarCons) ShiftCons) IdS) B S M This problem is similar to the one pointed out by Nederpelt for the # calculus extended with the # rule [36]. In that case, the confluence property holds on terms without type annotations in abstractions (# calculus in the Curry style) but does not on terms with annotated abstractions (# calculus in the Church style) In [11] Geuvers proposes a method to prove confluence for the ## reduction on ....

R. P. Nederpelt, Strong normalization in a typed lambda calculus with lambda structured types, Ph.D. thesis, Technical University Eindhoven, Eindhoven, 1973. 25

.... example, Qffi) Rffi) z ) P ffi) x ) y )N gfi (Rffi) z ) P ffi) x )fN [y : Q]g Surely this is clearer than writing ( z : x : y :N)P )RQ gfi ( z : x :N [y : Q]P )R Generalized reduction was first introduced by Nederpelt in 1973 to aid in proving the strong normalization of AUTOMATH [Ned73]. Kamareddine and Nederpelt have shown how generalized reduction makes more redexes visible, allowing flexibility in reducing a term [KN95] Bloo, Kamareddine, and Nederpelt show that with generalized reduction, one may indeed avoid size explosion without the cost of a longer reduction path; and, ....

....for calculi that do not have explicit substitutions. In fact, the whole idea of internal and external reduction and of skeletons is based around substitutions. It is also fair to say that generalized reduction did not play any role in the proof of PSN (despite its role in proofs of SN as shown in [KW95b, Klo80, Ned73]) 5 The Typed s and gs Calculi Our calculi of explicit substitutions s and gs possess a very nice property that other calculi of explicit substitutions do not possess: namely, the simply typed versions of s and gs are strongly normalizing. The oe calculus of [GL97] does not possess this ....

R. P. Nederpelt. Strong normalization for a typed lambda calculus with lambda structured types. PhD thesis, Technische Hogeschool Eindhoven, 1973.

....is not obvious. Indeed, Barendregt presented a conjecture at Typed LambdaCalculus and Applications 1995 stating that, for every pure type system, weak normalization implies strong normalization. The conjecture is also mentioned by Geuvers [11] and, in a less concrete form, by Klop. Nederpelt [19], Klop [18] Khasidashvili [17] Karr [15] de Groote [9] and Kfoury and Wells [16] present techniques to infer strong normalization from weak normalization. However, these techniques all infer strong normalization of fi reduction in a typed calculus from weak normalization of a more complicated ....

R. Nederpelt. Strong normalization for a typed lambda calculus with lambda structured types. PhD thesis, Eindhoven, 1973.

....redexes can easily be found also in the wide class of strongly sequential OTRSs [3] We develop a method for proving that the reductions constructed according to our perpetual strategy are indeed the longest, and for finding their lengths. Our method is similar to the Nederpelt s method [8] invented to reduce proofs of strong normalization to proofs of weak normalization (i.e. existence of a normal form) For any OTRS R, we define the corresponding non erasing OTRS R , called the extension of R. We add fresh function symbols n of arity n (n = 0; 1; in the alphabet of R. ....

Nederpelt R. P. Strong normalization for a typed lambda-calculus with lambda structured types. Ph. D. Thesis, Eindhoven, 1973.

....that the reductions constructed according to our perpetual strategy are indeed the longest and for computing their lengths. Our method, developed independently in [22] can be viewed as a refinement of the Nederpelt Klop method, which reduces proving strong normalization in a typed calculus [46] and orthogonal Combinatory Reduction Systems (OCRSs) in general [38] to proving weak normalization (i.e. existence of a normal form) We first consider orthogonal TRSs. For any OTRS R, we define the corresponding non erasing OTRS R , which contains special function symbols n , and show that ....

