Bruck Loops with Abelian Inner Mapping Groups

Loops with Abelian Inner Mapping Groups: An Application of Automated Deduction

“…This is indicated by "class ≤ 2" in the figure. The already known cases where cℓ(Q) ≤ 2 are AIM LCC loops [5] and AIM left Bruck loops [40].…”

“…They appeared already in the milestone paper of Knuth and Bendix [24], and interest in them has continued in the theoremproving community [12] [49]. In more recent years, mathematicians specializing in quasigroups and loops have been making significant use of automated deduction tools [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29] [ 36,37,38,39,40,41,42,43,44]. With the exception of [39] and [40], all of the aforementioned references used Bill McCune's Prover9 [31] or its predecessor Otter [30].…”

Automated theorem proving in quasigroup and loop theory

“…Statements about central nilpotence can be formalized using the associator-commutator calculus (instead of central series). Let's take an example from [45]. The neat English statement Bruck loops with abelian inner mapping group are centrally nilpotent of class 2.…”

“…To date, all but two computer solutions in loop theory have been obtained by Prover9 [33] or its predecessor Otter [32]; the sole exception are two theorems from [45], proved by Waldmeister in 2008 (see Section 4). One of the main goals of the present paper is to give guidelines regarding which prover should be selected for which problem (see the discussion in Section 5).…”

“…As noted above, all but one of these papers were assisted by Prover9 or Otter. Published proofs were almost always translated to human language and usually simplified (the exceptions are [47] and [45], for reasons explained below), hence none of the results relies on the soundness of Otter/Prover9. As far as we know, no automatically generated proof was found to be incorrect during translation (or verification, using tools such as Ivy).…”