  Completion is one of the most studied techniques in term rewriting and
fundamental to automated reasoning with equalities. In this paper we present
new correctness proofs of abstract completion, both for finite and infinite
runs. For the special case of ground completion we present a new proof based on
random descent. We moreover extend the results to ordered completion, an
important extension of completion that aims to produce ground-complete
presentations of the initial equations. We present new proofs concerning the
completeness of ordered completion for two settings. Moreover, we revisit and
extend results of M\'etivier concerning canonicity of rewrite systems. All
proofs presented in the paper have been formalized in Isabelle/HOL.
