A formal system for manipulating terms over a signature by means of rules. A set R of rules (each a rewrite rule) creates an abstract reduction system →R on the algebra T(Σ,X) of all terms over signature Σ and variables X. Usually, the rules are a set E of equations that determine a reduction system →E using rewrites based on equational logic.
Let E be a set of equations such that, for each t=t′ ∈ E, the left-hand side t is not a variable. The pair (Σ,E) is called an equational TRS. The equations of E are used in derivations of terms where the reduction t →Et′ requires substitutions to be made in some equation e ∈ E and the left-hand side of e is replaced by the right-hand side of e in t to obtain t′.
The first set of properties of a term rewriting system (Σ,E) is now obtained from the properties of abstract reduction systems. The following are examples.