SeCaV: A Sequent Calculus Verifier in Isabelle/HOL

We describe SeCaV, a sequent calculus verifier for first-order logic in Isabelle/HOL, and the SeCaV Unshortener, an online tool that expands succinct derivations into the full SeCaV syntax. We leverage the power of Isabelle/HOL as a proof checker for our SeCaV derivations. The interactive features of Isabelle/HOL make our system transparent. For instance, the user can simply click on a side condition to see its exact definition. Our formalized soundness and completeness proofs pertain exactly to the calculus as exposed to the user and not just to some model of our tool. Users can also write their derivations in the SeCaV Unshortener, which provides a lighter syntax, and expand them for later verification. We have used both tools in our teaching.Comment: In Proceedings LSFA 2021, arXiv:2204.0341

ProofBuddy: A Proof Assistant for Learning and Monitoring

Proof competence, i.e. the ability to write and check (mathematical) proofs, is an important skill in Computer Science, but for many students it represents a difficult challenge. The main issues are the correct use of formal language and the ascertainment of whether proofs, especially the students' own, are complete and correct. Many authors have suggested using proof assistants to assist in teaching proof competence, but the efficacy of the approach is unclear. To improve the state of affairs, we introduce ProofBuddy: a web-based tool using the Isabelle proof assistant which enables researchers to conduct studies of the efficacy of approaches to using proof assistants in education by collecting fine-grained data about the way students interact with proof assistants. We have performed a preliminary usability study of ProofBuddy at the Technical University of Denmark.Comment: In Proceedings TFPIE 2023, arXiv:2308.0611

Verifying a Sequent Calculus Prover for First-Order Logic with Functions in Isabelle/HOL

We describe the design, implementation and verification of an automated theorem prover for first-order logic with functions. The proof search procedure is based on sequent calculus and we formally verify its soundness and completeness in Isabelle/HOL using an existing abstract framework for coinductive proof trees. Our analytic completeness proof covers both open and closed formulas. Since our deterministic prover considers only the subset of terms relevant to proving a given sequent, we do so as well when building a countermodel from a failed proof. Finally, we formally connect our prover with the proof system and semantics of the existing SeCaV system. In particular, the prover can generate human-readable SeCaV proofs which are also machine-verifiable proof certificates

Teaching Functional Programmers Logic and Metatheory

We present a novel approach for teaching logic and the metatheory of logic to students who have some experience with functional programming. We define concepts in logic as a series of functional programs in the language of the proof assistant Isabelle/HOL. This allows us to make notions which are often unclear in textbooks precise, to experiment with definitions by executing them, and to prove metatheoretical theorems in full detail. We have surveyed student perceptions of our teaching approach to determine its usefulness and found that students felt that our formalizations helped them understand concepts in logic, and that they experimented with them as a learning tool. However, the approach was not enough to make students feel confident in their abilities to design and implement their own formal systems. Further studies are needed to confirm and generalize the results of our survey, but our initial results seem promising

Do repeated transurethral procedures under general anesthesia influence mortality in patients with non-invasive urothelial bladder cancer? A Danish national cohort study

Purpose: To investigate the effect of repeated transurethral procedures under general anesthesia on overall mortality in patients with non-invasive bladder cancer. Materials and methods: All Danish residents with non-invasive papillary urothelial carcinoma or primary urothelial carcinoma in situ diagnosed between 1 January 2000 and 1 January 2011 were included and followed until death or 31 March 2017. All transurethral procedures under general anesthesia, intravesical instillation therapy, recurrences and progression to invasive disease or cystectomy were recorded during follow-up. Associations between treatments and overall mortality were evaluated using multivariable regression analysis adjusted for age, gender, comorbidities and socioeconomic status. The effect of disease progression on mortality was removed by censoring patients at the time of progression or cystectomy. Results: Risk of death increased with the number of transurethral procedures under general anesthesia for Ta low- and high-grade tumors compared to patients who had only one procedure; after eight or more procedures the risk of death increased by 28% and 83%, respectively. There was no similar relationship for carcinomas in situ. In total, 36–52% of procedures under general anesthesia did not identify urothelial neoplasia. Conclusions: Repeated transurethral procedures under general anesthesia appear to be associated with increased risk of death in patients with primary non-invasive papillary urothelial carcinoma. Furthermore, a substantial number of procedures were without findings of neoplasia, indicating that too many patients are admitted for transurethral procedures under GA. Attempts should be taken to reduce unnecessary transurethral procedures under GA, e.g. by improved outpatient diagnosis of urothelial neoplasia