100 theorems

Freek Wiedijk maintains a list tracking progress of theorem provers in formalizing 100 classic theorems in mathematics as a way of comparing prominent theorem provers. Currently 63 of them are formalized in Lean. We also have a page with the theorems from the list not yet in mathlib.

1. The Irrationality of the Square Root of 2 #

Author: mathlib

docs, source

2. Fundamental Theorem of Algebra #

Author: Chris Hughes

theorem complex.exists_root {f : polynomial } (hf : 0 < f.degree) :
∃ (z : ), f.is_root z

docs, source

3. The Denumerability of the Rational Numbers #

Author: Chris Hughes

docs, source

4. Pythagorean Theorem #

Author: Joseph Myers

theorem euclidean_geometry.dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] (p1 p2 p3 : P) :
(dist p1 p3) * dist p1 p3 = (dist p1 p2) * dist p1 p2 + (dist p3 p2) * dist p3 p2 p1 p2 p3 = π / 2

docs, source

7. Law of Quadratic Reciprocity #

Author: Chris Hughes

theorem zmod.quadratic_reciprocity (p q : ) [fact (nat.prime p)] [fact (nat.prime q)] [hp1 : fact (p % 2 = 1)] [hq1 : fact (q % 2 = 1)] (hpq : p q) :
(zmod.legendre_sym p q) * zmod.legendre_sym q p = (-1) ^ (p / 2) * (q / 2)

docs, source

9. The Area of a Circle #

Authors: James Arthur, Benjamin Davidson, and Andrew Souther

mathlib archive

10. Euler’s Generalization of Fermat’s Little Theorem #

Author: Chris Hughes

theorem nat.modeq.pow_totient {x n : } (h : x.coprime n) :
x ^ n.totient 1 [MOD n]

docs, source

11. The Infinitude of Primes #

Author: Jeremy Avigad

theorem nat.exists_infinite_primes (n : ) :
∃ (p : ), n p nat.prime p

docs, source

14. Euler’s Summation of 1 + (1/2)^2 + (1/3)^2 + …. #

Author: Marc Masdeu

result

website

15. Fundamental Theorem of Integral Calculus #

Author: Yury G. Kudryashov (first) and Benjamin Davidson (second)

docs, source

theorem interval_integral.integral_eq_sub_of_has_deriv_right_of_le {E : Type u_4} [measurable_space E] [normed_group E] [topological_space.second_countable_topology E] [complete_space E] [normed_space E] [borel_space E] {f : → E} {a b : } {f' : → E} (hab : a b) (hcont : continuous_on f (set.Icc a b)) (hderiv : ∀ (x : ), x set.Ioo a bhas_deriv_within_at f (f' x) (set.Ioi x) x) (f'int : interval_integrable f' measure_theory.measure_space.volume a b) :
∫ (y : ) in a..b, f' y = f b - f a

docs, source

16. Insolvability of General Higher Degree Equations (Abel-Ruffini Theorem) #

Author: Thomas Browning

mathlib archive

17. De Moivre’s Formula #

Author: Abhimanyu Pallavi Sudhir

docs, source

18. Liouville’s Theorem and the Construction of Transcendental Numbers #

Author: Jujian Zhang

docs, source

19. Four Squares Theorem #

Author: Chris Hughes

theorem nat.sum_four_squares (n : ) :
∃ (a b c d : ), a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = n

docs, source

20. All Primes (1 mod 4) Equal the Sum of Two Squares #

Author: Chris Hughes

theorem nat.prime.sq_add_sq (p : ) [hp : fact (nat.prime p)] (hp1 : p % 4 = 1) :
∃ (a b : ), a ^ 2 + b ^ 2 = p

docs, source

22. The Non-Denumerability of the Continuum #

Author: Floris van Doorn

docs, source

23. Formula for Pythagorean Triples #

Author: Paul van Wamelen

theorem pythagorean_triple.classification {x y z : } :
pythagorean_triple x y z ∃ (k m n : ), (x = k * (m ^ 2 - n ^ 2) y = k * (2 * m) * n x = k * (2 * m) * n y = k * (m ^ 2 - n ^ 2)) (z = k * (m ^ 2 + n ^ 2) z = (-k) * (m ^ 2 + n ^ 2))

docs, source

24. The Independence of the Continuum Hypothesis #

Author: Jesse Michael Han and Floris van Doorn

result

website

see the README file in the linked repository.

