abstract
we prove the completeness of an axiomatization for differential equation invariants. first,
we show that the differential equation axioms in differential dynamic logic are complete for
all algebraic invariants. our proof exploits differential ghosts, which introduce additional variables that can be chosen to evolve freely along new differential equations. cleverly chosen
differential ghosts are the proof-theoretical counterpart of dark matter. they create new hypothetical state, whose relationship to the original state variables satisfies invariants that did not
exist before. the reflection of these new invariants in the original system enables its analysis.
we then show that extending the axiomatization with existence and uniqueness axioms
makes it complete for all local progress properties, and further extension with a real induction
axiom makes it complete for all real arithmetic invariants. this yields a parsimonious axiomatization, which serves as the logical foundation for reasoning about invariants of differential
equations. moreover, our approach is purely axiomatic, and so the axiomatization is suitable
for sound implementation in foundational theorem provers.
keywords: differential equation axiomatization, differential dynamic logic, differential ghosts