J.H. Geuvers and M.J. Nederhof, A modular proof of strong normalisation for the calculus of constructions. Journal of Functional Programming, vol 1 (2), pp 155-189.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is an open problem whether extending an arbitrary PTS with definitions preserves strong normalisation Worse still, proving strong normalisation for particular PTS s extended with definitions is already a problem. The strong normalisation proofs for particular type systems given in [Coq85] Luo89] [GN91] [Bar92] cannot be extended in any obvious way to prove strong normalisation of these systems extended with definitions. In this paper we show how strong normalisation of a PTS extended with definitions follows from strong normalisation of another (larger) PTS. This enables us to prove that for ....

Herman Geuvers and Mark-Jan Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1(2):155--189, 1991.

.... the Equational Theory of Non Normalising Pure Type Systems Gilles Barthe and Thierry Coquand June 28, 2001 1 Introduction Pure Type Systems [1, 8] provide a generic framework for the description of many typed calculi and logics that occur in the literature. In such type systems abstraction is usually explicitely typed. One writes x : A:e : Pix : A)B for the function of domain A which sends x to e. Another possible formulation however ....

H. Geuvers and M.J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1(2):155189, April 1991.

.... 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 ....

....= Rl [ f(s; s; F s ) j s 2 Sr F g The diagonal closure of S is the specication S Delta = Sr Delta ; Ax; Rl Delta ) 6 2.2 Results This subsection summarizes some properties of PTSs that are used crucially throughout the paper. Other properties can be found in standard texts on PTSs [3, 13, 14]. The rst lemma states that fi reduction is conAEuent. Lemma 7 (ConAEuence) The relation fi is conAEuent. Hence normal forms are unique. Corollary 8 If M 2 WN(fi) there exists exactly one N 2 NF(fi) denoted nf(M) such that M fi N . The second lemma provides a precise analysis of the ....

H. Geuvers and M.J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1(2):155189, April 1991.

....of subtyping is used to prove the following lemma, which is crucial to prove subject reduction. Lemma 8. If Pi x : A: B v Pi x : A 0 : B 0 then A 0 v A and B v B 0 . Using this lemma, subject reduction can be proved by adapting the standard proof for pure type systems (see for example [15]) to the case of pure type systems with subtying, as also done in [36] Note however that the use of reduction in the (conv) rule is crucial. Indeed, consider the term M = conseven0 (nil even) which has type list even. We have (using notation as in the denition of the typing rules) N 1 : C 1 and ....

....We have the conversion: Q (cons even 0 (nil even) Q (cons nat 0 (nil even) and hence by the conversion rule we have N 2 0 (nil even) Q (cons even 0 (nil even) Proposition 9 (Subject Reduction) If Gamma M : A and M fi M 0 then Gamma M 0 : A. Proof. The proof proceeds as in [15] by induction on the structure of derivations, proving simultaneously the following two implications: if Gamma M : A and M fi M 0 , then Gamma M 0 : A, and if Gamma M : A and Gamma fi Gamma 0 then Gamma 0 M : A. Here we only consider the case where the last typing rule ....

H. Geuvers and M.J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1(2):155189, April 1991.

....is again terminating. This result is generalized by Barbanera, Fern andez and Geuvers in a series of papers first to intersection type systems, then to higher order calculus and finally to the so called algebraic cube [1, 3, 4] Termination of the algebraic cube is proved along the lines of [18] by using a computability predicate and two reduction preserving translations. In [7] Barthe and Geuvers introduce the notion of algebraic type system and provide a general criterion for termination of an algebraic type system. The criterion is proved by a model construction based on saturated ....

H. Geuvers and M.-J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1:155--189, 1991.

.... if we assign type P (kak) to kek, then type P (kbk) can also be assigned to kek; when kak 6= fi kbk, this may not be possible since P (kak) and P (kbk) are two different types; kak 6= fi kbk often holds when a 6= b(try a = x:x) and b = a(a) Fortunately, there exist some methods in [12] and [11] to remove type dependencies on terms for certain typed calculi. A proof in [11] shows that j= SN fi implies C j= SN fi , where C stands for the construction of calculus. Following this example, we can verify that S j= WN fi if and only if S j= SN fi for every system S in cube [2] Again we ....

