Lean AI formalization leaderboard

lean-eval

Lean AI formalization leaderboard

Public results on a benchmark of hard Lean formalization problems, based on solutions submitted by external participants. Expand any row to inspect solved theorems, extracted statements, and links to public proofs when available.

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211

Leaderboard

Model rankings

Ranked by main benchmark problems solved. Internal test problems do not count toward the score.

1Seed Prover (ByteDance)141 solved

Problems uniquely solved by this model

Fáry–Milnor theorem (knot total curvature ≤ 4π implies unknotted)
fary_milnor

Verso theorem preview

theorem declaration uses `sorry`fary_milnor_total_curvature {r : ℝ → LeanEval.Geometry.FaryMilnorProblem.Space} (_hknot : LeanEval.Geometry.FaryMilnorProblem.IsSmoothKnot r) (_hK : LeanEval.Geometry.FaryMilnorProblem.totalCurvature r ≤ 4 * Real.pi) : LeanEval.Geometry.FaryMilnorProblem.IsUnknotted r := r:ℝ → Space_hknot:IsSmoothKnot r_hK:totalCurvature r ≤ 4 * π⊢ IsUnknotted r All goals completed! 🐙
#1
How produced

Automatically proved by Seed Prover.

Novikov's theorem: the word problem is undecidable for finitely presented groups
novikov_unsolvable

Verso theorem preview

theorem declaration uses `sorry`novikov_unsolvable : ∃ (n : ℕ) (rels : Set (FreeGroup (Fin n))), rels.Finite ∧ ¬ LeanEval.GroupTheory.NovikovUnsolvableProblem.WordProblemSolvable (PresentedGroup.mk rels) := ⊢ ∃ n rels, rels.Finite ∧ ¬WordProblemSolvable (PresentedGroup.mk rels) All goals completed! 🐙
#2
How produced

Automatically proved by Seed Prover.

Existence of a chiral oriented knot
exists_chiral_knot

Verso theorem preview

theorem declaration uses `sorry`exists_chiral_knot : ∃ K : LeanEval.KnotTheory.Knot, K.Chiral := ⊢ ∃ K, K.Chiral All goals completed! 🐙
#3
How produced

Automatically proved by Seed Prover.

Upper bound theorem for geometric simplicial spheres (Stanley 1975)
upper_bound_simplicial_spheres

Verso theorem preview

theorem declaration uses `sorry`upper_bound_theorem_simplicial_spheres {d n k : ℕ} (X : LeanEval.Combinatorics.UpperBoundSimplicialSpheresProblem.FiniteSimplicialSphere d) (_hn : LeanEval.Combinatorics.UpperBoundSimplicialSpheresProblem.faceCount X 0 = n) (_hk : k < d) : LeanEval.Combinatorics.UpperBoundSimplicialSpheresProblem.faceCount X k ≤ LeanEval.Combinatorics.UpperBoundSimplicialSpheresProblem.cyclicPolytopeFaceCount n d k := d:ℕn:ℕk:ℕX:FiniteSimplicialSphere d_hn:faceCount X 0 = n_hk:k < d⊢ faceCount X k ≤ cyclicPolytopeFaceCount n d k All goals completed! 🐙
#4
How produced

Automatically proved by Seed Prover.

Weak Morse inequalities
weak_morse_inequality

Verso theorem preview

theorem declaration uses `sorry`weak_morse_inequality {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] {M : Type} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] [CompactSpace M] [T2Space M] (f : M → ℝ) (_hf : LeanEval.Geometry.WeakMorseInequality.IsMorseFunction I f) (k : ℕ) : bettiNumber M k ≤ morseCount I f k := E:Type u_1inst✝⁹:NormedAddCommGroup Einst✝⁸:NormedSpace ℝ Einst✝⁷:FiniteDimensional ℝ EH:Type u_2inst✝⁶:TopologicalSpace HI:ModelWithCorners ℝ E Hinst✝⁵:I.BoundarylessM:Typeinst✝⁴:TopologicalSpace Minst✝³:ChartedSpace H Minst✝²:IsManifold I ∞ Minst✝¹:CompactSpace Minst✝:T2Space Mf:M → ℝ_hf:IsMorseFunction I fk:ℕ⊢ bettiNumber M k ≤ morseCount I f k All goals completed! 🐙
#5
How produced

Automatically proved by Seed Prover.

Morse inequalities
morse_inequality

Verso theorem preview

theorem declaration uses `sorry`morse_inequality {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} [I.Boundaryless] {M : Type} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I ∞ M] [CompactSpace M] [T2Space M] (f : M → ℝ) (_hf : LeanEval.Geometry.MorseInequalities.IsMorseFunction I f) (k : ℕ) : LeanEval.Geometry.MorseInequalities.alternatingPartialSum (bettiNumber M) k ≤ LeanEval.Geometry.MorseInequalities.alternatingPartialSum (morseCount I f) k := E:Type u_1inst✝⁹:NormedAddCommGroup Einst✝⁸:NormedSpace ℝ Einst✝⁷:FiniteDimensional ℝ EH:Type u_2inst✝⁶:TopologicalSpace HI:ModelWithCorners ℝ E Hinst✝⁵:I.BoundarylessM:Typeinst✝⁴:TopologicalSpace Minst✝³:ChartedSpace H Minst✝²:IsManifold I ∞ Minst✝¹:CompactSpace Minst✝:T2Space Mf:M → ℝ_hf:IsMorseFunction I fk:ℕ⊢ alternatingPartialSum (bettiNumber M) k ≤ alternatingPartialSum (morseCount I f) k All goals completed! 🐙
#6
How produced

Automatically proved by Seed Prover.

Mandelbrot set is connected (Douady–Hubbard)
mandelbrot_connected

Verso theorem preview

theorem declaration uses `sorry`mandelbrot_connected : IsConnected LeanEval.ComplexAnalysis.MandelbrotProblem.Mandelbrot := ⊢ IsConnected Mandelbrot All goals completed! 🐙
#7
How produced

Automatically proved by Seed Prover.

Dehn–Sommerville equations for simplicial spheres
dehn_sommerville

Verso theorem preview

theorem declaration uses `sorry`dehn_sommerville {d j : ℕ} (X : LeanEval.Combinatorics.DehnSommerville.FiniteSimplicialSphere d) (hj : j ≤ d) : hVector X j = hVector X (d - j) := d:ℕj:ℕX:FiniteSimplicialSphere dhj:j ≤ d⊢ hVector X j = hVector X (d - j) All goals completed! 🐙
#8
How produced

Automatically proved by Seed Prover.

Fatou–Julia / Cantor dichotomy
fatou_julia_dichotomy

Verso theorem preview

theorem declaration uses `sorry`julia_cantor_dichotomy (c : ℂ) : (c ∈ LeanEval.ComplexAnalysis.FatouJuliaProblem.Mandelbrot → IsConnected (LeanEval.ComplexAnalysis.FatouJuliaProblem.FilledJulia c)) ∧ (c ∉ LeanEval.ComplexAnalysis.FatouJuliaProblem.Mandelbrot → Nonempty ((LeanEval.ComplexAnalysis.FatouJuliaProblem.FilledJulia c) ≃ₜ (ℕ → Bool))) := c:ℂ⊢ (c ∈ Mandelbrot → IsConnected (FilledJulia c)) ∧ (c ∉ Mandelbrot → Nonempty (↑(FilledJulia c) ≃ₜ (ℕ → Bool))) All goals completed! 🐙
#9
How produced

Automatically proved by Seed Prover.

Gleason's theorem (finite-dimensional)
gleason_theorem_finite

Verso theorem preview

theorem declaration uses `sorry`gleason_theorem_finite {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] [FiniteDimensional ℂ H] (hdim : 3 ≤ Module.finrank ℂ H) (f : LeanEval.Analysis.FrameFunction H) : ∃! ρ : H →L[ℂ] H, ContinuousLinearMap.IsPositive ρ ∧ reTr ρ = 1 ∧ ∀ P : H →L[ℂ] H, LeanEval.Analysis.IsOrthProj P → f.μ P = reTr (ρ * P) := H:Type u_1inst✝³:NormedAddCommGroup Hinst✝²:InnerProductSpace ℂ Hinst✝¹:CompleteSpace Hinst✝:FiniteDimensional ℂ Hhdim:3 ≤ Module.finrank ℂ Hf:FrameFunction H⊢ ∃! ρ, ρ.IsPositive ∧ reTr ρ = 1 ∧ ∀ (P : H →L[ℂ] H), IsOrthProj P → f.μ P = reTr (ρ * P) All goals completed! 🐙
#10
How produced

Automatically proved by Seed Prover.

Gleason's theorem (separable Hilbert space)
gleason_theorem_separable

Lean theorem statement

/-- **Gleason's theorem**, separable Hilbert space version (Gleason's original 1957
formulation). For a separable complex Hilbert space `H` of dimension at least `3`, every
frame function on the unit sphere of `H` is given by `x ↦ re ⟨x, ρ x⟩` for some positive
bounded operator `ρ`. (The Lean conclusion does not separately assert trace-class /
`Tr ρ = 1`; see the file docstring.) -/
theorem gleason_theorem_separable
    {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
      [CompleteSpace H] [TopologicalSpace.SeparableSpace H]
    (hdim : 3 ≤ Module.rank ℂ H)
    (f : SphereFrameFunction H) :
    ∃ ρ : H →L[ℂ] H,
      ContinuousLinearMap.IsPositive ρ ∧
      ∀ x : Metric.sphere (0 : H) 1,
        f.f x = (inner ℂ (x : H) (ρ (x : H))).re := by
  sorry
#11
How produced

Automatically proved by Seed Prover.

Wieferich's theorem g(3) = 9
wieferich_g_three

Lean theorem statement

/-- **Wieferich's theorem `g(3) = 9`.** Every natural number is a sum of nine
cubes, and nine is necessary: some `n` (namely `23`) is not a sum of eight
cubes. -/
theorem wieferich_g_three :
    (∀ n : ℕ, IsSumOfCubes 9 n) ∧ ∃ n : ℕ, ¬ IsSumOfCubes 8 n := by
  sorry
#12
How produced

Automatically proved by Seed Prover.

Other solved problems

Onsager's 2D Ising phase transition
ising_2d_phase_transition

Verso theorem preview

theorem declaration uses `sorry`ising_2d_phase_transition : ∃ (F : ℝ → ℝ) (βc : ℝ), 0 < βc ∧ (∀ β : ℝ, Tendsto (fun n : ℕ => finiteIsingFreeEnergySeq n β) atTop (𝓝 (F β))) ∧ ¬ AnalyticAt ℝ F βc := ⊢ ∃ F βc, 0 < βc ∧ (∀ (β : ℝ), Tendsto (fun n => finiteIsingFreeEnergySeq n β) atTop (𝓝 (F β))) ∧ ¬AnalyticAt ℝ F βc All goals completed! 🐙
#13
How produced

Automatically proved by Seed Prover.

KAM persistence of an invariant curve
kam_invariant_curve

Lean theorem statement

/-- **KAM theorem (persistence of an invariant curve).** For real-analytic,
`1`-periodic, non-constant, mean-zero `f` and Diophantine `α`, for all small
`|c|` the twist-map functional equation
`q(t+α) − 2q(t) + q(t−α) = c·f(q(t))` has a smooth strictly increasing solution
`q` with `q − id` periodic — the `c = 0` curve `q₀(t) = t` persists as a smooth
invariant curve of rotation number `α`. -/
theorem kam_invariant_curve
    (α : ℝ) (_hα : IsDiophantine α)
    (f : ℝ → ℝ)
    (_hf_analytic : AnalyticOnNhd ℝ f Set.univ)
    (_hf_per : Function.Periodic f 1)
    (_hf_nonconst : ¬ ∃ k : ℝ, ∀ x, f x = k)
    (_hf_mean : ∫ x in (0 : ℝ)..1, f x = 0) :
    ∃ c₀ : ℝ, 0 < c₀ ∧ ∀ c : ℝ, |c| < c₀ →
      ∃ q : ℝ → ℝ,
        ContDiff ℝ ∞ q ∧ StrictMono q ∧
        Function.Periodic (fun t => q t - t) 1 ∧
        ∀ t : ℝ, q (t + α) - 2 * q t + q (t - α) = c * f (q t) := by
  sorry
#14
How produced

Automatically proved by Seed Prover.

Existence of a simple group of order 10200960 with a 22-dim irrep whose tensor square has 4 isotypic components
m23_irrep_tensor_square_decomp

Verso theorem preview

theorem declaration uses `sorry`m23_irrep_tensor_square_decomp : ∃ (G : Type) (_ : Group G) (_ : Fintype G), Fintype.card G = 10200960 ∧ IsSimpleGroup G ∧ ∃ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (ρ : Representation ℂ G V), Module.finrank ℂ V = 22 ∧ ρ.IsIrreducible ∧ (@isotypicComponents (MonoidAlgebra ℂ G) (V ⊗[ℂ] V) _ _ (Module.compHom (V ⊗[ℂ] V) (Representation.asAlgebraHom (ρ.tprod ρ)).toRingHom)).ncard = 4 := ⊢ ∃ G x x_1, Fintype.card G = 10200960 ∧ IsSimpleGroup G ∧ ∃ V x_2 x_3 ρ, Module.finrank ℂ V = 22 ∧ ρ.IsIrreducible ∧ (isotypicComponents (MonoidAlgebra ℂ G) (V ⊗[ℂ] V)).ncard = 4 All goals completed! 🐙
#15
How produced

Automatically proved by Seed Prover.

Thue–Siegel–Roth theorem (irrationality measure ≤ 2 for algebraic irrationals)
thue_siegel_roth

Verso theorem preview

theorem declaration uses `sorry`thueSiegelRoth (x : ℝ) (_h_irr : Irrational x) (_h_alg : IsAlgebraic ℤ x) : LeanEval.NumberTheory.ThueSiegelRothProblem.IsDiophantine x := x:ℝ_h_irr:Irrational x_h_alg:IsAlgebraic ℤ x⊢ IsDiophantine x All goals completed! 🐙
#16
How produced

Automatically proved by Seed Prover.

Commuting probabilities are closed
commProb_closed

Verso theorem preview

theorem declaration uses `sorry`commProb_closed : IsClosed ({p : ℝ | ∃ (G : Type) (hG : Group G), commProb G = p}) := ⊢ IsClosed {p | ∃ G hG, ↑(commProb G) = p} All goals completed! 🐙
#17
How produced

Automatically proved by Seed Prover.

Seventeen wallpaper groups (Pólya–Niggli 1924)
wallpaper_groups_17

Verso theorem preview

theorem declaration uses `sorry`there_are_17_wallpaper_groups : crystallographicCount 2 = 17 := ⊢ crystallographicCount 2 = 17 All goals completed! 🐙
#18
How produced

Automatically proved by Seed Prover.

Margulis–Ruelle inequality
margulis_ruelle

Verso theorem preview

theorem declaration uses `sorry`margulis_ruelle (T T_inv : LeanEval.Dynamics.EucPlane → LeanEval.Dynamics.EucPlane) (hT_smooth : ContDiff ℝ 2 T) (hT_inv_smooth : ContDiff ℝ 2 T_inv) (hT_left : Function.LeftInverse T_inv T) (hT_right : Function.RightInverse T_inv T) (K : Set LeanEval.Dynamics.EucPlane) (hK_compact : IsCompact K) (hK_inv : T '' K = K) (μ : Measure LeanEval.Dynamics.EucPlane) [IsProbabilityMeasure μ] (hμ_supp : μ Kᶜ = 0) (hμ_pres : MeasurePreserving T μ μ) (hμ_erg : Ergodic T μ) : kolmogorovSinaiEntropy μ T ≤ max 0 (∫ x, lyapunovUpperAt T x ∂μ) + max 0 (∫ x, lyapunovLowerAt T x ∂μ) := T:EucPlane → EucPlaneT_inv:EucPlane → EucPlanehT_smooth:ContDiff ℝ 2 ThT_inv_smooth:ContDiff ℝ 2 T_invhT_left:Function.LeftInverse T_inv ThT_right:Function.RightInverse T_inv TK:Set EucPlanehK_compact:IsCompact KhK_inv:T '' K = Kμ:Measure EucPlaneinst✝:IsProbabilityMeasure μhμ_supp:μ Kᶜ = 0hμ_pres:MeasurePreserving T μ μhμ_erg:Ergodic T μ⊢ kolmogorovSinaiEntropy μ T ≤ max 0 (∫ (x : EucPlane), lyapunovUpperAt T x ∂μ) + max 0 (∫ (x : EucPlane), lyapunovLowerAt T x ∂μ) All goals completed! 🐙
#19
How produced

Automatically proved by Seed Prover.

Poincaré–Bendixson theorem
poincare_bendixson

Verso theorem preview

theorem declaration uses `sorry`poincare_bendixson (F : LeanEval.Dynamics.Plane → LeanEval.Dynamics.Plane) (_hF : ContDiff ℝ 1 F) (γ : ℝ → LeanEval.Dynamics.Plane) (_hγ : IsIntegralCurveOn γ (fun _ x => F x) (Set.Ici 0)) : ¬ Bornology.IsBounded (γ '' Set.Ici 0) ∨ (∃ x₀, F x₀ = 0 ∧ x₀ ∈ ⋂ s : ℝ, closure (γ '' Set.Ici s)) ∨ (∃ T : ℝ, 0 < T ∧ ∃ β : ℝ → LeanEval.Dynamics.Plane, IsIntegralCurve β (fun _ x => F x) ∧ (∀ t, β (t + T) = β t) ∧ F (β 0) ≠ 0 ∧ (⋂ s : ℝ, closure (γ '' Set.Ici s)) = Set.range β) := F:Plane → Plane_hF:ContDiff ℝ 1 Fγ:ℝ → Plane_hγ:IsIntegralCurveOn γ (fun x x_1 => F x_1) (Ici 0)⊢ ¬Bornology.IsBounded (γ '' Ici 0) ∨ (∃ x₀, F x₀ = 0 ∧ x₀ ∈ ⋂ s, closure (γ '' Ici s)) ∨ ∃ T, 0 < T ∧ ∃ β, (IsIntegralCurve β fun x x_1 => F x_1) ∧ (∀ (t : ℝ), β (t + T) = β t) ∧ F (β 0) ≠ 0 ∧ ⋂ s, closure (γ '' Ici s) = range β All goals completed! 🐙
#20
How produced

Automatically proved by Seed Prover.

Mergelyan's theorem
mergelyan_theorem

Lean theorem statement

/-- **Mergelyan's theorem.** For a compact `K ⊆ ℂ` with connected
complement and `f : ℂ → ℂ` continuous on `K` and analytic on the
interior of `K`, every `ε > 0` admits a complex polynomial `p` with
`‖f z − p(z)‖ < ε` on `K`. -/
theorem mergelyan (K : Set ℂ) (_hK : IsCompact K) (_hKc : IsConnected (Kᶜ))
    (f : ℂ → ℂ) (_hfc : ContinuousOn f K) (_hfh : AnalyticOnNhd ℂ f (interior K))
    (ε : ℝ) (_hε : 0 < ε) :
    ∃ p : ℂ[X], ∀ z ∈ K, ‖f z - p.eval z‖ < ε := by
  sorry
#21
How produced

Automatically proved by Seed Prover.

pi_n of the n-sphere is Z
pin_sphere_n_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pin_sphere_n_mulEquiv_int (n : ℕ) (x : Metric.sphere (0 : EuclideanSpace ℝ (Fin (n + 2))) 1) : Nonempty (HomotopyGroup.Pi (n + 1) (Metric.sphere (0 : EuclideanSpace ℝ (Fin (n + 2))) 1) x ≃* Multiplicative ℤ) := n:ℕx:↑(Metric.sphere 0 1)⊢ Nonempty (HomotopyGroup.Pi (n + 1) (↑(Metric.sphere 0 1)) x ≃* Multiplicative ℤ) All goals completed! 🐙
#22
How produced

Automatically proved by Seed Prover.

pi_3 of the 2-sphere is Z
pi3_sphere_two_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi3_sphere_two_mulEquiv_int (x : Metric.sphere (0 : EuclideanSpace ℝ (Fin 3)) 1) : Nonempty (HomotopyGroup.Pi 3 (Metric.sphere (0 : EuclideanSpace ℝ (Fin 3)) 1) x ≃* Multiplicative ℤ) := x:↑(Metric.sphere 0 1)⊢ Nonempty (HomotopyGroup.Pi 3 (↑(Metric.sphere 0 1)) x ≃* Multiplicative ℤ) All goals completed! 🐙
#23
How produced

Automatically proved by Seed Prover.

Bézout's theorem (projective, with multiplicity)
bezout_projective_multiplicity

Verso theorem preview

theorem declaration uses `sorry`bezout_multiplicity [IsAlgClosed K] {n : ℕ} (f : Fin n → MvPolynomial (Fin (n + 1)) K) (d : Fin n → ℕ) (_hd : ∀ k, (f k).IsHomogeneous (d k)) (_hdeg : ∀ k, (f k).totalDegree = d k) (_hd_pos : ∀ k, 1 ≤ d k) (_hfin : (⋂ k, LeanEval.AlgebraicGeometry.vanishingSet (f k)).Finite) : ∑ᶠ p ∈ (⋂ k, LeanEval.AlgebraicGeometry.vanishingSet (f k)), intersectionMultiplicity f p = (∏ k, d k : ℕ∞) := K:Type u_1inst✝¹:Field Kinst✝:IsAlgClosed Kn:ℕf:Fin n → MvPolynomial (Fin (n + 1)) Kd:Fin n → ℕ_hd:∀ (k : Fin n), (f k).IsHomogeneous (d k)_hdeg:∀ (k : Fin n), (f k).totalDegree = d k_hd_pos:∀ (k : Fin n), 1 ≤ d k_hfin:(⋂ k, vanishingSet (f k)).Finite⊢ ∑ᶠ (p : ProjSpace K n) (_ : p ∈ ⋂ k, vanishingSet (f k)), intersectionMultiplicity f p = ∏ k, ↑(d k) All goals completed! 🐙
#24
How produced

Automatically proved by Seed Prover.

Jordan–Brouwer separation theorem
jordan_brouwer

Verso theorem preview

theorem declaration uses `sorry`jordan_brouwer (d : ℕ) (_hd : 2 ≤ d) (r : Metric.sphere (0 : EuclideanSpace ℝ (Fin d)) 1 → EuclideanSpace ℝ (Fin d)) (_hcont : Continuous r) (_hinj : Function.Injective r) : Nat.card (ConnectedComponents ((Set.range r)ᶜ : Set (EuclideanSpace ℝ (Fin d)))) = 2 := d:ℕ_hd:2 ≤ dr:↑(Metric.sphere 0 1) → EuclideanSpace ℝ (Fin d)_hcont:Continuous r_hinj:Function.Injective r⊢ Nat.card (ConnectedComponents ↑(Set.range r)ᶜ) = 2 All goals completed! 🐙
#25
How produced

Automatically proved by Seed Prover.

A 3-manifold group with no faithful representation into GL(4, ℝ)
nonlinear_three_manifold_group

Verso theorem preview

theorem declaration uses `sorry`nonlinear_three_manifold_group : ∃ (M : LeanEval.Topology.Closed3Manifold) (x : M.carrier), ∀ f : FundamentalGroup M.carrier x →* GL (Fin 4) ℝ, ¬ Function.Injective f := ⊢ ∃ M x, ∀ (f : FundamentalGroup M.carrier x →* GL (Fin 4) ℝ), ¬Function.Injective ⇑f All goals completed! 🐙
#26
How produced

Automatically proved by Seed Prover.

Chudnovsky formula for pi inverse
chudnovsky_formula_for_pi_inv

Verso theorem preview

theorem declaration uses `sorry`chudnovsky_formula_for_pi_inv : chudnovskySum = π⁻¹ := ⊢ chudnovskySum = π⁻¹ All goals completed! 🐙
#27
How produced

Automatically proved by Seed Prover.

The Hopf Umlaufsatz (theorem of turning tangents)
hopf_umlaufsatz

Verso theorem preview

theorem declaration uses `sorry`hopf_umlaufsatz {r : ℝ → LeanEval.Geometry.HopfUmlaufsatz.Plane} {α : ℝ → ℝ} (_hr : LeanEval.Geometry.HopfUmlaufsatz.IsPositiveSimpleClosedUnitSpeedCurve r) (_hα : LeanEval.Geometry.HopfUmlaufsatz.IsTangentAngleLift r α) : totalCurvature α = 2 * Real.pi := r:ℝ → Planeα:ℝ → ℝ_hr:IsPositiveSimpleClosedUnitSpeedCurve r_hα:IsTangentAngleLift r α⊢ totalCurvature α = 2 * Real.pi All goals completed! 🐙
#28
How produced

Automatically proved by Seed Prover.

Darboux's theorem (symplectic forms are locally standard)
darboux

Verso theorem preview

theorem declaration uses `sorry`darboux {n : ℕ} {U : Set (LeanEval.Geometry.Darboux.E n)} (_hU : IsOpen U) (α : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n [⋀^Fin 2]→L[ℝ] ℝ) (_hα : LeanEval.Geometry.Darboux.IsSymplecticOn α U) {x : LeanEval.Geometry.Darboux.E n} (_hx : x ∈ U) : ∃ φ : OpenPartialHomeomorph (LeanEval.Geometry.Darboux.E n) (LeanEval.Geometry.Darboux.E n), x ∈ φ.source ∧ φ.source ⊆ U ∧ ContDiffOn ℝ ∞ (φ : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n) φ.source ∧ ContDiffOn ℝ ∞ (φ.symm : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n) φ.target ∧ ∀ z ∈ φ.target, LeanEval.Geometry.Darboux.IsDarbouxNormal ((α (φ.symm z)).compContinuousLinearMap (fderiv ℝ (φ.symm : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n) z)) := n:ℕU:Set (E n)_hU:IsOpen Uα:E n → E n [⋀^Fin 2]→L[ℝ] ℝ_hα:IsSymplecticOn α Ux:E n_hx:x ∈ U⊢ ∃ φ, x ∈ φ.source ∧ φ.source ⊆ U ∧ ContDiffOn ℝ ∞ (↑φ) φ.source ∧ ContDiffOn ℝ ∞ (↑φ.symm) φ.target ∧ ∀ z ∈ φ.target, IsDarbouxNormal ((α (↑φ.symm z)).compContinuousLinearMap (fderiv ℝ (↑φ.symm) z)) All goals completed! 🐙
#29
How produced

Automatically proved by Seed Prover.

Morrison–Walker Lemma B.0.1: adapting families of maps to open covers
families_of_maps_b01

Verso theorem preview

/-- **Lemma B.0.1** of Morrison–Walker, *The Blob Complex* (arXiv:1009.5025, §B), continuous case. -/ theorem declaration uses `sorry`continuous {P : Set (Fin k → ℝ)} (_hP : IsPolyhedron P) [CompactSpace X] (U : ι → Set X) (_hUopen : ∀ α, IsOpen (U α)) (ρ : PartitionOfUnity ι X univ) (_hρ : ρ.IsSubordinate U) (f : C(P × X, T)) : ∃ F : C(I × P × X, T), (∀ p : P, ∀ x : X, F (0, p, x) = f (p, x)) ∧ (∃ K : Subdivision P, ∀ D ∈ K.complex.facets, AdaptedTo U k (fun q : closedCell P D × X => F (1, q.1.1, q.2))) ∧ (∀ S : Set X, Supported (f := f.toFun) S → Supported (fun q : (I × P) × X => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ Q : Set P, IsBoundarySubpolyhedron Q → ∀ S' : Set X, Supported (fun q : Q × X => f (q.1.1, q.2)) S' → Supported (fun q : (I × Q) × X => F (q.1.1, q.1.2.1, q.2)) S') := k:ℕι:Type u_1X:Type u_2T:Type u_3inst✝²:TopologicalSpace Xinst✝¹:TopologicalSpace TP:Set (Fin k → ℝ)_hP:IsPolyhedron Pinst✝:CompactSpace XU:ι → Set X_hUopen:∀ (α : ι), IsOpen[inst✝²] (U α)ρ:PartitionOfUnity ι X_hρ:ρ.IsSubordinate Uf:C(↑P × X, T)⊢ ∃ F, (∀ (p : ↑P) (x : X), F (0, p, x) = f (p, x)) ∧ (∃ K, ∀ D ∈ K.complex.facets, AdaptedTo U k fun q => F (1, ↑q.1, q.2)) ∧ (∀ (S : Set X), Supported f.toFun S → Supported (fun q => F (q.1.1, q.1.2, q.2)) S) ∧ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ (S' : Set X), Supported (fun q => f (↑q.1, q.2)) S' → Supported (fun q => F (q.1.1, ↑q.1.2, q.2)) S' All goals completed! 🐙
/-- **Lemma B.0.1**, bi-Lipschitz variant (part 4 of the paper). -/ theorem declaration uses `sorry`biLipschitz {X T : Type*} [MetricSpace X] [MetricSpace T] [CompactSpace X] {P : Set (Fin k → ℝ)} (_hP : IsPolyhedron P) {ι : Type*} (U : ι → Set X) (_hUopen : ∀ α, IsOpen (U α)) (ρ : PartitionOfUnity ι X univ) (_hρ : ρ.IsSubordinate U) (f : C(P × X, T)) (slice : P → (X ≃ₜ T)) (_h_slice_eq : ∀ p : P, ∀ x : X, f (p, x) = slice p x) (L : NNReal) (_hf_joint : LipschitzWith L f.toFun) (_hf_slice_inv : ∀ p : P, LipschitzWith L (slice p).symm) : ∃ F : C(I × P × X, T), ∃ L' : NNReal, ∃ Slice : I × P → (X ≃ₜ T), (∀ p : P, ∀ x : X, F (0, p, x) = f (p, x)) ∧ (∀ t : I, ∀ p : P, ∀ x : X, F (t, p, x) = Slice (t, p) x) ∧ (∃ K : Subdivision P, ∀ D ∈ K.complex.facets, AdaptedTo U k (fun q : closedCell P D × X => F (1, q.1.1, q.2))) ∧ (∀ S : Set X, Supported (f := f.toFun) S → Supported (fun q : (I × P) × X => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ Q : Set P, IsBoundarySubpolyhedron Q → ∀ S' : Set X, Supported (fun q : Q × X => f (q.1.1, q.2)) S' → Supported (fun q : (I × Q) × X => F (q.1.1, q.1.2.1, q.2)) S') ∧ (∀ tp : I × P, LipschitzWith L' (Slice tp)) ∧ (∀ tp : I × P, LipschitzWith L' (Slice tp).symm) := k:ℕX:Type u_4T:Type u_5inst✝²:MetricSpace Xinst✝¹:MetricSpace Tinst✝:CompactSpace XP:Set (Fin k → ℝ)_hP:IsPolyhedron Pι:Type u_6U:ι → Set X_hUopen:∀ (α : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U α)ρ:PartitionOfUnity ι X_hρ:ρ.IsSubordinate Uf:C(↑P × X, T)slice:↑P → X ≃ₜ T_h_slice_eq:∀ (p : ↑P) (x : X), f (p, x) = (slice p) xL:NNReal_hf_joint:LipschitzWith L f.toFun_hf_slice_inv:∀ (p : ↑P), LipschitzWith L ⇑(slice p).symm⊢ ∃ F L' Slice, (∀ (p : ↑P) (x : X), F (0, p, x) = f (p, x)) ∧ (∀ (t : ↑I) (p : ↑P) (x : X), F (t, p, x) = (Slice (t, p)) x) ∧ (∃ K, ∀ D ∈ K.complex.facets, AdaptedTo U k fun q => F (1, ↑q.1, q.2)) ∧ (∀ (S : Set X), Supported f.toFun S → Supported (fun q => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ (S' : Set X), Supported (fun q => f (↑q.1, q.2)) S' → Supported (fun q => F (q.1.1, ↑q.1.2, q.2)) S') ∧ (∀ (tp : ↑I × ↑P), LipschitzWith L' ⇑(Slice tp)) ∧ ∀ (tp : ↑I × ↑P), LipschitzWith L' ⇑(Slice tp).symm All goals completed! 🐙
#30
How produced

Automatically proved by Seed Prover.

Existence of a non-isotopic pair of oriented knots
exists_nonisotopic_knots

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_knots : ∃ K₁ K₂ : LeanEval.KnotTheory.Knot, ¬ K₁.Isotopic K₂ := ⊢ ∃ K₁ K₂, ¬K₁.Isotopic K₂ All goals completed! 🐙
#31
How produced

Automatically proved by Seed Prover.

Jordan curve theorem
jordan_curve

Verso theorem preview

theorem declaration uses `sorry`jordan_curve (r : Metric.sphere (0 : EuclideanSpace ℝ (Fin 2)) 1 → EuclideanSpace ℝ (Fin 2)) (_hcont : Continuous r) (_hinj : Function.Injective r) : Nat.card (ConnectedComponents ((Set.range r)ᶜ : Set (EuclideanSpace ℝ (Fin 2)))) = 2 := r:↑(Metric.sphere 0 1) → EuclideanSpace ℝ (Fin 2)_hcont:Continuous r_hinj:Function.Injective r⊢ Nat.card (ConnectedComponents ↑(Set.range r)ᶜ) = 2 All goals completed! 🐙
#32
How produced

Automatically proved by Seed Prover.

Radó's theorem on Riemann surfaces
rado_riemannSurface

Verso theorem preview

theorem declaration uses `sorry`rado_riemannSurface {X : Type*} [TopologicalSpace X] [T2Space X] [ConnectedSpace X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) 1 X] : SecondCountableTopology X := X:Type u_1inst✝⁴:TopologicalSpace Xinst✝³:T2Space Xinst✝²:ConnectedSpace Xinst✝¹:ChartedSpace ℂ Xinst✝:IsManifold (modelWithCornersSelf ℂ ℂ) 1 X⊢ SecondCountableTopology X All goals completed! 🐙
#33
How produced

Automatically proved by Seed Prover.

Cauchy–Kovalevskaya theorem
cauchy_kovalevskaya

Verso theorem preview

theorem declaration uses `sorry`cauchy_kovalevskaya {d : ℕ} (F : LeanEval.Analysis.E d × ℝ × ℝ → LeanEval.Analysis.E d) (f : LeanEval.Analysis.E d × ℝ × ℝ → ℝ) (u₀ : LeanEval.Analysis.E d → ℝ) (_hF : AnalyticOnNhd ℝ F univ) (_hf : AnalyticOnNhd ℝ f univ) (_hu₀ : AnalyticOnNhd ℝ u₀ univ) (x₀ : LeanEval.Analysis.E d) : ∃ (U : Set (LeanEval.Analysis.E d × ℝ)) (u : LeanEval.Analysis.E d × ℝ → ℝ), (x₀, (0 : ℝ)) ∈ U ∧ IsOpen U ∧ AnalyticOnNhd ℝ u U ∧ (∀ x : LeanEval.Analysis.E d, (x, (0 : ℝ)) ∈ U → u (x, 0) = u₀ x) ∧ (∀ p ∈ U, fderiv ℝ u p ((0 : LeanEval.Analysis.E d), (1 : ℝ)) = fderiv ℝ u p (F (p.1, p.2, u p), (0 : ℝ)) + f (p.1, p.2, u p)) ∧ (∀ v : LeanEval.Analysis.E d × ℝ → ℝ, AnalyticOnNhd ℝ v U → (∀ x : LeanEval.Analysis.E d, (x, (0 : ℝ)) ∈ U → v (x, 0) = u₀ x) → (∀ p ∈ U, fderiv ℝ v p ((0 : LeanEval.Analysis.E d), (1 : ℝ)) = fderiv ℝ v p (F (p.1, p.2, v p), (0 : ℝ)) + f (p.1, p.2, v p)) → ∀ p ∈ U, u p = v p) := d:ℕF:E d × ℝ × ℝ → E df:E d × ℝ × ℝ → ℝu₀:E d → ℝ_hF:AnalyticOnNhd ℝ F univ_hf:AnalyticOnNhd ℝ f univ_hu₀:AnalyticOnNhd ℝ u₀ univx₀:E d⊢ ∃ U u, (x₀, 0) ∈ U ∧ IsOpen U ∧ AnalyticOnNhd ℝ u U ∧ (∀ (x : E d), (x, 0) ∈ U → u (x, 0) = u₀ x) ∧ (∀ p ∈ U, (fderiv ℝ u p) (0, 1) = (fderiv ℝ u p) (F (p.1, p.2, u p), 0) + f (p.1, p.2, u p)) ∧ ∀ (v : E d × ℝ → ℝ), AnalyticOnNhd ℝ v U → (∀ (x : E d), (x, 0) ∈ U → v (x, 0) = u₀ x) → (∀ p ∈ U, (fderiv ℝ v p) (0, 1) = (fderiv ℝ v p) (F (p.1, p.2, v p), 0) + f (p.1, p.2, v p)) → ∀ p ∈ U, u p = v p All goals completed! 🐙
#34
How produced

Automatically proved by Seed Prover.

The Golod–Shafarevich inequality
golod_shafarevich_inequality

Verso theorem preview

theorem declaration uses `sorry`golod_shafarevich_inequality (p : ℕ) [Fact p.Prime] (Q : Type) [Group Q] [TopologicalSpace Q] [IsTopologicalGroup Q] [DiscreteTopology Q] [Finite Q] : IsPGroup p Q → Nontrivial Q → (generatorRank Q : ℝ) ^ 2 < 4 * (relationRank p Q : ℝ) := p:ℕinst✝⁵:Fact (Nat.Prime p)Q:Typeinst✝⁴:Group Qinst✝³:TopologicalSpace Qinst✝²:IsTopologicalGroup Qinst✝¹:DiscreteTopology Qinst✝:Finite Q⊢ IsPGroup p Q → Nontrivial Q → ↑(generatorRank Q) ^ 2 < 4 * ↑(relationRank p Q) All goals completed! 🐙
#35
How produced

Automatically proved by Seed Prover.

Halmos's generic weak-mixing theorem
halmos_generic_weak_mixing

Verso theorem preview

theorem declaration uses `sorry`generic_weakly_mixing [StandardBorelSpace X] (m : Measure X) [IsProbabilityMeasure m] [NoAtoms m] : (∃ G : Set (LeanEval.Dynamics.HalmosGenericWeakMixingProblem.Automorphism m), IsGδ G ∧ Dense G ∧ ∀ T ∈ G, LeanEval.Dynamics.HalmosGenericWeakMixingProblem.IsWeaklyMixing m T) ∧ (∀ T : LeanEval.Dynamics.HalmosGenericWeakMixingProblem.Automorphism m, LeanEval.Dynamics.HalmosGenericWeakMixingProblem.IsWeaklyMixing m T → Ergodic (T.toEquiv : X → X) m) := X:Type u_1inst✝³:MeasurableSpace Xinst✝²:StandardBorelSpace Xm:Measure Xinst✝¹:IsProbabilityMeasure minst✝:NoAtoms m⊢ (∃ G, IsGδ G ∧ Dense G ∧ ∀ T ∈ G, IsWeaklyMixing m T) ∧ ∀ (T : Automorphism m), IsWeaklyMixing m T → Ergodic (⇑T.toEquiv) m All goals completed! 🐙
#36
How produced

Automatically proved by Seed Prover.

Pick's theorem
pick

Verso theorem preview

theorem declaration uses `sorry`pick {n : ℕ} (hn : 3 ≤ n) (v : Fin n → ℤ × ℤ) (hsimple : LeanEval.Geometry.PicksTheorem.IsSimple (LeanEval.Geometry.PicksTheorem.latPoly v)) : area ((LeanEval.Geometry.PicksTheorem.latPoly v).boundary (R := ℝ)) = (interiorPts v : ℝ) + (boundaryPts v : ℝ) / 2 - 1 := n:ℕhn:3 ≤ nv:Fin n → ℤ × ℤhsimple:IsSimple (latPoly v)⊢ area (Polygon.boundary ℝ (latPoly v)) = ↑(interiorPts v) + ↑(boundaryPts v) / 2 - 1 All goals completed! 🐙
#37
How produced

Automatically proved by Seed Prover.

Riesz brothers' theorem
riesz_brothers_theorem

Lean theorem statement

/-- **Riesz brothers' theorem.** Let `μ` be a complex Borel measure on
the unit circle such that `∫ z^n dμ = 0` for every `n ≥ 1`. Then `μ` is
absolutely continuous with respect to Haar measure, and more precisely
`dμ = h dm` for some `h ∈ H^1`, i.e. some integrable density whose negative
Fourier coefficients vanish. -/
theorem riesz_brothers_theorem (μ : ComplexMeasure UnitAddCircle)
    (hμ : ∀ n : ℕ, 1 ≤ n → ∫ᵛ z, fourier n z ∂[ContinuousLinearMap.mul ℝ ℂ; μ] = 0) :
    μ ≪ᵥ AddCircle.haarAddCircle.toENNRealVectorMeasure := by
  sorry
#38
How produced

Automatically proved by Seed Prover.

Wigner semicircle law
wigner_semicircle

Verso theorem preview

theorem declaration uses `sorry`wigner_semicircle {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → ℕ → Ω → ℝ) (_hX_meas : ∀ i j, Measurable (X i j)) (_hX_indep : iIndepFun (fun ij : {p : ℕ × ℕ // p.1 ≤ p.2} => X ij.val.1 ij.val.2) μ) (_hX_iid : ∀ i j i' j', i ≤ j → i' ≤ j' → ProbabilityTheory.IdentDistrib (X i j) (X i' j') μ μ) (_hX_int : ∀ i j, i ≤ j → Integrable (X i j) μ) (_hX_sq_int : ∀ i j, i ≤ j → Integrable (fun ω => (X i j ω) ^ 2) μ) (_hX_mean : ∀ i j, i ≤ j → ∫ ω, X i j ω ∂μ = 0) (_hX_var : ∀ i j, i ≤ j → ∫ ω, (X i j ω) ^ 2 ∂μ = 1) : ∀ᵐ ω ∂μ, ∀ (f : ℝ → ℝ), Continuous f → (∃ M, ∀ x, ‖f x‖ ≤ M) → Tendsto (fun n : ℕ => ∫ x, f x ∂ (empiricalSpectralMeasureHerm (wignerMatrix_isHermitian X n ω)).map (fun x : ℝ => x / Real.sqrt n)) atTop (𝓝 (∫ x, f x ∂semicircleLaw)) := Ω:Type u_1inst✝¹:MeasurableSpace Ωμ:Measure Ωinst✝:IsProbabilityMeasure μX:ℕ → ℕ → Ω → ℝ_hX_meas:∀ (i j : ℕ), Measurable (X i j)_hX_indep:iIndepFun (fun ij => X (↑ij).1 (↑ij).2) μ_hX_iid:∀ (i j i' j' : ℕ), i ≤ j → i' ≤ j' → IdentDistrib (X i j) (X i' j') μ μ_hX_int:∀ (i j : ℕ), i ≤ j → Integrable (X i j) μ_hX_sq_int:∀ (i j : ℕ), i ≤ j → Integrable (fun ω => X i j ω ^ 2) μ_hX_mean:∀ (i j : ℕ), i ≤ j → ∫ (ω : Ω), X i j ω ∂μ = 0_hX_var:∀ (i j : ℕ), i ≤ j → ∫ (ω : Ω), X i j ω ^ 2 ∂μ = 1⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (f : ℝ → ℝ), Continuous f → (∃ M, ∀ (x : ℝ), ‖f x‖ ≤ M) → Tendsto (fun n => ∫ (x : ℝ), f x ∂Measure.map (fun x => x / √↑n) (empiricalSpectralMeasureHerm ⋯)) atTop (𝓝 (∫ (x : ℝ), f x ∂semicircleLaw)) All goals completed! 🐙
#39
How produced

Automatically proved by Seed Prover.

No bounded projection from L^1 onto H^1
H1_not_closedComplemented

Verso theorem preview

theorem declaration uses `sorry`H1_not_closedComplemented : ¬ LeanEval.Analysis.H1.ClosedComplemented := ⊢ ¬H1.ClosedComplemented All goals completed! 🐙
#40
How produced

Automatically proved by Seed Prover.

Wiener–Lévy theorem
wiener_levy_analytic_calculus

Verso theorem preview

theorem declaration uses `sorry`wiener_levy_analytic_calculus (f : C(AddCircle T, ℂ)) (φ : ℂ → ℂ) (U : Set ℂ) (hf : LeanEval.Analysis.WienerLevy.InWienerAlgebra f) (hU : IsOpen U) (hrange : range f ⊆ U) (hφ : AnalyticOnNhd ℂ φ U) : ∃ g : C(AddCircle T, ℂ), (∀ x, g x = φ (f x)) ∧ LeanEval.Analysis.WienerLevy.InWienerAlgebra g := T:ℝinst✝:Fact (0 < T)f:C(AddCircle T, ℂ)φ:ℂ → ℂU:Set ℂhf:InWienerAlgebra fhU:IsOpen Uhrange:range ⇑f ⊆ Uhφ:AnalyticOnNhd ℂ φ U⊢ ∃ g, (∀ (x : AddCircle T), g x = φ (f x)) ∧ InWienerAlgebra g All goals completed! 🐙
#41
How produced

Automatically proved by Seed Prover.

Radial symmetry for positive semilinear Poisson solutions
semilinear_poisson_radial_symmetry

Lean theorem statement

/-- **Radial symmetry theorem.** Let `u ∈ C^2(closedBall 0 1)` be a positive
solution of the semilinear Poisson problem `-Δ u = f(u)` in the open unit ball,
with zero Dirichlet boundary values on the unit sphere. If `f : ℝ → ℝ` is
Lipschitz, then `u` is radial: `u x = v ‖x‖` for a nonnegative strictly
decreasing radial profile `v` on `[0, 1]`. -/
theorem semilinear_poisson_radial_symmetry {n : ℕ} (hn : 0 < n)
    {f : ℝ → ℝ} (u : EuclideanSpace ℝ (Fin n) → ℝ)
    (hf_lipschitz : ∃ K : ℝ≥0, LipschitzWith K f)
    (hu_c2 : ContDiffOn ℝ 2 u (closedBall 0 1))
    (hu_solve : SolvesSemilinearPoisson f u)
    (hu_positive : ∀ x ∈ ball 0 1, 0 < u x) :
    ∃ v : ℝ → ℝ≥0,
      StrictAntiOn v (Set.Icc (0 : ℝ) 1) ∧
        ∀ x ∈ closedBall 0 1, u x = v ‖x‖ := by
  sorry
#42
How produced

Automatically proved by Seed Prover.

Fundamental theorem of Riemannian geometry (Levi-Civita)
levi_civita_exists_unique

Verso theorem preview

theorem declaration uses `sorry`levi_civita_exists_unique {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [CompleteSpace E] {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type*} [TopologicalSpace M] [T2Space M] [ChartedSpace H M] [IsManifold I ∞ M] [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)] [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)] : ∃ cov : CovariantDerivative I E (TangentSpace I (M := M)), (ContMDiffCovariantDerivative cov ∞ ∧ cov.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov) ∧ ∀ cov' : CovariantDerivative I E (TangentSpace I (M := M)), (ContMDiffCovariantDerivative cov' ∞ ∧ cov'.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov') → LeanEval.Geometry.LeviCivita.SameOnSmooth cov cov' := E:Type u_1inst✝¹⁰:NormedAddCommGroup Einst✝⁹:NormedSpace ℝ Einst✝⁸:FiniteDimensional ℝ Einst✝⁷:CompleteSpace EH:Type u_2inst✝⁶:TopologicalSpace HI:ModelWithCorners ℝ E HM:Type u_3inst✝⁵:TopologicalSpace Minst✝⁴:T2Space Minst✝³:ChartedSpace H Minst✝²:IsManifold I ∞ Minst✝¹:RiemannianBundle fun x => TangentSpace I xinst✝:IsContMDiffRiemannianBundle I ∞ E fun x => TangentSpace I x⊢ ∃ cov, (cov.ContMDiffCovariantDerivative ∞ ∧ cov.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov) ∧ ∀ (cov' : CovariantDerivative I E (TangentSpace I)), cov'.ContMDiffCovariantDerivative ∞ ∧ cov'.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov' → SameOnSmooth cov cov' All goals completed! 🐙
#43
How produced

Automatically proved by Seed Prover.

Strong Subadditivity of von Neumann Entropy
strong_subadditivity

Lean theorem statement

/-- Strong subadditivity of quantum entropy. We relax the common assumption that M is a normalized
 density matrix to the simpler statement that it's PSD, which holds since normalization just produces
 a positive affine transformation on the entropy. -/
theorem strong_subadditivity (M_ABC : Matrix (A × B × C) (A × B × C) ℂ) (h : M_ABC.PosSemidef) :
    let M_AB : Matrix (A × B) (A × B) ℂ :=
      .traceRight <| M_ABC.reindex (.symm <| .prodAssoc ..) (.symm <| .prodAssoc ..)
    let M_BC : Matrix (B × C) (B × C) ℂ := M_ABC.traceLeft
    let M_B : Matrix B B ℂ := M_BC.traceRight
    entropy M_ABC + entropy M_B ≤ entropy M_AB + entropy M_BC := by
  sorry
#44
How produced

Automatically proved by Seed Prover.

Brun's theorem (convergence of the twin-prime reciprocal sum)
brun_constant_converges

Lean theorem statement

/-- **Brun's theorem.** The reciprocal sum over twin-prime pairs converges. -/
theorem brun_constant_converges :
    Summable twinPrimeReciprocalTerm := by
  sorry
#45
How produced

Automatically proved by Seed Prover.

Frobenius determinant theorem
frobenius_group_determinant

Lean theorem statement

/-- **Frobenius determinant theorem** (§171). The group determinant factors as
a product of irreducible polynomials, each appearing to the power of its own
(total) degree `d_j = deg p_j`, with the factors pairwise non-associated
(*distinct*) and their number equal to the number of conjugacy classes of `G`.
-/
theorem frobenius_group_determinant
    (G : Type*) [Group G] [Fintype G] [DecidableEq G] :
    ∃ (r : ℕ) (p : Fin r → MvPolynomial G ℂ),
      r = Nat.card (ConjClasses G) ∧
      (∀ j, Irreducible (p j)) ∧
      (∀ i j, i ≠ j → ¬ Associated (p i) (p j)) ∧
      groupDeterminant G = ∏ j, (p j) ^ (p j).totalDegree := by
  sorry
#46
How produced

Automatically proved by Seed Prover.

General recursive equals Turing computable
turing_recursive_equiv

Lean theorem statement

/-- **General recursive = Turing computable** (total form). A total function
`f : ℕ → ℕ` is recursive (`Computable`, i.e. partial recursive as a partial
function) **iff** it is computed by some Turing machine (mathlib's `FinTM2`
model) under the standard binary encoding of `ℕ`. This is Knill's class
equality; the backward direction (TM-computable ⇒ recursive) is absent from
mathlib. -/
theorem turing_recursive_equiv (f : ℕ → ℕ) :
    Computable f ↔ Nonempty (TM2Computable encodeNat encodeNat f) := by
  sorry
#47
How produced

Automatically proved by Seed Prover.

Wiener's 1/f theorem
wiener_inverse_closed

Verso theorem preview

theorem declaration uses `sorry`wiener_inverse_closed (f : C(AddCircle T, ℂ)) (hf : LeanEval.Analysis.WienerOneOverF.InWienerAlgebra f) (hzero : ∀ x, f x ≠ 0) : ∃ g : C(AddCircle T, ℂ), (∀ x, g x = (f x)⁻¹) ∧ LeanEval.Analysis.WienerOneOverF.InWienerAlgebra g := T:ℝinst✝:Fact (0 < T)f:C(AddCircle T, ℂ)hf:InWienerAlgebra fhzero:∀ (x : AddCircle T), f x ≠ 0⊢ ∃ g, (∀ (x : AddCircle T), g x = (f x)⁻¹) ∧ InWienerAlgebra g All goals completed! 🐙
#48
How produced

Automatically proved by Seed Prover.

Normal spectral theorem
normal_spectral_theorem

Lean theorem statement

/-- **Spectral theorem** (§14). A complex matrix `A` is **normal**
(`Aᴴ A = A Aᴴ`, i.e. `IsStarNormal A`) **iff** it is **unitarily
diagonalizable**: there is a unitary `U` and a diagonal matrix `diagonal d`
with `A = U (diagonal d) Uᴴ`. -/
theorem normal_spectral_theorem (A : Matrix n n ℂ) :
    IsStarNormal A ↔
      ∃ U ∈ unitary (Matrix n n ℂ), ∃ d : n → ℂ,
        A = U * diagonal d * star U := by
  sorry
#49
How produced

Automatically proved by Seed Prover.

Peano existence theorem for ODEs
peano_existence

Lean theorem statement

/-- **Peano existence theorem.** Replacing the Lipschitz condition with mere
continuity still yields a local solution of `x' = f(x)`, `x(0) = x₀` — but
uniqueness may fail (e.g. `x' = √x`, `x(0) = 0`). Stated for a
finite-dimensional space: Peano's theorem requires local compactness and is
false in general (infinite-dimensional) Banach spaces. -/
theorem peano_existence
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
    {f : E → E} (hf : Continuous f) (x₀ : E) :
    ∃ a : ℝ, 0 < a ∧ ∃ α : ℝ → E, α 0 = x₀ ∧
      ∀ t ∈ Ioo (-a) a, HasDerivAt α (f (α t)) t := by
  sorry
#50
How produced

Automatically proved by Seed Prover.

Gauss-Wantzel constructible regular polygon theorem
gauss_wantzel_constructible_polygon

Lean theorem statement

/-- **Gauss-Wantzel constructible polygon theorem** (§174): a regular `n`-gon
is straightedge-and-compass constructible exactly for the Gauss-Wantzel
integers. -/
theorem gauss_wantzel_constructible_polygon (n : ℕ) (hn : 3 ≤ n) :
    IsConstructible (Real.cos (2 * Real.pi / n)) ↔ GaussWantzelNumber n := by
  sorry
#51
How produced

Automatically proved by Seed Prover.

Trace Cayley-Hamilton / Newton identity
trace_cayley_hamilton_newton

Lean theorem statement

/-- The trace Cayley-Hamilton / Newton identity:
`k c_k + ∑_{j=1}^k tr(A^j) c_{k-j} = 0`, where
`χ_A(X) = X^N + c₁ X^(N-1) + ... + c_N`.

For `k > N`, `c_k = 0`, and the remaining relation is the trace of
Cayley-Hamilton multiplied by a power of `A`. -/
theorem trace_cayley_hamilton_newton {R : Type*} [CommRing R]
    (A : Matrix n n R) {k : ℕ} (hk : 1 ≤ k) :
    (k : R) * charpolyDescendingCoeff A k +
        ∑ j ∈ Finset.Icc 1 k,
          trace (A ^ j) * charpolyDescendingCoeff A (k - j) = 0 := by
  sorry
#52
How produced

Automatically proved by Seed Prover.

Poincaré–Siegel linearisation theorem
poincare_siegel_linearisation

Lean theorem statement

/-- **Poincaré–Siegel linearisation theorem.** If `α` is Diophantine,
`λ = e^{2πiα}`, and `f` is holomorphic near `0` with `f 0 = 0` and
`f'(0) = λ`, then there is a holomorphic germ `u` with `u 0 = 0`,
`u'(0) = 1`, and `f(u z) = u(λ z)` for `z` near `0`. -/
theorem poincare_siegel
    (α : ℝ) (_hα : IsDiophantine α)
    (lam : ℂ) (_hlam : lam = Complex.exp (2 * Real.pi * Complex.I * (α : ℂ)))
    (f : ℂ → ℂ) (_hf : AnalyticAt ℂ f 0) (_hf0 : f 0 = 0)
    (_hmult : deriv f 0 = lam) :
    ∃ u : ℂ → ℂ, AnalyticAt ℂ u 0 ∧ u 0 = 0 ∧ deriv u 0 = 1 ∧
      ∀ᶠ z in nhds (0 : ℂ), f (u z) = u (lam * z) := by
  sorry
#53
How produced

Automatically proved by Seed Prover.

Shannon capacity of the pentagon
shannon_capacity_pentagon

Lean theorem statement

/-- **Lovász's theorem on Shannon capacity of the pentagon** (§238).
The Shannon capacity of the five-cycle is `√5`. -/
theorem shannon_capacity_pentagon :
    HasShannonCapacity (SimpleGraph.cycleGraph 5) (Real.sqrt 5) := by
  sorry
#54
How produced

Automatically proved by Seed Prover.

Nyquist–Shannon sampling theorem
nyquist_shannon_sampling

Lean theorem statement

/-- **Nyquist--Shannon sampling theorem / Whittaker--Shannon interpolation
formula** for Schwartz functions with Fourier support in the mathlib-convention
Nyquist band `[-1/2, 1/2]`.

