Programming via Rewriting

Programming via rewriting is a part of the declarative programming which is illustrated by the languages: OBJ, Maude, CafeOBJ, CASL and so on. Programming via rewriting is very close to the equational logic. In the equational logic a set Γ of axioms is given. Axioms are Horn clauses, called also conditional equations. We are looking for all universal quantified equalities which are consequences of these axioms. Equational logic gives us a sound and complete set of deduction rules for these consequences: reflexivity, symmetry, transitivity, compatibility with operations and substitution. Programming via rewriting tries more: to find a proof for each consequence of these axioms. In a lot of cases such a proof may be found.