# 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

theorem irrational_sqrt_two  :
##### 2. Fundamental Theorem of Algebra #

Author: Chris Hughes

theorem complex.exists_root {f : polynomial } (hf : 0 < f.degree) :
∃ (z : ), f.is_root z
##### 3. The Denumerability of the Rational Numbers #

Author: Chris Hughes

##### 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} [metric_space P] [ 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
##### 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) :
q) * = (-1) ^ (p / 2) * (q / 2)
##### 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]
##### 11. The Infinitude of Primes #

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

theorem interval_integral.integral_has_strict_deriv_at_of_tendsto_ae_right {E : Type u_4} [normed_group E] [ E] [borel_space E] {f : → E} {c : E} {a b : } (hmeas : (𝓝 b) measure_theory.measure_space.volume) (hb : (𝓝 c)) :
has_strict_deriv_at (λ (u : ), ∫ (x : ) in a..u, f x) c b
theorem interval_integral.integral_eq_sub_of_has_deriv_right_of_le {E : Type u_4} [normed_group E] [ E] [borel_space E] {f : → E} {a b : } {f' : → E} (hab : a b) (hcont : (set.Icc a b)) (hderiv : ∀ (x : ), x b (f' x) (set.Ioi x) x) (f'int : b) :
∫ (y : ) in a..b, f' y = f b - f a
##### 16. Insolvability of General Higher Degree Equations (Abel-Ruffini Theorem) #

Author: Thomas Browning

mathlib archive

##### 17. De Moivre’s Formula #

Author: Abhimanyu Pallavi Sudhir

theorem complex.cos_add_sin_mul_I_pow (n : ) (z : ) :
+ ^ n = complex.cos ((n) * z) + (complex.sin ((n) * z)) * complex.I
##### 18. Liouville’s Theorem and the Construction of Transcendental Numbers #

Author: Jujian Zhang

theorem liouville.transcendental {x : } (lx : liouville x) :
##### 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
##### 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
##### 22. The Non-Denumerability of the Continuum #

Author: Floris van Doorn

##### 23. Formula for Pythagorean Triples #

Author: Paul van Wamelen

theorem pythagorean_triple.classification {x y z : } :
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))
##### 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),
##### 26. Leibniz’s Series for Pi #

Author: Benjamin Davidson

theorem real.tendsto_sum_pi_div_four  :
filter.tendsto (λ (k : ), ∑ (i : ) in , (-1) ^ i / (2 * i + 1)) filter.at_top (𝓝 (π / 4))
##### 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} [metric_space P] [ P] {p1 p2 p3 : P} (h2 : p2 p1) (h3 : p3 p1) :
p1 p2 p3 + p2 p3 p1 + p3 p1 p2 = π
##### 31. Ramsey’s Theorem #

Author: Bhavik Mehta

result

##### 34. Divergence of the Harmonic Series #

Authors: Anatole Dedecker, Yury Kudryashov

##### 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
##### 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

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} (x y : R) (n : ) :
(x + y) ^ n = ∑ (m : ) in finset.range (n + 1), ((x ^ m) * y ^ (n - m)) * (n.choose m)
##### 49. The Cayley-Hamilton Theorem #

Author: Scott Morrison

theorem matrix.aeval_self_charpoly {R : Type u} [comm_ring R] {n : Type w} [decidable_eq n] [fintype n] (M : n R) :
= 0
##### 51. Wilson’s Lemma #

Author: Chris Hughes

theorem zmod.wilsons_lemma (p : ) [fact (nat.prime p)] :
(p - 1)! = -1
##### 52. The Number of Subsets of a Set #

Author: mathlib

theorem finset.card_powerset {α : Type u_1} (s : finset α) :
##### 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} {P : Type u_2} [metric_space P] [ 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
##### 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) :
s).card = s.card.choose n
theorem finset.mem_powerset_len {α : Type u_1} {n : } {s t : finset α} :
s s t s.card = n
##### 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
##### 63. Cantor’s Theorem #