....Theorem to fully extended OERSs. Because of a huge amount of work on strong normalization in typed and untyped calculi, as well as in TRSs, we review only some very closely related work. For other recent surveys on perpetuality and strong normalization, see [58,64] The memory method: Nederpelt [46] and Klop [38] have independently introduced the memory method: Nederpelt used it for proving strong normalization of a typed calculus equivalent to the simply typed calculus; and Klop used it for proving strong normalization of a labelled calculus, from which strong normalization of simply ....

R.P. Nederpelt, Strong normalization for a typed lambda-calculus with lambda structured types, Ph.D. Thesis, Eindhoven University, 1973.

....reduction eventually terminates in a normal form; the rewrite system is UN if every term is UN. Interest in criteria for UN arises, for example, in the proofs of strong normalization of typed calculi, as it relates to the work on reducing strong normalization proofs to proving weak normalization [Ned73, Klo80, dGr93, Kha94c, KW95]. Further, the question: Which classes of terms have the uniform normalization property is posed in [BI94] in connection with finding UN solutions to fixed point equations, and with representability of partial recursive functions by UN terms only, in the calculus. 1 The UN property is clearly ....

Nederpelt R. P. Strong Normalization for a typed lambda-calculus with lambda structured types. Ph.D. Thesis, Eindhoven, 1973.

....two main issues: the relationship between normalization and strong normalization, and the distinction of proof techniques into semantic and syntactic ones. Normalization and strong normalization. The relationship between normalization and strong normalization is studied by Nederpelt and Klop in [42] and [35] respectively, where it is used to give a strong normalization proof for simply typed calculus: an extended calculus is considered to deal with erasing redexes, i.e. those in which the argument is discarded during contraction. The same problem is tackled in a different way by De ....

....The same problem is tackled in a different way by De Groote [25] and Kfoury and Wells [32] introducing new notions of reduction. In all these works, the strong normalization problem is reduced to the Partially supported by MURST grants. 1 normalization problem with respect to a new calculus ([42, 35, 33]) or to a new notion of reduction ( 25, 32] More recently, continuations have been used by Xi and S rensen to reduce strong normalization to normalization [63, 54] for the same notion of reduction: in some sense, at least in the cases where the method of [63, 54] applies (e.g. in the simply ....

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R.P. Nederpelt. Strong normalization for a typed lambda calculus with lambda structured types. PhD thesis, Technische Hogeschool Eindhoven, 1973. 32

....latter part is the difficult part of the proof. Klop [38] shows strong normalization of a labeled calculus by an interpretation in I . Several of the above techniques also use translations from K to I . The technique by Klop was discovered independently from a similar technique by Nederpelt [47] and has been reinvented and extended by many researchers, e.g. Khasidashvili [32] Karr [29] de Groote [16] Kfoury and Wells [30] Xi [83, 86] and Srensen [69] the latter paper gives a survey of some of the variations on the technique. 6. Developments The preceding section analyzed ....

R. Nederpelt. Strong normalization for a typed lambda calculus with lambda structured types. PhD thesis, Eindhoven, 1973.

....where a rewrite system is said to be UN if each of its terms is so. Interest in the criteria for UN arises, for example, in the proofs of strong normalization of typed calculi, since these criteria are related to the work on reducing strong normalization proofs to proving weak normalization [50, 37, 23, 70, 17, 31, 24, 25, 65, 73, 49]. Furthermore, the question: Which classes of terms are UN is posed by Bohm and Intrigila [11] in connection with finding UN solutions to fixed point equations, and with the representability of partial recursive functions by UN PERPETUALITY AND UNIFORM NORMALIZATION 3 terms only, in the ....

Nederpelt, R. P. (1973), "Strong normalization for a typed lambda-calculus with lambda structured types," Ph.D. Thesis, Technische Hogeschool Eindhoven. 30 KHASIDASHVILI, OGAWA, AND VAN OOSTROM

....from Turing s work [Gan80] independently invented by Prawitz [Pra65] in proof theory. A detailed account of it can also be found in [And71] Several authors have invented between techniques to infer from this result strong normalisation of simply typed calculus and related systems, see [Ned73], Klo80] deGr93] and [KW94] Another proof, using a different characterisation of strongly normalising terms is given in [RS95] Definition 6.6 The complexity com(T ) of a type T is defined as follows. com(T ) 0 if T is atomic; maxf1 com(T 0 ) com(T 1 )g if T = T 0 T 1 . Let the ....

