Is this a Lamport signature? One-time public-private key pairs are used. The next public-private key pair to be used, is recorded with a proof c = sha3(nextPublicKey).
No, Lamport signature is a particular one-time signature scheme prescribing a particular method of verifying signatures, generating key pairs, and generating signatures.  You could use that technique to turn a one-time signature scheme such as Lamport's into a tremendously awkward stateful many-time signature scheme.