Tech • IA • Crypto
X-UPS Days 2026: "Formal Proof Verification" – Micaela Mayero 2
5/10TL;DR
The formalization of mathematics using proof assistants is progressing, but still faces technical, conceptual, and time-related challenges, particularly around handling real numbers.
KEY POINTS
A recent and demanding discipline
Formal proofs, developed over about 50 years, aim for complete rigor, where every assumption is made explicit. Assistants like Coq, Lean, or HOL Light do not fully automate proofs: they support the user, who must still master the reasoning. Automated tools exist, notably SMT solvers, but remain limited to simple cases.
A diversity of tools and libraries
No prover has emerged as a universal standard. Each system has advantages depending on the use case, whether for pure mathematics or program verification. Massive libraries, such as that of Mizar, illustrate the scale of accumulated work. Efforts are underway to translate proofs between systems, despite deep logical differences.
The high cost of formalization
Formalizing a theorem can take months or even years. For example, some analysis components required several months of initial development, and full projects can span over 20 years. This slowness is due to the need to rebuild entire mathematical foundations before tackling advanced results.
The limits of floating-point numbers
In computing, real numbers are often approximated by floating-point numbers, which introduces errors. Simple operations can lose fundamental properties like associativity. For example, calculations involving 1000 or 7.5 may yield different results depending on the order of operations. This approximation is incompatible with the requirements of mathematical proofs.
The complexity of defining real numbers
Real numbers are not directly representable on a computer. Several approaches coexist: classical, constructive, exact reals, or approximations such as interval arithmetic. Each choice depends on the goal, much like selecting an appropriate data structure in programming.
Axiomatization and technical trade-offs
In Coq, real numbers are defined using 17 axioms. Some functions, like inversion, are made total, allowing expressions such as 1/0, while deferring validity constraints to later in the proof. This trade-off improves usability without compromising logical consistency.
Classical vs constructive
Two perspectives coexist: classical logic, with principles like the law of excluded middle, and intuitionistic logic, where some statements differ. For example, the intermediate value theorem has different formulations depending on the chosen framework, illustrating deep divergences.
Evolving libraries
The standard library of Coq, notably enriched by Coquelicot in 2013, introduced more modern tools and better handling of functions and limits. In 2020, a common base helped bridge classical and constructive approaches while preserving their specificities.
From teaching to advanced research
Initially developed for research, these tools are now used in education, including at the high school level. They offer a new perspective on mathematics by enforcing a level of detail rarely reached in traditional approaches.
Industrial and scientific applications
Proof assistants are used to verify complex scientific programs, particularly in C++. Work includes validating numerical methods such as finite elements, combining mathematical proof with analysis of floating-point computation errors.
An ongoing endeavor
Despite progress, many components remain to be formalized, such as Sobolev spaces or certain advanced integration methods. Ongoing projects show that building a complete mathematical library is a long-term effort.
CONCLUSION
The formalization of mathematics is advancing steadily but remains demanding, balancing absolute rigor with practical constraints, and finding growing applications in research, education, and complex system verification.