Hybrid systems are a powerful formalism for modeling cyber-physical systems. Reachability analysis is a general method for checking safety properties, especially in the presence of uncertainty and non-determinism. Rigorous simulation is a convenient tool for reachability analysis of hybrid systems. However, to serve as proof tool, a rigorous simulator must be correct w.r.t. a clearly defined notion of reachability, which captures what is intuitively reachable in finite time. As a step towards addressing this challenge, this paper presents a rigorous simulator in the form of an operational semantics and a specification in the form of a denotational semantics. We show that, under certain conditions about the representation of enclosures, the rigorous simulator is correct. We also show that finding a representation satisfying these assumptions is non-trivial. © 2018, Springer International Publishing AG, part of Springer Nature.