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Local theory specifications in Isabelle/Isar
"... Recent versions of the proof assistant Isabelle have acquired a “local theory” concept that integrates a variety of mechanisms for structured specifications into a common framework. We explicitly separate a local theory “target” from its “body”, i.e. a fixed axiomatic specification (parameters and a ..."
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Recent versions of the proof assistant Isabelle have acquired a “local theory” concept that integrates a variety of mechanisms for structured specifications into a common framework. We explicitly separate a local theory “target” from its “body”, i.e. a fixed axiomatic specification (parameters and assumptions) vs. arbitrary definitional extensions (conclusions) depending on it. Body elements may be added incrementally, and admit local polymorphism according to HindleyMilner. The foundations of our local theories rest firmly on existing Isabelle/Isar principles, without having to invent new logics or module calculi. Particular target contexts and body elements may be implemented within the generic infrastructure. This results in a large combinatorial space of specification idioms available to the enduser. Here we introduce targets for Isabelle locales, typeclasses, and class instantiations. The available selection of body elements covers primitive definitions and theorems, inductive predicates and sets, and recursive functions. Porting such existing definitional packages is reasonably simple, and enables to reuse sophisticated tools in a variety of target contexts without further ado. For example, a recursive function may be defined depending on locale parameters and assumptions, or an inductive predicate definition may provide the witness in a typeclass instantiation.
Proof reconstruction for firstorder logic and settheoretical constructions
 Sixth International Workshop on Automated Verification of Critical Systems (AVOCS ’06) – Preliminary Proceedings
, 2006
"... Proof reconstruction is a technique that combines an interactive theorem prover and an automatic one in a sound way, so that users benefit from the expressiveness of the first tool and the automation of the latter. We present an implementation of proof reconstruction for firstorder logic and setth ..."
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Proof reconstruction is a technique that combines an interactive theorem prover and an automatic one in a sound way, so that users benefit from the expressiveness of the first tool and the automation of the latter. We present an implementation of proof reconstruction for firstorder logic and settheoretical constructions between the interactive theorem prover Isabelle and the automatic SMT prover haRVey. 1
Parametric linear arithmetic over ordered fields in Isabelle/HOL
"... We use higherorder logic to verify a quantifier elimination procedure for linear arithmetic over ordered fields, where the coefficients of variables are multivariate polynomials over another set of variables, we call parameters. The procedure generalizes Ferrante and Rackoff’s algorithm for the non ..."
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We use higherorder logic to verify a quantifier elimination procedure for linear arithmetic over ordered fields, where the coefficients of variables are multivariate polynomials over another set of variables, we call parameters. The procedure generalizes Ferrante and Rackoff’s algorithm for the nonparametric case. The formalization is based on axiomatic type classes and automatically carries over to e.g. the rational, real and nonstandard real numbers. It is executable, can be applied to HOL formulae by reflection and performs well on practical examples.
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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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.
SML with antiquotations embedded into Isabelle/Isar
"... Abstract. We report on some recent experiments with SML embedded into the Isabelle/Isar theory and proof language, such that the program text may again refer to formal logical entities via antiquotations. The meaning of our antiquotations within SML text observes the different logical environments a ..."
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Abstract. We report on some recent experiments with SML embedded into the Isabelle/Isar theory and proof language, such that the program text may again refer to formal logical entities via antiquotations. The meaning of our antiquotations within SML text observes the different logical environments at compile time, link time (of theory interpretations), and runtime (within proof procedures). As a general design principle we neither touch the logical foundations of Isabelle, nor the SML language implementation. Thus we achieve a modular composition of the programming language and the logic within the Isabelle/Isar framework. Our work should be understood as a continuation and elaboration of the original “LCF system approach”, which has introduced ML as a programming language for theorem proving in the first place. 1
α The Isabelle/Isar Reference Manual
"... The Isabelle system essentially provides a generic infrastructure for building deductive systems (programmed in Standard ML), with a special focus on interactive theorem proving in higherorder logics. Many years ago, even endusers would refer to certain ML functions (goal commands, tactics, tactica ..."
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The Isabelle system essentially provides a generic infrastructure for building deductive systems (programmed in Standard ML), with a special focus on interactive theorem proving in higherorder logics. Many years ago, even endusers would refer to certain ML functions (goal commands, tactics, tacticals etc.) to pursue their everyday theorem proving tasks. In contrast Isar provides an interpreted language environment of its own, which has been specifically tailored for the needs of theory and proof development. Compared to raw ML, the Isabelle/Isar toplevel provides a more robust and comfortable development platform, with proper support for theory development graphs, managed transactions with unlimited undo etc. In its pioneering times, the Isabelle/Isar version of the Proof General user interface [2, 3] has contributed to the success of for interactive theory and proof development in this advanced theorem proving environment, even though it was somewhat biased towards oldstyle proof scripts. The more recent Isabelle/jEdit Prover IDE [53] emphasizes the documentoriented approach
Under consideration for publication in Math. Struct. in Comp. Science Formalization of Real Analysis: A Survey of
"... In the recent years, numerous proof systems have improved enough to be used for formally verifying nontrivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on proper ..."
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In the recent years, numerous proof systems have improved enough to be used for formally verifying nontrivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability, and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPowerHOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, CCoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis.