Lecture Notes on Differential Equations & Proofs

Lecture 10 on Differential Equations & Differential Invariants introduced equational differential invariants of the form e = 0 for differential equations that are much more general than the ones supported by axiom [′] from Lecture 5 on Dynamical Systems & Dynamic Axioms. Axiom [′] replaces properties of differential equations with universally quantified properties of solutions, but is limited to differential equations that have explicit closed-form solutions whose arithmetic can be handled (mostly polynomials or rational functions). But axiom [′] works for any arbitrary postcondition. The equational differential invariant proof rule dI=0 supports general differential equations, but was limited to equational postconditions of the form e = 0. The goal of this lecture is to generalize the differential invariant proof rules to work for more general postconditions but retaining the flexibility with the differential equations that differential invariants provide. Indeed, the principles developed in Lecture 10 generalize beautifully to logical formulas other than the limited form e = 0. This lecture will establish generalizations that make the differential invariant proof rule work for formulas F of more general forms. The most important part will be soundly defining the total derivative F ′, because the basic shape of the differential invariants proof rule stays the same: ` [x′ := f(x)](F )′ F ` [x′ = f(x)]F