T. Coquand. Une Theorie des Constructions. PhD thesis, Universite Paris 7, January 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by Seldin. None of these results is very deep, but it seems useful to collect them in one place. Key words: Calculus of constructions, Typed # calculus, Pure typed systems 1991 MSC: 03B15, 03B40, 03B70, 68T15 1 Introduction Since Coquand first introduced the calculus of constructions in [1 5], there have been a number of di#erent versions of the system published. Of the Corresponding author. Email addresses: martin bunder uow.edu.au (M. W. Bunder) jonathan.seldin uleth.ca (Jonathan P. Seldin) URLs: http: www.uow.edu.au informatics maths staff bunder.html (M. W. Bunder) ....

....Seldin) Work supported in part by a grant from the Australian Research Council. Work supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada. Preprint submitted to Elsevier Science 6 April 2004 versions published by Coquand himself, one appears in [1,4,5], and another appears in [2] and still another appears in [3] One of the most distinctive versions in the literature is due to Seldin, which di#ers from the others in some important ways. Seldin had first learned of the calculus of constructions in early 1986, when he was working for Odyssey ....

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T. Coquand, Une theorie des constructions, Ph.D. thesis, University of Paris VII (1985).

....by P (x) Each unfolding of norm becomes an appeal to the induction step (1) Perhaps domains and partial objects are not essential even for di#cult proofs of termination. As a sample proof in his higher order theory of constructions, Thierry Coquand has proved the termination of normalization [2] (pages 46 48) He defines a predicate N(x) to mean x can be put into normal form, and proves #x. N(x) Translated from his formalism, the axioms are N(At (a) N(y) N(z) N(If (At (a) y,z) N(If (u, If (v, y, z) If (w,y,z) N(If (If (u, v, w) y,z) The connection between N(x) ....

T. Coquand, Une Theorie des Constructions,These de 3eme cycle (in French), University of Paris VII (1985).

.... pairs, unions, and homogeneous lists and trees [Reynolds 1985, BShm Berarducci 1985] Going beyond the polymorphic A calculus, in a A calculus with dependent function types and the type of all types we can apply the programming techniques of the more powerful Calculus of Constructions [Coquand 1985b, Mobring 1986, Coquand Huet 1988] For example, we can define a form of dependent pair type, also called a general sum or an existential type, that is useful for modelling abstract data types [Mitchell Plotkin 1985, MacQueen 1986] The combination of dependent function types and the type of ....

....recursion and in which recursion is not obviously definable. The positive forms of (4) 7) are enjoyed by a variety of typed A calculi, including the simply typed A calculus [Barendregt 1984, Appendix A] the polymorphic A calculus [Girard 1972, Reynolds 1974] and the Calculus of Constructions [Coquand 1985b, Coquand Huet 1988] That strong normalization fails (4) for languages like the A calculus has been known for some time; the central result of this thesis is that the normal form relation is undecidable (5) The undecidability of the equational theory (6) follows from (5) because, by the ....

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Thierry Coquand. Une ThEorie des Constructions. Ph.D. thesis, Universit de Paris, January 1985. Thse de troisime cycle.

....matters, and it 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 ....

Thierry Coquand. Une Theorie des Constructions. PhD thesis, Universit'e Paris VII, 1985.

....methods to mark logical parts in proofs and extract their algorithmic contents. The result is a correct program with respect to a specification. This paper focuses on the inverse problem : how to generate a proof from its specification. The framework is the Calculus of Inductive Constructions [Coq85] A notion of coherence is introduced between a specification and a program containing types but no logical proofs. This notion is based on the definition of an extraction function called the weak extraction. Such a program can give a method to reconstruct a set of logical properties needed to ....

....parties logiques des preuves et extraire leur contenu algorithmique. Il en r esulte un programme correct vis a vis d une sp ecification. Ce papier s int eresse au probl eme inverse : comment engendrer une preuve a partir de sa sp ecification. Le cadre est le Calcul des Constructions Inductives [Coq85] Une notion de coh erence entre une sp ecification et un programme contenant des types mais pas de preuves logiques est introduite. Cette notion est bas ee sur la d efinition d une fonction d extraction appel ee fonction d extraction faible. Un tel programme fournit une m ethode de ....