....the frameork of [RS95] but the arguments in our opinion are more involved in some cases. There exists a close similarity between . l relation and the perpetual strategies in [Bar76] and [BK82] Other ideas of transforming strong normalisation into weak normalisation can also be found in [Ned73], Klo80] deGr93] and [KW94] l relation brings out inner fi redexes or their residuals by leftmost reductions and subterm relations. It is often easier to prove H(t) 1 than to prove S(t) 1 for given terms t. Perpetual strategies spot the crucial places where fi reductions may change ....

R.P. Nederpelt (1973), Strong normalization in a typed lambda calculus with lambda structured types, Ph.D. thesis, Technische Hogeschool Eindhoven.

....example, Qffi) Rffi) z ) P ffi) x ) y )N gfi (Rffi) z ) P ffi) x )fN [y : Q]g Surely this is clearer than writing ( z : x : y :N)P )RQ gfi ( z : x :N [y : Q]P )R. Generalized reduction was first introduced by Nederpelt in 1973 to aid in proving the strong normalization of AUTOMATH [39]. Kamareddine and Nederpelt have shown how generalised reduction makes more redexes visible, allowing flexibility in reducing a term [21] 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 ....

....PSN for calculi that do not explicit substitutions. In fact, the whole idea of internal and external reduction and of skeletons is based around substitutions. It is also fair to say that generalised reduction did not play any role in the proof of PSN (despite its role in proofs of SN as shown in [30, 31, 39]) 5 The typed s and gs calculi Our calculi of explicit substitutions s and gs possess a very nice property that other calculi of explicit substitutions do not possess. Namely, the simply typed versions of s and gs are strongly normalising. The oe calculus of [15] does not possess this property ....

R. P. Nederpelt. Strong Normalization for a Typed Lambda Calculus with Lambda Structured Types. PhD thesis, Technische Hogeschool Eindhoven, 1973. Appears as a chapter in [38].

....f and g are used for grouping purposes, so that no confusion arises. For example, Qffi) Rffi) z ) P ffi) x ) y )N gfi (Rffi) z ) P ffi) x )fN [y : Q]g Surely this is clearer than writing ( z : x : y :N)P )RQ gfi ( z : x :N [y : Q]P )R Generalized reduction was first introduced by Nederpelt in 1973 to aid in proving the strong normalization of AUTOMATH [Ned73] Kamareddine and Nederpelt have shown how generalized reduction makes more redexes visible, allowing flexibility in reducing a term [KN95] Bloo, Kamareddine, and Nederpelt show that with generalized reduction, one may indeed avoid ....

.... example, Qffi) Rffi) z ) P ffi) x ) y )N gfi (Rffi) z ) P ffi) x )fN [y : Q]g Surely this is clearer than writing ( z : x : y :N)P )RQ gfi ( z : x :N [y : Q]P )R Generalized reduction was first introduced by Nederpelt in 1973 to aid in proving the strong normalization of AUTOMATH [Ned73] Kamareddine and Nederpelt have shown how generalized reduction makes more redexes visible, allowing flexibility in reducing a term [KN95] Bloo, Kamareddine, and Nederpelt show that with generalized reduction, one may indeed avoid size explosion without the cost of a longer reduction path; and, ....

[Article contains additional citation context not shown here]

R. P. Nederpelt. Strong normalization for a typed lambda calculus with lambda structured types. PhD thesis, Technische Hogeschool Eindhoven, 1973.