....when kak 6= fi kbk, this may not be possible since P (kak) and P (kbk) are two different types; kak 6= fi kbk often holds when a 6= b(try a = x:x) and b = a(a) Fortunately, there exist some methods in [12] and [11] to remove type dependencies on terms for certain typed calculi. A proof in [11] shows that j= SN fi implies C j= SN fi , where C stands for the construction of calculus. Following this example, we can verify that S j= WN fi if and 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 ....

H. Geuvers and M.J. Nederhof (1991), A modular proof of strong normalisation for the calculus of constructions, Journal of Functional Programming.

.... Institutionen for Datavetenskap, Chalmers Tekniska Hogskola, Goteborg, Sweden Departamento de Inform atica, Universidade do Minho, Braga, Portugal gillesb cs.chalmers.se gilles di.uminho.pt 1 Introduction Pure Type Systems (PTS fi s) provide a parametric framework for typed calculi a la Church [1, 2, 10, 11]. One important aspect of PTS fi s is to feature a definitional equality based on fi conversion. In some instances however, one desires a stronger definitional equality based on fij conversion. The need for such a strengthened definitional equality arises for example when using type theory as a ....

H. Geuvers and M.J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1(2):155--189, April 1991.

....terminating. This result is generalised by Barbanera, Fern andez and Geuvers in a series of papers first to intersection type systems, then to higher order calculus and finally to the so called algebraic cube [2, 4, 5] Strong normalisation of the algebraic cube is proved along the lines of [20] by using a computability predicate and two reduction preserving translations. In [9] Barthe and Geuvers introduce the notion of algebraic type system and provide a general criterion for strong normalisation of an algebraic type system. The criterion is proved by a model construction based on ....

H. Geuvers and M.-J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1:155--189, 1991.

....is a good reference about A translation. Recently, connections between A translation and Continuation Passing Style have been investigated. See for instance Murthy s Ph. D. thesis[10] We are going to generalise A translation to a large class of Pure Type Systems, introduced recently by Barendregt [1, 4]. This generalisation is motivated by the following problem: to extract constructive informations from paradoxes in inconsistent type systems. More specifically, let us define a looping combinator as being a term having the same Bohm tree as the fixed point combinator Y: It has been shown by ....

....in inconsistent Type Systems. The last section gives some examples of Type Systems containing looping combinators. We end by raising some questions suggested by our work. 1 Logical Pure Type Systems We use here the standard definition of Pure Type Systems from Barendregt and Geuvers Nederhof [1, 4]. In particular, we make fairly heavy implicit use of the general properties of Pure Types Systems as presented in [4] Definitions : A Pure Type System L is logical iff it is functional (see [4] and contains two distinguished sorts P rop and Type such that ffl P rop : T ype is an axiom of L ....

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H. Geuvers and M.-J. Nederhof. Modular proof of strong normalisation for the calculus of constructions. In the Journal of Functional Programming, Volume 1, Part 2, April 1991, pages 155 - 189.

.... 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] ....

....financial support from the Dutch Science Foundation (NWO) and from the European TMR programme. 2 Pure Type Systems In this section, we present the basics of Pure Type Systems. Only crucial properties are considered. Other properties can be found in standard texts on Pure Type Systems [2, 9, 10]. 2.1 Specifications Pure Type Systems provide a parametric framework for typed calculi a la Church. Parametricity is achieved through the notion of specification, which consists of a set of universes and two relations expressing abstract dependencies between them. Definition 1 (Specification) ....

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H. Geuvers and M.J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1(2):155--189, April 1991.

....and lays the foundations for the design of proof development systems with a computationally meaningful classical operator. This paper presents a uniform framework for classical calculi. The central notion of this paper, classical pure type system (CPTS) is based on the notion of pure type system [6,30,31] and offers a uniform formalism to define and study classical calculi. The formalism is minimal e.g. its only type constructor is the generalized function space Pi and yet al..lows for many interesting observations. In particular, it may be used to study for the first time it seems ....