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T. Coquand. Une th'eorie des constructions. PhD thesis, Universit'e Paris VII, 1985.

....According to fi reduction, terms of the Calculus of Inductive Constructions enjoy some fundamental properties such as confluence, strong normalization, subject reduction. These results are theoretically of great importance but we will not detail them here and refer the interested reader to [19]. reduction. A specific conversion rule is associated to the inductive objects in the environment. We shall give later on (section 6.5.4) the precise rules but it just says that a destructor applied to an object built from a constructor behaves as expected. This reduction is called reduction ....

Th. Coquand. Une Th'eorie des Constructions. PhD thesis, Universit'e Paris 7, January 1985.

.... universal quantification t represents negation t1 t2 represents conjunction t1 = t2 represents implication b = t1 j t2 represents conditional expression t1 t2 represents function application 4 The Coq System The Coq system [3] is based on the calculus of constructions [6, 5] enriched with inductive [12] and co inductive definitions [8] This is a higher typed Calculus in which definitions of mathematical objects can be given and proofs of propositions on these objects can be performed. Coq s logic is a higher order constructive logic which relies on the ....

T. Coquand. Une Th'eorie des Constructions. PhD thesis, Universit'e Paris 7, Janvier 1989.

....F. An instance of predicative type theory is Martin Lo . f s intuitionistic theory of types (ITT) Martin Lo . f82] In his theory, Martin Lo . f attempted to reconstruct the foundations of mathematics. His intuitionistic mathematics has become the theoretical basis of the system Nuprl [Constable86] in which one develops provably correct functional programs. Predicative as well as impredicative type theories can interpret first , second , and higher order logics. For example, Martin Lo . f type theories without universes interpret first order predicate logic. When extended with a ....

128 Coquand, Th., Une Theorie des Constructions, these de 3eme cycle, Paris VII (1985).

....and to type theories such as Constable s V3. 1 Introduction A number of formulations of intuitionistic type theory have been considered as a basis for studying machine assisted formal proof development, and as a theoretical foundation for the study of programming languages (see, for example, [16, 46, 34, 36, 37, 9, 10, 11, 14, 2, 4, 19], to name but a few. One such system, the Calculus of Constructions (CC) was introduced by Coquand and Huet as a comprehensive basis for the formalization of constructive mathematics. 11, 14] CC may be viewed as the calculus associated, via the propositions as types principle [24] with ....

....for the study of programming languages (see, for example, 16, 46, 34, 36, 37, 9, 10, 11, 14, 2, 4, 19] to name but a few. One such system, the Calculus of Constructions (CC) was introduced by Coquand and Huet as a comprehensive basis for the formalization of constructive mathematics. [11, 14]. CC may be viewed as the calculus associated, via the propositions as types principle [24] with natural deduction proofs in an extension of Church s higher order logic [6] The system has been proved both proof theoretically [11] and model theoretically [29, 17, 27] consistent, and the type ....

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Thierry Coquand. Une Th'eorie des Constructions. PhD thesis, Universit 'e Paris VII, January 1985.

....be found on URL http: www.cl.cam.ac.uk Research HVG atmproof . 8 Chapter 3 Structural Hardware Specifications in Type Theory 3. 1 Overview of the Coq system The Coq system is a tactic oriented theorem prover, in the style of LCF [24] It is an implementation of the Calculus of Constructions [13] [14] a higher order typed calculus, enriched with inductive definitions [40] and co inductive definitions [23] Coq s logic is a higher order constructive logic and relies on the Curry Howard isomorphism. The system [5] provides an unifying framework, since a proposition is a type and a proof ....

Thierry Coquand. Une Th'eorie des Constructions. PhD thesis, Universit'e Paris 7, Janvier 1989.

....toolbox implementing the automata theory. Then, in section 4, we present an application of our methodology to the ATM Switch Fabric. Finally, in the last section, we compare our study to other related work. 2 An Overview of Coq The Coq system [1] is based on the Calculus of Constructions [4] [3] enriched with inductive [14] and co inductive definitions [9] Coq s logic is a higher order constructive logic which relies on the Curry Howard isomorphism and which makes both objects and propositions to be terms of the Lamba Calculus. The rules for constructing terms are as follows : ....

T. Coquand. Une Th'eorie des Constructions. PhD thesis, Universit'e Paris 7, Janvier 1989.

....of LCF [15] Developments can be splitted into various parameterized modules to be separately verified. Thus, several developments can share modules that, being compiled once for all, are loaded fast. Coq s language implements a higher order typed lambda calculus, the Calculus of Constructions [7, 8], enriched with inductive definitions [21] Coq s logic is a higher order constructive logic and relies on the propositionsas types correspondence. In Coq, a proposition is a type and a proof is a term inhabiting this type. Such a system provides an elegant unifying framework, since there is no ....

T. Coquand. Une Th'eorie des Constructions. PhD thesis, Universit'e Paris 7, Janvier 1989.

.... United Kingdom 1 ideas were subsequently taken up by Paulson [37] for LCF) and Petersson [40] for Martin Lof s type theory) Coquand and Huet, inspired by the work of Girard, extended the language of AUTOMATH with impredicative features, and developed an interactive proof checker for it [10, 12, 13]. Building on the experience of AUTOMATH and LCF, the NuPRL system [9] is a full scale interactive proof development environment for type theory that provides support not only for interactive proof construction, but also notational extension, abbreviations, library management, and automated proof ....

....The latter are not 3 concerned with the problem of representation of arbitrary formal systems, but rather with developing the internal mathematics of a specific constructive set theory. For detailed explanations and illustrations, see the NuPRL book [9] and the papers of Coquand and Huet [10, 12, 13]. There is a much closer relationship between the present work and that of the AUTOMATH project [15] On the one hand the AUTOMATH project was concerned with formalizing traditional mathematical reasoning without foundational prejudice. In this regard the overall aims of LF are similar, and our ....

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Coquand, T. Une Th'eorie des Constructions. PhD thesis, Universit'e Paris VII, Jan. 1985.

....speed up execution. We also imagine finer type structures which make the resource use explicit; an example would be linear types which can be used to avoid garbage collection. 6 1.2. The Calculus of Constructions The Calculus of Constructions (CC) was introduced by T. Coquand and Huet ( CH88] Coq85] It can be viewed as a unification of Girard s impredicative system F and dependent types, which are the base of Martin Lof s Type Theory. When Martin Lof initially proposed a Type Theory [Mar71] he also attempted to capture Girard s system. However, it turned out that this system was ....

Thierry Coquand. Une th'eorie des constructions. PhD thesis, Universit 'e Paris VII, 1985.

....According to fi reduction, terms of the Calculus of Inductive Constructions enjoy some fundamental properties such as confluence, strong normalization, subject reduction. These results are theoretically of great importance but we will not detail them here and refer the interested reader to [13]. reduction. A specific conversion rule is associated to the inductive objects in the environment. We shall give later on (section 6.5.4) the precise rules but it just says that a destructor applied to an object built from a constructor behaves as expected. This reduction is called reduction ....

Th. Coquand. Une Th'eorie des Constructions. PhD thesis, Universit'e Paris 7, janvier 1985.

.... as Pure Type Systems (see, e.g. Barendregt, 1992; Laan, 1997) Pure Type Systems are a generalization of different systems of (explicitly) typed lambda calculus, such as the Automath Languages (De Bruijn, 1980) Constructive Type Theory (Martin L of, 1984) and the Calculus of Constructions (Coquand, 1985). Let us give four reasons for using a PTS as our framework. 1) Ahn Kolb (1990) show that PTSs can be seen as a higher order generalization of Discourse Representation Theory ( DRT; Kamp, 1981; Kamp Reyle, 1993) This means that the framework also covers other context dependent phenomena, ....

Coquand, T. (1985), Une theorie des constructions, Th ese de troisi eme cycle, Universit e de Paris--VII.