....only if S j= SN fi for every system S in cube [2] Again we point out that this is a result which can be formulated in the first order Peano arithmetic. 5 Related Work The research on deriving strong normalisation (SN) from weak normalisation (WN) has lasted for at least thirty years. Nederpelt[21], Klop[17] Karr[16] de Groot[7] and Kfoury and Wells[20] have all invented techniques to infer SN from WN. Their techniques all require introducing some notions of reduction different from fi reduction, deriving strong fi normalisation from weak normalisation of these newly introduced notions ....

R.P. Nederpelt (1973), Strong normalization in a typed lambda calculus with lambda structured types, Ph.D. thesis, Technische Hogeschool Eindhoven.

....Many readers have pointed out numerous methods for proving the fi SN property for the simply typed calculus that are not variations on the methods of Tait or Girard. The methods mentioned have included proofs by Gandy, Nederpelt, Klop, de Groote, de Vrijer, van de Pol and Schwichtenberg [Gan80b, Ned73, Klo80, dG93, dV87, vdPS95] Our intention was to discuss proofs of the fi SN property for more powerful type systems such as system F, the system of intersection types, and the system of positive recursive types. These are the usual polymorphic extensions of the simply typed calculus. Although ....

....fi reduction is even more restricted: only one instance of x is replaced instead of all of them 4 Relation to Nederpelt s and Klop s Method In Klop s extensive Ph.D. thesis [Klo80, Chap. 1, x 8] a simple proof is given for the fi SN property which is inspired by an earlier proof by Nederpelt [Ned73] Although this proof is not extended to polymorphic type systems, it has much in common with our method and de Groote s method. Our method and de Groote s method can be seen as ways of avoiding the erasure that occurs when K redexes are reduced. The proofs of Nederpelt and Klop do not avoid or ....

R. P. Nederpelt. Strong Normalization for a Typed Lambda Calculus with Lambda Structured Types. PhD thesis, Technische Hogeschool Eindhoven, 1973.

.... that are fi I normalizable but not strongly fi I normalizable as shown by the following counterexample: x : Omega : x) y) fi I y where Omega j ( x : y: xx) x : y: xx) Nevertheless, a version of Theorem 22 may be stated for pseudo terms by following an idea due to Nederpelt [17]. This idea consists in adapting the notion of reduction fi I as follows. fi T I : x : M:N ) O) x : M:N [x: O] O) if x 2 FV(M ) On the other hand, our notion of reduction fi S may be kept unchanged (except for the syntax) fi T S : x : M:N ) O)P ) x : M: N P ) O) if x 62 ....

....simply typed terms and is, therefore, quite dioeerent form our proof. Nevertheless, it is interesting to note that he transforms fi K redexes into fi I redexes. The untyped terms typable in Nederpelt s calculus correspond exactly to the simply typable terms [7] Therefore Nederpelt s proof [17] may also be seen as an arithmetizable proof for Church s simply typed calculus. As we explained in Section 6, our proof technique is close to the one of Nederpelt. The main dioeerence is that Nederpelt does not use the notion of reduction fi S , but generalizes further the notion of reduction ....

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R.P. Nederpelt. Strong normalization in a typed lambda calculus with lambda structured types. PhD thesis, Technische hogeschool Eindhoven, 1973.

....Unabsorbed redexes can easily be found also in the wide class of strongly sequential OTRSs [4] We develop a method for proving that the reductions constructed according to our perpetual strategy are indeed the longest, and for finding their lengths. Our method is similar to Nederpelt s method [11] by which proving strong normalization in a typed calculus gets reduced to proving weak normalization (i.e. existence of a normal form) Nederpelt s method was reinvented and used by Klop [9] for OCRSs. For any OTRS R, we define the corresponding nonerasing OTRS R which contain special ....

Nederpelt R. P. Strong normalization for a typed lambda-calculus with lambda structured types. Ph. D. Thesis, Eindhoven, 1973.

