T. Coquand and G. Huet. The calculus of Constructions. Inf. and Comp., 76(2/3):95--120, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in the term. ffl The notation provides a general vehicle for describing many type theories and calculi. This point has been elaborated in [24] where systems from Barendregt s cube are special instances of our own. Further, we showed there how theorem proving in the calculus of constructions (see [9]) could be more easily done in our framework. ffl Bound and free variables are easily accounted for as can be seen from Example 2.43. ffl Items and segments play an important role in many applications. As explained before, a item is the part joined to a term in an abstraction, and a ffi item is ....

T. Coquand and G. Huet, "The calculus of constructions", Informations and Computation 76 (1988), 95-120.

....between several requirements: 1. It must be expressive enough to develop signi cant bodies of mathematics. 2. It must be theoretically sound. 3. It must be as simple as possible. The original basis of Coq is the Calculus of Constructions (CC) originally introduced by Coquand and Huet [CH88]. CC is a Pure Type System with an impredicative sort called Prop and an untyped sort Type, the latter being the type of the former. CC ful lls requirements 2 and 3 but is not expressive enough: typing Type might be necessary. Coquand proposed an extension of CC with cumulative universes, ....

Thierry Coquand and Grard Huet. The Calculus of Constructions. Inf. Comp., 76:95120, 1988.

....amounts to strongly normalizing a partial application of the function. The other area where strong reduction is required, which prompted the work presented here, is type checking proof checking in type systems logics based on dependent types, such as LF or the Calculus of Constructions [14, 7, 21], which are at the basis of proof assistants such as Alf, Coq, Elf, Lego and NuPRL. In these systems, dependent types may contain arbitrary terms, and types are compared up to b equivalence of the terms they contain, as captured by the following conversion rule: a : t t t # (conv) a : t ....

T. Coquand and G. Huet. The calculus of Constructions. Inf. and Comp., 76(2/3):95--120, 1988.

....for doing both proof and extraction of programs. Keywords: lambda calcul, type, subtype MSC codes: 03B15, 03B40, 68N18 1 Introduction. The Curry Howard [7] isomorphism establishes a relation between proofs and programs. It has been extensively used to develop type systems (for instance COQ [2], AF2 [10] where it is possible to extract a program from a proof of the existence of the function computed by the program. However, they are programs that can not be extracted from a proof even when we can prove that these programs are correct [8] System ST is an attempt to enlarge the set of ....

T. Coquand and G. Huet. The calculus of construction. In Information and Computation, pages 241-262, 1988.

....has a su cient deductive power. We have formalized and used our axioms inside the Logical Framework Coq [BB 01] However, the axioms can be stated and worked with in a general constructive logical setting, because we do not need all the richness of the Calculus of Constructions [CH88] the logic beneath Coq. In particular we do not require the use of dependent inductive types and universes. On the contrary, we should have available a logical system that accommodates second order quanti cation (in order to axiomatize the existence of limit) and the Axiom of Choice (for de ning ....

T. Coquand and G. Huet. "The Calculus of Constructions". Information and Control 76, 1988.

....in type theory as dependently typed types and nding programs meeting these speci cations is reduced to nding inhabitants of the corresponding type. The target system in Typelab is the Extended Calculus of Constructions or ECC [17] ECC is an extension of the Calculus of Constructions [7] with dependent types and universes. This means that Typelab employs a dependently typed language with metavariables, explicit substitutions and dependent product and types. It has an in nite hierarchy of type universes P rop : Type 0 : Type 1 : Typen : 2.2.1 Description of ....

Thierry Coquand and Gerard Huet. The calculus of constructions. In Information and Computation, number 76(2/3), pages 95-120. 1988.

....an index free setting. A larger class of algebraic types is de ned. Logical aspects need more examination. But we already give a syntactic way for dealing with partial and total objects, leading to the notion of generic proof. 1 Introduction The Calculus of Constructions (CC for short) 8] 9] is a typed high order functional calculus which provides a nice formalism for constructive proofs in natural deduction style. It can also be seen as a high level functional programming language. Since F is embeddable in it, we already know that any fonction over integers is de nable on Nat ....

....M is of one of the following forms (modulo conversion) M [ x : P ] y : Q) M 0 R ) with M 0 ( z : T ]M 1 M 2 ) hz : T iM 1 constant or variable and S(M) S(M 0 ) S(M 1 ) Where x : M is for x 1 : M 1 . x k : M k , k 0. 2. 3 Basic results We refer to [9] for all the basic results on the original calculus CC. We point out 2.3.1 Proposition [ x is Church Rosser (CR) 2.3.2 Theorem CC is a conservative, hence consistent, extension of CC. Proof CC is clearly embeddable in CC . Moreover, given a total term M , we can extract a term ....

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Coquand T. and Huet G. (1988) The Calculus of Constructions. Inf. Comp. 76 p.95-120.

