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Isabelle/Isar  a generic framework for humanreadable proof documents
 UNIVERSITY OF BIA̷LYSTOK
, 2007
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Textbook proofs meet formal logic  the problem of underspecification and granularity
 Proceedings of MKM’05, volume 3863 of LNAI, IUB
, 2006
"... Abstract. Unlike computer algebra systems, automated theorem provers have not yet achieved considerable recognition and relevance in mathematical practice. A significant shortcoming of mathematical proof assistance systems is that they require the fully formal representation of mathematical content, ..."
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Cited by 7 (3 self)
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Abstract. Unlike computer algebra systems, automated theorem provers have not yet achieved considerable recognition and relevance in mathematical practice. A significant shortcoming of mathematical proof assistance systems is that they require the fully formal representation of mathematical content, whereas in mathematical practice an informal, naturallanguagelike representation where obvious parts are omitted is common. We aim to support mathematical paper writing by integrating a scientific text editor and mathematical assistance systems such that mathematical derivations authored by human beings in a mathematical document can be automatically checked. To this end, we first define a calculusindependent representation language for formal mathematics that allows for underspecified parts. Then we provide two systems of rules that check if a proof is correct and at an acceptable level of granularity. These checks are done by decomposing the proof into basic steps that are then passed on to proof assistance systems for formal verification. We illustrate our approach using an example textbook proof. 1
Structured induction proofs in Isabelle/Isar
 MATHEMATICAL KNOWLEDGE MANAGEMENT (MKM 2006), LNAI
, 2006
"... Isabelle/Isar is a generic framework for humanreadable formal proof documents, based on higherorder natural deduction. The Isar proof language provides general principles that may be instantiated to particular objectlogics and applications. We discuss specific Isar language elements that support ..."
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Cited by 4 (1 self)
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Isabelle/Isar is a generic framework for humanreadable formal proof documents, based on higherorder natural deduction. The Isar proof language provides general principles that may be instantiated to particular objectlogics and applications. We discuss specific Isar language elements that support complex induction patterns of practical importance. Despite the additional bookkeeping required for induction with local facts and parameters, definitions, simultaneous goals and multiple rules, the resulting Isar proof texts turn out wellstructured and readable. Our techniques can be applied to nonstandard variants of induction as well, such as coinduction and nominal induction. This demonstrates that Isar provides a viable platform for building domainspecific tools that support fullyformal mathematical proof composition.
Generating Counterexamples for Structural Inductions by Exploiting Nonstandard Models
"... Abstract. Induction proofs often fail because the stated theorem is noninductive, in which case the user must strengthen the theorem or prove auxiliary properties before performing the induction step. (Counter)model finders are useful for detecting nontheorems, but they will not find any counterexa ..."
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Cited by 1 (1 self)
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Abstract. Induction proofs often fail because the stated theorem is noninductive, in which case the user must strengthen the theorem or prove auxiliary properties before performing the induction step. (Counter)model finders are useful for detecting nontheorems, but they will not find any counterexamples for noninductive theorems. We explain how to apply a wellknown concept from firstorder logic, nonstandard models, to the detection of noninductive invariants. Our work was done in the context of the proof assistant Isabelle/HOL and the counterexample generator Nitpick. 1
Logicfree reasoning in Isabelle/Isar
"... Abstract. Traditionally a rigorous mathematical document consists of a sequence of definition – statement – proof. Taking this basic outline as starting point we investigate how these three categories of text can be represented adequately in the formal language of Isabelle/Isar. Proofs represented i ..."
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Abstract. Traditionally a rigorous mathematical document consists of a sequence of definition – statement – proof. Taking this basic outline as starting point we investigate how these three categories of text can be represented adequately in the formal language of Isabelle/Isar. Proofs represented in humanreadable form have been the initial motivation of Isar language design 10 years ago. The principles developed here allow to turn deductions of the Isabelle logical framework into a format that transcends the raw logical calculus, with more direct description of reasoning using pseudonatural language elements. Statements describe the main result of a theorem in an open format as a reasoning scheme, saying that in the context of certain parameters and assumptions certain conclusions can be derived. This idea of turning Isar context elements into rule statements has been recently refined to support the dual form of elimination rules as well. Definitions in their primitive form merely name existing elements of the logical environment, by stating a suitable equation or logical equivalence. Inductive definitions provide a convenient derived principle to describe a new predicate as the closure of given natural deduction rules. Again there is a direct connection to Isar principles, rules stemming from an inductive characterization are immediately available in structured reasoning. All three subcategories benefit from replacing raw logical encodings by native Isar language elements. The overall formality in the presented mathematical text is reduced. Instead of manipulating auxiliary logical connectives and quantifiers, the mathematical concepts are emphasized. 1
LIMITATIONS ON OUR UNDERSTANDING OF THE BEHAVIOR OF SIMPLIFIED PHYSICAL SYSTEMS
, 2008
"... Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of discrete dete ..."
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Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of discrete deterministic physical systems involving a few moving bodies (twelve point masses) in potentially infinite one dimensional space. There are two kinds of such limiting results that must be carefully distinguished. Results of the first kind state the nonexistence of any algorithm for determining whether any statement among a given set of statements is true or false. Results of the second kind are much deeper and present much greater challenges. They point to specific statements A, where we can neither prove nor refute A using accepted principles of mathematical reasoning. We give a brief survey of these limiting results. These include limiting results of the first kind: from number theory, group theory, and topology, in mathematics, and from idealized computing devices in theoretical computer science. We present a new limiting result of the first kind for simplified physical systems. We conjecture some related limiting results of the second kind, for simplified physical systems.
1 LIMITATIONS ON OUR UNDERSTANDING OF THE BEHAVIOR OF SIMPLIFIED PHYSICAL SYSTEMS
, 2008
"... Abstract. Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of disc ..."
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Abstract. Results going back to Turing and Gödel provide us with limitations on our ability to algorithmically decide the truth or falsity of mathematical assertions in a number of important mathematical contexts. Here we adapt some of this earlier work to very simplified mathematical models of discrete deterministic physical systems involving a few moving bodies (twelve point masses) in potentially infinite one dimensional space. There are two kinds of such limiting results that must be carefully distinguished. Results of the first kind state the nonexistence of any algorithm for determining whether any statement among a given set of statements is true or false. Results of the second kind are much deeper and present much greater challenges. They point to specific statements A, where we can neither prove nor refute A using accepted principles of mathematical reasoning. We give a brief survey of these limiting results. These include limiting results of the first kind: from number theory, group theory, and topology, in mathematics, and from idealized computing devices in theoretical computer science. We present a new limiting result of the first kind for simplified physical systems. We conjecture some related limiting results of the second kind, for simplified physical systems.