Formalizing 100 Theorems
There used to exist a "top 100" of mathematical theorems on the web,
which is a rather arbitrary list (and most of the theorems seem rather
elementary), but still is nice to look at.
On the current page I will keep track of which theorems from this list
have been formalized.
Currently the fraction that already has been formalized seems to be
99%
The page does not keep track of all formalizations of these theorems.
It just shows formalizations in systems that have formalized a significant number of theorems,
or that have formalized a theorem that none of the others have done.
The systems that this page refers to are (in order of the number of
theorems that have been formalized, so the more
interesting systems for mathematics are near the top):
Theorems in the list which have not been formalized yet
are in italics.
Formalizations of constructive proofs are in italics too.
The difficult proofs in the list (according to John all the others
are not a serious challenge "given a week or two") have been underlined.
The formalizations under a theorem are in the order of the list of systems,
and not in chronological order.
The List
- The Irrationality of the Square Root of 2
- Isabelle, Markus Wenzel: statement
- HOL Light, John Harrison: statement
- Coq, contrib, many versions: statement
- Metamath, Norman Megill: statement
- Lean, mathlib: statement
- Mizar, Freek Wiedijk: statement
- ProofPower, Rob Arthan: statement
- PVS, Bart Jacobs & John Rushby
- ACL2, Ruben Gamboa
- NuPRL, Stuart Allen, Paul Jackson
- Fundamental Theorem of Algebra
- The Denumerability of the Rational Numbers
- Isabelle, Stefan Richter & Benjamin Porter: statement
- HOL Light, John Harrison: statement
- Coq, contrib, Yves Bertot & Milad Niqui: statement
- Metamath, revised by Mario Carneiro: statement
- Lean, Chris Hughes: statement
- Mizar, Grzegorz Bancerek: statement
- ProofPower, Rob Arthan: statement
- PVS, Jerry James
- NuPRL, Stuart Allen
- Pythagorean Theorem
- Isabelle, Amine Chaieb: statement
- HOL Light, John Harrison: statement
- Coq, contrib, Frédérique Guilhot: statement
- Metamath, Norman Megill: statement
- Lean, Joseph Myers: statement
- Mizar, Akihiro Kubo & Yatsuka Nakamura: statement
- ProofPower, Rob Arthan: statement
- PVS, NASA library, David Lester
- Prime Number Theorem
- Gödel's Incompleteness Theorem
- Isabelle, Larry Paulson: statement
- HOL Light, John Harrison: statement
- Coq, contrib, Russell O'Connor: statement
- nqthm, Natarajan Shankar
- Law of Quadratic Reciprocity
- The Impossibility of Trisecting the Angle and Doubling the Cube
- The Area of a Circle
- Euler's Generalization of Fermat's Little Theorem
- The Infinitude of Primes
- Isabelle, Jacques D. Fleuriot: statement
- HOL Light, John Harrison: statement
- Coq, not in contribs, Russell O'Connor: statement
- Metamath, Norman Megill: statement
- Lean, Jeremy Avigad: statement
- Mizar, Rafal Kwiatek: statement
- ProofPower, Rob Arthan: statement
- PVS, NASA library, Ricky Butler
- ACL2, David Russinoff
- The Independence of the Parallel Postulate
- Polyhedron Formula
- Euler's Summation of 1 + (1/2)^2 + (1/3)^2 + ....
- Fundamental Theorem of Integral Calculus
- Isabelle, Jacques D. Fleuriot: statement
- HOL Light, John Harrison: statement
- Coq, C-CoRN, Luís Cruz-Filipe: statement
- Metamath, Mario Carneiro: statement
- Lean, Yury G. Kudryashov (first) and Benjamin Davidson (second): statement
- Mizar, Noboru Endou & Katsumi Wasaki & Yasunari Shidama: statement
- ProofPower, Rob Arthan: statement
- PVS, NASA library, Ricky Butler
- ACL2, Matt Kaufmann
- Insolvability of General Higher Degree Equations
- Coq, Sophie Bernard & Cyril Cohen & Assia Mahboubi & Pierre-Yves Strub, statement
- Lean, Thomas Browning: statement
- De Moivre's Theorem
- Liouville's Theorem and the Construction of Transcendental Numbers
- Four Squares Theorem
- All Primes (1 mod 4) Equal the Sum of Two Squares
- Green's Theorem
- Isabelle, Mohammad Abdulaziz & Larry Paulson: statement
- The Non-Denumerability of the Continuum
- Formula for Pythagorean Triples
- The Undecidability of the Continuum Hypothesis
- Isabelle, Emmanuel Gunther, Miguel Pagano, Pedro Sánchez Terraf & Matías Steinberg: statement
- Lean, Jesse Han & Floris van Doorn: statement
