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Locales are Isabelle's approach for dealing with parametric theories. They have been designed as a module system for a theorem prover that can adequately represent the complex inter-dependencies between structures found in abstract algebra, but have proven fruitful also in other applications -for example, software verification.Both design and implementation of locales have evolved considerably since Kammüller did his initial experiments. Today, locales are a simple yet powerful extension of the Isar proof language. The present tutorial covers all major facilities of locales. It is intended for locale novices; familiarity with Isabelle and Isar is presumed.

We present a complete mechanized proof of the result in homological algebra known as basic perturbation lemma. The proof has been carried out in the proof assistant Isabelle, more concretely, in the implementation of higher-order logic (HOL) available in the system. We report on the difficulties found when dealing with abstract algebra in HOL, and also on the ongoing stages of our project to give a certified version of some of the algorithms present in the Kenzo symbolic computation system.

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International audienceWe apply current theorem proving technology to certified code in the domain of abstract algebra. More concretely, based on a formal proof of the (a central result in homological algebra) in the prover Isabelle/HOL, we apply various code generation techniques, which lead to certified implementations of the associated algorithm in ML. In the formal proof, algebraic structures occurring in the Basic Perturbation Lemma are represented in a way, which is not directly amenable to code generation with the available tools. Interestingly, this representation is required in the proof, while for the algorithm simpler data structures are sufficient. Our approach is to establish a link between the non-executable setting of the proof and the executable representation in the algorithm, which is to be generated. This correspondence is established within the logical framework of Isabelle/HOL—that is, it is formally proved correct. The generated code is applied to and illustrated with a number of examples

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