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Formal verification of real-time systems is attractive because these systems often perform critical operations. Unlike non real-time systems, latency and response time guarantees are of critical importance in this setting, as much as functional correctness. Nevertheless, formal verification of real-time OSes usually stops the scheduling analysis at the policy level: they only prove that the scheduler (or its abstract model) satisfies some scheduling policy. In this paper, we go further and connect together Prosa, a verified schedulability analyzer, and RT-CertiKOS, a verified single-core sequential real-time OS kernel. Thus, we get a more general and extensible schedulability analysis proof for RT-CertiKOS, as well a concrete implementation validating Prosa models. It also showcases that it is realistic to connect two completely independent formal developments in a proof assistant. 

The reliability and security of safety-critical real-time systems are of utmost importance because the failure of these systems could incur severe consequences (e.g., loss of lives or failure of a mission). Such properties require strong isolation between components and they rely on enforcement mechanisms provided by the underlying operating system (OS) kernel. In addition to spatial isolation which is commonly provided by OS kernels to various extents, it also requires temporal isolation, that is, properties on the schedule of one component (e.g., schedulability) are independent of behaviors of other components. The strict isolation between components relies critically on algorithmic properties of theconcrete implementationof the scheduler, such as timely provision of time slots, obliviousness to preemption, etc. However, existing work either only reasons about an abstract model of the scheduler, or proves properties of the scheduler implementation that are not rich enough to establish the isolation between different components. In this paper, we present a novel compositional framework for reasoning about algorithmic properties of the concrete implementation of preemptive schedulers. In particular, we usevirtual timeline, a variant of the supply bound function used in real-time scheduling analysis, to specify and reason about the scheduling of each component in isolation. We show that the properties proved on this abstraction carry down to the generated assembly code of the OS kernel. Using this framework, we successfully verify a real-time OS kernel, which extends mCertiKOS, a single-processor non-preemptive kernel, with user-level preemption, a verified timer interrupt handler, and a verified real-time scheduler. We prove that in the absence of microarchitectural-level timing channels, this new kernel enjoys temporal and spatial isolation on top of the functional correctness guarantee. All the proofs are implemented in the Coq proof assistant.