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The Foundation of a Generic Theorem Prover
 Journal of Automated Reasoning
, 1989
"... Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a metalogic (or `logical framework') in which the objectlogics are formalized. Isabell ..."
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Isabelle [28, 30] is an interactive theorem prover that supports a variety of logics. It represents rules as propositions (not as functions) and builds proofs by combining rules. These operations constitute a metalogic (or `logical framework') in which the objectlogics are formalized. Isabelle is now based on higherorder logic  a precise and wellunderstood foundation. Examples illustrate use of this metalogic to formalize logics and proofs. Axioms for firstorder logic are shown sound and complete. Backwards proof is formalized by metareasoning about objectlevel entailment. Higherorder logic has several practical advantages over other metalogics. Many proof techniques are known, such as Huet's higherorder unification procedure. Key words: higherorder logic, higherorder unification, Isabelle, LCF, logical frameworks, metareasoning, natural deduction Contents 1 History and overview 2 2 The metalogic M 4 2.1 Syntax of the metalogic ......................... 4 2.2 ...
A Constructive Formalization of the Fundamental Theorem of Calculus
"... We have finished a constructive formalization in the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. In this formalization, we have closely followed Bishop's work ([4]). In this paper, we describe the formalizat ..."
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We have finished a constructive formalization in the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. In this formalization, we have closely followed Bishop's work ([4]). In this paper, we describe the formalization in some detail, focusing on how some of Bishop's original proofs had to be refined, adapted or redone from scratch.
A new implementation of Automath
 Journal of Automated Reasoning
"... Abstract. This paper presents aut, a modern Automath checker. It is a straightforward reimplementation of the Zandleven Automath checker from the seventies. It was implemented about five years ago, in the programming language C. It accepts both the AUT68 and AUTQE dialects of Automath. This progr ..."
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Abstract. This paper presents aut, a modern Automath checker. It is a straightforward reimplementation of the Zandleven Automath checker from the seventies. It was implemented about five years ago, in the programming language C. It accepts both the AUT68 and AUTQE dialects of Automath. This program was written to restore a damaged version of Jutting’s translation of Landau’s Grundlagen. Some notable features: − It is fast. On a 1GHz machine it will check the full Jutting formalization (736K of nonwhitespace Automath source) in 0.6 seconds. − Its implementation of λterms does not use named variables or de Bruijn indices (the two common approaches) but instead uses a graph representation. In this representation variables are represented by pointers to a binder. − The program can compile an Automath text into one big ‘Automath single line’ style λterm. It outputs such a term using de Bruijn indices. (These λterms cannot be checked by modern systems like Coq or Agda, because the λtyped λcalculi of de Bruijn are different from the Πtyped λcalculi of modern type theory.) The source of aut is freely available on the Web at the address
Formalizing Real Calculus in Coq
, 2002
"... We have finished a constructive formalization in the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. This formalization is built upon the library of constructive algebra created in the FTA (Fundamental Theorem of Alg ..."
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We have finished a constructive formalization in the theorem prover Coq of the Fundamental Theorem of Calculus, which states that differentiation and integration are inverse processes. This formalization is built upon the library of constructive algebra created in the FTA (Fundamental Theorem of Algebra) project, which is extended with results about the real numbers, namely about (power) series. Two important issues that arose in this formalization and which will be discussed in this paper are partial functions (different ways of dealing with this concept and the advantages of each different approach) and the high level tactics that were developed in parallel with the formalization (which automate several routine procedures involving results about realvalued functions).
Using Web Access to Formal Mathematics to Support Instruction in Computational Discrete Mathematics
, 2002
"... We have begun a project to produce interactive formallygrounded courseware for teaching mathematics and computing. 1 The courseware is created by a modern proof development system, Nuprl, based on its growing em reference library of formal computational mathematics. The project is supported by NSF ..."
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We have begun a project to produce interactive formallygrounded courseware for teaching mathematics and computing. 1 The courseware is created by a modern proof development system, Nuprl, based on its growing em reference library of formal computational mathematics. The project is supported by NSF and some results of the past eighteen months of work are available on the World Wide Web. 2 This proposal requests an increment of funding to supplement the continuing investment of Cornell resources. We are asking for funds to improve the educational value of the resources we have created. First, we want to add more targeted lessons as entry points to the large corpus of formal material. Second, we want to gather feedback on the existing lessons from a wider range of students and instructors. Third, we want to prepare for using the full Nuprl interactive capability when it becomes available on the Web in 1998 and then deploy it in 1999. This proposal reviews the technical and pedagogical case for the project, reports on current progress and future plans and explains our ideas for improving the educational value of the
and
, 1996
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, ..."