25. Schroeder-Bernstein Theorem #

Author: Mario Carneiro

theorem function.embedding.schroeder_bernstein {α : Type u} {β : Type v} {f : α → β} {g : β → α} (hf : function.injective f) (hg : function.injective g) :
∃ (h : α → β), function.bijective h

docs, source

26. Leibniz’s Series for Pi #

Author: Benjamin Davidson

theorem real.tendsto_sum_pi_div_four  :
filter.tendsto (λ (k : ), ∑ (i : ) in finset.range k, (-1) ^ i / (2 * i + 1)) filter.at_top (𝓝 (π / 4))

docs, source

27. Sum of the Angles of a Triangle #

Author: Joseph Myers

theorem euclidean_geometry.angle_add_angle_add_angle_eq_pi {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p1 p2 p3 : P} (h2 : p2 p1) (h3 : p3 p1) :
p1 p2 p3 + p2 p3 p1 + p3 p1 p2 = π

docs, source

31. Ramsey’s Theorem #

Author: Bhavik Mehta

result

34. Divergence of the Harmonic Series #

Authors: Anatole Dedecker, Yury Kudryashov

docs, source

38. Arithmetic Mean/Geometric Mean #

Author: Yury G. Kudryashov

theorem real.geom_mean_le_arith_mean_weighted {ι : Type u} (s : finset ι) (w z : ι → ) (hw : ∀ (i : ι), i s0 w i) (hw' : ∑ (i : ι) in s, w i = 1) (hz : ∀ (i : ι), i s0 z i) :
∏ (i : ι) in s, z i ^ w i ∑ (i : ι) in s, (w i) * z i

docs, source

39. Solutions to Pell’s Equation #

Author: Mario Carneiro

theorem pell.eq_pell {a : } (a1 : 1 < a) {x y : } (hp : x * x - ((d a1) * y) * y = 1) :
∃ (n : ), x = pell.xn a1 n y = pell.yn a1 n

docs, source

d is defined to be a*a - 1 for an arbitrary a > 1.

42. Sum of the Reciprocals of the Triangular Numbers #

Authors: Jalex Stark, Yury Kudryashov

mathlib archive

44. The Binomial Theorem #

Author: Chris Hughes

theorem add_pow {R : Type u_1} [comm_semiring R] (x y : R) (n : ) :
(x + y) ^ n = ∑ (m : ) in finset.range (n + 1), ((x ^ m) * y ^ (n - m)) * (n.choose m)

docs, source

49. The Cayley-Hamilton Theorem #

Author: Scott Morrison

theorem aeval_self_char_poly {R : Type u} [comm_ring R] {n : Type w} [decidable_eq n] [fintype n] (M : matrix n n R) :

docs, source

51. Wilson’s Lemma #

Author: Chris Hughes

theorem zmod.wilsons_lemma (p : ) [fact (nat.prime p)] :
(p - 1)! = -1

docs, source

52. The Number of Subsets of a Set #

Author: mathlib

theorem finset.card_powerset {α : Type u_1} (s : finset α) :

docs, source

55. Product of Segments of Chords #

Author: Manuel Candales

theorem euclidean_geometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_pi {V : Type u_1} [inner_product_space V] {P : Type u_2} [metric_space P] [normed_add_torsor V P] {a b c d p : P} (h : euclidean_geometry.cospherical {a, b, c, d}) (hapb : a p b = π) (hcpd : c p d = π) :
(dist a p) * dist b p = (dist c p) * dist d p

docs, source

57. Heron’s Formula #

Author: Matt Kempster

mathlib archive

58. Formula for the Number of Combinations #

Author: mathlib

theorem finset.card_powerset_len {α : Type u_1} (n : ) (s : finset α) :

docs, source

theorem finset.mem_powerset_len {α : Type u_1} {n : } {s t : finset α} :

docs, source

60. Bezout’s Theorem #

Author: mathlib

theorem nat.gcd_eq_gcd_ab (x y : ) :
(x.gcd y) = (x) * x.gcd_a y + (y) * x.gcd_b y