- Schroeder-Bernstein Theorem
- Leibniz's Series for Pi
- Sum of the Angles of a Triangle
- Pascal's Hexagon Theorem
- Feuerbach's Theorem
- The Ballot Problem
- Ramsey's Theorem
- Isabelle, Tom Ridge: statement
- HOL Light, John Harrison: statement
- Coq, Frédéric Blanqui: statement
- Metamath, Mario Carneiro: statement
- Lean, Bhavik Mehta: statement
- Mizar, Marco Riccardi: statement
- ProofPower, Rob Arthan & Roger Bishop Jones: statement
- PVS, NASA library, Natarajan Shankar
- nqthm, Matt Kaufmann
- NuPRL, David Basin
- The Four Color Problem
- Coq, not in contribs, Georges Gonthier: statement
- Fermat's Last Theorem
- Divergence of the Harmonic Series
- Taylor's Theorem
- Isabelle, Jacques D. Fleuriot & Lukas Bulwahn & Bernhard Häupler: statement
- HOL Light, John Harrison: statement
- Coq, C-CoRN, Luís Cruz-Filipe: statement
- Metamath, Mario Carneiro: statement
- Lean, Moritz Doll: statement
- Mizar, Yasunari Shidama: statement
- ProofPower, Rob Arthan: statement
- PVS, NASA library, Ricky Butler
- ACL2, Ruben Gamboa & Brittany Middleton & Jun Sawada
- Brouwer Fixed Point Theorem
- The Solution of a Cubic
- Arithmetic Mean/Geometric Mean
- Isabelle, Benjamin Porter: statement
- HOL Light, John Harrison: statement
- Coq, not in contribs, Jean-Marie Madiot: statement
- Metamath, Mario Carneiro: statement
- Lean, Yury G. Kudryashov: statement
- Mizar, Adam Grabowski: statement
- ProofPower, Rob Arthan: statement
- ACL2, Matt Kaufmann, Paolo Pecchiari
- Solutions to Pell's Equation
- Minkowski's Fundamental Theorem
- Puiseux's Theorem
- Isabelle, Manuel Eberl: statement
- Coq, not yet in contrib, Daniel de Rauglaudre: statement
- Sum of the Reciprocals of the Triangular Numbers
- The Isoperimetric Theorem
- The Binomial Theorem
- Isabelle, Tobias Nipkow: statement
- HOL Light, John Harrison: statement
- Coq, contrib, Laurent Théry: statement
- Metamath, Norman Megill: statement
- Lean, Chris Hughes: statement
- Mizar, Christoph Schwarzweller: statement
- ProofPower, Rob Arthan: statement
- ACL2, Ruben Gamboa
- NuPRL, Paul Jackson & Rich Eaton
- The Partition Theorem
- The Solution of the General Quartic Equation
- The Central Limit Theorem
- Dirichlet's Theorem
- The Cayley-Hamilton Theorem
- Isabelle, Stephan Adelsberger, Stefan Hetzl & Florian Pollak: statement
- HOL Light, John Harrison: statement
- Coq, contrib, Sidi Ould Biha: statement
- Metamath, Alexander van der Vekens: statement
- Lean, Scott Morrison: statement
- The Number of Platonic Solids
- Wilson's Theorem
- The Number of Subsets of a Set
- Pi is Transcendental
- Konigsberg Bridges Problem
- Product of Segments of Chords
- The Hermite-Lindemann Transcendence Theorem
- Heron's Formula
- Formula for the Number of Combinations
- The Laws of Large Numbers
- Bezout's Theorem
- Theorem of Ceva
- Fair Games Theorem
- Cantor's Theorem
- L'Hôpital's Rule
- Isosceles Triangle Theorem
- Sum of a Geometric Series
- Isabelle, Jacques D. Fleuriot: statements
- HOL Light, John Harrison: statements
- Coq, C-CoRN, Luís Cruz-Filipe: statement
- Metamath, Norman Megill: statement
- Lean, Sander R. Dahmen (finite) and Johannes Hölzl (infinite): statement
- Mizar, Konrad Raczkowski & Andrzej Nedzusiak: statement
- ProofPower, Rob Arthan: statement
- PVS, NASA library, Ricky Butler
- ACL2, Ruben Gamboa
- e is Transcendental
- Sum of an arithmetic series
- Isabelle, Benjamin Porter: statement
- HOL Light, John Harrison: statement
- Coq, not in contribs, many versions: statement
- Metamath, Frédéric Liné: statement
- Lean, Johannes Hölzl: statement
- Mizar, Ming Liang & Yuzhong Ding: statement
- ProofPower, Rob Arthan: statement
- PVS, NASA library, Tanner Slagel
- ACL2, Ruben Gamboa
- Greatest Common Divisor Algorithm
- The Perfect Number Theorem
- Order of a Subgroup
- Isabelle, Florian Kammüller: statement
- HOL Light, John Harrison: statement
- Coq, almost C-CoRN, Dan Synek & contrib, Laurent Théry: statement
- Metamath, Mario Carneiro: statement
- Lean, mathlib: statement
- Mizar, Wojciech Trybulec: statement
- ProofPower, Rob Arthan: statement
- PVS, NASA library, David Lester
- Sylow's Theorem
- Ascending or Descending Sequences
- The Principle of Mathematical Induction
- The Mean Value Theorem
