Tech • IA • Crypto
X-UPS Days 2026: "Formal Verification of Proofs" – Sébastien Gouzel 2
6/10TL;DR
An open-source formal mathematics library is expanding rapidly while grappling with how much generality and structure to impose to keep proofs consistent and usable.
KEY POINTS
Rise of formalized mathematics
Proof assistants such as Lean, Coq, and Isabelle are driving the development of large formal math libraries. These systems require every theorem and definition to be encoded precisely, forcing mathematicians to rely heavily on existing formalized results due to the sheer scale of modern mathematics.
Mathlib as a collaborative project
The library Mathlib stands out as a fully open-source initiative built by volunteers. Contributions are reviewed under strict coding and structural standards, making participation possible but demanding. Its growth has been incremental, with users adding missing theorems and expanding coverage over time.
Rapid expansion of mathematical coverage
Initially sparse, Mathlib now includes substantial material across algebra, analysis, topology, and probability. It covers results such as the Cayley–Hamilton theorem, matrix exponentials, and elements of group and measure theory, with gaps steadily shrinking as contributors extend the library.
Tension between generality and usability
A central challenge is choosing the right level of abstraction. For example, instead of focusing on specific cases like 3×3 matrices, the library prioritizes general notions such as linear maps over arbitrary structures. While this increases coherence and reuse, it can make simple computations more complex for users.
A deliberately “Bourbaki-style” approach
Mathlib adopts a highly abstract and unified style inspired by Bourbaki, aiming to define concepts once at maximum generality and reuse them everywhere. This allows different areas of mathematics to interoperate seamlessly but requires careful design decisions about definitions and structures.
Designing definitions for flexibility
Definitions are crafted to support broad applicability. For instance, derivatives are defined even for non-differentiable functions by assigning a default value, typically zero. This unconventional choice simplifies theorem statements and avoids repeatedly checking technical conditions.
Handling edge cases pragmatically
Similar conventions apply to operations like division by zero or integrals of partially defined functions. By assigning default values, the system mirrors how mathematicians informally work—writing expressions freely and verifying conditions only when necessary.
Non-uniqueness and domain issues
Allowing derivatives on restricted domains introduces complications such as non-uniqueness. Multiple linear approximations may coincide on a subset, making definitions less clean but necessary for broader applicability in analysis.
Stronger theorems with fewer assumptions
These design choices enable more general theorems. For example, a version of Rolle’s theorem can be stated without explicitly assuming differentiability everywhere, since problematic points are handled automatically within the framework.
Iterative development and refactoring
Definitions evolve over time. Concepts like Cⁿ functions have undergone multiple revisions over several years, reflecting practical obstacles encountered when proving new theorems. Proof assistants help by checking consistency after each change.
Limits of formalization
Formal systems impose rigidity that contrasts with the flexibility of traditional mathematical practice. While they ensure correctness, they can slow exploration, as informal reasoning often blends results from different frameworks without strict compatibility.
Need for diverse approaches
No single library can satisfy all use cases. While Mathlib emphasizes generality, other libraries may favor simplicity or domain-specific efficiency. Maintaining multiple approaches is seen as essential for both research and education.
CONCLUSION
The development of Mathlib highlights both the promise and the constraints of formalized mathematics, balancing rigor, generality, and usability in an evolving collaborative ecosystem.