Semantics of Mizar as an Isabelle Object Logic

Abstract

We formally define the foundations of the Mizar system as an object logic in the Isabelle logical framework. For this, we propose adequate mechanisms to represent the various components of Mizar. We express Mizar types in a uniform way, provide a common type intersection operation, allow reasoning about type inhabitation, and develop a type inference mechanism. We provide Mizar-like definition mechanisms which require the same proof obligations and provide same derived properties. Structures and set comprehension operators can be defined as definitional extensions. Re-formalized proofs from various parts of the Mizar Library show the practical usability of the specified foundations.

Experiences from Exporting Major Proof Assistant Libraries

“…The main challenges here are mimicking the highly human-friendly concrete syntax of Mizar and the automated reasoning capabilities. Promising first results have been accomplished in [30] on top of Isabelle. A modern reimplementation on top of an LF-like framework, e.g., implemented in Mmt, could also be promising: it would allow for supporting Mizar's dependent types more naturally but would hardly be able to match Isabelle's automated reasoning.…”

A Tale of Two Set Theories

“…The situation with FOTG is analogous to that of ZFC, and we conjecture FOTG extended with a global choice operator (the) is conservative over FOTG. Regardless of the truth of this conjecture, we take the proper foundation of the MML to be FOTG extended with a global choice operator (see [20]).…”

“…For example, the type morphism of a,b can be defined under the assumption that there exists a morphism from a to b. Without the assumption the definition of morphism of a,b would not be allowed since the type would not be provably inhabited (see [19,20]).…”

Combining the Syntactic and Semantic Representations of Mizar Proofs

“…V. Our representation as a next stage to cross-verify MML in Isabelle In this section we describe possibilities of our representation in relation to the needs of the Isabelle logical framework and in particular Isar reasoning patterns. In our previous work [18], we defined a unique and faithful equivalent of the Mizar dependent type system and higherorder concepts as an Isabelle object logic. This equivalent has been tested so far only on a manually reformalized part of the MML.…”

“…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.…”