H. Geuvers and M.-J. Nederhof. A modular proof of strong normalization for the Calculus of Constructions. Journal of Functional Programming, 1(2):155{ 189, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we present, without proof, the main syntactical results that Barendregt s cube enjoys, like the Church Rosser Property, subject reduction, strong normalization, and a list of properties that will be useful in the next chapters. A through investigation of this properties can be found in [Bar92, GN91] In Section 2.2, we present the main differences between the original presentation of some well known systems, namely the simply typed Church s calculus, and the second order polymorphic Girard s calculus, and the corresponding presentations in the Barendregt s cube, whereas Section 2.3 gives ....

....F of Girard are given in an equivalent constructive version. The system P is often called Logical Framework [HHP92] The system P (also called CC) is the Calculus of Construction of Coquand and Huet [CH88] 2.1. 1 Properties of Barendregt Cube The properties of this cube are proved in [Bar92, GN91] We list just a few of them, those that are used in the next chapters. Proposition 2.1.11 If Gamma t A : s 1 and Gamma t A : s 2 , then s 1 j s 2 . Property 2.1.12 A[B=b] C=c] j A[C=c] B[C=c] b] provided b 62 FV(C) Property 2.1.13 (Church Rosser Property for Typed Systems) A = fi B ....

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

....from weak normalization. Section 6 assesses the scope of the technique and reviews directions for further work. 2. Pure type systems This section presents some fundamental definitions. The first subsection reviews pure type systems, as presented by Barendregt, Geuvers, and Nederhof [2, 12, 11]. Throughout the paper we use implicitly numerous well known properties about pure type systems. The second subsection introduces some notation regarding normalization. The third subsection presents the new class of generalized non dependent pure type systems, in which types do not depend on ....

....S is weakly normalizing if all legal expressions are weakly normalizing, and strongly normalizing if all legal expressions are strongly normalizing. In this case we write S j= WN fi and S j= SN fi , respectively. 2.10. Example. All the systems of the cube are strongly normalizing see, e.g. [6, 2, 12, 11]. The system is the simplest PTS which is not strongly normalizing. The system U is is a natural extension of which, surprisingly, is not strongly normalizing. This result shows that, apparently, the fact that fails to be strongly normalizing is not merely a consequence of the cyclicity in its ....

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

....equivalence, denoted =R , as the re exive, symmetric and transitive closure of R . Proposition 1 (Church Rosser) The , and reduction are ChurchRosser. Notice that in the strict framework of Pure Type Systems, the reduction does not satisfy the Church Rosser property [4], due to the presence of a type annotation in the abstraction. However, such a problem does not arise in the implicit calculus, since we use a Curry style abstraction. This point has some importance, since the implicit calculus has a strong requirement on the reduction rule as we shall see ....

....Implicit Calculus of Constructions The meaning of the non dependent implicit product. The presence of the last rule called (Str) for strengthening may be surprising, since the corresponding rule is admissible in the (Extended) Calculus of Constructions, and more generally in all functional PTS [4]. In the implicit calculus, this is not the case, due to the presence of non dependent implicit product. The main consequence of rule (Str) an the reason for introducing it is the following: Lemma 2 (Non dependent implicit product) Let be a context, and let T and U be terms such that x = 2 FV ....

J. H. Geuvers and M. J. Nederhof. A modular proof of strong normalization for the calculus of constructions. In Journal of Functional Programming, volume 1,2(

....= Dependencies 6 6 2 P 6 P 6 P2 P Let S be one of these eight systems. We write S A : B to indicate that t A : B can be derived using only the rules for S. The properties of this cube are studied in [1, 8]. 1.2 The Cube of Type Assignment Systems In this subsection we will present the cube of type assignment systems as was rst presented in [10] The de nition of the type assignment cube is based on the de nition of an erasing function E that erases all type information from the typed terms. In ....

Geuvers, H. and Nederhof, M., Modular Proof of Strong Normalization for the Calculus of Constructions, Journal of Functional Programming, 1(2), 155-189, 1991.

