Infinite Runs in Abstract Completion

Completion is one of the first and most studied techniques in term rewriting and fundamental to automated reasoning with equalities. In an earlier paper we presented a new and formalized correctness proof of abstract completion for finite runs. In this paper we extend our analysis and our formalization to infinite runs, resulting in a new proof that fair infinite runs produce complete presentations of the initial equations. We further consider ordered completion - an important extension of completion that aims to produce ground-complete presentations of the initial equations. Moreover, we revisit and extend results of Métivier concerning canonicity of rewrite systems. All proofs presented in the paper have been formalized in Isabelle/HOL

Still another approach to the braid ordering

We develop a new approach to the linear ordering of the braid group $B\_n$, based on investigating its restriction to the set \Div(\Delta\_n^d) of all divisors of $\Delta\_n^d$ in the monoid $B\_\infty^+$, i.e., to positive $n$-braids whose normal form has length at most $d$. In the general case, we compute several numerical parameters attached with the finite orders (\Div(\Delta\_n^d), <). In the case of 3 strands, we moreover give a complete description of the increasing enumeration of (\Div(\Delta\_3^d), <). We deduce a new and specially direct construction of the ordering on $B\_3$, and a new proof of the result that its restriction to $B\_3^+$ is a well-ordering of ordinal type $\omega^\omega$