R. Nederpelt, J. Geuvers, and R. de Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....be completely sure that it is correct, except of course for paradoxes in the foundations of the logic and bugs in the kernel of the checking software. But that logic can be very simple and clear and that kernel can be very small. This dream has been dreamt many, many times (the Automath project [5], the Mizar project [7] the QED project [1] the MBase project [2] it has been dreamt many times) and until now nothing has happened. A favorite passtime of people who like to dream this dream is to guess how long it will take before the utopia arrives. Our guess is that when enough mathematics ....

R.P. Nederpelt, J.H. Geuvers, and R.C. de Vrijer. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. Elsevier Science, Amsterdam, 1994.

....prove mathematical properties in which functions and predicates can be defined by rewrite rules, hence enabling the automatic proof of equational problems. The Calculus of Constructions. The quest for such a language started with Girard s system F [19] on one hand and De Bruijn s Automath project [18] on the other hand. Later, Coquand and Huet combined both calculi into the Calculus of Constructions (CC) 10] As in system F, in CC, data structures are defined by using an impredicative encoding which is difficult to use in practice. Following Martin Lof s theory of types [24] Coquand and ....

H. Geuvers, R. Nederpelt, and R. de Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. NorthHolland, 1994.

....between the proof states and not the proof states themselves. 1 Also since the initial proof obligation is the nal statement to be proved, procedural proofs tend to run backwards , from the conclusion back to the assumptions. The other two proof checkers from the seventies, Automath [13] and Mizar [12, 17] both are of the declarative kind. Another system that is declarative is the Ontic system by David McAllester [11] In a declarative system, a proof doesn t consist of instructions to transform statements but of those statements themselves. Furthermore, the statements are ....

R.P. Nederpelt, J.H. Geuvers, and R.C. de Vrijer. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. Elsevier Science, Amsterdam, 1994.

....libraries provide a constructive theory. Section 9 Logic and the Peano Axioms Our approach to logic comes from Brouwer as formalized in Heyting [61] One of the most influential accounts historically is Howard [66] and also deBruijn [48, 49] for Automath. The Automath papers are collected in [85]. The connection between propositions and types has found its analogue in set theory as well. The set theory of Anthony P. Morse from 1986 equated sets and propositions. He asserted a set by the claim that it was non empty. Morse believed that every (mathematical) thing is a set. For him, ....

R. P. Nederpelt, J. H. Geuvers, and R. C. De Vrijer. Selected Papers in Automath, volume 133 of Studies in Logic and The Foundations of Mathematics. Elsevier, Amsterdam, 1994.

....proving the soundness and completeness of the algorithm. Finally there are other algorithms that are concerned with the smaller class of (weakly) injective PTSs [2, 6] These algorithms are simpler but do not cover all existing systems. For example some of the languages of the Automath family [4] and predicative F [5] are not weakly injective. The purpose of this paper is to present a new sound and complete typechecking algorithm for functional PTSs. The novelty of our algorithm is to remain within the framework of PTSs. It is an improvement over [3, 8, 9] our algorithm is ....

R. Nederpelt, H. Geuvers, and R. de Vrijer, editors. Selected papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1994.

.... the axiom of choice for setoids and follows from Diaconescu s construction, see e.g. 34] and also [32] The third item establishes that proof irrelevance, i.e. the property that all proofs of a proposition are equal (the property was rst considered by de Bruijn in the Automath project, see e.g. [38]) is derivable from the axiom of choice for setoids and can be established from [9] It should also be pointed that these results were probably discovered independently by authors not cited above, e.g. the results are also implicit in [27] Recall that a topos T is a cartesian closed category ....

R. Nederpelt, H. Geuvers, and R. de Vrijer, editors. Selected papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1994.

....the amount of detail present in the original text and the expressivity of the system used to do the formalization. For three speci c examples this factor turns out to be approximately equal to four. 1 Loss Factor In A survey of the project Automath de Bruijn wrote (p. 160 in section A. 5 of [9] which is a reprint from [1] A very important thing that can be concluded from all writing experiments is the constancy of the loss factor. The loss factor expresses what we loose in shortness when translating very meticulous ordinary mathematics into Automath. This factor may be quite big, ....

....of a proof compress similarly well. This means that the apparent and intrinsic de Bruijn factors turn out to be approximately the same. 3 Arithmetic in Automath The rst example for which we will calculate the de Bruijn factor is Jutting s classic Automath translation (see section D. 2 of [9], a reprint from [6] of Landau s Grundlagen der Analysis [7] a cute little book about the basic laws of arithmetic up to the complex numbers. To give an impression of the text and its translation, here is a small fragment of a proof (of Satz 27 on p. 37 of [7] in the Grundlagen: 1 geh ort ....

R.P. Nederpelt, J.H. Geuvers, and R.C. de Vrijer. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. Elsevier Science, Amsterdam, etc., 1994.

....of T . Example 1.1 below presents a formulation of Peano arithmetic, a famous axiomatic theory that represents the standard model of the natural numbers. The computer theorem proving framework is mechanized by a wide range of di erent kinds of computer theorem provers. Examples include Automath [41], Coq [2] eves [14] hol [31] imps [25] Isabelle [43] Mizar [46] Nqthm [5] Nuprl [13] Otter [38] and pvs [42] Most theorem provers are primarily used to prove conjectures in the context of an axiomatic theory. Other aspects of the mathematics process are usually not well supported. ....

