Jean-Yves Girard. Interpretation fonctionnelle et elimination des coupures dans l'arithmetique d'ordre superieur. These de Doctorat d'  Etat, Universite de Paris VII, 1972.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....from that of F . Section 6 gives the idea of an extension even to primitive recursion of rank 2 while many closure properties of monotone constructors are shown in section 7. A few examples are considered in section 8. 2 System F Girard s system of higher order parametric polymorphism [7] is an extension of system F where the set of types is extended to a simply typed calculus of constructors. The types of the constructors are called kinds and are built from the base kind (the universe of types) and the binary ) intended to form function kinds. We denote kinds by ....

.... z 8F:F mon F F 0 F :zF yz 1 x)Ly 0 (z 1 0 z 2 ) It is not too hard to check the translation of steps of MICC 2 into those of F . ut Corollary 1. System MICC 2 is strongly normalizing, i.e. B is well founded. Proof. It is well known that F is strongly normalizing ([7] only considered weak normalization although his proof method copes with strong normalization as well) and, certainly, strong normalization is inherited via embeddings. ut In this impredicative encoding, the previous section s concern becomes irrelevant since ( FF) 0 and ( FF) 0 are always ....

Jean-Yves Girard. Interpretation fonctionnelle et elimination des coupures dans l'arithmetique d'ordre superieur. These de Doctorat d'  Etat, Universite de Paris VII, 1972.

....to be highly parameterized. Functional abstraction is not enough for expressing the desirable forms of parameterization. Also a limited form of polymorphism, like that supported by SML, appears inadequate (see Section 6) Therefore, in MetaKlaim we have included polymorphic types a la system F [19]. However, this design choice and the need for decidable type checking at run time (since the input primitive performs dynamic type checking) imply that MetaKlaim programs have to include a lot of type information (at run time) For a description of the 2 types we refer to Remark 3.1. An ....

Girard, J., \Interpretation fonctionelle et elimination des coupures dans l'arithmetique d'ordre superieur," These de doctorat d'etat, University of Paris VII (1972).

....this to be supported with some automated checking of termination, which ensures that partially or totally unde ned proofs are not permitted. Consistency also depends on the strength of the type system itself; a su ciently powerful type system will be inconsistent as shown by Girard s paradox [11]. 5 Applications of an Integrated Logic Having identi ed a logic within Aldor, how can it be used There are various applications possible; we outline some here and for others one can refer to the number of implementations of type theories which already exist, including Nuprl [7] and Coq [8] ....

Jean-Yves Girard. Interpretation fonctionelle et elimination des coupures dans l'arithmetique d'ordre superieure. These d'Etat, Universite Paris VII, 1972.

....P.O.Box 513, 5600 MB Eindhoven, the Netherlands, r.p.nederpelt tue.nl Abstract. The Barendregt Cube (introduced in [3] is a framework in which eight important typed calculi are described in a uniform way. Moreover, many type systems (like Automath [18] LF [11] ML [17] and system F [10]) can be related to one of these eight systems. Furthermore, via the propositions as types principle, many logical systems can be described in the Barendregt Cube as well (see for instance [9] However, there are important systems (including Automath, LF and ML) that cannot be adequately ....

....and well known type systems are presented in a uniform way. This makes a detailed comparison of these systems possible. The weakest systems of the Cube is Church s simply typed calculus [7] and the strongest system is the Calculus of Constructions C [8] Girard s wellknown System F [10] gures on the Cube between and C. Moreover, via the Propositions as Types principle (see [13] many logical systems can be described in the systems of the Cube, see [9] In the Cube, we have in addition to the usual abstraction, a type forming operator . Brie y, if A is a type, and B is ....

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J.-Y. Girard. Interpretation fonctionelle et elimination des coupures dans l'arithmetique d'ordre superieur. PhD thesis, Universite Paris VII, 1972.

....various logics this observation become famous as the Curry Howard Isomorphism. It is central to a body of work which strives to create a foundational system for computer science, starting with the rst formulation (1971) of MartinL of s type theory. Due to an inconsistency discovered by Girard [27], it was substantially modi ed, giving rise to newer, predicative versions [47, 48] of Martin L of type theory on one hand and the so called Calculus of Constructions and its extensions [12, 45] on the other hand. A number of other foundational systems for computer science has been proposed. ....

