See the papers "Soft Contract Verification" and "Soft Contract Verification for Higher-Order Stateful Programs" (Nguyễn et al.) In general, there's a lot of academic work on the topic coming from the Racket community.
In short: yes, some classes of contracts can be statically proven. Not all of them, and it depends on the language's semantics, so it's probably not easy to retrofit into C++. But there's work being done on unifying types and contracts, moving back and forth between static proof and dynamic enforcement, and it's all pretty interesting. Proving contracts statically eliminates runtime overhead; generating contracts from types makes it possible for a gradually typed language to be sound.
In short: yes, some classes of contracts can be statically proven. Not all of them, and it depends on the language's semantics, so it's probably not easy to retrofit into C++. But there's work being done on unifying types and contracts, moving back and forth between static proof and dynamic enforcement, and it's all pretty interesting. Proving contracts statically eliminates runtime overhead; generating contracts from types makes it possible for a gradually typed language to be sound.