....9c; C[ c : C] the variables of Var( are di erent, FV(c : C) Var( and y : D 2 ) y : D. Moreover, if s : 2 and legal, then s : 2. The relation induces the abstract reduction: Furthermore, every PTS veri es a generation lemma similar to the one in [5] but replacing = by . ax) 2:4 2 : 4 2 A (var) A:s ;x:A x:A x 62 (weak) b:B A:s ;x:A b:B b 2 S [ V; x 62 (apl) f : x:A:F a:A fa:F [x: a] A:s 1 ;x:A B:s 2 ;x:A b:B x:A:b : x:A:B s1 : s2 : s3 2 R ( a:A A 0 :s a:A 0 A = A 0 ( a:A A 0 :s a:A 0 ....

.... ( r a : A 0 A If we include terms the cases ( 1 ) and ( 2 ) are trivial, but the case (pair) has a serious inconvenient. The S r property is very dicult to be proved 1 . Due to the equivalence EP S r we can try S r directly. Following an standard proof like [5] we nd an important inconvenient if the last rule applied is ( If we consider the 0 r system obtained replacing the ( rule by ( 0 ) x : A 0 r b : B 0 0 r x : A:B : s 0 r x : A:b : x : A 0 :B 0 B B 0 ; A A 0 it is easy to obtain 0 r and (by ....

H. Geuvers and M. Nederhof. Modular proof of strong normalization for the calculus of constructions. Journal of Functional Programming, 1:15-189, april 1991.

....in a framework that extends the scope of usual functional programming languages such as ML. In the following, we will only recall some basic definitions and results whose proofs are detailed in [10] 2. 2 Basic notations In this section, we shall assume that the reader is familiar with PTS [5]. The set of sorts used in the implicit calculus is defined by S = fProp; Setg [ fType i ; i 0g; where Prop and Set denote the impredicative sorts, and (Type i ) i 0 the usual universe hierarchy of ECC. Notice that here, we have two impredicative sorts, since it is convenient to distinguish a ....

J. H. Geuvers, M. J. Nederhof. A modular proof of Strong Normalization for the Calculus of Constructions. Journal of Functional Programming 1,2(1991), 155-189.

....then the consistency of Coqw . The translation from Coqw to coq preserves the underlying pure lambda terms (without any type information) We believe (but did not check precisely the details) that the method of translation from the pure Calculus of Constructions to F used by Geuvers and Nederhof [13] could also be adapted to Coqw and coq in order to justify strong normalization for Coqw . 4.4.2 Strong elimination To deal with strong elimination is more complicated because in a system with strong elimination we cannot anymore ignore dependencies with respect to programs. Also we know that a ....

H. Geuvers and M.-J. Nederhof. A modular proof of strong normalization for the Calculus of Constructions. Faculty of Mathematics and Informatics, Catholic University Nijmegen, 1989.

....interesting examples formalized in LEGO include program specification and data refinement [25] strong normalization of System F [1] synthetic domain theory [40, 41] and operational semantics for imperative programs [43, 23] 1.1. Why PTS have a beautiful meta theory, developed informally in [2, 6, 17, 48, 16]. 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 level of detail needed to ....

....because it shows that full weakening (weakening) is admissible in our system, justifying our use of atomic weakening in the definition of (section 4. 2) The subcontext relation is defined Gamma v Delta , 8b : PP Theta Trm : b 2 Gamma ) b 2 Delta This is the definition used informally in [2, 17, 48]; a much more complicated definition is required to express this property in a representation using de Bruijn indices for global variables. McKinnaPollack.tex; 7 12 1998; 17:47; p.30 Some Lambda Calculus and Type Theory Formalized 31 Now we can state (thinning lemma) Gamma M : A ) Gamma v ....

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Geuvers, H. and M.-J. Nederhof: 1991, `A Modular Proof of Strong Normalization for the Calculus of Constructions'. Journal of Functional Programming 1(2), 155--189.

