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

90%

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):

HOL Light 86
Mizar 61
Isabelle 55
Coq 50
   C-CoRN    10
ProofPower 42
Metamath 38
nqthm/ACL2 18
PVS 16
NuPRL/MetaPRL 8

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

  1. The Irrationality of the Square Root of 2
    • HOL Light, John Harrison: statement
    • Mizar, Freek Wiedijk: statement
    • Isabelle, Markus Wenzel: statement
    • Coq, contrib, many versions: statement
    • ProofPower, Rob Arthan: statement
    • ACL2, Ruben Gamboa
    • Metamath, Norman Megill: statement
    • PVS, Bart Jacobs & John Rushby
    • NuPRL, Stuart Allen, Paul Jackson
  2. Fundamental Theorem of Algebra
  3. The Denumerability of the Rational Numbers
    • HOL Light, John Harrison: statement
    • Mizar, Grzegorz Bancerek: statement
    • Isabelle, Stefan Richter & Benjamin Porter: statement
    • Coq, contrib, Yves Bertot & Milad Niqui: statement
    • ProofPower, Rob Arthan: statement
    • Metamath, revised by Mario Carneiro: statement
    • NuPRL, Stuart Allen
  4. Pythagorean Theorem
  5. Prime Number Theorem
  6. Gödel's Incompleteness Theorem
    • HOL Light, John Harrison: statement
    • Isabelle, Larry Paulson: statement
    • Coq, contrib, Russell O'Connor: statement
    • nqthm, Natarajan Shankar
  7. Law of Quadratic Reciprocity
    • HOL Light, John Harrison: statement
    • Mizar, Li Yan & Xiquan Liang & Junjie Zhao: statement
    • Isabelle, Jeremy Avigad et al.: statement
    • nqthm, David Russinoff
  8. The Impossibility of Trisecting the Angle and Doubling the Cube
  9. The Area of a Circle
  10. Euler's Generalization of Fermat's Little Theorem
    • HOL Light, John Harrison: statement
    • Mizar, Yoshinori Fujisawa & Yasushi Fuwa & Hidetaka Shimizu: statement
    • Isabelle, Thomas M. Rasmussen, Amine Chaieb: statement
    • Coq, contrib, Laurent Théry: statement
    • Metamath, Mario Carneiro: statement
  11. The Infinitude of Primes
    • HOL Light, John Harrison: statement
    • Mizar, Rafal Kwiatek: statement
    • Isabelle, Jacques D. Fleuriot: statement
    • Coq, not in contribs, Russell O'Connor: statement
    • ProofPower, Rob Arthan: statement
    • ACL2, David Russinoff
    • Metamath, Norman Megill: statement
    • PVS, NASA library, Ricky Butler
  12. The Independence of the Parallel Postulate
  13. Polyhedron Formula
  14. Euler's Summation of 1 + (1/2)^2 + (1/3)^2 + ....
  15. Fundamental Theorem of Integral Calculus
    • HOL Light, John Harrison: statement
    • Mizar, Noboru Endou & Katsumi Wasaki & Yasunari Shidama: statement
    • Isabelle, Jacques D. Fleuriot: statement
    • Coq, C-CoRN, Luís Cruz-Filipe: statement
    • ProofPower, Rob Arthan: statement
    • Metamath, Mario Carneiro: statement
    • ACL2, Matt Kaufmann
    • PVS, NASA library, Ricky Butler
  16. Insolvability of General Higher Degree Equations
  17. De Moivre's Theorem
  18. Liouville's Theorem and the Construction of Transcendental Numbers
  19. Four Squares Theorem
  20. All Primes (1 mod 4) Equal the Sum of Two Squares
  21. Green's Theorem
  22. The Non-Denumerability of the Continuum
  23. Formula for Pythagorean Triples
  24. The Undecidability of the Continuum Hypothesis