The explicit `Summable` conjunct records the convergence content of the
cardinal series, rather than relying only on Lean's total `tsum`.
-/
theorem nyquist_shannon_sampling (f : 𝓢(ℝ, ℂ)) (hf : FourierSupportedInNyquist f) :
    ∀ t : ℝ,
      Summable (fun n : ℤ ↦ f (n : ℝ) * sinc (Real.pi * ((n : ℝ) - t))) ∧
        f t =
          ∑' n : ℤ, f (n : ℝ) * sinc (Real.pi * ((n : ℝ) - t)) := by
  sorry
#55
How produced

Automatically proved by Seed Prover.

Complete reducibility for compact groups
compact_group_semisimple

Lean theorem statement

/-- **Representations of compact groups are semisimple** (complete
reducibility / the unitarian trick). A continuous representation of a compact
topological group on a finite-dimensional real vector space is semisimple:
every subrepresentation has a `G`-invariant complement, so the representation
decomposes as a direct sum of irreducible finite-dimensional
subrepresentations. -/
theorem compact_group_semisimple
    {G V : Type*} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G]
    [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V]
    (ρ : Representation ℝ G V)
    (hρ : Continuous fun p : G × V => ρ p.1 p.2) :
    ρ.IsSemisimpleRepresentation := by
  sorry
#56
How produced

Automatically proved by Seed Prover.

Kirk's normal-structure fixed point theorem
kirk_normal_structure

Lean theorem statement

/-- **Kirk's normal-structure fixed-point theorem** (§228).  A nonexpansive
self-map of a nonempty bounded closed convex subset of a reflexive Banach
space with normal structure has a fixed point. -/
theorem kirk_normal_structure [CompleteSpace E]
    (hE_reflexive : Function.Surjective (NormedSpace.inclusionInDoubleDual ℝ E))
    (K : Set E) (hK_nonempty : K.Nonempty) (hK_closed : IsClosed K)
    (hK_bounded : Bornology.IsBounded K) (hK_convex : Convex ℝ K)
    (hK_normal : HasNormalStructure K) (T : K → K)
    (hT : IsNonexpansiveSelfMap K T) :
    ∃ x : K, IsFixedPt T x := by
  sorry
#57
How produced

Automatically proved by Seed Prover.

Kolmogorov–Arnold superposition theorem (non-universal Lorentz form)
kolmogorov_arnold_superposition

Verso theorem preview

theorem declaration uses `sorry`kolmogorov_arnold (n : ℕ) (_hn : 1 ≤ n) (f : (Fin n → ℝ) → ℝ) (_hf : ContinuousOn f (Set.Icc 0 1)) : ∃ (g : ℝ → ℝ) (φ : Fin (2 * n + 1) → Fin n → ℝ → ℝ), Continuous g ∧ (∀ k l, Continuous (φ k l)) ∧ ∀ x ∈ Set.Icc (0 : Fin n → ℝ) 1, f x = ∑ k, g (∑ l, φ k l (x l)) := n:ℕ_hn:1 ≤ nf:(Fin n → ℝ) → ℝ_hf:ContinuousOn f (Set.Icc 0 1)⊢ ∃ g φ, Continuous g ∧ (∀ (k : Fin (2 * n + 1)) (l : Fin n), Continuous (φ k l)) ∧ ∀ x ∈ Set.Icc 0 1, f x = ∑ k, g (∑ l, φ k l (x l)) All goals completed! 🐙
#58
How produced

Automatically proved by Seed Prover.

Radon transform: Fourier-slice diagonalization and pseudo-inversion
radon_transform_inversion

Lean theorem statement

/-- **Radon's theorem (diagonalization + pseudo-inversion).** The Fourier slice
theorem diagonalizes the Radon transform, and the transform admits a left
inverse on the Schwartz space. -/
theorem radon_can_be_diagonalized_and_pseudo_inverted :
    (∀ φ : SchwartzMap (ℝ × ℝ) ℂ, ∀ θ k : ℝ,
        fourier1 (fun p => radon (φ : ℝ × ℝ → ℂ) (p, θ)) k =
          fourier2 (φ : ℝ × ℝ → ℂ) (k * Real.cos θ, k * Real.sin θ)) ∧
    (∃ Rinv : (ℝ × ℝ → ℂ) → (ℝ × ℝ → ℂ),
        ∀ φ : SchwartzMap (ℝ × ℝ) ℂ,
          Rinv (radon (φ : ℝ × ℝ → ℂ)) = (φ : ℝ × ℝ → ℂ)) := by
  sorry
#59
How produced

Automatically proved by Seed Prover.

Lindemann's theorem (e and π transcendental)
lindemann

Lean theorem statement

/-- **Lindemann's theorem.** Both `e = exp 1` and `π` are transcendental over
`ℤ`. -/
theorem lindemann :
    Transcendental ℤ (Real.exp 1) ∧ Transcendental ℤ Real.pi := by
  sorry
#60
How produced

Automatically proved by Seed Prover.

The Lindemann–Weierstrass theorem
lindemann_weierstrass

Lean theorem statement

/-- **Lindemann–Weierstrass theorem.** If `x₁, …, xₙ ∈ ℂ` are algebraic over `ℚ`
and ℚ-linearly independent, then `e^{x₁}, …, e^{xₙ}` are algebraically
independent over `ℚ`. -/
theorem lindemann_weierstrass {n : ℕ} (x : Fin n → ℂ)
    (h_alg : ∀ i, IsAlgebraic ℚ (x i))
    (h_lin : LinearIndependent ℚ x) :
    AlgebraicIndependent ℚ (fun i => Complex.exp (x i)) := by
  sorry
#61
How produced

Automatically proved by Seed Prover.

Bauer's uniqueness at extreme points
bauer_extreme_point_uniqueness

Lean theorem statement

/-- **Bauer's uniqueness at extreme points.** If `x` is an extreme point of a
compact convex set `K` and `μ` is a probability measure supported on `K`
(`μ Kᶜ = 0`) with barycenter `x = ∫ y, y ∂μ`, then `μ` is the Dirac mass at
`x`. (The support hypothesis is the weaker `μ Kᶜ = 0`, making this a
strengthening of the textbook statement: uniqueness among all ambient Borel
probability measures on `K`, not only those already supported on `ext K`.) -/
theorem bauer_unique [MeasurableSpace X] [BorelSpace X]
    (K : Set X) (hK_cpt : IsCompact K) (hK_cvx : Convex ℝ K)
    {x : X} (hx : x ∈ K.extremePoints ℝ)
    (μ : Measure X) [IsProbabilityMeasure μ]
    (hμ : μ Kᶜ = 0) (hbar : x = ∫ y, y ∂μ) :
    μ = Measure.dirac x := by
  sorry
#62
How produced

Automatically proved by Seed Prover.

Fundamental theorem of topos theory
fundamental_topos_theory

Verso theorem preview

theorem declaration uses `sorry`fundamental_topos_theory {E : Type*} [Category E] (hE : LeanEval.ToposTheory.IsTopos E) (X : E) : LeanEval.ToposTheory.IsTopos (Over X) := E:Type u_1inst✝:Category.{u_2, u_1} EhE:IsTopos EX:E⊢ IsTopos (Over X) All goals completed! 🐙
#63
How produced

Automatically proved by Seed Prover.

Lax's approximation theorem for toral homeomorphisms
lax_approximation

Lean theorem statement

/-- **Lax's approximation theorem.** Every toral dynamical system on `𝕋^d`
(`d ≥ 1`) is approximated arbitrarily well in the metric `δ` by cyclic cube
exchange transformations. -/
theorem lax_approximation {d : ℕ} (hd : 0 < d) (T : ToralDynamicalSystem d)
    {ε : ℝ≥0∞} (hε : 0 < ε) :
    ∃ (n : ℕ) (S : VolumePreservingEquiv d),
      IsCyclicCubeExchange S n ∧ deltaDist T.toVolumePreservingEquiv S < ε := by
  sorry
#64
How produced

Automatically proved by Seed Prover.

The Landsberg–Schaar relation
landsberg_schaar

Lean theorem statement

/-- **Landsberg–Schaar relation.** For positive odd integers `p, q`,
`S(2q, p) = e^{iπ/4} · S(−p, 2q)`. -/
theorem landsberg_schaar (p q : ℕ) (hp : Odd p) (hq : Odd q) :
    gaussS (2 * q : ℕ) p
      = Complex.exp ((Real.pi : ℂ) * Complex.I / 4) * gaussS (-(p : ℤ)) (2 * q) := by
  sorry
#65
How produced

Automatically proved by Seed Prover.

Moran's equality for affine-symmetric iterated function systems
moran_equality_affine

Lean theorem statement

/-- **Moran's equality for affine-symmetric IFS.** For an affine-symmetric IFS
on `ℝᵈ` with common contraction factor `λ ∈ (0,1)`, orthogonal linear parts, and
the open set condition, the Hausdorff dimension of the attractor is
`−log n / log λ` (positive since `λ < 1`). -/
theorem moran_equality_affine
    {d n : ℕ} (hn : 1 ≤ n)
    (f : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (lam : ℝ)
    (h_aff : IsAffineSymmetricIFS f lam)
    (h_osc : OpenSetCondition f)
    {S : Set (EuclideanSpace ℝ (Fin d))} (hS : IsAttractor f S) :
    dimH S = ENNReal.ofReal (- Real.log n / Real.log lam) := by
  sorry
#66
How produced

Automatically proved by Seed Prover.

The Hausdorff–Hildebrandt–Schoenberg moment theorem
hausdorff_hildebrandt_schoenberg

Lean theorem statement

/-- **Hausdorff–Hildebrandt–Schoenberg theorem.** A multi-indexed real sequence
is the moment sequence of a signed bounded-variation measure on the unit cube
`Iᵈ` iff its moments are Hausdorff bounded. -/
theorem hausdorff_hildebrandt_schoenberg {d : ℕ} (a : (Fin d → ℕ) → ℝ) :
    IsMomentConfiguration a ↔ HausdorffBounded a := by
  sorry
#67
How produced

Automatically proved by Seed Prover.

Hurewicz theorem in degree 1 (H₁ = abelianization of π₁)
hurewicz_h1_abelianization

Lean theorem statement

/-- **Hurewicz (n = 1).** For a path-connected space `X`, `H₁(X;ℤ)` is the
abelianization of `π₁(X, x)`. -/
theorem hurewicz_h1_abelianization
    (X : Type) [TopologicalSpace X] [PathConnectedSpace X] (x : X) :
    Nonempty (Additive (Abelianization (FundamentalGroup X x)) ≃+
      (IntegralHomology 1 X : Type)) := by
  sorry
#68
How produced

Automatically proved by Seed Prover.

Jordan normal form
jordan_normal_form

Lean theorem statement

/-- **Jordan normal form.** Over an algebraically closed field, every
endomorphism of `Kⁿ` admits a Jordan-chain basis. -/
theorem jordan_normal_form {K : Type*} [Field K] [IsAlgClosed K] (n : ℕ)
    (f : Module.End K (StdSpace K n)) :
    Nonempty (JordanChainBasis f) := by
  sorry
#69
How produced

Automatically proved by Seed Prover.

Pascal's theorem
pascal

Lean theorem statement

/-- **Pascal's theorem.** Six distinct points on a non-singular conic determine
three collinear intersection points `Aᵢ Bⱼ ∩ Aⱼ Bᵢ`. -/
theorem pascal
    (M : Matrix (Fin 3) (Fin 3) ℝ) (hMsymm : M.IsSymm) (hMdet : M.det ≠ 0)
    (a₁ a₂ a₃ b₁ b₂ b₃ : Fin 3 → ℝ)
    (ha₁ : a₁ ≠ 0) (ha₂ : a₂ ≠ 0) (ha₃ : a₃ ≠ 0)
    (hb₁ : b₁ ≠ 0) (hb₂ : b₂ ≠ 0) (hb₃ : b₃ ≠ 0)
    (hdist : [a₁, a₂, a₃, b₁, b₂, b₃].Pairwise (fun v w => ¬ SamePoint v w))
    (hA₁ : OnConic M a₁) (hA₂ : OnConic M a₂) (hA₃ : OnConic M a₃)
    (hB₁ : OnConic M b₁) (hB₂ : OnConic M b₂) (hB₃ : OnConic M b₃) :
    Collinear3 (meet a₁ b₂ a₂ b₁) (meet a₁ b₃ a₃ b₁) (meet a₂ b₃ a₃ b₂) := by
  sorry
#70
How produced

Automatically proved by Seed Prover.

Tverberg's theorem
tverberg_theorem

Lean theorem statement

/-- **Tverberg's theorem.** Any `(r-1)(d+1)+1` points in `ℝ^d` admit an
`r`-part Tverberg partition. -/
theorem tverberg_theorem (d r : ℕ) (hr : 1 ≤ r)
    (f : Fin ((r - 1) * (d + 1) + 1) → Space d) :
    HasTverbergPartition (r := r) f := by
  sorry
#71
How produced

Automatically proved by Seed Prover.

Boone–Higman theorem (easy direction)
boone_higman_embedding

Lean theorem statement

/-- **Boone–Higman theorem (easy direction).** If a finitely presented group `G`
embeds (via injective `f`) into a simple group `H`, which embeds (via injective
`g`) into a finitely presented group `K`, then the word problem of `G` is
solvable. -/
theorem boone_higman_embedding
    {G H K : Type*} [Group G] [Group H] [Group K]
    [IsSimpleGroup H] [Group.IsFinitelyPresented K]
    (f : G →* H) (hf : Function.Injective f)
    (g : H →* K) (hg : Function.Injective g)
    {n : ℕ} (φ : FreeGroup (Fin n) →* G)
    (hsurj : Function.Surjective φ)
    (hker : (MonoidHom.ker φ).IsNormalClosureFG) :
    WordProblemSolvable φ := by
  sorry
#72
How produced

Automatically proved by Seed Prover.

Choquet's representation theorem
choquet_representation_theorem

Lean theorem statement

/-- **Choquet's representation theorem.** Every point `x` of a compact convex
set `K` in a Banach space is the barycenter of a probability measure supported
on the extreme points of `K`: there is a probability measure `μ` with
`μ (ext K)ᶜ = 0` whose barycenter `∫ y, y ∂μ` equals `x`. -/
theorem choquet [MeasurableSpace X] [BorelSpace X]
    (K : Set X) (hK_cpt : IsCompact K) (hK_cvx : Convex ℝ K)
    {x : X} (hx : x ∈ K) :
    ∃ μ : Measure X, IsProbabilityMeasure μ ∧
      μ (K.extremePoints ℝ)ᶜ = 0 ∧
      x = ∫ y, y ∂μ := by
  sorry
#73
How produced

Automatically proved by Seed Prover.

Hausdorff moment problem: absolute-continuity criterion
hausdorff_absolute_continuity

Lean theorem statement

/-- **Absolute continuity criterion (Hausdorff moment problem).** A positive
probability measure `μ` on the cube is uniformly absolutely continuous w.r.t.
Lebesgue measure iff there is `C` with `(Δᵏμ)ₙ ≤ C·(Δᵏν)ₙ` for all `k ≤ n`. -/
theorem hausdorff_absolute_continuity {d : ℕ}
    (μ : Measure (EuclideanSpace ℝ (Fin d)))
    [IsProbabilityMeasure μ] (hμ : μ ((cube d)ᶜ) = 0) :
    UniformlyAbsolutelyContinuous μ (volume.restrict (cube d)) ↔
      ∃ C : ℝ, ∀ k n : Fin d → ℕ, k ≤ n →
        diff (momentOf μ) k n ≤ C * diff (momentOf (volume.restrict (cube d))) k n := by
  sorry
#74
How produced

Automatically proved by Seed Prover.

The Hausdorff positivity (complete-monotonicity) criterion
hausdorff_positivity_criterion

Lean theorem statement

/-- **Hausdorff positivity criterion** (Hausdorff 1921). A moment configuration
comes from a positive measure iff all its iterated backward differences are
nonnegative — i.e. the sequence is *completely monotone*: `(Δᵏa)ₙ ≥ 0` for all
`k ≤ n`. -/
theorem hausdorff_positivity {d : ℕ} (a : (Fin d → ℕ) → ℝ) :
    IsPositiveMomentConfiguration a ↔ ∀ k n : Fin d → ℕ, k ≤ n → 0 ≤ diff a k n := by
  sorry
#75
How produced

Automatically proved by Seed Prover.

Sard's regular-value corollary
regular_value_ae

Lean theorem statement

/-- **Regular value corollary (Sard).** For a smooth `f : ℝᵐ → ℝ`, almost every
`c ∈ ℝ` is a regular value. -/
theorem regular_value_ae {m : ℕ} (f : EuclideanSpace ℝ (Fin m) → ℝ)
    (hf : ContDiff ℝ ∞ f) :
    ∀ᵐ c ∂(volume : Measure ℝ), IsRegularValue f c := by
  sorry
#76
How produced

Automatically proved by Seed Prover.

Riesz's rising sun lemma
rising_sun_lemma

Lean theorem statement

/-- **Riesz's rising sun lemma.** Every continuous real function on a compact
interval has the rising-sun property. -/
theorem rising_sun_lemma {a b : ℝ} (hab : a < b) {f : ℝ → ℝ}
    (hf : ContinuousOn f (Icc a b)) :
    HasRisingSunProperty a b f := by
  sorry
#77
How produced

Automatically proved by Seed Prover.

Fang–Xia: tiling of the symmetric group by transpositions implies λ-transitivity
fang_xia_tiling_partition_transitive

Lean theorem statement

/-- **Fang–Xia, Theorem 1.4.** A tiling `(T_n, Y)` of `S_n` forces
λ-transitivity of `Y` for every partition `λ` of `n` whose Young-
diagram content sum is nonnegative. -/
theorem fang_xia_partition_transitive_of_tiling
    {n : ℕ} {Y : Set (Equiv.Perm (Fin n))}
    (_h : IsTiling (transpositionsWithOne n) Y) :
    ∀ lam : PartitionShape n, 0 ≤ lam.contentSum → IsPartitionTransitive Y lam := by
  sorry
#78
How produced

Automatically proved by Seed Prover.

Anosov–Bowen shadowing lemma
anosov_bowen_shadowing

Lean theorem statement

/-- **Anosov–Bowen shadowing lemma** (Anosov 1967; Bowen 1975). Every
compact hyperbolic invariant set has the shadowing property. -/
theorem hyperbolic_has_shadowing
    (T : E d ≃ₜ E d) (K : Set (E d))
    (_hKc : IsCompact K) (_hK : IsHyperbolic T K) :
    HasShadowing (T : E d → E d) K := by
  sorry
#79
How produced

Automatically proved by Seed Prover.

Sard's theorem (critical-set image has measure zero)
sard_theorem

Lean theorem statement

/-- **Sard's theorem** (Morse 1939 / Sard 1942), Knill's rank-
deficient form. The image of the rank-deficient locus of a smooth
map `f : ℝᵐ → ℝⁿ` has Lebesgue measure zero. -/
theorem sard {m n : ℕ} (f : E m → E n) (_hf : ContDiff ℝ ∞ f) :
    volume (criticalValues f) = 0 := by
  sorry
#80
How produced

Automatically proved by Seed Prover.

Ornstein–Weiss ℤᵈ Rokhlin lemma
ornstein_weiss_rokhlin

Verso theorem preview

theorem declaration uses `sorry`ornstein_weiss_rokhlin {Ω : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω] {d : ℕ} (_hd : 1 ≤ d) (μ : Measure Ω) [IsProbabilityMeasure μ] (T : (Fin d → ℤ) → Ω → Ω) (_hid : ∀ x, T 0 x = x) (_hT : ∀ v, MeasurePreserving (T v) μ μ) (_hgrp : ∀ u v x, T (u + v) x = T u (T v x)) (_hfree : LeanEval.Dynamics.IsFreeAction μ T) (N : ℕ) (_hN : 1 ≤ N) {ε : ENNReal} (_hε : 0 < ε) : ∃ B : Set Ω, MeasurableSet B ∧ ((boxShape d N : Finset (Fin d → ℤ)) : Set (Fin d → ℤ)).PairwiseDisjoint (fun v => T v '' B) ∧ μ (⋃ v ∈ boxShape d N, T v '' B) ≥ 1 - ε := Ω:Type u_1inst✝²:MeasurableSpace Ωinst✝¹:StandardBorelSpace Ωd:ℕ_hd:1 ≤ dμ:Measure Ωinst✝:IsProbabilityMeasure μT:(Fin d → ℤ) → Ω → Ω_hid:∀ (x : Ω), T 0 x = x_hT:∀ (v : Fin d → ℤ), MeasurePreserving (T v) μ μ_hgrp:∀ (u v : Fin d → ℤ) (x : Ω), T (u + v) x = T u (T v x)_hfree:IsFreeAction μ TN:ℕ_hN:1 ≤ Nε:ENNReal_hε:0 < ε⊢ ∃ B, MeasurableSet B ∧ ((↑(boxShape d N)).PairwiseDisjoint fun v => T v '' B) ∧ μ (⋃ v ∈ boxShape d N, T v '' B) ≥ 1 - ε All goals completed! 🐙
#81
How produced

Automatically proved by Seed Prover.

Sobolev embedding theorem (Morrey regime)
sobolev_embedding_morrey

Verso theorem preview

theorem declaration uses `sorry`sobolev_embedding {n k r : ℕ} {α p : ℝ} (_hp : (n : ℝ) < p) (_hα : 0 < α) (_hα1 : α ≤ 1) (_hgap : (r : ℝ) + α < (k : ℝ) - n / p) (f : LeanEval.Analysis.SobolevMorreyProblem.E n → ℝ) (_hf : LeanEval.Analysis.SobolevMorreyProblem.MemSobolevWk k (ENNReal.ofReal p) f) : ∃ g : LeanEval.Analysis.SobolevMorreyProblem.E n → ℝ, f =ᵐ[volume] g ∧ LeanEval.Analysis.SobolevMorreyProblem.MemHolder r α g := n:ℕk:ℕr:ℕα:ℝp:ℝ_hp:↑n < p_hα:0 < α_hα1:α ≤ 1_hgap:↑r + α < ↑k - ↑n / pf:E n → ℝ_hf:MemSobolevWk k (ENNReal.ofReal p) f⊢ ∃ g, f =ᵐ[volume] g ∧ LeanEval.Analysis.SobolevMorreyProblem.MemHolder r α g All goals completed! 🐙
#82
How produced

Automatically proved by Seed Prover.

Liouville–Arnold theorem on integrable systems
liouville_arnold

Verso theorem preview

theorem declaration uses `sorry`liouville_arnold {n : ℕ} (F : Fin n → LeanEval.Geometry.LiouvilleArnold.E n → ℝ) (U : Set (LeanEval.Geometry.LiouvilleArnold.E n)) (_hU : IsOpen U) (_hLI : LeanEval.Geometry.LiouvilleArnold.IsLiouvilleIntegrable F U) (c : Fin n → ℝ) (_hMc_sub : LeanEval.Geometry.LiouvilleArnold.levelSet F c ⊆ U) (_hMc_compact : IsCompact (LeanEval.Geometry.LiouvilleArnold.levelSet F c)) (_hMc_connected : IsConnected (LeanEval.Geometry.LiouvilleArnold.levelSet F c)) : Nonempty ((LeanEval.Geometry.LiouvilleArnold.levelSet F c) ≃ₜ (Fin n → AddCircle (1 : ℝ))) := n:ℕF:Fin n → E n → ℝU:Set (E n)_hU:IsOpen U_hLI:IsLiouvilleIntegrable F Uc:Fin n → ℝ_hMc_sub:levelSet F c ⊆ U_hMc_compact:IsCompact (levelSet F c)_hMc_connected:IsConnected (levelSet F c)⊢ Nonempty (↑(levelSet F c) ≃ₜ (Fin n → AddCircle 1)) All goals completed! 🐙
#83
How produced

Automatically proved by Seed Prover.

Rokhlin lemma
rokhlin_lemma

Lean theorem statement

/-- **Rokhlin lemma.** For every aperiodic measure-preserving
automorphism `T` of a standard Borel probability space `(Ω, μ)`, every
height `n ≥ 1`, and every `ε > 0`, there is a Rokhlin tower of height
`n` whose union has measure at least `1 − ε`. -/
theorem rokhlin_lemma {Ω : Type*} [MeasurableSpace Ω]
    [StandardBorelSpace Ω]
    (μ : Measure Ω) [IsProbabilityMeasure μ] (T : Ω → Ω)
    (_hT : MeasurePreserving T μ μ) (_hap : IsAperiodic T μ)
    (n : ℕ) (_hn : 1 ≤ n) {ε : ENNReal} (_hε : 0 < ε) :
    ∃ B : Set Ω, IsRokhlinTower T B n ∧
      μ (towerUnion T B n) ≥ 1 - ε := by
  sorry
#84
How produced

Automatically proved by Seed Prover.

Hippocrates' theorem on lunes
hippocrates_lunes

Verso theorem preview

theorem declaration uses `sorry`hippocrates_lunes (a b : ℝ) (ha : 0 < a) (hb : 0 < b) : volume (LeanEval.Geometry.HippocratesLunes.horizontalLune a b) + volume (LeanEval.Geometry.HippocratesLunes.verticalLune a b) = volume (LeanEval.Geometry.HippocratesLunes.rightTriangle a b) := a:ℝb:ℝha:0 < ahb:0 < b⊢ volume (horizontalLune a b) + volume (verticalLune a b) = volume (rightTriangle a b) All goals completed! 🐙
#85
How produced

Automatically proved by Seed Prover.

Morley's trisector theorem
morley_theorem

Verso theorem preview

theorem declaration uses `sorry`morley_theorem (A B C P Q R : LeanEval.Geometry.Morley.Plane) (h : LeanEval.Geometry.Morley.IsMorleyConfiguration A B C P Q R) : LeanEval.Geometry.Morley.IsEquilateralTriple P Q R := A:PlaneB:PlaneC:PlaneP:PlaneQ:PlaneR:Planeh:IsMorleyConfiguration A B C P Q R⊢ IsEquilateralTriple P Q R All goals completed! 🐙
#86
How produced

Automatically proved by Seed Prover.

Local stable/unstable sets at a hyperbolic fixed point (set-level Hadamard–Perron)
stable_unstable_manifolds

Verso theorem preview

theorem declaration uses `sorry`stable_unstable_manifolds_exist (n : ℕ) (f : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (x₀ : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (_hf : ContDiffAt ℝ 1 f x₀) (_hfix : f x₀ = x₀) (_hhyp : LeanEval.Dynamics.StableUnstableManifoldsProblem.IsHyperbolicLinear (fderiv ℝ f x₀)) (_hf_inv : (fderiv ℝ f x₀).IsInvertible) : ∃ U : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), Ws = {x | (∀ k : ℕ, f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y : ℕ → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n, y 0 = x ∧ (∀ k : ℕ, y k ∈ U) ∧ (∀ k : ℕ, f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} := n:ℕf:E n → E nx₀:E n_hf:ContDiffAt ℝ 1 f x₀_hfix:f x₀ = x₀_hhyp:IsHyperbolicLinear (fderiv ℝ f x₀)_hf_inv:(fderiv ℝ f x₀).IsInvertible⊢ ∃ U, IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu, Ws = {x | (∀ (k : ℕ), f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y, y 0 = x ∧ (∀ (k : ℕ), y k ∈ U) ∧ (∀ (k : ℕ), f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} All goals completed! 🐙
#87
How produced

Automatically proved by Seed Prover.

Mountain Pass Theorem (Ambrosetti–Rabinowitz 1973)
mountain_pass

Verso theorem preview

theorem declaration uses `sorry`mountain_pass (f : E → ℝ) (_hf : ContDiff ℝ 1 f) (_hps : LeanEval.Analysis.MountainPassProblem.PalaisSmale f) {a b : E} {ε r : ℝ} (_hmr : LeanEval.Analysis.MountainPassProblem.MountainRange f a b ε r) : ∃ x : E, LeanEval.Analysis.MountainPassProblem.IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace Ef:E → ℝ_hf:ContDiff ℝ 1 f_hps:PalaisSmale fa:Eb:Eε:ℝr:ℝ_hmr:MountainRange f a b ε r⊢ ∃ x, IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b All goals completed! 🐙
#88
How produced

Automatically proved by Seed Prover.

Fraser: Fourier decay for finite-field Kakeya sets is q^{-1} and sharp
fraser_kakeya_fourier_decay

Verso theorem preview

theorem declaration uses `sorry`fraser_kakeya_fourier_decay_and_sharp {d : ℕ} (_hd : 2 ≤ d) {K : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F d)} (_hK : LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K) (χ : AddChar F ℂ) (_hχ : χ ≠ 1) : (∃ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K μ ∧ ∀ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d, ξ ≠ 0 → ‖fourier χ μ ξ‖ ≤ (Fintype.card F : ℝ)⁻¹) ∧ (∀ κ : ℝ, 0 < κ → κ < 1 → ∃ Q : ℕ, ∀ (F' : Type*) [Field F'] [Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K' : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d), LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K' ∧ ∀ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K' μ → ∃ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d, ξ ≠ 0 ∧ κ * (Fintype.card F' : ℝ)⁻¹ ≤ ‖fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖) := F:Type u_1inst✝²:Field Finst✝¹:Fintype Finst✝:DecidableEq Fd:ℕ_hd:2 ≤ dK:Set (Space F d)_hK:IsKakeya Kχ:AddChar F ℂ_hχ:χ ≠ 1⊢ (∃ μ, IsProbabilityMeasureOn K μ ∧ ∀ (ξ : Space F d), ξ ≠ 0 → ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier χ μ ξ‖ ≤ (↑(Fintype.card F))⁻¹) ∧ ∀ (κ : ℝ), 0 < κ → κ < 1 → ∃ Q, ∀ (F' : Type u_2) [inst : Field F'] [inst_1 : Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K', IsKakeya K' ∧ ∀ (μ : Space F' d → ℝ), IsProbabilityMeasureOn K' μ → ∃ ξ, ξ ≠ 0 ∧ κ * (↑(Fintype.card F'))⁻¹ ≤ ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖ All goals completed! 🐙
#89
How produced

Automatically proved by Seed Prover.

Euler–Lagrange equation
euler_lagrange_equation

Verso theorem preview

theorem declaration uses `sorry`euler_lagrange_equation {a b : ℝ} (L : ℝ → ℝ → ℝ → ℝ) (x : ℝ → ℝ) (_hab : a < b) (_hL : ContDiff ℝ 2 (fun p : ℝ × ℝ × ℝ => L p.1 p.2.1 p.2.2)) (_hx : ContDiff ℝ 2 x) (_hxe : LeanEval.Analysis.IsVariationalExtremum a b L x) : ∀ t ∈ Set.Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t := a:ℝb:ℝL:ℝ → ℝ → ℝ → ℝx:ℝ → ℝ_hab:a < b_hL:ContDiff ℝ 2 fun p => L p.1 p.2.1 p.2.2_hx:ContDiff ℝ 2 x_hxe:IsVariationalExtremum a b L x⊢ ∀ t ∈ Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t All goals completed! 🐙
#90
How produced

Automatically proved by Seed Prover.

Monge–Kantorovich existence theorem
monge_kantorovich

Verso theorem preview

theorem declaration uses `sorry`monge_kantorovich_exists {X Y : Type*} [TopologicalSpace X] [PolishSpace X] [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [PolishSpace Y] [MeasurableSpace Y] [BorelSpace Y] (P : Measure X) (Q : Measure Y) [IsProbabilityMeasure P] [IsProbabilityMeasure Q] (c : X × Y → ENNReal) (_hc : Continuous c) : ∃ π ∈ LeanEval.Analysis.Couplings P Q, ∀ π' ∈ LeanEval.Analysis.Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' := X:Type u_1Y:Type u_2inst✝⁹:TopologicalSpace Xinst✝⁸:PolishSpace Xinst✝⁷:MeasurableSpace Xinst✝⁶:BorelSpace Xinst✝⁵:TopologicalSpace Yinst✝⁴:PolishSpace Yinst✝³:MeasurableSpace Yinst✝²:BorelSpace YP:Measure XQ:Measure Yinst✝¹:IsProbabilityMeasure Pinst✝:IsProbabilityMeasure Qc:X × Y → ENNReal_hc:Continuous c⊢ ∃ π ∈ Couplings P Q, ∀ π' ∈ Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' All goals completed! 🐙
#91
How produced

Automatically proved by Seed Prover.

Solvable extensions ↔ solvable groups (the missing converse in Abel–Ruffini)
solvable_by_radicals_converse

Verso theorem preview

theorem declaration uses `sorry`solvable_iff_solvableByRad (F : Type*) [Field F] [CharZero F] (p : F[X]) (_hp : p ≠ 0) : (∀ x : AlgebraicClosure F, aeval x p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal := F:Type u_1inst✝¹:Field Finst✝:CharZero Fp:F[X]_hp:p ≠ 0⊢ (∀ (x : AlgebraicClosure F), (aeval x) p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal All goals completed! 🐙
#92
How produced

Automatically proved by Seed Prover.

Furstenberg–Weiss topological multiple recurrence (single-transformation form)
furstenberg_topological

Verso theorem preview

theorem declaration uses `sorry`furstenberg_topological_recurrence {X : Type*} [MetricSpace X] [CompactSpace X] [Nonempty X] (T : X ≃ₜ X) : ∃ x : X, LeanEval.Dynamics.IsMultiplyRecurrent (T : X → X) x := X:Type u_1inst✝²:MetricSpace Xinst✝¹:CompactSpace Xinst✝:Nonempty XT:X ≃ₜ X⊢ ∃ x, IsMultiplyRecurrent (⇑T) x All goals completed! 🐙
#93
How produced

Automatically proved by Seed Prover.

Lidskii's inequality
lidskii_inequality

Verso theorem preview

theorem declaration uses `sorry`lidskii_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) {p : ℝ} (_hp : 1 ≤ p) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |(hB.sub hA).eigenvalues₀ j| ^ p := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitianp:ℝ_hp:1 ≤ p⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |⋯.eigenvalues₀ j| ^ p All goals completed! 🐙
#94
How produced

Automatically proved by Seed Prover.

Frobenius's theorem: the Frobenius kernel is normal
frobenius_kernel_isNormal

Lean theorem statement

theorem frobenius_kernel_isNormal
    (G X : Type) [Group G] [Fintype G] [Fintype X]
    [MulAction G X] [FaithfulSMul G X]
    (hcard : 2 ≤ Fintype.card X)
    (htrans : ∀ x y : X, ∃ g : G, g • x = y)
    (hstab : ∀ x : X, MulAction.stabilizer G x ≠ ⊥)
    (hfrob : ∀ g : G, g ≠ 1 → ∀ x y : X, g • x = x → g • y = y → x = y) :
    ∃ N : Subgroup G, N.Normal ∧
      (N : Set G) = {1} ∪ {g : G | ∀ x : X, g • x ≠ x} := by
  sorry
#95
How produced

Automatically proved by Seed Prover.

Lidskii–Last eigenvalue-perturbation theorem
lidskii_last

Verso theorem preview

theorem declaration uses `sorry`lidskii_last {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitian⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ All goals completed! 🐙
#96
How produced

Automatically proved by Seed Prover.

Independence of the parallel postulate
parallel_postulate_independent

Lean theorem statement

/-- **Independence of the parallel postulate** (Freek #12). The Euclidean
axiom `A10` is logically independent of Tarski's absolute axioms `A1`–`A9`
and `A11`: there is a model of the absolute axioms in which the parallel
postulate holds (the real coordinate plane) and one in which it fails (the
Klein–Beltrami disk, or any other hyperbolic-plane model). -/
theorem parallel_postulate_independent :
    (∃ (M : Type) (T : TarskiAbsolute M), Euclidean M T) ∧
    (∃ (M : Type) (T : TarskiAbsolute M), ¬ Euclidean M T) := by
  sorry
#97
How produced

Automatically proved by Seed Prover.

Sturm's theorem
sturm

Lean theorem statement

/-- **Sturm's theorem.** For a squarefree real polynomial `p` and an interval
`(a, b)` with `a < b` whose endpoints are not roots of `p`, the number of
distinct roots of `p` in `(a, b)` equals `σ(a) − σ(b)`. -/
theorem sturm (p : ℝ[X]) (hp : Squarefree p) {a b : ℝ} (hab : a < b)
    (ha : p.eval a ≠ 0) (hb : p.eval b ≠ 0) :
    ((p.roots.toFinset).filter (fun x => a < x ∧ x < b)).card =
      sigma p a - sigma p b := by
  sorry
#98
How produced

Automatically proved by Seed Prover.

Kakutani fixed-point theorem
kakutani_fixed_point

Lean theorem statement

/-- **Kakutani fixed-point theorem.** Every upper-hemicontinuous
correspondence `F` from a nonempty compact convex `K ⊆ ℝᵈ` to itself, with
nonempty convex closed values, has a fixed point `x ∈ F x`. -/
theorem kakutani_fixed_point {d : ℕ}
    {K : Set (EuclideanSpace ℝ (Fin d))}
    (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K)
    (_hK_nonempty : K.Nonempty)
    (F : EuclideanSpace ℝ (Fin d) → Set (EuclideanSpace ℝ (Fin d)))
    (_hF_uhc : IsUpperHemicontinuous F)
    (_hF_nonempty : ∀ x ∈ K, (F x).Nonempty)
    (_hF_convex : ∀ x ∈ K, Convex ℝ (F x))
    (_hF_closed : ∀ x ∈ K, IsClosed (F x))
    (_hF_maps : ∀ x ∈ K, F x ⊆ K) :
    ∃ x ∈ K, x ∈ F x := by
  sorry
#99
How produced

Automatically proved by Seed Prover.

Koszul formula
koszul_formula

Lean theorem statement

/-- **Koszul formula.** For any smooth torsion-free metric-compatible
covariant derivative `cov` on `TM`, `2 ⟨∇_X Y, Z⟩` equals the cyclic sum
of directional derivatives `X·⟨Y, Z⟩ + Y·⟨X, Z⟩ − Z·⟨X, Y⟩` minus the
Lie-bracket cyclic sum `⟨X, [Y, Z]⟩ + ⟨Y, [X, Z]⟩ − ⟨Z, [X, Y]⟩`. -/
theorem koszul_formula
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
      [FiniteDimensional ℝ E] [CompleteSpace E]
    {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H}
    {M : Type*} [TopologicalSpace M] [ChartedSpace H M]
      [IsManifold I ∞ M]
    [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)]
    [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)]
    (cov : CovariantDerivative I E (TangentSpace I (M := M)))
    [ContMDiffCovariantDerivative cov ∞]
    (_htor : cov.torsion = 0) (_hmet : IsMetricCompatible cov)
    (X Y Z : Π x : M, TangentSpace I x)
    (_hX : CMDiff ∞ (T% X)) (_hY : CMDiff ∞ (T% Y)) (_hZ : CMDiff ∞ (T% Z))
    (x : M) :
    2 * inner ℝ (cov Y x (X x)) (Z x) =
      mvfderiv I (fun y : M => inner ℝ (Y y) (Z y)) x (X x)
      + mvfderiv I (fun y : M => inner ℝ (X y) (Z y)) x (Y x)
      - mvfderiv I (fun y : M => inner ℝ (X y) (Y y)) x (Z x)
      - inner ℝ (X x) (mlieBracket I Y Z x)
      - inner ℝ (Y x) (mlieBracket I X Z x)
      + inner ℝ (Z x) (mlieBracket I X Y x) := by
  sorry
#100
How produced

Automatically proved by Seed Prover.

Nash equilibrium existence theorem
nash_equilibrium_exists

Lean theorem statement

/-- **Nash equilibrium existence theorem.** Every finite `n`-player game
with nonempty finite pure-strategy sets and arbitrary real payoffs admits
at least one mixed-strategy Nash equilibrium. -/
theorem nash_equilibrium_exists {n : ℕ} {S : Fin n → Type*}
    [∀ i, Fintype (S i)] [∀ i, Nonempty (S i)]
    (u : Fin n → StrategyProfile n S → ℝ) :
    ∃ σ : ∀ i, S i → ℝ, IsNashEquilibrium u σ := by
  sorry
#101
How produced

Automatically proved by Seed Prover.

Brauer–Fowler theorem
brauer_fowler

Lean theorem statement

/-- **Brauer–Fowler theorem.** There is a function bounding the order
of a finite nonabelian simple group by the order of any involution
centralizer. -/
theorem brauer_fowler :
    ∃ f : ℕ → ℕ, ∀ (G : Type) [Group G] [Finite G],
      IsSimpleGroup G → (∃ a b : G, a * b ≠ b * a) →
      ∀ t : G, orderOf t = 2 →
        Nat.card G ≤ f (Nat.card (Subgroup.centralizer ({t} : Set G))) := by
  sorry
#102
How produced

Automatically proved by Seed Prover.

Balanceable k-bounded partitions
balanceable_bounded_partitions

Lean theorem statement

/--
For any `k`, the smallest `n` such that any `k`-bounded partition of `n` is
also balanceable is given by `2 * lcm(1, ..., k)`.
-/
theorem minimal_balanceable_of_bounded (k : ℕ) (hk : 0 < k) :
    Minimal (fun n => 0 < n ∧ ∀ p : n.Partition, Bounded k p → Balanceable p) (2 * (Finset.Icc 1 k).lcm id) := by
  sorry
#103
Chen theorem for Markoff graphs
dvd_card_connectedComponent_markoffGraph

Lean theorem statement

/-- For prime `p > 3`, every connected component of the nonzero Markoff graph over `ZMod p`
has cardinality divisible by `p`. -/
theorem dvd_card_connectedComponent_markoffGraph
    {p : ℕ} (hp : Nat.Prime p) (hgt : 3 < p) :
    ∀ c : (markoffGraph p).ConnectedComponent, p ∣ Nat.card c := by
  sorry
#104
A competition programming problem about permuting a permutation to be unimodal
permute_to_unimodal

Lean theorem statement

/--
`minRearrange` correctly computes the smallest number of indices that need to be permuted in order to
turn `arr` into a unimodal permutation.
-/
theorem minRearrange_correct {arr : Array Nat} :
    arr.Perm (1...=arr.size).toArray →
      (∃ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x ∧ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector = minRearrange arr) ∧
      (∀ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x → minRearrange arr ≤ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector) := by
  sorry
#105
Kuznetsov's theorem: finitely presented simple groups have solvable word problem
boone_higman_simple

Verso theorem preview

theorem declaration uses `sorry`boone_higman_simple {G : Type*} [Group G] [IsSimpleGroup G] {n : ℕ} (φ : FreeGroup (Fin n) →* G) (_hsurj : Function.Surjective φ) (_hker : (MonoidHom.ker φ).IsNormalClosureFG) : LeanEval.GroupTheory.BooneHigmanSimpleProblem.WordProblemSolvable φ := G:Type u_1inst✝¹:Group Ginst✝:IsSimpleGroup Gn:ℕφ:FreeGroup (Fin n) →* G_hsurj:Function.Surjective ⇑φ_hker:φ.ker.IsNormalClosureFG⊢ WordProblemSolvable φ All goals completed! 🐙
#106
How produced

Automatically proved by Seed Prover.

Linear programming: maximum principle and vertex optimality
lp_maximum_principle

Verso theorem preview

/-- **Maximum principle for linear programming** (§101). A local maximiser of the LP objective on the feasible region is automatically a global maximiser; and whenever the objective is non-constant (`c ≠ 0`), the maximiser lies on the topological frontier of the feasible region. -/ theorem declaration uses `sorry`lp_maximum_principle {m n : ℕ} (lp : LinearProgram m n) (x : Fin m → ℝ) (_hx : x ∈ lp.feasible) (_hlocal : IsLocalMaxOn lp.objective lp.feasible x) : IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) := m:ℕn:ℕlp:LinearProgram m nx:Fin m → ℝ_hx:x ∈ lp.feasible_hlocal:IsLocalMaxOn lp.objective lp.feasible x⊢ IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) All goals completed! 🐙
/-- **Vertex optimality** (§101; the existence content of Dantzig's 1947 simplex algorithm). Every linear program with a nonempty bounded feasible region admits a global maximiser that is an extreme point (vertex) of the feasible region. -/ theorem declaration uses `sorry`simplex_algorithm {m n : ℕ} (lp : LinearProgram m n) (_hfeas : lp.feasible.Nonempty) (_hbdd : Bornology.IsBounded lp.feasible) : ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible := m:ℕn:ℕlp:LinearProgram m n_hfeas:lp.feasible.Nonempty_hbdd:Bornology.IsBounded lp.feasible⊢ ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible All goals completed! 🐙
#107
How produced

Automatically proved by Seed Prover.

Wiener's atom-detection formula
wiener_atom_detection

Verso theorem preview

theorem declaration uses `sorry`wiener_atom_detection (μ : Measure (AddCircle (2 * Real.pi))) [IsProbabilityMeasure μ] : Tendsto (fun N : ℕ => (1 / (N : ℝ)) * ∑ k ∈ Finset.Icc (1 : ℤ) N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' x : AddCircle (2 * Real.pi), ((μ {x}).toReal) ^ 2)) := μ:Measure (AddCircle (2 * π))inst✝:IsProbabilityMeasure μ⊢ Tendsto (fun N => 1 / ↑N * ∑ k ∈ Finset.Icc 1 ↑N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' (x : AddCircle (2 * π)), (μ {x}).toReal ^ 2)) All goals completed! 🐙
#108
How produced

Automatically proved by Seed Prover.

Bourbaki's locally convex extension of Banach–Alaoglu
banach_alaoglu_bourbaki

Verso theorem preview

theorem declaration uses `sorry`banach_alaoglu_bourbaki (E : Type*) [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] (U : Set E) (_hU : U ∈ 𝓝 (0 : E)) : IsCompact (LeanEval.Analysis.weakStarPolar E U) := E:Type u_1inst✝⁵:AddCommGroup Einst✝⁴:Module ℝ Einst✝³:TopologicalSpace Einst✝²:ContinuousAdd Einst✝¹:ContinuousSMul ℝ Einst✝:LocallyConvexSpace ℝ EU:Set E_hU:U ∈ 𝓝 0⊢ IsCompact (weakStarPolar E U) All goals completed! 🐙
#109
How produced

Automatically proved by Seed Prover.

Runge's theorem
runge_theorem

Verso theorem preview

theorem declaration uses `sorry`runge (K : Set ℂ) (_hK : IsCompact K) (U : Set ℂ) (_hU : IsOpen U) (_hKU : K ⊆ U) (f : ℂ → ℂ) (_hf : AnalyticOnNhd ℂ f U) (ε : ℝ) (_hε : 0 < ε) : ∃ p q : ℂ[X], (∀ z ∈ K, q.eval z ≠ 0) ∧ (∀ z ∈ K, ‖f z - p.eval z / q.eval z‖ < ε) := K:Set ℂ_hK:IsCompact KU:Set ℂ_hU:IsOpen U_hKU:K ⊆ Uf:ℂ → ℂ_hf:AnalyticOnNhd ℂ f Uε:ℝ_hε:0 < ε⊢ ∃ p q, (∀ z ∈ K, Polynomial.eval z q ≠ 0) ∧ ∀ z ∈ K, ‖f z - Polynomial.eval z p / Polynomial.eval z q‖ < ε All goals completed! 🐙
#110
How produced

Automatically proved by Seed Prover.

Baer–Suzuki theorem
baer_suzuki

Verso theorem preview

theorem declaration uses `sorry`baer_suzuki {G : Type*} [Group G] [Finite G] {p : ℕ} [Fact p.Prime] (x : G) : x ∈ LeanEval.GroupTheory.Defs.pCore p G ↔ ∀ g : G, IsPGroup p (Subgroup.closure ({x, g * x * g⁻¹} : Set G)) := G:Type u_1inst✝²:Group Ginst✝¹:Finite Gp:ℕinst✝:Fact (Nat.Prime p)x:G⊢ x ∈ pCore p G ↔ ∀ (g : G), IsPGroup p ↥(Subgroup.closure {x, g * x * g⁻¹}) All goals completed! 🐙
#111
How produced

Automatically proved by Seed Prover.

Schauder fixed-point theorem
schauder_fixed_point

Verso theorem preview

theorem declaration uses `sorry`schauder_fixed_point {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {K : Set E} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : E → E) (_hf_cont : ContinuousOn f K) (_hf_maps : Set.MapsTo f K K) : ∃ x ∈ K, f x = x := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace EK:Set E_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:E → E_hf_cont:ContinuousOn f K_hf_maps:Set.MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#112
How produced

Automatically proved by Seed Prover.

Brouwer fixed-point theorem
brouwer_fixed_point

Verso theorem preview

theorem declaration uses `sorry`brouwer_fixed_point {d : ℕ} {K : Set (EuclideanSpace ℝ (Fin d))} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (_hf_cont : ContinuousOn f K) (_hf_maps : MapsTo f K K) : ∃ x ∈ K, f x = x := d:ℕK:Set (EuclideanSpace ℝ (Fin d))_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)_hf_cont:ContinuousOn f K_hf_maps:MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#113
How produced

Automatically proved by Seed Prover.

Symplectic matrices have determinant 1
symplectic_matrix_det

Verso theorem preview

theorem declaration uses `sorry`symplectic_matrix_det {l R : Type*} [DecidableEq l] [Fintype l] [CommRing R] {A : Matrix (l ⊕ l) (l ⊕ l) R} (_hA : A ∈ Matrix.symplecticGroup l R) : A.det = 1 := l:Type u_1R:Type u_2inst✝²:DecidableEq linst✝¹:Fintype linst✝:CommRing RA:Matrix (l ⊕ l) (l ⊕ l) R_hA:A ∈ symplecticGroup l R⊢ A.det = 1 All goals completed! 🐙
#114
How produced

Automatically proved by Seed Prover.

Schur-Weyl duality: GL(V) image equals centralizer of S_k image
glAction_range_eq_centralizer_symAction

Verso theorem preview

theorem declaration uses `sorry`glAction_range_eq_centralizer_symAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(glAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(symAction R M k)) All goals completed! 🐙
#115
Schur-Weyl duality: S_k image equals centralizer of GL(V) image
symAction_range_eq_centralizer_glAction

Verso theorem preview

theorem declaration uses `sorry`symAction_range_eq_centralizer_glAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(symAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(glAction R M k)) All goals completed! 🐙
#116
Pointwise and Cesàro convergence of Fourier series (Dirichlet, Fejér)
fourier_dirichlet_fejer

Verso theorem preview

/-- **Dirichlet's pointwise convergence theorem** (§46). For every `C¹` 2π-periodic complex function `f`, the symmetric Fourier partial sums `S_N(f)(x)` converge to `f(x)` at every point `x ∈ ℝ`. -/ theorem declaration uses `sorry`dirichlet_pointwise {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hC1 : ContDiff ℝ 1 f) (x : ℝ) : Tendsto (fun N : ℕ => fourierPartialSum f N x) atTop (𝓝 (f x)) := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hC1:ContDiff ℝ 1 fx:ℝ⊢ Tendsto (fun N => fourierPartialSum f N x) atTop (𝓝 (f x)) All goals completed! 🐙
/-- **Fejér's theorem** (§46). For every *continuous* 2π-periodic complex function `f` — without the `C¹` hypothesis of Dirichlet's theorem — the Cesàro means `σ_N(f)` of the symmetric Fourier partial sums converge to `f` uniformly on `ℝ`. -/ theorem declaration uses `sorry`fejer {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hcont : Continuous f) : TendstoUniformly (fun N : ℕ => fourierCesaroMean f N) f atTop := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hcont:Continuous f⊢ TendstoUniformly (fun N => fourierCesaroMean f N) f atTop All goals completed! 🐙
#117
How produced

Automatically proved by Seed Prover.

Abel–Ruffini theorem
abel_ruffini

Verso theorem preview

theorem declaration uses `sorry`abel_ruffini (n : ℕ) (_hn : 1 ≤ n) : (∀ p : ℚ[X], p.natDegree = n → ∀ x : ℂ, aeval x p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 := n:ℕ_hn:1 ≤ n⊢ (∀ (p : ℚ[X]), p.natDegree = n → ∀ (x : ℂ), (aeval x) p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 All goals completed! 🐙
#118
How produced

Automatically proved by Seed Prover.

Pell solutions are convergents of √d
pell_solution_convergent

Verso theorem preview

theorem declaration uses `sorry`pell_solution_is_convergent (d : ℤ) (_hd : Squarefree d) (_hd0 : 0 < d) (x y : ℤ) (_hx : 0 < x) (_hy : 0 < y) (_hsol : x ^ 2 - d * y ^ 2 = 1) : ∃ n : ℕ, (GenContFract.of (Real.sqrt (d : ℝ))).convs n = (x : ℝ) / (y : ℝ) := d:ℤ_hd:Squarefree d_hd0:0 < dx:ℤy:ℤ_hx:0 < x_hy:0 < y_hsol:x ^ 2 - d * y ^ 2 = 1⊢ ∃ n, (GenContFract.of √↑d).convs n = ↑x / ↑y All goals completed! 🐙
#119
How produced

Automatically proved by Seed Prover.

Bing's house with two rooms is contractible
contractibleSpace_houseWithTwoRooms

Verso theorem preview

theorem declaration uses `sorry`contractibleSpace_houseWithTwoRooms : ContractibleSpace LeanEval.Topology.HouseWithTwoRooms := ⊢ ContractibleSpace ↑HouseWithTwoRooms All goals completed! 🐙
#120
Burnside p^a q^b theorem
finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow

Verso theorem preview

theorem declaration uses `sorry`finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow {G : Type*} [Group G] [Fintype G] {p q a b : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q) (hpq : p ≠ q) (hcard : Fintype.card G = p ^ a * q ^ b) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Fintype Gp:ℕq:ℕa:ℕb:ℕhp:Nat.Prime phq:Nat.Prime qhpq:p ≠ qhcard:Fintype.card G = p ^ a * q ^ b⊢ IsSolvable G All goals completed! 🐙
#121
Real cyclotomic integer with house at most 2
cyclotomic_integer_house_le_two

Verso theorem preview

theorem declaration uses `sorry`cyclotomic_integer_house_le_two {K : Type*} [Field K] [NumberField K] [Algebra ℚ K] (n : ℕ) [NeZero n] [IsCyclotomicExtension {n} ℚ K] {β : K} (hβ_int : IsIntegral ℤ β) (hβ_real : β ∈ NumberField.maximalRealSubfield K) : house β ≤ 2 → house β = 2 ∨ ∃ m : ℕ, 0 < m ∧ house β = 2 * Real.cos (Real.pi / m) := K:Type u_1inst✝⁴:Field Kinst✝³:NumberField Kinst✝²:Algebra ℚ Kn:ℕinst✝¹:NeZero ninst✝:IsCyclotomicExtension {n} ℚ Kβ:Khβ_int:IsIntegral ℤ βhβ_real:β ∈ maximalRealSubfield K⊢ house β ≤ 2 → house β = 2 ∨ ∃ m, 0 < m ∧ house β = 2 * Real.cos (Real.pi / ↑m) All goals completed! 🐙
#122
Existence of a non-isotopic pair of oriented two-component links
exists_nonisotopic_link

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_link : ∃ L₁ L₂ : LeanEval.KnotTheory.TwoLink, ¬ L₁.Isotopic L₂ := ⊢ ∃ L₁ L₂, ¬L₁.Isotopic L₂ All goals completed! 🐙
#123
von Neumann double commutant theorem
vonNeumann_doubleCommutant_tfae

Verso theorem preview

theorem declaration uses `sorry`vonNeumann_doubleCommutant_tfae {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : StarSubalgebra ℂ (H →L[ℂ] H)) : List.TFAE [ Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S , IsClosed (ContinuousLinearMapWOT.ofCLM '' (S : Set (H →L[ℂ] H))) , IsClosed (ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H '' (S : Set (H →L[ℂ] H))) ] := H:Type u_1inst✝²:NormedAddCommGroup Hinst✝¹:InnerProductSpace ℂ Hinst✝:CompleteSpace HS:StarSubalgebra ℂ (H →L[ℂ] H)⊢ [(↑S).centralizer.centralizer = ↑S, IsClosed (ContinuousLinearMapWOT.ofCLM '' ↑S), IsClosed (⇑(ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H) '' ↑S)].TFAE All goals completed! 🐙
#124
Perron-Frobenius for irreducible nonnegative matrices
irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius

Verso theorem preview

theorem declaration uses `sorry`irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius {n : Type*} [Fintype n] [DecidableEq n] [Nonempty n] (A : Matrix n n ℝ) (hA : A.IsIrreducible) : ∃ v : n → ℝ, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ (∀ i, 0 < v i) := n:Type u_1inst✝²:Fintype ninst✝¹:DecidableEq ninst✝:Nonempty nA:Matrix n n ℝhA:A.IsIrreducible⊢ ∃ v, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ ∀ (i : n), 0 < v i All goals completed! 🐙
#125
Linear ODE with negative-real-part eigenvalues is asymptotically stable
linear_ode_asymptotic_stability

Verso theorem preview

theorem declaration uses `sorry`linear_ode_asymptotic_stability (n : ℕ) (A : Matrix (Fin n) (Fin n) ℝ) (hA : ∀ μ : ℂ, Module.End.HasEigenvalue (Matrix.toLin' (A.map (algebraMap ℝ ℂ))) μ → μ.re < 0) (x : ℝ → (Fin n → ℝ)) (hx : ∀ t : ℝ, 0 < t → HasDerivAt x (A.mulVec (x t)) t) : Filter.Tendsto (fun t : ℝ => ‖x t‖) Filter.atTop (nhds 0) := n:ℕA:Matrix (Fin n) (Fin n) ℝhA:∀ (μ : ℂ), Module.End.HasEigenvalue (Matrix.toLin' (A.map ⇑(algebraMap ℝ ℂ))) μ → μ.re < 0x:ℝ → Fin n → ℝhx:∀ (t : ℝ), 0 < t → HasDerivAt x (A *ᵥ x t) t⊢ Filter.Tendsto (fun t => ‖x t‖) Filter.atTop (nhds 0) All goals completed! 🐙
#126
Character values of finite groups lie in cyclotomic fields
brauer_character_in_cyclotomic

Verso theorem preview

theorem declaration uses `sorry`brauer_character_in_cyclotomic (G : Type) [Group G] [Fintype G] : ∃ φ : CyclotomicField (Monoid.exponent G) ℚ →+* ℂ, ∀ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (_ : FiniteDimensional ℂ V) (ρ : Representation ℂ G V) (g : G), LinearMap.trace ℂ V (ρ g) ∈ φ.range := G:Typeinst✝¹:Group Ginst✝:Fintype G⊢ ∃ φ, ∀ (V : Type) (x : AddCommGroup V) (x_1 : Module ℂ V), FiniteDimensional ℂ V → ∀ (ρ : Representation ℂ G V) (g : G), (LinearMap.trace ℂ V) (ρ g) ∈ φ.range All goals completed! 🐙
#127
Rouche theorem via zero counting
rouche_zero_count_eq

Verso theorem preview

theorem declaration uses `sorry`rouche_zero_count_eq {f g : ℂ → ℂ} {R : ℝ} (hR : 0 < R) (hf : MeromorphicNFOn f Set.univ) (hg : AnalyticOn ℂ g Set.univ) (hbound : ∀ z : ℂ, ‖z‖ = R → ‖g z‖ < ‖f z‖) : (∑ᶠ z, ((divisor (f + g) (Metric.closedBall 0 R))⁺) z) = (∑ᶠ z, ((divisor f (Metric.closedBall 0 R))⁺) z) := f:ℂ → ℂg:ℂ → ℂR:ℝhR:0 < Rhf:MeromorphicNFOn f Set.univhg:AnalyticOn ℂ g Set.univhbound:∀ (z : ℂ), ‖z‖ = R → ‖g z‖ < ‖f z‖⊢ ∑ᶠ (z : ℂ), (divisor (f + g) (Metric.closedBall 0 R))⁺ z = ∑ᶠ (z : ℂ), (divisor f (Metric.closedBall 0 R))⁺ z All goals completed! 🐙
#128
Complementary polynomial on the unit circle
exists_complementary_polynomial_on_unit_circle

Verso theorem preview

theorem declaration uses `sorry`exists_complementary_polynomial_on_unit_circle (P : ℂ[X]) (hP : ∀ z : Circle, ‖P.eval (z : ℂ)‖ ≤ 1) : ∃ Q : ℂ[X], Q.natDegree ≤ P.natDegree ∧ ∀ z : Circle, ‖P.eval (z : ℂ)‖ ^ 2 + ‖Q.eval (z : ℂ)‖ ^ 2 = 1 := P:ℂ[X]hP:∀ (z : Circle), ‖eval (↑z) P‖ ≤ 1⊢ ∃ Q, Q.natDegree ≤ P.natDegree ∧ ∀ (z : Circle), ‖eval (↑z) P‖ ^ 2 + ‖eval (↑z) Q‖ ^ 2 = 1 All goals completed! 🐙
#129
Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2
dirichlet_eigenvalues_eq_nat_sq

Verso theorem preview

theorem declaration uses `sorry`dirichlet_eigenvalues_eq_nat_sq (lam : ℝ) : (∃ (y : ℝ → ℝ) (J : Set ℝ), IsOpen J ∧ Set.Icc (0 : ℝ) Real.pi ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y Real.pi = 0 ∧ ∃ x ∈ Set.Ioo (0 : ℝ) Real.pi, y x ≠ 0) ↔ ∃ n : ℕ, 0 < n ∧ lam = (n : ℝ) ^ 2 := lam:ℝ⊢ (∃ y J, IsOpen J ∧ Set.Icc 0 π ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y π = 0 ∧ ∃ x ∈ Set.Ioo 0 π, y x ≠ 0) ↔ ∃ n, 0 < n ∧ lam = ↑n ^ 2 All goals completed! 🐙
#130
Gaussian heat kernel solves the 1D heat equation
heat_kernel_solves_heat_equation

Verso theorem preview

theorem declaration uses `sorry`heat_kernel_solves_heat_equation (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_bdd : ∃ M : ℝ, ∀ x, |f x| ≤ M) : -- The PDE on (0, ∞) × ℝ. (∀ t : ℝ, 0 < t → ∀ x : ℝ, ∃ ux : ℝ → ℝ, ∃ uxx : ℝ, (∀ y : ℝ, HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ -- Initial condition recovered as a one-sided limit at t = 0. (∀ x : ℝ, Filter.Tendsto (fun t : ℝ => heatSolution f t x) (nhdsWithin (0 : ℝ) (Set.Ioi 0)) (nhds (f x))) := f:ℝ → ℝhf_cont:Continuous fhf_bdd:∃ M, ∀ (x : ℝ), |f x| ≤ M⊢ (∀ (t : ℝ), 0 < t → ∀ (x : ℝ), ∃ ux uxx, (∀ (y : ℝ), HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ ∀ (x : ℝ), Filter.Tendsto (fun t => heatSolution f t x) (nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) All goals completed! 🐙
#131
Oppenheim's inequality for Hadamard products
oppenheim_inequality

Verso theorem preview

theorem declaration uses `sorry`oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : A.det * ∏ i, B i i ≤ (A ⊙ B).det := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝB:Matrix n n ℝhA:A.PosSemidefhB:B.PosSemidef⊢ A.det * ∏ i, B i i ≤ (A ⊙ B).det All goals completed! 🐙
#132
pi_1 of the circle is Z
pi1_circle_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi1_circle_mulEquiv_int : Nonempty (HomotopyGroup.Pi 1 Circle (1 : Circle) ≃* Multiplicative ℤ) := ⊢ Nonempty (HomotopyGroup.Pi 1 Circle 1 ≃* Multiplicative ℤ) All goals completed! 🐙
#133
Entrywise exponential of a PSD matrix is PSD
posSemidef_map_exp

Verso theorem preview

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n] {A : Matrix n n ℝ} (hA : A.PosSemidef) : (A.map Real.exp).PosSemidef := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef All goals completed! 🐙
#134
Minkowski-Caratheodory theorem
mem_convexHull_finset_extremePoints_of_mem_compact_convex

Verso theorem preview

theorem declaration uses `sorry`mem_convexHull_finset_extremePoints_of_mem_compact_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hscomp : IsCompact s) (hsconv : Convex ℝ s) (hx : x ∈ s) : ∃ t : Finset E, (↑t : Set E) ⊆ s.extremePoints ℝ ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ convexHull ℝ (↑t : Set E) := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:FiniteDimensional ℝ Es:Set Ex:Ehscomp:IsCompact shsconv:Convex ℝ shx:x ∈ s⊢ ∃ t, ↑t ⊆ extremePoints ℝ s ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ (convexHull ℝ) ↑t All goals completed! 🐙
#135
Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#136
Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#137
Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#138
Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#139
Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#140
Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#141

Submission history

First submissionMay 20, 2026
Last submissionJul 24, 2026

Contributors

GanjinZero132hanwenzhu9
2Aristotle (Harmonic)124 solved

Other solved problems

Onsager's 2D Ising phase transition
ising_2d_phase_transition

Verso theorem preview

theorem declaration uses `sorry`ising_2d_phase_transition : ∃ (F : ℝ → ℝ) (βc : ℝ), 0 < βc ∧ (∀ β : ℝ, Tendsto (fun n : ℕ => finiteIsingFreeEnergySeq n β) atTop (𝓝 (F β))) ∧ ¬ AnalyticAt ℝ F βc := ⊢ ∃ F βc, 0 < βc ∧ (∀ (β : ℝ), Tendsto (fun n => finiteIsingFreeEnergySeq n β) atTop (𝓝 (F β))) ∧ ¬AnalyticAt ℝ F βc All goals completed! 🐙
#1
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Commuting probabilities are closed
commProb_closed

Verso theorem preview

theorem declaration uses `sorry`commProb_closed : IsClosed ({p : ℝ | ∃ (G : Type) (hG : Group G), commProb G = p}) := ⊢ IsClosed {p | ∃ G hG, ↑(commProb G) = p} All goals completed! 🐙
#2
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Poincaré–Bendixson theorem
poincare_bendixson

Verso theorem preview

theorem declaration uses `sorry`poincare_bendixson (F : LeanEval.Dynamics.Plane → LeanEval.Dynamics.Plane) (_hF : ContDiff ℝ 1 F) (γ : ℝ → LeanEval.Dynamics.Plane) (_hγ : IsIntegralCurveOn γ (fun _ x => F x) (Set.Ici 0)) : ¬ Bornology.IsBounded (γ '' Set.Ici 0) ∨ (∃ x₀, F x₀ = 0 ∧ x₀ ∈ ⋂ s : ℝ, closure (γ '' Set.Ici s)) ∨ (∃ T : ℝ, 0 < T ∧ ∃ β : ℝ → LeanEval.Dynamics.Plane, IsIntegralCurve β (fun _ x => F x) ∧ (∀ t, β (t + T) = β t) ∧ F (β 0) ≠ 0 ∧ (⋂ s : ℝ, closure (γ '' Set.Ici s)) = Set.range β) := F:Plane → Plane_hF:ContDiff ℝ 1 Fγ:ℝ → Plane_hγ:IsIntegralCurveOn γ (fun x x_1 => F x_1) (Ici 0)⊢ ¬Bornology.IsBounded (γ '' Ici 0) ∨ (∃ x₀, F x₀ = 0 ∧ x₀ ∈ ⋂ s, closure (γ '' Ici s)) ∨ ∃ T, 0 < T ∧ ∃ β, (IsIntegralCurve β fun x x_1 => F x_1) ∧ (∀ (t : ℝ), β (t + T) = β t) ∧ F (β 0) ≠ 0 ∧ ⋂ s, closure (γ '' Ici s) = range β All goals completed! 🐙
#3
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

pi_3 of the 2-sphere is Z
pi3_sphere_two_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi3_sphere_two_mulEquiv_int (x : Metric.sphere (0 : EuclideanSpace ℝ (Fin 3)) 1) : Nonempty (HomotopyGroup.Pi 3 (Metric.sphere (0 : EuclideanSpace ℝ (Fin 3)) 1) x ≃* Multiplicative ℤ) := x:↑(Metric.sphere 0 1)⊢ Nonempty (HomotopyGroup.Pi 3 (↑(Metric.sphere 0 1)) x ≃* Multiplicative ℤ) All goals completed! 🐙
#4
pi_n of the n-sphere is Z
pin_sphere_n_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pin_sphere_n_mulEquiv_int (n : ℕ) (x : Metric.sphere (0 : EuclideanSpace ℝ (Fin (n + 2))) 1) : Nonempty (HomotopyGroup.Pi (n + 1) (Metric.sphere (0 : EuclideanSpace ℝ (Fin (n + 2))) 1) x ≃* Multiplicative ℤ) := n:ℕx:↑(Metric.sphere 0 1)⊢ Nonempty (HomotopyGroup.Pi (n + 1) (↑(Metric.sphere 0 1)) x ≃* Multiplicative ℤ) All goals completed! 🐙
#5
Bézout's theorem (projective, with multiplicity)
bezout_projective_multiplicity

Verso theorem preview

theorem declaration uses `sorry`bezout_multiplicity [IsAlgClosed K] {n : ℕ} (f : Fin n → MvPolynomial (Fin (n + 1)) K) (d : Fin n → ℕ) (_hd : ∀ k, (f k).IsHomogeneous (d k)) (_hdeg : ∀ k, (f k).totalDegree = d k) (_hd_pos : ∀ k, 1 ≤ d k) (_hfin : (⋂ k, LeanEval.AlgebraicGeometry.vanishingSet (f k)).Finite) : ∑ᶠ p ∈ (⋂ k, LeanEval.AlgebraicGeometry.vanishingSet (f k)), intersectionMultiplicity f p = (∏ k, d k : ℕ∞) := K:Type u_1inst✝¹:Field Kinst✝:IsAlgClosed Kn:ℕf:Fin n → MvPolynomial (Fin (n + 1)) Kd:Fin n → ℕ_hd:∀ (k : Fin n), (f k).IsHomogeneous (d k)_hdeg:∀ (k : Fin n), (f k).totalDegree = d k_hd_pos:∀ (k : Fin n), 1 ≤ d k_hfin:(⋂ k, vanishingSet (f k)).Finite⊢ ∑ᶠ (p : ProjSpace K n) (_ : p ∈ ⋂ k, vanishingSet (f k)), intersectionMultiplicity f p = ∏ k, ↑(d k) All goals completed! 🐙
#6
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Jordan–Brouwer separation theorem
jordan_brouwer

Verso theorem preview

theorem declaration uses `sorry`jordan_brouwer (d : ℕ) (_hd : 2 ≤ d) (r : Metric.sphere (0 : EuclideanSpace ℝ (Fin d)) 1 → EuclideanSpace ℝ (Fin d)) (_hcont : Continuous r) (_hinj : Function.Injective r) : Nat.card (ConnectedComponents ((Set.range r)ᶜ : Set (EuclideanSpace ℝ (Fin d)))) = 2 := d:ℕ_hd:2 ≤ dr:↑(Metric.sphere 0 1) → EuclideanSpace ℝ (Fin d)_hcont:Continuous r_hinj:Function.Injective r⊢ Nat.card (ConnectedComponents ↑(Set.range r)ᶜ) = 2 All goals completed! 🐙
#7
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Thue–Siegel–Roth theorem (irrationality measure ≤ 2 for algebraic irrationals)
thue_siegel_roth

Verso theorem preview

theorem declaration uses `sorry`thueSiegelRoth (x : ℝ) (_h_irr : Irrational x) (_h_alg : IsAlgebraic ℤ x) : LeanEval.NumberTheory.ThueSiegelRothProblem.IsDiophantine x := x:ℝ_h_irr:Irrational x_h_alg:IsAlgebraic ℤ x⊢ IsDiophantine x All goals completed! 🐙
#8
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Margulis–Ruelle inequality
margulis_ruelle

Verso theorem preview

theorem declaration uses `sorry`margulis_ruelle (T T_inv : LeanEval.Dynamics.EucPlane → LeanEval.Dynamics.EucPlane) (hT_smooth : ContDiff ℝ 2 T) (hT_inv_smooth : ContDiff ℝ 2 T_inv) (hT_left : Function.LeftInverse T_inv T) (hT_right : Function.RightInverse T_inv T) (K : Set LeanEval.Dynamics.EucPlane) (hK_compact : IsCompact K) (hK_inv : T '' K = K) (μ : Measure LeanEval.Dynamics.EucPlane) [IsProbabilityMeasure μ] (hμ_supp : μ Kᶜ = 0) (hμ_pres : MeasurePreserving T μ μ) (hμ_erg : Ergodic T μ) : kolmogorovSinaiEntropy μ T ≤ max 0 (∫ x, lyapunovUpperAt T x ∂μ) + max 0 (∫ x, lyapunovLowerAt T x ∂μ) := T:EucPlane → EucPlaneT_inv:EucPlane → EucPlanehT_smooth:ContDiff ℝ 2 ThT_inv_smooth:ContDiff ℝ 2 T_invhT_left:Function.LeftInverse T_inv ThT_right:Function.RightInverse T_inv TK:Set EucPlanehK_compact:IsCompact KhK_inv:T '' K = Kμ:Measure EucPlaneinst✝:IsProbabilityMeasure μhμ_supp:μ Kᶜ = 0hμ_pres:MeasurePreserving T μ μhμ_erg:Ergodic T μ⊢ kolmogorovSinaiEntropy μ T ≤ max 0 (∫ (x : EucPlane), lyapunovUpperAt T x ∂μ) + max 0 (∫ (x : EucPlane), lyapunovLowerAt T x ∂μ) All goals completed! 🐙
#9
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Seventeen wallpaper groups (Pólya–Niggli 1924)
wallpaper_groups_17

Verso theorem preview

theorem declaration uses `sorry`there_are_17_wallpaper_groups : crystallographicCount 2 = 17 := ⊢ crystallographicCount 2 = 17 All goals completed! 🐙
#10
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Mergelyan's theorem
mergelyan_theorem

Lean theorem statement

/-- **Mergelyan's theorem.** For a compact `K ⊆ ℂ` with connected
complement and `f : ℂ → ℂ` continuous on `K` and analytic on the
interior of `K`, every `ε > 0` admits a complex polynomial `p` with
`‖f z − p(z)‖ < ε` on `K`. -/
theorem mergelyan (K : Set ℂ) (_hK : IsCompact K) (_hKc : IsConnected (Kᶜ))
    (f : ℂ → ℂ) (_hfc : ContinuousOn f K) (_hfh : AnalyticOnNhd ℂ f (interior K))
    (ε : ℝ) (_hε : 0 < ε) :
    ∃ p : ℂ[X], ∀ z ∈ K, ‖f z - p.eval z‖ < ε := by
  sorry
#11
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

KAM persistence of an invariant curve
kam_invariant_curve

Lean theorem statement

/-- **KAM theorem (persistence of an invariant curve).** For real-analytic,
`1`-periodic, non-constant, mean-zero `f` and Diophantine `α`, for all small
`|c|` the twist-map functional equation
`q(t+α) − 2q(t) + q(t−α) = c·f(q(t))` has a smooth strictly increasing solution
`q` with `q − id` periodic — the `c = 0` curve `q₀(t) = t` persists as a smooth
invariant curve of rotation number `α`. -/
theorem kam_invariant_curve
    (α : ℝ) (_hα : IsDiophantine α)
    (f : ℝ → ℝ)
    (_hf_analytic : AnalyticOnNhd ℝ f Set.univ)
    (_hf_per : Function.Periodic f 1)
    (_hf_nonconst : ¬ ∃ k : ℝ, ∀ x, f x = k)
    (_hf_mean : ∫ x in (0 : ℝ)..1, f x = 0) :
    ∃ c₀ : ℝ, 0 < c₀ ∧ ∀ c : ℝ, |c| < c₀ →
      ∃ q : ℝ → ℝ,
        ContDiff ℝ ∞ q ∧ StrictMono q ∧
        Function.Periodic (fun t => q t - t) 1 ∧
        ∀ t : ℝ, q (t + α) - 2 * q t + q (t - α) = c * f (q t) := by
  sorry
#12
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Darboux's theorem (symplectic forms are locally standard)
darboux

Verso theorem preview

theorem declaration uses `sorry`darboux {n : ℕ} {U : Set (LeanEval.Geometry.Darboux.E n)} (_hU : IsOpen U) (α : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n [⋀^Fin 2]→L[ℝ] ℝ) (_hα : LeanEval.Geometry.Darboux.IsSymplecticOn α U) {x : LeanEval.Geometry.Darboux.E n} (_hx : x ∈ U) : ∃ φ : OpenPartialHomeomorph (LeanEval.Geometry.Darboux.E n) (LeanEval.Geometry.Darboux.E n), x ∈ φ.source ∧ φ.source ⊆ U ∧ ContDiffOn ℝ ∞ (φ : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n) φ.source ∧ ContDiffOn ℝ ∞ (φ.symm : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n) φ.target ∧ ∀ z ∈ φ.target, LeanEval.Geometry.Darboux.IsDarbouxNormal ((α (φ.symm z)).compContinuousLinearMap (fderiv ℝ (φ.symm : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n) z)) := n:ℕU:Set (E n)_hU:IsOpen Uα:E n → E n [⋀^Fin 2]→L[ℝ] ℝ_hα:IsSymplecticOn α Ux:E n_hx:x ∈ U⊢ ∃ φ, x ∈ φ.source ∧ φ.source ⊆ U ∧ ContDiffOn ℝ ∞ (↑φ) φ.source ∧ ContDiffOn ℝ ∞ (↑φ.symm) φ.target ∧ ∀ z ∈ φ.target, IsDarbouxNormal ((α (↑φ.symm z)).compContinuousLinearMap (fderiv ℝ (↑φ.symm) z)) All goals completed! 🐙
#13
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Existence of a non-isotopic pair of oriented knots
exists_nonisotopic_knots

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_knots : ∃ K₁ K₂ : LeanEval.KnotTheory.Knot, ¬ K₁.Isotopic K₂ := ⊢ ∃ K₁ K₂, ¬K₁.Isotopic K₂ All goals completed! 🐙
#14
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Radó's theorem on Riemann surfaces
rado_riemannSurface

Verso theorem preview

theorem declaration uses `sorry`rado_riemannSurface {X : Type*} [TopologicalSpace X] [T2Space X] [ConnectedSpace X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) 1 X] : SecondCountableTopology X := X:Type u_1inst✝⁴:TopologicalSpace Xinst✝³:T2Space Xinst✝²:ConnectedSpace Xinst✝¹:ChartedSpace ℂ Xinst✝:IsManifold (modelWithCornersSelf ℂ ℂ) 1 X⊢ SecondCountableTopology X All goals completed! 🐙
#15
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Morrison–Walker Lemma B.0.1: adapting families of maps to open covers
families_of_maps_b01

Verso theorem preview

/-- **Lemma B.0.1** of Morrison–Walker, *The Blob Complex* (arXiv:1009.5025, §B), continuous case. -/ theorem declaration uses `sorry`continuous {P : Set (Fin k → ℝ)} (_hP : IsPolyhedron P) [CompactSpace X] (U : ι → Set X) (_hUopen : ∀ α, IsOpen (U α)) (ρ : PartitionOfUnity ι X univ) (_hρ : ρ.IsSubordinate U) (f : C(P × X, T)) : ∃ F : C(I × P × X, T), (∀ p : P, ∀ x : X, F (0, p, x) = f (p, x)) ∧ (∃ K : Subdivision P, ∀ D ∈ K.complex.facets, AdaptedTo U k (fun q : closedCell P D × X => F (1, q.1.1, q.2))) ∧ (∀ S : Set X, Supported (f := f.toFun) S → Supported (fun q : (I × P) × X => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ Q : Set P, IsBoundarySubpolyhedron Q → ∀ S' : Set X, Supported (fun q : Q × X => f (q.1.1, q.2)) S' → Supported (fun q : (I × Q) × X => F (q.1.1, q.1.2.1, q.2)) S') := k:ℕι:Type u_1X:Type u_2T:Type u_3inst✝²:TopologicalSpace Xinst✝¹:TopologicalSpace TP:Set (Fin k → ℝ)_hP:IsPolyhedron Pinst✝:CompactSpace XU:ι → Set X_hUopen:∀ (α : ι), IsOpen[inst✝²] (U α)ρ:PartitionOfUnity ι X_hρ:ρ.IsSubordinate Uf:C(↑P × X, T)⊢ ∃ F, (∀ (p : ↑P) (x : X), F (0, p, x) = f (p, x)) ∧ (∃ K, ∀ D ∈ K.complex.facets, AdaptedTo U k fun q => F (1, ↑q.1, q.2)) ∧ (∀ (S : Set X), Supported f.toFun S → Supported (fun q => F (q.1.1, q.1.2, q.2)) S) ∧ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ (S' : Set X), Supported (fun q => f (↑q.1, q.2)) S' → Supported (fun q => F (q.1.1, ↑q.1.2, q.2)) S' All goals completed! 🐙
/-- **Lemma B.0.1**, bi-Lipschitz variant (part 4 of the paper). -/ theorem declaration uses `sorry`biLipschitz {X T : Type*} [MetricSpace X] [MetricSpace T] [CompactSpace X] {P : Set (Fin k → ℝ)} (_hP : IsPolyhedron P) {ι : Type*} (U : ι → Set X) (_hUopen : ∀ α, IsOpen (U α)) (ρ : PartitionOfUnity ι X univ) (_hρ : ρ.IsSubordinate U) (f : C(P × X, T)) (slice : P → (X ≃ₜ T)) (_h_slice_eq : ∀ p : P, ∀ x : X, f (p, x) = slice p x) (L : NNReal) (_hf_joint : LipschitzWith L f.toFun) (_hf_slice_inv : ∀ p : P, LipschitzWith L (slice p).symm) : ∃ F : C(I × P × X, T), ∃ L' : NNReal, ∃ Slice : I × P → (X ≃ₜ T), (∀ p : P, ∀ x : X, F (0, p, x) = f (p, x)) ∧ (∀ t : I, ∀ p : P, ∀ x : X, F (t, p, x) = Slice (t, p) x) ∧ (∃ K : Subdivision P, ∀ D ∈ K.complex.facets, AdaptedTo U k (fun q : closedCell P D × X => F (1, q.1.1, q.2))) ∧ (∀ S : Set X, Supported (f := f.toFun) S → Supported (fun q : (I × P) × X => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ Q : Set P, IsBoundarySubpolyhedron Q → ∀ S' : Set X, Supported (fun q : Q × X => f (q.1.1, q.2)) S' → Supported (fun q : (I × Q) × X => F (q.1.1, q.1.2.1, q.2)) S') ∧ (∀ tp : I × P, LipschitzWith L' (Slice tp)) ∧ (∀ tp : I × P, LipschitzWith L' (Slice tp).symm) := k:ℕX:Type u_4T:Type u_5inst✝²:MetricSpace Xinst✝¹:MetricSpace Tinst✝:CompactSpace XP:Set (Fin k → ℝ)_hP:IsPolyhedron Pι:Type u_6U:ι → Set X_hUopen:∀ (α : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U α)ρ:PartitionOfUnity ι X_hρ:ρ.IsSubordinate Uf:C(↑P × X, T)slice:↑P → X ≃ₜ T_h_slice_eq:∀ (p : ↑P) (x : X), f (p, x) = (slice p) xL:NNReal_hf_joint:LipschitzWith L f.toFun_hf_slice_inv:∀ (p : ↑P), LipschitzWith L ⇑(slice p).symm⊢ ∃ F L' Slice, (∀ (p : ↑P) (x : X), F (0, p, x) = f (p, x)) ∧ (∀ (t : ↑I) (p : ↑P) (x : X), F (t, p, x) = (Slice (t, p)) x) ∧ (∃ K, ∀ D ∈ K.complex.facets, AdaptedTo U k fun q => F (1, ↑q.1, q.2)) ∧ (∀ (S : Set X), Supported f.toFun S → Supported (fun q => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ (S' : Set X), Supported (fun q => f (↑q.1, q.2)) S' → Supported (fun q => F (q.1.1, ↑q.1.2, q.2)) S') ∧ (∀ (tp : ↑I × ↑P), LipschitzWith L' ⇑(Slice tp)) ∧ ∀ (tp : ↑I × ↑P), LipschitzWith L' ⇑(Slice tp).symm All goals completed! 🐙
#16
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

The Golod–Shafarevich inequality
golod_shafarevich_inequality

Verso theorem preview

theorem declaration uses `sorry`golod_shafarevich_inequality (p : ℕ) [Fact p.Prime] (Q : Type) [Group Q] [TopologicalSpace Q] [IsTopologicalGroup Q] [DiscreteTopology Q] [Finite Q] : IsPGroup p Q → Nontrivial Q → (generatorRank Q : ℝ) ^ 2 < 4 * (relationRank p Q : ℝ) := p:ℕinst✝⁵:Fact (Nat.Prime p)Q:Typeinst✝⁴:Group Qinst✝³:TopologicalSpace Qinst✝²:IsTopologicalGroup Qinst✝¹:DiscreteTopology Qinst✝:Finite Q⊢ IsPGroup p Q → Nontrivial Q → ↑(generatorRank Q) ^ 2 < 4 * ↑(relationRank p Q) All goals completed! 🐙
#17
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Cauchy–Kovalevskaya theorem
cauchy_kovalevskaya

Verso theorem preview

theorem declaration uses `sorry`cauchy_kovalevskaya {d : ℕ} (F : LeanEval.Analysis.E d × ℝ × ℝ → LeanEval.Analysis.E d) (f : LeanEval.Analysis.E d × ℝ × ℝ → ℝ) (u₀ : LeanEval.Analysis.E d → ℝ) (_hF : AnalyticOnNhd ℝ F univ) (_hf : AnalyticOnNhd ℝ f univ) (_hu₀ : AnalyticOnNhd ℝ u₀ univ) (x₀ : LeanEval.Analysis.E d) : ∃ (U : Set (LeanEval.Analysis.E d × ℝ)) (u : LeanEval.Analysis.E d × ℝ → ℝ), (x₀, (0 : ℝ)) ∈ U ∧ IsOpen U ∧ AnalyticOnNhd ℝ u U ∧ (∀ x : LeanEval.Analysis.E d, (x, (0 : ℝ)) ∈ U → u (x, 0) = u₀ x) ∧ (∀ p ∈ U, fderiv ℝ u p ((0 : LeanEval.Analysis.E d), (1 : ℝ)) = fderiv ℝ u p (F (p.1, p.2, u p), (0 : ℝ)) + f (p.1, p.2, u p)) ∧ (∀ v : LeanEval.Analysis.E d × ℝ → ℝ, AnalyticOnNhd ℝ v U → (∀ x : LeanEval.Analysis.E d, (x, (0 : ℝ)) ∈ U → v (x, 0) = u₀ x) → (∀ p ∈ U, fderiv ℝ v p ((0 : LeanEval.Analysis.E d), (1 : ℝ)) = fderiv ℝ v p (F (p.1, p.2, v p), (0 : ℝ)) + f (p.1, p.2, v p)) → ∀ p ∈ U, u p = v p) := d:ℕF:E d × ℝ × ℝ → E df:E d × ℝ × ℝ → ℝu₀:E d → ℝ_hF:AnalyticOnNhd ℝ F univ_hf:AnalyticOnNhd ℝ f univ_hu₀:AnalyticOnNhd ℝ u₀ univx₀:E d⊢ ∃ U u, (x₀, 0) ∈ U ∧ IsOpen U ∧ AnalyticOnNhd ℝ u U ∧ (∀ (x : E d), (x, 0) ∈ U → u (x, 0) = u₀ x) ∧ (∀ p ∈ U, (fderiv ℝ u p) (0, 1) = (fderiv ℝ u p) (F (p.1, p.2, u p), 0) + f (p.1, p.2, u p)) ∧ ∀ (v : E d × ℝ → ℝ), AnalyticOnNhd ℝ v U → (∀ (x : E d), (x, 0) ∈ U → v (x, 0) = u₀ x) → (∀ p ∈ U, (fderiv ℝ v p) (0, 1) = (fderiv ℝ v p) (F (p.1, p.2, v p), 0) + f (p.1, p.2, v p)) → ∀ p ∈ U, u p = v p All goals completed! 🐙
#18
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Halmos's generic weak-mixing theorem
halmos_generic_weak_mixing

Verso theorem preview

theorem declaration uses `sorry`generic_weakly_mixing [StandardBorelSpace X] (m : Measure X) [IsProbabilityMeasure m] [NoAtoms m] : (∃ G : Set (LeanEval.Dynamics.HalmosGenericWeakMixingProblem.Automorphism m), IsGδ G ∧ Dense G ∧ ∀ T ∈ G, LeanEval.Dynamics.HalmosGenericWeakMixingProblem.IsWeaklyMixing m T) ∧ (∀ T : LeanEval.Dynamics.HalmosGenericWeakMixingProblem.Automorphism m, LeanEval.Dynamics.HalmosGenericWeakMixingProblem.IsWeaklyMixing m T → Ergodic (T.toEquiv : X → X) m) := X:Type u_1inst✝³:MeasurableSpace Xinst✝²:StandardBorelSpace Xm:Measure Xinst✝¹:IsProbabilityMeasure minst✝:NoAtoms m⊢ (∃ G, IsGδ G ∧ Dense G ∧ ∀ T ∈ G, IsWeaklyMixing m T) ∧ ∀ (T : Automorphism m), IsWeaklyMixing m T → Ergodic (⇑T.toEquiv) m All goals completed! 🐙
#19
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Radial symmetry for positive semilinear Poisson solutions
semilinear_poisson_radial_symmetry

Lean theorem statement

/-- **Radial symmetry theorem.** Let `u ∈ C^2(closedBall 0 1)` be a positive
solution of the semilinear Poisson problem `-Δ u = f(u)` in the open unit ball,
with zero Dirichlet boundary values on the unit sphere. If `f : ℝ → ℝ` is
Lipschitz, then `u` is radial: `u x = v ‖x‖` for a nonnegative strictly
decreasing radial profile `v` on `[0, 1]`. -/
theorem semilinear_poisson_radial_symmetry {n : ℕ} (hn : 0 < n)
    {f : ℝ → ℝ} (u : EuclideanSpace ℝ (Fin n) → ℝ)
    (hf_lipschitz : ∃ K : ℝ≥0, LipschitzWith K f)
    (hu_c2 : ContDiffOn ℝ 2 u (closedBall 0 1))
    (hu_solve : SolvesSemilinearPoisson f u)
    (hu_positive : ∀ x ∈ ball 0 1, 0 < u x) :
    ∃ v : ℝ → ℝ≥0,
      StrictAntiOn v (Set.Icc (0 : ℝ) 1) ∧
        ∀ x ∈ closedBall 0 1, u x = v ‖x‖ := by
  sorry
#21
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

No bounded projection from L^1 onto H^1
H1_not_closedComplemented

Verso theorem preview

theorem declaration uses `sorry`H1_not_closedComplemented : ¬ LeanEval.Analysis.H1.ClosedComplemented := ⊢ ¬H1.ClosedComplemented All goals completed! 🐙
#22
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Riesz brothers' theorem
riesz_brothers_theorem

Lean theorem statement

/-- **Riesz brothers' theorem.** Let `μ` be a complex Borel measure on
the unit circle such that `∫ z^n dμ = 0` for every `n ≥ 1`. Then `μ` is
absolutely continuous with respect to Haar measure, and more precisely
`dμ = h dm` for some `h ∈ H^1`, i.e. some integrable density whose negative
Fourier coefficients vanish. -/
theorem riesz_brothers_theorem (μ : ComplexMeasure UnitAddCircle)
    (hμ : ∀ n : ℕ, 1 ≤ n → ∫ᵛ z, fourier n z ∂[ContinuousLinearMap.mul ℝ ℂ; μ] = 0) :
    μ ≪ᵥ AddCircle.haarAddCircle.toENNRealVectorMeasure := by
  sorry
#23
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Pick's theorem
pick

Verso theorem preview

theorem declaration uses `sorry`pick {n : ℕ} (hn : 3 ≤ n) (v : Fin n → ℤ × ℤ) (hsimple : LeanEval.Geometry.PicksTheorem.IsSimple (LeanEval.Geometry.PicksTheorem.latPoly v)) : area ((LeanEval.Geometry.PicksTheorem.latPoly v).boundary (R := ℝ)) = (interiorPts v : ℝ) + (boundaryPts v : ℝ) / 2 - 1 := n:ℕhn:3 ≤ nv:Fin n → ℤ × ℤhsimple:IsSimple (latPoly v)⊢ area (Polygon.boundary ℝ (latPoly v)) = ↑(interiorPts v) + ↑(boundaryPts v) / 2 - 1 All goals completed! 🐙
#24
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Wiener–Lévy theorem
wiener_levy_analytic_calculus

Verso theorem preview

theorem declaration uses `sorry`wiener_levy_analytic_calculus (f : C(AddCircle T, ℂ)) (φ : ℂ → ℂ) (U : Set ℂ) (hf : LeanEval.Analysis.WienerLevy.InWienerAlgebra f) (hU : IsOpen U) (hrange : range f ⊆ U) (hφ : AnalyticOnNhd ℂ φ U) : ∃ g : C(AddCircle T, ℂ), (∀ x, g x = φ (f x)) ∧ LeanEval.Analysis.WienerLevy.InWienerAlgebra g := T:ℝinst✝:Fact (0 < T)f:C(AddCircle T, ℂ)φ:ℂ → ℂU:Set ℂhf:InWienerAlgebra fhU:IsOpen Uhrange:range ⇑f ⊆ Uhφ:AnalyticOnNhd ℂ φ U⊢ ∃ g, (∀ (x : AddCircle T), g x = φ (f x)) ∧ InWienerAlgebra g All goals completed! 🐙
#25
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Frobenius determinant theorem
frobenius_group_determinant

Lean theorem statement

/-- **Frobenius determinant theorem** (§171). The group determinant factors as
a product of irreducible polynomials, each appearing to the power of its own
(total) degree `d_j = deg p_j`, with the factors pairwise non-associated
(*distinct*) and their number equal to the number of conjugacy classes of `G`.
-/
theorem frobenius_group_determinant
    (G : Type*) [Group G] [Fintype G] [DecidableEq G] :
    ∃ (r : ℕ) (p : Fin r → MvPolynomial G ℂ),
      r = Nat.card (ConjClasses G) ∧
      (∀ j, Irreducible (p j)) ∧
      (∀ i j, i ≠ j → ¬ Associated (p i) (p j)) ∧
      groupDeterminant G = ∏ j, (p j) ^ (p j).totalDegree := by
  sorry
#26
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Wiener's 1/f theorem
wiener_inverse_closed

Verso theorem preview

theorem declaration uses `sorry`wiener_inverse_closed (f : C(AddCircle T, ℂ)) (hf : LeanEval.Analysis.WienerOneOverF.InWienerAlgebra f) (hzero : ∀ x, f x ≠ 0) : ∃ g : C(AddCircle T, ℂ), (∀ x, g x = (f x)⁻¹) ∧ LeanEval.Analysis.WienerOneOverF.InWienerAlgebra g := T:ℝinst✝:Fact (0 < T)f:C(AddCircle T, ℂ)hf:InWienerAlgebra fhzero:∀ (x : AddCircle T), f x ≠ 0⊢ ∃ g, (∀ (x : AddCircle T), g x = (f x)⁻¹) ∧ InWienerAlgebra g All goals completed! 🐙
#27
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Brun's theorem (convergence of the twin-prime reciprocal sum)
brun_constant_converges

Lean theorem statement

/-- **Brun's theorem.** The reciprocal sum over twin-prime pairs converges. -/
theorem brun_constant_converges :
    Summable twinPrimeReciprocalTerm := by
  sorry
#28
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Gauss-Wantzel constructible regular polygon theorem
gauss_wantzel_constructible_polygon

Lean theorem statement

/-- **Gauss-Wantzel constructible polygon theorem** (§174): a regular `n`-gon
is straightedge-and-compass constructible exactly for the Gauss-Wantzel
integers. -/
theorem gauss_wantzel_constructible_polygon (n : ℕ) (hn : 3 ≤ n) :
    IsConstructible (Real.cos (2 * Real.pi / n)) ↔ GaussWantzelNumber n := by
  sorry
#29
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Kirk's normal-structure fixed point theorem
kirk_normal_structure

Lean theorem statement

/-- **Kirk's normal-structure fixed-point theorem** (§228).  A nonexpansive
self-map of a nonempty bounded closed convex subset of a reflexive Banach
space with normal structure has a fixed point. -/
theorem kirk_normal_structure [CompleteSpace E]
    (hE_reflexive : Function.Surjective (NormedSpace.inclusionInDoubleDual ℝ E))
    (K : Set E) (hK_nonempty : K.Nonempty) (hK_closed : IsClosed K)
    (hK_bounded : Bornology.IsBounded K) (hK_convex : Convex ℝ K)
    (hK_normal : HasNormalStructure K) (T : K → K)
    (hT : IsNonexpansiveSelfMap K T) :
    ∃ x : K, IsFixedPt T x := by
  sorry
#30
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Normal spectral theorem
normal_spectral_theorem

Lean theorem statement

/-- **Spectral theorem** (§14). A complex matrix `A` is **normal**
(`Aᴴ A = A Aᴴ`, i.e. `IsStarNormal A`) **iff** it is **unitarily
diagonalizable**: there is a unitary `U` and a diagonal matrix `diagonal d`
with `A = U (diagonal d) Uᴴ`. -/
theorem normal_spectral_theorem (A : Matrix n n ℂ) :
    IsStarNormal A ↔
      ∃ U ∈ unitary (Matrix n n ℂ), ∃ d : n → ℂ,
        A = U * diagonal d * star U := by
  sorry
#31
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Trace Cayley-Hamilton / Newton identity
trace_cayley_hamilton_newton

Lean theorem statement

/-- The trace Cayley-Hamilton / Newton identity:
`k c_k + ∑_{j=1}^k tr(A^j) c_{k-j} = 0`, where
`χ_A(X) = X^N + c₁ X^(N-1) + ... + c_N`.

For `k > N`, `c_k = 0`, and the remaining relation is the trace of
Cayley-Hamilton multiplied by a power of `A`. -/
theorem trace_cayley_hamilton_newton {R : Type*} [CommRing R]
    (A : Matrix n n R) {k : ℕ} (hk : 1 ≤ k) :
    (k : R) * charpolyDescendingCoeff A k +
        ∑ j ∈ Finset.Icc 1 k,
          trace (A ^ j) * charpolyDescendingCoeff A (k - j) = 0 := by
  sorry
#32
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Peano existence theorem for ODEs
peano_existence

Lean theorem statement

/-- **Peano existence theorem.** Replacing the Lipschitz condition with mere
continuity still yields a local solution of `x' = f(x)`, `x(0) = x₀` — but
uniqueness may fail (e.g. `x' = √x`, `x(0) = 0`). Stated for a
finite-dimensional space: Peano's theorem requires local compactness and is
false in general (infinite-dimensional) Banach spaces. -/
theorem peano_existence
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
    {f : E → E} (hf : Continuous f) (x₀ : E) :
    ∃ a : ℝ, 0 < a ∧ ∃ α : ℝ → E, α 0 = x₀ ∧
      ∀ t ∈ Ioo (-a) a, HasDerivAt α (f (α t)) t := by
  sorry
#33
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

General recursive equals Turing computable
turing_recursive_equiv

Lean theorem statement

/-- **General recursive = Turing computable** (total form). A total function
`f : ℕ → ℕ` is recursive (`Computable`, i.e. partial recursive as a partial
function) **iff** it is computed by some Turing machine (mathlib's `FinTM2`
model) under the standard binary encoding of `ℕ`. This is Knill's class
equality; the backward direction (TM-computable ⇒ recursive) is absent from
mathlib. -/
theorem turing_recursive_equiv (f : ℕ → ℕ) :
    Computable f ↔ Nonempty (TM2Computable encodeNat encodeNat f) := by
  sorry
#34
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Shannon capacity of the pentagon
shannon_capacity_pentagon

Lean theorem statement

/-- **Lovász's theorem on Shannon capacity of the pentagon** (§238).
The Shannon capacity of the five-cycle is `√5`. -/
theorem shannon_capacity_pentagon :
    HasShannonCapacity (SimpleGraph.cycleGraph 5) (Real.sqrt 5) := by
  sorry
#35
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Complete reducibility for compact groups
compact_group_semisimple

Lean theorem statement

/-- **Representations of compact groups are semisimple** (complete
reducibility / the unitarian trick). A continuous representation of a compact
topological group on a finite-dimensional real vector space is semisimple:
every subrepresentation has a `G`-invariant complement, so the representation
decomposes as a direct sum of irreducible finite-dimensional
subrepresentations. -/
theorem compact_group_semisimple
    {G V : Type*} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G]
    [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V]
    (ρ : Representation ℝ G V)
    (hρ : Continuous fun p : G × V => ρ p.1 p.2) :
    ρ.IsSemisimpleRepresentation := by
  sorry
#36
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Nyquist–Shannon sampling theorem
nyquist_shannon_sampling

Lean theorem statement

/-- **Nyquist--Shannon sampling theorem / Whittaker--Shannon interpolation
formula** for Schwartz functions with Fourier support in the mathlib-convention
Nyquist band `[-1/2, 1/2]`.

The explicit `Summable` conjunct records the convergence content of the
cardinal series, rather than relying only on Lean's total `tsum`.
-/
theorem nyquist_shannon_sampling (f : 𝓢(ℝ, ℂ)) (hf : FourierSupportedInNyquist f) :
    ∀ t : ℝ,
      Summable (fun n : ℤ ↦ f (n : ℝ) * sinc (Real.pi * ((n : ℝ) - t))) ∧
        f t =
          ∑' n : ℤ, f (n : ℝ) * sinc (Real.pi * ((n : ℝ) - t)) := by
  sorry
#37
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Wigner semicircle law
wigner_semicircle

Verso theorem preview

theorem declaration uses `sorry`wigner_semicircle {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → ℕ → Ω → ℝ) (_hX_meas : ∀ i j, Measurable (X i j)) (_hX_indep : iIndepFun (fun ij : {p : ℕ × ℕ // p.1 ≤ p.2} => X ij.val.1 ij.val.2) μ) (_hX_iid : ∀ i j i' j', i ≤ j → i' ≤ j' → ProbabilityTheory.IdentDistrib (X i j) (X i' j') μ μ) (_hX_int : ∀ i j, i ≤ j → Integrable (X i j) μ) (_hX_sq_int : ∀ i j, i ≤ j → Integrable (fun ω => (X i j ω) ^ 2) μ) (_hX_mean : ∀ i j, i ≤ j → ∫ ω, X i j ω ∂μ = 0) (_hX_var : ∀ i j, i ≤ j → ∫ ω, (X i j ω) ^ 2 ∂μ = 1) : ∀ᵐ ω ∂μ, ∀ (f : ℝ → ℝ), Continuous f → (∃ M, ∀ x, ‖f x‖ ≤ M) → Tendsto (fun n : ℕ => ∫ x, f x ∂ (empiricalSpectralMeasureHerm (wignerMatrix_isHermitian X n ω)).map (fun x : ℝ => x / Real.sqrt n)) atTop (𝓝 (∫ x, f x ∂semicircleLaw)) := Ω:Type u_1inst✝¹:MeasurableSpace Ωμ:Measure Ωinst✝:IsProbabilityMeasure μX:ℕ → ℕ → Ω → ℝ_hX_meas:∀ (i j : ℕ), Measurable (X i j)_hX_indep:iIndepFun (fun ij => X (↑ij).1 (↑ij).2) μ_hX_iid:∀ (i j i' j' : ℕ), i ≤ j → i' ≤ j' → IdentDistrib (X i j) (X i' j') μ μ_hX_int:∀ (i j : ℕ), i ≤ j → Integrable (X i j) μ_hX_sq_int:∀ (i j : ℕ), i ≤ j → Integrable (fun ω => X i j ω ^ 2) μ_hX_mean:∀ (i j : ℕ), i ≤ j → ∫ (ω : Ω), X i j ω ∂μ = 0_hX_var:∀ (i j : ℕ), i ≤ j → ∫ (ω : Ω), X i j ω ^ 2 ∂μ = 1⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (f : ℝ → ℝ), Continuous f → (∃ M, ∀ (x : ℝ), ‖f x‖ ≤ M) → Tendsto (fun n => ∫ (x : ℝ), f x ∂Measure.map (fun x => x / √↑n) (empiricalSpectralMeasureHerm ⋯)) atTop (𝓝 (∫ (x : ℝ), f x ∂semicircleLaw)) All goals completed! 🐙
#38
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Fundamental theorem of Riemannian geometry (Levi-Civita)
levi_civita_exists_unique

Verso theorem preview

theorem declaration uses `sorry`levi_civita_exists_unique {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [CompleteSpace E] {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type*} [TopologicalSpace M] [T2Space M] [ChartedSpace H M] [IsManifold I ∞ M] [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)] [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)] : ∃ cov : CovariantDerivative I E (TangentSpace I (M := M)), (ContMDiffCovariantDerivative cov ∞ ∧ cov.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov) ∧ ∀ cov' : CovariantDerivative I E (TangentSpace I (M := M)), (ContMDiffCovariantDerivative cov' ∞ ∧ cov'.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov') → LeanEval.Geometry.LeviCivita.SameOnSmooth cov cov' := E:Type u_1inst✝¹⁰:NormedAddCommGroup Einst✝⁹:NormedSpace ℝ Einst✝⁸:FiniteDimensional ℝ Einst✝⁷:CompleteSpace EH:Type u_2inst✝⁶:TopologicalSpace HI:ModelWithCorners ℝ E HM:Type u_3inst✝⁵:TopologicalSpace Minst✝⁴:T2Space Minst✝³:ChartedSpace H Minst✝²:IsManifold I ∞ Minst✝¹:RiemannianBundle fun x => TangentSpace I xinst✝:IsContMDiffRiemannianBundle I ∞ E fun x => TangentSpace I x⊢ ∃ cov, (cov.ContMDiffCovariantDerivative ∞ ∧ cov.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov) ∧ ∀ (cov' : CovariantDerivative I E (TangentSpace I)), cov'.ContMDiffCovariantDerivative ∞ ∧ cov'.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov' → SameOnSmooth cov cov' All goals completed! 🐙
#39
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Strong Subadditivity of von Neumann Entropy
strong_subadditivity

Lean theorem statement

/-- Strong subadditivity of quantum entropy. We relax the common assumption that M is a normalized
 density matrix to the simpler statement that it's PSD, which holds since normalization just produces
 a positive affine transformation on the entropy. -/
theorem strong_subadditivity (M_ABC : Matrix (A × B × C) (A × B × C) ℂ) (h : M_ABC.PosSemidef) :
    let M_AB : Matrix (A × B) (A × B) ℂ :=
      .traceRight <| M_ABC.reindex (.symm <| .prodAssoc ..) (.symm <| .prodAssoc ..)
    let M_BC : Matrix (B × C) (B × C) ℂ := M_ABC.traceLeft
    let M_B : Matrix B B ℂ := M_BC.traceRight
    entropy M_ABC + entropy M_B ≤ entropy M_AB + entropy M_BC := by
  sorry
#40
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Fundamental theorem of topos theory
fundamental_topos_theory

Verso theorem preview

theorem declaration uses `sorry`fundamental_topos_theory {E : Type*} [Category E] (hE : LeanEval.ToposTheory.IsTopos E) (X : E) : LeanEval.ToposTheory.IsTopos (Over X) := E:Type u_1inst✝:Category.{u_2, u_1} EhE:IsTopos EX:E⊢ IsTopos (Over X) All goals completed! 🐙
#41
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Ornstein–Weiss ℤᵈ Rokhlin lemma
ornstein_weiss_rokhlin

Verso theorem preview

theorem declaration uses `sorry`ornstein_weiss_rokhlin {Ω : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω] {d : ℕ} (_hd : 1 ≤ d) (μ : Measure Ω) [IsProbabilityMeasure μ] (T : (Fin d → ℤ) → Ω → Ω) (_hid : ∀ x, T 0 x = x) (_hT : ∀ v, MeasurePreserving (T v) μ μ) (_hgrp : ∀ u v x, T (u + v) x = T u (T v x)) (_hfree : LeanEval.Dynamics.IsFreeAction μ T) (N : ℕ) (_hN : 1 ≤ N) {ε : ENNReal} (_hε : 0 < ε) : ∃ B : Set Ω, MeasurableSet B ∧ ((boxShape d N : Finset (Fin d → ℤ)) : Set (Fin d → ℤ)).PairwiseDisjoint (fun v => T v '' B) ∧ μ (⋃ v ∈ boxShape d N, T v '' B) ≥ 1 - ε := Ω:Type u_1inst✝²:MeasurableSpace Ωinst✝¹:StandardBorelSpace Ωd:ℕ_hd:1 ≤ dμ:Measure Ωinst✝:IsProbabilityMeasure μT:(Fin d → ℤ) → Ω → Ω_hid:∀ (x : Ω), T 0 x = x_hT:∀ (v : Fin d → ℤ), MeasurePreserving (T v) μ μ_hgrp:∀ (u v : Fin d → ℤ) (x : Ω), T (u + v) x = T u (T v x)_hfree:IsFreeAction μ TN:ℕ_hN:1 ≤ Nε:ENNReal_hε:0 < ε⊢ ∃ B, MeasurableSet B ∧ ((↑(boxShape d N)).PairwiseDisjoint fun v => T v '' B) ∧ μ (⋃ v ∈ boxShape d N, T v '' B) ≥ 1 - ε All goals completed! 🐙
#42
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

The Lindemann–Weierstrass theorem
lindemann_weierstrass

Lean theorem statement

/-- **Lindemann–Weierstrass theorem.** If `x₁, …, xₙ ∈ ℂ` are algebraic over `ℚ`
and ℚ-linearly independent, then `e^{x₁}, …, e^{xₙ}` are algebraically
independent over `ℚ`. -/
theorem lindemann_weierstrass {n : ℕ} (x : Fin n → ℂ)
    (h_alg : ∀ i, IsAlgebraic ℚ (x i))
    (h_lin : LinearIndependent ℚ x) :
    AlgebraicIndependent ℚ (fun i => Complex.exp (x i)) := by
  sorry
#43
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Radon transform: Fourier-slice diagonalization and pseudo-inversion
radon_transform_inversion

Lean theorem statement

/-- **Radon's theorem (diagonalization + pseudo-inversion).** The Fourier slice
theorem diagonalizes the Radon transform, and the transform admits a left
inverse on the Schwartz space. -/
theorem radon_can_be_diagonalized_and_pseudo_inverted :
    (∀ φ : SchwartzMap (ℝ × ℝ) ℂ, ∀ θ k : ℝ,
        fourier1 (fun p => radon (φ : ℝ × ℝ → ℂ) (p, θ)) k =
          fourier2 (φ : ℝ × ℝ → ℂ) (k * Real.cos θ, k * Real.sin θ)) ∧
    (∃ Rinv : (ℝ × ℝ → ℂ) → (ℝ × ℝ → ℂ),
        ∀ φ : SchwartzMap (ℝ × ℝ) ℂ,
          Rinv (radon (φ : ℝ × ℝ → ℂ)) = (φ : ℝ × ℝ → ℂ)) := by
  sorry
#44
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Hurewicz theorem in degree 1 (H₁ = abelianization of π₁)
hurewicz_h1_abelianization

Lean theorem statement

/-- **Hurewicz (n = 1).** For a path-connected space `X`, `H₁(X;ℤ)` is the
abelianization of `π₁(X, x)`. -/
theorem hurewicz_h1_abelianization
    (X : Type) [TopologicalSpace X] [PathConnectedSpace X] (x : X) :
    Nonempty (Additive (Abelianization (FundamentalGroup X x)) ≃+
      (IntegralHomology 1 X : Type)) := by
  sorry
#45
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Bauer's uniqueness at extreme points
bauer_extreme_point_uniqueness

Lean theorem statement

/-- **Bauer's uniqueness at extreme points.** If `x` is an extreme point of a
compact convex set `K` and `μ` is a probability measure supported on `K`
(`μ Kᶜ = 0`) with barycenter `x = ∫ y, y ∂μ`, then `μ` is the Dirac mass at
`x`. (The support hypothesis is the weaker `μ Kᶜ = 0`, making this a
strengthening of the textbook statement: uniqueness among all ambient Borel
probability measures on `K`, not only those already supported on `ext K`.) -/
theorem bauer_unique [MeasurableSpace X] [BorelSpace X]
    (K : Set X) (hK_cpt : IsCompact K) (hK_cvx : Convex ℝ K)
    {x : X} (hx : x ∈ K.extremePoints ℝ)
    (μ : Measure X) [IsProbabilityMeasure μ]
    (hμ : μ Kᶜ = 0) (hbar : x = ∫ y, y ∂μ) :
    μ = Measure.dirac x := by
  sorry
#46
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Boone–Higman theorem (easy direction)
boone_higman_embedding

