Simplification by Cooperating Decision Procedures
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Theorem Proving with the Real Numbers
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Given a theory T, a set of equations E, and a single equation e, the uniform word problem (UWP) is to determine if E ⇒ e in the theory T. We recall the classic Nelson-Oppen combination result for solving the UWP over combinations of theories and then present a constructive version of this result for equational theories. We present three applications of this constructive variant. First, we use it on the pure theory of equality (TEQ) and arrive at an algorithm for computing congruence closure of a set of ground term equations. Second, we use it on the theory of associativity and commutativity (TAC) and obtain a procedure for computing congruence closure modulo AC. Finally, we use it on the combination theory TEQ ∪ TAC ∪ TPR, where TPR is the theory of polynomial rings, to present a decision procedure for solving the UWP for this combination.