Formalizing Mathematics in Lean
•
0 j'aime•58 vues
"Formal verification has been used by computer scientists for decades to prevent software bugs. However, with a few exceptions, it has not been used by researchers working in most areas of mathematics (geometry, algebra, analysis, etc.). In this talk, we will discuss how this has changed in the past few years, and the possible implications to the future of mathematical research, teaching and communication. We will focus on the theorem prover Lean and its mathematical library mathlib, since this is currently the system most widely used by mathematicians. Lean is a functional programming language and interactive theorem prover based on dependent type theory, with proof irrelevance and non-cumulative universes. The mathlib library, open-source and designed as a basis for research level mathematics, is one of the largest collections of formalized mathematics. It allows classical reasoning, uses large- and small-scale automation, and is characterized by its decentralized nature with over 200 contributors, including both computer scientists and mathematicians."
Recommandé
Recommandé
Contenu connexe
Similaire à Formalizing Mathematics in Lean
Similaire à Formalizing Mathematics in Lean (20)
Plus de Facultad de Informática UCM
Plus de Facultad de Informática UCM (20)
Dernier
Dernier (20)
Formalizing Mathematics in Lean
- 1. Conferencias de Posgrado UCM Formalizing Mathematics in Lean María Inés de Frutos Fernández Formalizing Mathematicsin Lean 08/11/2022
- 2. Formalizing Mathematicsin Lean 08/11/2022 What? • Formal Mathematics Why? • Research • Teaching How? • Lean • Mathlib
- 4. Formal verification Software Hardware Mathematics Formalizing Mathematicsin Lean 08/11/2022
- 5. Formalizing Mathematicsin Lean 08/11/2022 Computation • Used for decades • Not what this talk is about Reasoning • Check proofs • Finding proofs/counterexamples? Computers in Mathematics
- 6. History Formalizing Mathematicsin Lean 08/11/2022 G. Gonthier (2005), A computer-checked proof of the four colour theorem. T. Hales et. al. (2017), A formal proof of the Kepler conjecture.
- 7. Formalizing perfectoid spaces • K. Buzzard, J. Commelin, and P. Massot (2020) "Formalising perfectoid spaces". https://doi.org/10.1145/3372885.33 73830 • More than 3000 definitions or statements used in this definition Formalizing Mathematicsin Lean 08/11/2022
- 8. The Liquid Tensor Experiment Formalizing Mathematicsin Lean 08/11/2022 2019 Condensed Mathematics (Clausen, Scholze) Dec. 2020 Liquid Tensor Experiment challenge (Scholze) June 2021 Key technical lemma formalized (Commelin et. al.) July 2022 Liquid Tensor Experiment completed (Commelin, Topaz et. al.)
- 12. Checking proofs Big proofs Technical parts Consistency at all scales Formalizing Mathematicsin Lean 08/11/2022
- 13. Creating mathematics Routine steps Assumptions/ dependencies Better understanding AI Formalizing Mathematicsin Lean 08/11/2022
- 15. Semantic Search Engines for Mathematics Formalizing Mathematicsin Lean 08/11/2022
- 16. Teaching Pros • Proof understanding • Precision • Immediate feedback Cons • Barrier to entry • Lack of graphical inferface/documentation • Tradition Formalizing Mathematicsin Lean 08/11/2022
- 17. A new kind of mathematical document Formalizing Mathematicsin Lean 08/11/2022
- 18. Formalizing Mathematicsin Lean 08/11/2022 Lean and mathlib
- 19. Formalizing Mathematicsin Lean 08/11/2022 Theorem provers • Coq • Isabelle/HOL • HOL Light • Agda • Metamath • Mizar • Lean • ...
- 20. Lean 2013 Lean 2017 Lean 3 mathlib 2022 Lean 4 Formalizing Mathematicsin Lean 08/11/2022 Interactive Theorem Prover • Dependent type theory • Proof irrelevance Microsoft Research • Leonardo de Moura
- 21. Dependent type theory Dependent type theory • Prop, Type, Type 1, ... , Type* • t : T Notation • λ x, f x • π, ∞ , ⊗, ∀, ∃, . . . Formalizing Mathematicsin Lean 08/11/2022
- 22. Formalizing Mathematicsin Lean 08/11/2022 Type class inference • We can declare type classes and instances. • Variables can be: (x) explicit {x} implicit [x] type class inference
- 23. Tactics Basic • intro • apply • rw • simp • ... Math-specific • continuity • linarith • ring • polyrith • ... Search • library_search • suggest • hint Formalizing Mathematicsin Lean 08/11/2022
- 28. Formalizing Mathematicsin Lean 08/11/2022 Why Lean? mathlib Unified library State of the art projects LTE Sphere eversion FLT ...
- 29. mathlib • Open source • Decentralized • Monolithic • Overview Formalizing Mathematicsin Lean 08/11/2022
- 30. Formalizing Mathematicsin Lean 08/11/2022 Mathlib's dependency graph: • By Eric Wieser • Interactive version
- 31. Lean projects LTE Sphere Eversion FLT ... Formalizing Mathematicsin Lean 08/11/2022
- 32. • Led by Patrick Massot • Turn a sphere inside-out in 3D space • Differential topology • Blueprint Sphere eversion Formalizing Mathematicsin Lean 08/11/2022
- 33. Formalizing Number Theory: guiding goals Formalizing Mathematicsin Lean 08/11/2022 Fermat's Last Theorem "If xn + yn = zn for n ≥ 3, then xyz = 0." • Formulated around 1637. • Proven by Wiles and Taylor in 1995. • Proof uses elliptic curves, modular forms, Galois representations, class field theory... The Langlands Program • Deep conjectures relating algebra and analysis, number theory and geometry. • Largest research program in modern mathematics.
- 34. My Contributions Formalizing Mathematicsin Lean 08/11/2022 Formalized • Adèles and idèles of global fields. • Stating Global Class Field Theory. • Extensions of norms. • The p-adic complex numbers. Ongoing • Local Class Field Theory (with Filippo Nuccio). • Divided powers (with Antoine Chambert-Loir). • Fontaine's period rings.
- 35. Other Number Theory projects Formalizing Mathematicsin Lean 08/11/2022 Formalized • p-adic numbers (R. Lewis). • Witt vectors (J. Commelin, R.Lewis). • Elliptic curves (K. Buzzard). • Modular forms (C. Birkbeck). • Galois cohomology (A. Livingston). Ongoing • FLT for regular primes (led by R. Brasca). • Iwasawa Theory (A. Narayanan). • Modularity Conjecture (K. Buzzard, M. Karatarakis).
- 36. Formalizing Mathematicsin Lean 08/11/2022 Thank you! Questions?