.... [Wong Toi Hoffmann 92] Control of walking robot s legs Assembly line control [Antoniotti 95] Challenges ffl More robots ffl Highway control ffl Flight control a hybrid system ffl Protocols in operating systems ffl Bus protocols References Functional Synthesis work [BD77] How80] MW80] Coq85] CAB 86] CH88] HN88] Smi90] MT91] Abr90] Closed system synthesis [MW81] CE81] EC82] MW84] Anu95] AE89] Open synchronous [Chu62] Trees [Rab72] EJ88] PR89b] SVW87] Saf88] R.R92] Games [GS53] BL69] McN93] Les95, Tho95] Dav64] Mar75] Control [RW89] TW94a, TW94b] MPS95] ....

Th. Coquand. Une Th'eorie des Constructions. PhD thesis, Universit'e Paris 7, 1985.

.... The inverse problem can be solved: it is possible to reconstruct proof obligations from a program and its specification [Par95a, Par95b] The framework is a type theory where a proof can be represented as a typed term [Bar91, ML84] and, particularly, the Calculus of Inductive Constructions [Coq85] This paper shows how programs can be simplified in order to be written in a much closer way to the ML one s. Indeed, proofs structures are often much more heavy than programs structures. The problem is consequently to consider natural programs (in a ML sense) and see how to retrieve natural ....

....methods exist to extract the computational part representing the program from a mathematical proof [Ber94, PM89b, PM89a] The correctness of the resulting functional program is then certified by construction. We focus on a particular framework that is the Calculus of Inductive Constructions [Coq85, Coq89] and a particular implementation that is the Coq system [CCF 94] Programs can be extracted from proofs, but an other possible way is to synthesize proofs from programs. This consists in inverting the program extraction of [PM89a] and has been detailed in [Par93, Par95a, Par95b] given ....

T. Coquand. Une th'eorie des constructions. PhD thesis, Universit 'e Paris VII, 1985.

....years after the AUTOMATH project, type theory has been further 1 The expression is due to H.Barendregt. It stems from de Bruijn s classification between genial mathematicians, mathematical craftsmen and mathematical engineers. developed (for some of the most important developments, see [7, 18, 35, 37, 41]) and a new generation of proof assistants based on type theory has appeared. In these new systems ( 17, 19, 36, 38] the task of the user has been simplified to such an extent that formalising large parts of mathematics is now perceived as feasible and, more importantly, as reasonable in term of ....

T.Coquand. Une theorie des constructions, Ph.D thesis, University Paris 7, 1985.

....as a source of inspiration in the construction of several languages designed with the proofs as algorithms paradigm in mind. A survey may be found in [NGdV94, pp. 8 12] For example, there is the calculus of constructions by Coquand and Huet, combining ideas from Girard s 2 and automath ([Coq85, CH88]) Luo ( Luo89] designed the extended calculus of constructions . For these languages implementations have been designed, such as lego and Coq, which can be used for interactive proof construction. Close to automath is LF ( Logical Framework ) developed in Edinburgh ( HHP87] this system has ....

T. Coquand. Une th'eorie des constructions. PhD thesis, Paris, Universit 'e Paris VII, 1985.

....= x : x. Thus, I is a polymorphic identity function, from which the specific identities I can be recovered by type application and fi reduction for types. The two forms of abstraction for types, with respect to objects and with respect to types, are merged in the Calculus of Construction [ Coquand, 1985; Coquand and Huet, 1985; Coquand and Huet, 1988 ] A further construct involving type abstraction is the recursion operator on types, which permits the introduction of user defined types. This is a definable construct in the Theory of Construction, just as the fixpoint operator is definable in ....

T. Coquand. Une Th'eorie des Constructions. Ph.D. Thesis, Universit'e Paris VII, January 1985.

....formalized. It is shown that ECC is strongly normalizing and has other nice prooftheoretic properties. An GammaSet (realizability) model is described to show how the essential properties of the calculus can be captured set theoretically. 1 Introduction The calculus of constructions [CH88][Coq85] is a typed higher order functional calculus which provides a nice formalism for constructive proofs in natural deduction style and can also be seen as a high level functional programming language. In this paper, we present an Extended Calculus of Constructions, ECC, which can be seen as an ....

