Differential temporal dynamic logic dTL2 is a logic to specify and verify temporal properties of hybrid systems. It extends differential dynamic logic (dL) with temporal operators that enable reasoning on intermediate states in both discrete and continuous dynamics. This paper presents an embedding of dTL2 in the Prototype Verification System (PVS). The embedding includes the formalization of a trace semantics as well as the logic and proof calculus of dTL, which have been enhanced to support the verification of universally quantified reachability properties. The embedding is fully functional and can be used to interactively verify hybrid programs in PVS using a combination of PVS proof commands and specialized proof strategies.

Semi-algebraic sets and real analytic functions are fundamental concepts in Real Algebraic Geometry and Real Analysis, respectively. These concepts appear in the study of Differential Equations, where the real analytic solution to a differential equation is known to enter or exit a semi-algebraic set in a predictable way. Motivated to enhance the capability to reason about differential equations in the Prototype Verification System (PVS), a formalization of multivariate polynomials, semi-algebraic sets, and real analytic functions is developed. The way that a real analytic function behaves in a neighborhood around a point where the function meets the boundary of a semi-algebraic set is described and verified. It is further shown that if the function is assumed to be smooth, a slightly weaker assumption than real analytic, the behavior around the boundary of a semi-algebraic set can be very different.