.... Vrijer [61] Other results on bounds on lengths of reductions are obtained for example by Schwichtenberg [52] for , using a method due to Howard [26] and by Springintveld [56] for two systems of the Barendregt s Cube [7] Springintveld uses Schwichtenberg s results and Geuvers and Nederhof s [22] translation of calculi in Barendregt s Cube into the simply typed calculus. In [22] a (syntactic) modular proof for the Calculus of Constructions [12] is given (see [23] for more references to normalization proofs for the Calculus of Constructions) Syntactic proof techniques. One of the rst ....

....by Schwichtenberg [52] for , using a method due to Howard [26] and by Springintveld [56] for two systems of the Barendregt s Cube [7] Springintveld uses Schwichtenberg s results and Geuvers and Nederhof s [22] translation of calculi in Barendregt s Cube into the simply typed calculus. In [22], a (syntactic) modular proof for the Calculus of Constructions [12] is given (see [23] for more references to normalization proofs for the Calculus of Constructions) Syntactic proof techniques. One of the rst syntactic proofs of strong normalization for simply typed calculus is due to van ....

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

....proof of normalization for F , defined as a Prawitz style natural deduction system. 6. Zhaohui Luo, 1989 [Luo90] There is apparently a proof of strong normalization for an extension of CC with universes, given in Luo s thesis. We do not have this document yet. 7. Geuvers and Nederhof, June 1989 [GN89]. The authors present what they call a modular proof of strong normalization, by reducing strong normalization in CC to strong normalization in Girard s F . This is accomplished by defining a mapping from CC to F , such that reduction of terms is preserved. Strong normalization for the terms ....

H. Geuvers and M.-J. Neherhof. A modular proof of strong normalization for the calculus of constructions. Journal of Functional Programming, pp. 38, June 1989. Submitted for publication.

....It is easy to see that the introduction of rewriting does not affect the structural properties shared by all the PTS, even though rewriting is defined on a larger class of terms. Hence we recall these properties without necessary providing their proof, that the interested reader can find them in [GN91] or [Bar93] Theorem 3.2 (PTS structural properties) GN91] Bar93] free variables) If Gamma a : b and Gamma = x 1 :c 1 ; xn :c n then all the variables x i are distinct, FV (a) FV (b) fx 1 ; xng and FV (c i ) fx 1 ; x i Gamma1 g (1 i n) substitution) If ....

....affect the structural properties shared by all the PTS, even though rewriting is defined on a larger class of terms. Hence we recall these properties without necessary providing their proof, that the interested reader can find them in [GN91] or [Bar93] Theorem 3. 2 (PTS structural properties) GN91] Bar93] free variables) If Gamma a : b and Gamma = x 1 :c 1 ; xn :c n then all the variables x i are distinct, FV (a) FV (b) fx 1 ; xng and FV (c i ) fx 1 ; x i Gamma1 g (1 i n) substitution) If Gamma; x:a; Delta b : c and Gamma d : a then Gamma; ....

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

....of the formal syntax and derivation rules in our own denotation, since that differs from the one commonly used; this should enable the appreciation of the presentation of our cube of type assignment systems in the next subsection. For a complete development of Barendregt s cube, we refer to [4, 11]. Definition 1.1 i) f; 2g is the set of sorts. ii) The sets of typed terms ( t ) typed constructors (Cons t ) and typed kinds (Kind t ) are mutually defined by the following grammar, where M , OE, and K are metavariables for terms, constructors and kinds respectively: M : x j x:OE:M j MM j ....

....Note that the behaviour of E is such that, in the image of E, terms are completely untyped, while constructors and kinds are partially typed. The notions of free variables, subterms and fi reduction, to be defined below, are similar to their fully typed counterparts as can be found in [4, 11], but slightly modified, according to the untyped term syntax. Definition 1.2 FV (A) the set of free variables of A, and ST (A) the set of subterms of A, are inductively defined by: FV ( FV (a) fag; FV (BC) FV (B) FV (C) FV ( Pia:B:C) FV (B) FV (C) n fag) FV (a:B:C) ....

H. Geuvers and M. Nederhof. Modular Proof of Strong Normalization for the Calculus of Constructions. Journal of Functional Programming, 1(2):155--189, 1991.

