# Smart Proofs via Smart Contracts: Succinct and Informative Mathematical Derivations via Decentralized Markets

Sylvain Carré, Franck Gabriel, Clément Hongler, Gustavo Lacerda, Gloria Capano

Sylvain Carré, Franck Gabriel, Clément Hongler, Gustavo Lacerda, Gloria Capano

**5/2/21**Published in : arXiv:2102.03044

Modern mathematics is built on the idea that proofs should be translatable into formal proofs, whose validity is an objective question, decidable by a computer. Yet, in practice, proofs are informal and may omit many details. An agent considers a proof valid if they trust that it could be expanded into a machine-verifiable proof. A proof's validity can thus become a subjective matter and lead to a debate, which may be difficult to settle. Hence, while the concept of valid proof is well-defined, the process to establish validity is itself a complex multi-agent problem.

We introduce the SPRIG protocol. SPRIG allows agents to propose and verify succinct and informative proofs in a decentralized fashion; the trust is established by agents being able to request more details in the proof steps; debates, if they arise, must isolate details of proofs and, if they persist, go down to machine-level details, where they are automatically settled. A structure of bounties and stakes is set to incentivize agents to act in good faith.

We propose a game-theoretic discussion of SPRIG, showing how agents with various types of information interact, leading to a proof tree with an appropriate level of detail and to the invalidation of wrong proofs, and we discuss resilience against various attacks. We then analyze a simplified model, characterize its equilibria and compute the agents' level of trust.

SPRIG is designed to run as a smart contract on a blockchain platform. This allows anonymous agents to participate in the verification debate, and to contribute with their information. The smart contract mediates the interactions, settles debates, and guarantees that bounties and stakes are paid as specified.

SPRIG enables new applications, such as the issuance of bounties for open problems, and the creation of derivatives markets, allowing agents to inject more information pertaining to proofs.