The following are the theorems from this list proved so far in the Isabelle proof assistant.

If you have proved additional ones or know of any, please so I can add them here.

If the theorem is not part of the Isabelle distribution, the entry will usually contain a link to the repository that does. The list does not automatically track the most recent version of each theorem. If you find one that that is out of date and would like me to update it, let me know.

1. Square Root of 2 is Irrational

theorem sqrt_prime_irrational: "prime p  sqrt (real p)  "

corollary sqrt_2_not_rat: "sqrt 2 ∉ ℚ"

https://isabelle.in.tum.de/dist/library/HOL/HOL-ex/Sqrt.html

2. Fundamental Theorem of Algebra

lemma fundamental_theorem_of_algebra:
  assumes nc: "¬ constant (poly p)"
  shows "z::complex. poly p z = 0"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Computational_Algebra/Fundamental_Theorem_Algebra.html

3. Denumerability of the Rational Numbers

theorem rat_denum: "∃f :: nat ⇒ rat. surj f"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Library/Countable.html

4. Pythagorean Theorem

lemma pythagoras:
  fixes a b c :: "'a :: real_inner"
  assumes "orthogonal (b - a) (c - a)"
  shows   "dist b c ^ 2 = dist a b ^ 2 + dist a c ^ 2"

5. Prime Number Theorem

Avigad et al. formalised the elementary Erdős–Selberg proof:

definition pi :: "nat ⇒ real"
  where "pi x = real (card {y::nat. y ≤ x ∧ prime y})"

lemma PrimeNumberTheorem: "(λx. pi x * ln (real x) / real x) ⇢ 1"

http://www.andrew.cmu.edu/user/avigad/isabelle/NumberTheory/PrimeNumberTheorem.html


A formalisation by Eberl and Paulson of the shorter analytic proof is available in the AFP:

definition primes_pi :: "real ⇒ real" where
  "primes_pi x = real (card {p::nat. prime p ∧ p ≤ x})"

corollary prime_number_theorem: "primes_pi ∼[at_top] (λx. x / ln x)"

https://www.isa-afp.org/entries/Prime_Number_Theorem.html

6. Gödel's Incompleteness Theorem

theorem Goedel_I:
  assumes "¬ {}  Fls"
  obtains δ where
      "{}  δ IFF Neg (PfP ⌈δ⌉)"
      "¬ {}  δ"
      "¬ {}  Neg δ"
      "eval_fm e δ"
      "ground_fm δ"

theorem Goedel_II:
  assumes "¬ {}  Fls"
    shows "¬ {}  Neg (PfP Fls)"

https://isa-afp.org/entries/Incompleteness.shtml

7. Law of Quadratic Reciprocity

theorem Quadratic_Reciprocity:
  assumes "prime p" "2 < p" "prime q" "2 < q" "p ≠ q"
  shows "Legendre p q * Legendre q p = (-1) ^ ((p - 1) div 2 * ((q - 1) div 2))"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Number_Theory/Quadratic_Reciprocity.html

8. The Impossibility of Trisecting the Angle and Doubling the Cube

theorem impossibility_of_doubling_the_cube:
  "x^3 = 2  (Point x 0)  constructible"

theorem impossibility_of_trisecting_angle_pi_over_3:
  "Point (cos (pi / 9)) 0  constructible"

https://isa-afp.org/entries/Impossible_Geometry.shtml

9. The Area of a Circle

theorem content_ball:
  fixes c :: "'a :: euclidean_space"
  assumes "r ≥ 0"
  shows   "content (ball c r) = pi powr (DIM('a) / 2) / Gamma (DIM('a) / 2 + 1) * r ^ DIM('a)"

corollary content_circle:
  "r ≥ 0 ⟹ content (ball c r :: (real ^ 2) set) = r ^ 2 * pi"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Ball_Volume.html