....is derivable for some A. We shall write Gamma M :A for Gamma M :A is derivable , and Gamma M N ( Gamma M N ) for M and N are well typed under Gamma and M N (M N ) respectively. This completes our formal presentation of ECC. ECC extends the calculus of constructions [CH88] [Coq85] by adding Sigma types and a cumulative type hierarchy. It can also be seen as an extension of the core of Martin Lof s type theory (with infinite type universes) ML84] by adding a lowest impredicative level of propositions (the types of type P rop) The propositions, which stand for the ....

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Th.Coquand, `Une Theorie des Constructions ', PhD thesis, University of Paris VII.

....and it is an open problem whether extending an arbitrary PTS with defintions preserves strong normalization or not. Worse still, proving strong normalization for particular PTS s extended with definitions is already a problem. The strong normalization proofs for particular type systems given in [Coq85], Luo89a] GN91] Bar92b] cannot be extended in any obvious way. In this chapter, we show how strong normalization of a PTS extended with definitions follows from strong normalization of another (larger) PTS. This enables us to prove that for all strongly normalizing PTS s that we know the ....

Thierry Coquand. Une Theorie des Constructions. PhD thesis, Universit'e Paris VII, 1985.

....as types (see the following explanations) The infinite type hierarchy in ECC is similar to that presented in [Coq86a] and that of MartinL of s type theory [ML73,84] but is fully cumulative in the following sense. First, unlike the original presentations of the calculi of constructions [CH88][Coq85][Coq86a] the propositions at the lowest impredicative level are lifted as higher level types (of their proofs) This lifting is essential for Sigmatypes in ECC to play their role as an abstraction mechanism, and it solves the technical difficulty mentioned above. Secondly, type inclusions ....

.... (e.g. abstract algebras and categories) Viewing intuitively types as sets and : as the membership relation, we have Prop 2 Type 0 2 Type 1 2 : by (Ax) and (T ) Prop Type 0 Type 1 : by (cum) In particular, unlike the original presentations of the calculi of constructions [CH88][Coq85][Coq86a] propositions are lifted to higher level types (Prop Type 0 ) It might appear that this would propagate the impredicativity at the level of propositions to the higher levels. For instance, we can derive Pix:T ype j PiB:T ype j Prop:Bx : Type j . However, the type hierarchy, ....

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Th. Coquand, `Une Theorie des Constructions', PhD thesis, University of Paris VII.

....with their types. Such a view forms a basic starting point of type theories and strongly typed programming languages. This thesis presents and studies a unifying theory of dependent types, ECC Extended Calculus of Constructions. ECC integrates Coquand Huet s calculus of constructions [CH88] Coq85] and Martin Lof s type theory with universes [ML73,84] and turns out to be a strong expressive calculus for formalization of mathematics, structured proof development and program specification. In this introduction, we first give a general and brief discussion about type theories as logical ....

....extension can be translated. In the other direction, one may consider weakening the impredicative polymorphism into stratified polymorphism, as considered by Leivant [Lei89] Coquand Huet s calculus of constructions The calculus of constructions (CC) was introduced by Coquand and Huet [CH88] Coq85] based on ideas from Martin Lof s type theory, Girard s higher order polymorphic calculus and de Bruijn s Automath [dB80] Like Martin Lof s type theory, it uses judgements with contexts and has dependent product as the basic type constructor. Like F , it is impredicative as one can ....

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Th. Coquand, `Une Theorie des Constructions', PhD thesis, University of Paris VII.

....to be correct. We focus on the inverse problem : is it possible to reconstruct proof obligations from a program and its specification The framework is the type theory where a proof can be represented as a typed term [Con86, NPS90] and particularly the Calculus of Inductive Constructions [Coq85] A notion of coherence is introduced between a specification and a program containing annotations as in the Hoare sense. This notion is based on the definition of an extraction function called the weak extraction. Such an annotated program can give a method to reconstruct a set of proof ....

