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Packaging mathematical structures
 THEOREM PROVING IN HIGHER ORDER LOGICS 5674
, 2009
"... This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated ..."
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Cited by 41 (10 self)
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This paper proposes generic design patterns to define and combine algebraic structures, using dependent records, coercions and type inference, inside the Coq system. This alternative to telescopes in particular supports multiple inheritance, maximal sharing of notations and theories, and automated structure inference. Our methodology is robust enough to handle a hierarchy comprising a broad variety of algebraic structures, from types with a choice operator to algebraically closed fields. Interfaces for the structures enjoy the convenience of a classical setting, without requiring any axiom. Finally, we present two applications of our proof techniques: a key lemma for characterising the discrete logarithm, and a matrix decomposition problem.
Locales: a Module System for Mathematical Theories
"... Locales are a module system for managing theory hierarchies in a theorem prover through theory interpretation. They are available for the theorem prover Isabelle. In this paper, their semantics is defined in terms of local theories and morphisms. Locales aim at providing flexible means of extension ..."
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Locales are a module system for managing theory hierarchies in a theorem prover through theory interpretation. They are available for the theorem prover Isabelle. In this paper, their semantics is defined in terms of local theories and morphisms. Locales aim at providing flexible means of extension and reuse. Theory modules (which are called locales) may be extended by definitions and theorems. Interpretation to Isabelle’s global theories and proof contexts is possible via morphisms. Even the locale hierarchy may be changed if declared relations between locales do not adequately reflect logical relations, which are implied by the locales’ specifications. By discussing their design and relating it to more commonly known structuring mechanisms of programming languages and provers, locales are made accessible to a wider audience beyond the users of Isabelle. The discussed mechanisms include MLstyle functors, type classes and mixins (the latter are found in modern objectoriented languages). 1
B.: Type classes and filters for mathematical analysis in Isabelle/HOL
 Interactive theorem proving. Lecture Notes in Computer Science
, 2013
"... Abstract. The theory of analysis in Isabelle/HOL derives from earlier formalizations that were limited to specific concrete types: R, C and Rn. Isabelle’s new analysis theory unifies and generalizes these earlier efforts. The improvements are centered on two primary contributions: a generic theory o ..."
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Abstract. The theory of analysis in Isabelle/HOL derives from earlier formalizations that were limited to specific concrete types: R, C and Rn. Isabelle’s new analysis theory unifies and generalizes these earlier efforts. The improvements are centered on two primary contributions: a generic theory of limits based on filters, and a new hierarchy of type classes that includes various topological, metric, vector, and algebraic spaces. These let us apply many results in multivariate analysis to types which are not Euclidean spaces, such as the extended real numbers, bounded continuous functions, or finite maps.
Numerical Analysis of Ordinary Differential Equations
, 2013
"... Since many ordinary differential equations (ODEs) do not have a closed solution, approximating them is an important problem in numerical analysis. This work formalizes a method to approximate solutions of ODEs in Isabelle/HOL. We formalize initial value problems (IVPs) of ODEs and prove the existenc ..."
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Since many ordinary differential equations (ODEs) do not have a closed solution, approximating them is an important problem in numerical analysis. This work formalizes a method to approximate solutions of ODEs in Isabelle/HOL. We formalize initial value problems (IVPs) of ODEs and prove the existence of a unique solution, i.e. the PicardLindelöf theorem. We introduce general onestep methods for numerical approximation of the solution and provide an analysis regarding the local and global error of onestep methods. We give an executable specification of the Euler method to approximate the solution of IVPs. With usersupplied proofs for bounds of the differential equation we can prove an explicit bound for the global error. We use arbitraryprecision floatingpoint numbers and also handle rounding errors when we truncate the numbers for efficiency reasons. 1 Relations to the paper Our paper [1] is structured roughly according to the sources you find here. In the following list we show which notions of the paper correspond to which parts of the source code:
Tutorial to Locales and Locale Interpretation
"... Locales are Isabelle’s mechanism to deal with parametric theories. We present typical examples of locale specifications, along with interpretations between locales to change their hierarchic dependencies and interpretations to reuse locales in theory contexts and proofs. This tutorial is intended fo ..."