R. P. Nederpelt, J. H. Geuvers, and R. C. De Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and The Foundations of Mathematics. North Holland, 1994.

....of type theory. 1 Introduction 1.1 Problem Until a few decades ago mathematics was something that was done in human heads, on the blackboard or on paper. Only since the seventies have systems been developed that verify mathematics with the computer. The rst of these was the Automath system [13] from the Netherlands. Other early systems of this kind were the Mizar system [12, 20] from Poland and the LCF system [6] from the UK. Recently this kind of system has become widely used (mostly because of applications in computer science) Currently the most popular is the PVS system [14] from a ....

....just meaningless. Our paper integrates the type theoretical way to model partial functions with the PVS approach of having correctness conditions on the side. The type theoretical approach of having proof terms as an argument to model partial functions already dates from the Automath project (see [13] page 710) For a discussion of type correctness conditions in PVS see [18] 1.4 Contribution Our main contribution is that we stay close to ordinary rst order logic. Our method allows one to reason in rst order logic with total functions without translation of the statements and still know ....

R.P. Nederpelt, J.H. Geuvers, and R.C. de Vrijer. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. Elsevier Science, Amsterdam, 1994.

....in a formal language using a restricted set of proof construction commands in order to verify them. A human user is required to perform the formalisation task. Basically, two main approaches towards the formalisation and veri cation of proofs were taken. In the rst, the Automath Mizar approach [20, 24], the user is required to give a full and explicit construction of a formal proof. The proof component then checks the proof for correctness (the compiler approach) A well known example of a large formalisation is van Benthem Jutting s translation of Landau s Grundlagen der Analysis into ....

R. P. Nederpelt, J.H. Geuvers, and R.C. de Vrijer, editors. Selected papers on Automath, volume 133 of Studies in Logic and the foundations of Mathematics. North-Holland, 1994.

....analysis of the Calculus of Constructions [11] in particular by relating it to other important calculi such as the polymorphic calculus (a.k.a. System F) of Girard [16] and Reynolds [25] the polymorphic higher order calculus (a.k.a. System F ) of Girard [16] and the Logical Frameworks [17, 21]. Most denitions and theorems concerning Pure Type Systems rely on some induction principle. Typically, the induction proceeds on the structure of terms or the structure of derivations. In some instances however, these induction principles prove inadequate, and alternative induction principles ....

R. Nederpelt, H. Geuvers, and R. de Vrijer, editors. Selected papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1994. 47

....Fn . 2 First steps The first task is to select and or develop an appropriate formalism for the machine representation and generalisation of a proof. There are various systems in which inference rules and proofs may be represented and generated, such as Isabelle [37] Prolog[35] ELF[40] Automath[36], LF[25] Lego[41] and various other sequent calculus, natural deduction [10] typed terms [10] tableau based, resolutionbased, matrix[46] hybrid proof systems. Each approach has its own advantages as well as limitations. Analytic proof systems [39] such as resolution, matrix or connection ....

Nederpelt R.P., Geuvers J.H. and de Vrijer R.C., editors Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics, North-Holland, 1994.

....language using a restricted set of proof construction commands in order to verify them. A human user is required to fulfill the formalization task. Basically, two main approaches towards the formalization and verification of proofs were taken. In the first approach, the Automath Mizar approach [14, 15], the user is required to give a full and explicit construction of a proof. The proof component then checks the proof for correctness (the compiler approach) A well known example of a larger formalization task for Automath is van Benthem Jutting s translation of Landau s Grundlagen der Analysis ....

....structuring and refining. For representing textbook proofs and proof plans, we proposed to use augmented discourse representation structures. We think that an extended DRT formalism is better suited for the process of formalizing mathematics than earlier languages proposed for this purpose [14, 15, 9] because it allows to handle phenomena that occur in natural language proofs. Being able to process textbook proofs allows us to close the gap between the language of mathematicians and the language of proofs systems. Reaching this goal will enable us to build AI systems that really assist ....

R. P. Nederpelt, J.H. Geuvers, and R.C. de Vrijer, editors. Selected papers on Automath, volume 133 of Studies in Logic and the foundations of Mathematics. North-Holland, 1994.

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H. Geuvers, R. Nederpelt, and R. de Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1994.

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R.P. Nederpelt, J.H. Geuvers, and R.C. de Vrijer. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1994.

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R. Nederpelt, J. Geuvers, and R. de Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1994.

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R. Nederpelt, J. Geuvers, and R. de Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1994.

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H. Geuvers, R. Nederpelt, and R. de Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. NorthHolland, 1994.

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H. Geuvers, R. Nederpelt, and R. de Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1994.

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R. P. Nederpelt, J. H. Geuvers, and R. C. de Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1994.

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R. P. Nederpelt, J. H. Geuvers, and R. C. De Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and The Foundations of Mathematics. North Holland, 1994. 25

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R.P. Nederpelt, J.H. Geuvers, and R.C. de Vrijer. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. Elsevier Science, Amsterdam, 1994.

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R. P. Nederpelt, J. H. Geuvers, and R. C. De Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and The Foundations of Mathematics. North Holland, 1994. 36

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R. P. Nederpelt, J. H. Geuvers, and Roel C. de Vrijer, editors. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, 1994. Cited on page 116.

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R.P. Nederpelt, J.H. Geuvers, and R.C. de Vrijer. Selected Papers on Automath, volume 133 of Studies in Logic and the Foundations of Mathematics. Elsevier Science, Amsterdam, 1994.

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