Jean-Yves Girard. Interpretation fonctionnelle et elimination des coupures dans l'arithmetique d'ordre superieur. These d'  Etat, Universite de Paris VII, Paris, France, 1972. 35

.... outside SAT; note that we may assume that does not occur in ) The de nition of SC [ is a variant of the computability predicate de nition in the famous [Tai67] its relativization to a candidate assignment and the big intersection in the de nition of SC 8 [ have been invented in [Gir72] and only today seem to be the straightforward extension of Tait s ideas. It has to be stressed that exactly this big intersection shows the impredicativity of the system of universal types: We need the intersection over any saturated set M in order to de ne a speci c saturated set, namely SC ....

Jean-Yves Girard. Interpretation fonctionnelle et elimination des coupures dans l'arithmetique d'ordre superieur. These de Doctorat d'  Etat, Universite de Paris VII, 1972.

.... 1986) In this case, for the example above, applying the term P ( P a) P b) to the term x (R x x) we get ( P ( P a) P b) x (R x x) and we use the axioms to deduce ( R a a) R b b) The system HOL is well known to be consistent and to enjoy cut elimination (Girard, 1970; Girard, 1972). In HOL , equality needs not to be primitive. It can be de ned as Leibniz equality i.e. x y 8 p ( p x) p y) In this case, the propositions 8 f 8 g 8 x (f = g ) f x) g x) 8 x 8 y 8 f (x = y ) f x) f y) are provable. But the proposition 8 ....

Girard, J.-Y. (1972). Interpretation fonctionnelle et elimination des coupures dans l'arithmetique d'ordre superieur. PhD thesis, Paris VII.

....: t n must necessarily be a function type. In the rest of this section we show that polyhti is indeed welltyped. To this end we must rst x the target language we are compiling polytypic values to. Since we require rst class polymorphism, we will use a variant of the polymorphic lambda calculus (Girard, 1972), F , augmented by a polymorphic xpoint operator. Note that a similar language is also used as the internal language of the Glasgow Haskell Compiler (Peyton Jones, 1996) Similar to the type language we annotate value constants and variables with their types. If (s ; t) is a constant or a ....

Girard, Jean-Yves. (1972). Interpretation fonctionelle et elimination des coupures dans l'arithmetique d'ordre superieur. Ph.D. thesis, Universite Paris VII.

....t n must necessarily be a functional type. In the rest of this section we show that polyhti is indeed well typed. To this end we must rst x the target language we are compiling polytypic values to. Since we require rst class polymorphism, we will use a variant of the polymorphic lambda calculus [6], F , augmented by a polymorphic xpoint operator. A similar language is also used as the internal language of the Glasgow Haskell Compiler [26] Similar to the type language we annotate value constants and variables with their types. If (s ; t) is a constant or a variable, we de ne type (s ; t) ....

Jean-Yves Girard. Interpretation fonctionelle et elimination des coupures dans l'arithmetique d'ordre superieur. PhD thesis, Universite Paris VII, 1972.

.... ; Xn 2 TyVar 1 i n X i 2 Ty(X) Sum) 1 ; n 2 Ty(X) 2 Ty(X) Prod) 1 ; n 2 Ty(X) 2 Ty(X) Arr) 2 Ty( 2 Ty(X) 2 Ty(X) Mu) 2 Ty(X ; X) X: 2 Ty(X) We can restrict the free type variables to strictly positive ones because unlike Girard s System F [Gir72] we have no polymorphic types and therefore need type variables only to construct inductive types. Notation. In the following we write (X) to express 2 Ty(X) Then and (X) are synonyms. We also abbreviate the set of closed types Ty( by Ty. For binary sums ( we write , for ....

Jean-Yves Girard. Interpretation fonctionnelle et elimination des coupures dans l'arithmetique d'ordre superieur. These de Doctorat d'  Etat, Universite de Paris VII, 1972.

....Informatik der Universit at M unchen, Oettingenstra e 67, D 80538 M unchen, Germany; email: matthes informatik.uni muenchen.de c EDP Sciences 1999 2 RALPH MATTHES 1. Introduction Our goal is to establish relations between extensions of system F (the polymorphic calculus due to Girard [4] and Reynolds [10] with inductive types, coinductive types and xed point types. Why do we need to extend system F by these type constructs The main problem is that not (full) primitive recursion but only iteration on inductive types is modelled by the impredicative encoding of ....