docs, source

63. Cantor’s Theorem #

Author: mathlib

theorem cardinal.cantor (a : cardinal) :
a < 2 ^ a

docs, source

64. L’Hopital’s Rule #

Author: Anatole Dedecker

theorem deriv.lhopital_zero_nhds {a : } {l : filter } {f g : } (hdf : ∀ᶠ (x : ) in 𝓝 a, differentiable_at f x) (hg' : ∀ᶠ (x : ) in 𝓝 a, deriv g x 0) (hfa : filter.tendsto f (𝓝 a) (𝓝 0)) (hga : filter.tendsto g (𝓝 a) (𝓝 0)) (hdiv : filter.tendsto (λ (x : ), deriv f x / deriv g x) (𝓝 a) l) :
filter.tendsto (λ (x : ), f x / g x) (𝓝[set.univ \ {a}] a) l

docs, source

65. Isosceles Triangle Theorem #

Author: Joseph Myers

theorem euclidean_geometry.angle_eq_angle_of_dist_eq {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] {p1 p2 p3 : P} (h : dist p1 p2 = dist p1 p3) :
p1 p2 p3 = p1 p3 p2

docs, source

66. Sum of a Geometric Series #

Author: Sander R. Dahmen (finite) and Johannes Hölzl (infinite)

def geom_sum {α : Type u} [semiring α] (x : α) (n : ) :
α

docs, source

theorem nnreal.has_sum_geometric {r : ℝ≥0} (hr : r < 1) :
has_sum (λ (n : ), r ^ n) (1 - r)⁻¹

docs, source

67. e is Transcendental #

Author: Jujian Zhang

result

website

68. Sum of an arithmetic series #

Author: Johannes Hölzl

theorem finset.sum_range_id (n : ) :
∑ (i : ) in finset.range n, i = n * (n - 1) / 2

docs, source

69. Greatest Common Divisor Algorithm #

Author: mathlib

def euclidean_domain.gcd {R : Type u} [euclidean_domain R] [decidable_eq R] :
R → R → R

docs, source

docs, source

theorem euclidean_domain.dvd_gcd {R : Type u} [euclidean_domain R] [decidable_eq R] {a b c : R} :
c ac bc euclidean_domain.gcd a b

docs, source

70. The Perfect Number Theorem #

Author: Aaron Anderson

mathlib archive

71. Order of a Subgroup #

Author: mathlib

theorem subgroup.card_subgroup_dvd_card {α : Type u_1} [group α] [fintype α] (s : subgroup α) [fintype s] :

docs, source

72. Sylow’s Theorem #

Author: Chris Hughes

theorem sylow.exists_subgroup_card_pow_prime {G : Type u} [group G] [fintype G] (p : ) {n : } [fact (nat.prime p)] (hdvd : p ^ n fintype.card G) :
∃ (K : subgroup G), fintype.card K = p ^ n

docs, source

sylow_conjugate

card_sylow_dvd

card_sylow_modeq_one

73. Ascending or Descending Sequences (Erdős–Szekeres Theorem) #

Author: Bhavik Mehta

mathlib archive

74. The Principle of Mathematical Induction #

Author: Leonardo de Moura

inductive nat  :
Type

docs, source

Automatically generated when defining the natural numbers

75. The Mean Value Theorem #

Author: Yury G. Kudryashov

theorem exists_deriv_eq_slope (f : ) {a b : } (hab : a < b) (hfc : continuous_on f (set.Icc a b)) (hfd : differentiable_on f (set.Ioo a b)) :
∃ (c : ) (H : c set.Ioo a b), deriv f c = (f b - f a) / (b - a)

docs, source

77. Sum of kth powers #

Authors: mathlib (Moritz Firsching, Fabian Kruse, Ashvni Narayanan)

theorem sum_range_pow (n p : ) :
∑ (k : ) in finset.range n, k ^ p = ∑ (i : ) in finset.range (p + 1), ((bernoulli i) * ((p + 1).choose i)) * n ^ (p + 1 - i) / (p + 1)

docs, source

theorem sum_Ico_pow (n p : ) :
∑ (k : ) in finset.Ico 1 (n + 1), k ^ p = ∑ (i : ) in finset.range (p + 1), ((bernoulli' i) * ((p + 1).choose i)) * n ^ (p + 1 - i) / (p + 1)

docs, source

78. The Cauchy-Schwarz Inequality #

Author: Zhouhang Zhou

theorem inner_mul_inner_self_le {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

docs, source

theorem abs_inner_le_norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [inner_product_space 𝕜 E] (x y : E) :

docs, source

79. The Intermediate Value Theorem #

Author: mathlib (Rob Lewis and Chris Hughes)

