Formal Verification

From Axioms to Algorithms: Mechanized Proofs of the vNM Utility Theorem ArXiv ID: 2506.07066 “View on arXiv” Authors: Li Jingyuan Abstract This paper presents a comprehensive formalization of the von Neumann-Morgenstern (vNM) expected utility theorem using the Lean 4 interactive theorem prover. We implement the classical axioms of preference-completeness, transitivity, continuity, and independence-enabling machine-verified proofs of both the existence and uniqueness of utility representations. Our formalization captures the mathematical structure of preference relations over lotteries, verifying that preferences satisfying the vNM axioms can be represented by expected utility maximization. Our contributions include a granular implementation of the independence axiom, formally verified proofs of fundamental claims about mixture lotteries, constructive demonstrations of utility existence, and computational experiments validating the results. We prove equivalence to classical presentations while offering greater precision at decision boundaries. This formalization provides a rigorous foundation for applications in economic modeling, AI alignment, and management decision systems, bridging the gap between theoretical decision theory and computational implementation. ...

Geometric Formalization of First-Order Stochastic Dominance in $N$ Dimensions: A Tractable Path to Multi-Dimensional Economic Decision Analysis ArXiv ID: 2505.12840 “View on arXiv” Authors: Jingyuan Li Abstract This paper introduces and formally verifies a novel geometric framework for first-order stochastic dominance (FSD) in $N$ dimensions using the Lean 4 theorem prover. Traditional analytical approaches to multi-dimensional stochastic dominance rely heavily on complex measure theory and multivariate calculus, creating significant barriers to formalization in proof assistants. Our geometric approach characterizes $N$-dimensional FSD through direct comparison of survival probabilities in upper-right orthants, bypassing the need for complex integration theory. We formalize key definitions and prove the equivalence between traditional FSD requirements and our geometric characterization. This approach achieves a more tractable and intuitive path to formal verification while maintaining mathematical rigor. We demonstrate how this framework directly enables formal analysis of multi-dimensional economic problems in portfolio selection, risk management, and welfare analysis. The work establishes a foundation for further development of verified decision-making tools in economics and finance, particularly for high-stakes domains requiring rigorous guarantees. ...

Efficient and Verified Continuous Double Auctions ArXiv ID: 2412.08624 “View on arXiv” Authors: Unknown Abstract Continuous double auctions are commonly used to match orders at currency, stock, and commodities exchanges. A verified implementation of continuous double auctions is a useful tool for market regulators as they give rise to automated checkers that are guaranteed to detect errors in the trade logs of an existing exchange if they contain trades that violate the matching rules. We provide an efficient and formally verified implementation of continuous double auctions that takes $O(n \log n)$ time to match $n$ orders. This improves an earlier $O(n^2)$ verified implementation. We also prove a matching $Ω(n\log n)$ lower bound on the running time for continuous double auctions. Our new implementation takes only a couple of minutes to run on ten million randomly generated orders as opposed to a few days taken by the earlier implementation. Our new implementation gives rise to an efficient automatic checker. We use the Coq proof assistant for verifying our implementation and extracting a verified OCaml program. While using Coq’s standard library implementation of red-black trees to obtain our improvement, we observed that its specification has serious gaps, which we fill in this work; this might be of independent interest. ...

Double Auctions: Formalization and Automated Checkers ArXiv ID: 2410.18751 “View on arXiv” Authors: Unknown Abstract Double auctions are widely used in financial markets, such as those for stocks, derivatives, currencies, and commodities, to match demand and supply. Once all buyers and sellers have placed their trade requests, the exchange determines how these requests are to be matched. The two most common objectives for determining the matching are maximizing trade volume at a uniform price and maximizing trade volume through dynamic pricing. Prior research has primarily focused on single-quantity trade requests. In this work, we extend the framework to handle multiple-quantity trade requests and present fully formalized matching algorithms for double auctions, along with their correctness proofs. We establish new uniqueness theorems, enabling automatic detection of violations in exchange systems by comparing their output to that of a verified program. All proofs are formalized in the Coq Proof Assistant, and we extract verified OCaml and Haskell programs that could serve as a resource for exchanges and market regulators. We demonstrate the practical applicability of our work by running the verified program on real market data from an exchange to automatically check for violations in the exchange algorithm. ...