Towards a Mizar environment for Isabelle: foundations and language

Abstract

In this paper we explore the possibility of emulating the Mizar environment as close as possible inside the Isabelle logical framework. We introduce adaptations to the Isabelle/FOL object logic that correspond to the logic of Mizar, as well as Isar inner syntax notations that correspond to these of the Mizar language. We show how Isabelle types can be used to differentiate between the syntactic categories of the Mizar language, such as sets and Mizar types including modes and attributes, and show how they interact with the basic constructs of the Tarski-Grothendieck set theory. We discuss Mizar definitions and provide simple abbreviations that allow the introduction of Mizar predicates, functions, attributes and modes using the Isabelle/Pure language elements for introducing definitions and theorems. We finally consider the definite and indefinite description operators in Mizar and their use to introduce definitions by "means" and "equals". We demonstrate the usability of the environment on a sample Mizar-style formalization, with cluster inferences and "by" steps performed manually.

Combining the Syntactic and Semantic Representations of Mizar Proofs

“…where impMI connects uncurry and impI; conjMI is a modification of conjI; ballI, bexI are bounded quantifier introduction and elimination rules which apart from the condition ensure that the given Mizar types are inhabited. These correctly correspond to the Mizar foundations (see [17]). Note that conjMI corresponds to the Mizar conclusion where a given proposition is a conjunct of the current thesis and impI separates a list of conjunctions in an assumption to give them independent labels as follows:…”

“…There has been a lot of progress on the first goal. The environment [17], [18] has an equivalent of the Mizar dependent type system including Mizar-like structures [19], as well as higher-order concepts, such as set comprehensions and schemes [20]. We have recently developed an automated translation of the statements; however, a large gap between the syntactic and semantic Mizar representations significantly hinders work on an automatic export of the MML proofs.…”

Progress in the Independent Certification of Mizar Mathematical Library in Isabelle

“…We defined an object logic that provides Mizar-like foundations in [14]. Here, we briefly remind its construction.…”

“…We make use of the Isabelle logical framework [13] to specify the foundations of Mizar [14]. We further define a number of mechanisms that help to translate the Mizar definitions and proofs [15].…”

Mizar Set Comprehension in Isabelle Framework

“…Isabelle/Mizar is a project whose main goal is an automatic translation of the Mizar proof scripts from the MML to the Isabelle framework, enabling cross-verification of the obtained scripts, but in contrast to the existing translations it tries to preserve types, commands and the structure of proofs originally used [11], [12]. The project is also a unique from the point of view of the order of logic.…”