What does it mean for an algebraic rewrite rule to subsume another rule (that may then be called a subrule)? We view subsumptions as rule morphisms such that the simultaneous application of a rule and a subrule (i.e. the application of a subsumption morphism) yields the same result as a single application of the subsuming rule. Simultaneous applications of categories of rules are obtained by Global Coherent Transformations and illustrated on graphs in the DPO approach. Other approaches are possible since these transformations are formulated in an abstract Rewriting Environment, and such environments exist for various approaches to Algebraic Rewriting, including DPO, SqPO and PBPO.

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