With the potential arrival of quantum computers, communication protocols are now being updated at a fast pace to be secure even against attackers with access to such a computer. A core issue is that we need to update the existing tools used to verify security protocols, as classical security proofs do not always carry over to quantum attackers. In this work, we prove the post-quantum soundness of the CryptoVerif prover, a tool used to semi-automatically obtain computational security guarantees over cryptographic constructions. It required an update of the whole semantics in order to define it for any black-box interactive attacker and not just probabilistic Turing machines. We also had to validate the soundness of all its proof techniques, the so-called game transformations. We used this new post-quantum sound CryptoVerif to obtain the first formal security guarantees over two IETF draft proposals designing post-quantum variants of SSH and TLS.