....of x. We denote the re exive transitive closure of by . It is well known that F has subject reduction, i.e. if r : and r r 0 , then r 0 : This will also be the case for all of the systems to be de ned in the sequel. Strong normalization of F is a famous result by Girard [4] (eta reduction requires but a modi cation of the proof for beta only or, alternatively, the result follows from strong beta normalization by eta postponement) As mentioned in the introduction, con uence is easily established by Takahashi s method [11] The type former is assumed to associate ....

Jean-Yves Girard. Interpretation fonctionnelle et elimination des coupures dans l'arithmetique d'ordre superieur. These de Doctorat d'  Etat, Universite de Paris VII, 1972.

....For a careful de nition yielding normal terms map see [Mat98, 5.1.1] Now we may set for those types: E s) map ; x :xE s) Theorem 2 The term rewrite system of positive inductive types is strongly normalising. Proof Reduction preserving embedding into Girard s [Gir72] (and Reynolds [Rey74] system F. System F is the polymorphic lambda calculus. We do not have inductive types but universal types 8 (for any type and any type variable ) The corresponding term formation rules are as follows: If r is a term of type , then r is a term of type 8 , ....

Jean-Yves Girard. Interpretation fonctionnelle et elimination des coupures dans l'arithmetique d'ordre superieur. These de Doctorat d'  Etat, Universite de Paris VII, 1972.

....to use ( I) and ( E) in the de nition of map. 10 Now we may set for those types: 7 ) x :xE s) map 7 ; x :xE s) Theorem 2 The term rewrite system of positive inductive types is strongly normalizing. Proof Reduction preserving embedding into Girard s [Gir72] (and Reynolds [Rey74] system F. System F is the polymorphic lambda calculus. We do not have inductive types but universal types 8 (for any type and any type variable ) The corresponding term formation rules are as follows: If r is a term of type , then r is a term of type 8 , ....

Jean-Yves Girard. Interpretation fonctionnelle et elimination des coupures dans l'arithmetique d'ordre superieur. These de Doctorat d'  Etat, Universite de Paris VII, 1972.

....For example, a principal type scheme of K is . Polymorphism Note that in one has I : for all 2 T: In the polymorphic lambda calculus this quanti cation can be internalized by stating I : 8 : The resulting system is the polymorphic of second order lambda calculus due to Girard (1972) and Reynolds (1974) 5.15. Definition. The set of types of 2 (notation T = Type( 2) is speci ed by the syntax T = V j B j T T j 8V:T: 5.16. Definition. The rules of type assignment are those of , plus M : 8 : M : M : M : 8 : In the latter rule, the type variable may ....

....: M : M : 8 : In the latter rule, the type variable may not occur free in any assumption on which the premiss M : depends. 5.17. Example. i) I : 8 : ii) De ne Nat 8 : Then for the Church numerals c n fx:f n (x) we have c n : Nat. The following is due to Girard (1972). 5.18. Theorem. i) The Subject Reduction property holds for 2. ii) 2 is strongly normalizing. Typability in 2 is not decidable; see Wells (1994) 40 Introduction to Lambda Calculus Exercises 5.1. i) Give a derivation of SK : ii) Give a derivation of KI : ....

Girard, J.-Y. (1972). Interpretation fonctionelle et elimination des coupures dans l'arithmetique d'ordre superieur, Dissertation, Universite Paris VII.

....Martin L of s Intuitionistic Type Theory (ITT) with the hierarchy of universes [10] One of the aims of the restriction is to provide a program synthesis system which directly generates functional programs with the program constructs such as if then else and pairing. Unlike CoC, F [5] and F [4], we do not use Prawitz coding of logical connectives [15] because, if we use the coding, the constructs such as if then else and pairing are not primitives but the functionals de ned in higher order lambda terms. Another aim of the restriction is to nd a subset of existing higher order systems ....

J.-Y. Girard. Interpretation fonctionnelle et elimination des coupures dans l'arithmetique d'order superieur. These d'etat. Universite Paris 7, 1972.

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Gir72. J.Y. Girard. Interpretation Fonctionnelle et  Elimination des Coupures dans l'Arithmetique d'Ordre Superieur. PhD thesis, Universite Paris VII, 1972.

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