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Yes, things like mentioned in the post can be expressed and checked statically, as demonstrated by languages like Idris and ATS. ATS might be even more relevant as it's an imperative language too, it can get rather low-level (like talking about properties of C runtime functions) while proving required properties statically, and it includes a solver for certain amount of arithmetics so that you don't need to prove obvious mathematical identities to the compiler. http://www.ats-lang.org/
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