Bruijn, N.G. de, (1970) The mathematical language AUTOMATH, its usage and some of its extensions, in: Symposium on Automatic Demonstration, IRIA, Versailles, 1968, Lecture Notes in Mathematics, 125, 29-61, Springer.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....alike. Such a treatment is necessary since many of the principles that govern terms govern types too. In fact it is to be noted that in the more general type systems, types and terms are treated alike. This is for example the case, in the Automath systems and in the calculus of constructions (see [deB70] and [CoH88] 2. Omega Gamma the set of operators contains many s and ffi s and contains substitution, typing and many more operators. In fact, the more general type theories use more than one as an abstraction operator. For example, in the Pure Type Systems of Barendregt in [Bar92] we ....

Bruijn, N.G. de, The mathematical language AUTOMATH, its usage and some of its extensions, in: Symposium on Automatic Demonstration, IRIA, Versailles, 1968, Lecture Notes in Mathematics, 125, 29-61, Springer, 1970.

.... address the question above of precision , and the automation of formal reasoning, to address the question of correctness (logical frameworks and proof assistants [HP91, HP93] In the last 20 years, several systems for implementing and checking formal reasoning (proof assistants like AUTOMATH [Bru68, Bru80], no longer used, COQ [CH85, CH87] Mizar [Rud92, TR93] PVS [ORS92] have been 1 developed with quite notable results. This fact has testified the importance of the functionalities and the applications offered by Information Technologies, for developing and maintaining the mathematical knowledge ....

N. G. de Bruijn. The Mathematical Language AUTOMATH, its usage and some of its extensions. In Symposium on Automatic Demonstration, volume 125 of Lectures Notes in Mathematics, pages 29-61. INRIA, Springer-Verlag. Versailles, France, 1968.

....is done within the well known system of second order predicate logic. Unlike in previous work, no complex semantic formalism or specialized logic is required. The Curry Howard isomorphism has informed the development of powerful lambda calculi with dependent types, such as de Bruijn s Automath [deB70], Martin Lof s type theory [Mar82] Constable s Nuprl [Con86] Coquand and 2 Foo Syntax Type variables X; Y; Z Individual variables x; y; z Types A; B : X j A B j 8X: B Typed terms s; t; u : x j x:A: u j s t j X: u j s A Contexts Gamma : x1 : A1 ; xn : An Rules Id x1 : A1 ....

N. G. de Bruijn, The mathematical language of AUTOMATH, its usage and some of its extensions, Proceedings of the Symposium on Automatic Demonstration, LNCS 125, Springer-Verlag, 1970.

....is done within the well known system of second order predicate logic. Unlike in previous work, no complex semantic formalism or specialized logic is required. The Curry Howard isomorphism has informed the development of powerful lamdba calculi with dependent types, such as de Bruijn s Automath [deB70], Martin Lof s type theory [Mar82] Constable s Nuprl [Con86] Coquand and Huet s calculus of constructions [CH88] and Barendregt s lambda cube [Bar91] Each of these calculi introduces dependent types (types that depend upon values) to map first order quantifiers into the type system. In ....

N. G. de Bruijn, The mathematical language of AUTOMATH, its usage and some of its extensions, Proceedings of the Symposium on Automatic Demonstration, LNCS 125, Springer-Verlag, 1970.

....for propositional logic, but also for intuitionistic arithmetic. By FAT(B) normalization in # # is the same as in natural deduction. 10 N.G. de Bruijn There is an independent discovery of the formulas as types idea in the form FAT(C) towards the end of the sixties, by de Bruijn [8]. De Bruijn and his group developed a language AUTOMATH, designed for the mechanical checking of mathematical proofs. De Bruijn s only clue was the intuitionistic interpretation of implication, as formulated by Heyting for his proof interpretation. For a long time the results on AUTOMATH were not ....

N.G. de Bruijn, The mathematical language AUTOMATH, its usage and some of its extensions, in: M. Laudet and D. Lacombe, M. Schuetzenberger, (Eds.), Symp. on Automatic Demonstration, Lecture Notes in Mathematics, vol. 125, Springer, Berlin, 1970, pp. 29--61.

....structure as necessary. First, Nuprl has been able to define all the basic concepts for number theory, finite set theory, algebra and basic analysis. We are looking at geometry and topology as well. The language was designed based on the foundational type theories studied for much of the century [23,25,26,33,34, 76, 97, 100] and believed to be adequate for all of mathematics. Based on this experience with Nuprl, we know that type theory is a good basis for meeting the above goals. The language can express mathematical problems and their algorithmic solutions. The Nuprl group has been able to check the type ....

N. G. de Bruijn. The mathematical language Automath, its usage and some of its extensions. In Symposium on Automatic Demonstration, volume 125 of Lecture Notes in Mathematics, pages 29--61, Berlin, New York, 1970. Springer-Verlag.

....explanation can be extended to full predicate logic, thus providing types that can be read as propositions involving quantifiers. The crucial extension of ML s type system that will make this possible is the provision of dependent types. A pioneering work on proof checking is de Bruijn s AUTOMATH [13, 14]. This is a computer system for checking ordinary mathematical proofs and, for this purpose, de Bruijn was not satisfied with a traditional formalization of mathematics in set theory expressed in predicate logic; instead he was led to a type theoretic way of expressing proofs, that is, as objects ....

