Systémy pro formální matematiku

Abstract

Na'zev pra'ce: Syste'my pro forma'lnı' matematiku Autor: Ondrěj Kuncˇar Katedra (u'stav): Katedra teoreticke' informatiky a matematicke' logiky Vedoucı' diplomove' pra'ce: Mgr. Josef Urban, Ph.D. e-mail vedoucı'ho: josef.urban@gmail.com Abstrakt: Typovy' syste'm Mizaru je pomeřneˇ sofistikovany' - obsahuje celou rˇadu vlastnostı', jako jsou za'visle' typy, atributy, prětežˇova'nı', konstrukci podtypu˚, struktury a dalsˇı'. Tyto vlastnosti nemalou meřou prˇispı'vajı' k tomu, zě forma- lizace matematiky je v syste'mu Mizar mnohem intuitivneˇjsˇı' nezˇ v ostatnı'ch syste'mech. Za'rovenˇ vyvsta'va' potrěba verifikovat matematicke' poznatky forma- lizovane' v Mizaru i v jiny'ch syste'mech, aby se zvy'sˇila jistota, zě syste'm Mizar pracuje korektneˇ. Je tedy prˇirozene' pokusit se rekonstruovat tento typovy' syste'm v jiny'ch syste'mech pro formalizaci matematiky. Tato pra'ce navrhuje takovou rekonstrukci do syste'mu HOL Light. Idea rekon- strukce je reprezentovat typy Mizaru jako predika'ty v tomto syste'mu (HOL Light). Soucˇa'stı' te'to pra'ce je i pokus o co nejprěsneˇjsˇı' zachycenı' relevantnı' cˇa'sti typove'ho syste'mu Mizaru. V za'veřu jsou pak uvedeny postrěhy, ktere' byly ucˇi- neňy prˇi na'vrhu a cˇa'stecňe' implementaci navrzěne' rekonstrukce. Klı'cˇova' slova: typovy' syste'm, Mizar, HOL...

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: josef.urban@gmail.com 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 1