The Near-Horn Prolog procedures have been proposed as effective procedures in the area of disjunctive logic programming, an extension of logic programming to the (first-order) non-Horn clause domain. In this paper, we show that these case-analysis based procedures may be viewed as members of a class of procedures called the ``ancestry family,'''' which also includes Model Elimination (and its variants), the Positive Refinement of Model Elimination, and SLWV. The common feature which binds these procedures is the extension of SLD-resolution to full first-order logic with the addition of an ancestor cancellation rule. Procedures in the ancestry family are all sound and complete first-order procedures that can be seen to vary along three parameters: (1) the number of clause contrapositives required, (2) the amount of ancestor checking that must occur, and (3) the use (if any) of a Restart rule. Using a sequent-style presentation format for all procedures, we highlight the close relationships between these procedures and compare their relative advantages.