10. Euler's Generalisation of Fermat's Little Theorem

lemma euler_theorem:
  fixes a m :: nat
  assumes "coprime a m"
  shows "[a ^ totient m = 1] (mod m)"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Number_Theory/Residues.html

11. The Infinitude of Primes

lemma primes_infinite: "¬finite {p::nat. prime p}"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Computational_Algebra/Primes.html

12. The Independence of the Parallel Postulate

https://isa-afp.org/entries/Tarskis_Geometry.shtml

14. Euler's Summation of 1 + (1/2)^2 + (1/3)^2 + ....

theorem inverse_squares_sums: "(λn. 1 / (n + 1)2) sums (pi2 / 6)"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Gamma_Function.html

15. Fundamental Theorem of Integral Calculus

theorem fundamental_theorem_of_calculus:
  fixes f :: "real ⇒ 'a::banach"
  assumes "a ≤ b" 
    and   "⋀x. x ∈ {a..b} ⟹ (f has_vector_derivative f' x) (at x within {a..b})"
  shows   "(f' has_integral (f b - f a)) {a..b}"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Henstock_Kurzweil_Integration.html

17. DeMoivre's Theorem

lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"

lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Complex.html

18. Liouville's Theorem and the Construction of Transcendental Numbers

corollary transcendental_liouville_constant:
  "¬algebraic (standard_liouville (λ_. 1) 10)"

https://isa-afp.org/entries/Liouville_Numbers.shtml

19. Lagrange's Four-Square Theorem

theorem four_squares: "∃a b c d::nat. n = a2 + b2 + c2 + d2"

https://www.isa-afp.org/entries/SumSquares.html

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

theorem two_squares:
  assumes "prime (p :: nat)"
  shows   "(∃a b. p = a2 + b2) = [p ≠ 3] (mod 4)

https://www.isa-afp.org/entries/SumSquares.html

21. Green's theorem

lemma GreenThm_typeI_typeII_divisible_region:
      assumes valid_typeI_div: "valid_typeI_division s twoChain_typeI" and
              valid_typeII_div: "valid_typeII_division s twoChain_typeII" and
              f_analytically_valid: "twoC  twoChain_typeI. analytically_valid (cubeImage twoC) (%x. (F x)  i) j"
                                    "twoC  twoChain_typeII. analytically_valid (cubeImage twoC) (%x. (F x)  j) i" and
              only_vertical_division:
              "only_vertical_division one_chain_typeI twoChain_typeI"
              "boundary_chain one_chain_typeI" and
              only_horizontal_division:
              "only_horizontal_division one_chain_typeII twoChain_typeII"
              "boundary_chain one_chain_typeII" and
              typeI_and_typII_one_chains_have_common_subdiv:
              "common_boundary_sudivision_exists one_chain_typeI one_chain_typeII"
      shows "integral s (%x. partial_vector_derivative (%x. (F x)  j) i x - partial_vector_derivative (%x. (F x)  i) j x) = one_chain_line_integral F {i, j} one_chain_typeI"
            "integral s (%x. partial_vector_derivative (%x. (F x)  j) i x - partial_vector_derivative (%x. (F x)  i) j x) = one_chain_line_integral F {i, j} one_chain_typeII"

https://www.isa-afp.org/entries/Green.html

22. The Non-Denumerability of the Continuum

theorem real_non_denum: "f :: nat  real. surj f"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Continuum_Not_Denumerable.html

23. Formula for Pythagorean Triples

theorem nat_euclid_pyth_triples:
  fixes a b c :: nat
  assumes "coprime a b" "odd a" "a⇧2 + b⇧2 = c⇧2"
  shows   "∃p q. a = p⇧2 - q⇧2 ∧ b = 2 * p * q ∧ c = p⇧2 + q⇧2 ∧ coprime p q"

https://www.isa-afp.org/entries/Fermat3_4.html

25. Schroeder-Bernstein Theorem

theorem Schroeder_Bernstein:
  fixes f :: "'a ⇒ 'b" and g :: "'b ⇒ 'a"
    and A :: "'a set" and B :: "'b set"
  assumes "inj_on f A" and "f ` A ⊆ B"
    and   "inj_on g B" and "g ` B ⊆ A"
  shows   "∃h. bij_betw h A B"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Inductive.html