Lean theorem statement

/-- **Boone–Higman theorem (easy direction).** If a finitely presented group `G`
embeds (via injective `f`) into a simple group `H`, which embeds (via injective
`g`) into a finitely presented group `K`, then the word problem of `G` is
solvable. -/
theorem boone_higman_embedding
    {G H K : Type*} [Group G] [Group H] [Group K]
    [IsSimpleGroup H] [Group.IsFinitelyPresented K]
    (f : G →* H) (hf : Function.Injective f)
    (g : H →* K) (hg : Function.Injective g)
    {n : ℕ} (φ : FreeGroup (Fin n) →* G)
    (hsurj : Function.Surjective φ)
    (hker : (MonoidHom.ker φ).IsNormalClosureFG) :
    WordProblemSolvable φ := by
  sorry
#47
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Choquet's representation theorem
choquet_representation_theorem

Lean theorem statement

/-- **Choquet's representation theorem.** Every point `x` of a compact convex
set `K` in a Banach space is the barycenter of a probability measure supported
on the extreme points of `K`: there is a probability measure `μ` with
`μ (ext K)ᶜ = 0` whose barycenter `∫ y, y ∂μ` equals `x`. -/
theorem choquet [MeasurableSpace X] [BorelSpace X]
    (K : Set X) (hK_cpt : IsCompact K) (hK_cvx : Convex ℝ K)
    {x : X} (hx : x ∈ K) :
    ∃ μ : Measure X, IsProbabilityMeasure μ ∧
      μ (K.extremePoints ℝ)ᶜ = 0 ∧
      x = ∫ y, y ∂μ := by
  sorry
#48
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Hausdorff moment problem: absolute-continuity criterion
hausdorff_absolute_continuity

Lean theorem statement

/-- **Absolute continuity criterion (Hausdorff moment problem).** A positive
probability measure `μ` on the cube is uniformly absolutely continuous w.r.t.
Lebesgue measure iff there is `C` with `(Δᵏμ)ₙ ≤ C·(Δᵏν)ₙ` for all `k ≤ n`. -/
theorem hausdorff_absolute_continuity {d : ℕ}
    (μ : Measure (EuclideanSpace ℝ (Fin d)))
    [IsProbabilityMeasure μ] (hμ : μ ((cube d)ᶜ) = 0) :
    UniformlyAbsolutelyContinuous μ (volume.restrict (cube d)) ↔
      ∃ C : ℝ, ∀ k n : Fin d → ℕ, k ≤ n →
        diff (momentOf μ) k n ≤ C * diff (momentOf (volume.restrict (cube d))) k n := by
  sorry
#49
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

The Hausdorff–Hildebrandt–Schoenberg moment theorem
hausdorff_hildebrandt_schoenberg

Lean theorem statement

/-- **Hausdorff–Hildebrandt–Schoenberg theorem.** A multi-indexed real sequence
is the moment sequence of a signed bounded-variation measure on the unit cube
`Iᵈ` iff its moments are Hausdorff bounded. -/
theorem hausdorff_hildebrandt_schoenberg {d : ℕ} (a : (Fin d → ℕ) → ℝ) :
    IsMomentConfiguration a ↔ HausdorffBounded a := by
  sorry
#50
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

The Hausdorff positivity (complete-monotonicity) criterion
hausdorff_positivity_criterion

Lean theorem statement

/-- **Hausdorff positivity criterion** (Hausdorff 1921). A moment configuration
comes from a positive measure iff all its iterated backward differences are
nonnegative — i.e. the sequence is *completely monotone*: `(Δᵏa)ₙ ≥ 0` for all
`k ≤ n`. -/
theorem hausdorff_positivity {d : ℕ} (a : (Fin d → ℕ) → ℝ) :
    IsPositiveMomentConfiguration a ↔ ∀ k n : Fin d → ℕ, k ≤ n → 0 ≤ diff a k n := by
  sorry
#51
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Jordan normal form
jordan_normal_form

Lean theorem statement

/-- **Jordan normal form.** Over an algebraically closed field, every
endomorphism of `Kⁿ` admits a Jordan-chain basis. -/
theorem jordan_normal_form {K : Type*} [Field K] [IsAlgClosed K] (n : ℕ)
    (f : Module.End K (StdSpace K n)) :
    Nonempty (JordanChainBasis f) := by
  sorry
#52
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Kolmogorov–Arnold superposition theorem (non-universal Lorentz form)
kolmogorov_arnold_superposition

Verso theorem preview

theorem declaration uses `sorry`kolmogorov_arnold (n : ℕ) (_hn : 1 ≤ n) (f : (Fin n → ℝ) → ℝ) (_hf : ContinuousOn f (Set.Icc 0 1)) : ∃ (g : ℝ → ℝ) (φ : Fin (2 * n + 1) → Fin n → ℝ → ℝ), Continuous g ∧ (∀ k l, Continuous (φ k l)) ∧ ∀ x ∈ Set.Icc (0 : Fin n → ℝ) 1, f x = ∑ k, g (∑ l, φ k l (x l)) := n:ℕ_hn:1 ≤ nf:(Fin n → ℝ) → ℝ_hf:ContinuousOn f (Set.Icc 0 1)⊢ ∃ g φ, Continuous g ∧ (∀ (k : Fin (2 * n + 1)) (l : Fin n), Continuous (φ k l)) ∧ ∀ x ∈ Set.Icc 0 1, f x = ∑ k, g (∑ l, φ k l (x l)) All goals completed! 🐙
#53
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

The Landsberg–Schaar relation
landsberg_schaar

Lean theorem statement

/-- **Landsberg–Schaar relation.** For positive odd integers `p, q`,
`S(2q, p) = e^{iπ/4} · S(−p, 2q)`. -/
theorem landsberg_schaar (p q : ℕ) (hp : Odd p) (hq : Odd q) :
    gaussS (2 * q : ℕ) p
      = Complex.exp ((Real.pi : ℂ) * Complex.I / 4) * gaussS (-(p : ℤ)) (2 * q) := by
  sorry
#54
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Lax's approximation theorem for toral homeomorphisms
lax_approximation

Lean theorem statement

/-- **Lax's approximation theorem.** Every toral dynamical system on `𝕋^d`
(`d ≥ 1`) is approximated arbitrarily well in the metric `δ` by cyclic cube
exchange transformations. -/
theorem lax_approximation {d : ℕ} (hd : 0 < d) (T : ToralDynamicalSystem d)
    {ε : ℝ≥0∞} (hε : 0 < ε) :
    ∃ (n : ℕ) (S : VolumePreservingEquiv d),
      IsCyclicCubeExchange S n ∧ deltaDist T.toVolumePreservingEquiv S < ε := by
  sorry
#55
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Lindemann's theorem (e and π transcendental)
lindemann

Lean theorem statement

/-- **Lindemann's theorem.** Both `e = exp 1` and `π` are transcendental over
`ℤ`. -/
theorem lindemann :
    Transcendental ℤ (Real.exp 1) ∧ Transcendental ℤ Real.pi := by
  sorry
#56
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Moran's equality for affine-symmetric iterated function systems
moran_equality_affine

Lean theorem statement

/-- **Moran's equality for affine-symmetric IFS.** For an affine-symmetric IFS
on `ℝᵈ` with common contraction factor `λ ∈ (0,1)`, orthogonal linear parts, and
the open set condition, the Hausdorff dimension of the attractor is
`−log n / log λ` (positive since `λ < 1`). -/
theorem moran_equality_affine
    {d n : ℕ} (hn : 1 ≤ n)
    (f : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (lam : ℝ)
    (h_aff : IsAffineSymmetricIFS f lam)
    (h_osc : OpenSetCondition f)
    {S : Set (EuclideanSpace ℝ (Fin d))} (hS : IsAttractor f S) :
    dimH S = ENNReal.ofReal (- Real.log n / Real.log lam) := by
  sorry
#57
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Pascal's theorem
pascal

Lean theorem statement

/-- **Pascal's theorem.** Six distinct points on a non-singular conic determine
three collinear intersection points `Aᵢ Bⱼ ∩ Aⱼ Bᵢ`. -/
theorem pascal
    (M : Matrix (Fin 3) (Fin 3) ℝ) (hMsymm : M.IsSymm) (hMdet : M.det ≠ 0)
    (a₁ a₂ a₃ b₁ b₂ b₃ : Fin 3 → ℝ)
    (ha₁ : a₁ ≠ 0) (ha₂ : a₂ ≠ 0) (ha₃ : a₃ ≠ 0)
    (hb₁ : b₁ ≠ 0) (hb₂ : b₂ ≠ 0) (hb₃ : b₃ ≠ 0)
    (hdist : [a₁, a₂, a₃, b₁, b₂, b₃].Pairwise (fun v w => ¬ SamePoint v w))
    (hA₁ : OnConic M a₁) (hA₂ : OnConic M a₂) (hA₃ : OnConic M a₃)
    (hB₁ : OnConic M b₁) (hB₂ : OnConic M b₂) (hB₃ : OnConic M b₃) :
    Collinear3 (meet a₁ b₂ a₂ b₁) (meet a₁ b₃ a₃ b₁) (meet a₂ b₃ a₃ b₂) := by
  sorry
#58
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Sard's regular-value corollary
regular_value_ae

Lean theorem statement

/-- **Regular value corollary (Sard).** For a smooth `f : ℝᵐ → ℝ`, almost every
`c ∈ ℝ` is a regular value. -/
theorem regular_value_ae {m : ℕ} (f : EuclideanSpace ℝ (Fin m) → ℝ)
    (hf : ContDiff ℝ ∞ f) :
    ∀ᵐ c ∂(volume : Measure ℝ), IsRegularValue f c := by
  sorry
#59
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Riesz's rising sun lemma
rising_sun_lemma

Lean theorem statement

/-- **Riesz's rising sun lemma.** Every continuous real function on a compact
interval has the rising-sun property. -/
theorem rising_sun_lemma {a b : ℝ} (hab : a < b) {f : ℝ → ℝ}
    (hf : ContinuousOn f (Icc a b)) :
    HasRisingSunProperty a b f := by
  sorry
#60
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Tverberg's theorem
tverberg_theorem

Lean theorem statement

/-- **Tverberg's theorem.** Any `(r-1)(d+1)+1` points in `ℝ^d` admit an
`r`-part Tverberg partition. -/
theorem tverberg_theorem (d r : ℕ) (hr : 1 ≤ r)
    (f : Fin ((r - 1) * (d + 1) + 1) → Space d) :
    HasTverbergPartition (r := r) f := by
  sorry
#61
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Poincaré–Siegel linearisation theorem
poincare_siegel_linearisation

Lean theorem statement

/-- **Poincaré–Siegel linearisation theorem.** If `α` is Diophantine,
`λ = e^{2πiα}`, and `f` is holomorphic near `0` with `f 0 = 0` and
`f'(0) = λ`, then there is a holomorphic germ `u` with `u 0 = 0`,
`u'(0) = 1`, and `f(u z) = u(λ z)` for `z` near `0`. -/
theorem poincare_siegel
    (α : ℝ) (_hα : IsDiophantine α)
    (lam : ℂ) (_hlam : lam = Complex.exp (2 * Real.pi * Complex.I * (α : ℂ)))
    (f : ℂ → ℂ) (_hf : AnalyticAt ℂ f 0) (_hf0 : f 0 = 0)
    (_hmult : deriv f 0 = lam) :
    ∃ u : ℂ → ℂ, AnalyticAt ℂ u 0 ∧ u 0 = 0 ∧ deriv u 0 = 1 ∧
      ∀ᶠ z in nhds (0 : ℂ), f (u z) = u (lam * z) := by
  sorry
#62
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Anosov–Bowen shadowing lemma
anosov_bowen_shadowing

Lean theorem statement

/-- **Anosov–Bowen shadowing lemma** (Anosov 1967; Bowen 1975). Every
compact hyperbolic invariant set has the shadowing property. -/
theorem hyperbolic_has_shadowing
    (T : E d ≃ₜ E d) (K : Set (E d))
    (_hKc : IsCompact K) (_hK : IsHyperbolic T K) :
    HasShadowing (T : E d → E d) K := by
  sorry
#63
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Sobolev embedding theorem (Morrey regime)
sobolev_embedding_morrey

Verso theorem preview

theorem declaration uses `sorry`sobolev_embedding {n k r : ℕ} {α p : ℝ} (_hp : (n : ℝ) < p) (_hα : 0 < α) (_hα1 : α ≤ 1) (_hgap : (r : ℝ) + α < (k : ℝ) - n / p) (f : LeanEval.Analysis.SobolevMorreyProblem.E n → ℝ) (_hf : LeanEval.Analysis.SobolevMorreyProblem.MemSobolevWk k (ENNReal.ofReal p) f) : ∃ g : LeanEval.Analysis.SobolevMorreyProblem.E n → ℝ, f =ᵐ[volume] g ∧ LeanEval.Analysis.SobolevMorreyProblem.MemHolder r α g := n:ℕk:ℕr:ℕα:ℝp:ℝ_hp:↑n < p_hα:0 < α_hα1:α ≤ 1_hgap:↑r + α < ↑k - ↑n / pf:E n → ℝ_hf:MemSobolevWk k (ENNReal.ofReal p) f⊢ ∃ g, f =ᵐ[volume] g ∧ LeanEval.Analysis.SobolevMorreyProblem.MemHolder r α g All goals completed! 🐙
#64
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Rokhlin lemma
rokhlin_lemma

Lean theorem statement

/-- **Rokhlin lemma.** For every aperiodic measure-preserving
automorphism `T` of a standard Borel probability space `(Ω, μ)`, every
height `n ≥ 1`, and every `ε > 0`, there is a Rokhlin tower of height
`n` whose union has measure at least `1 − ε`. -/
theorem rokhlin_lemma {Ω : Type*} [MeasurableSpace Ω]
    [StandardBorelSpace Ω]
    (μ : Measure Ω) [IsProbabilityMeasure μ] (T : Ω → Ω)
    (_hT : MeasurePreserving T μ μ) (_hap : IsAperiodic T μ)
    (n : ℕ) (_hn : 1 ≤ n) {ε : ENNReal} (_hε : 0 < ε) :
    ∃ B : Set Ω, IsRokhlinTower T B n ∧
      μ (towerUnion T B n) ≥ 1 - ε := by
  sorry
#65
Fang–Xia: tiling of the symmetric group by transpositions implies λ-transitivity
fang_xia_tiling_partition_transitive

Lean theorem statement

/-- **Fang–Xia, Theorem 1.4.** A tiling `(T_n, Y)` of `S_n` forces
λ-transitivity of `Y` for every partition `λ` of `n` whose Young-
diagram content sum is nonnegative. -/
theorem fang_xia_partition_transitive_of_tiling
    {n : ℕ} {Y : Set (Equiv.Perm (Fin n))}
    (_h : IsTiling (transpositionsWithOne n) Y) :
    ∀ lam : PartitionShape n, 0 ≤ lam.contentSum → IsPartitionTransitive Y lam := by
  sorry
#66
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Sard's theorem (critical-set image has measure zero)
sard_theorem

Lean theorem statement

/-- **Sard's theorem** (Morse 1939 / Sard 1942), Knill's rank-
deficient form. The image of the rank-deficient locus of a smooth
map `f : ℝᵐ → ℝⁿ` has Lebesgue measure zero. -/
theorem sard {m n : ℕ} (f : E m → E n) (_hf : ContDiff ℝ ∞ f) :
    volume (criticalValues f) = 0 := by
  sorry
#67
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Liouville–Arnold theorem on integrable systems
liouville_arnold

Verso theorem preview

theorem declaration uses `sorry`liouville_arnold {n : ℕ} (F : Fin n → LeanEval.Geometry.LiouvilleArnold.E n → ℝ) (U : Set (LeanEval.Geometry.LiouvilleArnold.E n)) (_hU : IsOpen U) (_hLI : LeanEval.Geometry.LiouvilleArnold.IsLiouvilleIntegrable F U) (c : Fin n → ℝ) (_hMc_sub : LeanEval.Geometry.LiouvilleArnold.levelSet F c ⊆ U) (_hMc_compact : IsCompact (LeanEval.Geometry.LiouvilleArnold.levelSet F c)) (_hMc_connected : IsConnected (LeanEval.Geometry.LiouvilleArnold.levelSet F c)) : Nonempty ((LeanEval.Geometry.LiouvilleArnold.levelSet F c) ≃ₜ (Fin n → AddCircle (1 : ℝ))) := n:ℕF:Fin n → E n → ℝU:Set (E n)_hU:IsOpen U_hLI:IsLiouvilleIntegrable F Uc:Fin n → ℝ_hMc_sub:levelSet F c ⊆ U_hMc_compact:IsCompact (levelSet F c)_hMc_connected:IsConnected (levelSet F c)⊢ Nonempty (↑(levelSet F c) ≃ₜ (Fin n → AddCircle 1)) All goals completed! 🐙
#68
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Morley's trisector theorem
morley_theorem

Verso theorem preview

theorem declaration uses `sorry`morley_theorem (A B C P Q R : LeanEval.Geometry.Morley.Plane) (h : LeanEval.Geometry.Morley.IsMorleyConfiguration A B C P Q R) : LeanEval.Geometry.Morley.IsEquilateralTriple P Q R := A:PlaneB:PlaneC:PlaneP:PlaneQ:PlaneR:Planeh:IsMorleyConfiguration A B C P Q R⊢ IsEquilateralTriple P Q R All goals completed! 🐙
#69
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Mountain Pass Theorem (Ambrosetti–Rabinowitz 1973)
mountain_pass

Verso theorem preview

theorem declaration uses `sorry`mountain_pass (f : E → ℝ) (_hf : ContDiff ℝ 1 f) (_hps : LeanEval.Analysis.MountainPassProblem.PalaisSmale f) {a b : E} {ε r : ℝ} (_hmr : LeanEval.Analysis.MountainPassProblem.MountainRange f a b ε r) : ∃ x : E, LeanEval.Analysis.MountainPassProblem.IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace Ef:E → ℝ_hf:ContDiff ℝ 1 f_hps:PalaisSmale fa:Eb:Eε:ℝr:ℝ_hmr:MountainRange f a b ε r⊢ ∃ x, IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b All goals completed! 🐙
#70
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Lidskii's inequality
lidskii_inequality

Verso theorem preview

theorem declaration uses `sorry`lidskii_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) {p : ℝ} (_hp : 1 ≤ p) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |(hB.sub hA).eigenvalues₀ j| ^ p := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitianp:ℝ_hp:1 ≤ p⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |⋯.eigenvalues₀ j| ^ p All goals completed! 🐙
#71
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with prompting and submission by John Jennings. No human review of the proofs was performed; correctness was checked only through the benchmark comparator.

Lidskii–Last eigenvalue-perturbation theorem
lidskii_last

Verso theorem preview

theorem declaration uses `sorry`lidskii_last {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitian⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ All goals completed! 🐙
#72
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with prompting and submission by John Jennings. No human review of the proofs was performed; correctness was checked only through the benchmark comparator.

Solvable extensions ↔ solvable groups (the missing converse in Abel–Ruffini)
solvable_by_radicals_converse

Verso theorem preview

theorem declaration uses `sorry`solvable_iff_solvableByRad (F : Type*) [Field F] [CharZero F] (p : F[X]) (_hp : p ≠ 0) : (∀ x : AlgebraicClosure F, aeval x p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal := F:Type u_1inst✝¹:Field Finst✝:CharZero Fp:F[X]_hp:p ≠ 0⊢ (∀ (x : AlgebraicClosure F), (aeval x) p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal All goals completed! 🐙
#73
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Hippocrates' theorem on lunes
hippocrates_lunes

Verso theorem preview

theorem declaration uses `sorry`hippocrates_lunes (a b : ℝ) (ha : 0 < a) (hb : 0 < b) : volume (LeanEval.Geometry.HippocratesLunes.horizontalLune a b) + volume (LeanEval.Geometry.HippocratesLunes.verticalLune a b) = volume (LeanEval.Geometry.HippocratesLunes.rightTriangle a b) := a:ℝb:ℝha:0 < ahb:0 < b⊢ volume (horizontalLune a b) + volume (verticalLune a b) = volume (rightTriangle a b) All goals completed! 🐙
#74
How produced

Autoformalized using Aristotle (Harmonic), orchestrated by Amogh Parab.

Local stable/unstable sets at a hyperbolic fixed point (set-level Hadamard–Perron)
stable_unstable_manifolds

Verso theorem preview

theorem declaration uses `sorry`stable_unstable_manifolds_exist (n : ℕ) (f : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (x₀ : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (_hf : ContDiffAt ℝ 1 f x₀) (_hfix : f x₀ = x₀) (_hhyp : LeanEval.Dynamics.StableUnstableManifoldsProblem.IsHyperbolicLinear (fderiv ℝ f x₀)) (_hf_inv : (fderiv ℝ f x₀).IsInvertible) : ∃ U : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), Ws = {x | (∀ k : ℕ, f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y : ℕ → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n, y 0 = x ∧ (∀ k : ℕ, y k ∈ U) ∧ (∀ k : ℕ, f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} := n:ℕf:E n → E nx₀:E n_hf:ContDiffAt ℝ 1 f x₀_hfix:f x₀ = x₀_hhyp:IsHyperbolicLinear (fderiv ℝ f x₀)_hf_inv:(fderiv ℝ f x₀).IsInvertible⊢ ∃ U, IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu, Ws = {x | (∀ (k : ℕ), f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y, y 0 = x ∧ (∀ (k : ℕ), y k ∈ U) ∧ (∀ (k : ℕ), f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} All goals completed! 🐙
#75
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Fraser: Fourier decay for finite-field Kakeya sets is q^{-1} and sharp
fraser_kakeya_fourier_decay

Verso theorem preview

theorem declaration uses `sorry`fraser_kakeya_fourier_decay_and_sharp {d : ℕ} (_hd : 2 ≤ d) {K : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F d)} (_hK : LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K) (χ : AddChar F ℂ) (_hχ : χ ≠ 1) : (∃ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K μ ∧ ∀ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d, ξ ≠ 0 → ‖fourier χ μ ξ‖ ≤ (Fintype.card F : ℝ)⁻¹) ∧ (∀ κ : ℝ, 0 < κ → κ < 1 → ∃ Q : ℕ, ∀ (F' : Type*) [Field F'] [Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K' : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d), LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K' ∧ ∀ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K' μ → ∃ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d, ξ ≠ 0 ∧ κ * (Fintype.card F' : ℝ)⁻¹ ≤ ‖fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖) := F:Type u_1inst✝²:Field Finst✝¹:Fintype Finst✝:DecidableEq Fd:ℕ_hd:2 ≤ dK:Set (Space F d)_hK:IsKakeya Kχ:AddChar F ℂ_hχ:χ ≠ 1⊢ (∃ μ, IsProbabilityMeasureOn K μ ∧ ∀ (ξ : Space F d), ξ ≠ 0 → ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier χ μ ξ‖ ≤ (↑(Fintype.card F))⁻¹) ∧ ∀ (κ : ℝ), 0 < κ → κ < 1 → ∃ Q, ∀ (F' : Type u_2) [inst : Field F'] [inst_1 : Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K', IsKakeya K' ∧ ∀ (μ : Space F' d → ℝ), IsProbabilityMeasureOn K' μ → ∃ ξ, ξ ≠ 0 ∧ κ * (↑(Fintype.card F'))⁻¹ ≤ ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖ All goals completed! 🐙
#76
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Euler–Lagrange equation
euler_lagrange_equation

Verso theorem preview

theorem declaration uses `sorry`euler_lagrange_equation {a b : ℝ} (L : ℝ → ℝ → ℝ → ℝ) (x : ℝ → ℝ) (_hab : a < b) (_hL : ContDiff ℝ 2 (fun p : ℝ × ℝ × ℝ => L p.1 p.2.1 p.2.2)) (_hx : ContDiff ℝ 2 x) (_hxe : LeanEval.Analysis.IsVariationalExtremum a b L x) : ∀ t ∈ Set.Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t := a:ℝb:ℝL:ℝ → ℝ → ℝ → ℝx:ℝ → ℝ_hab:a < b_hL:ContDiff ℝ 2 fun p => L p.1 p.2.1 p.2.2_hx:ContDiff ℝ 2 x_hxe:IsVariationalExtremum a b L x⊢ ∀ t ∈ Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t All goals completed! 🐙
#77
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Monge–Kantorovich existence theorem
monge_kantorovich

Verso theorem preview

theorem declaration uses `sorry`monge_kantorovich_exists {X Y : Type*} [TopologicalSpace X] [PolishSpace X] [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [PolishSpace Y] [MeasurableSpace Y] [BorelSpace Y] (P : Measure X) (Q : Measure Y) [IsProbabilityMeasure P] [IsProbabilityMeasure Q] (c : X × Y → ENNReal) (_hc : Continuous c) : ∃ π ∈ LeanEval.Analysis.Couplings P Q, ∀ π' ∈ LeanEval.Analysis.Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' := X:Type u_1Y:Type u_2inst✝⁹:TopologicalSpace Xinst✝⁸:PolishSpace Xinst✝⁷:MeasurableSpace Xinst✝⁶:BorelSpace Xinst✝⁵:TopologicalSpace Yinst✝⁴:PolishSpace Yinst✝³:MeasurableSpace Yinst✝²:BorelSpace YP:Measure XQ:Measure Yinst✝¹:IsProbabilityMeasure Pinst✝:IsProbabilityMeasure Qc:X × Y → ENNReal_hc:Continuous c⊢ ∃ π ∈ Couplings P Q, ∀ π' ∈ Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' All goals completed! 🐙
#78
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Furstenberg–Weiss topological multiple recurrence (single-transformation form)
furstenberg_topological

Verso theorem preview

theorem declaration uses `sorry`furstenberg_topological_recurrence {X : Type*} [MetricSpace X] [CompactSpace X] [Nonempty X] (T : X ≃ₜ X) : ∃ x : X, LeanEval.Dynamics.IsMultiplyRecurrent (T : X → X) x := X:Type u_1inst✝²:MetricSpace Xinst✝¹:CompactSpace Xinst✝:Nonempty XT:X ≃ₜ X⊢ ∃ x, IsMultiplyRecurrent (⇑T) x All goals completed! 🐙
#79
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Kakutani fixed-point theorem
kakutani_fixed_point

Lean theorem statement

/-- **Kakutani fixed-point theorem.** Every upper-hemicontinuous
correspondence `F` from a nonempty compact convex `K ⊆ ℝᵈ` to itself, with
nonempty convex closed values, has a fixed point `x ∈ F x`. -/
theorem kakutani_fixed_point {d : ℕ}
    {K : Set (EuclideanSpace ℝ (Fin d))}
    (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K)
    (_hK_nonempty : K.Nonempty)
    (F : EuclideanSpace ℝ (Fin d) → Set (EuclideanSpace ℝ (Fin d)))
    (_hF_uhc : IsUpperHemicontinuous F)
    (_hF_nonempty : ∀ x ∈ K, (F x).Nonempty)
    (_hF_convex : ∀ x ∈ K, Convex ℝ (F x))
    (_hF_closed : ∀ x ∈ K, IsClosed (F x))
    (_hF_maps : ∀ x ∈ K, F x ⊆ K) :
    ∃ x ∈ K, x ∈ F x := by
  sorry
#80
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with prompting and submission by John Jennings. No human review of the proofs was performed; correctness was checked only through the benchmark comparator. The Brouwer fixed-point theorem formalization component of this proof was extracted from Aristotle's proof `brouwer_fixed_point`.

Nash equilibrium existence theorem
nash_equilibrium_exists

Lean theorem statement

/-- **Nash equilibrium existence theorem.** Every finite `n`-player game
with nonempty finite pure-strategy sets and arbitrary real payoffs admits
at least one mixed-strategy Nash equilibrium. -/
theorem nash_equilibrium_exists {n : ℕ} {S : Fin n → Type*}
    [∀ i, Fintype (S i)] [∀ i, Nonempty (S i)]
    (u : Fin n → StrategyProfile n S → ℝ) :
    ∃ σ : ∀ i, S i → ℝ, IsNashEquilibrium u σ := by
  sorry
#81
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Independence of the parallel postulate
parallel_postulate_independent

Lean theorem statement

/-- **Independence of the parallel postulate** (Freek #12). The Euclidean
axiom `A10` is logically independent of Tarski's absolute axioms `A1`–`A9`
and `A11`: there is a model of the absolute axioms in which the parallel
postulate holds (the real coordinate plane) and one in which it fails (the
Klein–Beltrami disk, or any other hyperbolic-plane model). -/
theorem parallel_postulate_independent :
    (∃ (M : Type) (T : TarskiAbsolute M), Euclidean M T) ∧
    (∃ (M : Type) (T : TarskiAbsolute M), ¬ Euclidean M T) := by
  sorry
#82
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Sturm's theorem
sturm

Lean theorem statement

/-- **Sturm's theorem.** For a squarefree real polynomial `p` and an interval
`(a, b)` with `a < b` whose endpoints are not roots of `p`, the number of
distinct roots of `p` in `(a, b)` equals `σ(a) − σ(b)`. -/
theorem sturm (p : ℝ[X]) (hp : Squarefree p) {a b : ℝ} (hab : a < b)
    (ha : p.eval a ≠ 0) (hb : p.eval b ≠ 0) :
    ((p.roots.toFinset).filter (fun x => a < x ∧ x < b)).card =
      sigma p a - sigma p b := by
  sorry
#83
How produced

Solved autonomously by aristotle. Manually bumped to 4.30.0-rc2.

Brauer–Fowler theorem
brauer_fowler

Lean theorem statement

/-- **Brauer–Fowler theorem.** There is a function bounding the order
of a finite nonabelian simple group by the order of any involution
centralizer. -/
theorem brauer_fowler :
    ∃ f : ℕ → ℕ, ∀ (G : Type) [Group G] [Finite G],
      IsSimpleGroup G → (∃ a b : G, a * b ≠ b * a) →
      ∀ t : G, orderOf t = 2 →
        Nat.card G ≤ f (Nat.card (Subgroup.centralizer ({t} : Set G))) := by
  sorry
#84
How produced

Auto-formalized by Aristotle

Frobenius's theorem: the Frobenius kernel is normal
frobenius_kernel_isNormal

Lean theorem statement

theorem frobenius_kernel_isNormal
    (G X : Type) [Group G] [Fintype G] [Fintype X]
    [MulAction G X] [FaithfulSMul G X]
    (hcard : 2 ≤ Fintype.card X)
    (htrans : ∀ x y : X, ∃ g : G, g • x = y)
    (hstab : ∀ x : X, MulAction.stabilizer G x ≠ ⊥)
    (hfrob : ∀ g : G, g ≠ 1 → ∀ x y : X, g • x = x → g • y = y → x = y) :
    ∃ N : Subgroup G, N.Normal ∧
      (N : Set G) = {1} ∪ {g : G | ∀ x : X, g • x ≠ x} := by
  sorry
#85
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Koszul formula
koszul_formula

Lean theorem statement

/-- **Koszul formula.** For any smooth torsion-free metric-compatible
covariant derivative `cov` on `TM`, `2 ⟨∇_X Y, Z⟩` equals the cyclic sum
of directional derivatives `X·⟨Y, Z⟩ + Y·⟨X, Z⟩ − Z·⟨X, Y⟩` minus the
Lie-bracket cyclic sum `⟨X, [Y, Z]⟩ + ⟨Y, [X, Z]⟩ − ⟨Z, [X, Y]⟩`. -/
theorem koszul_formula
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
      [FiniteDimensional ℝ E] [CompleteSpace E]
    {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H}
    {M : Type*} [TopologicalSpace M] [ChartedSpace H M]
      [IsManifold I ∞ M]
    [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)]
    [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)]
    (cov : CovariantDerivative I E (TangentSpace I (M := M)))
    [ContMDiffCovariantDerivative cov ∞]
    (_htor : cov.torsion = 0) (_hmet : IsMetricCompatible cov)
    (X Y Z : Π x : M, TangentSpace I x)
    (_hX : CMDiff ∞ (T% X)) (_hY : CMDiff ∞ (T% Y)) (_hZ : CMDiff ∞ (T% Z))
    (x : M) :
    2 * inner ℝ (cov Y x (X x)) (Z x) =
      mvfderiv I (fun y : M => inner ℝ (Y y) (Z y)) x (X x)
      + mvfderiv I (fun y : M => inner ℝ (X y) (Z y)) x (Y x)
      - mvfderiv I (fun y : M => inner ℝ (X y) (Y y)) x (Z x)
      - inner ℝ (X x) (mlieBracket I Y Z x)
      - inner ℝ (Y x) (mlieBracket I X Z x)
      + inner ℝ (Z x) (mlieBracket I X Y x) := by
  sorry
#86
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Balanceable k-bounded partitions
balanceable_bounded_partitions

Lean theorem statement

/--
For any `k`, the smallest `n` such that any `k`-bounded partition of `n` is
also balanceable is given by `2 * lcm(1, ..., k)`.
-/
theorem minimal_balanceable_of_bounded (k : ℕ) (hk : 0 < k) :
    Minimal (fun n => 0 < n ∧ ∀ p : n.Partition, Bounded k p → Balanceable p) (2 * (Finset.Icc 1 k).lcm id) := by
  sorry
#87
A competition programming problem about permuting a permutation to be unimodal
permute_to_unimodal

Lean theorem statement

/--
`minRearrange` correctly computes the smallest number of indices that need to be permuted in order to
turn `arr` into a unimodal permutation.
-/
theorem minRearrange_correct {arr : Array Nat} :
    arr.Perm (1...=arr.size).toArray →
      (∃ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x ∧ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector = minRearrange arr) ∧
      (∀ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x → minRearrange arr ≤ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector) := by
  sorry
#89
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Chen theorem for Markoff graphs
dvd_card_connectedComponent_markoffGraph

Lean theorem statement

/-- For prime `p > 3`, every connected component of the nonzero Markoff graph over `ZMod p`
has cardinality divisible by `p`. -/
theorem dvd_card_connectedComponent_markoffGraph
    {p : ℕ} (hp : Nat.Prime p) (hgt : 3 < p) :
    ∀ c : (markoffGraph p).ConnectedComponent, p ∣ Nat.card c := by
  sorry
#90
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration by Stefano Rocca and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Baer–Suzuki theorem
baer_suzuki

Verso theorem preview

theorem declaration uses `sorry`baer_suzuki {G : Type*} [Group G] [Finite G] {p : ℕ} [Fact p.Prime] (x : G) : x ∈ LeanEval.GroupTheory.Defs.pCore p G ↔ ∀ g : G, IsPGroup p (Subgroup.closure ({x, g * x * g⁻¹} : Set G)) := G:Type u_1inst✝²:Group Ginst✝¹:Finite Gp:ℕinst✝:Fact (Nat.Prime p)x:G⊢ x ∈ pCore p G ↔ ∀ (g : G), IsPGroup p ↥(Subgroup.closure {x, g * x * g⁻¹}) All goals completed! 🐙
#91
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with prompting and submission by John Jennings. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Runge's theorem
runge_theorem

Verso theorem preview

theorem declaration uses `sorry`runge (K : Set ℂ) (_hK : IsCompact K) (U : Set ℂ) (_hU : IsOpen U) (_hKU : K ⊆ U) (f : ℂ → ℂ) (_hf : AnalyticOnNhd ℂ f U) (ε : ℝ) (_hε : 0 < ε) : ∃ p q : ℂ[X], (∀ z ∈ K, q.eval z ≠ 0) ∧ (∀ z ∈ K, ‖f z - p.eval z / q.eval z‖ < ε) := K:Set ℂ_hK:IsCompact KU:Set ℂ_hU:IsOpen U_hKU:K ⊆ Uf:ℂ → ℂ_hf:AnalyticOnNhd ℂ f Uε:ℝ_hε:0 < ε⊢ ∃ p q, (∀ z ∈ K, Polynomial.eval z q ≠ 0) ∧ ∀ z ∈ K, ‖f z - Polynomial.eval z p / Polynomial.eval z q‖ < ε All goals completed! 🐙
#92
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Kuznetsov's theorem: finitely presented simple groups have solvable word problem
boone_higman_simple

Verso theorem preview

theorem declaration uses `sorry`boone_higman_simple {G : Type*} [Group G] [IsSimpleGroup G] {n : ℕ} (φ : FreeGroup (Fin n) →* G) (_hsurj : Function.Surjective φ) (_hker : (MonoidHom.ker φ).IsNormalClosureFG) : LeanEval.GroupTheory.BooneHigmanSimpleProblem.WordProblemSolvable φ := G:Type u_1inst✝¹:Group Ginst✝:IsSimpleGroup Gn:ℕφ:FreeGroup (Fin n) →* G_hsurj:Function.Surjective ⇑φ_hker:φ.ker.IsNormalClosureFG⊢ WordProblemSolvable φ All goals completed! 🐙
#93
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Linear programming: maximum principle and vertex optimality
lp_maximum_principle

Verso theorem preview

/-- **Maximum principle for linear programming** (§101). A local maximiser of the LP objective on the feasible region is automatically a global maximiser; and whenever the objective is non-constant (`c ≠ 0`), the maximiser lies on the topological frontier of the feasible region. -/ theorem declaration uses `sorry`lp_maximum_principle {m n : ℕ} (lp : LinearProgram m n) (x : Fin m → ℝ) (_hx : x ∈ lp.feasible) (_hlocal : IsLocalMaxOn lp.objective lp.feasible x) : IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) := m:ℕn:ℕlp:LinearProgram m nx:Fin m → ℝ_hx:x ∈ lp.feasible_hlocal:IsLocalMaxOn lp.objective lp.feasible x⊢ IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) All goals completed! 🐙
/-- **Vertex optimality** (§101; the existence content of Dantzig's 1947 simplex algorithm). Every linear program with a nonempty bounded feasible region admits a global maximiser that is an extreme point (vertex) of the feasible region. -/ theorem declaration uses `sorry`simplex_algorithm {m n : ℕ} (lp : LinearProgram m n) (_hfeas : lp.feasible.Nonempty) (_hbdd : Bornology.IsBounded lp.feasible) : ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible := m:ℕn:ℕlp:LinearProgram m n_hfeas:lp.feasible.Nonempty_hbdd:Bornology.IsBounded lp.feasible⊢ ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible All goals completed! 🐙
#94
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Bourbaki's locally convex extension of Banach–Alaoglu
banach_alaoglu_bourbaki

Verso theorem preview

theorem declaration uses `sorry`banach_alaoglu_bourbaki (E : Type*) [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] (U : Set E) (_hU : U ∈ 𝓝 (0 : E)) : IsCompact (LeanEval.Analysis.weakStarPolar E U) := E:Type u_1inst✝⁵:AddCommGroup Einst✝⁴:Module ℝ Einst✝³:TopologicalSpace Einst✝²:ContinuousAdd Einst✝¹:ContinuousSMul ℝ Einst✝:LocallyConvexSpace ℝ EU:Set E_hU:U ∈ 𝓝 0⊢ IsCompact (weakStarPolar E U) All goals completed! 🐙
#95
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Wiener's atom-detection formula
wiener_atom_detection

Verso theorem preview

theorem declaration uses `sorry`wiener_atom_detection (μ : Measure (AddCircle (2 * Real.pi))) [IsProbabilityMeasure μ] : Tendsto (fun N : ℕ => (1 / (N : ℝ)) * ∑ k ∈ Finset.Icc (1 : ℤ) N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' x : AddCircle (2 * Real.pi), ((μ {x}).toReal) ^ 2)) := μ:Measure (AddCircle (2 * π))inst✝:IsProbabilityMeasure μ⊢ Tendsto (fun N => 1 / ↑N * ∑ k ∈ Finset.Icc 1 ↑N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' (x : AddCircle (2 * π)), (μ {x}).toReal ^ 2)) All goals completed! 🐙
#96
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Schauder fixed-point theorem
schauder_fixed_point

Verso theorem preview

theorem declaration uses `sorry`schauder_fixed_point {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {K : Set E} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : E → E) (_hf_cont : ContinuousOn f K) (_hf_maps : Set.MapsTo f K K) : ∃ x ∈ K, f x = x := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace EK:Set E_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:E → E_hf_cont:ContinuousOn f K_hf_maps:Set.MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#97
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with prompting and submission by John Jennings. No human review of the proofs was performed; correctness was checked only through the benchmark comparator. The Brouwer fixed-point theorem formalization component of the `schauder_fixed_point` proof was extracted from Aristotle's proof of `brouwer_fixed_point`.

Symplectic matrices have determinant 1
symplectic_matrix_det

Verso theorem preview

theorem declaration uses `sorry`symplectic_matrix_det {l R : Type*} [DecidableEq l] [Fintype l] [CommRing R] {A : Matrix (l ⊕ l) (l ⊕ l) R} (_hA : A ∈ Matrix.symplecticGroup l R) : A.det = 1 := l:Type u_1R:Type u_2inst✝²:DecidableEq linst✝¹:Fintype linst✝:CommRing RA:Matrix (l ⊕ l) (l ⊕ l) R_hA:A ∈ symplecticGroup l R⊢ A.det = 1 All goals completed! 🐙
#98
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with prompting and submission by John Jennings. No human review of the proofs was performed; correctness was checked only through the benchmark comparator. The Brouwer fixed-point theorem formalization component of the `schauder_fixed_point` proof was extracted from Aristotle's proof of `brouwer_fixed_point`.

Brouwer fixed-point theorem
brouwer_fixed_point

Verso theorem preview

theorem declaration uses `sorry`brouwer_fixed_point {d : ℕ} {K : Set (EuclideanSpace ℝ (Fin d))} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (_hf_cont : ContinuousOn f K) (_hf_maps : MapsTo f K K) : ∃ x ∈ K, f x = x := d:ℕK:Set (EuclideanSpace ℝ (Fin d))_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)_hf_cont:ContinuousOn f K_hf_maps:MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#99
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with prompting and submission by John Jennings. No human review of the proofs was performed; correctness was checked only through the benchmark comparator. Most of this proof was extracted from an intermediate result that Aristotle used to prove `nash_equilibrium_exists`.

Schur-Weyl duality: GL(V) image equals centralizer of S_k image
glAction_range_eq_centralizer_symAction

Verso theorem preview

theorem declaration uses `sorry`glAction_range_eq_centralizer_symAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(glAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(symAction R M k)) All goals completed! 🐙
#100
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Schur-Weyl duality: S_k image equals centralizer of GL(V) image
symAction_range_eq_centralizer_glAction

Verso theorem preview

theorem declaration uses `sorry`symAction_range_eq_centralizer_glAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(symAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(glAction R M k)) All goals completed! 🐙
#101
Pointwise and Cesàro convergence of Fourier series (Dirichlet, Fejér)
fourier_dirichlet_fejer

Verso theorem preview

/-- **Dirichlet's pointwise convergence theorem** (§46). For every `C¹` 2π-periodic complex function `f`, the symmetric Fourier partial sums `S_N(f)(x)` converge to `f(x)` at every point `x ∈ ℝ`. -/ theorem declaration uses `sorry`dirichlet_pointwise {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hC1 : ContDiff ℝ 1 f) (x : ℝ) : Tendsto (fun N : ℕ => fourierPartialSum f N x) atTop (𝓝 (f x)) := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hC1:ContDiff ℝ 1 fx:ℝ⊢ Tendsto (fun N => fourierPartialSum f N x) atTop (𝓝 (f x)) All goals completed! 🐙
/-- **Fejér's theorem** (§46). For every *continuous* 2π-periodic complex function `f` — without the `C¹` hypothesis of Dirichlet's theorem — the Cesàro means `σ_N(f)` of the symmetric Fourier partial sums converge to `f` uniformly on `ℝ`. -/ theorem declaration uses `sorry`fejer {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hcont : Continuous f) : TendstoUniformly (fun N : ℕ => fourierCesaroMean f N) f atTop := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hcont:Continuous f⊢ TendstoUniformly (fun N => fourierCesaroMean f N) f atTop All goals completed! 🐙
#102
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Abel–Ruffini theorem
abel_ruffini

Verso theorem preview

theorem declaration uses `sorry`abel_ruffini (n : ℕ) (_hn : 1 ≤ n) : (∀ p : ℚ[X], p.natDegree = n → ∀ x : ℂ, aeval x p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 := n:ℕ_hn:1 ≤ n⊢ (∀ (p : ℚ[X]), p.natDegree = n → ∀ (x : ℂ), (aeval x) p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 All goals completed! 🐙
#103
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Pell solutions are convergents of √d
pell_solution_convergent

Verso theorem preview

theorem declaration uses `sorry`pell_solution_is_convergent (d : ℤ) (_hd : Squarefree d) (_hd0 : 0 < d) (x y : ℤ) (_hx : 0 < x) (_hy : 0 < y) (_hsol : x ^ 2 - d * y ^ 2 = 1) : ∃ n : ℕ, (GenContFract.of (Real.sqrt (d : ℝ))).convs n = (x : ℝ) / (y : ℝ) := d:ℤ_hd:Squarefree d_hd0:0 < dx:ℤy:ℤ_hx:0 < x_hy:0 < y_hsol:x ^ 2 - d * y ^ 2 = 1⊢ ∃ n, (GenContFract.of √↑d).convs n = ↑x / ↑y All goals completed! 🐙
#104
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Bing's house with two rooms is contractible
contractibleSpace_houseWithTwoRooms

Verso theorem preview

theorem declaration uses `sorry`contractibleSpace_houseWithTwoRooms : ContractibleSpace LeanEval.Topology.HouseWithTwoRooms := ⊢ ContractibleSpace ↑HouseWithTwoRooms All goals completed! 🐙
#105
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Burnside p^a q^b theorem
finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow

Verso theorem preview

theorem declaration uses `sorry`finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow {G : Type*} [Group G] [Fintype G] {p q a b : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q) (hpq : p ≠ q) (hcard : Fintype.card G = p ^ a * q ^ b) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Fintype Gp:ℕq:ℕa:ℕb:ℕhp:Nat.Prime phq:Nat.Prime qhpq:p ≠ qhcard:Fintype.card G = p ^ a * q ^ b⊢ IsSolvable G All goals completed! 🐙
#106
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Real cyclotomic integer with house at most 2
cyclotomic_integer_house_le_two

Verso theorem preview

theorem declaration uses `sorry`cyclotomic_integer_house_le_two {K : Type*} [Field K] [NumberField K] [Algebra ℚ K] (n : ℕ) [NeZero n] [IsCyclotomicExtension {n} ℚ K] {β : K} (hβ_int : IsIntegral ℤ β) (hβ_real : β ∈ NumberField.maximalRealSubfield K) : house β ≤ 2 → house β = 2 ∨ ∃ m : ℕ, 0 < m ∧ house β = 2 * Real.cos (Real.pi / m) := K:Type u_1inst✝⁴:Field Kinst✝³:NumberField Kinst✝²:Algebra ℚ Kn:ℕinst✝¹:NeZero ninst✝:IsCyclotomicExtension {n} ℚ Kβ:Khβ_int:IsIntegral ℤ βhβ_real:β ∈ maximalRealSubfield K⊢ house β ≤ 2 → house β = 2 ∨ ∃ m, 0 < m ∧ house β = 2 * Real.cos (Real.pi / ↑m) All goals completed! 🐙
#107
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Existence of a non-isotopic pair of oriented two-component links
exists_nonisotopic_link

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_link : ∃ L₁ L₂ : LeanEval.KnotTheory.TwoLink, ¬ L₁.Isotopic L₂ := ⊢ ∃ L₁ L₂, ¬L₁.Isotopic L₂ All goals completed! 🐙
#108
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

von Neumann double commutant theorem
vonNeumann_doubleCommutant_tfae

Verso theorem preview

theorem declaration uses `sorry`vonNeumann_doubleCommutant_tfae {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : StarSubalgebra ℂ (H →L[ℂ] H)) : List.TFAE [ Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S , IsClosed (ContinuousLinearMapWOT.ofCLM '' (S : Set (H →L[ℂ] H))) , IsClosed (ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H '' (S : Set (H →L[ℂ] H))) ] := H:Type u_1inst✝²:NormedAddCommGroup Hinst✝¹:InnerProductSpace ℂ Hinst✝:CompleteSpace HS:StarSubalgebra ℂ (H →L[ℂ] H)⊢ [(↑S).centralizer.centralizer = ↑S, IsClosed (ContinuousLinearMapWOT.ofCLM '' ↑S), IsClosed (⇑(ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H) '' ↑S)].TFAE All goals completed! 🐙
#109
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Perron-Frobenius for irreducible nonnegative matrices
irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius

Verso theorem preview

theorem declaration uses `sorry`irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius {n : Type*} [Fintype n] [DecidableEq n] [Nonempty n] (A : Matrix n n ℝ) (hA : A.IsIrreducible) : ∃ v : n → ℝ, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ (∀ i, 0 < v i) := n:Type u_1inst✝²:Fintype ninst✝¹:DecidableEq ninst✝:Nonempty nA:Matrix n n ℝhA:A.IsIrreducible⊢ ∃ v, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ ∀ (i : n), 0 < v i All goals completed! 🐙
#110
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Linear ODE with negative-real-part eigenvalues is asymptotically stable
linear_ode_asymptotic_stability

Verso theorem preview

theorem declaration uses `sorry`linear_ode_asymptotic_stability (n : ℕ) (A : Matrix (Fin n) (Fin n) ℝ) (hA : ∀ μ : ℂ, Module.End.HasEigenvalue (Matrix.toLin' (A.map (algebraMap ℝ ℂ))) μ → μ.re < 0) (x : ℝ → (Fin n → ℝ)) (hx : ∀ t : ℝ, 0 < t → HasDerivAt x (A.mulVec (x t)) t) : Filter.Tendsto (fun t : ℝ => ‖x t‖) Filter.atTop (nhds 0) := n:ℕA:Matrix (Fin n) (Fin n) ℝhA:∀ (μ : ℂ), Module.End.HasEigenvalue (Matrix.toLin' (A.map ⇑(algebraMap ℝ ℂ))) μ → μ.re < 0x:ℝ → Fin n → ℝhx:∀ (t : ℝ), 0 < t → HasDerivAt x (A *ᵥ x t) t⊢ Filter.Tendsto (fun t => ‖x t‖) Filter.atTop (nhds 0) All goals completed! 🐙
#111
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Rouche theorem via zero counting
rouche_zero_count_eq

Verso theorem preview

theorem declaration uses `sorry`rouche_zero_count_eq {f g : ℂ → ℂ} {R : ℝ} (hR : 0 < R) (hf : MeromorphicNFOn f Set.univ) (hg : AnalyticOn ℂ g Set.univ) (hbound : ∀ z : ℂ, ‖z‖ = R → ‖g z‖ < ‖f z‖) : (∑ᶠ z, ((divisor (f + g) (Metric.closedBall 0 R))⁺) z) = (∑ᶠ z, ((divisor f (Metric.closedBall 0 R))⁺) z) := f:ℂ → ℂg:ℂ → ℂR:ℝhR:0 < Rhf:MeromorphicNFOn f Set.univhg:AnalyticOn ℂ g Set.univhbound:∀ (z : ℂ), ‖z‖ = R → ‖g z‖ < ‖f z‖⊢ ∑ᶠ (z : ℂ), (divisor (f + g) (Metric.closedBall 0 R))⁺ z = ∑ᶠ (z : ℂ), (divisor f (Metric.closedBall 0 R))⁺ z All goals completed! 🐙
#112
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Complementary polynomial on the unit circle
exists_complementary_polynomial_on_unit_circle

Verso theorem preview

theorem declaration uses `sorry`exists_complementary_polynomial_on_unit_circle (P : ℂ[X]) (hP : ∀ z : Circle, ‖P.eval (z : ℂ)‖ ≤ 1) : ∃ Q : ℂ[X], Q.natDegree ≤ P.natDegree ∧ ∀ z : Circle, ‖P.eval (z : ℂ)‖ ^ 2 + ‖Q.eval (z : ℂ)‖ ^ 2 = 1 := P:ℂ[X]hP:∀ (z : Circle), ‖eval (↑z) P‖ ≤ 1⊢ ∃ Q, Q.natDegree ≤ P.natDegree ∧ ∀ (z : Circle), ‖eval (↑z) P‖ ^ 2 + ‖eval (↑z) Q‖ ^ 2 = 1 All goals completed! 🐙
#113
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Character values of finite groups lie in cyclotomic fields
brauer_character_in_cyclotomic

Verso theorem preview

theorem declaration uses `sorry`brauer_character_in_cyclotomic (G : Type) [Group G] [Fintype G] : ∃ φ : CyclotomicField (Monoid.exponent G) ℚ →+* ℂ, ∀ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (_ : FiniteDimensional ℂ V) (ρ : Representation ℂ G V) (g : G), LinearMap.trace ℂ V (ρ g) ∈ φ.range := G:Typeinst✝¹:Group Ginst✝:Fintype G⊢ ∃ φ, ∀ (V : Type) (x : AddCommGroup V) (x_1 : Module ℂ V), FiniteDimensional ℂ V → ∀ (ρ : Representation ℂ G V) (g : G), (LinearMap.trace ℂ V) (ρ g) ∈ φ.range All goals completed! 🐙
#114
How produced

Solved by the Aristotle automated theorem prover (Harmonic) via the `aristotle` CLI, with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Minkowski-Caratheodory theorem
mem_convexHull_finset_extremePoints_of_mem_compact_convex

Verso theorem preview

theorem declaration uses `sorry`mem_convexHull_finset_extremePoints_of_mem_compact_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hscomp : IsCompact s) (hsconv : Convex ℝ s) (hx : x ∈ s) : ∃ t : Finset E, (↑t : Set E) ⊆ s.extremePoints ℝ ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ convexHull ℝ (↑t : Set E) := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:FiniteDimensional ℝ Es:Set Ex:Ehscomp:IsCompact shsconv:Convex ℝ shx:x ∈ s⊢ ∃ t, ↑t ⊆ extremePoints ℝ s ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ (convexHull ℝ) ↑t All goals completed! 🐙
#115
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2
dirichlet_eigenvalues_eq_nat_sq

Verso theorem preview

theorem declaration uses `sorry`dirichlet_eigenvalues_eq_nat_sq (lam : ℝ) : (∃ (y : ℝ → ℝ) (J : Set ℝ), IsOpen J ∧ Set.Icc (0 : ℝ) Real.pi ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y Real.pi = 0 ∧ ∃ x ∈ Set.Ioo (0 : ℝ) Real.pi, y x ≠ 0) ↔ ∃ n : ℕ, 0 < n ∧ lam = (n : ℝ) ^ 2 := lam:ℝ⊢ (∃ y J, IsOpen J ∧ Set.Icc 0 π ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y π = 0 ∧ ∃ x ∈ Set.Ioo 0 π, y x ≠ 0) ↔ ∃ n, 0 < n ∧ lam = ↑n ^ 2 All goals completed! 🐙
#116
How produced

Solved by the Aristotle automated theorem prover (Harmonic) via the `aristotle` CLI, with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Gaussian heat kernel solves the 1D heat equation
heat_kernel_solves_heat_equation

Verso theorem preview

theorem declaration uses `sorry`heat_kernel_solves_heat_equation (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_bdd : ∃ M : ℝ, ∀ x, |f x| ≤ M) : -- The PDE on (0, ∞) × ℝ. (∀ t : ℝ, 0 < t → ∀ x : ℝ, ∃ ux : ℝ → ℝ, ∃ uxx : ℝ, (∀ y : ℝ, HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ -- Initial condition recovered as a one-sided limit at t = 0. (∀ x : ℝ, Filter.Tendsto (fun t : ℝ => heatSolution f t x) (nhdsWithin (0 : ℝ) (Set.Ioi 0)) (nhds (f x))) := f:ℝ → ℝhf_cont:Continuous fhf_bdd:∃ M, ∀ (x : ℝ), |f x| ≤ M⊢ (∀ (t : ℝ), 0 < t → ∀ (x : ℝ), ∃ ux uxx, (∀ (y : ℝ), HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ ∀ (x : ℝ), Filter.Tendsto (fun t => heatSolution f t x) (nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) All goals completed! 🐙
#117
How produced

Solved by the Aristotle automated theorem prover (Harmonic) via the `aristotle` CLI, with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Oppenheim's inequality for Hadamard products
oppenheim_inequality

Verso theorem preview

theorem declaration uses `sorry`oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : A.det * ∏ i, B i i ≤ (A ⊙ B).det := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝB:Matrix n n ℝhA:A.PosSemidefhB:B.PosSemidef⊢ A.det * ∏ i, B i i ≤ (A ⊙ B).det All goals completed! 🐙
#118
How produced

Solved by the Aristotle automated theorem prover (Harmonic) via the `aristotle` CLI, with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

pi_1 of the circle is Z
pi1_circle_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi1_circle_mulEquiv_int : Nonempty (HomotopyGroup.Pi 1 Circle (1 : Circle) ≃* Multiplicative ℤ) := ⊢ Nonempty (HomotopyGroup.Pi 1 Circle 1 ≃* Multiplicative ℤ) All goals completed! 🐙
#119
How produced

Solved by the Aristotle automated theorem prover (Harmonic) via the `aristotle` CLI, with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Entrywise exponential of a PSD matrix is PSD
posSemidef_map_exp

Verso theorem preview

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n] {A : Matrix n n ℝ} (hA : A.PosSemidef) : (A.map Real.exp).PosSemidef := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef All goals completed! 🐙
#120
How produced

Solved by the Aristotle automated theorem prover (Harmonic) via the `aristotle` CLI v1.0.1, with submission orchestration by Kim Morrison. The proof was generated against Aristotle's default Lean v4.28.0 / Mathlib v4.28.0 target; submitted here against the benchmark's pinned Lean v4.30.0-rc2 + pinned Mathlib commit. Comparator's verdict applies — proofs that exploit features added to Mathlib after v4.28.0 will compile; those that rely on tactic behaviour that changed are at the mercy of forward compatibility.

Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#121
How produced

Solved by the Aristotle automated theorem prover (Harmonic), with orchestration and submission by Lorenzo Luccioli. No human review or supervision of the proofs was performed; correctness was checked only through the benchmark comparator.

Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#124
How produced

Solved by the Aristotle automated theorem prover (Harmonic) via the `aristotle` CLI v1.0.1, with submission orchestration by Kim Morrison. The proof was generated against Aristotle's default Lean v4.28.0 / Mathlib v4.28.0 target; submitted here against the benchmark's pinned Lean v4.30.0-rc2 + pinned Mathlib commit. Comparator's verdict applies — proofs that exploit features added to Mathlib after v4.28.0 will compile; those that rely on tactic behaviour that changed are at the mercy of forward compatibility.

Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#125
How produced

Solved by the Aristotle automated theorem prover (Harmonic) via the `aristotle` CLI v1.0.1, with submission orchestration by Kim Morrison. The proof was generated against Aristotle's default Lean v4.28.0 / Mathlib v4.28.0 target; submitted here against the benchmark's pinned Lean v4.30.0-rc2 + pinned Mathlib commit. Comparator's verdict applies — proofs that exploit features added to Mathlib after v4.28.0 will compile; those that rely on tactic behaviour that changed are at the mercy of forward compatibility.

Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#128
How produced

Solved by the Aristotle automated theorem prover (Harmonic) via the `aristotle` CLI v1.0.1, with submission orchestration by Kim Morrison. The proof was generated against Aristotle's default Lean v4.28.0 / Mathlib v4.28.0 target; submitted here against the benchmark's pinned Lean v4.30.0-rc2 + pinned Mathlib commit. Comparator's verdict applies — proofs that exploit features added to Mathlib after v4.28.0 will compile; those that rely on tactic behaviour that changed are at the mercy of forward compatibility.

Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#129
How produced

Solved by the Aristotle automated theorem prover (Harmonic) via the `aristotle` CLI v1.0.1, with submission orchestration by Kim Morrison. The proof was generated against Aristotle's default Lean v4.28.0 / Mathlib v4.28.0 target; submitted here against the benchmark's pinned Lean v4.30.0-rc2 + pinned Mathlib commit. Comparator's verdict applies — proofs that exploit features added to Mathlib after v4.28.0 will compile; those that rely on tactic behaviour that changed are at the mercy of forward compatibility.

Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#130
How produced

Solved by the Aristotle automated theorem prover (Harmonic) via the `aristotle` CLI v1.0.1, with submission orchestration by Kim Morrison. The proof was generated against Aristotle's default Lean v4.28.0 / Mathlib v4.28.0 target; submitted here against the benchmark's pinned Lean v4.30.0-rc2 + pinned Mathlib commit. Comparator's verdict applies — proofs that exploit features added to Mathlib after v4.28.0 will compile; those that rely on tactic behaviour that changed are at the mercy of forward compatibility.

Test problems: noncomputable_hole_example, variable_binder_example, def_hole_example, instance_hole_example, list_append_singleton_length, ci_regenerate_main_check, two_plus_two (7 / 8 solved)

Submission history

First submissionMay 1, 2026
Last submissionJul 18, 2026

Contributors

LorenzoLuccioli108sqrt-of-212kim-em10JohnEdwardJennings7parabamoghv4adrianmartir3Parcly-Taxel1
3Stealth Model118 solved

Other solved problems

Spencer-Szemerédi-Trotter unit-distance upper bound
unit_distance_upper_bound

Verso theorem preview

theorem declaration uses `sorry`unit_distance_upper_bound : ∃ C : ℝ, 0 < C ∧ ∀ P : Finset (EuclideanSpace ℝ (Fin 2)), (unitDist P : ℝ) ≤ C * (P.card : ℝ) ^ ((4 : ℝ) / 3) := ⊢ ∃ C, 0 < C ∧ ∀ (P : Finset (EuclideanSpace ℝ (Fin 2))), ↑(unitDist P) ≤ C * ↑P.card ^ (4 / 3) All goals completed! 🐙
#1
A 3-manifold group with no faithful representation into GL(4, ℝ)
nonlinear_three_manifold_group

Verso theorem preview

theorem declaration uses `sorry`nonlinear_three_manifold_group : ∃ (M : LeanEval.Topology.Closed3Manifold) (x : M.carrier), ∀ f : FundamentalGroup M.carrier x →* GL (Fin 4) ℝ, ¬ Function.Injective f := ⊢ ∃ M x, ∀ (f : FundamentalGroup M.carrier x →* GL (Fin 4) ℝ), ¬Function.Injective ⇑f All goals completed! 🐙
#2
How produced

No Description

The Hopf Umlaufsatz (theorem of turning tangents)
hopf_umlaufsatz

Verso theorem preview

theorem declaration uses `sorry`hopf_umlaufsatz {r : ℝ → LeanEval.Geometry.HopfUmlaufsatz.Plane} {α : ℝ → ℝ} (_hr : LeanEval.Geometry.HopfUmlaufsatz.IsPositiveSimpleClosedUnitSpeedCurve r) (_hα : LeanEval.Geometry.HopfUmlaufsatz.IsTangentAngleLift r α) : totalCurvature α = 2 * Real.pi := r:ℝ → Planeα:ℝ → ℝ_hr:IsPositiveSimpleClosedUnitSpeedCurve r_hα:IsTangentAngleLift r α⊢ totalCurvature α = 2 * Real.pi All goals completed! 🐙
#3
How produced

No Description

Chudnovsky formula for pi inverse
chudnovsky_formula_for_pi_inv

Verso theorem preview

theorem declaration uses `sorry`chudnovsky_formula_for_pi_inv : chudnovskySum = π⁻¹ := ⊢ chudnovskySum = π⁻¹ All goals completed! 🐙
#4
How produced

No Description

Morrison–Walker Lemma B.0.1: adapting families of maps to open covers
families_of_maps_b01

Verso theorem preview

/-- **Lemma B.0.1** of Morrison–Walker, *The Blob Complex* (arXiv:1009.5025, §B), continuous case. -/ theorem declaration uses `sorry`continuous {P : Set (Fin k → ℝ)} (_hP : IsPolyhedron P) [CompactSpace X] (U : ι → Set X) (_hUopen : ∀ α, IsOpen (U α)) (ρ : PartitionOfUnity ι X univ) (_hρ : ρ.IsSubordinate U) (f : C(P × X, T)) : ∃ F : C(I × P × X, T), (∀ p : P, ∀ x : X, F (0, p, x) = f (p, x)) ∧ (∃ K : Subdivision P, ∀ D ∈ K.complex.facets, AdaptedTo U k (fun q : closedCell P D × X => F (1, q.1.1, q.2))) ∧ (∀ S : Set X, Supported (f := f.toFun) S → Supported (fun q : (I × P) × X => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ Q : Set P, IsBoundarySubpolyhedron Q → ∀ S' : Set X, Supported (fun q : Q × X => f (q.1.1, q.2)) S' → Supported (fun q : (I × Q) × X => F (q.1.1, q.1.2.1, q.2)) S') := k:ℕι:Type u_1X:Type u_2T:Type u_3inst✝²:TopologicalSpace Xinst✝¹:TopologicalSpace TP:Set (Fin k → ℝ)_hP:IsPolyhedron Pinst✝:CompactSpace XU:ι → Set X_hUopen:∀ (α : ι), IsOpen[inst✝²] (U α)ρ:PartitionOfUnity ι X_hρ:ρ.IsSubordinate Uf:C(↑P × X, T)⊢ ∃ F, (∀ (p : ↑P) (x : X), F (0, p, x) = f (p, x)) ∧ (∃ K, ∀ D ∈ K.complex.facets, AdaptedTo U k fun q => F (1, ↑q.1, q.2)) ∧ (∀ (S : Set X), Supported f.toFun S → Supported (fun q => F (q.1.1, q.1.2, q.2)) S) ∧ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ (S' : Set X), Supported (fun q => f (↑q.1, q.2)) S' → Supported (fun q => F (q.1.1, ↑q.1.2, q.2)) S' All goals completed! 🐙
/-- **Lemma B.0.1**, bi-Lipschitz variant (part 4 of the paper). -/ theorem declaration uses `sorry`biLipschitz {X T : Type*} [MetricSpace X] [MetricSpace T] [CompactSpace X] {P : Set (Fin k → ℝ)} (_hP : IsPolyhedron P) {ι : Type*} (U : ι → Set X) (_hUopen : ∀ α, IsOpen (U α)) (ρ : PartitionOfUnity ι X univ) (_hρ : ρ.IsSubordinate U) (f : C(P × X, T)) (slice : P → (X ≃ₜ T)) (_h_slice_eq : ∀ p : P, ∀ x : X, f (p, x) = slice p x) (L : NNReal) (_hf_joint : LipschitzWith L f.toFun) (_hf_slice_inv : ∀ p : P, LipschitzWith L (slice p).symm) : ∃ F : C(I × P × X, T), ∃ L' : NNReal, ∃ Slice : I × P → (X ≃ₜ T), (∀ p : P, ∀ x : X, F (0, p, x) = f (p, x)) ∧ (∀ t : I, ∀ p : P, ∀ x : X, F (t, p, x) = Slice (t, p) x) ∧ (∃ K : Subdivision P, ∀ D ∈ K.complex.facets, AdaptedTo U k (fun q : closedCell P D × X => F (1, q.1.1, q.2))) ∧ (∀ S : Set X, Supported (f := f.toFun) S → Supported (fun q : (I × P) × X => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ Q : Set P, IsBoundarySubpolyhedron Q → ∀ S' : Set X, Supported (fun q : Q × X => f (q.1.1, q.2)) S' → Supported (fun q : (I × Q) × X => F (q.1.1, q.1.2.1, q.2)) S') ∧ (∀ tp : I × P, LipschitzWith L' (Slice tp)) ∧ (∀ tp : I × P, LipschitzWith L' (Slice tp).symm) := k:ℕX:Type u_4T:Type u_5inst✝²:MetricSpace Xinst✝¹:MetricSpace Tinst✝:CompactSpace XP:Set (Fin k → ℝ)_hP:IsPolyhedron Pι:Type u_6U:ι → Set X_hUopen:∀ (α : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U α)ρ:PartitionOfUnity ι X_hρ:ρ.IsSubordinate Uf:C(↑P × X, T)slice:↑P → X ≃ₜ T_h_slice_eq:∀ (p : ↑P) (x : X), f (p, x) = (slice p) xL:NNReal_hf_joint:LipschitzWith L f.toFun_hf_slice_inv:∀ (p : ↑P), LipschitzWith L ⇑(slice p).symm⊢ ∃ F L' Slice, (∀ (p : ↑P) (x : X), F (0, p, x) = f (p, x)) ∧ (∀ (t : ↑I) (p : ↑P) (x : X), F (t, p, x) = (Slice (t, p)) x) ∧ (∃ K, ∀ D ∈ K.complex.facets, AdaptedTo U k fun q => F (1, ↑q.1, q.2)) ∧ (∀ (S : Set X), Supported f.toFun S → Supported (fun q => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ (S' : Set X), Supported (fun q => f (↑q.1, q.2)) S' → Supported (fun q => F (q.1.1, ↑q.1.2, q.2)) S') ∧ (∀ (tp : ↑I × ↑P), LipschitzWith L' ⇑(Slice tp)) ∧ ∀ (tp : ↑I × ↑P), LipschitzWith L' ⇑(Slice tp).symm All goals completed! 🐙
#5
The Golod–Shafarevich inequality
golod_shafarevich_inequality

Verso theorem preview

theorem declaration uses `sorry`golod_shafarevich_inequality (p : ℕ) [Fact p.Prime] (Q : Type) [Group Q] [TopologicalSpace Q] [IsTopologicalGroup Q] [DiscreteTopology Q] [Finite Q] : IsPGroup p Q → Nontrivial Q → (generatorRank Q : ℝ) ^ 2 < 4 * (relationRank p Q : ℝ) := p:ℕinst✝⁵:Fact (Nat.Prime p)Q:Typeinst✝⁴:Group Qinst✝³:TopologicalSpace Qinst✝²:IsTopologicalGroup Qinst✝¹:DiscreteTopology Qinst✝:Finite Q⊢ IsPGroup p Q → Nontrivial Q → ↑(generatorRank Q) ^ 2 < 4 * ↑(relationRank p Q) All goals completed! 🐙
#6
Halmos's generic weak-mixing theorem
halmos_generic_weak_mixing

Verso theorem preview

theorem declaration uses `sorry`generic_weakly_mixing [StandardBorelSpace X] (m : Measure X) [IsProbabilityMeasure m] [NoAtoms m] : (∃ G : Set (LeanEval.Dynamics.HalmosGenericWeakMixingProblem.Automorphism m), IsGδ G ∧ Dense G ∧ ∀ T ∈ G, LeanEval.Dynamics.HalmosGenericWeakMixingProblem.IsWeaklyMixing m T) ∧ (∀ T : LeanEval.Dynamics.HalmosGenericWeakMixingProblem.Automorphism m, LeanEval.Dynamics.HalmosGenericWeakMixingProblem.IsWeaklyMixing m T → Ergodic (T.toEquiv : X → X) m) := X:Type u_1inst✝³:MeasurableSpace Xinst✝²:StandardBorelSpace Xm:Measure Xinst✝¹:IsProbabilityMeasure minst✝:NoAtoms m⊢ (∃ G, IsGδ G ∧ Dense G ∧ ∀ T ∈ G, IsWeaklyMixing m T) ∧ ∀ (T : Automorphism m), IsWeaklyMixing m T → Ergodic (⇑T.toEquiv) m All goals completed! 🐙
#7
Existence of a non-isotopic pair of oriented knots
exists_nonisotopic_knots

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_knots : ∃ K₁ K₂ : LeanEval.KnotTheory.Knot, ¬ K₁.Isotopic K₂ := ⊢ ∃ K₁ K₂, ¬K₁.Isotopic K₂ All goals completed! 🐙
#8
How produced

No Description

Cauchy–Kovalevskaya theorem
cauchy_kovalevskaya

Verso theorem preview

theorem declaration uses `sorry`cauchy_kovalevskaya {d : ℕ} (F : LeanEval.Analysis.E d × ℝ × ℝ → LeanEval.Analysis.E d) (f : LeanEval.Analysis.E d × ℝ × ℝ → ℝ) (u₀ : LeanEval.Analysis.E d → ℝ) (_hF : AnalyticOnNhd ℝ F univ) (_hf : AnalyticOnNhd ℝ f univ) (_hu₀ : AnalyticOnNhd ℝ u₀ univ) (x₀ : LeanEval.Analysis.E d) : ∃ (U : Set (LeanEval.Analysis.E d × ℝ)) (u : LeanEval.Analysis.E d × ℝ → ℝ), (x₀, (0 : ℝ)) ∈ U ∧ IsOpen U ∧ AnalyticOnNhd ℝ u U ∧ (∀ x : LeanEval.Analysis.E d, (x, (0 : ℝ)) ∈ U → u (x, 0) = u₀ x) ∧ (∀ p ∈ U, fderiv ℝ u p ((0 : LeanEval.Analysis.E d), (1 : ℝ)) = fderiv ℝ u p (F (p.1, p.2, u p), (0 : ℝ)) + f (p.1, p.2, u p)) ∧ (∀ v : LeanEval.Analysis.E d × ℝ → ℝ, AnalyticOnNhd ℝ v U → (∀ x : LeanEval.Analysis.E d, (x, (0 : ℝ)) ∈ U → v (x, 0) = u₀ x) → (∀ p ∈ U, fderiv ℝ v p ((0 : LeanEval.Analysis.E d), (1 : ℝ)) = fderiv ℝ v p (F (p.1, p.2, v p), (0 : ℝ)) + f (p.1, p.2, v p)) → ∀ p ∈ U, u p = v p) := d:ℕF:E d × ℝ × ℝ → E df:E d × ℝ × ℝ → ℝu₀:E d → ℝ_hF:AnalyticOnNhd ℝ F univ_hf:AnalyticOnNhd ℝ f univ_hu₀:AnalyticOnNhd ℝ u₀ univx₀:E d⊢ ∃ U u, (x₀, 0) ∈ U ∧ IsOpen U ∧ AnalyticOnNhd ℝ u U ∧ (∀ (x : E d), (x, 0) ∈ U → u (x, 0) = u₀ x) ∧ (∀ p ∈ U, (fderiv ℝ u p) (0, 1) = (fderiv ℝ u p) (F (p.1, p.2, u p), 0) + f (p.1, p.2, u p)) ∧ ∀ (v : E d × ℝ → ℝ), AnalyticOnNhd ℝ v U → (∀ (x : E d), (x, 0) ∈ U → v (x, 0) = u₀ x) → (∀ p ∈ U, (fderiv ℝ v p) (0, 1) = (fderiv ℝ v p) (F (p.1, p.2, v p), 0) + f (p.1, p.2, v p)) → ∀ p ∈ U, u p = v p All goals completed! 🐙
#9
How produced

No Description

Radó's theorem on Riemann surfaces
rado_riemannSurface

Verso theorem preview

theorem declaration uses `sorry`rado_riemannSurface {X : Type*} [TopologicalSpace X] [T2Space X] [ConnectedSpace X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) 1 X] : SecondCountableTopology X := X:Type u_1inst✝⁴:TopologicalSpace Xinst✝³:T2Space Xinst✝²:ConnectedSpace Xinst✝¹:ChartedSpace ℂ Xinst✝:IsManifold (modelWithCornersSelf ℂ ℂ) 1 X⊢ SecondCountableTopology X All goals completed! 🐙
#10
How produced

No Description

Jordan curve theorem
jordan_curve

Verso theorem preview

theorem declaration uses `sorry`jordan_curve (r : Metric.sphere (0 : EuclideanSpace ℝ (Fin 2)) 1 → EuclideanSpace ℝ (Fin 2)) (_hcont : Continuous r) (_hinj : Function.Injective r) : Nat.card (ConnectedComponents ((Set.range r)ᶜ : Set (EuclideanSpace ℝ (Fin 2)))) = 2 := r:↑(Metric.sphere 0 1) → EuclideanSpace ℝ (Fin 2)_hcont:Continuous r_hinj:Function.Injective r⊢ Nat.card (ConnectedComponents ↑(Set.range r)ᶜ) = 2 All goals completed! 🐙
#11
How produced

no_response

Jacobian of a compact Riemann surface (Buzzard challenge)
jacobian_challenge_diffgeo

Verso theorem preview

def declaration uses `sorry`genus (X : Type u) [TopologicalSpace X] [T2Space X] [CompactSpace X] [ConnectedSpace X] [Nonempty X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) ω X] : ℕ := sorry
theorem declaration uses `sorry`genus_eq_zero_iff_homeo : genus X = 0 ↔ Nonempty (X ≃ₜ (Metric.sphere (0 : EuclideanSpace ℝ (Fin 3)) 1)) := sorry
def declaration uses `sorry`Jacobian (X : Type u) [TopologicalSpace X] [T2Space X] [CompactSpace X] [ConnectedSpace X] [Nonempty X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) ω X] : Type u := sorry
instance declaration uses `sorry`instAddCommGroup : AddCommGroup (Jacobian X) := sorry
instance declaration uses `sorry`instTopologicalSpace : TopologicalSpace (Jacobian X) := sorry
instance declaration uses `sorry`instT2Space : T2Space (Jacobian X) := sorry
instance declaration uses `sorry`instCompactSpace : CompactSpace (Jacobian X) := sorry
instance declaration uses `sorry`instChartedSpace : ChartedSpace (Fin (genus X) → ℂ) (Jacobian X) := sorry
instance declaration uses `sorry`instIsManifold : IsManifold (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (Jacobian X) := sorry
instance declaration uses `sorry`instLieAddGroup : LieAddGroup (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (Jacobian X) := sorry
def declaration uses `sorry`ofCurve (P : X) : X → Jacobian X := sorry
theorem declaration uses `sorry`ofCurve_contMDiff (P : X) : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (ofCurve P) := sorry
theorem declaration uses `sorry`ofCurve_self (P : X) : ofCurve P P = 0 := sorry
theorem declaration uses `sorry`ofCurve_inj (P : X) (h : 0 < genus X) : Function.Injective (ofCurve P) := sorry
def declaration uses `sorry`pushforward (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : Jacobian X →ₜ+ Jacobian Y := sorry
theorem declaration uses `sorry`pushforward_contMDiff (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ContMDiff (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) (modelWithCornersSelf ℂ (Fin (genus Y) → ℂ)) ω (pushforward f hf) := sorry
theorem declaration uses `sorry`pushforward_id_apply (P : Jacobian X) : pushforward id contMDiff_id P = P := sorry
theorem declaration uses `sorry`pushforward_comp_apply (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (g : Y → Z) (hg : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω g) (P : Jacobian X) : pushforward (g ∘ f) (hg.comp hf) P = pushforward g hg (pushforward f hf P) := sorry
def declaration uses `sorry`pullback (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : Jacobian Y →ₜ+ Jacobian X := sorry
theorem declaration uses `sorry`pullback_contMDiff (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ContMDiff (modelWithCornersSelf ℂ (Fin (genus Y) → ℂ)) (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (pullback f hf) := sorry
theorem declaration uses `sorry`pullback_id_apply (P : Jacobian X) : pullback id contMDiff_id P = P := sorry
theorem declaration uses `sorry`pullback_comp_apply (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (g : Y → Z) (hg : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω g) (P : Jacobian Z) : pullback (g.comp f) (hg.comp hf) P = pullback f hf (pullback g hg P) := sorry
def declaration uses `sorry`degree (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ℕ := sorry -- 0 for constant case
theorem declaration uses `sorry`pushforward_pullback (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (P : Jacobian Y) : pushforward f hf (pullback f hf P) = (degree f hf) • P := sorry
#14
Feit–Thompson odd-order theorem
feit_thompson

Verso theorem preview

theorem declaration uses `sorry`feit_thompson {G : Type*} [Group G] [Finite G] (_h : Odd (Nat.card G)) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Finite G_h:Odd (Nat.card G)⊢ IsSolvable G All goals completed! 🐙
#15
Wigner semicircle law
wigner_semicircle

Verso theorem preview

theorem declaration uses `sorry`wigner_semicircle {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → ℕ → Ω → ℝ) (_hX_meas : ∀ i j, Measurable (X i j)) (_hX_indep : iIndepFun (fun ij : {p : ℕ × ℕ // p.1 ≤ p.2} => X ij.val.1 ij.val.2) μ) (_hX_iid : ∀ i j i' j', i ≤ j → i' ≤ j' → ProbabilityTheory.IdentDistrib (X i j) (X i' j') μ μ) (_hX_int : ∀ i j, i ≤ j → Integrable (X i j) μ) (_hX_sq_int : ∀ i j, i ≤ j → Integrable (fun ω => (X i j ω) ^ 2) μ) (_hX_mean : ∀ i j, i ≤ j → ∫ ω, X i j ω ∂μ = 0) (_hX_var : ∀ i j, i ≤ j → ∫ ω, (X i j ω) ^ 2 ∂μ = 1) : ∀ᵐ ω ∂μ, ∀ (f : ℝ → ℝ), Continuous f → (∃ M, ∀ x, ‖f x‖ ≤ M) → Tendsto (fun n : ℕ => ∫ x, f x ∂ (empiricalSpectralMeasureHerm (wignerMatrix_isHermitian X n ω)).map (fun x : ℝ => x / Real.sqrt n)) atTop (𝓝 (∫ x, f x ∂semicircleLaw)) := Ω:Type u_1inst✝¹:MeasurableSpace Ωμ:Measure Ωinst✝:IsProbabilityMeasure μX:ℕ → ℕ → Ω → ℝ_hX_meas:∀ (i j : ℕ), Measurable (X i j)_hX_indep:iIndepFun (fun ij => X (↑ij).1 (↑ij).2) μ_hX_iid:∀ (i j i' j' : ℕ), i ≤ j → i' ≤ j' → IdentDistrib (X i j) (X i' j') μ μ_hX_int:∀ (i j : ℕ), i ≤ j → Integrable (X i j) μ_hX_sq_int:∀ (i j : ℕ), i ≤ j → Integrable (fun ω => X i j ω ^ 2) μ_hX_mean:∀ (i j : ℕ), i ≤ j → ∫ (ω : Ω), X i j ω ∂μ = 0_hX_var:∀ (i j : ℕ), i ≤ j → ∫ (ω : Ω), X i j ω ^ 2 ∂μ = 1⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (f : ℝ → ℝ), Continuous f → (∃ M, ∀ (x : ℝ), ‖f x‖ ≤ M) → Tendsto (fun n => ∫ (x : ℝ), f x ∂Measure.map (fun x => x / √↑n) (empiricalSpectralMeasureHerm ⋯)) atTop (𝓝 (∫ (x : ℝ), f x ∂semicircleLaw)) All goals completed! 🐙
#16
Tverberg's theorem
tverberg_theorem

Lean theorem statement

/-- **Tverberg's theorem.** Any `(r-1)(d+1)+1` points in `ℝ^d` admit an
`r`-part Tverberg partition. -/
theorem tverberg_theorem (d r : ℕ) (hr : 1 ≤ r)
    (f : Fin ((r - 1) * (d + 1) + 1) → Space d) :
    HasTverbergPartition (r := r) f := by
  sorry
#17
How produced

No Description

Strong Subadditivity of von Neumann Entropy
strong_subadditivity

Lean theorem statement

/-- Strong subadditivity of quantum entropy. We relax the common assumption that M is a normalized
 density matrix to the simpler statement that it's PSD, which holds since normalization just produces
 a positive affine transformation on the entropy. -/
theorem strong_subadditivity (M_ABC : Matrix (A × B × C) (A × B × C) ℂ) (h : M_ABC.PosSemidef) :
    let M_AB : Matrix (A × B) (A × B) ℂ :=
      .traceRight <| M_ABC.reindex (.symm <| .prodAssoc ..) (.symm <| .prodAssoc ..)
    let M_BC : Matrix (B × C) (B × C) ℂ := M_ABC.traceLeft
    let M_B : Matrix B B ℂ := M_BC.traceRight
    entropy M_ABC + entropy M_B ≤ entropy M_AB + entropy M_BC := by
  sorry
#18
How produced

No Description

Sobolev embedding theorem (Morrey regime)
sobolev_embedding_morrey

Verso theorem preview

theorem declaration uses `sorry`sobolev_embedding {n k r : ℕ} {α p : ℝ} (_hp : (n : ℝ) < p) (_hα : 0 < α) (_hα1 : α ≤ 1) (_hgap : (r : ℝ) + α < (k : ℝ) - n / p) (f : LeanEval.Analysis.SobolevMorreyProblem.E n → ℝ) (_hf : LeanEval.Analysis.SobolevMorreyProblem.MemSobolevWk k (ENNReal.ofReal p) f) : ∃ g : LeanEval.Analysis.SobolevMorreyProblem.E n → ℝ, f =ᵐ[volume] g ∧ LeanEval.Analysis.SobolevMorreyProblem.MemHolder r α g := n:ℕk:ℕr:ℕα:ℝp:ℝ_hp:↑n < p_hα:0 < α_hα1:α ≤ 1_hgap:↑r + α < ↑k - ↑n / pf:E n → ℝ_hf:MemSobolevWk k (ENNReal.ofReal p) f⊢ ∃ g, f =ᵐ[volume] g ∧ LeanEval.Analysis.SobolevMorreyProblem.MemHolder r α g All goals completed! 🐙
#19
How produced

No Description

Radial symmetry for positive semilinear Poisson solutions
semilinear_poisson_radial_symmetry

Lean theorem statement

/-- **Radial symmetry theorem.** Let `u ∈ C^2(closedBall 0 1)` be a positive
solution of the semilinear Poisson problem `-Δ u = f(u)` in the open unit ball,
with zero Dirichlet boundary values on the unit sphere. If `f : ℝ → ℝ` is
Lipschitz, then `u` is radial: `u x = v ‖x‖` for a nonnegative strictly
decreasing radial profile `v` on `[0, 1]`. -/
theorem semilinear_poisson_radial_symmetry {n : ℕ} (hn : 0 < n)
    {f : ℝ → ℝ} (u : EuclideanSpace ℝ (Fin n) → ℝ)
    (hf_lipschitz : ∃ K : ℝ≥0, LipschitzWith K f)
    (hu_c2 : ContDiffOn ℝ 2 u (closedBall 0 1))
    (hu_solve : SolvesSemilinearPoisson f u)
    (hu_positive : ∀ x ∈ ball 0 1, 0 < u x) :
    ∃ v : ℝ → ℝ≥0,
      StrictAntiOn v (Set.Icc (0 : ℝ) 1) ∧
        ∀ x ∈ closedBall 0 1, u x = v ‖x‖ := by
  sorry
#20
How produced

No Description

Fundamental theorem of topos theory
fundamental_topos_theory

Verso theorem preview

theorem declaration uses `sorry`fundamental_topos_theory {E : Type*} [Category E] (hE : LeanEval.ToposTheory.IsTopos E) (X : E) : LeanEval.ToposTheory.IsTopos (Over X) := E:Type u_1inst✝:Category.{u_2, u_1} EhE:IsTopos EX:E⊢ IsTopos (Over X) All goals completed! 🐙
#21
How produced

No Description

Frobenius determinant theorem
frobenius_group_determinant

Lean theorem statement

/-- **Frobenius determinant theorem** (§171). The group determinant factors as
a product of irreducible polynomials, each appearing to the power of its own
(total) degree `d_j = deg p_j`, with the factors pairwise non-associated
(*distinct*) and their number equal to the number of conjugacy classes of `G`.
-/
theorem frobenius_group_determinant
    (G : Type*) [Group G] [Fintype G] [DecidableEq G] :
    ∃ (r : ℕ) (p : Fin r → MvPolynomial G ℂ),
      r = Nat.card (ConjClasses G) ∧
      (∀ j, Irreducible (p j)) ∧
      (∀ i j, i ≠ j → ¬ Associated (p i) (p j)) ∧
      groupDeterminant G = ∏ j, (p j) ^ (p j).totalDegree := by
  sorry
#22
How produced

No Description

Brun's theorem (convergence of the twin-prime reciprocal sum)
brun_constant_converges

Lean theorem statement

/-- **Brun's theorem.** The reciprocal sum over twin-prime pairs converges. -/
theorem brun_constant_converges :
    Summable twinPrimeReciprocalTerm := by
  sorry
#23
How produced

No Description

Sard's theorem (critical-set image has measure zero)
sard_theorem

Lean theorem statement

/-- **Sard's theorem** (Morse 1939 / Sard 1942), Knill's rank-
deficient form. The image of the rank-deficient locus of a smooth
map `f : ℝᵐ → ℝⁿ` has Lebesgue measure zero. -/
theorem sard {m n : ℕ} (f : E m → E n) (_hf : ContDiff ℝ ∞ f) :
    volume (criticalValues f) = 0 := by
  sorry
#24
How produced

No Description

Poincaré–Siegel linearisation theorem
poincare_siegel_linearisation

Lean theorem statement

/-- **Poincaré–Siegel linearisation theorem.** If `α` is Diophantine,
`λ = e^{2πiα}`, and `f` is holomorphic near `0` with `f 0 = 0` and
`f'(0) = λ`, then there is a holomorphic germ `u` with `u 0 = 0`,
`u'(0) = 1`, and `f(u z) = u(λ z)` for `z` near `0`. -/
theorem poincare_siegel
    (α : ℝ) (_hα : IsDiophantine α)
    (lam : ℂ) (_hlam : lam = Complex.exp (2 * Real.pi * Complex.I * (α : ℂ)))
    (f : ℂ → ℂ) (_hf : AnalyticAt ℂ f 0) (_hf0 : f 0 = 0)
    (_hmult : deriv f 0 = lam) :
    ∃ u : ℂ → ℂ, AnalyticAt ℂ u 0 ∧ u 0 = 0 ∧ deriv u 0 = 1 ∧
      ∀ᶠ z in nhds (0 : ℂ), f (u z) = u (lam * z) := by
  sorry
#25
How produced

No Description

Kolmogorov–Arnold superposition theorem (non-universal Lorentz form)
kolmogorov_arnold_superposition

Verso theorem preview

theorem declaration uses `sorry`kolmogorov_arnold (n : ℕ) (_hn : 1 ≤ n) (f : (Fin n → ℝ) → ℝ) (_hf : ContinuousOn f (Set.Icc 0 1)) : ∃ (g : ℝ → ℝ) (φ : Fin (2 * n + 1) → Fin n → ℝ → ℝ), Continuous g ∧ (∀ k l, Continuous (φ k l)) ∧ ∀ x ∈ Set.Icc (0 : Fin n → ℝ) 1, f x = ∑ k, g (∑ l, φ k l (x l)) := n:ℕ_hn:1 ≤ nf:(Fin n → ℝ) → ℝ_hf:ContinuousOn f (Set.Icc 0 1)⊢ ∃ g φ, Continuous g ∧ (∀ (k : Fin (2 * n + 1)) (l : Fin n), Continuous (φ k l)) ∧ ∀ x ∈ Set.Icc 0 1, f x = ∑ k, g (∑ l, φ k l (x l)) All goals completed! 🐙
#26
How produced

No Description

Hurewicz theorem in degree 1 (H₁ = abelianization of π₁)
hurewicz_h1_abelianization

Lean theorem statement

/-- **Hurewicz (n = 1).** For a path-connected space `X`, `H₁(X;ℤ)` is the
abelianization of `π₁(X, x)`. -/
theorem hurewicz_h1_abelianization
    (X : Type) [TopologicalSpace X] [PathConnectedSpace X] (x : X) :
    Nonempty (Additive (Abelianization (FundamentalGroup X x)) ≃+
      (IntegralHomology 1 X : Type)) := by
  sorry
#27
How produced

No Description

No bounded projection from L^1 onto H^1
H1_not_closedComplemented

Verso theorem preview

theorem declaration uses `sorry`H1_not_closedComplemented : ¬ LeanEval.Analysis.H1.ClosedComplemented := ⊢ ¬H1.ClosedComplemented All goals completed! 🐙
#28
How produced

No Description

Shannon capacity of the pentagon
shannon_capacity_pentagon

Lean theorem statement

/-- **Lovász's theorem on Shannon capacity of the pentagon** (§238).
The Shannon capacity of the five-cycle is `√5`. -/
theorem shannon_capacity_pentagon :
    HasShannonCapacity (SimpleGraph.cycleGraph 5) (Real.sqrt 5) := by
  sorry
#29
Peano existence theorem for ODEs
peano_existence

Lean theorem statement

/-- **Peano existence theorem.** Replacing the Lipschitz condition with mere
continuity still yields a local solution of `x' = f(x)`, `x(0) = x₀` — but
uniqueness may fail (e.g. `x' = √x`, `x(0) = 0`). Stated for a
finite-dimensional space: Peano's theorem requires local compactness and is
false in general (infinite-dimensional) Banach spaces. -/
theorem peano_existence
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
    {f : E → E} (hf : Continuous f) (x₀ : E) :
    ∃ a : ℝ, 0 < a ∧ ∃ α : ℝ → E, α 0 = x₀ ∧
      ∀ t ∈ Ioo (-a) a, HasDerivAt α (f (α t)) t := by
  sorry
#30
Jordan normal form
jordan_normal_form

Lean theorem statement

/-- **Jordan normal form.** Over an algebraically closed field, every
endomorphism of `Kⁿ` admits a Jordan-chain basis. -/
theorem jordan_normal_form {K : Type*} [Field K] [IsAlgClosed K] (n : ℕ)
    (f : Module.End K (StdSpace K n)) :
    Nonempty (JordanChainBasis f) := by
  sorry
#31
Wiener–Lévy theorem
wiener_levy_analytic_calculus

Verso theorem preview

theorem declaration uses `sorry`wiener_levy_analytic_calculus (f : C(AddCircle T, ℂ)) (φ : ℂ → ℂ) (U : Set ℂ) (hf : LeanEval.Analysis.WienerLevy.InWienerAlgebra f) (hU : IsOpen U) (hrange : range f ⊆ U) (hφ : AnalyticOnNhd ℂ φ U) : ∃ g : C(AddCircle T, ℂ), (∀ x, g x = φ (f x)) ∧ LeanEval.Analysis.WienerLevy.InWienerAlgebra g := T:ℝinst✝:Fact (0 < T)f:C(AddCircle T, ℂ)φ:ℂ → ℂU:Set ℂhf:InWienerAlgebra fhU:IsOpen Uhrange:range ⇑f ⊆ Uhφ:AnalyticOnNhd ℂ φ U⊢ ∃ g, (∀ (x : AddCircle T), g x = φ (f x)) ∧ InWienerAlgebra g All goals completed! 🐙
#32
The Lindemann–Weierstrass theorem
lindemann_weierstrass

Lean theorem statement

/-- **Lindemann–Weierstrass theorem.** If `x₁, …, xₙ ∈ ℂ` are algebraic over `ℚ`
and ℚ-linearly independent, then `e^{x₁}, …, e^{xₙ}` are algebraically
independent over `ℚ`. -/
theorem lindemann_weierstrass {n : ℕ} (x : Fin n → ℂ)
    (h_alg : ∀ i, IsAlgebraic ℚ (x i))
    (h_lin : LinearIndependent ℚ x) :
    AlgebraicIndependent ℚ (fun i => Complex.exp (x i)) := by
  sorry
#33
Liouville–Arnold theorem on integrable systems
liouville_arnold

Verso theorem preview

theorem declaration uses `sorry`liouville_arnold {n : ℕ} (F : Fin n → LeanEval.Geometry.LiouvilleArnold.E n → ℝ) (U : Set (LeanEval.Geometry.LiouvilleArnold.E n)) (_hU : IsOpen U) (_hLI : LeanEval.Geometry.LiouvilleArnold.IsLiouvilleIntegrable F U) (c : Fin n → ℝ) (_hMc_sub : LeanEval.Geometry.LiouvilleArnold.levelSet F c ⊆ U) (_hMc_compact : IsCompact (LeanEval.Geometry.LiouvilleArnold.levelSet F c)) (_hMc_connected : IsConnected (LeanEval.Geometry.LiouvilleArnold.levelSet F c)) : Nonempty ((LeanEval.Geometry.LiouvilleArnold.levelSet F c) ≃ₜ (Fin n → AddCircle (1 : ℝ))) := n:ℕF:Fin n → E n → ℝU:Set (E n)_hU:IsOpen U_hLI:IsLiouvilleIntegrable F Uc:Fin n → ℝ_hMc_sub:levelSet F c ⊆ U_hMc_compact:IsCompact (levelSet F c)_hMc_connected:IsConnected (levelSet F c)⊢ Nonempty (↑(levelSet F c) ≃ₜ (Fin n → AddCircle 1)) All goals completed! 🐙
#34
Fang–Xia: tiling of the symmetric group by transpositions implies λ-transitivity
fang_xia_tiling_partition_transitive

Lean theorem statement

/-- **Fang–Xia, Theorem 1.4.** A tiling `(T_n, Y)` of `S_n` forces
λ-transitivity of `Y` for every partition `λ` of `n` whose Young-
diagram content sum is nonnegative. -/
theorem fang_xia_partition_transitive_of_tiling
    {n : ℕ} {Y : Set (Equiv.Perm (Fin n))}
    (_h : IsTiling (transpositionsWithOne n) Y) :
    ∀ lam : PartitionShape n, 0 ≤ lam.contentSum → IsPartitionTransitive Y lam := by
  sorry
#35
How produced

No Description

Riesz brothers' theorem
riesz_brothers_theorem

Lean theorem statement

/-- **Riesz brothers' theorem.** Let `μ` be a complex Borel measure on
the unit circle such that `∫ z^n dμ = 0` for every `n ≥ 1`. Then `μ` is
absolutely continuous with respect to Haar measure, and more precisely
`dμ = h dm` for some `h ∈ H^1`, i.e. some integrable density whose negative
Fourier coefficients vanish. -/
theorem riesz_brothers_theorem (μ : ComplexMeasure UnitAddCircle)
    (hμ : ∀ n : ℕ, 1 ≤ n → ∫ᵛ z, fourier n z ∂[ContinuousLinearMap.mul ℝ ℂ; μ] = 0) :
    μ ≪ᵥ AddCircle.haarAddCircle.toENNRealVectorMeasure := by
  sorry
#36
How produced

No Description

Pascal's theorem
pascal

Lean theorem statement

/-- **Pascal's theorem.** Six distinct points on a non-singular conic determine
three collinear intersection points `Aᵢ Bⱼ ∩ Aⱼ Bᵢ`. -/
theorem pascal
    (M : Matrix (Fin 3) (Fin 3) ℝ) (hMsymm : M.IsSymm) (hMdet : M.det ≠ 0)
    (a₁ a₂ a₃ b₁ b₂ b₃ : Fin 3 → ℝ)
    (ha₁ : a₁ ≠ 0) (ha₂ : a₂ ≠ 0) (ha₃ : a₃ ≠ 0)
    (hb₁ : b₁ ≠ 0) (hb₂ : b₂ ≠ 0) (hb₃ : b₃ ≠ 0)
    (hdist : [a₁, a₂, a₃, b₁, b₂, b₃].Pairwise (fun v w => ¬ SamePoint v w))
    (hA₁ : OnConic M a₁) (hA₂ : OnConic M a₂) (hA₃ : OnConic M a₃)
    (hB₁ : OnConic M b₁) (hB₂ : OnConic M b₂) (hB₃ : OnConic M b₃) :
    Collinear3 (meet a₁ b₂ a₂ b₁) (meet a₁ b₃ a₃ b₁) (meet a₂ b₃ a₃ b₂) := by
  sorry
#37
How produced

No Description

Radon transform: Fourier-slice diagonalization and pseudo-inversion
radon_transform_inversion

Lean theorem statement

/-- **Radon's theorem (diagonalization + pseudo-inversion).** The Fourier slice
theorem diagonalizes the Radon transform, and the transform admits a left
inverse on the Schwartz space. -/
theorem radon_can_be_diagonalized_and_pseudo_inverted :
    (∀ φ : SchwartzMap (ℝ × ℝ) ℂ, ∀ θ k : ℝ,
        fourier1 (fun p => radon (φ : ℝ × ℝ → ℂ) (p, θ)) k =
          fourier2 (φ : ℝ × ℝ → ℂ) (k * Real.cos θ, k * Real.sin θ)) ∧
    (∃ Rinv : (ℝ × ℝ → ℂ) → (ℝ × ℝ → ℂ),
        ∀ φ : SchwartzMap (ℝ × ℝ) ℂ,
          Rinv (radon (φ : ℝ × ℝ → ℂ)) = (φ : ℝ × ℝ → ℂ)) := by
  sorry
#38
How produced

No Description

Nyquist–Shannon sampling theorem
nyquist_shannon_sampling

Lean theorem statement

/-- **Nyquist--Shannon sampling theorem / Whittaker--Shannon interpolation
formula** for Schwartz functions with Fourier support in the mathlib-convention
Nyquist band `[-1/2, 1/2]`.

The explicit `Summable` conjunct records the convergence content of the
cardinal series, rather than relying only on Lean's total `tsum`.
-/
theorem nyquist_shannon_sampling (f : 𝓢(ℝ, ℂ)) (hf : FourierSupportedInNyquist f) :
    ∀ t : ℝ,
      Summable (fun n : ℤ ↦ f (n : ℝ) * sinc (Real.pi * ((n : ℝ) - t))) ∧
        f t =
          ∑' n : ℤ, f (n : ℝ) * sinc (Real.pi * ((n : ℝ) - t)) := by
  sorry
#39
How produced

No Description

Moran's equality for affine-symmetric iterated function systems
moran_equality_affine

Lean theorem statement

/-- **Moran's equality for affine-symmetric IFS.** For an affine-symmetric IFS
on `ℝᵈ` with common contraction factor `λ ∈ (0,1)`, orthogonal linear parts, and
the open set condition, the Hausdorff dimension of the attractor is
`−log n / log λ` (positive since `λ < 1`). -/
theorem moran_equality_affine
    {d n : ℕ} (hn : 1 ≤ n)
    (f : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (lam : ℝ)
    (h_aff : IsAffineSymmetricIFS f lam)
    (h_osc : OpenSetCondition f)
    {S : Set (EuclideanSpace ℝ (Fin d))} (hS : IsAttractor f S) :
    dimH S = ENNReal.ofReal (- Real.log n / Real.log lam) := by
  sorry
#40
How produced

No Description

The Landsberg–Schaar relation
landsberg_schaar

Lean theorem statement

/-- **Landsberg–Schaar relation.** For positive odd integers `p, q`,
`S(2q, p) = e^{iπ/4} · S(−p, 2q)`. -/
theorem landsberg_schaar (p q : ℕ) (hp : Odd p) (hq : Odd q) :
    gaussS (2 * q : ℕ) p
      = Complex.exp ((Real.pi : ℂ) * Complex.I / 4) * gaussS (-(p : ℤ)) (2 * q) := by
  sorry
#41
How produced

No Description

Kirk's normal-structure fixed point theorem
kirk_normal_structure

Lean theorem statement

/-- **Kirk's normal-structure fixed-point theorem** (§228).  A nonexpansive
self-map of a nonempty bounded closed convex subset of a reflexive Banach
space with normal structure has a fixed point. -/
theorem kirk_normal_structure [CompleteSpace E]
    (hE_reflexive : Function.Surjective (NormedSpace.inclusionInDoubleDual ℝ E))
    (K : Set E) (hK_nonempty : K.Nonempty) (hK_closed : IsClosed K)
    (hK_bounded : Bornology.IsBounded K) (hK_convex : Convex ℝ K)
    (hK_normal : HasNormalStructure K) (T : K → K)
    (hT : IsNonexpansiveSelfMap K T) :
    ∃ x : K, IsFixedPt T x := by
  sorry
#42
How produced

No Description

Boone–Higman theorem (easy direction)
boone_higman_embedding

Lean theorem statement

/-- **Boone–Higman theorem (easy direction).** If a finitely presented group `G`
embeds (via injective `f`) into a simple group `H`, which embeds (via injective
`g`) into a finitely presented group `K`, then the word problem of `G` is
solvable. -/
theorem boone_higman_embedding
    {G H K : Type*} [Group G] [Group H] [Group K]
    [IsSimpleGroup H] [Group.IsFinitelyPresented K]
    (f : G →* H) (hf : Function.Injective f)
    (g : H →* K) (hg : Function.Injective g)
    {n : ℕ} (φ : FreeGroup (Fin n) →* G)
    (hsurj : Function.Surjective φ)
    (hker : (MonoidHom.ker φ).IsNormalClosureFG) :
    WordProblemSolvable φ := by
  sorry
#43
Choquet's representation theorem
choquet_representation_theorem

Lean theorem statement

/-- **Choquet's representation theorem.** Every point `x` of a compact convex
set `K` in a Banach space is the barycenter of a probability measure supported
on the extreme points of `K`: there is a probability measure `μ` with
`μ (ext K)ᶜ = 0` whose barycenter `∫ y, y ∂μ` equals `x`. -/
theorem choquet [MeasurableSpace X] [BorelSpace X]
    (K : Set X) (hK_cpt : IsCompact K) (hK_cvx : Convex ℝ K)
    {x : X} (hx : x ∈ K) :
    ∃ μ : Measure X, IsProbabilityMeasure μ ∧
      μ (K.extremePoints ℝ)ᶜ = 0 ∧
      x = ∫ y, y ∂μ := by
  sorry
#44
Gauss-Wantzel constructible regular polygon theorem
gauss_wantzel_constructible_polygon

Lean theorem statement

/-- **Gauss-Wantzel constructible polygon theorem** (§174): a regular `n`-gon
is straightedge-and-compass constructible exactly for the Gauss-Wantzel
integers. -/
theorem gauss_wantzel_constructible_polygon (n : ℕ) (hn : 3 ≤ n) :
    IsConstructible (Real.cos (2 * Real.pi / n)) ↔ GaussWantzelNumber n := by
  sorry
#45
Anosov–Bowen shadowing lemma
anosov_bowen_shadowing

Lean theorem statement

/-- **Anosov–Bowen shadowing lemma** (Anosov 1967; Bowen 1975). Every
compact hyperbolic invariant set has the shadowing property. -/
theorem hyperbolic_has_shadowing
    (T : E d ≃ₜ E d) (K : Set (E d))
    (_hKc : IsCompact K) (_hK : IsHyperbolic T K) :
    HasShadowing (T : E d → E d) K := by
  sorry
#46
Ornstein–Weiss ℤᵈ Rokhlin lemma
ornstein_weiss_rokhlin

Verso theorem preview

theorem declaration uses `sorry`ornstein_weiss_rokhlin {Ω : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω] {d : ℕ} (_hd : 1 ≤ d) (μ : Measure Ω) [IsProbabilityMeasure μ] (T : (Fin d → ℤ) → Ω → Ω) (_hid : ∀ x, T 0 x = x) (_hT : ∀ v, MeasurePreserving (T v) μ μ) (_hgrp : ∀ u v x, T (u + v) x = T u (T v x)) (_hfree : LeanEval.Dynamics.IsFreeAction μ T) (N : ℕ) (_hN : 1 ≤ N) {ε : ENNReal} (_hε : 0 < ε) : ∃ B : Set Ω, MeasurableSet B ∧ ((boxShape d N : Finset (Fin d → ℤ)) : Set (Fin d → ℤ)).PairwiseDisjoint (fun v => T v '' B) ∧ μ (⋃ v ∈ boxShape d N, T v '' B) ≥ 1 - ε := Ω:Type u_1inst✝²:MeasurableSpace Ωinst✝¹:StandardBorelSpace Ωd:ℕ_hd:1 ≤ dμ:Measure Ωinst✝:IsProbabilityMeasure μT:(Fin d → ℤ) → Ω → Ω_hid:∀ (x : Ω), T 0 x = x_hT:∀ (v : Fin d → ℤ), MeasurePreserving (T v) μ μ_hgrp:∀ (u v : Fin d → ℤ) (x : Ω), T (u + v) x = T u (T v x)_hfree:IsFreeAction μ TN:ℕ_hN:1 ≤ Nε:ENNReal_hε:0 < ε⊢ ∃ B, MeasurableSet B ∧ ((↑(boxShape d N)).PairwiseDisjoint fun v => T v '' B) ∧ μ (⋃ v ∈ boxShape d N, T v '' B) ≥ 1 - ε All goals completed! 🐙
#47
Fundamental theorem of Riemannian geometry (Levi-Civita)
levi_civita_exists_unique

Verso theorem preview

theorem declaration uses `sorry`levi_civita_exists_unique {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [CompleteSpace E] {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type*} [TopologicalSpace M] [T2Space M] [ChartedSpace H M] [IsManifold I ∞ M] [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)] [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)] : ∃ cov : CovariantDerivative I E (TangentSpace I (M := M)), (ContMDiffCovariantDerivative cov ∞ ∧ cov.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov) ∧ ∀ cov' : CovariantDerivative I E (TangentSpace I (M := M)), (ContMDiffCovariantDerivative cov' ∞ ∧ cov'.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov') → LeanEval.Geometry.LeviCivita.SameOnSmooth cov cov' := E:Type u_1inst✝¹⁰:NormedAddCommGroup Einst✝⁹:NormedSpace ℝ Einst✝⁸:FiniteDimensional ℝ Einst✝⁷:CompleteSpace EH:Type u_2inst✝⁶:TopologicalSpace HI:ModelWithCorners ℝ E HM:Type u_3inst✝⁵:TopologicalSpace Minst✝⁴:T2Space Minst✝³:ChartedSpace H Minst✝²:IsManifold I ∞ Minst✝¹:RiemannianBundle fun x => TangentSpace I xinst✝:IsContMDiffRiemannianBundle I ∞ E fun x => TangentSpace I x⊢ ∃ cov, (cov.ContMDiffCovariantDerivative ∞ ∧ cov.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov) ∧ ∀ (cov' : CovariantDerivative I E (TangentSpace I)), cov'.ContMDiffCovariantDerivative ∞ ∧ cov'.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov' → SameOnSmooth cov cov' All goals completed! 🐙
#48
Complete reducibility for compact groups
compact_group_semisimple

Lean theorem statement

/-- **Representations of compact groups are semisimple** (complete
reducibility / the unitarian trick). A continuous representation of a compact
topological group on a finite-dimensional real vector space is semisimple:
every subrepresentation has a `G`-invariant complement, so the representation
decomposes as a direct sum of irreducible finite-dimensional
subrepresentations. -/
theorem compact_group_semisimple
    {G V : Type*} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G]
    [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V]
    (ρ : Representation ℝ G V)
    (hρ : Continuous fun p : G × V => ρ p.1 p.2) :
    ρ.IsSemisimpleRepresentation := by
  sorry
#49
How produced

No Description

Lindemann's theorem (e and π transcendental)
lindemann

Lean theorem statement

/-- **Lindemann's theorem.** Both `e = exp 1` and `π` are transcendental over
`ℤ`. -/
theorem lindemann :
    Transcendental ℤ (Real.exp 1) ∧ Transcendental ℤ Real.pi := by
  sorry
#50
Normal spectral theorem
normal_spectral_theorem

Lean theorem statement

/-- **Spectral theorem** (§14). A complex matrix `A` is **normal**
(`Aᴴ A = A Aᴴ`, i.e. `IsStarNormal A`) **iff** it is **unitarily
diagonalizable**: there is a unitary `U` and a diagonal matrix `diagonal d`
with `A = U (diagonal d) Uᴴ`. -/
theorem normal_spectral_theorem (A : Matrix n n ℂ) :
    IsStarNormal A ↔
      ∃ U ∈ unitary (Matrix n n ℂ), ∃ d : n → ℂ,
        A = U * diagonal d * star U := by
  sorry
#51
How produced

No Description

Trace Cayley-Hamilton / Newton identity
trace_cayley_hamilton_newton

Lean theorem statement

/-- The trace Cayley-Hamilton / Newton identity:
`k c_k + ∑_{j=1}^k tr(A^j) c_{k-j} = 0`, where
`χ_A(X) = X^N + c₁ X^(N-1) + ... + c_N`.

