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We propose a new specification language for the proof-based approach to verification of graph programs by introducing μ-conditions as an alternative to existing formalisms which can express many non-local properties of interest. The contributions of this paper are the lifting of constructions from nested conditions to the new, more expressive conditions and a proof calculus for partial correctness relative to μ-conditions. Most importantly, we prove the correctness of a new construction to compute weakest preconditions with respect to finite graph programs.

Proving correctness of graph programs relative to recursively nested conditions