....4 (Atomic terms) A term is said to be atomic if it is of the form app xn :An:Bn ( app x 1 :A 1 :B 1 (P; Q 1 ) Q n ) with P of one of the following forms: s, x, x : A)B. We write AT for the set of atomic terms. The following is essentially Tait s (and Krivine s and other s) [11, 5] version of reducibility candidates [6] Definition 5 (Saturated sets) A set C of terms is said to be saturated, if and only if 1. C ae SN 2. SN AT ) ae C 3. if (A; B; P ) 2 SN 3 and app xn :An:Bn ( app x 1 :A 1 :B 1 (M [x n P ] Q 1 ) Q n ) 2 C then app xn :An:Bn ( ....

....Y ) Now let M be a strongly normalizable atomic term; we have 8ff X : 8N j= X ff : 8A; B 2 SN : app x:A:B (M; N) 2 SN AT and hence app x:A:B (M; N) j= Y ff Y ff : Which is sufficient for M j= Pi(X;Y ) f . 3. The proof is easy and simillar to its counterpart for saturated sets. See [1, 5] for example. Definition 12 (E relation product) Let A 1 and A 2 be two E set. Let there be elements X and X 0 of A 1 , and two families (Y ff ) ff2X 0 and (Y 0 ff 0 ) ff 0 2X 0 0 pof sets elements of A 2 indexed over X 0 and X 0 0 . The following definitions extend the i A ....

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

....Dependencies 6 ae ae ae ae 6 2 ae ae ae ae P ae ae ae ae 6 P 6 P2 ae ae ae ae P Let S be one of these eight systems. We write Gamma S A : B to indicate that Gamma t A : B can be derived using only the rules for S. The properties of this cube are studied in [1, 8]. 1.2 The Cube of Type Assignment Systems In this subsection we will present the cube of type assignment systems as was first presented in [10] The definition of the type assignment cube is based on the definition of an erasing function E that erases all type information from the typed terms. ....

Geuvers, H. and Nederhof, M., Modular Proof of Strong Normalization for the Calculus of Constructions, Journal of Functional Programming, 1(2), 155--189, 1991.

....gives a better development of the basic metatheory because no case of the Generation Lemma except start depends on the Thinning Lemma. The basic metatheory of this presentation of PTS has been formalized in LEGO 2. 1 Basic Theorems The basic meta theoretic properties of PTS are presented in [Bar92, Ber90, GN91, vBJ92]. The ones we will explicitly use are summarized here Lemma 3 (Thinning Lemma) If Gamma M : A, Gamma Delta, and Delta Valid, then Delta M : A. Lemma 4 (Generation Lemma) 1) If Gamma c : A then there exists s such that Ax(c:s) and A s. 2) If Gamma x : A then there exists A ....

Herman Geuvers and Mark-Jan Nederhof. A modular proof of strong normalization for the calculus of constructions. Journal of Functional Programming, 1(2):155--189, April 1991.

....0 . Then Gamma[x : T ] v : T 0 : s and x does not occur in v # nor in T 0 , thus, by proposition 30 x does not occur in v, which is contradictory. 2.5 Normalization In this section we prove that each well typed marked term has a normal form. We use the an adaptation of the method of [14, 11], i.e. we associate to each marked term t an unmarked term t ffi that mimics all its reductions. Definition 15 (Marked terms translation) We define by simultaneous induction two translations from marked terms to unmarked terms. We use a fresh variable o (of type P rop) ffl If P is a term of ....

H. Geuvers, M.J. Nederhof, A modular proof of strong normalization for the Calculus of constructions, Journal of Functional Programming, I, 2 (1991) pp. 155-189.

....some (s 1 ; s 2 ; s 3 ) 2R; s 3 ; s)2A. Proof Let Gamma M : D : s and use in each case generation, making use of Church Rosser, subject reduction, and preservation of sorts where necessary. The following fundamental property is similar to results proved by Geuvers (1993) Berardi (1990) and Geuvers and Nederhof (1991). Corollary 24 (Classification Lemma) Let S be injective and preserve sorts. Then for all sorts s 6j s 0 , Term s Term s 0 = Type s Type s 0 = Proof Use Lemmas 22 and 23 to prove M 2 Term s Term s 0 ) s j s 0 M 2 Type s Type s 0 ) s j s 0 simultaneously ....