....property (CR) for Pure Type Systems with fij reduction. For Pure Type Systems with only fi reduction, CR on well typed terms follows immediately from CR on the so called pseudoterms and subject reduction. For fij reduction, CR on the set of pseudoterms is just false, as was shown by [Nederpelt 1973]. Here we prove that CR (for fij) on the well typed terms of a fixed type holds, which is the maximum we can expect in view of Nederpelts counterexample. The proof is given for a large class of Pure Type systems that contains e.g. LF (for which CR for fij was proved by [Salvesen 1989] and ....

....fi M 00 , then 9N 2 Term :M 0 Gamma Gamma fi N M 00 Gamma Gamma fi N . If one is interested in the combination of fi and j reduction, the situation is a little bit more complicated, because we have the following counterexample to CR for fij reduction on T . Originally due to [Nederpelt 1973]. Take M : x:oe: y: y)x; with oe 6= Then M Gamma fi x:oe:x and M Gamma j x: x and both are in normal form. This problem can be overcome by noticing that the only problematic overlapping of redexes in T is when we have a subterm x:oe: y: M )x (with x = 2 FV(M ) and by proving ....

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R.P. Nederpelt, Strong normalization in a typed lambda calculus with lambda structured types. Ph.D. thesis, Eindhoven Technological University, The Netherlands.

....like f(A) g(A; B) which are usually excluded a priori. This is only useful when the system is conditional. 3 Orthogonal CERSs We define orthogonal CERSs (OCERSs) and sketch our proof of Finite Developments for them, implying confluence. The FD proof is based on Nederpelt Klop s method [Ned73,Klo80] for reducing strong normalization to weak normalization. It is similar in structure to, but simpler than Klop s original confluence proof for orthogonal CRSs [Klo80] and we think not more difficult than other existing confluence proofs [vR93,Nip93,OR94,Mel93] The idea of orthogonality is that ....

Nederpelt R.P. Strong normalization for a typed lambda-calculus with lambda structured types. Ph.D. Thesis, Eindhoven, 1973.

.... Its proof remains valid for OERSs if one uses the limit strategy instead of the perpetual strategy of [2] Our method for proving that the reductions constructed according to our perpetual strategy are the longest, and for computing their lengths, is a refinement of Nederpelt Klop method [14, 10], used to reduce proofs of strong normalization to proofs of weak normalization. For any OERS R, we define the corresponding non erasing OERS R , called the extension of R. We add fresh function symbols n in the alphabet of R. For any R rule r : t s, we keep the erased arguments of t in ....

Nederpelt R. P. Strong Normalization for a typed lambda-calculus with lambda structured types. Ph.D. Thesis, Eindhoven, 1973.

....like f(A) g(A; B) which are usually excluded a priori. This is only useful when the system is conditional. 3 Orthogonal CERSs We define orthogonal CERSs (OCERSs) and sketch our proof of Finite Developments for them, implying confluence. The FD proof is based on Nederpelt Klop s method [Ned73, Klo80] for reducing strong normalization to weak normalization. It is similar in structure to, but simpler than Klop s original confluence proof for orthogonal CRSs [Klo80] and we think not more difficult than other existing confluence proofs [vR93, Nip93, OR94, Mel93] The idea of orthogonality is that ....

Nederpelt R.P. Strong normalization for a typed lambda-calculus with lambda structured types. Ph.D. Thesis, Eindhoven, 1973.

....for fi reduction in simply typed calculus by giving an explicit measure which decreases in every step of a certain fi reduction sequence. Prawitz [55] independently uses the same technique to prove weak normalization for reduction of natural deduction derivations in predicate logic. Nederpelt [50], Klop [41] Khasidashvili [38] Karr [32] de Groote [14] and Kfoury and Wells [36] have invented techniques to infer strong normalization from weak normalization. However, these techniques all infer strong normalization of one notion of reduction from weak normalization of a more complicated ....

