© 2019 IEEE. Blockchain is a prime example of disruptive technology in multiple levels. With the advent of blockchains becomes obsolete the need for a mutually trusted third party acting as intermediary between agents which do not necessarily trust each other in transactions of any kind, including political or shareholder voting, crowdfunding, financial deals, logistics and supply chain management, and contract formulation. An integral part of the blockchain stack is the proof system, namely the mechanism efficiently verifying the claims of various blockchain stakeholders. Thus, trust is effectively established in a literally trustless environment with purely computational means. This is especially critical in the digital formulation of smart contracts where clauses are to be strictly upheld by intelligent agents. The most prominent proof systems recently proposed in the scientific literature are reviewed. Additionally, the applications of blockchain technology to smart contracts is discussed. The latter allows clause re-negotiation, increasing thus the flexibility factor in transactions. As a concrete example, a simple smart contract written in Solidity, a high level language for the Ethereum Virtual Machine, is presented.