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Mizar Light for HOL Light
 Theorem Proving in Higher Order Logics: TPHOLs 2001, LNCS 2152
, 2001
"... There are two dierent approaches to formalizing proofs in a computer: the procedural approach (which is the one of the HOL system) and the declarative approach (which is the one of the Mizar system). ..."
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There are two dierent approaches to formalizing proofs in a computer: the procedural approach (which is the one of the HOL system) and the declarative approach (which is the one of the Mizar system).
ComputerAssisted Mathematics at Work  The HahnBanach Theorem in Isabelle/Isar
 TYPES FOR PROOFS AND PROGRAMS: TYPES’99, LNCS
, 2000
"... We present a complete formalization of the HahnBanach theorem in the simplytyped settheory of Isabelle/HOL, such that both the modeling of the underlying mathematical notions and the full proofs are intelligible to human readers. This is achieved by means of the Isar environment, which provides ..."
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We present a complete formalization of the HahnBanach theorem in the simplytyped settheory of Isabelle/HOL, such that both the modeling of the underlying mathematical notions and the full proofs are intelligible to human readers. This is achieved by means of the Isar environment, which provides a framework for highlevel reasoning based on natural deduction. The final result is presented as a readable formal proof document, following usual presentations in mathematical textbooks quite closely. Our case study demonstrates that Isabelle/Isar is capable to support this kind of application of formal logic very well, while being open for an even larger scope.
A Partial Translation Path from MathLang to Isabelle
, 2011
"... thesis or use of any of the information contained in it must acknowledge this thesis as the source of the quotation or information. This dissertation describes certain developments in computer techniques for managingmathematical knowledge. Computers currently assist mathematicians in presenting and ..."
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thesis or use of any of the information contained in it must acknowledge this thesis as the source of the quotation or information. This dissertation describes certain developments in computer techniques for managingmathematical knowledge. Computers currently assist mathematicians in presenting and archiving mathematics, as well as performing calculation and verification tasks. MathLang is a framework for computerising mathematical documents which features new approaches to these issues. In this dissertation, several extensions to MathLang are described: a system and notation for annotating text; improved methods for annotating complex mathematical expressions; and a method for creating rules to translate document annotations. A typical MathLang work flow for document annotation and computerisation is demonstrated, showing how writing style can complicate the annotation process and how these may be resolved. This workflow is compared with the standard process for producing formal computer theories in a computer proof assistant (Isabelle is the system we choose). The rules for translation are further discussed as a way of producing text in the syntax of Isabelle (without a deep knowledge of the system), with possible use cases of providing a text which can be used either as an aid to learning Isabelle, or as a skeleton framework to be used as a starting point for a formal document. i ii For my family both present and future
A Formal Proof Of The Riesz Representation Theorem
"... This paper presents a formal proof of the Riesz representation theorem in the PVS theorem prover. The Riemann Stieltjes integral was defined in PVS, and the theorem relies on this integral. In order to prove the Riesz representation theorem, it was necessary to prove that continuous functions on a c ..."
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This paper presents a formal proof of the Riesz representation theorem in the PVS theorem prover. The Riemann Stieltjes integral was defined in PVS, and the theorem relies on this integral. In order to prove the Riesz representation theorem, it was necessary to prove that continuous functions on a closed interval are Riemann Stieltjes integrable with respect to any function of bounded variation. This result follows from the equivalence of the Riemann Stieltjes and Darboux Stieltjes integrals, which would have been a lengthy result to prove in PVS, so a simpler lemma was proved that captures the underlying concept of this integral equivalence. In order to prove the Riesz theorem, the Hahn Banach theorem was proved in the case where the normed linear spaces are the continuous and bounded functions on a closed interval. The proof of the Riesz theorem follows the proof in Haaser and Sullivan’s book Real Analysis. The formal proof of this result in PVS revealed an error in textbook’s proof. Indeed, the proof of the Riesz representation theorem is constructive, and the function constructed in the textbook does not satisfy a key property. This error illustrates the ability of formal verification to find logical errors. A specific counterexample is given to the proof in the textbook. Finally, a corrected proof of the Riesz representation theorem is presented.