We show that our ground completion modulo E always admits a finite ground convergent (modulo E) rewrite system, which allows us to obtain the decidability of the word problem of ground theories modulo E. We also present a completion and ground completion method for rewriting modulo a finite set of permutation equations E using our ordering modulo E. Any of the ways we can arrange things, where the order is important. The number of non negative integral solutions for the equation x1 + x2 + x3 + x4.xr n, given the variable cannot have equal values. It is an E-compatible reduction ordering on terms with the subterm property and is E-total on ground terms. Permutations are commonly denoted in lexicographic or transposition order. Given a finite set of permutation equations E, we present a new RPO-based ordering modulo E using (permutation) group actions and their associated orbits. Any permutation is also a product of transpositions. Rewriting modulo equations has been researched for several decades but due to the lack of suitable orderings, there are some limitations to rewriting modulo permutation equations.
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