....standard proof technique for strong normalization, namely the technique of reductionpreserving mappings. More precisely, we examine the Harper Honsell Plotkin translation [36] from the PTS core of Edinburgh s Logical Frameworks to simply typed calculus and the Geuvers Nederhof translation [31] from Coquand s Calculus of Constructions to Girard s higher order calculus. It turns out that these translations lift to the framework of DFCPTSs but not to that of CPTSs. In the latter case, this failure is once more due to the unorthodox behavior of Delta reduction. ffl Section 7: Classical ....

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H. Geuvers and M.-J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1:155--189, 1991.

.... 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 ....

.... Delta = R [ f(s; s; F s ) j s 2 SF g The diagonal closure of S is the specification S Delta = S Delta ; A; R Delta ) 2.2 Results This subsection summarises some properties of PTSs that are used crucially throughout the paper. Other properties can be found in standard texts on PTSs [3, 13, 14]. The first lemma provides a precise analysis of the possible ways in which a legal judgement is derived. Lemma 7 (Generation) 1. Gamma s : C ) 9(s; s 0 ) 2 A : C = fi s 0 2. Gamma x : C ) 9s 2 S; D 2 T : C = fi D; x : D) 2 Gamma; Gamma D : s 3. Gamma x:A: b : C ) 9s 2 S; B 2 ....

H. Geuvers and M.J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1(2):155--189, April 1991.

....Definition 3.7] By Proposition 3, it is enough to prove: 1. if A; B 2 C [ K and jAj j jBj then [ A] B] 2. if A; B 2 C [ K are legal and A fi B then [ A] B] Both properties are proved by induction on the structure of A. One can also prove strong normalisation along the lines of [17]. Again we only need to change the Adequacy Lemma, embodied in Remark 47 and Lemma 49 of [17] Consistency is derived from normalization and subject reduction in the usual way. For the sake of brevity, we omit a more general consistency criterion and concentrate on cPICC. In the sequel we write ....

....then [ A] B] 2. if A; B 2 C [ K are legal and A fi B then [ A] B] Both properties are proved by induction on the structure of A. One can also prove strong normalisation along the lines of [17] Again we only need to change the Adequacy Lemma, embodied in Remark 47 and Lemma 49 of [17]. Consistency is derived from normalization and subject reduction in the usual way. For the sake of brevity, we omit a more general consistency criterion and concentrate on cPICC. In the sequel we write Con(S) if there is no M 2 T such that hi M : Pi ff: ff. Lemma 4. Con(cPICC) Proof. Show ....

H. Geuvers and M.J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1(2):155--189, April 1991.

....by jxj (or by jPj) and x i and P i are denoted by (x) i and (P) i , respectively. As usual, we write P Q for the term 5x : P:Q when x = 2 FV(Q) We also write P Q for (P) 0 (P) 1 : P)n01 Q where n = jPj. The following properties of PTS s are well known in literature (cf. Bar92] [GN91], vBJ93] 1. Confluence) If P 1 = fi P 2 , then 9P [P 1 fi P fi P 2 ] 2. Weakening lemma) If 0 P : Q, 0 0 is a legal basis, and 0 0 0 (i.e. every pair x : P in 0 is in 0 0 ) then 0 0 P : Q. 3. Substitution lemma) If 0; x : P; 0 0 Q : R and 0 M : P , then 0; 0 0 ....

....the content of lemma 4.3 by using j expansion, and we applied it in characterizing derivations in cube. However Each system in cube is known to be (strongly) normalizing. It is also known that if 0 P : Q and 0 P : Q 0 (where Q;Q 0 are in fi normal form) in cube, then Q j Q 0 [Bar92][GN91]. When 0 K : 2 (or 0 A : 3, respectively) the term K (or A) is called a kind (or a type) When 0 C : K : 2, the term C is called a constructor of kind K. When 0 a : A : 3, the term a is called an object of type A. We use metavariables K; J; for kinds, C; D; for constructors, ....