26. Leibnitz's Series for Pi

theorem pi_series:
  "pi / 4 = ( k. (-1)^k * 1 / real (k*2+1))"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Transcendental.html

27. Sum of the Angles of a Triangle

lemma angle_sum_triangle:
  assumes "a ≠ b ∨ b ≠ c ∨ a ≠ c"
  shows   "angle c a b + angle a b c + angle b c a = pi"

https://isa-afp.org/entries/Triangle.shtml

30. The Ballot Problem

lemma
  "valid_countings a b = (if a  b then (if b = 0 then 1 else 0) else (a - b) / (a + b) * all_countings a b)"

https://isabelle.in.tum.de/dist/library/HOL/HOL-ex/Ballot.html

31. Ramsey's Theorem

lemma ramsey: "
  (s::nat) (r::nat) (YY::'a set) (f::'a set => nat).
  infinite YY
   (X. X <= YY  finite X  card X = r  f X < s)
   (Y' t'.
  Y' <= YY
   infinite Y'
   t' < s
   (X. X <= Y'  finite X  card X = r  f X = t'))"

https://isa-afp.org/entries/Ramsey-Infinite.shtml

34. Divergence of the Harmonic Series

theorem not_summable_harmonic:
  shows "¬summable (λn. 1 / real (n + 1))"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Summation_Tests.html

35. Taylor's Theorem

theorem taylor:
  fixes a :: real and n :: nat and f :: "real ⇒ real"
  assumes "n > 0" and "diff 0 = f"
    and   "∀m t. m < n ∧ t ∈ {a..b} ⟶ (diff m has_real_derivative diff (m + 1) t) (at t)"
    and   "c ∈ {a..b}" and "x ∈ {a..b} - {c}"
  shows "∃t. t ∈ open_segment x c ∧
             f x = (∑m<n. (diff m c / fact m) * (x - c) ^ m) + (diff n t / fact n) * (x - c) ^ n"

https://isabelle.in.tum.de/dist/library/HOL/HOL/MacLaurin.html

36. Brouwer Fixed Point Theorem

lemma brouwer:
  fixes f :: "'a::euclidean_space  'a"
  assumes "compact S" and "convex S" and "S  {}"
    and "continuous_on S f"
    and "f ` S  S"
  obtains x where "x  S" and "f x = x"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Brouwer_Fixpoint.html

37. The Solution of a Cubic

lemma cubic:
  assumes a0: "a ≠ 0"
  shows "
  let p = (3 * a * c - b^2) / (9 * a^2) ;
      q = (9 * a * b * c - 2 * b^3 - 27 * a^2 * d) / (54 * a^3);
      s = csqrt(q^2 + p^3);
      s1 = (if p = 0 then ccbrt(2 * q) else ccbrt(q + s));
      s2 = -s1 * (1 + ii * csqrt 3) / 2;
      s3 = -s1 * (1 - ii * csqrt 3) / 2
  in if p = 0 then
       a * x^3 + b * x^2 + c * x + d = 0 ⟷
           x = s1 - b / (3 * a) ∨
           x = s2 - b / (3 * a) ∨
           x = s3 - b / (3 * a)
      else
        s1 ≠ 0 ∧
        (a * x^3 + b * x^2 + c * x + d = 0 ⟷
            x = s1 - p / s1 - b / (3 * a) ∨
            x = s2 - p / s2 - b / (3 * a) ∨
            x = s3 - p / s3 - b / (3 * a))"

https://isabelle.in.tum.de/dist/library/HOL/HOL-ex/Cubic_Quartic.html

38. Arithmetic Mean / Geometric Mean

theorem CauchysMeanTheorem_Eq:
  fixes z::"real list"
  assumes "pos z"
  shows "gmean z = mean z  het z = 0"

