Tech • IA • Crypto
X-UPS Days 2026: "Formal Verification of Proofs" – Micaela Mayero 1
7/10TL;DR
Proof assistants such as Coq/Rocq are emerging tools that use formal logic and type theory to mathematically verify programs and theorems, with growing applications in industry and research.
KEY POINTS
From ancient logic to modern verification
Formal reasoning traces back to Aristotle and early mathematical practices, but the systematic formalization of logic accelerated between the 17th and 20th centuries with figures like Leibniz, Hilbert, and Gödel. This long evolution culminated in the idea that reasoning itself could be encoded and checked mechanically. The development of computing in the 19th and 20th centuries, including Turing machines, enabled the automation of logical reasoning.
Birth of proof assistants in the late 20th century
Modern proof assistants emerged around the 1970s, grounded in mathematical logic and computer science. Systems such as Coq (recently renamed Rocq) rely on frameworks like Martin-Löf type theory and the Calculus of Inductive Constructions. These tools allow users to construct machine-checked proofs, ensuring correctness through formal verification rather than testing.
Two major applications: software safety and mathematics
Proof assistants are primarily used in two domains. First, they verify critical software systems, producing “zero-defect” guarantees by proving correctness mathematically. Second, they support the formalization of mathematics, including large-scale theorem verification and exploration of unresolved problems, increasingly intersecting with artificial intelligence research.
Real-world impact in infrastructure and industry
Formal methods have already been deployed in high-stakes environments. The automated Paris Metro Line 14 relied on formally verified software using the B-Method. Aerospace systems verified with tools like PVS, and certified compilers for languages such as C, demonstrate the industrial relevance. Failures in numerical software, such as those contributing to structural collapses, highlight the need for rigorous verification.
Logic foundations: classical vs intuitionistic
Proof assistants are built on different logical systems, notably classical logic and intuitionistic logic, historically debated by figures like Hilbert and Brouwer. Intuitionistic logic requires constructive proofs, meaning that existence claims must provide explicit examples. This property enables extraction of executable programs from proofs via the Curry–Howard correspondence.
From proofs to programs
A key innovation is the equivalence between proofs and programs, where proving a statement corresponds to constructing a program. For example, proving “A implies B” yields a function transforming proofs of A into proofs of B. This duality allows verified software to be directly generated from mathematical arguments.
Challenges in numerical reasoning
Representing real numbers and floating-point arithmetic in proof assistants remains complex. Real numbers are continuous and cannot be fully represented in machines, while floating-point numbers are approximations governed by standards like IEEE 754. Formalizing numerical analysis therefore requires careful handling of approximation errors and computational limits.
CONCLUSION
Proof assistants bridge mathematics and computing by transforming logical reasoning into verifiable programs, offering powerful tools for both scientific rigor and industrial reliability.