Abstract. We extended Isabelle/HOL with a pair of definitional commands for datatypes and codatatypes. They support mutual and nested (co)recursion through well-behaved type constructors, including mixed recursion-corecursion, and are complemented by syntaxes for introducing primitive (co)recursive functions and by a general proof method for reasoning coinductively. As a case study, we ported Isabelle's Coinductive library to use the new commands, eliminating the need for tedious ad hoc constructions.

Interactive theorem provers based on higher-order logic (HOL) traditionally follow the definitional approach, reducing high-level specifications to logical primitives. This also applies to the support for datatype definitions. However, the internal datatype construction used in HOL4, HOL Light, and Isabelle/ HOL is fundamentally noncompositional, limiting its efficiency and flexibility, and it does not cater for codatatypes.We present a fully modular framework for constructing (co)datatypes in HOL, with support for mixed mutual and nested (co)recursion. Mixed (co)recursion enables type definitions involving both datatypes and codatatypes, such as the type of finitely branching trees of possibly infinite depth. Our framework draws heavily from category theory. The key notion is that of a bounded natural functor-an enriched type constructor satisfying specific properties preserved by interesting categorical operations. Our ideas are implemented as a definitional package in Isabelle, addressing a frequent request from users.

Over the last 15 years Runtime Verification (RV) has grown into a diverse and active field, which has stimulated the development of numerous theoretical frameworks and tools. Many of the tools are at first sight very different and challenging to compare. Yet, there are similarities. In this work, we classify RV tools within a high-level taxonomy of concepts. We first present this taxonomy and discuss the different dimensions. Then, we survey RV tools and classify them according to the taxonomy. This paper constitutes a snapshot of the current state of the art and enables a comparison of existing tools. 1 Introduction Runtime Verification (RV) [7, 28, 29, 38] (or runtime monitoring) is (broadly) the study of methods to analyze the dynamic behavior of computational systems. The most typical analysis is to check whether a given run of a system satisfies a given specification and it is this general setting (and its variants) that we consider in this paper. Whilst topics such as specification mining or trace visualization are generally considered to be within this broad field, we do not include them in our discussion. This paper presents a taxonomy of RV frameworks and tools and uses this to classify 20 selected tools. This work is timely for a number of reasons. Firstly, after more than 15 years of maturing, the field has reached a point where such a general view is needed. The last significant attempt at a taxonomy was in 2004 [24] and had a distinctly different focus to our own. Secondly, a number of activities, such as the runtime verification competitions [4, 6, 30, 48], the RV-CuBES workshop [46, 49], two schools dedicated to RV [16], and a COST action [1] (including the development of a tutorial book on the topic [5]), have put the development of runtime verification tools into focus. Terminology. The field of RV is broad and the used terminology is not yet unified. For the sake of clarity, let us fix the following terms:-Monitored system. The system consisting of software, hardware, or a combination of the two, that is being monitored. Its behavior is usually abstracted as a trace object. The authors warmly thank Martin Leucker for the early discussions on the taxonomy and mind map representation. This article is based upon work from COST Action ARVI IC1402, supported by COST (European Cooperation in Science and Technology). In particular, the taxonomy and classification benefited from discussions within working groups one and two of this action. We would also like to acknowledge input from participants of Dagstuhl seminar 17462 [34].

This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under "friendly" operations, including constructors. Friendly corecursive functions can be registered as such, thereby increasing the corecursor's expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic.

We present a formalization of the first half of Bachmair and Ganzinger's chapter on resolution theorem proving in Isabelle/HOL, culminating with a refutationally complete first-order prover based on ordered resolution with literal selection. We develop general infrastructure and methodology that can form the basis of completeness proofs for related calculi, including superposition. Our work clarifies several of the fine points in the chapter's text, emphasizing the value of formal proofs in the field of automated reasoning.