H. Geuvers and M.J. Nederhof. Modular proof of strong normalisation for the calculus of constructions. Journal of Functional Programming, 1(2):155--189, 1991.

....(N ) M ) Hence OE E is well founded. Theorem 21 gives a very powerful criterion to prove strong normalisation of a classical pure type system. One may however wonder if other, less powerful techniques may extend to classical pure type systems. We discuss some of the techniques here. ffl In [29], Geuvers and Nederhof prove strong normalization of the Calculus of Constructions by defining a reduction and derivation preserving translation from C to . Our attempts to generalize the translation from C to failed as the obvious translation of abstraction does not preserve reduction and ....

H. Geuvers and M.-J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1:155--189, 1991.

....and Erik Barendsen were helpful in tracking down references. 2 The systems and 2 In this section we introduce two typed lambda calculi: the simply typed lambda calculus with one base type O, and 2, the polymorphic lambda calculus. These systems are presented in the PTS format (see [1] or [11]) Definition 2.1 (Terms and reductions) Pseudo terms are given by the following abstract syntax: T : C j V j T T j V:T :T j PiV:T :T : Here C is an infinite set of constants and V is an infinite set of variables; x, y, y 0 , y 1 , range over V. Among the constants, three elements ....

.... Gamma A 1 : Pix:B 1 :B 2 Gamma A 2 : B 1 Gamma A 1 A 2 : B 2 [x : A 2 ] Abstraction Gamma; Qx:A 1 A 2 : B 2 Gamma Pix:A 1 :B 2 : s Gamma x:A 1 :A 2 : Pix:A 1 :B 2 Table 1: The rules The following meta theoretic properties hold in and . The results are taken from [1] [11], and [10] Theorem 2.4. 1. Substitutivity) If A i fij A 0 , then A[x : B] i fij A 0 [x : B] 2. Substitution Lemma) Suppose Gamma; Qx : C; Delta ffi A : B and Gamma ffi D : C. Then Gamma; Delta[x : D] ffi A[x : D] B[x : D] 3. Strong Normalization) If Gamma ffi A ....

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J.H. Geuvers and M.-J. Nederhof. Modular proof of strong normalisation for the calculus of constructions. Journal of Functional Programming, 1:155--189, 1989.

.... dependent types and type constructors (see [3] Unfortunately, while second order matching is decidable in CoC ( 4] 6] 7] third order matching is undecidable ( 4] 5] Since CoC forms the starting point for many type theoretical studies as well as for concrete implementations (e.g. [13], 15] 11] 16] it is important to understand why third order and hence higher order matching is undecidable in CoC: is it a specific feature (polymorphism, dependent types, type constructors) that is responsible for the undecidability or is the undecidability caused by the interaction of ....

.... lambda calculi: the simply typed lambda calculus with one base type O, and , the simply typed lambda calculus with type variables and type constructors (we consider this system with the Conversion Rule for fij conversion) For more information on these systems the reader is referred to [2] or [13]. Definition 2.1 (Terms and reductions) Pseudo terms are given by the following abstract syntax: T : C j V j T T j V:T :T j T T : Here C is an infinite set of constants and V is an infinite set of variables; x, y, y 0 , y 1 , range over V. Among the constants, three elements are ....

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J.H. Geuvers and M.-J. Nederhof. Modular proof of strong normalisation for the calculus of constructions. Journal of Functional Programming, 1:155--189, 1989.

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J.H. Geuvers and M.J. Nederhof, A modular proof of strong normalisation for the calculus of constructions. Journal of Functional Programming, vol 1 (2), pp 155-189.

....and the Calculus of Constructions. 2.1 Properties of Pure Type Systems An important motivation for the definition of Pure Type Systems is that many important properties can be proved for all PTSs at once. Here we list the most important properties and discuss them briefly. Proofs can be found in [Geuvers and Nederhof 1991] and [Barendregt 1992] Here we only mention the ones that are needed for the proof of conservativity of the extension of a PTS with a fixed point combinator. In the following, unless explicitly stated otherwise, refers to derivability in an arbitrary PTS. Furthermore, Gamma is a correct ....