For `k > N`, `c_k = 0`, and the remaining relation is the trace of
Cayley-Hamilton multiplied by a power of `A`. -/
theorem trace_cayley_hamilton_newton {R : Type*} [CommRing R]
    (A : Matrix n n R) {k : ℕ} (hk : 1 ≤ k) :
    (k : R) * charpolyDescendingCoeff A k +
        ∑ j ∈ Finset.Icc 1 k,
          trace (A ^ j) * charpolyDescendingCoeff A (k - j) = 0 := by
  sorry
#52
How produced

No Description

Wiener's 1/f theorem
wiener_inverse_closed

Verso theorem preview

theorem declaration uses `sorry`wiener_inverse_closed (f : C(AddCircle T, ℂ)) (hf : LeanEval.Analysis.WienerOneOverF.InWienerAlgebra f) (hzero : ∀ x, f x ≠ 0) : ∃ g : C(AddCircle T, ℂ), (∀ x, g x = (f x)⁻¹) ∧ LeanEval.Analysis.WienerOneOverF.InWienerAlgebra g := T:ℝinst✝:Fact (0 < T)f:C(AddCircle T, ℂ)hf:InWienerAlgebra fhzero:∀ (x : AddCircle T), f x ≠ 0⊢ ∃ g, (∀ (x : AddCircle T), g x = (f x)⁻¹) ∧ InWienerAlgebra g All goals completed! 🐙
#53
How produced

No Description

General recursive equals Turing computable
turing_recursive_equiv

Lean theorem statement

/-- **General recursive = Turing computable** (total form). A total function
`f : ℕ → ℕ` is recursive (`Computable`, i.e. partial recursive as a partial
function) **iff** it is computed by some Turing machine (mathlib's `FinTM2`
model) under the standard binary encoding of `ℕ`. This is Knill's class
equality; the backward direction (TM-computable ⇒ recursive) is absent from
mathlib. -/
theorem turing_recursive_equiv (f : ℕ → ℕ) :
    Computable f ↔ Nonempty (TM2Computable encodeNat encodeNat f) := by
  sorry
#54
How produced

_no_response_

Pick's theorem
pick

Verso theorem preview

theorem declaration uses `sorry`pick {n : ℕ} (hn : 3 ≤ n) (v : Fin n → ℤ × ℤ) (hsimple : LeanEval.Geometry.PicksTheorem.IsSimple (LeanEval.Geometry.PicksTheorem.latPoly v)) : area ((LeanEval.Geometry.PicksTheorem.latPoly v).boundary (R := ℝ)) = (interiorPts v : ℝ) + (boundaryPts v : ℝ) / 2 - 1 := n:ℕhn:3 ≤ nv:Fin n → ℤ × ℤhsimple:IsSimple (latPoly v)⊢ area (Polygon.boundary ℝ (latPoly v)) = ↑(interiorPts v) + ↑(boundaryPts v) / 2 - 1 All goals completed! 🐙
#55
How produced

no_response

Rokhlin lemma
rokhlin_lemma

Lean theorem statement

/-- **Rokhlin lemma.** For every aperiodic measure-preserving
automorphism `T` of a standard Borel probability space `(Ω, μ)`, every
height `n ≥ 1`, and every `ε > 0`, there is a Rokhlin tower of height
`n` whose union has measure at least `1 − ε`. -/
theorem rokhlin_lemma {Ω : Type*} [MeasurableSpace Ω]
    [StandardBorelSpace Ω]
    (μ : Measure Ω) [IsProbabilityMeasure μ] (T : Ω → Ω)
    (_hT : MeasurePreserving T μ μ) (_hap : IsAperiodic T μ)
    (n : ℕ) (_hn : 1 ≤ n) {ε : ENNReal} (_hε : 0 < ε) :
    ∃ B : Set Ω, IsRokhlinTower T B n ∧
      μ (towerUnion T B n) ≥ 1 - ε := by
  sorry
#56
How produced

_no_response_

Lax's approximation theorem for toral homeomorphisms
lax_approximation

Lean theorem statement

/-- **Lax's approximation theorem.** Every toral dynamical system on `𝕋^d`
(`d ≥ 1`) is approximated arbitrarily well in the metric `δ` by cyclic cube
exchange transformations. -/
theorem lax_approximation {d : ℕ} (hd : 0 < d) (T : ToralDynamicalSystem d)
    {ε : ℝ≥0∞} (hε : 0 < ε) :
    ∃ (n : ℕ) (S : VolumePreservingEquiv d),
      IsCyclicCubeExchange S n ∧ deltaDist T.toVolumePreservingEquiv S < ε := by
  sorry
#57
How produced

_no_response_

Hausdorff moment problem: absolute-continuity criterion
hausdorff_absolute_continuity

Lean theorem statement

/-- **Absolute continuity criterion (Hausdorff moment problem).** A positive
probability measure `μ` on the cube is uniformly absolutely continuous w.r.t.
Lebesgue measure iff there is `C` with `(Δᵏμ)ₙ ≤ C·(Δᵏν)ₙ` for all `k ≤ n`. -/
theorem hausdorff_absolute_continuity {d : ℕ}
    (μ : Measure (EuclideanSpace ℝ (Fin d)))
    [IsProbabilityMeasure μ] (hμ : μ ((cube d)ᶜ) = 0) :
    UniformlyAbsolutelyContinuous μ (volume.restrict (cube d)) ↔
      ∃ C : ℝ, ∀ k n : Fin d → ℕ, k ≤ n →
        diff (momentOf μ) k n ≤ C * diff (momentOf (volume.restrict (cube d))) k n := by
  sorry
#58
How produced

_no_response_

The Hausdorff–Hildebrandt–Schoenberg moment theorem
hausdorff_hildebrandt_schoenberg

Lean theorem statement

/-- **Hausdorff–Hildebrandt–Schoenberg theorem.** A multi-indexed real sequence
is the moment sequence of a signed bounded-variation measure on the unit cube
`Iᵈ` iff its moments are Hausdorff bounded. -/
theorem hausdorff_hildebrandt_schoenberg {d : ℕ} (a : (Fin d → ℕ) → ℝ) :
    IsMomentConfiguration a ↔ HausdorffBounded a := by
  sorry
#59
How produced

_no_response_

The Hausdorff positivity (complete-monotonicity) criterion
hausdorff_positivity_criterion

Lean theorem statement

/-- **Hausdorff positivity criterion** (Hausdorff 1921). A moment configuration
comes from a positive measure iff all its iterated backward differences are
nonnegative — i.e. the sequence is *completely monotone*: `(Δᵏa)ₙ ≥ 0` for all
`k ≤ n`. -/
theorem hausdorff_positivity {d : ℕ} (a : (Fin d → ℕ) → ℝ) :
    IsPositiveMomentConfiguration a ↔ ∀ k n : Fin d → ℕ, k ≤ n → 0 ≤ diff a k n := by
  sorry
#60
How produced

_no_response_

Sard's regular-value corollary
regular_value_ae

Lean theorem statement

/-- **Regular value corollary (Sard).** For a smooth `f : ℝᵐ → ℝ`, almost every
`c ∈ ℝ` is a regular value. -/
theorem regular_value_ae {m : ℕ} (f : EuclideanSpace ℝ (Fin m) → ℝ)
    (hf : ContDiff ℝ ∞ f) :
    ∀ᵐ c ∂(volume : Measure ℝ), IsRegularValue f c := by
  sorry
#61
Riesz's rising sun lemma
rising_sun_lemma

Lean theorem statement

/-- **Riesz's rising sun lemma.** Every continuous real function on a compact
interval has the rising-sun property. -/
theorem rising_sun_lemma {a b : ℝ} (hab : a < b) {f : ℝ → ℝ}
    (hf : ContinuousOn f (Icc a b)) :
    HasRisingSunProperty a b f := by
  sorry
#62
Bauer's uniqueness at extreme points
bauer_extreme_point_uniqueness

Lean theorem statement

/-- **Bauer's uniqueness at extreme points.** If `x` is an extreme point of a
compact convex set `K` and `μ` is a probability measure supported on `K`
(`μ Kᶜ = 0`) with barycenter `x = ∫ y, y ∂μ`, then `μ` is the Dirac mass at
`x`. (The support hypothesis is the weaker `μ Kᶜ = 0`, making this a
strengthening of the textbook statement: uniqueness among all ambient Borel
probability measures on `K`, not only those already supported on `ext K`.) -/
theorem bauer_unique [MeasurableSpace X] [BorelSpace X]
    (K : Set X) (hK_cpt : IsCompact K) (hK_cvx : Convex ℝ K)
    {x : X} (hx : x ∈ K.extremePoints ℝ)
    (μ : Measure X) [IsProbabilityMeasure μ]
    (hμ : μ Kᶜ = 0) (hbar : x = ∫ y, y ∂μ) :
    μ = Measure.dirac x := by
  sorry
#63
Morley's trisector theorem
morley_theorem

Verso theorem preview

theorem declaration uses `sorry`morley_theorem (A B C P Q R : LeanEval.Geometry.Morley.Plane) (h : LeanEval.Geometry.Morley.IsMorleyConfiguration A B C P Q R) : LeanEval.Geometry.Morley.IsEquilateralTriple P Q R := A:PlaneB:PlaneC:PlaneP:PlaneQ:PlaneR:Planeh:IsMorleyConfiguration A B C P Q R⊢ IsEquilateralTriple P Q R All goals completed! 🐙
#64
How produced

No Description

Mountain Pass Theorem (Ambrosetti–Rabinowitz 1973)
mountain_pass

Verso theorem preview

theorem declaration uses `sorry`mountain_pass (f : E → ℝ) (_hf : ContDiff ℝ 1 f) (_hps : LeanEval.Analysis.MountainPassProblem.PalaisSmale f) {a b : E} {ε r : ℝ} (_hmr : LeanEval.Analysis.MountainPassProblem.MountainRange f a b ε r) : ∃ x : E, LeanEval.Analysis.MountainPassProblem.IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace Ef:E → ℝ_hf:ContDiff ℝ 1 f_hps:PalaisSmale fa:Eb:Eε:ℝr:ℝ_hmr:MountainRange f a b ε r⊢ ∃ x, IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b All goals completed! 🐙
#65
How produced

No Description

Fraser: Fourier decay for finite-field Kakeya sets is q^{-1} and sharp
fraser_kakeya_fourier_decay

Verso theorem preview

theorem declaration uses `sorry`fraser_kakeya_fourier_decay_and_sharp {d : ℕ} (_hd : 2 ≤ d) {K : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F d)} (_hK : LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K) (χ : AddChar F ℂ) (_hχ : χ ≠ 1) : (∃ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K μ ∧ ∀ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d, ξ ≠ 0 → ‖fourier χ μ ξ‖ ≤ (Fintype.card F : ℝ)⁻¹) ∧ (∀ κ : ℝ, 0 < κ → κ < 1 → ∃ Q : ℕ, ∀ (F' : Type*) [Field F'] [Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K' : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d), LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K' ∧ ∀ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K' μ → ∃ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d, ξ ≠ 0 ∧ κ * (Fintype.card F' : ℝ)⁻¹ ≤ ‖fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖) := F:Type u_1inst✝²:Field Finst✝¹:Fintype Finst✝:DecidableEq Fd:ℕ_hd:2 ≤ dK:Set (Space F d)_hK:IsKakeya Kχ:AddChar F ℂ_hχ:χ ≠ 1⊢ (∃ μ, IsProbabilityMeasureOn K μ ∧ ∀ (ξ : Space F d), ξ ≠ 0 → ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier χ μ ξ‖ ≤ (↑(Fintype.card F))⁻¹) ∧ ∀ (κ : ℝ), 0 < κ → κ < 1 → ∃ Q, ∀ (F' : Type u_2) [inst : Field F'] [inst_1 : Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K', IsKakeya K' ∧ ∀ (μ : Space F' d → ℝ), IsProbabilityMeasureOn K' μ → ∃ ξ, ξ ≠ 0 ∧ κ * (↑(Fintype.card F'))⁻¹ ≤ ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖ All goals completed! 🐙
#66
How produced

No Description

Monge–Kantorovich existence theorem
monge_kantorovich

Verso theorem preview

theorem declaration uses `sorry`monge_kantorovich_exists {X Y : Type*} [TopologicalSpace X] [PolishSpace X] [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [PolishSpace Y] [MeasurableSpace Y] [BorelSpace Y] (P : Measure X) (Q : Measure Y) [IsProbabilityMeasure P] [IsProbabilityMeasure Q] (c : X × Y → ENNReal) (_hc : Continuous c) : ∃ π ∈ LeanEval.Analysis.Couplings P Q, ∀ π' ∈ LeanEval.Analysis.Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' := X:Type u_1Y:Type u_2inst✝⁹:TopologicalSpace Xinst✝⁸:PolishSpace Xinst✝⁷:MeasurableSpace Xinst✝⁶:BorelSpace Xinst✝⁵:TopologicalSpace Yinst✝⁴:PolishSpace Yinst✝³:MeasurableSpace Yinst✝²:BorelSpace YP:Measure XQ:Measure Yinst✝¹:IsProbabilityMeasure Pinst✝:IsProbabilityMeasure Qc:X × Y → ENNReal_hc:Continuous c⊢ ∃ π ∈ Couplings P Q, ∀ π' ∈ Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' All goals completed! 🐙
#67
Solvable extensions ↔ solvable groups (the missing converse in Abel–Ruffini)
solvable_by_radicals_converse

Verso theorem preview

theorem declaration uses `sorry`solvable_iff_solvableByRad (F : Type*) [Field F] [CharZero F] (p : F[X]) (_hp : p ≠ 0) : (∀ x : AlgebraicClosure F, aeval x p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal := F:Type u_1inst✝¹:Field Finst✝:CharZero Fp:F[X]_hp:p ≠ 0⊢ (∀ (x : AlgebraicClosure F), (aeval x) p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal All goals completed! 🐙
#68
Local stable/unstable sets at a hyperbolic fixed point (set-level Hadamard–Perron)
stable_unstable_manifolds

Verso theorem preview

theorem declaration uses `sorry`stable_unstable_manifolds_exist (n : ℕ) (f : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (x₀ : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (_hf : ContDiffAt ℝ 1 f x₀) (_hfix : f x₀ = x₀) (_hhyp : LeanEval.Dynamics.StableUnstableManifoldsProblem.IsHyperbolicLinear (fderiv ℝ f x₀)) (_hf_inv : (fderiv ℝ f x₀).IsInvertible) : ∃ U : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), Ws = {x | (∀ k : ℕ, f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y : ℕ → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n, y 0 = x ∧ (∀ k : ℕ, y k ∈ U) ∧ (∀ k : ℕ, f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} := n:ℕf:E n → E nx₀:E n_hf:ContDiffAt ℝ 1 f x₀_hfix:f x₀ = x₀_hhyp:IsHyperbolicLinear (fderiv ℝ f x₀)_hf_inv:(fderiv ℝ f x₀).IsInvertible⊢ ∃ U, IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu, Ws = {x | (∀ (k : ℕ), f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y, y 0 = x ∧ (∀ (k : ℕ), y k ∈ U) ∧ (∀ (k : ℕ), f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} All goals completed! 🐙
#69
Lidskii's inequality
lidskii_inequality

Verso theorem preview

theorem declaration uses `sorry`lidskii_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) {p : ℝ} (_hp : 1 ≤ p) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |(hB.sub hA).eigenvalues₀ j| ^ p := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitianp:ℝ_hp:1 ≤ p⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |⋯.eigenvalues₀ j| ^ p All goals completed! 🐙
#70
How produced

_no_response_

Euler–Lagrange equation
euler_lagrange_equation

Verso theorem preview

theorem declaration uses `sorry`euler_lagrange_equation {a b : ℝ} (L : ℝ → ℝ → ℝ → ℝ) (x : ℝ → ℝ) (_hab : a < b) (_hL : ContDiff ℝ 2 (fun p : ℝ × ℝ × ℝ => L p.1 p.2.1 p.2.2)) (_hx : ContDiff ℝ 2 x) (_hxe : LeanEval.Analysis.IsVariationalExtremum a b L x) : ∀ t ∈ Set.Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t := a:ℝb:ℝL:ℝ → ℝ → ℝ → ℝx:ℝ → ℝ_hab:a < b_hL:ContDiff ℝ 2 fun p => L p.1 p.2.1 p.2.2_hx:ContDiff ℝ 2 x_hxe:IsVariationalExtremum a b L x⊢ ∀ t ∈ Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t All goals completed! 🐙
#71
Chen theorem for Markoff graphs
dvd_card_connectedComponent_markoffGraph

Lean theorem statement

/-- For prime `p > 3`, every connected component of the nonzero Markoff graph over `ZMod p`
has cardinality divisible by `p`. -/
theorem dvd_card_connectedComponent_markoffGraph
    {p : ℕ} (hp : Nat.Prime p) (hgt : 3 < p) :
    ∀ c : (markoffGraph p).ConnectedComponent, p ∣ Nat.card c := by
  sorry
#72
How produced

CI repair resubmission after extracting failed workflow logs. Root-workspace commit rebuilt locally with `lake build Submission`; forbidden-token scan clean; source tarball under 10 MiB.

A competition programming problem about permuting a permutation to be unimodal
permute_to_unimodal

Lean theorem statement

/--
`minRearrange` correctly computes the smallest number of indices that need to be permuted in order to
turn `arr` into a unimodal permutation.
-/
theorem minRearrange_correct {arr : Array Nat} :
    arr.Perm (1...=arr.size).toArray →
      (∃ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x ∧ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector = minRearrange arr) ∧
      (∀ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x → minRearrange arr ≤ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector) := by
  sorry
#73
How produced

_no_response_

Hippocrates' theorem on lunes
hippocrates_lunes

Verso theorem preview

theorem declaration uses `sorry`hippocrates_lunes (a b : ℝ) (ha : 0 < a) (hb : 0 < b) : volume (LeanEval.Geometry.HippocratesLunes.horizontalLune a b) + volume (LeanEval.Geometry.HippocratesLunes.verticalLune a b) = volume (LeanEval.Geometry.HippocratesLunes.rightTriangle a b) := a:ℝb:ℝha:0 < ahb:0 < b⊢ volume (horizontalLune a b) + volume (verticalLune a b) = volume (rightTriangle a b) All goals completed! 🐙
#74
How produced

_no_response_

Lidskii–Last eigenvalue-perturbation theorem
lidskii_last

Verso theorem preview

theorem declaration uses `sorry`lidskii_last {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitian⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ All goals completed! 🐙
#75
Frobenius's theorem: the Frobenius kernel is normal
frobenius_kernel_isNormal

Lean theorem statement

theorem frobenius_kernel_isNormal
    (G X : Type) [Group G] [Fintype G] [Fintype X]
    [MulAction G X] [FaithfulSMul G X]
    (hcard : 2 ≤ Fintype.card X)
    (htrans : ∀ x y : X, ∃ g : G, g • x = y)
    (hstab : ∀ x : X, MulAction.stabilizer G x ≠ ⊥)
    (hfrob : ∀ g : G, g ≠ 1 → ∀ x y : X, g • x = x → g • y = y → x = y) :
    ∃ N : Subgroup G, N.Normal ∧
      (N : Set G) = {1} ∪ {g : G | ∀ x : X, g • x ≠ x} := by
  sorry
#76
Brauer–Fowler theorem
brauer_fowler

Lean theorem statement

/-- **Brauer–Fowler theorem.** There is a function bounding the order
of a finite nonabelian simple group by the order of any involution
centralizer. -/
theorem brauer_fowler :
    ∃ f : ℕ → ℕ, ∀ (G : Type) [Group G] [Finite G],
      IsSimpleGroup G → (∃ a b : G, a * b ≠ b * a) →
      ∀ t : G, orderOf t = 2 →
        Nat.card G ≤ f (Nat.card (Subgroup.centralizer ({t} : Set G))) := by
  sorry
#77
How produced

No description

Sturm's theorem
sturm

Lean theorem statement

/-- **Sturm's theorem.** For a squarefree real polynomial `p` and an interval
`(a, b)` with `a < b` whose endpoints are not roots of `p`, the number of
distinct roots of `p` in `(a, b)` equals `σ(a) − σ(b)`. -/
theorem sturm (p : ℝ[X]) (hp : Squarefree p) {a b : ℝ} (hab : a < b)
    (ha : p.eval a ≠ 0) (hb : p.eval b ≠ 0) :
    ((p.roots.toFinset).filter (fun x => a < x ∧ x < b)).card =
      sigma p a - sigma p b := by
  sorry
#78
Independence of the parallel postulate
parallel_postulate_independent

Lean theorem statement

/-- **Independence of the parallel postulate** (Freek #12). The Euclidean
axiom `A10` is logically independent of Tarski's absolute axioms `A1`–`A9`
and `A11`: there is a model of the absolute axioms in which the parallel
postulate holds (the real coordinate plane) and one in which it fails (the
Klein–Beltrami disk, or any other hyperbolic-plane model). -/
theorem parallel_postulate_independent :
    (∃ (M : Type) (T : TarskiAbsolute M), Euclidean M T) ∧
    (∃ (M : Type) (T : TarskiAbsolute M), ¬ Euclidean M T) := by
  sorry
#79
Balanceable k-bounded partitions
balanceable_bounded_partitions

Lean theorem statement

/--
For any `k`, the smallest `n` such that any `k`-bounded partition of `n` is
also balanceable is given by `2 * lcm(1, ..., k)`.
-/
theorem minimal_balanceable_of_bounded (k : ℕ) (hk : 0 < k) :
    Minimal (fun n => 0 < n ∧ ∀ p : n.Partition, Bounded k p → Balanceable p) (2 * (Finset.Icc 1 k).lcm id) := by
  sorry
#80
Furstenberg–Weiss topological multiple recurrence (single-transformation form)
furstenberg_topological

Verso theorem preview

theorem declaration uses `sorry`furstenberg_topological_recurrence {X : Type*} [MetricSpace X] [CompactSpace X] [Nonempty X] (T : X ≃ₜ X) : ∃ x : X, LeanEval.Dynamics.IsMultiplyRecurrent (T : X → X) x := X:Type u_1inst✝²:MetricSpace Xinst✝¹:CompactSpace Xinst✝:Nonempty XT:X ≃ₜ X⊢ ∃ x, IsMultiplyRecurrent (⇑T) x All goals completed! 🐙
#81
Nash equilibrium existence theorem
nash_equilibrium_exists

Lean theorem statement

/-- **Nash equilibrium existence theorem.** Every finite `n`-player game
with nonempty finite pure-strategy sets and arbitrary real payoffs admits
at least one mixed-strategy Nash equilibrium. -/
theorem nash_equilibrium_exists {n : ℕ} {S : Fin n → Type*}
    [∀ i, Fintype (S i)] [∀ i, Nonempty (S i)]
    (u : Fin n → StrategyProfile n S → ℝ) :
    ∃ σ : ∀ i, S i → ℝ, IsNashEquilibrium u σ := by
  sorry
#82
Koszul formula
koszul_formula

Lean theorem statement

/-- **Koszul formula.** For any smooth torsion-free metric-compatible
covariant derivative `cov` on `TM`, `2 ⟨∇_X Y, Z⟩` equals the cyclic sum
of directional derivatives `X·⟨Y, Z⟩ + Y·⟨X, Z⟩ − Z·⟨X, Y⟩` minus the
Lie-bracket cyclic sum `⟨X, [Y, Z]⟩ + ⟨Y, [X, Z]⟩ − ⟨Z, [X, Y]⟩`. -/
theorem koszul_formula
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
      [FiniteDimensional ℝ E] [CompleteSpace E]
    {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H}
    {M : Type*} [TopologicalSpace M] [ChartedSpace H M]
      [IsManifold I ∞ M]
    [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)]
    [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)]
    (cov : CovariantDerivative I E (TangentSpace I (M := M)))
    [ContMDiffCovariantDerivative cov ∞]
    (_htor : cov.torsion = 0) (_hmet : IsMetricCompatible cov)
    (X Y Z : Π x : M, TangentSpace I x)
    (_hX : CMDiff ∞ (T% X)) (_hY : CMDiff ∞ (T% Y)) (_hZ : CMDiff ∞ (T% Z))
    (x : M) :
    2 * inner ℝ (cov Y x (X x)) (Z x) =
      mvfderiv I (fun y : M => inner ℝ (Y y) (Z y)) x (X x)
      + mvfderiv I (fun y : M => inner ℝ (X y) (Z y)) x (Y x)
      - mvfderiv I (fun y : M => inner ℝ (X y) (Y y)) x (Z x)
      - inner ℝ (X x) (mlieBracket I Y Z x)
      - inner ℝ (Y x) (mlieBracket I X Z x)
      + inner ℝ (Z x) (mlieBracket I X Y x) := by
  sorry
#83
Kakutani fixed-point theorem
kakutani_fixed_point

Lean theorem statement

/-- **Kakutani fixed-point theorem.** Every upper-hemicontinuous
correspondence `F` from a nonempty compact convex `K ⊆ ℝᵈ` to itself, with
nonempty convex closed values, has a fixed point `x ∈ F x`. -/
theorem kakutani_fixed_point {d : ℕ}
    {K : Set (EuclideanSpace ℝ (Fin d))}
    (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K)
    (_hK_nonempty : K.Nonempty)
    (F : EuclideanSpace ℝ (Fin d) → Set (EuclideanSpace ℝ (Fin d)))
    (_hF_uhc : IsUpperHemicontinuous F)
    (_hF_nonempty : ∀ x ∈ K, (F x).Nonempty)
    (_hF_convex : ∀ x ∈ K, Convex ℝ (F x))
    (_hF_closed : ∀ x ∈ K, IsClosed (F x))
    (_hF_maps : ∀ x ∈ K, F x ⊆ K) :
    ∃ x ∈ K, x ∈ F x := by
  sorry
#84
Wiener's atom-detection formula
wiener_atom_detection

Verso theorem preview

theorem declaration uses `sorry`wiener_atom_detection (μ : Measure (AddCircle (2 * Real.pi))) [IsProbabilityMeasure μ] : Tendsto (fun N : ℕ => (1 / (N : ℝ)) * ∑ k ∈ Finset.Icc (1 : ℤ) N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' x : AddCircle (2 * Real.pi), ((μ {x}).toReal) ^ 2)) := μ:Measure (AddCircle (2 * π))inst✝:IsProbabilityMeasure μ⊢ Tendsto (fun N => 1 / ↑N * ∑ k ∈ Finset.Icc 1 ↑N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' (x : AddCircle (2 * π)), (μ {x}).toReal ^ 2)) All goals completed! 🐙
#86
Kuznetsov's theorem: finitely presented simple groups have solvable word problem
boone_higman_simple

Verso theorem preview

theorem declaration uses `sorry`boone_higman_simple {G : Type*} [Group G] [IsSimpleGroup G] {n : ℕ} (φ : FreeGroup (Fin n) →* G) (_hsurj : Function.Surjective φ) (_hker : (MonoidHom.ker φ).IsNormalClosureFG) : LeanEval.GroupTheory.BooneHigmanSimpleProblem.WordProblemSolvable φ := G:Type u_1inst✝¹:Group Ginst✝:IsSimpleGroup Gn:ℕφ:FreeGroup (Fin n) →* G_hsurj:Function.Surjective ⇑φ_hker:φ.ker.IsNormalClosureFG⊢ WordProblemSolvable φ All goals completed! 🐙
#87
How produced

_no_response_

Runge's theorem
runge_theorem

Verso theorem preview

theorem declaration uses `sorry`runge (K : Set ℂ) (_hK : IsCompact K) (U : Set ℂ) (_hU : IsOpen U) (_hKU : K ⊆ U) (f : ℂ → ℂ) (_hf : AnalyticOnNhd ℂ f U) (ε : ℝ) (_hε : 0 < ε) : ∃ p q : ℂ[X], (∀ z ∈ K, q.eval z ≠ 0) ∧ (∀ z ∈ K, ‖f z - p.eval z / q.eval z‖ < ε) := K:Set ℂ_hK:IsCompact KU:Set ℂ_hU:IsOpen U_hKU:K ⊆ Uf:ℂ → ℂ_hf:AnalyticOnNhd ℂ f Uε:ℝ_hε:0 < ε⊢ ∃ p q, (∀ z ∈ K, Polynomial.eval z q ≠ 0) ∧ ∀ z ∈ K, ‖f z - Polynomial.eval z p / Polynomial.eval z q‖ < ε All goals completed! 🐙
#88
How produced

No Description

Schur-Weyl duality: S_k image equals centralizer of GL(V) image
symAction_range_eq_centralizer_glAction

Verso theorem preview

theorem declaration uses `sorry`symAction_range_eq_centralizer_glAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(symAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(glAction R M k)) All goals completed! 🐙
#89
How produced

_no_response_

Schur-Weyl duality: GL(V) image equals centralizer of S_k image
glAction_range_eq_centralizer_symAction

Verso theorem preview

theorem declaration uses `sorry`glAction_range_eq_centralizer_symAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(glAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(symAction R M k)) All goals completed! 🐙
#90
How produced

_no_response_

Baer–Suzuki theorem
baer_suzuki

Verso theorem preview

theorem declaration uses `sorry`baer_suzuki {G : Type*} [Group G] [Finite G] {p : ℕ} [Fact p.Prime] (x : G) : x ∈ LeanEval.GroupTheory.Defs.pCore p G ↔ ∀ g : G, IsPGroup p (Subgroup.closure ({x, g * x * g⁻¹} : Set G)) := G:Type u_1inst✝²:Group Ginst✝¹:Finite Gp:ℕinst✝:Fact (Nat.Prime p)x:G⊢ x ∈ pCore p G ↔ ∀ (g : G), IsPGroup p ↥(Subgroup.closure {x, g * x * g⁻¹}) All goals completed! 🐙
#91
How produced

_no_response_

Bourbaki's locally convex extension of Banach–Alaoglu
banach_alaoglu_bourbaki

Verso theorem preview

theorem declaration uses `sorry`banach_alaoglu_bourbaki (E : Type*) [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] (U : Set E) (_hU : U ∈ 𝓝 (0 : E)) : IsCompact (LeanEval.Analysis.weakStarPolar E U) := E:Type u_1inst✝⁵:AddCommGroup Einst✝⁴:Module ℝ Einst✝³:TopologicalSpace Einst✝²:ContinuousAdd Einst✝¹:ContinuousSMul ℝ Einst✝:LocallyConvexSpace ℝ EU:Set E_hU:U ∈ 𝓝 0⊢ IsCompact (weakStarPolar E U) All goals completed! 🐙
#92
How produced

No Description

Linear programming: maximum principle and vertex optimality
lp_maximum_principle

Verso theorem preview

/-- **Maximum principle for linear programming** (§101). A local maximiser of the LP objective on the feasible region is automatically a global maximiser; and whenever the objective is non-constant (`c ≠ 0`), the maximiser lies on the topological frontier of the feasible region. -/ theorem declaration uses `sorry`lp_maximum_principle {m n : ℕ} (lp : LinearProgram m n) (x : Fin m → ℝ) (_hx : x ∈ lp.feasible) (_hlocal : IsLocalMaxOn lp.objective lp.feasible x) : IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) := m:ℕn:ℕlp:LinearProgram m nx:Fin m → ℝ_hx:x ∈ lp.feasible_hlocal:IsLocalMaxOn lp.objective lp.feasible x⊢ IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) All goals completed! 🐙
/-- **Vertex optimality** (§101; the existence content of Dantzig's 1947 simplex algorithm). Every linear program with a nonempty bounded feasible region admits a global maximiser that is an extreme point (vertex) of the feasible region. -/ theorem declaration uses `sorry`simplex_algorithm {m n : ℕ} (lp : LinearProgram m n) (_hfeas : lp.feasible.Nonempty) (_hbdd : Bornology.IsBounded lp.feasible) : ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible := m:ℕn:ℕlp:LinearProgram m n_hfeas:lp.feasible.Nonempty_hbdd:Bornology.IsBounded lp.feasible⊢ ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible All goals completed! 🐙
#93
Symplectic matrices have determinant 1
symplectic_matrix_det

Verso theorem preview

theorem declaration uses `sorry`symplectic_matrix_det {l R : Type*} [DecidableEq l] [Fintype l] [CommRing R] {A : Matrix (l ⊕ l) (l ⊕ l) R} (_hA : A ∈ Matrix.symplecticGroup l R) : A.det = 1 := l:Type u_1R:Type u_2inst✝²:DecidableEq linst✝¹:Fintype linst✝:CommRing RA:Matrix (l ⊕ l) (l ⊕ l) R_hA:A ∈ symplecticGroup l R⊢ A.det = 1 All goals completed! 🐙
#94
Brouwer fixed-point theorem
brouwer_fixed_point

Verso theorem preview

theorem declaration uses `sorry`brouwer_fixed_point {d : ℕ} {K : Set (EuclideanSpace ℝ (Fin d))} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (_hf_cont : ContinuousOn f K) (_hf_maps : MapsTo f K K) : ∃ x ∈ K, f x = x := d:ℕK:Set (EuclideanSpace ℝ (Fin d))_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)_hf_cont:ContinuousOn f K_hf_maps:MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#95
Schauder fixed-point theorem
schauder_fixed_point

Verso theorem preview

theorem declaration uses `sorry`schauder_fixed_point {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {K : Set E} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : E → E) (_hf_cont : ContinuousOn f K) (_hf_maps : Set.MapsTo f K K) : ∃ x ∈ K, f x = x := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace EK:Set E_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:E → E_hf_cont:ContinuousOn f K_hf_maps:Set.MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#96
Abel–Ruffini theorem
abel_ruffini

Verso theorem preview

theorem declaration uses `sorry`abel_ruffini (n : ℕ) (_hn : 1 ≤ n) : (∀ p : ℚ[X], p.natDegree = n → ∀ x : ℂ, aeval x p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 := n:ℕ_hn:1 ≤ n⊢ (∀ (p : ℚ[X]), p.natDegree = n → ∀ (x : ℂ), (aeval x) p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 All goals completed! 🐙
#97
von Neumann double commutant theorem
vonNeumann_doubleCommutant_tfae

Verso theorem preview

theorem declaration uses `sorry`vonNeumann_doubleCommutant_tfae {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : StarSubalgebra ℂ (H →L[ℂ] H)) : List.TFAE [ Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S , IsClosed (ContinuousLinearMapWOT.ofCLM '' (S : Set (H →L[ℂ] H))) , IsClosed (ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H '' (S : Set (H →L[ℂ] H))) ] := H:Type u_1inst✝²:NormedAddCommGroup Hinst✝¹:InnerProductSpace ℂ Hinst✝:CompleteSpace HS:StarSubalgebra ℂ (H →L[ℂ] H)⊢ [(↑S).centralizer.centralizer = ↑S, IsClosed (ContinuousLinearMapWOT.ofCLM '' ↑S), IsClosed (⇑(ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H) '' ↑S)].TFAE All goals completed! 🐙
#98
How produced

_no_response_

Pointwise and Cesàro convergence of Fourier series (Dirichlet, Fejér)
fourier_dirichlet_fejer

Verso theorem preview

/-- **Dirichlet's pointwise convergence theorem** (§46). For every `C¹` 2π-periodic complex function `f`, the symmetric Fourier partial sums `S_N(f)(x)` converge to `f(x)` at every point `x ∈ ℝ`. -/ theorem declaration uses `sorry`dirichlet_pointwise {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hC1 : ContDiff ℝ 1 f) (x : ℝ) : Tendsto (fun N : ℕ => fourierPartialSum f N x) atTop (𝓝 (f x)) := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hC1:ContDiff ℝ 1 fx:ℝ⊢ Tendsto (fun N => fourierPartialSum f N x) atTop (𝓝 (f x)) All goals completed! 🐙
/-- **Fejér's theorem** (§46). For every *continuous* 2π-periodic complex function `f` — without the `C¹` hypothesis of Dirichlet's theorem — the Cesàro means `σ_N(f)` of the symmetric Fourier partial sums converge to `f` uniformly on `ℝ`. -/ theorem declaration uses `sorry`fejer {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hcont : Continuous f) : TendstoUniformly (fun N : ℕ => fourierCesaroMean f N) f atTop := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hcont:Continuous f⊢ TendstoUniformly (fun N => fourierCesaroMean f N) f atTop All goals completed! 🐙
#99
How produced

_no_response_

Bing's house with two rooms is contractible
contractibleSpace_houseWithTwoRooms

Verso theorem preview

theorem declaration uses `sorry`contractibleSpace_houseWithTwoRooms : ContractibleSpace LeanEval.Topology.HouseWithTwoRooms := ⊢ ContractibleSpace ↑HouseWithTwoRooms All goals completed! 🐙
#100
How produced

_no_response_

Existence of a non-isotopic pair of oriented two-component links
exists_nonisotopic_link

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_link : ∃ L₁ L₂ : LeanEval.KnotTheory.TwoLink, ¬ L₁.Isotopic L₂ := ⊢ ∃ L₁ L₂, ¬L₁.Isotopic L₂ All goals completed! 🐙
#101
How produced

_no_response_

Pell solutions are convergents of √d
pell_solution_convergent

Verso theorem preview

theorem declaration uses `sorry`pell_solution_is_convergent (d : ℤ) (_hd : Squarefree d) (_hd0 : 0 < d) (x y : ℤ) (_hx : 0 < x) (_hy : 0 < y) (_hsol : x ^ 2 - d * y ^ 2 = 1) : ∃ n : ℕ, (GenContFract.of (Real.sqrt (d : ℝ))).convs n = (x : ℝ) / (y : ℝ) := d:ℤ_hd:Squarefree d_hd0:0 < dx:ℤy:ℤ_hx:0 < x_hy:0 < y_hsol:x ^ 2 - d * y ^ 2 = 1⊢ ∃ n, (GenContFract.of √↑d).convs n = ↑x / ↑y All goals completed! 🐙
#102
Burnside p^a q^b theorem
finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow

Verso theorem preview

theorem declaration uses `sorry`finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow {G : Type*} [Group G] [Fintype G] {p q a b : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q) (hpq : p ≠ q) (hcard : Fintype.card G = p ^ a * q ^ b) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Fintype Gp:ℕq:ℕa:ℕb:ℕhp:Nat.Prime phq:Nat.Prime qhpq:p ≠ qhcard:Fintype.card G = p ^ a * q ^ b⊢ IsSolvable G All goals completed! 🐙
#103
Real cyclotomic integer with house at most 2
cyclotomic_integer_house_le_two

Verso theorem preview

theorem declaration uses `sorry`cyclotomic_integer_house_le_two {K : Type*} [Field K] [NumberField K] [Algebra ℚ K] (n : ℕ) [NeZero n] [IsCyclotomicExtension {n} ℚ K] {β : K} (hβ_int : IsIntegral ℤ β) (hβ_real : β ∈ NumberField.maximalRealSubfield K) : house β ≤ 2 → house β = 2 ∨ ∃ m : ℕ, 0 < m ∧ house β = 2 * Real.cos (Real.pi / m) := K:Type u_1inst✝⁴:Field Kinst✝³:NumberField Kinst✝²:Algebra ℚ Kn:ℕinst✝¹:NeZero ninst✝:IsCyclotomicExtension {n} ℚ Kβ:Khβ_int:IsIntegral ℤ βhβ_real:β ∈ maximalRealSubfield K⊢ house β ≤ 2 → house β = 2 ∨ ∃ m, 0 < m ∧ house β = 2 * Real.cos (Real.pi / ↑m) All goals completed! 🐙
#104
How produced

Corrected model label for the comparator-accepted private submission of cyclotomic_integer_house_le_two. Same verified pinned commit as issue #234; the previous submission used the wrong model label.

Perron-Frobenius for irreducible nonnegative matrices
irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius

Verso theorem preview

theorem declaration uses `sorry`irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius {n : Type*} [Fintype n] [DecidableEq n] [Nonempty n] (A : Matrix n n ℝ) (hA : A.IsIrreducible) : ∃ v : n → ℝ, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ (∀ i, 0 < v i) := n:Type u_1inst✝²:Fintype ninst✝¹:DecidableEq ninst✝:Nonempty nA:Matrix n n ℝhA:A.IsIrreducible⊢ ∃ v, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ ∀ (i : n), 0 < v i All goals completed! 🐙
#105
Linear ODE with negative-real-part eigenvalues is asymptotically stable
linear_ode_asymptotic_stability

Verso theorem preview

theorem declaration uses `sorry`linear_ode_asymptotic_stability (n : ℕ) (A : Matrix (Fin n) (Fin n) ℝ) (hA : ∀ μ : ℂ, Module.End.HasEigenvalue (Matrix.toLin' (A.map (algebraMap ℝ ℂ))) μ → μ.re < 0) (x : ℝ → (Fin n → ℝ)) (hx : ∀ t : ℝ, 0 < t → HasDerivAt x (A.mulVec (x t)) t) : Filter.Tendsto (fun t : ℝ => ‖x t‖) Filter.atTop (nhds 0) := n:ℕA:Matrix (Fin n) (Fin n) ℝhA:∀ (μ : ℂ), Module.End.HasEigenvalue (Matrix.toLin' (A.map ⇑(algebraMap ℝ ℂ))) μ → μ.re < 0x:ℝ → Fin n → ℝhx:∀ (t : ℝ), 0 < t → HasDerivAt x (A *ᵥ x t) t⊢ Filter.Tendsto (fun t => ‖x t‖) Filter.atTop (nhds 0) All goals completed! 🐙
#106
How produced

_no_response_

Rouche theorem via zero counting
rouche_zero_count_eq

Verso theorem preview

theorem declaration uses `sorry`rouche_zero_count_eq {f g : ℂ → ℂ} {R : ℝ} (hR : 0 < R) (hf : MeromorphicNFOn f Set.univ) (hg : AnalyticOn ℂ g Set.univ) (hbound : ∀ z : ℂ, ‖z‖ = R → ‖g z‖ < ‖f z‖) : (∑ᶠ z, ((divisor (f + g) (Metric.closedBall 0 R))⁺) z) = (∑ᶠ z, ((divisor f (Metric.closedBall 0 R))⁺) z) := f:ℂ → ℂg:ℂ → ℂR:ℝhR:0 < Rhf:MeromorphicNFOn f Set.univhg:AnalyticOn ℂ g Set.univhbound:∀ (z : ℂ), ‖z‖ = R → ‖g z‖ < ‖f z‖⊢ ∑ᶠ (z : ℂ), (divisor (f + g) (Metric.closedBall 0 R))⁺ z = ∑ᶠ (z : ℂ), (divisor f (Metric.closedBall 0 R))⁺ z All goals completed! 🐙
#107
Complementary polynomial on the unit circle
exists_complementary_polynomial_on_unit_circle

Verso theorem preview

theorem declaration uses `sorry`exists_complementary_polynomial_on_unit_circle (P : ℂ[X]) (hP : ∀ z : Circle, ‖P.eval (z : ℂ)‖ ≤ 1) : ∃ Q : ℂ[X], Q.natDegree ≤ P.natDegree ∧ ∀ z : Circle, ‖P.eval (z : ℂ)‖ ^ 2 + ‖Q.eval (z : ℂ)‖ ^ 2 = 1 := P:ℂ[X]hP:∀ (z : Circle), ‖eval (↑z) P‖ ≤ 1⊢ ∃ Q, Q.natDegree ≤ P.natDegree ∧ ∀ (z : Circle), ‖eval (↑z) P‖ ^ 2 + ‖eval (↑z) Q‖ ^ 2 = 1 All goals completed! 🐙
#108
How produced

No Description

Character values of finite groups lie in cyclotomic fields
brauer_character_in_cyclotomic

Verso theorem preview

theorem declaration uses `sorry`brauer_character_in_cyclotomic (G : Type) [Group G] [Fintype G] : ∃ φ : CyclotomicField (Monoid.exponent G) ℚ →+* ℂ, ∀ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (_ : FiniteDimensional ℂ V) (ρ : Representation ℂ G V) (g : G), LinearMap.trace ℂ V (ρ g) ∈ φ.range := G:Typeinst✝¹:Group Ginst✝:Fintype G⊢ ∃ φ, ∀ (V : Type) (x : AddCommGroup V) (x_1 : Module ℂ V), FiniteDimensional ℂ V → ∀ (ρ : Representation ℂ G V) (g : G), (LinearMap.trace ℂ V) (ρ g) ∈ φ.range All goals completed! 🐙
#109
How produced

_no_response_

Gaussian heat kernel solves the 1D heat equation
heat_kernel_solves_heat_equation

Verso theorem preview

theorem declaration uses `sorry`heat_kernel_solves_heat_equation (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_bdd : ∃ M : ℝ, ∀ x, |f x| ≤ M) : -- The PDE on (0, ∞) × ℝ. (∀ t : ℝ, 0 < t → ∀ x : ℝ, ∃ ux : ℝ → ℝ, ∃ uxx : ℝ, (∀ y : ℝ, HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ -- Initial condition recovered as a one-sided limit at t = 0. (∀ x : ℝ, Filter.Tendsto (fun t : ℝ => heatSolution f t x) (nhdsWithin (0 : ℝ) (Set.Ioi 0)) (nhds (f x))) := f:ℝ → ℝhf_cont:Continuous fhf_bdd:∃ M, ∀ (x : ℝ), |f x| ≤ M⊢ (∀ (t : ℝ), 0 < t → ∀ (x : ℝ), ∃ ux uxx, (∀ (y : ℝ), HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ ∀ (x : ℝ), Filter.Tendsto (fun t => heatSolution f t x) (nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) All goals completed! 🐙
#110
How produced

No Description

Oppenheim's inequality for Hadamard products
oppenheim_inequality

Verso theorem preview

theorem declaration uses `sorry`oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : A.det * ∏ i, B i i ≤ (A ⊙ B).det := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝB:Matrix n n ℝhA:A.PosSemidefhB:B.PosSemidef⊢ A.det * ∏ i, B i i ≤ (A ⊙ B).det All goals completed! 🐙
#111
How produced

No Description

Minkowski-Caratheodory theorem
mem_convexHull_finset_extremePoints_of_mem_compact_convex

Verso theorem preview

theorem declaration uses `sorry`mem_convexHull_finset_extremePoints_of_mem_compact_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hscomp : IsCompact s) (hsconv : Convex ℝ s) (hx : x ∈ s) : ∃ t : Finset E, (↑t : Set E) ⊆ s.extremePoints ℝ ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ convexHull ℝ (↑t : Set E) := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:FiniteDimensional ℝ Es:Set Ex:Ehscomp:IsCompact shsconv:Convex ℝ shx:x ∈ s⊢ ∃ t, ↑t ⊆ extremePoints ℝ s ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ (convexHull ℝ) ↑t All goals completed! 🐙
#112
How produced

No Description

Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2
dirichlet_eigenvalues_eq_nat_sq

Verso theorem preview

theorem declaration uses `sorry`dirichlet_eigenvalues_eq_nat_sq (lam : ℝ) : (∃ (y : ℝ → ℝ) (J : Set ℝ), IsOpen J ∧ Set.Icc (0 : ℝ) Real.pi ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y Real.pi = 0 ∧ ∃ x ∈ Set.Ioo (0 : ℝ) Real.pi, y x ≠ 0) ↔ ∃ n : ℕ, 0 < n ∧ lam = (n : ℝ) ^ 2 := lam:ℝ⊢ (∃ y J, IsOpen J ∧ Set.Icc 0 π ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y π = 0 ∧ ∃ x ∈ Set.Ioo 0 π, y x ≠ 0) ↔ ∃ n, 0 < n ∧ lam = ↑n ^ 2 All goals completed! 🐙
#113
How produced

_no_response_

Entrywise exponential of a PSD matrix is PSD
posSemidef_map_exp

Verso theorem preview

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n] {A : Matrix n n ℝ} (hA : A.PosSemidef) : (A.map Real.exp).PosSemidef := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef All goals completed! 🐙
#114
How produced

_no_response_

pi_1 of the circle is Z
pi1_circle_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi1_circle_mulEquiv_int : Nonempty (HomotopyGroup.Pi 1 Circle (1 : Circle) ≃* Multiplicative ℤ) := ⊢ Nonempty (HomotopyGroup.Pi 1 Circle 1 ≃* Multiplicative ℤ) All goals completed! 🐙
#115
Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#116
How produced

_no_response_

Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#117
Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#120
How produced

_no_response_

Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#123
Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#124
Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#125
How produced

No Description

Test problems: multi_hole_helpers_example, noncomputable_hole_example, variable_binder_example, def_hole_example, instance_hole_example, list_append_singleton_length, ci_regenerate_main_check, two_plus_two (8 / 8 solved)

Submission history

First submissionMay 8, 2026
Last submissionJul 22, 2026

Contributors

rishistyping128
4GPT-5.6112 solved

Other solved problems

A 3-manifold group with no faithful representation into GL(4, ℝ)
nonlinear_three_manifold_group

Verso theorem preview

theorem declaration uses `sorry`nonlinear_three_manifold_group : ∃ (M : LeanEval.Topology.Closed3Manifold) (x : M.carrier), ∀ f : FundamentalGroup M.carrier x →* GL (Fin 4) ℝ, ¬ Function.Injective f := ⊢ ∃ M x, ∀ (f : FundamentalGroup M.carrier x →* GL (Fin 4) ℝ), ¬Function.Injective ⇑f All goals completed! 🐙
#1
The Hopf Umlaufsatz (theorem of turning tangents)
hopf_umlaufsatz

Verso theorem preview

theorem declaration uses `sorry`hopf_umlaufsatz {r : ℝ → LeanEval.Geometry.HopfUmlaufsatz.Plane} {α : ℝ → ℝ} (_hr : LeanEval.Geometry.HopfUmlaufsatz.IsPositiveSimpleClosedUnitSpeedCurve r) (_hα : LeanEval.Geometry.HopfUmlaufsatz.IsTangentAngleLift r α) : totalCurvature α = 2 * Real.pi := r:ℝ → Planeα:ℝ → ℝ_hr:IsPositiveSimpleClosedUnitSpeedCurve r_hα:IsTangentAngleLift r α⊢ totalCurvature α = 2 * Real.pi All goals completed! 🐙
#2
Morrison–Walker Lemma B.0.1: adapting families of maps to open covers
families_of_maps_b01

Verso theorem preview

/-- **Lemma B.0.1** of Morrison–Walker, *The Blob Complex* (arXiv:1009.5025, §B), continuous case. -/ theorem declaration uses `sorry`continuous {P : Set (Fin k → ℝ)} (_hP : IsPolyhedron P) [CompactSpace X] (U : ι → Set X) (_hUopen : ∀ α, IsOpen (U α)) (ρ : PartitionOfUnity ι X univ) (_hρ : ρ.IsSubordinate U) (f : C(P × X, T)) : ∃ F : C(I × P × X, T), (∀ p : P, ∀ x : X, F (0, p, x) = f (p, x)) ∧ (∃ K : Subdivision P, ∀ D ∈ K.complex.facets, AdaptedTo U k (fun q : closedCell P D × X => F (1, q.1.1, q.2))) ∧ (∀ S : Set X, Supported (f := f.toFun) S → Supported (fun q : (I × P) × X => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ Q : Set P, IsBoundarySubpolyhedron Q → ∀ S' : Set X, Supported (fun q : Q × X => f (q.1.1, q.2)) S' → Supported (fun q : (I × Q) × X => F (q.1.1, q.1.2.1, q.2)) S') := k:ℕι:Type u_1X:Type u_2T:Type u_3inst✝²:TopologicalSpace Xinst✝¹:TopologicalSpace TP:Set (Fin k → ℝ)_hP:IsPolyhedron Pinst✝:CompactSpace XU:ι → Set X_hUopen:∀ (α : ι), IsOpen[inst✝²] (U α)ρ:PartitionOfUnity ι X_hρ:ρ.IsSubordinate Uf:C(↑P × X, T)⊢ ∃ F, (∀ (p : ↑P) (x : X), F (0, p, x) = f (p, x)) ∧ (∃ K, ∀ D ∈ K.complex.facets, AdaptedTo U k fun q => F (1, ↑q.1, q.2)) ∧ (∀ (S : Set X), Supported f.toFun S → Supported (fun q => F (q.1.1, q.1.2, q.2)) S) ∧ ∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ (S' : Set X), Supported (fun q => f (↑q.1, q.2)) S' → Supported (fun q => F (q.1.1, ↑q.1.2, q.2)) S' All goals completed! 🐙
/-- **Lemma B.0.1**, bi-Lipschitz variant (part 4 of the paper). -/ theorem declaration uses `sorry`biLipschitz {X T : Type*} [MetricSpace X] [MetricSpace T] [CompactSpace X] {P : Set (Fin k → ℝ)} (_hP : IsPolyhedron P) {ι : Type*} (U : ι → Set X) (_hUopen : ∀ α, IsOpen (U α)) (ρ : PartitionOfUnity ι X univ) (_hρ : ρ.IsSubordinate U) (f : C(P × X, T)) (slice : P → (X ≃ₜ T)) (_h_slice_eq : ∀ p : P, ∀ x : X, f (p, x) = slice p x) (L : NNReal) (_hf_joint : LipschitzWith L f.toFun) (_hf_slice_inv : ∀ p : P, LipschitzWith L (slice p).symm) : ∃ F : C(I × P × X, T), ∃ L' : NNReal, ∃ Slice : I × P → (X ≃ₜ T), (∀ p : P, ∀ x : X, F (0, p, x) = f (p, x)) ∧ (∀ t : I, ∀ p : P, ∀ x : X, F (t, p, x) = Slice (t, p) x) ∧ (∃ K : Subdivision P, ∀ D ∈ K.complex.facets, AdaptedTo U k (fun q : closedCell P D × X => F (1, q.1.1, q.2))) ∧ (∀ S : Set X, Supported (f := f.toFun) S → Supported (fun q : (I × P) × X => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ Q : Set P, IsBoundarySubpolyhedron Q → ∀ S' : Set X, Supported (fun q : Q × X => f (q.1.1, q.2)) S' → Supported (fun q : (I × Q) × X => F (q.1.1, q.1.2.1, q.2)) S') ∧ (∀ tp : I × P, LipschitzWith L' (Slice tp)) ∧ (∀ tp : I × P, LipschitzWith L' (Slice tp).symm) := k:ℕX:Type u_4T:Type u_5inst✝²:MetricSpace Xinst✝¹:MetricSpace Tinst✝:CompactSpace XP:Set (Fin k → ℝ)_hP:IsPolyhedron Pι:Type u_6U:ι → Set X_hUopen:∀ (α : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (U α)ρ:PartitionOfUnity ι X_hρ:ρ.IsSubordinate Uf:C(↑P × X, T)slice:↑P → X ≃ₜ T_h_slice_eq:∀ (p : ↑P) (x : X), f (p, x) = (slice p) xL:NNReal_hf_joint:LipschitzWith L f.toFun_hf_slice_inv:∀ (p : ↑P), LipschitzWith L ⇑(slice p).symm⊢ ∃ F L' Slice, (∀ (p : ↑P) (x : X), F (0, p, x) = f (p, x)) ∧ (∀ (t : ↑I) (p : ↑P) (x : X), F (t, p, x) = (Slice (t, p)) x) ∧ (∃ K, ∀ D ∈ K.complex.facets, AdaptedTo U k fun q => F (1, ↑q.1, q.2)) ∧ (∀ (S : Set X), Supported f.toFun S → Supported (fun q => F (q.1.1, q.1.2, q.2)) S) ∧ (∀ (Q : Set ↑P), IsBoundarySubpolyhedron Q → ∀ (S' : Set X), Supported (fun q => f (↑q.1, q.2)) S' → Supported (fun q => F (q.1.1, ↑q.1.2, q.2)) S') ∧ (∀ (tp : ↑I × ↑P), LipschitzWith L' ⇑(Slice tp)) ∧ ∀ (tp : ↑I × ↑P), LipschitzWith L' ⇑(Slice tp).symm All goals completed! 🐙
#3
The Golod–Shafarevich inequality
golod_shafarevich_inequality

Verso theorem preview

theorem declaration uses `sorry`golod_shafarevich_inequality (p : ℕ) [Fact p.Prime] (Q : Type) [Group Q] [TopologicalSpace Q] [IsTopologicalGroup Q] [DiscreteTopology Q] [Finite Q] : IsPGroup p Q → Nontrivial Q → (generatorRank Q : ℝ) ^ 2 < 4 * (relationRank p Q : ℝ) := p:ℕinst✝⁵:Fact (Nat.Prime p)Q:Typeinst✝⁴:Group Qinst✝³:TopologicalSpace Qinst✝²:IsTopologicalGroup Qinst✝¹:DiscreteTopology Qinst✝:Finite Q⊢ IsPGroup p Q → Nontrivial Q → ↑(generatorRank Q) ^ 2 < 4 * ↑(relationRank p Q) All goals completed! 🐙
#4
Halmos's generic weak-mixing theorem
halmos_generic_weak_mixing

Verso theorem preview

theorem declaration uses `sorry`generic_weakly_mixing [StandardBorelSpace X] (m : Measure X) [IsProbabilityMeasure m] [NoAtoms m] : (∃ G : Set (LeanEval.Dynamics.HalmosGenericWeakMixingProblem.Automorphism m), IsGδ G ∧ Dense G ∧ ∀ T ∈ G, LeanEval.Dynamics.HalmosGenericWeakMixingProblem.IsWeaklyMixing m T) ∧ (∀ T : LeanEval.Dynamics.HalmosGenericWeakMixingProblem.Automorphism m, LeanEval.Dynamics.HalmosGenericWeakMixingProblem.IsWeaklyMixing m T → Ergodic (T.toEquiv : X → X) m) := X:Type u_1inst✝³:MeasurableSpace Xinst✝²:StandardBorelSpace Xm:Measure Xinst✝¹:IsProbabilityMeasure minst✝:NoAtoms m⊢ (∃ G, IsGδ G ∧ Dense G ∧ ∀ T ∈ G, IsWeaklyMixing m T) ∧ ∀ (T : Automorphism m), IsWeaklyMixing m T → Ergodic (⇑T.toEquiv) m All goals completed! 🐙
#5
Cauchy–Kovalevskaya theorem
cauchy_kovalevskaya

Verso theorem preview

theorem declaration uses `sorry`cauchy_kovalevskaya {d : ℕ} (F : LeanEval.Analysis.E d × ℝ × ℝ → LeanEval.Analysis.E d) (f : LeanEval.Analysis.E d × ℝ × ℝ → ℝ) (u₀ : LeanEval.Analysis.E d → ℝ) (_hF : AnalyticOnNhd ℝ F univ) (_hf : AnalyticOnNhd ℝ f univ) (_hu₀ : AnalyticOnNhd ℝ u₀ univ) (x₀ : LeanEval.Analysis.E d) : ∃ (U : Set (LeanEval.Analysis.E d × ℝ)) (u : LeanEval.Analysis.E d × ℝ → ℝ), (x₀, (0 : ℝ)) ∈ U ∧ IsOpen U ∧ AnalyticOnNhd ℝ u U ∧ (∀ x : LeanEval.Analysis.E d, (x, (0 : ℝ)) ∈ U → u (x, 0) = u₀ x) ∧ (∀ p ∈ U, fderiv ℝ u p ((0 : LeanEval.Analysis.E d), (1 : ℝ)) = fderiv ℝ u p (F (p.1, p.2, u p), (0 : ℝ)) + f (p.1, p.2, u p)) ∧ (∀ v : LeanEval.Analysis.E d × ℝ → ℝ, AnalyticOnNhd ℝ v U → (∀ x : LeanEval.Analysis.E d, (x, (0 : ℝ)) ∈ U → v (x, 0) = u₀ x) → (∀ p ∈ U, fderiv ℝ v p ((0 : LeanEval.Analysis.E d), (1 : ℝ)) = fderiv ℝ v p (F (p.1, p.2, v p), (0 : ℝ)) + f (p.1, p.2, v p)) → ∀ p ∈ U, u p = v p) := d:ℕF:E d × ℝ × ℝ → E df:E d × ℝ × ℝ → ℝu₀:E d → ℝ_hF:AnalyticOnNhd ℝ F univ_hf:AnalyticOnNhd ℝ f univ_hu₀:AnalyticOnNhd ℝ u₀ univx₀:E d⊢ ∃ U u, (x₀, 0) ∈ U ∧ IsOpen U ∧ AnalyticOnNhd ℝ u U ∧ (∀ (x : E d), (x, 0) ∈ U → u (x, 0) = u₀ x) ∧ (∀ p ∈ U, (fderiv ℝ u p) (0, 1) = (fderiv ℝ u p) (F (p.1, p.2, u p), 0) + f (p.1, p.2, u p)) ∧ ∀ (v : E d × ℝ → ℝ), AnalyticOnNhd ℝ v U → (∀ (x : E d), (x, 0) ∈ U → v (x, 0) = u₀ x) → (∀ p ∈ U, (fderiv ℝ v p) (0, 1) = (fderiv ℝ v p) (F (p.1, p.2, v p), 0) + f (p.1, p.2, v p)) → ∀ p ∈ U, u p = v p All goals completed! 🐙
#6
Existence of a non-isotopic pair of oriented knots
exists_nonisotopic_knots

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_knots : ∃ K₁ K₂ : LeanEval.KnotTheory.Knot, ¬ K₁.Isotopic K₂ := ⊢ ∃ K₁ K₂, ¬K₁.Isotopic K₂ All goals completed! 🐙
#7
Feit–Thompson odd-order theorem
feit_thompson

Verso theorem preview

theorem declaration uses `sorry`feit_thompson {G : Type*} [Group G] [Finite G] (_h : Odd (Nat.card G)) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Finite G_h:Odd (Nat.card G)⊢ IsSolvable G All goals completed! 🐙
#8
How produced

The solution was produced using GPT-5.6 Sol in Codex, based on the existing formalization of the theorem in Coq. The LLM worked autonomously for several days, with minimal steering to keep the work productive and on target. The entire work was paid for with a ChatGPT Pro subscription (with 2 available usage resets). The repo also includes submissions to `baer_suzuki` and `brauer_character_in_cyclotomic` as derivatives of the main effort on `feit_thompson`.