N. G. de Bruijn. The Mathematical Language AUTOMATH, its usage and some of its extensions. In Symposium on Automatic Demonstration, volume 125 of Lecture Notes in Mathematics, pages 29--61, Versailles, France, 1968. IRIA, Springer-Verlag.

....the BCI calculus. Furthermore, we present another proof for the fact that each BCK term in normal form is completely determined by its principal pairs. 1 Introduction The connection between systems of calculus and logical systems, already suggested in [4] was established in 1968 (see [9] and [5]) for the K calculus and intuitionistic logic 1 . In fact the types of stratified, i.e. typable, K combinators form the set of provable formulas in intuitionistic logic. The same correspondence exists between the I calculus and the system R of relevance logic (see [1] as well as between the ....

N. G. de Bruijn. The mathematical language AUTOMATH, its usage and some of its extensions. In Symposium on automatic demonstration, volume 125 of Lecture Notes in Mathematics, pages 29--61. Springer Verlag, 1970.

....Apparently critical remarks as types for propositional logic, but also for intuitionistic arithmetic. By FAT(B) normalization in is the same as in natural deduction 9 . N.G. de Bruijn There is an independent discovery of the formulas as types idea in the form FAT(C) by N.G. de Bruijn ([Bru70]) De Bruijn and his group developed a language automath, designed for the mechanical checking of mathematical proofs. De Bruijn s only clue was the intuitionistic interpretation of implication, as formulated by Heyting for his proof interpretation. For a long time the results on automath were not ....

N. G. de Bruijn. The mathematical language automath, its usage and some of its extensions. In M. Laudet, D. Lacombe, and M. Schuetzenberger, editors, Symposium on Automatic Demonstration, Lecture Notes in Mathematics, 125, pages 29--61. Springer-Verlag, Berlin, Heidelberg, New York, 1970.

....while the degree of rigour in formal methods is much higher than would be normally expected in mathematics. An example, highlighting the difference between rigorous and formal mathematics, is the formalization of Landau s [9] analysis text book in the AUTOMATH 1 system of N.G. de Bruijn [10] (see [11] Landau s text book was already of a much higher degree of rigour than would normally be expected in standard mathematical texts, but when the book was formalized in AUTOMATH, its length was increased ten fold. A historical example, the Principia Mathematica of Russell and Whitehead ....

N.G. de Bruijn. The mathematical language AUTOMATH, its usage, and some of its extensions. In M. Laudet, editor, Proceedings of the Symposium on Automatic Demonstration, pages 29--61, Versailles, France, December 1968. Springer-Verlag LNM 125.

....about normalization and unicity of the representation of data have no equivalent in other systems. Moreover, the class of data types we consider is very large with some unusual ones. 1 Introduction Since the work of Curry, a lot of type systems have been created (e.g. De Bruijn s Automath [4]; Girard s system F [5] Martin Lof s type theory [10] Coquand Huet s Calculus of construction [3] etc) One of their purposes is program extraction via the CurryHoward isomorphism [6] which establishes a correspondence between programs and proofs of specifications. One of these systems is AF 2 ....

N. de Bruijn. The mathematical language automath, its usage and some of its extensions. In Symp. on automatic demonstration, pages 29--61. Springer Verlag, 1970. Lecture Notes in Mathematics Vol. 125.

....the way mathematics is approached in the undergraduate curricula. 2. 4 Mechanical Theorem Proving Mechanization of mathematical reasoning is an interesting topic in itself, independent of any applications (see for example [30] for an interesting historical account) Systems such as Automath [19] and Mizar [57] have been developed to this end. Mechanization raises the question what a mathematical proof really is, and how one can gain confidence in a proof produced by a computer [53] Theorem provers can be classified in several more or less independent ways. In the following, we discuss ....

N. G. de Bruijn. The mathematical language AUTOMATH, its usage, and some of its extensions. In M. Laudet, editor, Proceedings of the Symposium on Automatic Demonstration, pages 29--61, Versailles, France, December 1968. Springer-Verlag LNM 125.

....Such derivations easily become long and tedious and, hence, error prone; so, it is essential to formalize the proofs and to have computerized tools to check them. There are several examples of computer implementations of proof checkers for formal logics. An early example is the AUTOMATH system [11, 12] which was designed by de Bruijn to check proofs of mathematical theorems. Quite large proofs were checked by the system, for example the proofs in Landau s book Grundlagen der Analysis [24] Another system, which is more intended as a proof assistant, is the Edinburgh (Cambridge) LCF system [19, ....

N. G. de Bruijn. The Mathematical Language AUTOMATH, its usage and some of its extensions. In Symposium on Automatic Demonstration, volume 125 of Lecture Notes in Mathematics, pages 2961, Versailles, France, 1968. IRIA, Springer-Verlag.

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Bruijn, N.G. de, (1970) The mathematical language AUTOMATH, its usage and some of its extensions, in: Symposium on Automatic Demonstration, IRIA, Versailles, 1968, Lecture Notes in Mathematics, 125, 29-61, Springer.

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N.G. de Bruijn, The mathematical language automath, its usage and some of its extension, in [30], 29--61

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N.G. de Bruijn, The mathematical language automath, its usage and some of its extension, in

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