Adaptive Assertion - Level Proofs ( Extended Abstract ) ∗

For nearly two decades human-oriented theorem proving techniques have been in the focus of interest of the ΩMEGA project [1] at Saarland University. And since one decade their application to tutorial natural language dialog on mathematical proofs has been studied in the interdisciplinary project DIALOG [3]. The focus on user-orientedness and mathematical practice in these projects resulted in developments such as assertion level proofs, achievements in the field of tactic proof search and proof planning, and investigations into qualitative aspects of the generated proofs. The notion of the assertion level proofs was originally devised by Huang (cf. [1]). It characterizes proofs where each inference step is justified by a mathematical fact, such as a definition, theorem or a lemma. Initially, assertion level proofs were generated in a post-processing step from natural deduction proofs. The new ΩMEGA-CoRe system [1] supports reasoning directly at the assertion level. The ΩMEGA project (among others) has pioneered hierarchical proofs via tactics and proof planning as a human-oriented way of organizing proof search. The notion of island proofs was introduced for proof sketches deliberately omitting proof parts. Users are enabled to formulate proofs by indicating only a subset of relevant intermediary formulas, and automated proof search is used to fill the logical gaps in the argumentation. Traditionally, proof scripts and tactics specify a sequence of actions to be carried out in order to obtain a proof. In contrast to that, the advantage of island proofs (or declarative proof plans/sketches) is that they as such constitute a meaningful and human-readable proof representation, by making the intermediate stages of the proof explicit. This prompted the development of declarative proof scripts (cf. [2]) to enable tactic reasoning at a user-friendly level of representation. Hierarchical proofs, such as generated by tactics, enable proofs at different levels of abstraction. However, in a user-oriented setting such as e-learning for mathematical proofs, we need to ensure that the step size (granularity) of proofs is appropriate in the given context. To address this issue we have proposed and investigated metrics for appropriate proof step size, which we align with empirical standards as determined by experiments [5, 6]. Initially, we devise a catalog of criteria of (singleor multi-inference) proof steps that we deem potentially relevant for judging their granularity in context. This process of identifying relevant criteria is supported by a corpus of students’ proof steps, collected in the DIALOG project, which are annotated with granularity judgments by human expert judges. The question thus is, what distinguishes steps that were judged as appropriate from those considered inappropriate? We investigate a list of criteria: