Granularity-Adaptive Proof Presentation

Granularity matters in mathematics. For example, in introductory textbooks intermediate proof steps are often skipped, when this seems appropriate. Such a situation is given in the elementary proof in basic set theory reproduced in Figure 1. Whereas most of the proof steps consist of the application of exactly one mathematical fact (a definition or a lemma), the step from assertion 9 to assertion 10 applies several inference steps at once, namely the application of the definition of ∩ twice, and the distributivity of and over or. Similar observations were made in the empirical studies within the Dialog project (cf. [8]). Systems like Ωmega [1] and HiProofs [4] are capable of structuring proofs hierarchically, the problem remains though how to identify a suitable level of granularity. Autexier and Fiedler have proposed one particular level of granularity [2], which they call what-you-need-is-what-you-stated granularity. Their rigid solution, however, fails to fully model the proof in Figure 1. We present a flexible approach to proof presentation that dynamically adapts to specific levels of granularity in context. Different models for granularity can be learned in our framework from samples using machine learning techniques. More information on the work sketched here is available in a technical report [8].