....computation which is very similar to the above definition of partial recursive functions. As a result proofs that particular functions are computable are relatively straightforward. 3 An Overview of Coq The Coq system is an implementation in CAML of the Calculus of Inductive Constructions (CIC) [1], a variant of type theory related to Martin Lof s Intuitionistic Type Theory [7, 8] and Girard s polymorphic calculus F [4] Terms in CIC are typed and types are also terms. Such a type theory can be treated as a logic through the Curry Howard isomorphism (see [11, 8] for introductions of the ....

....in l i : Rarr f l i h[x i 0 ; x i 1 ; x i n Gamma1 ]i # r. 8 5. 6 Arithmetic Identity N (x) x Identity = Proj 0 Constants C n ( v) n Constant 0 = Zero Constant (S n) Subl Succ [Constant n] Addition 0 x 1 = x 1 Add = Rec Identity S(x 0 ) x 1 = S(x 0 x 1 ) Rarr Succ [1] Multiplication 0 Theta x 1 = 0 Multiply = Rec Zero S(x 0 ) Theta x 1 = x 1 (x 0 Theta x 1 ) Rarr Add [2, 1] Power x 0 1 = 1 Power = Rec (Constant 1) x S(x0 ) 1 = x 1 Theta (x x0 1 ) Rarr Multiply [2, 1] Power = Rarr Power [1, 0] Note that Power h[x 0 ; x 1 ]i # x x0 1 ....

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Thierry Coquand and G'erard Huet. The calculus of constructions. Rapport de Recherche 530, INRIA, Rocquencourt, France, May 1986.

....or batch mode. It is based on variants of two earlier systems, FOL, 90] and XCHECK, 83] that evolved during the 1970 s and early 1980 s. As such, NAP incorporates no novel concept, and it lacks many of the capabilities found in modern reasoning systems such as the Calculus of Constructions, [19], Deva, 78] 5] and [88] ISABELLE, 61] HOL, 38] LP, 37] Nqthm, 8] and many more. NAP was chosen because it was already implemented by the first author, it runs on his Macintosh computer, and it has a syntax compatible with SETL. We were also vaguely skeptical that the capabilities ....

Coquand, T., and Huet, G., The calculus of constructions, Inform. and Comput. 76, 1988, pp. 95--120.

....of module reductions in the system itself helps bringing the study of module systems back to the study of typed lambda calculi. Moreover, it seems to provide a firm basis for its use in proofs systems. In this respect, we are currently working on its adaptation to the Calculus of Constructions [CH88, CCF 95] which should be quite easy (despite the fact there is no distinction between types and terms) 12 in order to have a modular proof language well suited to proving modular programs. Since the Calculus of Construction is both a programming language and a proof language, this would ....

T. Coquand and G. Huet. The calculus of constructions. Inf. Comp., 76:95--120, 1988.

....constructors as needed. The Logical Framework can also be used to encode the syntax of other logical systems such as predicate logic and modal logic. The interested reader is referred to (Harper, Honsell, and Plotkin 1993) 2.2. 3 The Calculus of Constructions The Calculus of Constructions (CC) (Coquand and Huet 1988) is a type theory with Pi types and a universe closed under impredicative quantification (U Impr) The universe is traditionally denoted by Prop and the corresponding El operator is either written Prf ( Gamma) or omitted. The idea is that the universe Prop corresponds to a type of propositions ....

....The point is that Nat itself is of the form Prf ( Gamma) and so can serve as the argument c to an element of Nat . Also other inductive datatypes like lists or trees can be defined in this way. Similarly, logical connectives can be defined on the type Prop by their usual higher order encodings (Coquand and Huet 1988). The encoding of datatypes inside Prop proved insufficient as no internal induction principles (like R N ) are available for these. This gave rise to two extensions of the pure Calculus of Constructions: Luo s Extended Calculus of Constructions (ECC) implemented in the Lego system (Luo 1994; ....

Coquand, T. and G. Huet (1988). The Calculus of Constructions. Information and Computation 76, 95--120.

.... of the Calculus of Construction Extended by Strong Sums and Recursive Definitions Dieter Spreen Fachbereich Mathematik, Theoretische Informatik Universitat GH Siegen, D 57068 Siegen, Germany Abstract We present a purely domain theoretic model of Coquand and Huet s Calculus of Construction [3], which is one of the most powerful type systems proposed in the literature. The well formed expressions of its language are divided into three levels: Terms, Types, and Orders. Terms are the elements of Types, while the elements of Orders are called Operators. There is a special Order constant ....

T. Coquand and G. Huet, The calculus of constructions, Inform. and Comput. 76 (1988) 95--120.

....is based on type theory, all derived from the Automath family of de Bruijn (1970) These systems have 1 so called canonical public proof objects, that can be verified locally by other groups. Modern prototype systems for proof development based on type theory are for example Coq, see Coquand and Huet (1988)) and Lego, see Luo and Pollack (1992) Systems of computer algebra can deal very successfully with some parts of mathematics, namely symbolic equational reasoning. But systems of computer algebra do not have a notion of proof. One can imagine that these systems be extended with proofs of the ....

Coquand, T. and G. Huet (1988). The calculus of constructions, Information and Computation 76, pp. 95--120. Available at URL : http://pauillac.inria.fr/ coq/coq-eng.html.

....module systems to Pure Type Systems raises the problem of dealing with fi equivalence that appears in the conversion rule of PTS. In this paper, we give an adaptation of the system of [Cou96] to Pure Type Systems. This system applies to the LF logical framework, the Calculus of Construction [CH88] the Calculus of Constructions extended with universes [Luo89] We do not deal with the problem of adding inductive types to these systems, but the addition of inductive types as firstclass objects should not raise any problem as our proposal is quite orthogonal to the base language: as few ....

T. Coquand and G. Huet. The calculus of constructions. Inf. Comp., 76:95--120, 1988.

....sensej. A snapshot of the current implementation can be found on the Web at http: www.cs.chalmers.se augustss cayenne . 7 Related work There are many logical frameworks (proof checking systems) that are based on dependent types. Some examples, among many, are ALF [MN94,Nor93,ACN90] CoC [CH86,CH88] ELF [Pfe89,Fra91,HHP93] Lego [Pol94] and NuPRL [Con86] All these systems are primarily designed for making (constructive) proofs even if many of them can also execute the resulting proofs or extract a program from them. Our approach is dioeerent in that we want to make a programming ....

Thierry Coquand and G#rard Huet. The Calculus of Constructions. Technical Report 530, INRIA, Centre de Rocquencourt, 1986.

....a calculus based on but where de Bruijn s indices and explicit substitution are used. For this, we start by introducing de Bruijn s indices. Such indices have the practical advantages that they avoid all the need to deal with variable renaming in terms (see [de Bruijn 72] Abadi et al. 91] CH 88] and [KN 93] The calculus based on and on de Bruijn s indices will be called Omega Xi for Xi being the set of variables which are de Bruijn s indices together with a special variable. In the first instance, Omega is taken to be f; ffig. In order to accommodate substitution explicitly and ....

Coquand T., and Huet G., (1988) The calculus of constructions, Information and Computation 76, 95-120.

....in the term. ffl The notation provides a general vehicle for describing many type theories and calculi. This point has been elaborated in [24] where systems from Barendregt s cube are special instances of our own. Further, we showed there how theorem proving in the calculus of constructions (see [9]) could be more easily done in our framework. ffl Bound and free variables are easily accounted for as can be seen from Example 2.43. ffl Items and segments play an important role in many applications. As explained before, a item is the part joined to a term in an abstraction, and a ffi item ....

T. Coquand and G. Huet, "The calculus of constructions", Informations and Computation 76 (1988), 95-120.

.... Furthermore, expressions can also be seen as types denoting a set of values and the refinement relation can be compared with the subtype relation in [Car88] Rey85] and with the containment relation between types in [Mit88] In the same research line, there are several systems [ML79] CABea86] [CH88] [LB88] in which types and values are so intertwined that types become program specifications and programs become constructive proofs that such specifications are satisfiable. For instance, in the Calculus of Constructions [CH88] expressions have been used to represent both values and types. In ....

....In the same research line, there are several systems [ML79] CABea86] CH88] LB88] in which types and values are so intertwined that types become program specifications and programs become constructive proofs that such specifications are satisfiable. For instance, in the Calculus of Constructions [CH88], expressions have been used to represent both values and types. In that paper and in [ML79] however, value expressions and type expressions are rigorously distinguished and a type relation between values and types is formalized. Less rigorous is the distinction in Nuprl [CABea86] and in Pebble ....

T. Coquand and G. Huet. "The Calculus of Constructions". Information and Computation, 76:95--120, 1988.

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T. Coquand and G. Huet. The calculus of Constructions. Inf. and Comp., 76(2/3):95--120, 1988.

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Thierry Coquand and Gerard Huet. The Calculus of Constructions. Technical Report 530, INRIA, Centre de Rocquencourt, 1986. 61

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T. Coquand and G. Huet. The Calculus of Constructions. In Information and Control 76, 1988.

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Coquand T., and Huet G., (1988) The calculus of constructions, Information and Computation 76, 95-120.

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T.Coquand, G.Huet: "The Calculus of Constructions" Information and Computation 76, pp.95-120 (1988)

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Thiery Coquand and Gerard Huet. The Calculus of Constructions. ########### ### ###########, 76:96-120, 1988.

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T. Coquand and G. Huet, "The calculus of constructions", Information and Computation, 76 (1988) pp. 95-120.

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