Author: mathlib

theorem cardinal.cantor (a : cardinal) :
a < 2 ^ a
##### 64. L’Hopital’s Rule #

Author: Anatole Dedecker

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

Author: Joseph Myers

theorem euclidean_geometry.angle_eq_angle_of_dist_eq {V : Type u_1} {P : Type u_2} [metric_space P] [ P] {p1 p2 p3 : P} (h : dist p1 p2 = dist p1 p3) :
p1 p2 p3 = p1 p3 p2
##### 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
theorem nnreal.has_sum_geometric {r : ℝ≥0} (hr : r < 1) :
has_sum (λ (n : ), r ^ n) (1 - r)⁻¹
##### 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 , i = n * (n - 1) / 2
##### 69. Greatest Common Divisor Algorithm #

Author: mathlib

def euclidean_domain.gcd {R : Type u} [decidable_eq R] :
R → R → R
theorem euclidean_domain.gcd_dvd {R : Type u} [decidable_eq R] (a b : R) :
theorem euclidean_domain.dvd_gcd {R : Type u} [decidable_eq R] {a b c : R} :
c ac b
##### 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] :
##### 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 ) :
∃ (K : subgroup G), = p ^ n

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

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 : (set.Icc a b)) (hfd : (set.Ioo a b)) :
∃ (c : ) (H : c b), c = (f b - f a) / (b - a)
##### 77. Sum of kth powers #

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

theorem sum_range_pow (n p : ) :
∑ (k : ) in , k ^ p = ∑ (i : ) in finset.range (p + 1), ((bernoulli i) * ((p + 1).choose i)) * n ^ (p + 1 - i) / (p + 1)
theorem sum_Ico_pow (n p : ) :
∑ (k : ) in (n + 1), k ^ p = ∑ (i : ) in finset.range (p + 1), ((bernoulli' i) * ((p + 1).choose i)) * n ^ (p + 1 - i) / (p + 1)
##### 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] [ E] (x y : E) :
theorem abs_inner_le_norm {𝕜 : Type u_1} {E : Type u_2} [is_R_or_C 𝕜] [ E] (x y : E) :
##### 79. The Intermediate Value Theorem #

Author: mathlib (Rob Lewis and Chris Hughes)

theorem intermediate_value_Icc {α : Type u} {δ : Type u_1} [linear_order δ] {a b : α} (hab : a b) {f : α → δ} (hf : (set.Icc a b)) :
set.Icc (f a) (f b) f '' b
##### 80. The Fundamental Theorem of Arithmetic #

Author: mathlib (Chris Hughes)

theorem nat.factors_unique {n : } {l : list } (h₁ : l.prod = n) (h₂ : ∀ (p : ), p l) :
structure unique_factorization_monoid (α : Type u_2)  :
Prop
theorem unique_factorization_monoid.factors_unique {α : Type u_1} {f g : multiset α} :
(∀ (x : α), x f(∀ (x : α), x g g.prod

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
##### 86. Lebesgue Measure and Integration #

Author: Johannes Hölzl

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

Author: Henry Swanson

theorem card_derangements_eq_num_derangements (α : Type u_1) [fintype α] [decidable_eq α] :
theorem num_derangements_sum (n : ) :
= ∑ (k : ) in finset.range (n + 1), ((-1) ^ k) * (k.asc_factorial (n - k))
##### 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} :
theorem polynomial.mod_X_sub_C_eq_C_eval {R : Type u} [field R] (p : polynomial R) (a : R) :
p % =
##### 91. The Triangle Inequality #

Author: Zhouhang Zhou

theorem norm_add_le {E : Type u_3} (g h : E) :
##### 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] :
fintype.card β) =

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} [metric_space P] [ 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)
##### 95. Ptolemy’s Theorem #

Author: Manuel Candales

theorem euclidean_geometry.mul_dist_add_mul_dist_eq_mul_dist_of_cospherical {V : Type u_1} {P : Type u_2} [metric_space P] [ 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
##### 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 : n α) (b : n → α) :
A.mul_vec ((A.cramer) b) = A.det b