A Real-Analytic Approach to Differential-Algebraic Dynamic Logic

This paper introduces a proof calculus for real-analytic differential-algebraic dynamic logic, enabling correct transformations of differential-algebraic equations. Applications include index reductions from differential-algebraic equations to ordinary differential equations. The calculus ensures compatibility between differential-algebraic equation proof principles and (differential-form) differential dynamic logic for hybrid systems. One key contribution is ghost switching which establishes precise conditions that decompose multi-modal systems into hybrid systems, thereby correctly hybridizing sophisticated differential-algebraic dynamics. The calculus is demonstrated in a proof of equivalence for a Euclidean pendulum to index reduced form.