  25. Schroeder-Bernstein Theorem
  26. Leibnitz's Series for Pi
    • HOL Light, John Harrison: statement
    • Isabelle, Johannes Hoelzl: statement
    • Coq, standard library & C-CoRN, Luís Cruz-Filipe: statement
    • PVS, NASA library, David Lester
  27. Sum of the Angles of a Triangle
    • HOL Light, John Harrison: statement
    • Mizar, Wenpai Chang, Yatsuka Nakamura & Piotr Rudnicki: statement
    • Coq, contrib, Frédérique Guilhot: statement
    • Metamath, Mario Carneiro: statement
  28. Pascal's Hexagon Theorem
  29. Feuerbach's Theorem
  30. The Ballot Problem
  31. Ramsey's Theorem
    • HOL Light, John Harrison: statement
    • Mizar, Marco Riccardi: statement
    • Isabelle, Tom Ridge: statement
    • ProofPower, Rob Arthan & Roger Bishop Jones: statement
    • PVS, NASA library, Natarajan Shankar
    • nqthm, Matt Kaufmann
    • NuPRL, David Basin
  32. The Four Color Problem
    • Coq, not in contribs, Georges Gonthier: statement
  33. Fermat's Last Theorem
  34. Divergence of the Harmonic Series
  35. Taylor's Theorem
    • HOL Light, John Harrison: statement
    • Mizar, Yasunari Shidama: statement
    • Isabelle, Jacques D. Fleuriot & Lukas Bulwahn & Bernhard Häupler: statement
    • Coq, C-CoRN, Luís Cruz-Filipe: statement
    • ProofPower, Rob Arthan: statement
    • ACL2, Ruben Gamboa & Brittany Middleton & Jun Sawada
    • PVS, NASA library, Ricky Butler
  36. Brouwer Fixed Point Theorem
  37. The Solution of a Cubic
  38. Arithmetic Mean/Geometric Mean
  39. Solutions to Pell's Equation
  40. Minkowski's Fundamental Theorem
  41. Puiseux's Theorem
    • Coq, not yet in contrib, Daniel de Rauglaudre: statement
  42. Sum of the Reciprocals of the Triangular Numbers
  43. The Isoperimetric Theorem
  44. The Binomial Theorem
    • HOL Light, John Harrison: statement
    • Mizar, Christoph Schwarzweller: statement
    • Isabelle, Tobias Nipkow: statement
    • Coq, contrib, Laurent Théry: statement
    • ProofPower, Rob Arthan: statement
    • ACL2, Ruben Gamboa
    • Metamath, Norman Megill: statement
    • NuPRL, Paul Jackson & Rich Eaton
  45. The Partition Theorem
  46. The Solution of the General Quartic Equation
  47. The Central Limit Theorem
  48. Dirichlet's Theorem
  49. The Cayley-Hamilton Theorem
    • HOL Light, John Harrison: statement
    • Coq, contrib, Sidi Ould Biha: statement
    • Isabelle, Stephan Adelsberger, Stefan Hetzl & Florian Pollak: statement
  50. The Number of Platonic Solids
  51. Wilson's Theorem
  52. The Number of Subsets of a Set
  53. Pi is Transcendental
  54. Konigsberg Bridges Problem
  55. Product of Segments of Chords
  56. The Hermite-Lindemann Transcendence Theorem
  57. Heron's Formula
  58. Formula for the Number of Combinations
  59. The Laws of Large Numbers
  60. Bezout's Theorem
  61. Theorem of Ceva
  62. Fair Games Theorem
  63. Cantor's Theorem
  64. L'Hôpital's Rule
  65. Isosceles Triangle Theorem
  66. Sum of a Geometric Series
    • HOL Light, John Harrison: statements
    • Mizar, Konrad Raczkowski & Andrzej Nedzusiak: statement
    • Isabelle, Jacques D. Fleuriot: statements
    • Coq, C-CoRN, Luís Cruz-Filipe: statement
    • ProofPower, Rob Arthan: statement
    • ACL2, Ruben Gamboa
    • Metamath, Norman Megill: statement
    • PVS, NASA library, Ricky Butler
