This textbook offers a unified and self-contained introduction to the field of term rewriting. It covers all the basic material (abstract reduction systems, termination, confluence, completion, and combination problems), but also some important and closely connected subjects: universal algebra, unification theory, Gröbner bases and Buchberger's algorithm. The main algorithms are presented both informally and as programs in the functional language Standard ML (an appendix contains a quick and easy introduction to ML). Certain crucial algorithms like unification and congruence closure are covered in more depth and Pascal programs are developed. The book contains many examples and over 170 exercises. This text is also an ideal reference book for professional researchers: results that have been spread over many conference and journal articles are collected together in a unified notation, proofs of almost all theorems are provided, and each chapter closes with a guide to the literature.

We introduce Jinja, a Java-like programming language with a formal semantics designed to exhibit core features of the Java language architecture. Jinja is a compromise between the realism of the language and the tractability and clarity of its formal semantics. The following aspects are formalised: a big and a small step operational semantics for Jinja and a proof of their equivalence, a type system and a definite initialisation analysis, a type safety proof of the small step semantics, a virtual machine (JVM), its operational semantics and its type system, a type safety proof for the JVM; a bytecode verifier, that is, a data flow analyser for the JVM, a correctness proof of the bytecode verifier with respect to the type system, and a compiler and a proof that it preserves semantics and well-typedness. The emphasis of this work is not on particular language features but on providing a unified model of the source language, the virtual machine, and the compiler. The whole development has been carried out in the theorem prover Isabelle/HOL.

This article describes a formal proof of the Kepler conjecture on dense sphere packings in a combination of the HOL Light and Isabelle proof assistants. This paper constitutes the official published account of the now completed Flyspeck project.

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Abstract. We present the meta-theory behind the code generation facilities of Isabelle/HOL. To bridge the gap between the source (higherorder logic with type classes) and the many possible targets (functional programming languages), we introduce an intermediate language, MiniHaskell. To relate the source and the intermediate language, both are given a semantics in terms of higher-order rewrite systems (HRSs). In a second step, type classes are removed from Mini-Haskell programs by means of a dictionary translation; we prove the correctness of this step. Building on equational logic also directly supports a simple but powerful algorithm and data refinement concept.

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Sledgehammer, a component of the interactive theorem prover Isabelle, finds proofs in higher-order logic by calling the automated provers for first-order logic E, SPASS and Vampire. This paper is the largest and most detailed empirical evaluation of such a link to date. Our test data consists of 1240 proof goals arising in 7 diverse Isabelle theories, thus representing typical Isabelle proof obligations. We measure the effectiveness of Sledgehammer and many other parameters such as run time and complexity of proofs. A facility for minimizing the number of facts needed to prove a goal is presented and analyzed.