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Locales are Isabelle’s mechanism to deal with parametric theories. We present typical examples of locale specifications, along with interpretations between locales to change their hierarchic dependencies and interpretations to reuse locales in theory contexts and proofs. This tutorial is intended for locale novices; familiarity with Isabelle and Isar is presumed. 1
α The Isabelle/Isar Implementation
"... We describe the key concepts underlying the Isabelle/Isar implementation, including ML references for the most important functions. The aim is to give some insight into the overall system architecture, and provide clues on implementing applications within this framework. Isabelle was not designed; i ..."
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Cited by 1 (0 self)
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We describe the key concepts underlying the Isabelle/Isar implementation, including ML references for the most important functions. The aim is to give some insight into the overall system architecture, and provide clues on implementing applications within this framework. Isabelle was not designed; it evolved. Not everyone likes this idea. Specification experts rightly abhor trialanderror programming. They suggest that no one should write a program without first writing a complete formal specification. But university departments are not software houses. Programs like Isabelle are not products: when they have served their purpose, they are discarded. Lawrence C. Paulson, “Isabelle: The Next 700 Theorem Provers” As I did 20 years ago, I still fervently believe that the only way to make software secure, reliable, and fast is to make it small. Fight features.
Priority Inheritance Protocol Proved Correct
"... Abstract. In realtime systems with threads, resource locking and priority scheduling, one faces the problem of Priority Inversion. This problem can make the behaviour of threads unpredictable and the resulting bugs can be hard to find. The Priority Inheritance Protocol is one solution implemented ..."
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Abstract. In realtime systems with threads, resource locking and priority scheduling, one faces the problem of Priority Inversion. This problem can make the behaviour of threads unpredictable and the resulting bugs can be hard to find. The Priority Inheritance Protocol is one solution implemented in many systems for solving this problem, but the correctness of this solution has never been formally verified in a theorem prover. As already pointed out in the literature, the original informal investigation of the Property Inheritance Protocol presents a correctness “proof ” for an incorrect algorithm. In this paper we fix the problem of this proof by making all notions precise and implementing a variant of a solution proposed earlier. We also generalise the original informal proof to the practically relevant case where critical sections can overlap. Our formalisation in Isabelle/HOL not just uncovers facts not mentioned in the literature, but also helps with implementing efficiently this protocol. Earlier correct implementations were criticised as too inefficient. Our formalisation is based on Paulson’s inductive approach to verifying protocols; our implementation builds on top of the small PINTOS operating system used for teaching. 1
Reading an Algebra Textbook
"... Abstract. We report on a formalisation experiment where excerpts from an algebra textbook are compared to their translation into formal texts of the Isabelle/Isar prover, and where an attempt is made in the formal text to stick as closely as possible with the structure of the informal counterpart. T ..."
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Abstract. We report on a formalisation experiment where excerpts from an algebra textbook are compared to their translation into formal texts of the Isabelle/Isar prover, and where an attempt is made in the formal text to stick as closely as possible with the structure of the informal counterpart. The purpose of the exercise is to gain understanding on how adequately a modern algebra text can be represented using the module facilities of Isabelle. Our initial results are promising. 1
<inria00368403v2>
, 2009
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Packaging mathematical structures
An Evaluation of Automated Theorem Proving in Regular Algebra
"... Introduction The Isabelle/HOL environment [8] combines the power of automated reasoning with higherorder features for theory engineering and proof management. Its builtin Sledgehammer tool integrates state of the art ATP and SMT tools, allowing for powerful automated reasoning in proofs [2]. The ..."
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Introduction The Isabelle/HOL environment [8] combines the power of automated reasoning with higherorder features for theory engineering and proof management. Its builtin Sledgehammer tool integrates state of the art ATP and SMT tools, allowing for powerful automated reasoning in proofs [2]. Theory engineering features such as typeclasses and locales support the effective