https://isa-afp.org/entries/Cauchy.shtml

39. Solutions to Pell's Equation

theorem pell_solutions:
  fixes D :: nat
  assumes "k. D = k2"
  obtains "x0" "y0" :: nat
  where   "(x::int) (y::int).
             x2 - D * y2 = 1 
            (n::nat. nat ¦x¦ + sqrt D * nat ¦y¦ = (x0 + sqrt D * y0) ^ n)"

corollary pell_solutions_infinite:
  fixes D :: nat
  assumes "k. D = k2"
  shows   "infinite {(x :: int, y :: int). x2 - D * y2 = 1}"

https://www.isa-afp.org/entries/Pell.html

40. Minkowski's Fundamental Theorem

theorem minkowski:
  fixes B :: "(real ^ 'n) set"
  assumes "convex B" and symmetric: "uminus ` B ⊆ B"
  assumes meas_B [measurable]: "B ∈ sets lebesgue"
  assumes measure_B: "emeasure lebesgue B > 2 ^ CARD('n)"
  obtains x where "x ∈ B" and "x ≠ 0" and "⋀i. x $ i ∈ ℤ"

https://www.isa-afp.org/entries/Minkowskis_Theorem.shtml

44. The Binomial Theorem

theorem binomial_ring:
  fixes a b :: "'a :: comm_ring_1"
  shows "(a + b) ^ n = (∑k=0..n. of_nat (n choose k) * a ^ k * b ^ (n - k))"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Binomial.html

45. The Partition Theorem

theorem Euler_partition_theorem:
  "card {p. p partitions n ∧ (∀i. p i ≤ 1)} = card {p. p partitions n ∧ (∀i. p i ≠ 0  odd i)}"

https://isa-afp.org/entries/Euler_Partition.shtml

46. The Solution of the General Quartic Equation

lemma quartic:
 "(y::real)^3 - b * y^2 + (a * c - 4 * d) * y - a^2 * d + 4 * b * d - c^2 = 0 
  R^2 = a^2 / 4 - b + y 
  s^2 = y^2 - 4 * d 
  (D^2 = (if R = 0 then 3 * a^2 / 4 - 2 * b + 2 * s
                   else 3 * a^2 / 4 - R^2 - 2 * b + (4 * a * b - 8 * c - a^3) / (4 * R))) 
  (E^2 = (if R = 0 then 3 * a^2 / 4 - 2 * b - 2 * s
                   else 3 * a^2 / 4 - R^2 - 2 * b - (4 * a * b - 8 * c - a^3) / (4 * R)))
   x^4 + a * x^3 + b * x^2 + c * x + d = 0 
      x = -a / 4 + R / 2 + D / 2 
      x = -a / 4 + R / 2 - D / 2 
      x = -a / 4 - R / 2 + E / 2 
      x = -a / 4 - R / 2 - E / 2"

https://isabelle.in.tum.de/dist/library/HOL/HOL-ex/Cubic_Quartic.html

47. The Central Limit Theorem

theorem (in prob_space) central_limit_theorem:
  fixes
    X :: "nat  'a  real" and
    μ :: "real measure" and
    σ :: real and
    S :: "nat  'a  real"
  assumes
    X_indep: "indep_vars (λi. borel) X UNIV" and
    X_integrable: "n. integrable M (X n)" and
    X_mean_0: "n. expectation (X n) = 0" and
    σ_pos: "σ > 0" and
    X_square_integrable: "n. integrable M (λx. (X n x)2)" and
    X_variance: "n. variance (X n) = σ2" and
    X_distrib: "n. distr M borel (X n) = μ"
  defines
    "S n  λx. i<n. X i x"
  shows
    "weak_conv_m (λn. distr M borel (λx. S n x / sqrt (n * σ2)))
        (density lborel std_normal_density)"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Probability/Central_Limit_Theorem.html