theorem intermediate_value_Icc {α : Type u} [conditionally_complete_linear_order α] [topological_space α] [order_topology α] [densely_ordered α] {δ : Type u_1} [linear_order δ] [topological_space δ] [order_closed_topology δ] {a b : α} (hab : a b) {f : α → δ} (hf : continuous_on f (set.Icc a b)) :
set.Icc (f a) (f b) f '' set.Icc a b

docs, source

80. The Fundamental Theorem of Arithmetic #

Author: mathlib (Chris Hughes)

theorem nat.factors_unique {n : } {l : list } (h₁ : l.prod = n) (h₂ : ∀ (p : ), p lnat.prime p) :

docs, source

docs, source

docs, source

structure unique_factorization_monoid (α : Type u_2) [comm_cancel_monoid_with_zero α] :
Prop

docs, source

theorem unique_factorization_monoid.factors_unique {α : Type u_1} [comm_cancel_monoid_with_zero α] [unique_factorization_monoid α] {f g : multiset α} :
(∀ (x : α), x firreducible x)(∀ (x : α), x girreducible x)associated f.prod g.prodmultiset.rel associated f g

docs, source

it also has a generalized version, by showing that every Euclidean domain is a unique factorization domain, and showing that the integers form a Euclidean domain.

82. Dissection of Cubes (J.E. Littlewood’s ‘elegant’ proof) #

Author: Floris van Doorn

mathlib archive

83. The Friendship Theorem #

Authors: Aaron Anderson, Jalex Stark, Kyle Miller

mathlib archive

85. Divisibility by 3 Rule #

Author: Scott Morrison

theorem nat.three_dvd_iff (n : ) :
3 n 3 (10.digits n).sum

docs, source

86. Lebesgue Measure and Integration #

Author: Johannes Hölzl

def measure_theory.lintegral {α : Type u_1} {m : measurable_space α} (μ : measure_theory.measure α) (f : α → ℝ≥0∞) :

docs, source

88. Derangements Formula #

Author: Henry Swanson

docs, source

theorem num_derangements_sum (n : ) :
(num_derangements n) = ∑ (k : ) in finset.range (n + 1), ((-1) ^ k) * (k.asc_factorial (n - k))

docs, source

89. The Factor and Remainder Theorems #

Author: Chris Hughes

theorem polynomial.dvd_iff_is_root {R : Type u} {a : R} [comm_ring R] {p : polynomial R} :

docs, source

docs, source

91. The Triangle Inequality #

Author: Zhouhang Zhou

theorem norm_add_le {α : Type u_1} [semi_normed_group α] (g h : α) :

docs, source

93. The Birthday Problem #

Author: Eric Rodriguez

theorem fintype.card_embedding_eq {α : Type u_1} {β : Type u_2} [fintype α] [fintype β] [decidable_eq α] [decidable_eq β] :

docs, source

mathlib archive

94. The Law of Cosines #

Author: Joseph Myers

theorem euclidean_geometry.dist_sq_eq_dist_sq_add_dist_sq_sub_two_mul_dist_mul_dist_mul_cos_angle {V : Type u_1} {P : Type u_2} [inner_product_space V] [metric_space P] [normed_add_torsor V P] (p1 p2 p3 : P) :
(dist p1 p3) * dist p1 p3 = (dist p1 p2) * dist p1 p2 + (dist p3 p2) * dist p3 p2 - ((2 * dist p1 p2) * dist p3 p2) * real.cos ( p1 p2 p3)

docs, source

95. Ptolemy’s Theorem #

Author: Manuel Candales

theorem euclidean_geometry.mul_dist_add_mul_dist_eq_mul_dist_of_cospherical {V : Type u_1} [inner_product_space V] {P : Type u_2} [metric_space P] [normed_add_torsor V P] {a b c d p : P} (h : euclidean_geometry.cospherical {a, b, c, d}) (hapc : a p c = π) (hbpd : b p d = π) :
(dist a b) * dist c d + (dist b c) * dist d a = (dist a c) * dist b d

docs, source

96. Principle of Inclusion/Exclusion #

Author: Neil Strickland

github

97. Cramer’s Rule #

Author: Anne Baanen

theorem matrix.mul_vec_cramer {n : Type u} [decidable_eq n] [fintype n] {α : Type v} [comm_ring α] (A : matrix n n α) (b : n → α) :
A.mul_vec ((A.cramer) b) = A.det b

docs, source