....2 K , M) 2 SN fi ) M 2 SN fi Proof. By induction on M prove (M) M . This gives the lemma. ut 2.7. Main Lemma (Klop [41] For all M 2 K , M) 2 SN fi ) M 2 SN fi Proof. By Lemmas 2.5 and 2.6. ut 3. Variations on Klop s technique Klop s technique [41] was inspired by Nederpelt s [50] technique, and is also related to the later techniques by Khasidashvili [38] Karr [32] de Groote [14] and Kfoury and Wells [36] The similarity between the different approaches is sometimes blurred because each technique is described in a particular context in terms of labeled or typed terms. ....

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R. Nederpelt. Strong normalization for a typed lambda calculus with lambda structured types. PhD thesis, Eindhoven, 1973.

....For example, in the term (x:y:L) M N it is clear that the occurrences of y in L will be replaced by N , but that substitution will not be possible until we have first replaced x with M . Nederpelt proposed a notion of generalised fi reduction , which allows this contraction to occur at once (Nederpelt, 1973): x 1 :x 2 :L) M 1 M 2 , x 1 :L[x 2 : M 2 ] M 1 (x 1 :x 2 :x 3 :L) M 1 M 2 M 3 , x 1 :x 2 :L[x 3 : M 3 ] M 1 M 2 and so on. The manipulation made explicit by our (C) rule is implicit in Nederpelt s rule, appearing only when necessary for a beta like contraction to occur, but Nederpelt ....

Nederpelt, R. P. (1973). Strong Normalization in a Typed Lambda Calculus with Lambda Structured Typed. Doctoral thesis, Dept. of Mathematics and Computer Science, Eindhoven University of Technology.

....the fi SN question into a question of weak normalization, which merely asks whether some reduction sequence termi nates, not whether all of them do. This is the style used in this paper. The first proof in this style was by Nederpelt for a system equivalent to the simplytyped calculus [Ned73]. Klop devised a variant of this proof for the simply typed calculus [Klo80, Chap. 1, x 8] and then created a more general method which works for many combinatory reduction systems (sometimes called higher order rewrite systems , unrelated to the polymorphic extensions of the calculus) ....

R. P. Nederpelt. Strong Normalization for a Typed Lambda Calculus with Lambda Structured Types. PhD thesis, Technische Hogeschool Eindhoven, 1973.

....uses the same technique to prove weak normalization for reduction of natural deduction derivations in predicate logic. Since weak normalization is sometimes easier to establish than strong normalization, it is natural to develop techniques to infer the latter from the former. Indeed, Nederpelt [43], Klop [38] Khasidashvili [37] Karr [35] de Groote [17] and Kfoury and Wells [36] have invented such techniques. However, these techniques all infer strong normalization of one notion of reduction from weak normalization of a more complicated notion of reduction. S rensen [56] and Xi [61] ....

R. Nederpelt. Strong normalization for a typed lambda calculus with lambda structured types. PhD thesis, Eindhoven, 1973.

....formal developments. These two approaches should be joined in the future and some ideas how this can be achieved are mentioned in [AJS94] The formal system we will use as a basis is called Deva and is another descendant of a member of the Automath family of languages, namely Nederpelt s (cf. Ned73] In principle, it is a higher order functional language (see Section 3 below for a quick overview) Deva was originally developed in the context of the Esprit project ToolUse (cf. Gro90, Sin80, Web91b] in order to study formal software development methods. Several case studies on formalized ....

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

....reduction eventually terminates in a normal form; the rewrite system is UN if every term is UN. Interest in criteria for UN arises, for example, in the proofs of strong normalization of typed calculi, as it relates to the work on reducing strong normalization proofs to proving weak normalization [Ned73, Klo80, dVr87, dGr93, Kha94c]. Further, the question: Which classes of terms have the uniform normalization property is posed in [BI94] in connection with finding UN solutions to fixed point equations, and with representability of partial recursive functions by UN terms only, in the calculus. 1 The UN property is ....

Nederpelt R. P. Strong Normalization for a typed lambda-calculus with lambda structured types. Ph.D. Thesis, Eindhoven, 1973.