48. Dirichlet's Theorem

theorem Dirichlet:
  assumes "n > 1" and "coprime h n"
  shows   "infinite {p. prime p ∧ [p = h] (mod n)}"

https://isa-afp.org/entries/Dirichlet_L.html

49. Cayley-Hamilton Theorem

theorem Cayley_Hamilton:
  fixes A :: "'acomm_ring_1^'nfinite^'n"
  shows "evalmat (charpoly A) A = 0"

https://isa-afp.org/entries/Cayley_Hamilton.shtml

51. Wilson's Theorem

lemma wilson_theorem:
  assumes "prime p"
  shows   "[fact (p - 1) = -1] (mod p)"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Number_Theory/Residues.html

52. The Number of Subsets of a Set

lemma card_Pow:
  "finite A  card (Pow A) = 2 ^ card A"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Power.html

53. Pi is Transcendental

theorem transcendental_pi: "¬algebraic pi"

https://www.isa-afp.org/entries/Pi_Transcendental.html

54. The Koenigsberg Bridges Problem

lemma eulerian_split:
  assumes "nodes G1  nodes G2 = {}" "edges G1  edges G2={}"
    "valid_unMultigraph G1" "valid_unMultigraph G2"
    "valid_unMultigraph.is_Eulerian_trail  G1 v1 ps1 v1'"
    "valid_unMultigraph.is_Eulerian_trail  G2 v2 ps2 v2'"
  shows "valid_unMultigraph.is_Eulerian_trail nodes=nodes G1  nodes G2,
          edges=edges G1  edges G2  {(v1',w,v2),(v2,w,v1')} v1 (ps1@(v1',w,v2)#ps2) v2'"

https://isa-afp.org/entries/Koenigsberg_Friendship.shtml

55. Product of Segments of Chords

theorem product_of_chord_segments:
  fixes S1 T1 S2 T2 X C :: "'a :: euclidean_space"
  assumes "between (S1, T1) X" "between (S2, T2) X"
  assumes "dist C S1 = r" "dist C T1 = r"
  assumes "dist C S2 = r" "dist C T2 = r"
  shows "dist S1 X * dist X T1 = dist S2 X * dist X T2"

https://www.isa-afp.org/entries/Chord_Segments.shtml

57. Heron's formula

theorem heron:
  fixes A B C :: "real ^ 2"
  defines "a ≡ dist B C" and "b ≡ dist A C" and "c ≡ dist A B"
  defines "s ≡ (a + b + c) / 2"
  shows   "content (convex hull {A, B, C}) = sqrt (s * (s - a) * (s - b) * (s - c))"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Simplex_Content.html

58. Formula for the Number of Combinations

theorem n_subsets:
  "finite A  card {B. B  A  card B = k} = (card A choose k)"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Binomial.html

60. Bezout's Theorem

lemma bezout_gcd_nat:
  fixes a b :: nat
  shows "∃x y. a * x - b * y = gcd a b ∨ b * x - a * y = gcd a b"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Computational_Algebra/Primes.html

63. Cantor's Theorem

lemma Cantors_paradox: f. f ` A = Pow A

https://isabelle.in.tum.de/dist/library/HOL/HOL/Fun.html

64. L'Hôpital's Rule

lemma lhopital:
  fixes f g f' g' :: "real ⇒ real"
  assumes "f ─x→ 0" and "g ─x→ 0"
  assumes "∀⇩F u in at x. g u ≠ 0" 
  assumes "∀⇩F u in at x. g' u ≠ 0"
  assumes "∀⇩F u in at x. (f has_real_derivative f' u) (at u)"
  assumes "∀⇩F u in at x. (g has_real_derivative g' u) (at u)"
  assumes "filterlim (λx. f' x / g' x) F (at x)"
  shows   "filterlim (λx. f x / g x) F (at x)"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Deriv.html

