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CDSAT (Conflict-Driven Satisfiability) is a paradigm for theory combination that works by coordinating theory modules to reason in the union of the theories in a conflict-driven manner. We generalize CDSAT to the case of nondisjoint theories by presenting a new CDSAT theory module for a theory of arrays with abstract length, which is an abstraction of the theory of arrays with length. The length function is a bridging function as it forces theories to share symbols, but the proposed abstraction limits the sharing to one predicate symbol. The CDSAT framework handles shared predicates with minimal changes, and the new module satisfies the CDSAT requirements, so that completeness is preserved.

Search-based satisfiability procedures try to build a model of the input formula by simultaneously proposing candidate models and deriving new formulae implied by the input.Conflict-drivenprocedures perform non-trivial inferences only when resolving conflicts between formulæ and assignments representing the candidate model. CDSAT (Conflict-Driven SATisfiability) is a method for conflict-driven reasoning inunions of theories. It combines inference systems for individual theories astheory moduleswithin a solver for the union of the theories. This article augments CDSAT with a more generallemma learningcapability and withproof generation. Furthermore, theory modules for several theories of practical interest are shown to fulfill the requirements forcompletenessandterminationof CDSAT. Proof generation is accomplished by aproof-carryingversion of the CDSAT transition system that producesproof objectsin memory accommodating multiple proof formats. Alternatively, one can apply to CDSAT theLCF approach to proofsfrom interactive theorem proving, by defining a kernel of reasoning primitives that guarantees the correctness by construction of CDSAT proofs.