Canonical Systems and Inference (ACSI) were introduced by N. Dershowitz and C. Kirchner to formalize the intuitive notion of good proof and good inference appearing typically in first-order logic or in Knuth-Bendix like completion procedures. Since this abstract framework, based on proof orderings, is intended to be generic, it is of fundamental interest to show its adequacy to represent the main systems of interest. It is here done for minimal propositional natural deduction, and for the standard (Knuth-Bendix) completion, closing an open question. For the first proof system, a generalisation of the ACSI is needed. We provide here a conservative one, in the sense that all results of the original framework still hold. For the second one, two proof representations, proof terms and proofs by replacement, are compared to built a proof ordering that provides an instantiation adapted to the ACSI framework.