- Isabelle, Jacques D. Fleuriot: statement
- HOL Light, John Harrison: statement
- Coq, C-CoRN, Luís Cruz-Filipe: statement
- Metamath, Mario Carneiro: statement
- Lean, Yury G. Kudryashov: statement
- Mizar, Jaroslaw Kotowicz & Konrad Raczkowski & Pawel Sadowski: statement
- ProofPower, Rob Arthan: statement
- PVS, NASA library, Bruno Dutertre
- ACL2, Ruben Gamboa
- Fourier Series
- Sum of kth powers
- The Cauchy-Schwarz Inequality
- The Intermediate Value Theorem
- Isabelle, Jacques D. Fleuriot: statement
- HOL Light, John Harrison: statement
- Coq, C-CoRN, Herman Geuvers et al.: statement
- Metamath, Paul Chapman: statement
- Lean, mathlib: statement
- Mizar, Yatsuka Nakamura & Andrzej Trybulec: statement
- ProofPower, Rob Arthan: statement
- PVS, NASA library, Bruno Dutertre
- ACL2, Ruben Gamboa
- The Fundamental Theorem of Arithmetic
- Isabelle, Thomas M. Rasmussen: statement
- HOL Light, John Harrison: statement
- Coq, contrib, Martijn Oostdijk: statement
- Metamath, Paul Chapman: statement
- Lean, mathlib: statement
- Mizar, Artur Korniłowicz, Piotr Rudnicki & Hiroyuki Okazaki, Yasunari Shidama: statements
- ProofPower, Rob Arthan: statement
- PVS, NASA library, Ricky Butler
- ACL2, John Cowles
- NuPRL, Stuart Allen & Paul Jackson
- Divergence of the Prime Reciprocal Series
- Dissection of Cubes (J.E. Littlewood's "elegant" proof)
- The Friendship Theorem
- Morley's Theorem
- Divisibility by 3 Rule
- Lebesgue Measure and Integration
- Isabelle, Stefan Richter: statement
- HOL Light, John Harrison: statement
- Coq, Sylvie Boldo, François Clément, Florian Faissole, Vincent Martin & Micaela Mayero, and Reynald Affeldt & Cyril Cohen: statement
- Metamath, Mario Carneiro: statement
- Lean, Johannes Hölzl: statement
- Mizar, Noboru Endou & Yasunari Shidama: statement
- Desargues's Theorem
- Derangements Formula
- The Factor and Remainder Theorems
- Stirling's Formula
- Isabelle, Manuel Eberl: statements
- HOL Light, John Harrison: statement
- Coq, not in contribs, Coqtail team: statement
- Metamath, Glauco Siliprandi: statement
- Lean, mathlib (Moritz Firsching, Fabian Kruse, Nikolas Kuhn, Heather Macbeth): statement
- The Triangle Inequality
- Isabelle, Steven Obua: statement
- HOL Light, John Harrison: statement
- Coq, Frédérique Guilhot: statement
- Metamath, Norman Megill: statement
- Lean, Zhouhang Zhou: statement
- Mizar, Czeslaw Bylinski: statement
- ProofPower, Rob Arthan: statement
- PVS, NASA library, Ricky Butler & Cesar Munoz
- ACL2, Ruben Gamboa
- Pick's Theorem
- The Birthday Problem
- The Law of Cosines
- Ptolemy's Theorem
- Principle of Inclusion/Exclusion
- Cramer's Rule
- Bertrand's Postulate
- Buffon Needle Problem
- Descartes Rule of Signs
This list (which I did not make!) has some obvious omissions.
I will list here, in alphabetical order, the theorems that people have
complained to me are missing.
(Some of them might just be missing because they are "too new",
as the list is from 1999.)
- Atiyah-Singer Index Theorem
- Baker's Theorem on Linear Forms in Logarithms
- Black-Scholes Formula
- Borsuk-Ulam Theorem
- Cantor-Bendixson theorem
- Cauchy's Integral Theorem
- Cauchy's Residue Theorem
- Chen's theorem
- every vector space is free
- Classification of Finite Simple Groups
- Classification of semisimple Lie groups (Killing, Cartan, Dynkin)
- Sophie Germain's theorem
- Gödel's Completeness Theorem
- Gödel's Second Incompleteness Theorem
- Green-Tao Theorem
- Feit-Thompson Theorem
- Fundamental Theorem of Galois Theory
- Heine-Borel Theorem
- Hessenberg's Theorem = "Pappus → Desargues"
- Hilbert Basis Theorem
- Hilbert Nullstellensatz
- Hilbert-Waring theorem
- Invariance of Dimension
- IP=PSPACE
- Jordan Curve Theorem
- Kepler Conjecture
- Lie's work relating Algebras and Groups
- Nash's Theorem
- Perelman-Hamilton-Thurston theorem classifying compact 3-manifolds
- Poincaré Conjecture
- Riemann Mapping Theorem
- sorting takes Θ(N log N) steps
- Stokes' Theorem
- Stone-Weierstrass Theorem
- Thales' Theorem
- Thue-Siegel-Roth theorem about approximating algebraic numbers by rationals
- Yoneda lemma
- ζ(-1) = -1 ⁄ 12
- existence of ZF+DC+IHC model in which all real subsets are Lebesque measurable, have the perfect set property, and are Baire
(last modification 2022-12-14)