Differential Invariants & Proof Theory

Synopsis

This advanced chapter studies some meta-properties of differential equations proving. It investigates aspects of the proof theory of differential equations, i.e., the theory of proofs about differential equations. While the primary focus in this chapter is on their theoretical significance, it also provides insights into the practical questions of what types of differential invariants to search for under which circumstances. The primary tool is the proof-theoretical device of relative deductive power, i.e., the question of whether all properties provable with technique $\mathcal{A}$ are also provable with technique $\mathcal{B}$. These results leverage appropriate insights about properties of real arithmetic and of differential equations.