J.H. Geuvers and M.J. Nederhof, A modular proof of strong normalisation for the calculus of constructions. Journal of Functional Programming, vol 1 (2), pp 155-189.

.... fiR B or B fiR A Note that the abstraction and product rules have a slightly more general presentation than usual (see [3] for example) For pure type systems, the two presentations can be shown to be equivalent; in fact, this is a simple consequence of the permutation lemma and strengthening ([17]) In an algebraic pure type system, the reduction relation is not confluent on the set of pseudo terms; as a result, the usual proofs of subject reduction and of other results relying on subject reduction, such as strenghtening cannot be extended. This motivates the following definition. ....

H.Geuvers and M-J. Nederhof. A modular proof of strong normalisation for the calculus of constructions. Journal of Functional Programming, 1:155--189, 1991.

....and Computer Science, Eindhoven University of Technology The Netherlands 1. Introduction In the literature there are several different proofs of Strong Normalization (SN) for the Calculus of Constructions (CC) Some of them are of purely syntactical nature (like the ones in [Coquand 1985] [Geuvers and Nederhof 1991] and in [Coquand and Gallier 1990] while others give a proof of normalization by describing an appropriate semantics (like [Ong and Ritter 1994] and [Altenkirch 1993] who describe an denotational semantics, but also [Goguen 1994] who describes a typed operational semantics) Apart from these, ....

....of typed terms instead of untyped terms. This is done, for example, in [Berardi 1988] and [Coquand and Gallier 1990] Another possibility is to reduce the question e mail: herman win.tue.nl of SN for a system with type dependency to SN for a system without type dependency. This is done in [Geuvers and Nederhof 1991]. Both approaches lead to rather involved proofs that consist of putting several steps together. Furthermore, these proofs do not easily scale up to extensions of CC with other type constructors. The approach that we use here is based on saturated sets. It yields a (relatively short) direct proof ....

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J.H. Geuvers and M.J. Nederhof, A modular proof of strong normalisation for the calculus of constructions. Journal of Functional Programming, vol 1 (2), pp 155-189.

....a general discription of a large class of typed lambda calculi and makes it possible to derive a lot of meta theoretic properties in a generic way. We shall not go into details about meta theory nor do we give a list of examples of systems in the form of a PTS but refer to [Barendregt 199 ] and [Geuvers and Nederhof 1991]. Here we just repeat the definition and the main meta theoretic properties. Definition 2.5 For S a set, A ae S Theta S and R ae S Theta S Theta S, S; A; R) is the typed lambda calculus with the following deduction rules. sort) s 1 : s 2 if (s 1 ; s 2 ) 2 A (var) Gamma A : s Gamma; ....

....B or Gamma B : A is derivable. The set of typable terms of (S; A; R)is denoted by TERM( S; A; R) A practical purpose for the use of the PTS framework is that many properties can be proved once and for all for the whole class of PTSs. We list the most important ones. Proofs can be found in [Geuvers and Nederhof 1991] or [Barendregt 199 ] In most cases the proofs are not essentially different from the proof for the Calculus of Constructions. First, the reduction relation Gamma Gamma fi is Church Rosser on T. That is, if M Gamma Gamma fi M 1 and M Gamma Gamma fi M 2 then M 1 Gamma Gamma fi N ....

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J.H. Geuvers and M.J. Nederhof, A modular proof of strong normalisation for the calculus of constructions. Journal of Functional Programming, vol 1 (2), pp 155-189.

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H. Geuvers and M.J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1:155189, April 1991.

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H. Geuvers and M.J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1:155--189, April 1991.

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H. Geuvers and M.J. Nederhof. A modular proof of strong normalisation for the Calculus of Constructions. Journal of Functional Programming, 1:155--189, April 1991.

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