65. Isosceles Triangle Theorem

lemma isosceles_triangle:
  assumes "dist a c = dist b c"
  shows   "angle b a c = angle a b c"

https://isa-afp.org/entries/Triangle.shtml

66. Sum of a Geometric Series

lemma geometric_sums:
  "norm c < 1  (λn. c^n) sums (1 / (1 - c))"

lemma suminf_geometric:
  "norm c < 1  (∑n. c ^ n) = 1 / (1 - c)"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Series.html

67. e is Transcendental

corollary e_transcendental_real: "¬ algebraic (exp 1 :: real)"

https://www.isa-afp.org/entries/E_Transcendental.shtml

68. Sum of an Arithmetic Series

lemma double_arith_series:
  fixes a d :: "'a :: comm_semiring_1"
  shows "2 * (∑i=0..n. a + of_nat i * d) = (of_nat n + 1) * (2 * a + of_nat n * d)"

69. Greatest Common Divisor Algorithm

The greatest common divisor is defined in the semiring_gcd typeclass. Instances are provided for all the basic types, such as naturals, integers, and polynomials. The most important properties are:

lemma gcd_dvd1:     "gcd a b dvd a"
  and gcd_dvd2:     "gcd a b dvd b"
  and gcd_greatest: "c dvd a ⟹ c dvd b ⟹ c dvd gcd a b"

https://isabelle.in.tum.de/dist/library/HOL/HOL/GCD.html

70. Perfect Number Theorem

theorem perfect_number_theorem:
  assumes even: "even m" and perfect: "perfect m"
  shows " n . m = 2^n*(2^(n+1) - 1)  prime ((2::nat)^(n+1) - 1)"

https://isa-afp.org/entries/Perfect-Number-Thm.shtml

71. Order of a Subgroup

proposition (in group) lagrange_finite:
  assumes "finite (carrier G)" and "subgroup H G"
  shows "card (rcosets H) * card H = order G"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Algebra/Coset.html

72. Sylow's Theorem

theorem sylow_thm:
  assumes "prime p" and "group G" and "order G = p ^ a * m" and "finite (carrier G)"
  obtains H where "subgroup H G" and "card H = p ^ a"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Algebra/Sylow.html

73. Ascending or Descending Sequences

lemma Erdoes_Szekeres:
  fixes f :: "_  'a::linorder"
  shows "(S. S  {0..m * n}  card S = m + 1  mono_on f (op ) S) 
         (S. S  {0..m * n}  card S = n + 1  mono_on f (op ) S)"

https://isabelle.in.tum.de/dist/library/HOL/HOL-ex/Erdoes_Szekeres.html

74. The Principle of Mathematical Induction

theorem nat_induct:
  assumes "P 0" and "⋀n. P n ⟹ P (Suc n)"
  shows   "P n"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Nat.html

75. The Mean Value Theorem

theorem MVT:
  fixes a b :: real and f :: "real ⇒ real"
  assumes "a < b"
      and "∀x∈{a..b}. isCont f x"
      and "∀x∈{a<..<b}. f differentiable (at x)"
  shows "l z::real. z ∈ {a<..<b} ∧ (f has_real_derivative l) (at z) ∧
                         f b - f a = (b - a) * l"

lemma MVT2:
  fixes a b :: real and f f' :: "real ⇒ real"
  assumes "a < b"
      and "∀x∈{a..b}. (f has_real_derivative f' x) (at x)"
  shows "z::real. z ∈ {a<..<b} ∧ f b - f a = (b - a) * f' z"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Deriv.html

77. Sum of kth powers

lemma sum_of_powers:
  fixes m n :: nat
  shows "(k=0..n. k ^ m) = (bernpoly (m + 1) (n + 1) - bernpoly (m + 1) 0) / (m + 1)"

https://www.isa-afp.org/entries/Bernoulli.html

78. The Cauchy-Schwarz Inequality

theorem CauchySchwarzReal:
  fixes x::vector
  assumes "vlen x = vlen y"
  shows "¦xy¦  x*y"

https://isa-afp.org/entries/Cauchy.shtml


An alternative version is available in the standard library:

lemma Cauchy_Schwarz_ineq2:
  "¦inner x y¦ ≤ norm x * norm y"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Inner_Product.html