  67. e is Transcendental
  68. Sum of an arithmetic series
  69. Greatest Common Divisor Algorithm
  70. The Perfect Number Theorem
  71. Order of a Subgroup
    • HOL Light, John Harrison: statement
    • Mizar, Wojciech Trybulec: statement
    • Isabelle, Florian Kammüller: statement
    • Coq, almost C-CoRN, Dan Synek & contrib, Laurent Théry: statement
    • ProofPower, Rob Arthan: statement
    • Metamath, Mario Carneiro: statement
    • PVS, NASA library, David Lester
  72. Sylow's Theorem
  73. Ascending or Descending Sequences
  74. The Principle of Mathematical Induction
  75. The Mean Value Theorem
    • HOL Light, John Harrison: statement
    • Mizar, Jaroslaw Kotowicz & Konrad Raczkowski & Pawel Sadowski: statement
    • Isabelle, Jacques D. Fleuriot: statement
    • Coq, C-CoRN, Luís Cruz-Filipe: statement
    • ProofPower, Rob Arthan: statement
    • Metamath, Mario Carneiro: statement
    • ACL2, Ruben Gamboa
    • PVS, NASA library, Bruno Dutertre
  76. Fourier Series
  77. Sum of kth powers
  78. The Cauchy-Schwarz Inequality
  79. The Intermediate Value Theorem
    • HOL Light, John Harrison: statement
    • Mizar, Yatsuka Nakamura & Andrzej Trybulec: statement
    • Isabelle, Jacques D. Fleuriot: statement
    • Coq, C-CoRN, Herman Geuvers et al.: statement
    • ProofPower, Rob Arthan: statement
    • ACL2, Ruben Gamboa
    • Metamath, Paul Chapman: statement
    • PVS, NASA library, Bruno Dutertre
  80. The Fundamental Theorem of Arithmetic
    • HOL Light, John Harrison: statement
    • Mizar, Artur Korniłowicz, Piotr Rudnicki & Hiroyuki Okazaki, Yasunari Shidama: statements
    • Isabelle, Thomas M. Rasmussen: statement
    • Coq, contrib, Martijn Oostdijk: statement
    • ProofPower, Rob Arthan: statement
    • ACL2, John Cowles
    • Metamath, Paul Chapman: statement
    • PVS, NASA library, Ricky Butler
    • NuPRL, Stuart Allen & Paul Jackson
  81. Divergence of the Prime Reciprocal Series
  82. Dissection of Cubes (J.E. Littlewood's "elegant" proof)
  83. The Friendship Theorem
  84. Morley's Theorem
  85. Divisibility by 3 Rule
  86. Lebesgue Measure and Integration
  87. Desargues's Theorem
    • HOL Light, John Harrison: statement
    • Mizar, Wojciech Leonczuk & Krzysztof Prazmowski: statement
    • Coq, contrib, Frédérique Guilhot, Julien Narboux: statement
    • Metamath, Norman Megill: statement
  88. Derangements Formula
  89. The Factor and Remainder Theorems
  90. Stirling's Formula
  91. The Triangle Inequality
    • HOL Light, John Harrison: statement
    • Mizar, Czeslaw Bylinski: statement
    • Isabelle, Steven Obua: statement
    • ProofPower, Rob Arthan: statement
    • ACL2, Ruben Gamboa
    • Metamath, Norman Megill: statement
    • PVS, NASA library, Ricky Butler & Cesar Munoz
  92. Pick's Theorem
  93. The Birthday Problem
  94. The Law of Cosines
    • HOL Light, John Harrison: statement
    • Mizar, Marco Riccardi: statement
    • Coq, contrib, Frédérique Guilhot: statement
    • PVS, NASA library, Cesar Munoz & Ricky Butler
  95. Ptolemy's Theorem
  96. Principle of Inclusion/Exclusion
  97. Cramer's Rule
  98. Bertrand's Postulate
  99. Buffon Needle Problem
  100. 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.)

(last modification 2014-09-29)