Computable set theory
Introduction to set constraint-based program analysis
Science of Computer Programming
Simplification by Cooperating Decision Procedures
ACM Transactions on Programming Languages and Systems (TOPLAS)
Set theory for computing: from decision procedures to declarative programming with sets
Set theory for computing: from decision procedures to declarative programming with sets
Cooperation of Background Reasoners in Theory Reasoning by Residue Sharing
Journal of Automated Reasoning
Unions of non-disjoint theories and combinations of satisfiability procedures
Theoretical Computer Science
Set Constraints: Results, Applications, and Future Directions
PPCP '94 Proceedings of the Second International Workshop on Principles and Practice of Constraint Programming
FroCoS '02 Proceedings of the 4th International Workshop on Frontiers of Combining Systems
A Tableau Calculus for Integrating First-Order and Elementary Set Theory Reasoning
TABLEAUX '00 Proceedings of the International Conference on Automated Reasoning with Analytic Tableaux and Related Methods
Decision Problems for Tarski and Presburger Arithmetics Extended With Sets
CSL '90 Proceedings of the 4th Workshop on Computer Science Logic
Combining Multisets with Integers
CADE-18 Proceedings of the 18th International Conference on Automated Deduction
Towards Efficient Satisfiability Checking for Boolean Algebra with Presburger Arithmetic
CADE-21 Proceedings of the 21st international conference on Automated Deduction: Automated Deduction
Polynomial constraints for sets with cardinality bounds
FOSSACS'07 Proceedings of the 10th international conference on Foundations of software science and computational structures
PLDI '10 Proceedings of the 2010 ACM SIGPLAN conference on Programming language design and implementation
Theory-specific automated reasoning
A 25-year perspective on logic programming
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We introduce a quantifier-free set-theoretic language for combining sets with elements in the presence of the cardinality operator. We prove that the language is decidable by providing a combination method specifically tailored to the combination domain of sets, cardinal numbers, and elements. Our method uses as black boxes a decision procedure for the elements and a decision procedure for cardinal numbers. To be correct, our method requires that the theory of elements be stably infinite. However, we show that if we restrict set variables to range over finite sets only, then one can modify our method so that it works even when the theory of the elements is not stably infinite.