.... 9q; r(n 1 = b q r) b r) g let (q,r) div n b in f (n 1 = b q r) b r) g if b = r 1) then f (n 1 = b (q 1) 0) b 0) g (q 1,0) else f (n 1 = b q r 1) b r 1) g (q,r 1) We choose a particular framework that is the Calculus of Inductive Constructions [Coq85, PM89b] which is a typed calculus with polymorphism, higherorder and dependent types. We focus on the Coq [DFH 93, CCF 94] system which is an implementation of this calculus. It is a system for formalizing and checking the mathematical reasoning [Bar91, ML84] It contains a specification ....

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T. Coquand. Une th'eorie des constructions. PhD thesis, Universit'e Paris VII, 1985.

....None of the languages considered in this thesis have a notion of fully typed terms, however, so the left hand side of this diagram is the one that is important here. Higher order polymorphic calculi sometimes blur the syntactic distinction between ordinary applications and type applications [47, 48]. In such situations, type reconstruction can be generalized to a notion of argument synthesis [109] On the other hand, some second order calculi with subtyping, such as F and F , require fully explicit type abstractions and applications, but allow the types of terms to be implicitly promoted to ....

Thierry Coquand. Une Th eorie des Constructions. PhD thesis, University Paris VII, January 1985.

....in the theory of constructions are strongly normalizing (under fi reduction) The main novelty of this proof is that it uses a Kripkelike interpretation of the types and kinds, and that it does not use infinite contexts. The idea used for avoiding infinite contexts comes from Coquand s thesis [Coq85]. Our proof yields as a corollary another proof of strong normalization (under fi reduction) of well formed terms of LF . In fact, it is easy to see that this proof does not use the candidates of reducibility at all. We are unaware of similar proofs (using reducibility a la Tait ) for LF . 1 ....

....on raw terms, which is definitely more convenient than using equality judgments, many important properties of CC make use of the Church Rosser property. The propositions listed below consist of the translation in English and in our terminology of properties 1 7 in Chapter 1 of Coquand s thesis [Coq85]. In some cases, these proofs require some amplification. First, we need the following definitions, which are translations in our terminology of Coquand s definitions. Definition 4.2 K is a kind iff K is of the form or ( Pix 1 : A 1 ) Pix m : Am ) and Delta.K : kind for some context ....

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Thierry Coquand. Une Th'eorie Des Constructions. PhD thesis, Universit'e Paris VII, January 1985. Th`ese de 3 eme Cycle. 20

....to see that, conversely, the existence of a paradox entails the existence of non normalisable terms. This term represents here the explicit writing of the Burali Forti paradox in natural deduction s style. We give two applications: the inconsistency of the extension of the construction calculus [9] with four levels, and the inconsistency of the extension of the construction calculus with a strong notion of sums a la Martin Lof. We then explain why these results appear as a first step in the analysis of the Curry Howard analogy between propositions and types. We developp a general argument ....

....: Type)p is provable, i.e. with the definition of truth, we get a closed term of type ( Pip : Type)p. Such a term cannot be in head normal form, hence it is not normalisable. So we must give up the identification of propositions and types. A possible solution is the calculus of constructions [9] , where we keep only the identification of a proposition with the type of its proof, but we no longer identify every type with a proposition. There have been proposed some programming or specification languages which contains the idea of a type Type of all types, together with the fact that this ....

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Th. Coquand. "Une Th'eorie des Constructions." these de 3eme cycle, Paris VII (1985).

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T. Coquand. Une Theorie des Constructions. PhD thesis, Universite Paris 7, January 1985.

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Th. Coquand. "Une th'eorie des constructions." Th`ese de troisi`eme cycle, Universit'e Paris VII (Jan. 85).

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Th. Coquand. "Une th'eorie des constructions." Th`ese de troisi`eme cycle, Universit'e Paris VII (Jan. 85).

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T.Coquand. Une Th'eorie des Constructions. Thesis University Paris 7; 1985.

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