Ornstein–Weiss ℤᵈ Rokhlin lemma
ornstein_weiss_rokhlin

Verso theorem preview

theorem declaration uses `sorry`ornstein_weiss_rokhlin {Ω : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω] {d : ℕ} (_hd : 1 ≤ d) (μ : Measure Ω) [IsProbabilityMeasure μ] (T : (Fin d → ℤ) → Ω → Ω) (_hid : ∀ x, T 0 x = x) (_hT : ∀ v, MeasurePreserving (T v) μ μ) (_hgrp : ∀ u v x, T (u + v) x = T u (T v x)) (_hfree : LeanEval.Dynamics.IsFreeAction μ T) (N : ℕ) (_hN : 1 ≤ N) {ε : ENNReal} (_hε : 0 < ε) : ∃ B : Set Ω, MeasurableSet B ∧ ((boxShape d N : Finset (Fin d → ℤ)) : Set (Fin d → ℤ)).PairwiseDisjoint (fun v => T v '' B) ∧ μ (⋃ v ∈ boxShape d N, T v '' B) ≥ 1 - ε := Ω:Type u_1inst✝²:MeasurableSpace Ωinst✝¹:StandardBorelSpace Ωd:ℕ_hd:1 ≤ dμ:Measure Ωinst✝:IsProbabilityMeasure μT:(Fin d → ℤ) → Ω → Ω_hid:∀ (x : Ω), T 0 x = x_hT:∀ (v : Fin d → ℤ), MeasurePreserving (T v) μ μ_hgrp:∀ (u v : Fin d → ℤ) (x : Ω), T (u + v) x = T u (T v x)_hfree:IsFreeAction μ TN:ℕ_hN:1 ≤ Nε:ENNReal_hε:0 < ε⊢ ∃ B, MeasurableSet B ∧ ((↑(boxShape d N)).PairwiseDisjoint fun v => T v '' B) ∧ μ (⋃ v ∈ boxShape d N, T v '' B) ≥ 1 - ε All goals completed! 🐙
#9
Wigner semicircle law
wigner_semicircle

Verso theorem preview

theorem declaration uses `sorry`wigner_semicircle {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → ℕ → Ω → ℝ) (_hX_meas : ∀ i j, Measurable (X i j)) (_hX_indep : iIndepFun (fun ij : {p : ℕ × ℕ // p.1 ≤ p.2} => X ij.val.1 ij.val.2) μ) (_hX_iid : ∀ i j i' j', i ≤ j → i' ≤ j' → ProbabilityTheory.IdentDistrib (X i j) (X i' j') μ μ) (_hX_int : ∀ i j, i ≤ j → Integrable (X i j) μ) (_hX_sq_int : ∀ i j, i ≤ j → Integrable (fun ω => (X i j ω) ^ 2) μ) (_hX_mean : ∀ i j, i ≤ j → ∫ ω, X i j ω ∂μ = 0) (_hX_var : ∀ i j, i ≤ j → ∫ ω, (X i j ω) ^ 2 ∂μ = 1) : ∀ᵐ ω ∂μ, ∀ (f : ℝ → ℝ), Continuous f → (∃ M, ∀ x, ‖f x‖ ≤ M) → Tendsto (fun n : ℕ => ∫ x, f x ∂ (empiricalSpectralMeasureHerm (wignerMatrix_isHermitian X n ω)).map (fun x : ℝ => x / Real.sqrt n)) atTop (𝓝 (∫ x, f x ∂semicircleLaw)) := Ω:Type u_1inst✝¹:MeasurableSpace Ωμ:Measure Ωinst✝:IsProbabilityMeasure μX:ℕ → ℕ → Ω → ℝ_hX_meas:∀ (i j : ℕ), Measurable (X i j)_hX_indep:iIndepFun (fun ij => X (↑ij).1 (↑ij).2) μ_hX_iid:∀ (i j i' j' : ℕ), i ≤ j → i' ≤ j' → IdentDistrib (X i j) (X i' j') μ μ_hX_int:∀ (i j : ℕ), i ≤ j → Integrable (X i j) μ_hX_sq_int:∀ (i j : ℕ), i ≤ j → Integrable (fun ω => X i j ω ^ 2) μ_hX_mean:∀ (i j : ℕ), i ≤ j → ∫ (ω : Ω), X i j ω ∂μ = 0_hX_var:∀ (i j : ℕ), i ≤ j → ∫ (ω : Ω), X i j ω ^ 2 ∂μ = 1⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (f : ℝ → ℝ), Continuous f → (∃ M, ∀ (x : ℝ), ‖f x‖ ≤ M) → Tendsto (fun n => ∫ (x : ℝ), f x ∂Measure.map (fun x => x / √↑n) (empiricalSpectralMeasureHerm ⋯)) atTop (𝓝 (∫ (x : ℝ), f x ∂semicircleLaw)) All goals completed! 🐙
#10
Rokhlin lemma
rokhlin_lemma

Lean theorem statement

/-- **Rokhlin lemma.** For every aperiodic measure-preserving
automorphism `T` of a standard Borel probability space `(Ω, μ)`, every
height `n ≥ 1`, and every `ε > 0`, there is a Rokhlin tower of height
`n` whose union has measure at least `1 − ε`. -/
theorem rokhlin_lemma {Ω : Type*} [MeasurableSpace Ω]
    [StandardBorelSpace Ω]
    (μ : Measure Ω) [IsProbabilityMeasure μ] (T : Ω → Ω)
    (_hT : MeasurePreserving T μ μ) (_hap : IsAperiodic T μ)
    (n : ℕ) (_hn : 1 ≤ n) {ε : ENNReal} (_hε : 0 < ε) :
    ∃ B : Set Ω, IsRokhlinTower T B n ∧
      μ (towerUnion T B n) ≥ 1 - ε := by
  sorry
#11
Sobolev embedding theorem (Morrey regime)
sobolev_embedding_morrey

Verso theorem preview

theorem declaration uses `sorry`sobolev_embedding {n k r : ℕ} {α p : ℝ} (_hp : (n : ℝ) < p) (_hα : 0 < α) (_hα1 : α ≤ 1) (_hgap : (r : ℝ) + α < (k : ℝ) - n / p) (f : LeanEval.Analysis.SobolevMorreyProblem.E n → ℝ) (_hf : LeanEval.Analysis.SobolevMorreyProblem.MemSobolevWk k (ENNReal.ofReal p) f) : ∃ g : LeanEval.Analysis.SobolevMorreyProblem.E n → ℝ, f =ᵐ[volume] g ∧ LeanEval.Analysis.SobolevMorreyProblem.MemHolder r α g := n:ℕk:ℕr:ℕα:ℝp:ℝ_hp:↑n < p_hα:0 < α_hα1:α ≤ 1_hgap:↑r + α < ↑k - ↑n / pf:E n → ℝ_hf:MemSobolevWk k (ENNReal.ofReal p) f⊢ ∃ g, f =ᵐ[volume] g ∧ LeanEval.Analysis.SobolevMorreyProblem.MemHolder r α g All goals completed! 🐙
#12
Liouville–Arnold theorem on integrable systems
liouville_arnold

Verso theorem preview

theorem declaration uses `sorry`liouville_arnold {n : ℕ} (F : Fin n → LeanEval.Geometry.LiouvilleArnold.E n → ℝ) (U : Set (LeanEval.Geometry.LiouvilleArnold.E n)) (_hU : IsOpen U) (_hLI : LeanEval.Geometry.LiouvilleArnold.IsLiouvilleIntegrable F U) (c : Fin n → ℝ) (_hMc_sub : LeanEval.Geometry.LiouvilleArnold.levelSet F c ⊆ U) (_hMc_compact : IsCompact (LeanEval.Geometry.LiouvilleArnold.levelSet F c)) (_hMc_connected : IsConnected (LeanEval.Geometry.LiouvilleArnold.levelSet F c)) : Nonempty ((LeanEval.Geometry.LiouvilleArnold.levelSet F c) ≃ₜ (Fin n → AddCircle (1 : ℝ))) := n:ℕF:Fin n → E n → ℝU:Set (E n)_hU:IsOpen U_hLI:IsLiouvilleIntegrable F Uc:Fin n → ℝ_hMc_sub:levelSet F c ⊆ U_hMc_compact:IsCompact (levelSet F c)_hMc_connected:IsConnected (levelSet F c)⊢ Nonempty (↑(levelSet F c) ≃ₜ (Fin n → AddCircle 1)) All goals completed! 🐙
#13
Hurewicz theorem in degree 1 (H₁ = abelianization of π₁)
hurewicz_h1_abelianization

Lean theorem statement

/-- **Hurewicz (n = 1).** For a path-connected space `X`, `H₁(X;ℤ)` is the
abelianization of `π₁(X, x)`. -/
theorem hurewicz_h1_abelianization
    (X : Type) [TopologicalSpace X] [PathConnectedSpace X] (x : X) :
    Nonempty (Additive (Abelianization (FundamentalGroup X x)) ≃+
      (IntegralHomology 1 X : Type)) := by
  sorry
#14
Strong Subadditivity of von Neumann Entropy
strong_subadditivity

Lean theorem statement

/-- Strong subadditivity of quantum entropy. We relax the common assumption that M is a normalized
 density matrix to the simpler statement that it's PSD, which holds since normalization just produces
 a positive affine transformation on the entropy. -/
theorem strong_subadditivity (M_ABC : Matrix (A × B × C) (A × B × C) ℂ) (h : M_ABC.PosSemidef) :
    let M_AB : Matrix (A × B) (A × B) ℂ :=
      .traceRight <| M_ABC.reindex (.symm <| .prodAssoc ..) (.symm <| .prodAssoc ..)
    let M_BC : Matrix (B × C) (B × C) ℂ := M_ABC.traceLeft
    let M_B : Matrix B B ℂ := M_BC.traceRight
    entropy M_ABC + entropy M_B ≤ entropy M_AB + entropy M_BC := by
  sorry
#15
Lax's approximation theorem for toral homeomorphisms
lax_approximation

Lean theorem statement

/-- **Lax's approximation theorem.** Every toral dynamical system on `𝕋^d`
(`d ≥ 1`) is approximated arbitrarily well in the metric `δ` by cyclic cube
exchange transformations. -/
theorem lax_approximation {d : ℕ} (hd : 0 < d) (T : ToralDynamicalSystem d)
    {ε : ℝ≥0∞} (hε : 0 < ε) :
    ∃ (n : ℕ) (S : VolumePreservingEquiv d),
      IsCyclicCubeExchange S n ∧ deltaDist T.toVolumePreservingEquiv S < ε := by
  sorry
#16
Jordan normal form
jordan_normal_form

Lean theorem statement

/-- **Jordan normal form.** Over an algebraically closed field, every
endomorphism of `Kⁿ` admits a Jordan-chain basis. -/
theorem jordan_normal_form {K : Type*} [Field K] [IsAlgClosed K] (n : ℕ)
    (f : Module.End K (StdSpace K n)) :
    Nonempty (JordanChainBasis f) := by
  sorry
#17
Anosov–Bowen shadowing lemma
anosov_bowen_shadowing

Lean theorem statement

/-- **Anosov–Bowen shadowing lemma** (Anosov 1967; Bowen 1975). Every
compact hyperbolic invariant set has the shadowing property. -/
theorem hyperbolic_has_shadowing
    (T : E d ≃ₜ E d) (K : Set (E d))
    (_hKc : IsCompact K) (_hK : IsHyperbolic T K) :
    HasShadowing (T : E d → E d) K := by
  sorry
#18
General recursive equals Turing computable
turing_recursive_equiv

Lean theorem statement

/-- **General recursive = Turing computable** (total form). A total function
`f : ℕ → ℕ` is recursive (`Computable`, i.e. partial recursive as a partial
function) **iff** it is computed by some Turing machine (mathlib's `FinTM2`
model) under the standard binary encoding of `ℕ`. This is Knill's class
equality; the backward direction (TM-computable ⇒ recursive) is absent from
mathlib. -/
theorem turing_recursive_equiv (f : ℕ → ℕ) :
    Computable f ↔ Nonempty (TM2Computable encodeNat encodeNat f) := by
  sorry
#19
Wiener–Lévy theorem
wiener_levy_analytic_calculus

Verso theorem preview

theorem declaration uses `sorry`wiener_levy_analytic_calculus (f : C(AddCircle T, ℂ)) (φ : ℂ → ℂ) (U : Set ℂ) (hf : LeanEval.Analysis.WienerLevy.InWienerAlgebra f) (hU : IsOpen U) (hrange : range f ⊆ U) (hφ : AnalyticOnNhd ℂ φ U) : ∃ g : C(AddCircle T, ℂ), (∀ x, g x = φ (f x)) ∧ LeanEval.Analysis.WienerLevy.InWienerAlgebra g := T:ℝinst✝:Fact (0 < T)f:C(AddCircle T, ℂ)φ:ℂ → ℂU:Set ℂhf:InWienerAlgebra fhU:IsOpen Uhrange:range ⇑f ⊆ Uhφ:AnalyticOnNhd ℂ φ U⊢ ∃ g, (∀ (x : AddCircle T), g x = φ (f x)) ∧ InWienerAlgebra g All goals completed! 🐙
#20
Tverberg's theorem
tverberg_theorem

Lean theorem statement

/-- **Tverberg's theorem.** Any `(r-1)(d+1)+1` points in `ℝ^d` admit an
`r`-part Tverberg partition. -/
theorem tverberg_theorem (d r : ℕ) (hr : 1 ≤ r)
    (f : Fin ((r - 1) * (d + 1) + 1) → Space d) :
    HasTverbergPartition (r := r) f := by
  sorry
#21
Shannon capacity of the pentagon
shannon_capacity_pentagon

Lean theorem statement

/-- **Lovász's theorem on Shannon capacity of the pentagon** (§238).
The Shannon capacity of the five-cycle is `√5`. -/
theorem shannon_capacity_pentagon :
    HasShannonCapacity (SimpleGraph.cycleGraph 5) (Real.sqrt 5) := by
  sorry
#22
Radial symmetry for positive semilinear Poisson solutions
semilinear_poisson_radial_symmetry

Lean theorem statement

/-- **Radial symmetry theorem.** Let `u ∈ C^2(closedBall 0 1)` be a positive
solution of the semilinear Poisson problem `-Δ u = f(u)` in the open unit ball,
with zero Dirichlet boundary values on the unit sphere. If `f : ℝ → ℝ` is
Lipschitz, then `u` is radial: `u x = v ‖x‖` for a nonnegative strictly
decreasing radial profile `v` on `[0, 1]`. -/
theorem semilinear_poisson_radial_symmetry {n : ℕ} (hn : 0 < n)
    {f : ℝ → ℝ} (u : EuclideanSpace ℝ (Fin n) → ℝ)
    (hf_lipschitz : ∃ K : ℝ≥0, LipschitzWith K f)
    (hu_c2 : ContDiffOn ℝ 2 u (closedBall 0 1))
    (hu_solve : SolvesSemilinearPoisson f u)
    (hu_positive : ∀ x ∈ ball 0 1, 0 < u x) :
    ∃ v : ℝ → ℝ≥0,
      StrictAntiOn v (Set.Icc (0 : ℝ) 1) ∧
        ∀ x ∈ closedBall 0 1, u x = v ‖x‖ := by
  sorry
#23
Sard's theorem (critical-set image has measure zero)
sard_theorem

Lean theorem statement

/-- **Sard's theorem** (Morse 1939 / Sard 1942), Knill's rank-
deficient form. The image of the rank-deficient locus of a smooth
map `f : ℝᵐ → ℝⁿ` has Lebesgue measure zero. -/
theorem sard {m n : ℕ} (f : E m → E n) (_hf : ContDiff ℝ ∞ f) :
    volume (criticalValues f) = 0 := by
  sorry
#24
Sard's regular-value corollary
regular_value_ae

Lean theorem statement

/-- **Regular value corollary (Sard).** For a smooth `f : ℝᵐ → ℝ`, almost every
`c ∈ ℝ` is a regular value. -/
theorem regular_value_ae {m : ℕ} (f : EuclideanSpace ℝ (Fin m) → ℝ)
    (hf : ContDiff ℝ ∞ f) :
    ∀ᵐ c ∂(volume : Measure ℝ), IsRegularValue f c := by
  sorry
#25
Poincaré–Siegel linearisation theorem
poincare_siegel_linearisation

Lean theorem statement

/-- **Poincaré–Siegel linearisation theorem.** If `α` is Diophantine,
`λ = e^{2πiα}`, and `f` is holomorphic near `0` with `f 0 = 0` and
`f'(0) = λ`, then there is a holomorphic germ `u` with `u 0 = 0`,
`u'(0) = 1`, and `f(u z) = u(λ z)` for `z` near `0`. -/
theorem poincare_siegel
    (α : ℝ) (_hα : IsDiophantine α)
    (lam : ℂ) (_hlam : lam = Complex.exp (2 * Real.pi * Complex.I * (α : ℂ)))
    (f : ℂ → ℂ) (_hf : AnalyticAt ℂ f 0) (_hf0 : f 0 = 0)
    (_hmult : deriv f 0 = lam) :
    ∃ u : ℂ → ℂ, AnalyticAt ℂ u 0 ∧ u 0 = 0 ∧ deriv u 0 = 1 ∧
      ∀ᶠ z in nhds (0 : ℂ), f (u z) = u (lam * z) := by
  sorry
#26
Peano existence theorem for ODEs
peano_existence

Lean theorem statement

/-- **Peano existence theorem.** Replacing the Lipschitz condition with mere
continuity still yields a local solution of `x' = f(x)`, `x(0) = x₀` — but
uniqueness may fail (e.g. `x' = √x`, `x(0) = 0`). Stated for a
finite-dimensional space: Peano's theorem requires local compactness and is
false in general (infinite-dimensional) Banach spaces. -/
theorem peano_existence
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
    {f : E → E} (hf : Continuous f) (x₀ : E) :
    ∃ a : ℝ, 0 < a ∧ ∃ α : ℝ → E, α 0 = x₀ ∧
      ∀ t ∈ Ioo (-a) a, HasDerivAt α (f (α t)) t := by
  sorry
#27
The Lindemann–Weierstrass theorem
lindemann_weierstrass

Lean theorem statement

/-- **Lindemann–Weierstrass theorem.** If `x₁, …, xₙ ∈ ℂ` are algebraic over `ℚ`
and ℚ-linearly independent, then `e^{x₁}, …, e^{xₙ}` are algebraically
independent over `ℚ`. -/
theorem lindemann_weierstrass {n : ℕ} (x : Fin n → ℂ)
    (h_alg : ∀ i, IsAlgebraic ℚ (x i))
    (h_lin : LinearIndependent ℚ x) :
    AlgebraicIndependent ℚ (fun i => Complex.exp (x i)) := by
  sorry
#28
Fundamental theorem of Riemannian geometry (Levi-Civita)
levi_civita_exists_unique

Verso theorem preview

theorem declaration uses `sorry`levi_civita_exists_unique {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [CompleteSpace E] {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type*} [TopologicalSpace M] [T2Space M] [ChartedSpace H M] [IsManifold I ∞ M] [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)] [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)] : ∃ cov : CovariantDerivative I E (TangentSpace I (M := M)), (ContMDiffCovariantDerivative cov ∞ ∧ cov.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov) ∧ ∀ cov' : CovariantDerivative I E (TangentSpace I (M := M)), (ContMDiffCovariantDerivative cov' ∞ ∧ cov'.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov') → LeanEval.Geometry.LeviCivita.SameOnSmooth cov cov' := E:Type u_1inst✝¹⁰:NormedAddCommGroup Einst✝⁹:NormedSpace ℝ Einst✝⁸:FiniteDimensional ℝ Einst✝⁷:CompleteSpace EH:Type u_2inst✝⁶:TopologicalSpace HI:ModelWithCorners ℝ E HM:Type u_3inst✝⁵:TopologicalSpace Minst✝⁴:T2Space Minst✝³:ChartedSpace H Minst✝²:IsManifold I ∞ Minst✝¹:RiemannianBundle fun x => TangentSpace I xinst✝:IsContMDiffRiemannianBundle I ∞ E fun x => TangentSpace I x⊢ ∃ cov, (cov.ContMDiffCovariantDerivative ∞ ∧ cov.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov) ∧ ∀ (cov' : CovariantDerivative I E (TangentSpace I)), cov'.ContMDiffCovariantDerivative ∞ ∧ cov'.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov' → SameOnSmooth cov cov' All goals completed! 🐙
#29
Kolmogorov–Arnold superposition theorem (non-universal Lorentz form)
kolmogorov_arnold_superposition

Verso theorem preview

theorem declaration uses `sorry`kolmogorov_arnold (n : ℕ) (_hn : 1 ≤ n) (f : (Fin n → ℝ) → ℝ) (_hf : ContinuousOn f (Set.Icc 0 1)) : ∃ (g : ℝ → ℝ) (φ : Fin (2 * n + 1) → Fin n → ℝ → ℝ), Continuous g ∧ (∀ k l, Continuous (φ k l)) ∧ ∀ x ∈ Set.Icc (0 : Fin n → ℝ) 1, f x = ∑ k, g (∑ l, φ k l (x l)) := n:ℕ_hn:1 ≤ nf:(Fin n → ℝ) → ℝ_hf:ContinuousOn f (Set.Icc 0 1)⊢ ∃ g φ, Continuous g ∧ (∀ (k : Fin (2 * n + 1)) (l : Fin n), Continuous (φ k l)) ∧ ∀ x ∈ Set.Icc 0 1, f x = ∑ k, g (∑ l, φ k l (x l)) All goals completed! 🐙
#30
The Hausdorff–Hildebrandt–Schoenberg moment theorem
hausdorff_hildebrandt_schoenberg

Lean theorem statement

/-- **Hausdorff–Hildebrandt–Schoenberg theorem.** A multi-indexed real sequence
is the moment sequence of a signed bounded-variation measure on the unit cube
`Iᵈ` iff its moments are Hausdorff bounded. -/
theorem hausdorff_hildebrandt_schoenberg {d : ℕ} (a : (Fin d → ℕ) → ℝ) :
    IsMomentConfiguration a ↔ HausdorffBounded a := by
  sorry
#31
No bounded projection from L^1 onto H^1
H1_not_closedComplemented

Verso theorem preview

theorem declaration uses `sorry`H1_not_closedComplemented : ¬ LeanEval.Analysis.H1.ClosedComplemented := ⊢ ¬H1.ClosedComplemented All goals completed! 🐙
#32
Fundamental theorem of topos theory
fundamental_topos_theory

Verso theorem preview

theorem declaration uses `sorry`fundamental_topos_theory {E : Type*} [Category E] (hE : LeanEval.ToposTheory.IsTopos E) (X : E) : LeanEval.ToposTheory.IsTopos (Over X) := E:Type u_1inst✝:Category.{u_2, u_1} EhE:IsTopos EX:E⊢ IsTopos (Over X) All goals completed! 🐙
#33
Frobenius determinant theorem
frobenius_group_determinant

Lean theorem statement

/-- **Frobenius determinant theorem** (§171). The group determinant factors as
a product of irreducible polynomials, each appearing to the power of its own
(total) degree `d_j = deg p_j`, with the factors pairwise non-associated
(*distinct*) and their number equal to the number of conjugacy classes of `G`.
-/
theorem frobenius_group_determinant
    (G : Type*) [Group G] [Fintype G] [DecidableEq G] :
    ∃ (r : ℕ) (p : Fin r → MvPolynomial G ℂ),
      r = Nat.card (ConjClasses G) ∧
      (∀ j, Irreducible (p j)) ∧
      (∀ i j, i ≠ j → ¬ Associated (p i) (p j)) ∧
      groupDeterminant G = ∏ j, (p j) ^ (p j).totalDegree := by
  sorry
#34
Choquet's representation theorem
choquet_representation_theorem

Lean theorem statement

/-- **Choquet's representation theorem.** Every point `x` of a compact convex
set `K` in a Banach space is the barycenter of a probability measure supported
on the extreme points of `K`: there is a probability measure `μ` with
`μ (ext K)ᶜ = 0` whose barycenter `∫ y, y ∂μ` equals `x`. -/
theorem choquet [MeasurableSpace X] [BorelSpace X]
    (K : Set X) (hK_cpt : IsCompact K) (hK_cvx : Convex ℝ K)
    {x : X} (hx : x ∈ K) :
    ∃ μ : Measure X, IsProbabilityMeasure μ ∧
      μ (K.extremePoints ℝ)ᶜ = 0 ∧
      x = ∫ y, y ∂μ := by
  sorry
#35
Brun's theorem (convergence of the twin-prime reciprocal sum)
brun_constant_converges

Lean theorem statement

/-- **Brun's theorem.** The reciprocal sum over twin-prime pairs converges. -/
theorem brun_constant_converges :
    Summable twinPrimeReciprocalTerm := by
  sorry
#36
Wiener's 1/f theorem
wiener_inverse_closed

Verso theorem preview

theorem declaration uses `sorry`wiener_inverse_closed (f : C(AddCircle T, ℂ)) (hf : LeanEval.Analysis.WienerOneOverF.InWienerAlgebra f) (hzero : ∀ x, f x ≠ 0) : ∃ g : C(AddCircle T, ℂ), (∀ x, g x = (f x)⁻¹) ∧ LeanEval.Analysis.WienerOneOverF.InWienerAlgebra g := T:ℝinst✝:Fact (0 < T)f:C(AddCircle T, ℂ)hf:InWienerAlgebra fhzero:∀ (x : AddCircle T), f x ≠ 0⊢ ∃ g, (∀ (x : AddCircle T), g x = (f x)⁻¹) ∧ InWienerAlgebra g All goals completed! 🐙
#37
Riesz brothers' theorem
riesz_brothers_theorem

Lean theorem statement

/-- **Riesz brothers' theorem.** Let `μ` be a complex Borel measure on
the unit circle such that `∫ z^n dμ = 0` for every `n ≥ 1`. Then `μ` is
absolutely continuous with respect to Haar measure, and more precisely
`dμ = h dm` for some `h ∈ H^1`, i.e. some integrable density whose negative
Fourier coefficients vanish. -/
theorem riesz_brothers_theorem (μ : ComplexMeasure UnitAddCircle)
    (hμ : ∀ n : ℕ, 1 ≤ n → ∫ᵛ z, fourier n z ∂[ContinuousLinearMap.mul ℝ ℂ; μ] = 0) :
    μ ≪ᵥ AddCircle.haarAddCircle.toENNRealVectorMeasure := by
  sorry
#38
Riesz's rising sun lemma
rising_sun_lemma

Lean theorem statement

/-- **Riesz's rising sun lemma.** Every continuous real function on a compact
interval has the rising-sun property. -/
theorem rising_sun_lemma {a b : ℝ} (hab : a < b) {f : ℝ → ℝ}
    (hf : ContinuousOn f (Icc a b)) :
    HasRisingSunProperty a b f := by
  sorry
#39
Gauss-Wantzel constructible regular polygon theorem
gauss_wantzel_constructible_polygon

Lean theorem statement

/-- **Gauss-Wantzel constructible polygon theorem** (§174): a regular `n`-gon
is straightedge-and-compass constructible exactly for the Gauss-Wantzel
integers. -/
theorem gauss_wantzel_constructible_polygon (n : ℕ) (hn : 3 ≤ n) :
    IsConstructible (Real.cos (2 * Real.pi / n)) ↔ GaussWantzelNumber n := by
  sorry
#40
Hausdorff moment problem: absolute-continuity criterion
hausdorff_absolute_continuity

Lean theorem statement

/-- **Absolute continuity criterion (Hausdorff moment problem).** A positive
probability measure `μ` on the cube is uniformly absolutely continuous w.r.t.
Lebesgue measure iff there is `C` with `(Δᵏμ)ₙ ≤ C·(Δᵏν)ₙ` for all `k ≤ n`. -/
theorem hausdorff_absolute_continuity {d : ℕ}
    (μ : Measure (EuclideanSpace ℝ (Fin d)))
    [IsProbabilityMeasure μ] (hμ : μ ((cube d)ᶜ) = 0) :
    UniformlyAbsolutelyContinuous μ (volume.restrict (cube d)) ↔
      ∃ C : ℝ, ∀ k n : Fin d → ℕ, k ≤ n →
        diff (momentOf μ) k n ≤ C * diff (momentOf (volume.restrict (cube d))) k n := by
  sorry
#41
Fang–Xia: tiling of the symmetric group by transpositions implies λ-transitivity
fang_xia_tiling_partition_transitive

Lean theorem statement

/-- **Fang–Xia, Theorem 1.4.** A tiling `(T_n, Y)` of `S_n` forces
λ-transitivity of `Y` for every partition `λ` of `n` whose Young-
diagram content sum is nonnegative. -/
theorem fang_xia_partition_transitive_of_tiling
    {n : ℕ} {Y : Set (Equiv.Perm (Fin n))}
    (_h : IsTiling (transpositionsWithOne n) Y) :
    ∀ lam : PartitionShape n, 0 ≤ lam.contentSum → IsPartitionTransitive Y lam := by
  sorry
#42
Normal spectral theorem
normal_spectral_theorem

Lean theorem statement

/-- **Spectral theorem** (§14). A complex matrix `A` is **normal**
(`Aᴴ A = A Aᴴ`, i.e. `IsStarNormal A`) **iff** it is **unitarily
diagonalizable**: there is a unitary `U` and a diagonal matrix `diagonal d`
with `A = U (diagonal d) Uᴴ`. -/
theorem normal_spectral_theorem (A : Matrix n n ℂ) :
    IsStarNormal A ↔
      ∃ U ∈ unitary (Matrix n n ℂ), ∃ d : n → ℂ,
        A = U * diagonal d * star U := by
  sorry
#43
The Landsberg–Schaar relation
landsberg_schaar

Lean theorem statement

/-- **Landsberg–Schaar relation.** For positive odd integers `p, q`,
`S(2q, p) = e^{iπ/4} · S(−p, 2q)`. -/
theorem landsberg_schaar (p q : ℕ) (hp : Odd p) (hq : Odd q) :
    gaussS (2 * q : ℕ) p
      = Complex.exp ((Real.pi : ℂ) * Complex.I / 4) * gaussS (-(p : ℤ)) (2 * q) := by
  sorry
#44
Moran's equality for affine-symmetric iterated function systems
moran_equality_affine

Lean theorem statement

/-- **Moran's equality for affine-symmetric IFS.** For an affine-symmetric IFS
on `ℝᵈ` with common contraction factor `λ ∈ (0,1)`, orthogonal linear parts, and
the open set condition, the Hausdorff dimension of the attractor is
`−log n / log λ` (positive since `λ < 1`). -/
theorem moran_equality_affine
    {d n : ℕ} (hn : 1 ≤ n)
    (f : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (lam : ℝ)
    (h_aff : IsAffineSymmetricIFS f lam)
    (h_osc : OpenSetCondition f)
    {S : Set (EuclideanSpace ℝ (Fin d))} (hS : IsAttractor f S) :
    dimH S = ENNReal.ofReal (- Real.log n / Real.log lam) := by
  sorry
#45
The Hausdorff positivity (complete-monotonicity) criterion
hausdorff_positivity_criterion

Lean theorem statement

/-- **Hausdorff positivity criterion** (Hausdorff 1921). A moment configuration
comes from a positive measure iff all its iterated backward differences are
nonnegative — i.e. the sequence is *completely monotone*: `(Δᵏa)ₙ ≥ 0` for all
`k ≤ n`. -/
theorem hausdorff_positivity {d : ℕ} (a : (Fin d → ℕ) → ℝ) :
    IsPositiveMomentConfiguration a ↔ ∀ k n : Fin d → ℕ, k ≤ n → 0 ≤ diff a k n := by
  sorry
#46
Radon transform: Fourier-slice diagonalization and pseudo-inversion
radon_transform_inversion

Lean theorem statement

/-- **Radon's theorem (diagonalization + pseudo-inversion).** The Fourier slice
theorem diagonalizes the Radon transform, and the transform admits a left
inverse on the Schwartz space. -/
theorem radon_can_be_diagonalized_and_pseudo_inverted :
    (∀ φ : SchwartzMap (ℝ × ℝ) ℂ, ∀ θ k : ℝ,
        fourier1 (fun p => radon (φ : ℝ × ℝ → ℂ) (p, θ)) k =
          fourier2 (φ : ℝ × ℝ → ℂ) (k * Real.cos θ, k * Real.sin θ)) ∧
    (∃ Rinv : (ℝ × ℝ → ℂ) → (ℝ × ℝ → ℂ),
        ∀ φ : SchwartzMap (ℝ × ℝ) ℂ,
          Rinv (radon (φ : ℝ × ℝ → ℂ)) = (φ : ℝ × ℝ → ℂ)) := by
  sorry
#47
Nyquist–Shannon sampling theorem
nyquist_shannon_sampling

Lean theorem statement

/-- **Nyquist--Shannon sampling theorem / Whittaker--Shannon interpolation
formula** for Schwartz functions with Fourier support in the mathlib-convention
Nyquist band `[-1/2, 1/2]`.

The explicit `Summable` conjunct records the convergence content of the
cardinal series, rather than relying only on Lean's total `tsum`.
-/
theorem nyquist_shannon_sampling (f : 𝓢(ℝ, ℂ)) (hf : FourierSupportedInNyquist f) :
    ∀ t : ℝ,
      Summable (fun n : ℤ ↦ f (n : ℝ) * sinc (Real.pi * ((n : ℝ) - t))) ∧
        f t =
          ∑' n : ℤ, f (n : ℝ) * sinc (Real.pi * ((n : ℝ) - t)) := by
  sorry
#48
Kirk's normal-structure fixed point theorem
kirk_normal_structure

Lean theorem statement

/-- **Kirk's normal-structure fixed-point theorem** (§228).  A nonexpansive
self-map of a nonempty bounded closed convex subset of a reflexive Banach
space with normal structure has a fixed point. -/
theorem kirk_normal_structure [CompleteSpace E]
    (hE_reflexive : Function.Surjective (NormedSpace.inclusionInDoubleDual ℝ E))
    (K : Set E) (hK_nonempty : K.Nonempty) (hK_closed : IsClosed K)
    (hK_bounded : Bornology.IsBounded K) (hK_convex : Convex ℝ K)
    (hK_normal : HasNormalStructure K) (T : K → K)
    (hT : IsNonexpansiveSelfMap K T) :
    ∃ x : K, IsFixedPt T x := by
  sorry
#49
Trace Cayley-Hamilton / Newton identity
trace_cayley_hamilton_newton

Lean theorem statement

/-- The trace Cayley-Hamilton / Newton identity:
`k c_k + ∑_{j=1}^k tr(A^j) c_{k-j} = 0`, where
`χ_A(X) = X^N + c₁ X^(N-1) + ... + c_N`.

For `k > N`, `c_k = 0`, and the remaining relation is the trace of
Cayley-Hamilton multiplied by a power of `A`. -/
theorem trace_cayley_hamilton_newton {R : Type*} [CommRing R]
    (A : Matrix n n R) {k : ℕ} (hk : 1 ≤ k) :
    (k : R) * charpolyDescendingCoeff A k +
        ∑ j ∈ Finset.Icc 1 k,
          trace (A ^ j) * charpolyDescendingCoeff A (k - j) = 0 := by
  sorry
#50
Pascal's theorem
pascal

Lean theorem statement

/-- **Pascal's theorem.** Six distinct points on a non-singular conic determine
three collinear intersection points `Aᵢ Bⱼ ∩ Aⱼ Bᵢ`. -/
theorem pascal
    (M : Matrix (Fin 3) (Fin 3) ℝ) (hMsymm : M.IsSymm) (hMdet : M.det ≠ 0)
    (a₁ a₂ a₃ b₁ b₂ b₃ : Fin 3 → ℝ)
    (ha₁ : a₁ ≠ 0) (ha₂ : a₂ ≠ 0) (ha₃ : a₃ ≠ 0)
    (hb₁ : b₁ ≠ 0) (hb₂ : b₂ ≠ 0) (hb₃ : b₃ ≠ 0)
    (hdist : [a₁, a₂, a₃, b₁, b₂, b₃].Pairwise (fun v w => ¬ SamePoint v w))
    (hA₁ : OnConic M a₁) (hA₂ : OnConic M a₂) (hA₃ : OnConic M a₃)
    (hB₁ : OnConic M b₁) (hB₂ : OnConic M b₂) (hB₃ : OnConic M b₃) :
    Collinear3 (meet a₁ b₂ a₂ b₁) (meet a₁ b₃ a₃ b₁) (meet a₂ b₃ a₃ b₂) := by
  sorry
#51
Lindemann's theorem (e and π transcendental)
lindemann

Lean theorem statement

/-- **Lindemann's theorem.** Both `e = exp 1` and `π` are transcendental over
`ℤ`. -/
theorem lindemann :
    Transcendental ℤ (Real.exp 1) ∧ Transcendental ℤ Real.pi := by
  sorry
#52
Complete reducibility for compact groups
compact_group_semisimple

Lean theorem statement

/-- **Representations of compact groups are semisimple** (complete
reducibility / the unitarian trick). A continuous representation of a compact
topological group on a finite-dimensional real vector space is semisimple:
every subrepresentation has a `G`-invariant complement, so the representation
decomposes as a direct sum of irreducible finite-dimensional
subrepresentations. -/
theorem compact_group_semisimple
    {G V : Type*} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G]
    [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V]
    (ρ : Representation ℝ G V)
    (hρ : Continuous fun p : G × V => ρ p.1 p.2) :
    ρ.IsSemisimpleRepresentation := by
  sorry
#53
Bauer's uniqueness at extreme points
bauer_extreme_point_uniqueness

Lean theorem statement

/-- **Bauer's uniqueness at extreme points.** If `x` is an extreme point of a
compact convex set `K` and `μ` is a probability measure supported on `K`
(`μ Kᶜ = 0`) with barycenter `x = ∫ y, y ∂μ`, then `μ` is the Dirac mass at
`x`. (The support hypothesis is the weaker `μ Kᶜ = 0`, making this a
strengthening of the textbook statement: uniqueness among all ambient Borel
probability measures on `K`, not only those already supported on `ext K`.) -/
theorem bauer_unique [MeasurableSpace X] [BorelSpace X]
    (K : Set X) (hK_cpt : IsCompact K) (hK_cvx : Convex ℝ K)
    {x : X} (hx : x ∈ K.extremePoints ℝ)
    (μ : Measure X) [IsProbabilityMeasure μ]
    (hμ : μ Kᶜ = 0) (hbar : x = ∫ y, y ∂μ) :
    μ = Measure.dirac x := by
  sorry
#54
Boone–Higman theorem (easy direction)
boone_higman_embedding

Lean theorem statement

/-- **Boone–Higman theorem (easy direction).** If a finitely presented group `G`
embeds (via injective `f`) into a simple group `H`, which embeds (via injective
`g`) into a finitely presented group `K`, then the word problem of `G` is
solvable. -/
theorem boone_higman_embedding
    {G H K : Type*} [Group G] [Group H] [Group K]
    [IsSimpleGroup H] [Group.IsFinitelyPresented K]
    (f : G →* H) (hf : Function.Injective f)
    (g : H →* K) (hg : Function.Injective g)
    {n : ℕ} (φ : FreeGroup (Fin n) →* G)
    (hsurj : Function.Surjective φ)
    (hker : (MonoidHom.ker φ).IsNormalClosureFG) :
    WordProblemSolvable φ := by
  sorry
#55
Independence of the parallel postulate
parallel_postulate_independent

Lean theorem statement

/-- **Independence of the parallel postulate** (Freek #12). The Euclidean
axiom `A10` is logically independent of Tarski's absolute axioms `A1`–`A9`
and `A11`: there is a model of the absolute axioms in which the parallel
postulate holds (the real coordinate plane) and one in which it fails (the
Klein–Beltrami disk, or any other hyperbolic-plane model). -/
theorem parallel_postulate_independent :
    (∃ (M : Type) (T : TarskiAbsolute M), Euclidean M T) ∧
    (∃ (M : Type) (T : TarskiAbsolute M), ¬ Euclidean M T) := by
  sorry
#56
Local stable/unstable sets at a hyperbolic fixed point (set-level Hadamard–Perron)
stable_unstable_manifolds

Verso theorem preview

theorem declaration uses `sorry`stable_unstable_manifolds_exist (n : ℕ) (f : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (x₀ : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (_hf : ContDiffAt ℝ 1 f x₀) (_hfix : f x₀ = x₀) (_hhyp : LeanEval.Dynamics.StableUnstableManifoldsProblem.IsHyperbolicLinear (fderiv ℝ f x₀)) (_hf_inv : (fderiv ℝ f x₀).IsInvertible) : ∃ U : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), Ws = {x | (∀ k : ℕ, f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y : ℕ → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n, y 0 = x ∧ (∀ k : ℕ, y k ∈ U) ∧ (∀ k : ℕ, f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} := n:ℕf:E n → E nx₀:E n_hf:ContDiffAt ℝ 1 f x₀_hfix:f x₀ = x₀_hhyp:IsHyperbolicLinear (fderiv ℝ f x₀)_hf_inv:(fderiv ℝ f x₀).IsInvertible⊢ ∃ U, IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu, Ws = {x | (∀ (k : ℕ), f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y, y 0 = x ∧ (∀ (k : ℕ), y k ∈ U) ∧ (∀ (k : ℕ), f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} All goals completed! 🐙
#57
Nash equilibrium existence theorem
nash_equilibrium_exists

Lean theorem statement

/-- **Nash equilibrium existence theorem.** Every finite `n`-player game
with nonempty finite pure-strategy sets and arbitrary real payoffs admits
at least one mixed-strategy Nash equilibrium. -/
theorem nash_equilibrium_exists {n : ℕ} {S : Fin n → Type*}
    [∀ i, Fintype (S i)] [∀ i, Nonempty (S i)]
    (u : Fin n → StrategyProfile n S → ℝ) :
    ∃ σ : ∀ i, S i → ℝ, IsNashEquilibrium u σ := by
  sorry
#58
Lidskii's inequality
lidskii_inequality

Verso theorem preview

theorem declaration uses `sorry`lidskii_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) {p : ℝ} (_hp : 1 ≤ p) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |(hB.sub hA).eigenvalues₀ j| ^ p := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitianp:ℝ_hp:1 ≤ p⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |⋯.eigenvalues₀ j| ^ p All goals completed! 🐙
#59
Kakutani fixed-point theorem
kakutani_fixed_point

Lean theorem statement

/-- **Kakutani fixed-point theorem.** Every upper-hemicontinuous
correspondence `F` from a nonempty compact convex `K ⊆ ℝᵈ` to itself, with
nonempty convex closed values, has a fixed point `x ∈ F x`. -/
theorem kakutani_fixed_point {d : ℕ}
    {K : Set (EuclideanSpace ℝ (Fin d))}
    (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K)
    (_hK_nonempty : K.Nonempty)
    (F : EuclideanSpace ℝ (Fin d) → Set (EuclideanSpace ℝ (Fin d)))
    (_hF_uhc : IsUpperHemicontinuous F)
    (_hF_nonempty : ∀ x ∈ K, (F x).Nonempty)
    (_hF_convex : ∀ x ∈ K, Convex ℝ (F x))
    (_hF_closed : ∀ x ∈ K, IsClosed (F x))
    (_hF_maps : ∀ x ∈ K, F x ⊆ K) :
    ∃ x ∈ K, x ∈ F x := by
  sorry
#60
Frobenius's theorem: the Frobenius kernel is normal
frobenius_kernel_isNormal

Lean theorem statement

theorem frobenius_kernel_isNormal
    (G X : Type) [Group G] [Fintype G] [Fintype X]
    [MulAction G X] [FaithfulSMul G X]
    (hcard : 2 ≤ Fintype.card X)
    (htrans : ∀ x y : X, ∃ g : G, g • x = y)
    (hstab : ∀ x : X, MulAction.stabilizer G x ≠ ⊥)
    (hfrob : ∀ g : G, g ≠ 1 → ∀ x y : X, g • x = x → g • y = y → x = y) :
    ∃ N : Subgroup G, N.Normal ∧
      (N : Set G) = {1} ∪ {g : G | ∀ x : X, g • x ≠ x} := by
  sorry
#61
Balanceable k-bounded partitions
balanceable_bounded_partitions

Lean theorem statement

/--
For any `k`, the smallest `n` such that any `k`-bounded partition of `n` is
also balanceable is given by `2 * lcm(1, ..., k)`.
-/
theorem minimal_balanceable_of_bounded (k : ℕ) (hk : 0 < k) :
    Minimal (fun n => 0 < n ∧ ∀ p : n.Partition, Bounded k p → Balanceable p) (2 * (Finset.Icc 1 k).lcm id) := by
  sorry
#62
Morley's trisector theorem
morley_theorem

Verso theorem preview

theorem declaration uses `sorry`morley_theorem (A B C P Q R : LeanEval.Geometry.Morley.Plane) (h : LeanEval.Geometry.Morley.IsMorleyConfiguration A B C P Q R) : LeanEval.Geometry.Morley.IsEquilateralTriple P Q R := A:PlaneB:PlaneC:PlaneP:PlaneQ:PlaneR:Planeh:IsMorleyConfiguration A B C P Q R⊢ IsEquilateralTriple P Q R All goals completed! 🐙
#63
A competition programming problem about permuting a permutation to be unimodal
permute_to_unimodal

Lean theorem statement

/--
`minRearrange` correctly computes the smallest number of indices that need to be permuted in order to
turn `arr` into a unimodal permutation.
-/
theorem minRearrange_correct {arr : Array Nat} :
    arr.Perm (1...=arr.size).toArray →
      (∃ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x ∧ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector = minRearrange arr) ∧
      (∀ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x → minRearrange arr ≤ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector) := by
  sorry
#64
Hippocrates' theorem on lunes
hippocrates_lunes

Verso theorem preview

theorem declaration uses `sorry`hippocrates_lunes (a b : ℝ) (ha : 0 < a) (hb : 0 < b) : volume (LeanEval.Geometry.HippocratesLunes.horizontalLune a b) + volume (LeanEval.Geometry.HippocratesLunes.verticalLune a b) = volume (LeanEval.Geometry.HippocratesLunes.rightTriangle a b) := a:ℝb:ℝha:0 < ahb:0 < b⊢ volume (horizontalLune a b) + volume (verticalLune a b) = volume (rightTriangle a b) All goals completed! 🐙
#65
Mountain Pass Theorem (Ambrosetti–Rabinowitz 1973)
mountain_pass

Verso theorem preview

theorem declaration uses `sorry`mountain_pass (f : E → ℝ) (_hf : ContDiff ℝ 1 f) (_hps : LeanEval.Analysis.MountainPassProblem.PalaisSmale f) {a b : E} {ε r : ℝ} (_hmr : LeanEval.Analysis.MountainPassProblem.MountainRange f a b ε r) : ∃ x : E, LeanEval.Analysis.MountainPassProblem.IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace Ef:E → ℝ_hf:ContDiff ℝ 1 f_hps:PalaisSmale fa:Eb:Eε:ℝr:ℝ_hmr:MountainRange f a b ε r⊢ ∃ x, IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b All goals completed! 🐙
#66
Fraser: Fourier decay for finite-field Kakeya sets is q^{-1} and sharp
fraser_kakeya_fourier_decay

Verso theorem preview

theorem declaration uses `sorry`fraser_kakeya_fourier_decay_and_sharp {d : ℕ} (_hd : 2 ≤ d) {K : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F d)} (_hK : LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K) (χ : AddChar F ℂ) (_hχ : χ ≠ 1) : (∃ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K μ ∧ ∀ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d, ξ ≠ 0 → ‖fourier χ μ ξ‖ ≤ (Fintype.card F : ℝ)⁻¹) ∧ (∀ κ : ℝ, 0 < κ → κ < 1 → ∃ Q : ℕ, ∀ (F' : Type*) [Field F'] [Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K' : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d), LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K' ∧ ∀ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K' μ → ∃ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d, ξ ≠ 0 ∧ κ * (Fintype.card F' : ℝ)⁻¹ ≤ ‖fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖) := F:Type u_1inst✝²:Field Finst✝¹:Fintype Finst✝:DecidableEq Fd:ℕ_hd:2 ≤ dK:Set (Space F d)_hK:IsKakeya Kχ:AddChar F ℂ_hχ:χ ≠ 1⊢ (∃ μ, IsProbabilityMeasureOn K μ ∧ ∀ (ξ : Space F d), ξ ≠ 0 → ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier χ μ ξ‖ ≤ (↑(Fintype.card F))⁻¹) ∧ ∀ (κ : ℝ), 0 < κ → κ < 1 → ∃ Q, ∀ (F' : Type u_2) [inst : Field F'] [inst_1 : Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K', IsKakeya K' ∧ ∀ (μ : Space F' d → ℝ), IsProbabilityMeasureOn K' μ → ∃ ξ, ξ ≠ 0 ∧ κ * (↑(Fintype.card F'))⁻¹ ≤ ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖ All goals completed! 🐙
#67
Chen theorem for Markoff graphs
dvd_card_connectedComponent_markoffGraph

Lean theorem statement

/-- For prime `p > 3`, every connected component of the nonzero Markoff graph over `ZMod p`
has cardinality divisible by `p`. -/
theorem dvd_card_connectedComponent_markoffGraph
    {p : ℕ} (hp : Nat.Prime p) (hgt : 3 < p) :
    ∀ c : (markoffGraph p).ConnectedComponent, p ∣ Nat.card c := by
  sorry
#68
Solvable extensions ↔ solvable groups (the missing converse in Abel–Ruffini)
solvable_by_radicals_converse

Verso theorem preview

theorem declaration uses `sorry`solvable_iff_solvableByRad (F : Type*) [Field F] [CharZero F] (p : F[X]) (_hp : p ≠ 0) : (∀ x : AlgebraicClosure F, aeval x p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal := F:Type u_1inst✝¹:Field Finst✝:CharZero Fp:F[X]_hp:p ≠ 0⊢ (∀ (x : AlgebraicClosure F), (aeval x) p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal All goals completed! 🐙
#69
Euler–Lagrange equation
euler_lagrange_equation

Verso theorem preview

theorem declaration uses `sorry`euler_lagrange_equation {a b : ℝ} (L : ℝ → ℝ → ℝ → ℝ) (x : ℝ → ℝ) (_hab : a < b) (_hL : ContDiff ℝ 2 (fun p : ℝ × ℝ × ℝ => L p.1 p.2.1 p.2.2)) (_hx : ContDiff ℝ 2 x) (_hxe : LeanEval.Analysis.IsVariationalExtremum a b L x) : ∀ t ∈ Set.Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t := a:ℝb:ℝL:ℝ → ℝ → ℝ → ℝx:ℝ → ℝ_hab:a < b_hL:ContDiff ℝ 2 fun p => L p.1 p.2.1 p.2.2_hx:ContDiff ℝ 2 x_hxe:IsVariationalExtremum a b L x⊢ ∀ t ∈ Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t All goals completed! 🐙
#70
Sturm's theorem
sturm

Lean theorem statement

/-- **Sturm's theorem.** For a squarefree real polynomial `p` and an interval
`(a, b)` with `a < b` whose endpoints are not roots of `p`, the number of
distinct roots of `p` in `(a, b)` equals `σ(a) − σ(b)`. -/
theorem sturm (p : ℝ[X]) (hp : Squarefree p) {a b : ℝ} (hab : a < b)
    (ha : p.eval a ≠ 0) (hb : p.eval b ≠ 0) :
    ((p.roots.toFinset).filter (fun x => a < x ∧ x < b)).card =
      sigma p a - sigma p b := by
  sorry
#71
Monge–Kantorovich existence theorem
monge_kantorovich

Verso theorem preview

theorem declaration uses `sorry`monge_kantorovich_exists {X Y : Type*} [TopologicalSpace X] [PolishSpace X] [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [PolishSpace Y] [MeasurableSpace Y] [BorelSpace Y] (P : Measure X) (Q : Measure Y) [IsProbabilityMeasure P] [IsProbabilityMeasure Q] (c : X × Y → ENNReal) (_hc : Continuous c) : ∃ π ∈ LeanEval.Analysis.Couplings P Q, ∀ π' ∈ LeanEval.Analysis.Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' := X:Type u_1Y:Type u_2inst✝⁹:TopologicalSpace Xinst✝⁸:PolishSpace Xinst✝⁷:MeasurableSpace Xinst✝⁶:BorelSpace Xinst✝⁵:TopologicalSpace Yinst✝⁴:PolishSpace Yinst✝³:MeasurableSpace Yinst✝²:BorelSpace YP:Measure XQ:Measure Yinst✝¹:IsProbabilityMeasure Pinst✝:IsProbabilityMeasure Qc:X × Y → ENNReal_hc:Continuous c⊢ ∃ π ∈ Couplings P Q, ∀ π' ∈ Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' All goals completed! 🐙
#72
Lidskii–Last eigenvalue-perturbation theorem
lidskii_last

Verso theorem preview

theorem declaration uses `sorry`lidskii_last {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitian⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ All goals completed! 🐙
#73
Koszul formula
koszul_formula