79. The Intermediate Value Theorem

lemma IVT':
  fixes f :: "real ⇒ real"
  assumes "a ≤ b" and "y ∈ {f a..f b}" and  "continuous_on {a..b} f"
  obtains x where "x ∈ {a..b}" and "f x = y"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Topological_Spaces.html

80. Fundamental Theorem of Arithmetic

The function prime_factorization is defined for any factorial semiring. It returns the factorization as a multiset that fulfils the following properties:

lemma in_prime_factors_iff:
  "p ∈ set_mset (prime_factors x) ⟷ x ≠ 0 ∧ p dvd x ∧ prime p"

lemma prod_mset_prime_factorization:
  assumes "x ≠ 0"
  shows   "prod_mset (prime_factorization x) = normalize x"

lemma prime_factorization_unique:
  assumes "x ≠ 0" "y ≠ 0"
  shows   "prime_factorization x = prime_factorization y ⟷ normalize x = normalize y"

The normalize function is required because associated elements (like -3 and 3) have the same factorization; for natural numbers, it is the identity.

https://isabelle.in.tum.de/dist/library/HOL/HOL-Computational_Algebra/Factorial_Ring.html

81. Divergence of the Prime Reciprocal Series

corollary prime_harmonic_series_diverges:
  "¬convergent (λn. ∑pprimes_upto n. 1 / p)"

https://isa-afp.org/entries/Prime_Harmonic_Series.shtml

The more precise asymptotic estimate given by Mertens' Second Theorem is also available:

theorem mertens_second_theorem:
  "(λx. (∑p | real p ≤ x ∧ prime p. 1 / p) - ln (ln x) - meissel_mertens) ∈ O(λx. 1 / ln x)"

https://www.isa-afp.org/entries/Prime_Number_Theorem.html

83. The Friendship Theorem

theorem (in valid_unSimpGraph) friendship_thm:
  assumes "v u. vV  uV  vu  ! n. adjacent v n  adjacent u n" and "finite V"
  shows "v. nV. nv  adjacent v n"

https://isa-afp.org/entries/Koenigsberg_Friendship.shtml

85. Divisibility by Three Rule

theorem three_divides_nat: "3 dvd n ⟷ 3 dvd sumdig n"

https://isabelle.in.tum.de/dist/library/HOL/HOL-ex/ThreeDivides.html

86. Lebesgue Measure and Integration

https://isa-afp.org/entries/Integration.shtml

A more recent and more extensive library of the Lebesgue Measure and Lebesgue integration is now also in the standard distribution:

https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Lebesgue_Measure.html

https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Nonnegative_Lebesgue_Integration.html

88. Derangements Formula

theorem derangements_formula:
  assumes "n  0" and "finite S" and "card S = n"
  shows "card (derangements S) = round (fact n / exp 1)"

https://isa-afp.org/entries/Derangements.shtml

89. The Factor and Remainder Theorems

lemma long_div_theorem:
  assumes "g  carrier P" and "f  carrier P" and "g  𝟬P"
  shows "q r (k::nat). (q  carrier P)  (r  carrier P) 
        (lcoeff g)(^)Rk P f = g P q P r 
        (r = 𝟬P | deg R r < deg R g)"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Algebra/UnivPoly.html

Independently, HOL-Computational_Algebra provides notions of division and remainder in Euclidean rings (such as naturals, integers, polynomials):

https://isabelle.in.tum.de/dist/library/HOL/HOL-Computational_Algebra/Euclidean_Algorithm.html

