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The authors provide an automatic verification for a fragment of FOL quantifier-free logic with zero, successor and equality. They use BDD representation of such formulas and to verify them, they first introduce a (complete) term rewrite system to generate an equivalent Ordered (0, S,=)-BDD from any given (0, S,=)-BDD. Having the ordered representation of the BDDs, one can verify the original formula in constant time. Then, to have this transformation automatically, they provide an algorithm which will do the whole process.
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