Geuvers, J.H. & Nederhof, M.J. (1991). A Modular Proof of Strong Normalization for the Calculus of Constructions. Journal of Functional Programming, 1(2), 155--189.

....proof of simply typed calculus to Barendregt s cube. The ideas that we have developed are immediately applicable to the direct successors of , namely 2, and P. To put these ideas together into a modular proof of strong normalization for the calculus of Constructions (in the spirit of [11]) will be the subject of future work. ....

H. Geuvers and M.-J. Nederhof. Modular proof of strong normalization for the calculus of construction. Journal of Functional Programming, 1(2):155189, 1991.

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

....systems for correct programs. For instance, treatment of exceptions, unbounded loops or recursive definitions are widely used in all programming languages. So, connections between logic and programming need to be refined, maybe even beyond the Curry Howard correspondence. Griffin s paper [10] emphasized the interactions between continuations (in Scheme) and classical reasoning. This starting point led to a convincing typed system, recently introduced by Parigot [14] which showed how Curry Howard correspondence could be extrapolated to capture more programming constructs. Thus, we are ....

....and Mendler N.P. 1985) Recursive definitions in Type Theory. In Logics of Programs. Parikh R. editor. LNCS 193, Springer Verlag, 1985. 9] Crole R.L. and Pitts A.M. 1990) New foundations for Fixpoint Computations. In Proceedings of the 5th Conference on Logic In Computer Science. IEEE 1990. [10] Griffin T. 1990) A formulae as types notion of control. In Conference Record of the Seventeenth Annual ACM Synposium on Principles of Programming Languages, 1990. 11] Mendler N. 1987) Recursive Types and type Constraints in Second Order Lambda Calculus. In 2nd Conf. on Logic in Comp. Science. ....

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H. Geuvers, M.J. Nederhof, A Modular Proof of Strong Normalization for the Calculus of Constructions, Journal of Functional Programming, I, 2, 1991, pp. 155-189. 129

....Systems As has already been mentioned, an important motivation for the de nition of the general framework of Pure Type Systems is the fact that many important properties can be proved for all PTSs at once. Here, we list the most important properties and discuss them brie y. Proofs can be found in Geuvers and Nederhof [1991] and Barendregt [1992] In the following, unless explicitly stated otherwise, refers to derivability in an arbitrary PTS. As in HOL, we de ne a context to be wellformed if M : A for some M and A. Two basic properties are Thinning, saying that typing judgments remain valid in an extended ....

....T 0 , if M : A in T , then f( f(M) f(A) in T 0 . Not all PTSs are Strongly Normalizing. We have the following well known theorem. 3.32. Theorem. The Calculus of Constructions, CC, is Strongly Normalizing. 52 H. Barendregt and H. Geuvers The proof is rather dicult and can be found in Geuvers and Nederhof [1991], Coquand and Gallier [1990] Berardi [1990] As a consequence we nd that many other PTSs are Strongly Normalizing as well. This comprises all the sub systems of CC and also all systems T for which there is a PTS morphism from T to CC. Note that a PTS morphism preserves in nite reduction ....

Geuvers J. and Nederhof M. [1991], `A modular proof of strong normalization for the Calculus of Constructions', Journal of Functional Programming 1(2), 155-189.

....5 focuses on the applications of the result to existing systems. The last section contains some final remarks about the work as well as directions for future research. We assume the reader to be reasonably familiar with pure type systems and their basic meta theory, as presented for example in [15], 4] or [16] 2 Combining higher order rewriting systems and pure type systems 2.1 Higher order rewriting systems In this section, we introduce higher order rewriting systems. The framework we consider is slightly less general than the one of [3, 12, 19] and has been chosen for clarity of ....

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

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

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

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GN91. H. Geuvers and M.J. Nederhof. A Modular Proof of Strong Normalization for the Calculus of Constructions. J. of Functional Programming, 1(2):155-189, April 1991.

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