We present a framework in Isabelle for verifying asymptotic time complexity of imperative programs. We build upon an extension of Imperative HOL and its separation logic to include running time. Our framework is able to handle advanced techniques for time complexity analysis, such as the use of the Akra-Bazzi theorem and amortized analysis. Various automation is built and incorporated into the auto2 prover to reason about separation logic with time credits, and to derive asymptotic behaviour of functions. As case studies, we verify the asymptotic time complexity (in addition to functional correctness) of imperative algorithms and data structures such as median of medians selection, Karatsuba's algorithm, and splay trees.
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