Abstract. There are many automatic tools to prove termination of term rewrite systems, nowadays. Most of these tools use a combination of many complex termination criteria. Hence generated proofs may be of tremendous size, which makes it very tedious (if not impossible) for humans to check those proofs for correctness. In this paper we use the theorem prover Isabelle/HOL to automatically certify termination proofs. To this end, we first formalized the required theory of term rewriting including three major termination criteria: dependency pairs, dependency graphs, and reduction pairs. Second, for each of these techniques we developed an executable check which guarantees the correct application of that technique as it occurs in the generated proofs. Moreover, if a proof is not accepted, a readable error message is displayed. Finally, we used Isabelle's code generation facilities to generate a highly efficient and certified Haskell program, CeTA, which can be used to certify termination proofs without even having Isabelle installed.

This paper describes the second edition of the Tyrolean Termination Tool -a fully automatic termination analyzer for first-order term rewrite systems. The main features of this tool are its (non-)termination proving power, its speed, its flexibility due to a strategy language, and the fact that the source code of the whole project is freely available. The clean design together with a stand-alone OCaml library for term rewriting, make it a perfect starting point for other tools concerned with rewriting as well as experimental implementations of new termination methods.

The termination and complexity competition (termCOMP) focuses on automated termination and complexity analysis for various kinds of programming paradigms, including categories for term rewriting, integer transition systems, imperative programming, logic programming, and functional programming. In all categories, the competition also welcomes the participation of tools providing certifiable output. The goal of the competition is to demonstrate the power and advances of the stateof-the-art tools in each of these areas.

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This paper describes the second edition of the Tyrolean Termination Tool-a fully automatic termination analyzer for first-order term rewrite systems. The main features of this tool are its (non-)termination proving power, its speed, its flexibility due to a strategy language, and the fact that the source code of the whole project is freely available. The clean design together with a stand-alone OCaml library for term rewriting, make it a perfect starting point for other tools concerned with rewriting as well as experimental implementations of new termination methods.

We provide an overview of CPF, the certification problem format, and explain some design decisions. Whereas CPF was originally invented to combine three different formats for termination proofs into a single one, in the meanwhile proofs for several other properties of term rewrite systems are also expressible: like confluence, complexity, and completion. As a consequence, the format is already supported by several tools and certifiers. Its acceptance is also demonstrated in international competitions: the certified tracks of both the termination and the confluence competition utilized CPF as exchange format between automated tools and trusted certifiers.

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Abstract. We describe a line of work that started in 2011 towards enriching Isabelle/HOL's language with coinductive datatypes, which allow infinite values, and with a more expressive notion of inductive datatype than previously supported by any system based on higher-order logic. These (co)datatypes are complemented by definitional principles for (co)recursive functions and reasoning principles for (co)induction. In contrast with other systems offering codatatypes, no additional axioms or logic extensions are necessary with our approach.