Safety and conservativity of definitions in HOL and Isabelle/HOL

Definitions are traditionally considered to be a safe mechanism for introducing concepts on top of a logic known to be consistent. In contrast to arbitrary axioms, definitions should in principle be treatable as a form of abbreviation, and thus compiled away from the theory without losing provability. In particular, definitions should form a conservative extension of the pure logic. These properties are crucial for modern interactive theorem provers, since they ensure the consistency of the logic, as well as a valid environment for total/certified functional programming. We prove these properties, namely, safety and conservativity, for Higher-Order Logic (HOL), a logic implemented in several mainstream theorem provers and relied upon by thousands of users. Some unique features of HOL, such as the requirement to give non-emptiness proofs when defining new types and the impossibility to unfold type definitions, make the proof of these properties, and also the very formulation of safety, nontrivial. Our study also factors in the essential variation of HOL definitions featured by Isabelle/HOL, a popular member of the HOL-based provers family. The current work improves on recent results which showed a weaker property, consistency of Isabelle/HOL’s definitions

Systémy pro formální matematiku

Title: Systems for formal mathematics Author: Ondrěj Kuncˇar Department: Dep. of Theoretical Computer Science and Mathematical Logic Supervisor: Mgr. Josef Urban, Ph.D. Supervisor's e-mail address: [email protected] Abstract: The Mizar type system is a relatively sophisticated system as it allows for many properties, such as independent types, attributes, overloading, subty- ping, structures and many others. All these properties make formalization of mathematics more intuitive in Mizar that in other systems. However, there is a need to verify mathematical results formalized in Mizar in other systems, so that belief in consistency of Mizar system is strengthened. Attempts at recon- struction of this type system in other mathematics formalization systems follow directly from this requisite. The present work seeks to reconstruct Mizar type system in HOL Light system. The basic idea here is to represent Mizar types as predicates in this system (HOL Light). The present work also aims at precise description of relevant parts of Mi- zar type system. The thesis concludes by reviewing some of the insights that were arrived at in the course of designing and implementing suggested reconstruction. Keywords: type system, Mizar, HOL Light

Systémy pro formální matematiku

Title: Systems for formal mathematics Author: Ondrěj Kuncˇar Department: Dep. of Theoretical Computer Science and Mathematical Logic Supervisor: Mgr. Josef Urban, Ph.D. Supervisor's e-mail address: [email protected] Abstract: The Mizar type system is a relatively sophisticated system as it allows for many properties, such as independent types, attributes, overloading, subty- ping, structures and many others. All these properties make formalization of mathematics more intuitive in Mizar that in other systems. However, there is a need to verify mathematical results formalized in Mizar in other systems, so that belief in consistency of Mizar system is strengthened. Attempts at recon- struction of this type system in other mathematics formalization systems follow directly from this requisite. The present work seeks to reconstruct Mizar type system in HOL Light system. The basic idea here is to represent Mizar types as predicates in this system (HOL Light). The present work also aims at precise description of relevant parts of Mi- zar type system. The thesis concludes by reviewing some of the insights that were arrived at in the course of designing and implementing suggested reconstruction. Keywords: type system, Mizar, HOL Light

Systems for formal mathematics

The Mizar type system is a relatively sophisticated system as it allows for many properties, such as independent types, attributes, overloading, subtyping, structures and many others. All these properties make formalization of mathematics more intuitive in Mizar that in other systems. However, there is a need to verify mathematical results formalized in Mizar in other systems, so that belief in consistency of Mizar system is strengthened. Attempts at reconstruction of this type system in other mathematics formalization systems follow directly from this requisite. The present work seeks to reconstruct Mizar type system in HOL Light system. The basic idea here is to represent Mizar types as predicates in this system (HOL Light). The present work also aims at precise description of relevant parts of Mizar type system. The thesis concludes by reviewing some of the insights that were arrived at in the course of designing and implementing suggested reconstruction

SVN Proxy

SVN Proxy is an application that behaves like a proxy of a SVN repository. It contains a local repository, which is used by a user for local commits. The local repository enables to synchronize itself with a remote SVN repository on demand. Local revisions can be numbered in a way they never become unsynchronized with the remote repository. The data of the local repository is stored in XML. The application contains the implementation of a diff algorithm and enables to merge versions and to detect conflicts. SVN Proxy is a command line client and its interface is derived from the common SVN client interface. The main emphasis is laid on the easy future development in the design of the application. The design enables to add other new connections to the repository and create other user interfaces including GUI. The application is intended for source text developers who do not want or are not able to send every commit into the remote repository

Perron-Frobenius Theorem for Spectral Radius Analysis

The spectral radius of a matrixAis the maximum norm of alleigenvalues ofA. In previous work we already formalized that for acomplex matrixA, the values inAngrow polynomially innif andonly if the spectral radius is at most one. One problem with the abovecharacterization is the determination of allcomplexeigenvalues. In caseAcontains only non-negative real values, a simplification is possiblewith the help of the Perron-Frobenius theorem, which tells us that itsuffices to consider only therealeigenvalues ofA, i.e., applying Sturmsmethod can decide the polynomial growth ofAn.We formalize the Perron-Frobenius theorem based on a proof viaBrouwers fixpoint theorem, which is available in the HOL multivari-ate analysis (HMA) library. Since the results on the spectral radius isbased on matrices in the Jordan normal form (JNF) library, we fur-ther develop a connection which allows us to easily transfer theoremsbetween HMA and JNF. With this connection we derive the combinedresult: ifAis a non-negative real matrix, and no real eigenvalue ofAis strictly larger than one, thenAnis polynomially bounded in

Foundational (co)datatypes and (co)recursion for higher-order logic

We describe a line of work that started in 2011 towards enriching Isabelle/HOL’s language with coinductive datatypes, which allow infinite values, and with a more expressive notion of inductive datatype than previously supported by any system based on higher-order logic. These (co)datatypes are complemented by definitional principles for (co)recursive functions and reasoning principles for (co)induction. In contrast with other systems offering codatatypes, no additional axioms or logic extensions are necessary with our approach.</p