## What is Isabelle?

Isabelle is a generic proof assistant. It allows mathematical formulas to be expressed in a formal language and provides tools for proving those formulas in a logical calculus. The main application is the formalization of mathematical proofs and in particular *formal verification*, which includes proving the correctness of computer hardware or software and proving properties of computer languages and protocols.

## A bluffer's glance at Isabelle

The most widespread instance of Isabelle nowadays is *Isabelle/HOL*, which provides a higher-order logic theorem proving environment that is ready to use for big applications.

*Isabelle/HOL* includes powerful specification tools, e.g. for (co)datatypes, (co)inductive definitions and recursive functions with complex pattern matching.

Proofs are conducted in the structured proof language *Isar*, allowing for proof text naturally understandable for both humans *and* computers.

For proofs, Isabelle incorporates some tools to improve the user's productivity. In particular, Isabelle's *classical reasoner* can perform long chains of reasoning steps to prove formulas. The *simplifier* can reason with and about equations. Linear *arithmetic* facts are proved automatically, various *algebraic* decision procedures are provided. External *first-order provers* can be invoked through *sledgehammer*.

Abstract specifications are supported by a module system (known as locales), of which type classes are a special case.

Isabelle provides excellent notational support: new notations can be introduced, using normal mathematical symbols. Definitions and proofs may include LaTeX source, from which Isabelle can automatically generate typeset documents (papers, books, theses).

*Isabelle/HOL* allows to turn executable specifications directly into code in SML, OCaml, Haskell, and Scala.

Isabelle comes with a large theory library of formally verified mathematics, including elementary number theory (for example, Gauss's law of quadratic reciprocity), analysis (basic properties of limits, derivatives and integrals), algebra (up to Sylow's theorem) and set theory (the relative consistency of the Axiom of Choice). Also provided are numerous examples arising from research into formal verification. A vast collection of applications is accessible via the Archive of Formal Proofs, stemming both from mathematics and software engineering.

Isabelle/jEdit is the default user interface and Prover IDE for Isabelle. It is based on jEdit and Isabelle/Scala. It provides a metaphor of continuous proof checking of a versioned collection of theory sources, with instantaneous feedback in real-time and rich semantic markup for the formal text.

Isabelle may serve as a generic framework for rapid prototyping of deductive systems. These are formulated within Isabelle's logical framework *Isabelle/Pure*, which is suitable for a variety of formal calculi (e.g. axiomatic set theory). Instantiating the generic infrastructure to a particular calculus usually requires only minimal setup in the Isabelle implementation language ML. One may also write arbitrary proof procedures or even theory extension packages in ML, without breaking system soundness (Isabelle follows the well-known *LCF system approach* to achieve a secure system).