Abstract not found

We discuss the general concept of calculational reasoning within Isabelle/Isar, which provides a framework for high-level natural deduction proofs that may be written in a human-readable fashion. Setting out from a few basic logical concepts of the underlying meta-logical framework of Isabelle, such as higher-order unification and resolution, calculational commands are added to the basic Isar proof language in a flexible and non-intrusive manner. Thus calculational proof style may be combined with the remaining natural deduction proof language in a liberal manner, resulting in many useful proof patterns. A case-study on formalizing Computational Tree Logic (CTL) in simply-typed set-theory demonstrates common calculational idioms in practice.

We present a complete formalization of the Hahn-Banach theorem in the simply-typed set-theory of Isabelle/HOL, such that both the modeling of the underlying mathematical notions and the full proofs are intelligible to human readers. This is achieved by means of the Isar environment, which provides a framework for high-level reasoning based on natural deduction. The final result is presented as a readable formal proof document, following usual presentations in mathematical textbooks quite closely. Our case study demonstrates that Isabelle/Isar is capable to support this kind of application of formal logic very well, while being open for an even larger scope.

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 higher-order 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 top-level 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 old-style proof scripts. The more recent Isabelle/jEdit Prover IDE [51] emphasizes the document-oriented approach