Florian Rabe

Thomas Traversié, Florian Rabe

Representation theorems for formal systems often take the form of an inductive translation that satisfies certain invariants, which are proved inductively. Theory morphisms and logical relations are common patterns of such inductive constructions. They allow representing the translation and the proofs of the invariants as a set of translation rules, corresponding to the cases of the inductions. Importantly, establishing the invariants is reduced to checking a finite set of, typically decidable, statements. Therefore, in a framework supporting theory morphisms and logical relations, translations that fit one of these patterns become much easier to formalize and to verify. The lambdaPi-calculus modulo rewriting is a logical framework designed for representing and translating between formal systems that has previously not systematically supported such patterns. In this paper, we extend it with theory morphisms and logical relations. We apply these to define and verify invariants for a number of translations between formal systems. In doing so, we identify some best practices that enable us to obtain elegant novel formalizations of some challenging translations, in particular sort-erasure translations from sorted to unsorted languages.

Rhea Ranalter, Florian Rabe, Cezary Kaliszyk

DHOL is an extensional, classical logic that equips the well-known higher-order logic (HOL) with dependent types. This allows for concise encodings of important domains like size-bounded data structures, category theory, or proof theory. Automation support is obtained by translating DHOL to HOL, for which powerful modern automated theorem provers are available. However, a critically missing feature of DHOL is polymorphism. We develop the syntax and semantics of polymorphic DHOL and extend the translation accordingly. We implement the translation in the logic-embedding tool and evaluate it on a range of TPTP formalizations. The logic-embedding tool, together with an off-the-shelf HOL theorem prover easily creates a PDHOL theorem prover for experimenting.

Dominic Lohr, Marc Berges, Michael Kohlhase et al.

Students' answers to tasks provide a valuable source of information in teaching as they result from applying cognitive processes to a learning content addressed in the task. Due to steadily increasing course sizes, analyzing student answers is frequently the only means of obtaining evidence about student performance. However, in many cases, resources are limited, and when evaluating exams, the focus is solely on identifying correct or incorrect answers. This overlooks the value of analyzing incorrect answers, which can help improve teaching strategies or identify misconceptions to be addressed in the next cohort. In teacher training for secondary education, assessment guidelines are mandatory for every exam, including anticipated errors and misconceptions. We applied this concept to a university exam with 462 students and 41 tasks. For each task, the instructors developed answer classes -- classes of expected responses, to which student answers were mapped during the exam correction process. The experiment resulted in a shift in mindset among the tutors and instructors responsible for the course: after initially having great reservations about whether the significant additional effort would yield an appropriate benefit, the procedure was subsequently found to be extremely valuable. The concept presented, and the experience gained from the experiment were cast into a system with which it is possible to correct paper-based exams on the basis of answer classes. This updated version of the paper provides an overview and new potential in the course of using the digital version of the approach.

Colin Rothgang, Florian Rabe

The recently introduced dependent typed higher-order logic (DHOL) offers an interesting compromise between expressiveness and automation support. It sacrifices the decidability of its type system in order to significantly extend its expressiveness over standard HOL. Yet it retains strong automated theorem proving support via a sound and complete translation to HOL. We leverage this design to extend DHOL with refinement and quotient types. Both of these are commonly requested by practitioners but rarely provided by automated theorem provers. This is because they inherently require undecidable typing and thus are very difficult to retrofit to decidable type systems. But with DHOL already doing the heavy lifting, adding them is not only possible but elegant and simple. Concretely, we add refinement and quotient types as special cases of subtyping. This turns the associated canonical inclusion resp. projection maps into identity maps and thus avoids costly changes in representation. We present the syntax, semantics, and translation to HOL for the extended language, including the proofs of soundness and completeness.

Colin Rothgang, Florian Rabe, Christoph Benzmüller

Higher-order logic HOL offers a very simple syntax and semantics for representing and reasoning about typed data structures. But its type system lacks advanced features where types may depend on terms. Dependent type theory offers such a rich type system, but has rather substantial conceptual differences to HOL, as well as comparatively poor proof automation support. We introduce a dependently-typed extension DHOL of HOL that retains the style and conceptual framework of HOL. Moreover, we build a translation from DHOL to HOL and implement it as a preprocessor to a HOL theorem prover, thereby obtaining a theorem prover for DHOL.

Michael Kohlhase, Florian Rabe

The interoperability of proof assistants and the integration of their libraries is a highly valued but elusive goal in the field of theorem proving. As a preparatory step, in previous work, we translated the libraries of multiple proof assistants, specifically the ones of Coq, HOL Light, IMPS, Isabelle, Mizar, and PVS into a universal format: OMDoc/MMT. Each translation presented tremendous theoretical, technical, and social challenges, some universal and some system-specific, some solvable and some still open. In this paper, we survey these challenges and compare and evaluate the solutions we chose. We believe similar library translations will be an essential part of any future system interoperability solution and our experiences will prove valuable to others undertaking such efforts.

Richard Marcus, Michael Kohlhase, Florian Rabe

We describe TGView3D, an interactive 3D graph viewer optimized for exploring theory graphs. To exploit the three spatial dimensions, it extends a force-directed layout with a hierarchical component. Because of the limitations of regular displays, the system also supports the use of a head-mounted display and utilizes several virtual realty interaction concepts.

Jacques Carette, William M. Farmer, Michael Kohlhase et al.

Over the last decades, a class of important mathematical results have required an ever increasing amount of human effort to carry out. For some, the help of computers is now indispensable. We analyze the implications of this trend towards "big mathematics", its relation to human cognition, and how machine support for big math can be organized. The central contribution of this position paper is an information model for "doing mathematics", which posits that humans very efficiently integrate four aspects: inference, computation, tabulation, and narration around a well-organized core of mathematical knowledge. The challenge for mathematical software systems is that these four aspects need to be integrated as well. We briefly survey the state of the art.

Florian Rabe

The author's MMT system provides a framework for defining and implementing logical systems. By combining MMT with the jEdit text editor, we obtain a logic-independent IDE. The IDE functionality includes advanced features such as context-sensitive auto-completion, search, and change management.