....on typable terms and the Church Rosser property. The equivalence of the two versions follows from the Church Rosser property for typable terms in CC with (conv fij ) However, with fij reduction the Church Rosser property on the pseudoterms T is invalid. The well known counterexample is due to [Nederpelt 1973]: For A 6= fij B and x = 2 FV(M ) x:A: y:B:My)x can be reduced by a fi step and an j step to two terms that have no common reduct. This complicates matters quite a lot because some meta theorems depend on the Church Rosser property (normalization proofs usually require it. Further it not clear ....

R.P. Nederpelt, Strong normalization in a typed lambda calculus with lambda structured types. Ph.D. thesis, Eindhoven Technological University, The Netherlands.

....for fi reduction in simply typed calculus by giving an explicit measure which decreases in every step of a certain fi reduction sequence. Prawitz [71] independently uses the same technique to prove weak normalization for reduction of natural deduction derivations in predicate logic. Nederpelt [63], Klop [55] Khasidashvili [54] Karr [52] de Groote [26] and Kfoury and Wells [53] have invented techniques to infer strong normalization from weak normalization. However, these techniques all infer strong normalization of one notion of reduction from weak normalization of a more complicated ....

R. Nederpelt. Strong normalization for a typed lambda calculus with lambda structured types. PhD thesis, Eindhoven, 1973.

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R.P. Nederpelt. Strong Normalization in a Typed Lambda Calculus with Lambda Structured Types. PhD thesis, Eindhoven University of Technology, 1973.

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R. Nederpelt. Strong normalization in a typed lambda calculus with lambda structured types. PhD thesis, Eindhoven University (Netherlands), 1973.

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Nederpelt, R.: Strong normalization in a typed lambda calculus with lambda structured types, Ph.D. Thesis, Technische Universiteit Eindhoven, The Netherlands, 1973.

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R. Nederpelt. Strong normalization in a typed lambda calculus with lambda structured types. PhD thesis, Technische Universiteit Eindhoven, The Netherlands, 1973.

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R. P. Nederpelt. Strong Normalization for a Typed Lambda Calculus with Lambda Structured Types. PhD thesis, Technische Hogeschool Eindhoven, 1973.

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Nederpelt, R., Strong Normalization in a Typed Lambda Calculus With Lambda Structured Types, PhD thesis, Eindhoven 1973.

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R. Nederpelt, Strong normalization in a typed lambda calculus with lambda structured types, Ph.D. thesis, Eindhoven (1973).

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Nederpelt, R., Strong Normalization in a Typed Lambda Calculus With Lambda Structured Types, PhD thesis, Eindhoven 1973.

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R. Nederpelt. Strong Normalization in a Typed Lambda Calculus With Lambda Structured Types. PhD thesis, Eindhoven, 1973. Cited on page 2.

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R. P. Nederpelt. Strong Normalization in a Typed Lambda Calculus with Lambda Structured Types. PhD thesis, Technical University of Eindhoven, 1973.

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R.P. Nederpelt. Strong Normalization for a typed lambda calculus with lambda structured types. Phd. Thesis, Eindhoven, Netherlands, 1973.

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Nederpelt R.P., Strong normalization for a typed lambda-calculus with lambda structured types. Ph.D. Thesis, Technische Hogeschool Eindhoven, 1973.

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Nederpelt R.P., Strong normalization for a typed lambda-calculus with lambda structured types. Ph.D. Thesis, Eindhoven, 1973.

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R. P. Nederpelt. Strong Normalization in a Typed Lambda Calculus with Lambda Structured Types. PhD thesis, Technical University of Eindhoven, 1973.

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R. P. Nederpelt. Strong Normalization in a Typed Lambda Calculus with Lambda Structured Types. PhD thesis, Technische Hogeschool Eindhoven, June 1973.

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R.P. Nederpelt. Strong Normalization for a Typed Lambda-calculus with Lambda Structured Types. PhD thesis, Technische Universiteit Eindhoven, 1973.

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