The Maximal Extractible Value (MEV) is a central notion in smart contracts for decentralized finance. When an adversary proposes a new block of transactions, they can choose what to include in the block, and in which order. This privilege can be sometimes exploited to obtain a cryptocurrency gain: the MEV.

We formalize MEV in Lean 4. We provide a general framework providing the main definitions and related theorems, which can be applied to arbitrary smart contracts. We then put our tools to use, and establish the MEV of a few use cases.

This helper library defines a type for positive reals and related results. It is akin to the NNReal library in Mathlib. Most results in this library are taken from this repository.

• Basic.lean

  Contains coercions to NNReal and Real, and related properties.

• Order.lean

  Contains basic propositions about inequalities of PReal values.

• Sqrt.lean

  Defines the square root of a PReal value, and a few related propositions.

• SqrtCalc.lean

  Contains ad hoc lemmas that simplify MEV calculations for AMM contracts.

• Subtraction.lean

  Defines subtraction for PReal values, and related propositions.

The core library.

• System.lean

  Defines a generic smart contract system, as well as MEV. Also contains our main theorem, MEV.characterization which allows us to establish the MEV of a contract.

• Airdrop.lean

  Contains the simplest example of application of our theory, an Airdrop contract.

• CoinPusher.lean

  Contains an implementation of a coin pushing game, and establishes its MEV when the blockchain mempool is empty, or when it has a single transaction.

• AMM.Lean

  Defines our main example contract, a constant product Automated Market Maker.

• AMMEmptyMempool.lean

  Establishes the MEV of an AMM with an empty mempool.

• AMMOneTxMempool.lean

  Establishes the MEV of an AMM whose mempool has a single transaction.