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In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, the three subjects of calculus, logic and type theory, got separated due to the appearence of the paradoxes. Moreover, the existence of various versions of each topic is due to the need to get back to the lost paradise which allowed a great freedom in mixing expressivity and logic. In any case, the presence of such a variety of systems calls for a framework to unify them all. Barendregt's cube for example, is an attempt to unify various type systems and his associated logic cube is an attempt to nd connections between type theories and logic. We devise a new notation which enables categorising most of the known systems in a unied way. More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from Automathlike systems to singlytyped Pure Type Systems. The system We are grateful for Erik Poll who has read the paper carefully and for his productive comments. We are also grateful for discussions with Henk Barendregt, Inge Bethke, Tijn Borghuis and for the helpful remarks received from them. Furthermore, we are indebted to the anonymous referee for his/her useful suggestions and remarks. y
and
, 1996
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, ..."
Abstract
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In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, the three subjects of calculus, logic and type theory, got separated due to the appearence of the paradoxes. Moreover, the existence of various versions of each topic is due to the need to get back to the lost paradise which allowed a great freedom in mixing expressivity and logic. In any case, the presence of such a variety of systems calls for a framework to unify them all. Barendregt's cube for example, is an attempt to unify various type systems and his associated logic cube is an attempt to nd connections between type theories and logic. We devise a new notation which enables categorising most of the known systems in a unied way. More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from Automathlike systems to singlytyped Pure Type Systems. The system We are grateful for Erik Poll who has read the paper carefully and for his productive comments. We are also grateful for discussions with Henk Barendregt, Inge Bethke, Tijn Borghuis and for the helpful remarks received from them. Furthermore, we are indebted to the anonymous referee for his/her useful suggestions and remarks. y
and
, 1996
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, ..."
Abstract
 Add to MetaCart
In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, the three subjects of calculus, logic and type theory, got separated due to the appearence of the paradoxes. Moreover, the existence of various versions of each topic is due to the need to get back to the lost paradise which allowed a great freedom in mixing expressivity and logic. In any case, the presence of such a variety of systems calls for a framework to unify them all. Barendregt's cube for example, is an attempt to unify various type systems and his associated logic cube is an attempt to nd connections between type theories and logic. We devise a new notation which enables categorising most of the known systems in a unied way. More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from Automathlike systems to singlytyped Pure Type Systems. The system We are grateful for Erik Poll who has read the paper carefully and for his productive comments. We are also grateful for discussions with Henk Barendregt, Inge Bethke, Tijn Borghuis and for the helpful remarks received from them. Furthermore, we are indebted to the anonymous referee for his/her useful suggestions and remarks. y
and
, 1996
"... In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, ..."
Abstract
 Add to MetaCart
In the area of foundations of mathematics and computer science, three related topics dominate. These are calculus, type theory and logic. There are moreover, many versions of calculi and type theories. In these versions, the presence of logic ranges from the nonexistant to the dominant. In fact, the three subjects of calculus, logic and type theory, got separated due to the appearence of the paradoxes. Moreover, the existence of various versions of each topic is due to the need to get back to the lost paradise which allowed a great freedom in mixing expressivity and logic. In any case, the presence of such a variety of systems calls for a framework to unify them all. Barendregt's cube for example, is an attempt to unify various type systems and his associated logic cube is an attempt to nd connections between type theories and logic. We devise a new notation which enables categorising most of the known systems in a unied way. More precisely, we sketch the general structure of a system of typed lambda calculus and show that this system has enough expressive power for the description of various existing systems, ranging from Automathlike systems to singlytyped Pure Type Systems. The system We are grateful for Erik Poll who has read the paper carefully and for his productive comments. We are also grateful for discussions with Henk Barendregt, Inge Bethke, Tijn Borghuis and for the helpful remarks received from them. Furthermore, we are indebted to the anonymous referee for his/her useful suggestions and remarks. y