90. Stirling's Formula

theorem Gamma_asymp_equiv:
  "Gamma ∼ (λx. sqrt (2*pi/x) * (x / exp 1) powr x :: real)"

theorem fact_asymp_equiv:
  "fact ∼ (λn. sqrt (2*pi*n) * (n / exp 1) ^ n :: real)"

https://www.isa-afp.org/entries/Stirling_Formula.shtml

91. The Triangle Inequality

class ordered_ab_group_add_abs = ordered_ab_group_add + abs +
  assumes abs_ge_zero: "¦a¦  0"
    and abs_ge_self: "a  ¦a¦"
    and abs_leI: "a  b  - a  b  ¦a¦  b"
    and abs_minus_cancel: "¦-a¦ = ¦a¦"
    and abs_triangle_ineq: "¦a + b¦  ¦a¦ + ¦b¦"

The triangle inequality is a type class property in Isabelle. Real numbers, integers, etc are instances of this type class:

lemma abs_triangle_ineq_real: "¦(a::real) + b¦  ¦a¦ + ¦b¦"
lemma abs_triangle_ineq_int: "¦(a::int) + b¦  ¦a¦ + ¦b¦"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Groups.html

93. The Birthday Problem

lemma birthday_paradox:
  assumes "card S = 23" "card T = 365"
  shows "2 * card {f  S→E S T. ¬ inj_on f S}  card (S →E T)"

https://isabelle.in.tum.de/dist/library/HOL/HOL-ex/Birthday_Paradox.html

94. The Law of Cosines

lemma cosine_law_triangle:
  "dist b c ^ 2 = dist a b ^ 2 + dist a c ^ 2 - 2 * dist a b * dist a c * cos (angle b a c)"

https://isa-afp.org/entries/Triangle.shtml

95. Ptolemy's Theorem

theorem ptolemy:
  fixes A B C D center :: "real ^ 2"
  assumes "dist center A = radius" and "dist center B = radius"
  assumes "dist center C = radius" and "dist center D = radius"
  assumes ordering_of_points:
    "radiant_of (A - center) ≤ radiant_of (B - center)"
    "radiant_of (B - center) ≤ radiant_of (C - center)"
    "radiant_of (C - center) ≤ radiant_of (D - center)"
  shows "dist A C * dist B D = dist A B * dist C D + dist A D * dist B C"

https://www.isa-afp.org/entries/Ptolemys_Theorem.shtml

96. Principle of Inclusion/Exclusion

lemma card_UNION:
  assumes "finite A" and "k  A. finite k"
  shows "card (A) = nat (I | I  A  I  {}. (- 1) ^ (card I + 1) * int (card (I)))"

https://isabelle.in.tum.de/dist/library/HOL/HOL/Binomial.html

97. Cramer's Rule

lemma cramer:
  fixes A ::"real^'n^'n"
  assumes d0: "det A  0"
  shows "A *v x = b  x = (χ k. det(χ i j. if j=k then b$i else A$i$j) / det A)"

https://isabelle.in.tum.de/dist/library/HOL/HOL-Analysis/Determinants.html

98. Bertrand's Postulate

theorem bertrand: "n > 1 ⟹ ∃p∈{n<..<2*n}. prime p"

https://www.isa-afp.org/entries/Bertrands_Postulate.shtml

99. Buffon Needle Problem

theorem buffon_short:
assumes "l ≤ d"
shows   "emeasure (buffon l d) {True} = 2 * l / (d * pi)"

theorem buffon_long:
assumes "l ≥ d"
shows   "emeasure (buffon l d) {True} = 2 / pi * ((l / d) - sqrt ((l / d)⇧2 - 1) + arccos (d / l))"

https://www.isa-afp.org/entries/Buffons_Needle.shtml

100. Descartes Rule of Signs

theorem descartes_sign_rule:
  fixes p :: "real poly"
  assumes "p ≠ 0"
  shows "∃d. even d ∧ coeff_sign_changes p = count_pos_roots p + d"

https://isa-afp.org/entries/Descartes_Sign_Rule.shtml