Lean theorem statement

/-- **Koszul formula.** For any smooth torsion-free metric-compatible
covariant derivative `cov` on `TM`, `2 ⟨∇_X Y, Z⟩` equals the cyclic sum
of directional derivatives `X·⟨Y, Z⟩ + Y·⟨X, Z⟩ − Z·⟨X, Y⟩` minus the
Lie-bracket cyclic sum `⟨X, [Y, Z]⟩ + ⟨Y, [X, Z]⟩ − ⟨Z, [X, Y]⟩`. -/
theorem koszul_formula
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
      [FiniteDimensional ℝ E] [CompleteSpace E]
    {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H}
    {M : Type*} [TopologicalSpace M] [ChartedSpace H M]
      [IsManifold I ∞ M]
    [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)]
    [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)]
    (cov : CovariantDerivative I E (TangentSpace I (M := M)))
    [ContMDiffCovariantDerivative cov ∞]
    (_htor : cov.torsion = 0) (_hmet : IsMetricCompatible cov)
    (X Y Z : Π x : M, TangentSpace I x)
    (_hX : CMDiff ∞ (T% X)) (_hY : CMDiff ∞ (T% Y)) (_hZ : CMDiff ∞ (T% Z))
    (x : M) :
    2 * inner ℝ (cov Y x (X x)) (Z x) =
      mvfderiv I (fun y : M => inner ℝ (Y y) (Z y)) x (X x)
      + mvfderiv I (fun y : M => inner ℝ (X y) (Z y)) x (Y x)
      - mvfderiv I (fun y : M => inner ℝ (X y) (Y y)) x (Z x)
      - inner ℝ (X x) (mlieBracket I Y Z x)
      - inner ℝ (Y x) (mlieBracket I X Z x)
      + inner ℝ (Z x) (mlieBracket I X Y x) := by
  sorry
#74
Furstenberg–Weiss topological multiple recurrence (single-transformation form)
furstenberg_topological

Verso theorem preview

theorem declaration uses `sorry`furstenberg_topological_recurrence {X : Type*} [MetricSpace X] [CompactSpace X] [Nonempty X] (T : X ≃ₜ X) : ∃ x : X, LeanEval.Dynamics.IsMultiplyRecurrent (T : X → X) x := X:Type u_1inst✝²:MetricSpace Xinst✝¹:CompactSpace Xinst✝:Nonempty XT:X ≃ₜ X⊢ ∃ x, IsMultiplyRecurrent (⇑T) x All goals completed! 🐙
#75
Brauer–Fowler theorem
brauer_fowler

Lean theorem statement

/-- **Brauer–Fowler theorem.** There is a function bounding the order
of a finite nonabelian simple group by the order of any involution
centralizer. -/
theorem brauer_fowler :
    ∃ f : ℕ → ℕ, ∀ (G : Type) [Group G] [Finite G],
      IsSimpleGroup G → (∃ a b : G, a * b ≠ b * a) →
      ∀ t : G, orderOf t = 2 →
        Nat.card G ≤ f (Nat.card (Subgroup.centralizer ({t} : Set G))) := by
  sorry
#76
Baer–Suzuki theorem
baer_suzuki

Verso theorem preview

theorem declaration uses `sorry`baer_suzuki {G : Type*} [Group G] [Finite G] {p : ℕ} [Fact p.Prime] (x : G) : x ∈ LeanEval.GroupTheory.Defs.pCore p G ↔ ∀ g : G, IsPGroup p (Subgroup.closure ({x, g * x * g⁻¹} : Set G)) := G:Type u_1inst✝²:Group Ginst✝¹:Finite Gp:ℕinst✝:Fact (Nat.Prime p)x:G⊢ x ∈ pCore p G ↔ ∀ (g : G), IsPGroup p ↥(Subgroup.closure {x, g * x * g⁻¹}) All goals completed! 🐙
#77
Brouwer fixed-point theorem
brouwer_fixed_point

Verso theorem preview

theorem declaration uses `sorry`brouwer_fixed_point {d : ℕ} {K : Set (EuclideanSpace ℝ (Fin d))} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (_hf_cont : ContinuousOn f K) (_hf_maps : MapsTo f K K) : ∃ x ∈ K, f x = x := d:ℕK:Set (EuclideanSpace ℝ (Fin d))_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)_hf_cont:ContinuousOn f K_hf_maps:MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#78
Schauder fixed-point theorem
schauder_fixed_point

Verso theorem preview

theorem declaration uses `sorry`schauder_fixed_point {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {K : Set E} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : E → E) (_hf_cont : ContinuousOn f K) (_hf_maps : Set.MapsTo f K K) : ∃ x ∈ K, f x = x := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace EK:Set E_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:E → E_hf_cont:ContinuousOn f K_hf_maps:Set.MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#79
Runge's theorem
runge_theorem

Verso theorem preview

theorem declaration uses `sorry`runge (K : Set ℂ) (_hK : IsCompact K) (U : Set ℂ) (_hU : IsOpen U) (_hKU : K ⊆ U) (f : ℂ → ℂ) (_hf : AnalyticOnNhd ℂ f U) (ε : ℝ) (_hε : 0 < ε) : ∃ p q : ℂ[X], (∀ z ∈ K, q.eval z ≠ 0) ∧ (∀ z ∈ K, ‖f z - p.eval z / q.eval z‖ < ε) := K:Set ℂ_hK:IsCompact KU:Set ℂ_hU:IsOpen U_hKU:K ⊆ Uf:ℂ → ℂ_hf:AnalyticOnNhd ℂ f Uε:ℝ_hε:0 < ε⊢ ∃ p q, (∀ z ∈ K, Polynomial.eval z q ≠ 0) ∧ ∀ z ∈ K, ‖f z - Polynomial.eval z p / Polynomial.eval z q‖ < ε All goals completed! 🐙
#80
Linear programming: maximum principle and vertex optimality
lp_maximum_principle

Verso theorem preview

/-- **Maximum principle for linear programming** (§101). A local maximiser of the LP objective on the feasible region is automatically a global maximiser; and whenever the objective is non-constant (`c ≠ 0`), the maximiser lies on the topological frontier of the feasible region. -/ theorem declaration uses `sorry`lp_maximum_principle {m n : ℕ} (lp : LinearProgram m n) (x : Fin m → ℝ) (_hx : x ∈ lp.feasible) (_hlocal : IsLocalMaxOn lp.objective lp.feasible x) : IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) := m:ℕn:ℕlp:LinearProgram m nx:Fin m → ℝ_hx:x ∈ lp.feasible_hlocal:IsLocalMaxOn lp.objective lp.feasible x⊢ IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) All goals completed! 🐙
/-- **Vertex optimality** (§101; the existence content of Dantzig's 1947 simplex algorithm). Every linear program with a nonempty bounded feasible region admits a global maximiser that is an extreme point (vertex) of the feasible region. -/ theorem declaration uses `sorry`simplex_algorithm {m n : ℕ} (lp : LinearProgram m n) (_hfeas : lp.feasible.Nonempty) (_hbdd : Bornology.IsBounded lp.feasible) : ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible := m:ℕn:ℕlp:LinearProgram m n_hfeas:lp.feasible.Nonempty_hbdd:Bornology.IsBounded lp.feasible⊢ ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible All goals completed! 🐙
#81
Symplectic matrices have determinant 1
symplectic_matrix_det

Verso theorem preview

theorem declaration uses `sorry`symplectic_matrix_det {l R : Type*} [DecidableEq l] [Fintype l] [CommRing R] {A : Matrix (l ⊕ l) (l ⊕ l) R} (_hA : A ∈ Matrix.symplecticGroup l R) : A.det = 1 := l:Type u_1R:Type u_2inst✝²:DecidableEq linst✝¹:Fintype linst✝:CommRing RA:Matrix (l ⊕ l) (l ⊕ l) R_hA:A ∈ symplecticGroup l R⊢ A.det = 1 All goals completed! 🐙
#82
Schur-Weyl duality: S_k image equals centralizer of GL(V) image
symAction_range_eq_centralizer_glAction

Verso theorem preview

theorem declaration uses `sorry`symAction_range_eq_centralizer_glAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(symAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(glAction R M k)) All goals completed! 🐙
#83
Schur-Weyl duality: GL(V) image equals centralizer of S_k image
glAction_range_eq_centralizer_symAction

Verso theorem preview

theorem declaration uses `sorry`glAction_range_eq_centralizer_symAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(glAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(symAction R M k)) All goals completed! 🐙
#84
Wiener's atom-detection formula
wiener_atom_detection

Verso theorem preview

theorem declaration uses `sorry`wiener_atom_detection (μ : Measure (AddCircle (2 * Real.pi))) [IsProbabilityMeasure μ] : Tendsto (fun N : ℕ => (1 / (N : ℝ)) * ∑ k ∈ Finset.Icc (1 : ℤ) N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' x : AddCircle (2 * Real.pi), ((μ {x}).toReal) ^ 2)) := μ:Measure (AddCircle (2 * π))inst✝:IsProbabilityMeasure μ⊢ Tendsto (fun N => 1 / ↑N * ∑ k ∈ Finset.Icc 1 ↑N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' (x : AddCircle (2 * π)), (μ {x}).toReal ^ 2)) All goals completed! 🐙
#85
Kuznetsov's theorem: finitely presented simple groups have solvable word problem
boone_higman_simple

Verso theorem preview

theorem declaration uses `sorry`boone_higman_simple {G : Type*} [Group G] [IsSimpleGroup G] {n : ℕ} (φ : FreeGroup (Fin n) →* G) (_hsurj : Function.Surjective φ) (_hker : (MonoidHom.ker φ).IsNormalClosureFG) : LeanEval.GroupTheory.BooneHigmanSimpleProblem.WordProblemSolvable φ := G:Type u_1inst✝¹:Group Ginst✝:IsSimpleGroup Gn:ℕφ:FreeGroup (Fin n) →* G_hsurj:Function.Surjective ⇑φ_hker:φ.ker.IsNormalClosureFG⊢ WordProblemSolvable φ All goals completed! 🐙
#86
Bourbaki's locally convex extension of Banach–Alaoglu
banach_alaoglu_bourbaki

Verso theorem preview

theorem declaration uses `sorry`banach_alaoglu_bourbaki (E : Type*) [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] (U : Set E) (_hU : U ∈ 𝓝 (0 : E)) : IsCompact (LeanEval.Analysis.weakStarPolar E U) := E:Type u_1inst✝⁵:AddCommGroup Einst✝⁴:Module ℝ Einst✝³:TopologicalSpace Einst✝²:ContinuousAdd Einst✝¹:ContinuousSMul ℝ Einst✝:LocallyConvexSpace ℝ EU:Set E_hU:U ∈ 𝓝 0⊢ IsCompact (weakStarPolar E U) All goals completed! 🐙
#87
Bing's house with two rooms is contractible
contractibleSpace_houseWithTwoRooms

Verso theorem preview

theorem declaration uses `sorry`contractibleSpace_houseWithTwoRooms : ContractibleSpace LeanEval.Topology.HouseWithTwoRooms := ⊢ ContractibleSpace ↑HouseWithTwoRooms All goals completed! 🐙
#88
Abel–Ruffini theorem
abel_ruffini

Verso theorem preview

theorem declaration uses `sorry`abel_ruffini (n : ℕ) (_hn : 1 ≤ n) : (∀ p : ℚ[X], p.natDegree = n → ∀ x : ℂ, aeval x p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 := n:ℕ_hn:1 ≤ n⊢ (∀ (p : ℚ[X]), p.natDegree = n → ∀ (x : ℂ), (aeval x) p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 All goals completed! 🐙
#89
Existence of a non-isotopic pair of oriented two-component links
exists_nonisotopic_link

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_link : ∃ L₁ L₂ : LeanEval.KnotTheory.TwoLink, ¬ L₁.Isotopic L₂ := ⊢ ∃ L₁ L₂, ¬L₁.Isotopic L₂ All goals completed! 🐙
#90
von Neumann double commutant theorem
vonNeumann_doubleCommutant_tfae

Verso theorem preview

theorem declaration uses `sorry`vonNeumann_doubleCommutant_tfae {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : StarSubalgebra ℂ (H →L[ℂ] H)) : List.TFAE [ Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S , IsClosed (ContinuousLinearMapWOT.ofCLM '' (S : Set (H →L[ℂ] H))) , IsClosed (ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H '' (S : Set (H →L[ℂ] H))) ] := H:Type u_1inst✝²:NormedAddCommGroup Hinst✝¹:InnerProductSpace ℂ Hinst✝:CompleteSpace HS:StarSubalgebra ℂ (H →L[ℂ] H)⊢ [(↑S).centralizer.centralizer = ↑S, IsClosed (ContinuousLinearMapWOT.ofCLM '' ↑S), IsClosed (⇑(ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H) '' ↑S)].TFAE All goals completed! 🐙
#91
Burnside p^a q^b theorem
finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow

Verso theorem preview

theorem declaration uses `sorry`finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow {G : Type*} [Group G] [Fintype G] {p q a b : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q) (hpq : p ≠ q) (hcard : Fintype.card G = p ^ a * q ^ b) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Fintype Gp:ℕq:ℕa:ℕb:ℕhp:Nat.Prime phq:Nat.Prime qhpq:p ≠ qhcard:Fintype.card G = p ^ a * q ^ b⊢ IsSolvable G All goals completed! 🐙
#92
Pointwise and Cesàro convergence of Fourier series (Dirichlet, Fejér)
fourier_dirichlet_fejer

Verso theorem preview

/-- **Dirichlet's pointwise convergence theorem** (§46). For every `C¹` 2π-periodic complex function `f`, the symmetric Fourier partial sums `S_N(f)(x)` converge to `f(x)` at every point `x ∈ ℝ`. -/ theorem declaration uses `sorry`dirichlet_pointwise {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hC1 : ContDiff ℝ 1 f) (x : ℝ) : Tendsto (fun N : ℕ => fourierPartialSum f N x) atTop (𝓝 (f x)) := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hC1:ContDiff ℝ 1 fx:ℝ⊢ Tendsto (fun N => fourierPartialSum f N x) atTop (𝓝 (f x)) All goals completed! 🐙
/-- **Fejér's theorem** (§46). For every *continuous* 2π-periodic complex function `f` — without the `C¹` hypothesis of Dirichlet's theorem — the Cesàro means `σ_N(f)` of the symmetric Fourier partial sums converge to `f` uniformly on `ℝ`. -/ theorem declaration uses `sorry`fejer {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hcont : Continuous f) : TendstoUniformly (fun N : ℕ => fourierCesaroMean f N) f atTop := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hcont:Continuous f⊢ TendstoUniformly (fun N => fourierCesaroMean f N) f atTop All goals completed! 🐙
#93
Real cyclotomic integer with house at most 2
cyclotomic_integer_house_le_two

Verso theorem preview

theorem declaration uses `sorry`cyclotomic_integer_house_le_two {K : Type*} [Field K] [NumberField K] [Algebra ℚ K] (n : ℕ) [NeZero n] [IsCyclotomicExtension {n} ℚ K] {β : K} (hβ_int : IsIntegral ℤ β) (hβ_real : β ∈ NumberField.maximalRealSubfield K) : house β ≤ 2 → house β = 2 ∨ ∃ m : ℕ, 0 < m ∧ house β = 2 * Real.cos (Real.pi / m) := K:Type u_1inst✝⁴:Field Kinst✝³:NumberField Kinst✝²:Algebra ℚ Kn:ℕinst✝¹:NeZero ninst✝:IsCyclotomicExtension {n} ℚ Kβ:Khβ_int:IsIntegral ℤ βhβ_real:β ∈ maximalRealSubfield K⊢ house β ≤ 2 → house β = 2 ∨ ∃ m, 0 < m ∧ house β = 2 * Real.cos (Real.pi / ↑m) All goals completed! 🐙
#94
Pell solutions are convergents of √d
pell_solution_convergent

Verso theorem preview

theorem declaration uses `sorry`pell_solution_is_convergent (d : ℤ) (_hd : Squarefree d) (_hd0 : 0 < d) (x y : ℤ) (_hx : 0 < x) (_hy : 0 < y) (_hsol : x ^ 2 - d * y ^ 2 = 1) : ∃ n : ℕ, (GenContFract.of (Real.sqrt (d : ℝ))).convs n = (x : ℝ) / (y : ℝ) := d:ℤ_hd:Squarefree d_hd0:0 < dx:ℤy:ℤ_hx:0 < x_hy:0 < y_hsol:x ^ 2 - d * y ^ 2 = 1⊢ ∃ n, (GenContFract.of √↑d).convs n = ↑x / ↑y All goals completed! 🐙
#95
Linear ODE with negative-real-part eigenvalues is asymptotically stable
linear_ode_asymptotic_stability

Verso theorem preview

theorem declaration uses `sorry`linear_ode_asymptotic_stability (n : ℕ) (A : Matrix (Fin n) (Fin n) ℝ) (hA : ∀ μ : ℂ, Module.End.HasEigenvalue (Matrix.toLin' (A.map (algebraMap ℝ ℂ))) μ → μ.re < 0) (x : ℝ → (Fin n → ℝ)) (hx : ∀ t : ℝ, 0 < t → HasDerivAt x (A.mulVec (x t)) t) : Filter.Tendsto (fun t : ℝ => ‖x t‖) Filter.atTop (nhds 0) := n:ℕA:Matrix (Fin n) (Fin n) ℝhA:∀ (μ : ℂ), Module.End.HasEigenvalue (Matrix.toLin' (A.map ⇑(algebraMap ℝ ℂ))) μ → μ.re < 0x:ℝ → Fin n → ℝhx:∀ (t : ℝ), 0 < t → HasDerivAt x (A *ᵥ x t) t⊢ Filter.Tendsto (fun t => ‖x t‖) Filter.atTop (nhds 0) All goals completed! 🐙
#96
Perron-Frobenius for irreducible nonnegative matrices
irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius

Verso theorem preview

theorem declaration uses `sorry`irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius {n : Type*} [Fintype n] [DecidableEq n] [Nonempty n] (A : Matrix n n ℝ) (hA : A.IsIrreducible) : ∃ v : n → ℝ, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ (∀ i, 0 < v i) := n:Type u_1inst✝²:Fintype ninst✝¹:DecidableEq ninst✝:Nonempty nA:Matrix n n ℝhA:A.IsIrreducible⊢ ∃ v, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ ∀ (i : n), 0 < v i All goals completed! 🐙
#97
Complementary polynomial on the unit circle
exists_complementary_polynomial_on_unit_circle

Verso theorem preview

theorem declaration uses `sorry`exists_complementary_polynomial_on_unit_circle (P : ℂ[X]) (hP : ∀ z : Circle, ‖P.eval (z : ℂ)‖ ≤ 1) : ∃ Q : ℂ[X], Q.natDegree ≤ P.natDegree ∧ ∀ z : Circle, ‖P.eval (z : ℂ)‖ ^ 2 + ‖Q.eval (z : ℂ)‖ ^ 2 = 1 := P:ℂ[X]hP:∀ (z : Circle), ‖eval (↑z) P‖ ≤ 1⊢ ∃ Q, Q.natDegree ≤ P.natDegree ∧ ∀ (z : Circle), ‖eval (↑z) P‖ ^ 2 + ‖eval (↑z) Q‖ ^ 2 = 1 All goals completed! 🐙
#98
Rouche theorem via zero counting
rouche_zero_count_eq

Verso theorem preview

theorem declaration uses `sorry`rouche_zero_count_eq {f g : ℂ → ℂ} {R : ℝ} (hR : 0 < R) (hf : MeromorphicNFOn f Set.univ) (hg : AnalyticOn ℂ g Set.univ) (hbound : ∀ z : ℂ, ‖z‖ = R → ‖g z‖ < ‖f z‖) : (∑ᶠ z, ((divisor (f + g) (Metric.closedBall 0 R))⁺) z) = (∑ᶠ z, ((divisor f (Metric.closedBall 0 R))⁺) z) := f:ℂ → ℂg:ℂ → ℂR:ℝhR:0 < Rhf:MeromorphicNFOn f Set.univhg:AnalyticOn ℂ g Set.univhbound:∀ (z : ℂ), ‖z‖ = R → ‖g z‖ < ‖f z‖⊢ ∑ᶠ (z : ℂ), (divisor (f + g) (Metric.closedBall 0 R))⁺ z = ∑ᶠ (z : ℂ), (divisor f (Metric.closedBall 0 R))⁺ z All goals completed! 🐙
#99
Character values of finite groups lie in cyclotomic fields
brauer_character_in_cyclotomic

Verso theorem preview

theorem declaration uses `sorry`brauer_character_in_cyclotomic (G : Type) [Group G] [Fintype G] : ∃ φ : CyclotomicField (Monoid.exponent G) ℚ →+* ℂ, ∀ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (_ : FiniteDimensional ℂ V) (ρ : Representation ℂ G V) (g : G), LinearMap.trace ℂ V (ρ g) ∈ φ.range := G:Typeinst✝¹:Group Ginst✝:Fintype G⊢ ∃ φ, ∀ (V : Type) (x : AddCommGroup V) (x_1 : Module ℂ V), FiniteDimensional ℂ V → ∀ (ρ : Representation ℂ G V) (g : G), (LinearMap.trace ℂ V) (ρ g) ∈ φ.range All goals completed! 🐙
#100
Entrywise exponential of a PSD matrix is PSD
posSemidef_map_exp

Verso theorem preview

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n] {A : Matrix n n ℝ} (hA : A.PosSemidef) : (A.map Real.exp).PosSemidef := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef All goals completed! 🐙
#101
Oppenheim's inequality for Hadamard products
oppenheim_inequality

Verso theorem preview

theorem declaration uses `sorry`oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : A.det * ∏ i, B i i ≤ (A ⊙ B).det := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝB:Matrix n n ℝhA:A.PosSemidefhB:B.PosSemidef⊢ A.det * ∏ i, B i i ≤ (A ⊙ B).det All goals completed! 🐙
#102
Gaussian heat kernel solves the 1D heat equation
heat_kernel_solves_heat_equation

Verso theorem preview

theorem declaration uses `sorry`heat_kernel_solves_heat_equation (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_bdd : ∃ M : ℝ, ∀ x, |f x| ≤ M) : -- The PDE on (0, ∞) × ℝ. (∀ t : ℝ, 0 < t → ∀ x : ℝ, ∃ ux : ℝ → ℝ, ∃ uxx : ℝ, (∀ y : ℝ, HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ -- Initial condition recovered as a one-sided limit at t = 0. (∀ x : ℝ, Filter.Tendsto (fun t : ℝ => heatSolution f t x) (nhdsWithin (0 : ℝ) (Set.Ioi 0)) (nhds (f x))) := f:ℝ → ℝhf_cont:Continuous fhf_bdd:∃ M, ∀ (x : ℝ), |f x| ≤ M⊢ (∀ (t : ℝ), 0 < t → ∀ (x : ℝ), ∃ ux uxx, (∀ (y : ℝ), HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ ∀ (x : ℝ), Filter.Tendsto (fun t => heatSolution f t x) (nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) All goals completed! 🐙
#103
pi_1 of the circle is Z
pi1_circle_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi1_circle_mulEquiv_int : Nonempty (HomotopyGroup.Pi 1 Circle (1 : Circle) ≃* Multiplicative ℤ) := ⊢ Nonempty (HomotopyGroup.Pi 1 Circle 1 ≃* Multiplicative ℤ) All goals completed! 🐙
#104
Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2
dirichlet_eigenvalues_eq_nat_sq

Verso theorem preview

theorem declaration uses `sorry`dirichlet_eigenvalues_eq_nat_sq (lam : ℝ) : (∃ (y : ℝ → ℝ) (J : Set ℝ), IsOpen J ∧ Set.Icc (0 : ℝ) Real.pi ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y Real.pi = 0 ∧ ∃ x ∈ Set.Ioo (0 : ℝ) Real.pi, y x ≠ 0) ↔ ∃ n : ℕ, 0 < n ∧ lam = (n : ℝ) ^ 2 := lam:ℝ⊢ (∃ y J, IsOpen J ∧ Set.Icc 0 π ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y π = 0 ∧ ∃ x ∈ Set.Ioo 0 π, y x ≠ 0) ↔ ∃ n, 0 < n ∧ lam = ↑n ^ 2 All goals completed! 🐙
#105
Minkowski-Caratheodory theorem
mem_convexHull_finset_extremePoints_of_mem_compact_convex

Verso theorem preview

theorem declaration uses `sorry`mem_convexHull_finset_extremePoints_of_mem_compact_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hscomp : IsCompact s) (hsconv : Convex ℝ s) (hx : x ∈ s) : ∃ t : Finset E, (↑t : Set E) ⊆ s.extremePoints ℝ ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ convexHull ℝ (↑t : Set E) := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:FiniteDimensional ℝ Es:Set Ex:Ehscomp:IsCompact shsconv:Convex ℝ shx:x ∈ s⊢ ∃ t, ↑t ⊆ extremePoints ℝ s ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ (convexHull ℝ) ↑t All goals completed! 🐙
#106
Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#107
Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#108
Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#109
Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#110
Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#111
Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#112

Submission history

First submissionJul 9, 2026
Last submissionJul 21, 2026

Contributors

Morgan-Griffiths111eohjelle3
5Tau (caj.al)104 solved

Other solved problems

Darboux's theorem (symplectic forms are locally standard)
darboux

Verso theorem preview

theorem declaration uses `sorry`darboux {n : ℕ} {U : Set (LeanEval.Geometry.Darboux.E n)} (_hU : IsOpen U) (α : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n [⋀^Fin 2]→L[ℝ] ℝ) (_hα : LeanEval.Geometry.Darboux.IsSymplecticOn α U) {x : LeanEval.Geometry.Darboux.E n} (_hx : x ∈ U) : ∃ φ : OpenPartialHomeomorph (LeanEval.Geometry.Darboux.E n) (LeanEval.Geometry.Darboux.E n), x ∈ φ.source ∧ φ.source ⊆ U ∧ ContDiffOn ℝ ∞ (φ : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n) φ.source ∧ ContDiffOn ℝ ∞ (φ.symm : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n) φ.target ∧ ∀ z ∈ φ.target, LeanEval.Geometry.Darboux.IsDarbouxNormal ((α (φ.symm z)).compContinuousLinearMap (fderiv ℝ (φ.symm : LeanEval.Geometry.Darboux.E n → LeanEval.Geometry.Darboux.E n) z)) := n:ℕU:Set (E n)_hU:IsOpen Uα:E n → E n [⋀^Fin 2]→L[ℝ] ℝ_hα:IsSymplecticOn α Ux:E n_hx:x ∈ U⊢ ∃ φ, x ∈ φ.source ∧ φ.source ⊆ U ∧ ContDiffOn ℝ ∞ (↑φ) φ.source ∧ ContDiffOn ℝ ∞ (↑φ.symm) φ.target ∧ ∀ z ∈ φ.target, IsDarbouxNormal ((α (↑φ.symm z)).compContinuousLinearMap (fderiv ℝ (↑φ.symm) z)) All goals completed! 🐙
#1
Kolmogorov–Arnold superposition theorem (non-universal Lorentz form)
kolmogorov_arnold_superposition

Verso theorem preview

theorem declaration uses `sorry`kolmogorov_arnold (n : ℕ) (_hn : 1 ≤ n) (f : (Fin n → ℝ) → ℝ) (_hf : ContinuousOn f (Set.Icc 0 1)) : ∃ (g : ℝ → ℝ) (φ : Fin (2 * n + 1) → Fin n → ℝ → ℝ), Continuous g ∧ (∀ k l, Continuous (φ k l)) ∧ ∀ x ∈ Set.Icc (0 : Fin n → ℝ) 1, f x = ∑ k, g (∑ l, φ k l (x l)) := n:ℕ_hn:1 ≤ nf:(Fin n → ℝ) → ℝ_hf:ContinuousOn f (Set.Icc 0 1)⊢ ∃ g φ, Continuous g ∧ (∀ (k : Fin (2 * n + 1)) (l : Fin n), Continuous (φ k l)) ∧ ∀ x ∈ Set.Icc 0 1, f x = ∑ k, g (∑ l, φ k l (x l)) All goals completed! 🐙
#4
Sobolev embedding theorem (Morrey regime)
sobolev_embedding_morrey

Verso theorem preview

theorem declaration uses `sorry`sobolev_embedding {n k r : ℕ} {α p : ℝ} (_hp : (n : ℝ) < p) (_hα : 0 < α) (_hα1 : α ≤ 1) (_hgap : (r : ℝ) + α < (k : ℝ) - n / p) (f : LeanEval.Analysis.SobolevMorreyProblem.E n → ℝ) (_hf : LeanEval.Analysis.SobolevMorreyProblem.MemSobolevWk k (ENNReal.ofReal p) f) : ∃ g : LeanEval.Analysis.SobolevMorreyProblem.E n → ℝ, f =ᵐ[volume] g ∧ LeanEval.Analysis.SobolevMorreyProblem.MemHolder r α g := n:ℕk:ℕr:ℕα:ℝp:ℝ_hp:↑n < p_hα:0 < α_hα1:α ≤ 1_hgap:↑r + α < ↑k - ↑n / pf:E n → ℝ_hf:MemSobolevWk k (ENNReal.ofReal p) f⊢ ∃ g, f =ᵐ[volume] g ∧ LeanEval.Analysis.SobolevMorreyProblem.MemHolder r α g All goals completed! 🐙
#5
Fundamental theorem of topos theory
fundamental_topos_theory

Verso theorem preview

theorem declaration uses `sorry`fundamental_topos_theory {E : Type*} [Category E] (hE : LeanEval.ToposTheory.IsTopos E) (X : E) : LeanEval.ToposTheory.IsTopos (Over X) := E:Type u_1inst✝:Category.{u_2, u_1} EhE:IsTopos EX:E⊢ IsTopos (Over X) All goals completed! 🐙
#6
Wigner semicircle law
wigner_semicircle

Verso theorem preview

theorem declaration uses `sorry`wigner_semicircle {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] (X : ℕ → ℕ → Ω → ℝ) (_hX_meas : ∀ i j, Measurable (X i j)) (_hX_indep : iIndepFun (fun ij : {p : ℕ × ℕ // p.1 ≤ p.2} => X ij.val.1 ij.val.2) μ) (_hX_iid : ∀ i j i' j', i ≤ j → i' ≤ j' → ProbabilityTheory.IdentDistrib (X i j) (X i' j') μ μ) (_hX_int : ∀ i j, i ≤ j → Integrable (X i j) μ) (_hX_sq_int : ∀ i j, i ≤ j → Integrable (fun ω => (X i j ω) ^ 2) μ) (_hX_mean : ∀ i j, i ≤ j → ∫ ω, X i j ω ∂μ = 0) (_hX_var : ∀ i j, i ≤ j → ∫ ω, (X i j ω) ^ 2 ∂μ = 1) : ∀ᵐ ω ∂μ, ∀ (f : ℝ → ℝ), Continuous f → (∃ M, ∀ x, ‖f x‖ ≤ M) → Tendsto (fun n : ℕ => ∫ x, f x ∂ (empiricalSpectralMeasureHerm (wignerMatrix_isHermitian X n ω)).map (fun x : ℝ => x / Real.sqrt n)) atTop (𝓝 (∫ x, f x ∂semicircleLaw)) := Ω:Type u_1inst✝¹:MeasurableSpace Ωμ:Measure Ωinst✝:IsProbabilityMeasure μX:ℕ → ℕ → Ω → ℝ_hX_meas:∀ (i j : ℕ), Measurable (X i j)_hX_indep:iIndepFun (fun ij => X (↑ij).1 (↑ij).2) μ_hX_iid:∀ (i j i' j' : ℕ), i ≤ j → i' ≤ j' → IdentDistrib (X i j) (X i' j') μ μ_hX_int:∀ (i j : ℕ), i ≤ j → Integrable (X i j) μ_hX_sq_int:∀ (i j : ℕ), i ≤ j → Integrable (fun ω => X i j ω ^ 2) μ_hX_mean:∀ (i j : ℕ), i ≤ j → ∫ (ω : Ω), X i j ω ∂μ = 0_hX_var:∀ (i j : ℕ), i ≤ j → ∫ (ω : Ω), X i j ω ^ 2 ∂μ = 1⊢ ∀ᵐ (ω : Ω) ∂μ, ∀ (f : ℝ → ℝ), Continuous f → (∃ M, ∀ (x : ℝ), ‖f x‖ ≤ M) → Tendsto (fun n => ∫ (x : ℝ), f x ∂Measure.map (fun x => x / √↑n) (empiricalSpectralMeasureHerm ⋯)) atTop (𝓝 (∫ (x : ℝ), f x ∂semicircleLaw)) All goals completed! 🐙
#7
Wiener–Lévy theorem
wiener_levy_analytic_calculus

Verso theorem preview

theorem declaration uses `sorry`wiener_levy_analytic_calculus (f : C(AddCircle T, ℂ)) (φ : ℂ → ℂ) (U : Set ℂ) (hf : LeanEval.Analysis.WienerLevy.InWienerAlgebra f) (hU : IsOpen U) (hrange : range f ⊆ U) (hφ : AnalyticOnNhd ℂ φ U) : ∃ g : C(AddCircle T, ℂ), (∀ x, g x = φ (f x)) ∧ LeanEval.Analysis.WienerLevy.InWienerAlgebra g := T:ℝinst✝:Fact (0 < T)f:C(AddCircle T, ℂ)φ:ℂ → ℂU:Set ℂhf:InWienerAlgebra fhU:IsOpen Uhrange:range ⇑f ⊆ Uhφ:AnalyticOnNhd ℂ φ U⊢ ∃ g, (∀ (x : AddCircle T), g x = φ (f x)) ∧ InWienerAlgebra g All goals completed! 🐙
#8
Wiener's 1/f theorem
wiener_inverse_closed

Verso theorem preview

theorem declaration uses `sorry`wiener_inverse_closed (f : C(AddCircle T, ℂ)) (hf : LeanEval.Analysis.WienerOneOverF.InWienerAlgebra f) (hzero : ∀ x, f x ≠ 0) : ∃ g : C(AddCircle T, ℂ), (∀ x, g x = (f x)⁻¹) ∧ LeanEval.Analysis.WienerOneOverF.InWienerAlgebra g := T:ℝinst✝:Fact (0 < T)f:C(AddCircle T, ℂ)hf:InWienerAlgebra fhzero:∀ (x : AddCircle T), f x ≠ 0⊢ ∃ g, (∀ (x : AddCircle T), g x = (f x)⁻¹) ∧ InWienerAlgebra g All goals completed! 🐙
#9
Fundamental theorem of Riemannian geometry (Levi-Civita)
levi_civita_exists_unique

Verso theorem preview

theorem declaration uses `sorry`levi_civita_exists_unique {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [CompleteSpace E] {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type*} [TopologicalSpace M] [T2Space M] [ChartedSpace H M] [IsManifold I ∞ M] [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)] [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)] : ∃ cov : CovariantDerivative I E (TangentSpace I (M := M)), (ContMDiffCovariantDerivative cov ∞ ∧ cov.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov) ∧ ∀ cov' : CovariantDerivative I E (TangentSpace I (M := M)), (ContMDiffCovariantDerivative cov' ∞ ∧ cov'.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov') → LeanEval.Geometry.LeviCivita.SameOnSmooth cov cov' := E:Type u_1inst✝¹⁰:NormedAddCommGroup Einst✝⁹:NormedSpace ℝ Einst✝⁸:FiniteDimensional ℝ Einst✝⁷:CompleteSpace EH:Type u_2inst✝⁶:TopologicalSpace HI:ModelWithCorners ℝ E HM:Type u_3inst✝⁵:TopologicalSpace Minst✝⁴:T2Space Minst✝³:ChartedSpace H Minst✝²:IsManifold I ∞ Minst✝¹:RiemannianBundle fun x => TangentSpace I xinst✝:IsContMDiffRiemannianBundle I ∞ E fun x => TangentSpace I x⊢ ∃ cov, (cov.ContMDiffCovariantDerivative ∞ ∧ cov.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov) ∧ ∀ (cov' : CovariantDerivative I E (TangentSpace I)), cov'.ContMDiffCovariantDerivative ∞ ∧ cov'.torsion = 0 ∧ LeanEval.Geometry.LeviCivita.IsMetricCompatible cov' → SameOnSmooth cov cov' All goals completed! 🐙
#10
General recursive equals Turing computable
turing_recursive_equiv

Lean theorem statement

/-- **General recursive = Turing computable** (total form). A total function
`f : ℕ → ℕ` is recursive (`Computable`, i.e. partial recursive as a partial
function) **iff** it is computed by some Turing machine (mathlib's `FinTM2`
model) under the standard binary encoding of `ℕ`. This is Knill's class
equality; the backward direction (TM-computable ⇒ recursive) is absent from
mathlib. -/
theorem turing_recursive_equiv (f : ℕ → ℕ) :
    Computable f ↔ Nonempty (TM2Computable encodeNat encodeNat f) := by
  sorry
#11
Brun's theorem (convergence of the twin-prime reciprocal sum)
brun_constant_converges

Lean theorem statement

/-- **Brun's theorem.** The reciprocal sum over twin-prime pairs converges. -/
theorem brun_constant_converges :
    Summable twinPrimeReciprocalTerm := by
  sorry
#12
Radial symmetry for positive semilinear Poisson solutions
semilinear_poisson_radial_symmetry

Lean theorem statement

/-- **Radial symmetry theorem.** Let `u ∈ C^2(closedBall 0 1)` be a positive
solution of the semilinear Poisson problem `-Δ u = f(u)` in the open unit ball,
with zero Dirichlet boundary values on the unit sphere. If `f : ℝ → ℝ` is
Lipschitz, then `u` is radial: `u x = v ‖x‖` for a nonnegative strictly
decreasing radial profile `v` on `[0, 1]`. -/
theorem semilinear_poisson_radial_symmetry {n : ℕ} (hn : 0 < n)
    {f : ℝ → ℝ} (u : EuclideanSpace ℝ (Fin n) → ℝ)
    (hf_lipschitz : ∃ K : ℝ≥0, LipschitzWith K f)
    (hu_c2 : ContDiffOn ℝ 2 u (closedBall 0 1))
    (hu_solve : SolvesSemilinearPoisson f u)
    (hu_positive : ∀ x ∈ ball 0 1, 0 < u x) :
    ∃ v : ℝ → ℝ≥0,
      StrictAntiOn v (Set.Icc (0 : ℝ) 1) ∧
        ∀ x ∈ closedBall 0 1, u x = v ‖x‖ := by
  sorry
#13
Strong Subadditivity of von Neumann Entropy
strong_subadditivity

Lean theorem statement

/-- Strong subadditivity of quantum entropy. We relax the common assumption that M is a normalized
 density matrix to the simpler statement that it's PSD, which holds since normalization just produces
 a positive affine transformation on the entropy. -/
theorem strong_subadditivity (M_ABC : Matrix (A × B × C) (A × B × C) ℂ) (h : M_ABC.PosSemidef) :
    let M_AB : Matrix (A × B) (A × B) ℂ :=
      .traceRight <| M_ABC.reindex (.symm <| .prodAssoc ..) (.symm <| .prodAssoc ..)
    let M_BC : Matrix (B × C) (B × C) ℂ := M_ABC.traceLeft
    let M_B : Matrix B B ℂ := M_BC.traceRight
    entropy M_ABC + entropy M_B ≤ entropy M_AB + entropy M_BC := by
  sorry
#14
No bounded projection from L^1 onto H^1
H1_not_closedComplemented

Verso theorem preview

theorem declaration uses `sorry`H1_not_closedComplemented : ¬ LeanEval.Analysis.H1.ClosedComplemented := ⊢ ¬H1.ClosedComplemented All goals completed! 🐙
#15
Anosov–Bowen shadowing lemma
anosov_bowen_shadowing

Lean theorem statement

/-- **Anosov–Bowen shadowing lemma** (Anosov 1967; Bowen 1975). Every
compact hyperbolic invariant set has the shadowing property. -/
theorem hyperbolic_has_shadowing
    (T : E d ≃ₜ E d) (K : Set (E d))
    (_hKc : IsCompact K) (_hK : IsHyperbolic T K) :
    HasShadowing (T : E d → E d) K := by
  sorry
#16
Frobenius determinant theorem
frobenius_group_determinant

Lean theorem statement

/-- **Frobenius determinant theorem** (§171). The group determinant factors as
a product of irreducible polynomials, each appearing to the power of its own
(total) degree `d_j = deg p_j`, with the factors pairwise non-associated
(*distinct*) and their number equal to the number of conjugacy classes of `G`.
-/
theorem frobenius_group_determinant
    (G : Type*) [Group G] [Fintype G] [DecidableEq G] :
    ∃ (r : ℕ) (p : Fin r → MvPolynomial G ℂ),
      r = Nat.card (ConjClasses G) ∧
      (∀ j, Irreducible (p j)) ∧
      (∀ i j, i ≠ j → ¬ Associated (p i) (p j)) ∧
      groupDeterminant G = ∏ j, (p j) ^ (p j).totalDegree := by
  sorry
#17
Poincaré–Siegel linearisation theorem
poincare_siegel_linearisation

Lean theorem statement

/-- **Poincaré–Siegel linearisation theorem.** If `α` is Diophantine,
`λ = e^{2πiα}`, and `f` is holomorphic near `0` with `f 0 = 0` and
`f'(0) = λ`, then there is a holomorphic germ `u` with `u 0 = 0`,
`u'(0) = 1`, and `f(u z) = u(λ z)` for `z` near `0`. -/
theorem poincare_siegel
    (α : ℝ) (_hα : IsDiophantine α)
    (lam : ℂ) (_hlam : lam = Complex.exp (2 * Real.pi * Complex.I * (α : ℂ)))
    (f : ℂ → ℂ) (_hf : AnalyticAt ℂ f 0) (_hf0 : f 0 = 0)
    (_hmult : deriv f 0 = lam) :
    ∃ u : ℂ → ℂ, AnalyticAt ℂ u 0 ∧ u 0 = 0 ∧ deriv u 0 = 1 ∧
      ∀ᶠ z in nhds (0 : ℂ), f (u z) = u (lam * z) := by
  sorry
#18
Riesz brothers' theorem
riesz_brothers_theorem

Lean theorem statement

/-- **Riesz brothers' theorem.** Let `μ` be a complex Borel measure on
the unit circle such that `∫ z^n dμ = 0` for every `n ≥ 1`. Then `μ` is
absolutely continuous with respect to Haar measure, and more precisely
`dμ = h dm` for some `h ∈ H^1`, i.e. some integrable density whose negative
Fourier coefficients vanish. -/
theorem riesz_brothers_theorem (μ : ComplexMeasure UnitAddCircle)
    (hμ : ∀ n : ℕ, 1 ≤ n → ∫ᵛ z, fourier n z ∂[ContinuousLinearMap.mul ℝ ℂ; μ] = 0) :
    μ ≪ᵥ AddCircle.haarAddCircle.toENNRealVectorMeasure := by
  sorry
#19
Kirk's normal-structure fixed point theorem
kirk_normal_structure

Lean theorem statement

/-- **Kirk's normal-structure fixed-point theorem** (§228).  A nonexpansive
self-map of a nonempty bounded closed convex subset of a reflexive Banach
space with normal structure has a fixed point. -/
theorem kirk_normal_structure [CompleteSpace E]
    (hE_reflexive : Function.Surjective (NormedSpace.inclusionInDoubleDual ℝ E))
    (K : Set E) (hK_nonempty : K.Nonempty) (hK_closed : IsClosed K)
    (hK_bounded : Bornology.IsBounded K) (hK_convex : Convex ℝ K)
    (hK_normal : HasNormalStructure K) (T : K → K)
    (hT : IsNonexpansiveSelfMap K T) :
    ∃ x : K, IsFixedPt T x := by
  sorry
#20
Nyquist–Shannon sampling theorem
nyquist_shannon_sampling

Lean theorem statement

/-- **Nyquist--Shannon sampling theorem / Whittaker--Shannon interpolation
formula** for Schwartz functions with Fourier support in the mathlib-convention
Nyquist band `[-1/2, 1/2]`.

The explicit `Summable` conjunct records the convergence content of the
cardinal series, rather than relying only on Lean's total `tsum`.
-/
theorem nyquist_shannon_sampling (f : 𝓢(ℝ, ℂ)) (hf : FourierSupportedInNyquist f) :
    ∀ t : ℝ,
      Summable (fun n : ℤ ↦ f (n : ℝ) * sinc (Real.pi * ((n : ℝ) - t))) ∧
        f t =
          ∑' n : ℤ, f (n : ℝ) * sinc (Real.pi * ((n : ℝ) - t)) := by
  sorry
#21
Gauss-Wantzel constructible regular polygon theorem
gauss_wantzel_constructible_polygon

Lean theorem statement

/-- **Gauss-Wantzel constructible polygon theorem** (§174): a regular `n`-gon
is straightedge-and-compass constructible exactly for the Gauss-Wantzel
integers. -/
theorem gauss_wantzel_constructible_polygon (n : ℕ) (hn : 3 ≤ n) :
    IsConstructible (Real.cos (2 * Real.pi / n)) ↔ GaussWantzelNumber n := by
  sorry
#22
Trace Cayley-Hamilton / Newton identity
trace_cayley_hamilton_newton

Lean theorem statement

/-- The trace Cayley-Hamilton / Newton identity:
`k c_k + ∑_{j=1}^k tr(A^j) c_{k-j} = 0`, where
`χ_A(X) = X^N + c₁ X^(N-1) + ... + c_N`.

For `k > N`, `c_k = 0`, and the remaining relation is the trace of
Cayley-Hamilton multiplied by a power of `A`. -/
theorem trace_cayley_hamilton_newton {R : Type*} [CommRing R]
    (A : Matrix n n R) {k : ℕ} (hk : 1 ≤ k) :
    (k : R) * charpolyDescendingCoeff A k +
        ∑ j ∈ Finset.Icc 1 k,
          trace (A ^ j) * charpolyDescendingCoeff A (k - j) = 0 := by
  sorry
#23
Normal spectral theorem
normal_spectral_theorem

Lean theorem statement

/-- **Spectral theorem** (§14). A complex matrix `A` is **normal**
(`Aᴴ A = A Aᴴ`, i.e. `IsStarNormal A`) **iff** it is **unitarily
diagonalizable**: there is a unitary `U` and a diagonal matrix `diagonal d`
with `A = U (diagonal d) Uᴴ`. -/
theorem normal_spectral_theorem (A : Matrix n n ℂ) :
    IsStarNormal A ↔
      ∃ U ∈ unitary (Matrix n n ℂ), ∃ d : n → ℂ,
        A = U * diagonal d * star U := by
  sorry
#24
Shannon capacity of the pentagon
shannon_capacity_pentagon

Lean theorem statement

/-- **Lovász's theorem on Shannon capacity of the pentagon** (§238).
The Shannon capacity of the five-cycle is `√5`. -/
theorem shannon_capacity_pentagon :
    HasShannonCapacity (SimpleGraph.cycleGraph 5) (Real.sqrt 5) := by
  sorry
#25
Hurewicz theorem in degree 1 (H₁ = abelianization of π₁)
hurewicz_h1_abelianization

Lean theorem statement

/-- **Hurewicz (n = 1).** For a path-connected space `X`, `H₁(X;ℤ)` is the
abelianization of `π₁(X, x)`. -/
theorem hurewicz_h1_abelianization
    (X : Type) [TopologicalSpace X] [PathConnectedSpace X] (x : X) :
    Nonempty (Additive (Abelianization (FundamentalGroup X x)) ≃+
      (IntegralHomology 1 X : Type)) := by
  sorry
#26
Complete reducibility for compact groups
compact_group_semisimple

Lean theorem statement

/-- **Representations of compact groups are semisimple** (complete
reducibility / the unitarian trick). A continuous representation of a compact
topological group on a finite-dimensional real vector space is semisimple:
every subrepresentation has a `G`-invariant complement, so the representation
decomposes as a direct sum of irreducible finite-dimensional
subrepresentations. -/
theorem compact_group_semisimple
    {G V : Type*} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G]
    [NormedAddCommGroup V] [NormedSpace ℝ V] [FiniteDimensional ℝ V]
    (ρ : Representation ℝ G V)
    (hρ : Continuous fun p : G × V => ρ p.1 p.2) :
    ρ.IsSemisimpleRepresentation := by
  sorry
#27
Peano existence theorem for ODEs
peano_existence

Lean theorem statement

/-- **Peano existence theorem.** Replacing the Lipschitz condition with mere
continuity still yields a local solution of `x' = f(x)`, `x(0) = x₀` — but
uniqueness may fail (e.g. `x' = √x`, `x(0) = 0`). Stated for a
finite-dimensional space: Peano's theorem requires local compactness and is
false in general (infinite-dimensional) Banach spaces. -/
theorem peano_existence
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
    {f : E → E} (hf : Continuous f) (x₀ : E) :
    ∃ a : ℝ, 0 < a ∧ ∃ α : ℝ → E, α 0 = x₀ ∧
      ∀ t ∈ Ioo (-a) a, HasDerivAt α (f (α t)) t := by
  sorry
#28
Sard's theorem (critical-set image has measure zero)
sard_theorem

Lean theorem statement

/-- **Sard's theorem** (Morse 1939 / Sard 1942), Knill's rank-
deficient form. The image of the rank-deficient locus of a smooth
map `f : ℝᵐ → ℝⁿ` has Lebesgue measure zero. -/
theorem sard {m n : ℕ} (f : E m → E n) (_hf : ContDiff ℝ ∞ f) :
    volume (criticalValues f) = 0 := by
  sorry
#29
The Hausdorff–Hildebrandt–Schoenberg moment theorem
hausdorff_hildebrandt_schoenberg

Lean theorem statement

/-- **Hausdorff–Hildebrandt–Schoenberg theorem.** A multi-indexed real sequence
is the moment sequence of a signed bounded-variation measure on the unit cube
`Iᵈ` iff its moments are Hausdorff bounded. -/
theorem hausdorff_hildebrandt_schoenberg {d : ℕ} (a : (Fin d → ℕ) → ℝ) :
    IsMomentConfiguration a ↔ HausdorffBounded a := by
  sorry
#30
The Lindemann–Weierstrass theorem
lindemann_weierstrass

Lean theorem statement

/-- **Lindemann–Weierstrass theorem.** If `x₁, …, xₙ ∈ ℂ` are algebraic over `ℚ`
and ℚ-linearly independent, then `e^{x₁}, …, e^{xₙ}` are algebraically
independent over `ℚ`. -/
theorem lindemann_weierstrass {n : ℕ} (x : Fin n → ℂ)
    (h_alg : ∀ i, IsAlgebraic ℚ (x i))
    (h_lin : LinearIndependent ℚ x) :
    AlgebraicIndependent ℚ (fun i => Complex.exp (x i)) := by
  sorry
#31
Lax's approximation theorem for toral homeomorphisms
lax_approximation

Lean theorem statement

/-- **Lax's approximation theorem.** Every toral dynamical system on `𝕋^d`
(`d ≥ 1`) is approximated arbitrarily well in the metric `δ` by cyclic cube
exchange transformations. -/
theorem lax_approximation {d : ℕ} (hd : 0 < d) (T : ToralDynamicalSystem d)
    {ε : ℝ≥0∞} (hε : 0 < ε) :
    ∃ (n : ℕ) (S : VolumePreservingEquiv d),
      IsCyclicCubeExchange S n ∧ deltaDist T.toVolumePreservingEquiv S < ε := by
  sorry
#32
Sard's regular-value corollary
regular_value_ae

Lean theorem statement

/-- **Regular value corollary (Sard).** For a smooth `f : ℝᵐ → ℝ`, almost every
`c ∈ ℝ` is a regular value. -/
theorem regular_value_ae {m : ℕ} (f : EuclideanSpace ℝ (Fin m) → ℝ)
    (hf : ContDiff ℝ ∞ f) :
    ∀ᵐ c ∂(volume : Measure ℝ), IsRegularValue f c := by
  sorry
#33
The Hausdorff positivity (complete-monotonicity) criterion
hausdorff_positivity_criterion

Lean theorem statement

/-- **Hausdorff positivity criterion** (Hausdorff 1921). A moment configuration
comes from a positive measure iff all its iterated backward differences are
nonnegative — i.e. the sequence is *completely monotone*: `(Δᵏa)ₙ ≥ 0` for all
`k ≤ n`. -/
theorem hausdorff_positivity {d : ℕ} (a : (Fin d → ℕ) → ℝ) :
    IsPositiveMomentConfiguration a ↔ ∀ k n : Fin d → ℕ, k ≤ n → 0 ≤ diff a k n := by
  sorry
#34
Lindemann's theorem (e and π transcendental)
lindemann

Lean theorem statement

/-- **Lindemann's theorem.** Both `e = exp 1` and `π` are transcendental over
`ℤ`. -/
theorem lindemann :
    Transcendental ℤ (Real.exp 1) ∧ Transcendental ℤ Real.pi := by
  sorry
#35
Pascal's theorem
pascal

Lean theorem statement

/-- **Pascal's theorem.** Six distinct points on a non-singular conic determine
three collinear intersection points `Aᵢ Bⱼ ∩ Aⱼ Bᵢ`. -/
theorem pascal
    (M : Matrix (Fin 3) (Fin 3) ℝ) (hMsymm : M.IsSymm) (hMdet : M.det ≠ 0)
    (a₁ a₂ a₃ b₁ b₂ b₃ : Fin 3 → ℝ)
    (ha₁ : a₁ ≠ 0) (ha₂ : a₂ ≠ 0) (ha₃ : a₃ ≠ 0)
    (hb₁ : b₁ ≠ 0) (hb₂ : b₂ ≠ 0) (hb₃ : b₃ ≠ 0)
    (hdist : [a₁, a₂, a₃, b₁, b₂, b₃].Pairwise (fun v w => ¬ SamePoint v w))
    (hA₁ : OnConic M a₁) (hA₂ : OnConic M a₂) (hA₃ : OnConic M a₃)
    (hB₁ : OnConic M b₁) (hB₂ : OnConic M b₂) (hB₃ : OnConic M b₃) :
    Collinear3 (meet a₁ b₂ a₂ b₁) (meet a₁ b₃ a₃ b₁) (meet a₂ b₃ a₃ b₂) := by
  sorry
#36
Radon transform: Fourier-slice diagonalization and pseudo-inversion
radon_transform_inversion

Lean theorem statement

/-- **Radon's theorem (diagonalization + pseudo-inversion).** The Fourier slice
theorem diagonalizes the Radon transform, and the transform admits a left
inverse on the Schwartz space. -/
theorem radon_can_be_diagonalized_and_pseudo_inverted :
    (∀ φ : SchwartzMap (ℝ × ℝ) ℂ, ∀ θ k : ℝ,
        fourier1 (fun p => radon (φ : ℝ × ℝ → ℂ) (p, θ)) k =
          fourier2 (φ : ℝ × ℝ → ℂ) (k * Real.cos θ, k * Real.sin θ)) ∧
    (∃ Rinv : (ℝ × ℝ → ℂ) → (ℝ × ℝ → ℂ),
        ∀ φ : SchwartzMap (ℝ × ℝ) ℂ,
          Rinv (radon (φ : ℝ × ℝ → ℂ)) = (φ : ℝ × ℝ → ℂ)) := by
  sorry
#37
Tverberg's theorem
tverberg_theorem

Lean theorem statement

/-- **Tverberg's theorem.** Any `(r-1)(d+1)+1` points in `ℝ^d` admit an
`r`-part Tverberg partition. -/
theorem tverberg_theorem (d r : ℕ) (hr : 1 ≤ r)
    (f : Fin ((r - 1) * (d + 1) + 1) → Space d) :
    HasTverbergPartition (r := r) f := by
  sorry
#38
Bauer's uniqueness at extreme points
bauer_extreme_point_uniqueness

Lean theorem statement

/-- **Bauer's uniqueness at extreme points.** If `x` is an extreme point of a
compact convex set `K` and `μ` is a probability measure supported on `K`
(`μ Kᶜ = 0`) with barycenter `x = ∫ y, y ∂μ`, then `μ` is the Dirac mass at
`x`. (The support hypothesis is the weaker `μ Kᶜ = 0`, making this a
strengthening of the textbook statement: uniqueness among all ambient Borel
probability measures on `K`, not only those already supported on `ext K`.) -/
theorem bauer_unique [MeasurableSpace X] [BorelSpace X]
    (K : Set X) (hK_cpt : IsCompact K) (hK_cvx : Convex ℝ K)
    {x : X} (hx : x ∈ K.extremePoints ℝ)
    (μ : Measure X) [IsProbabilityMeasure μ]
    (hμ : μ Kᶜ = 0) (hbar : x = ∫ y, y ∂μ) :
    μ = Measure.dirac x := by
  sorry
#39
Boone–Higman theorem (easy direction)
boone_higman_embedding

Lean theorem statement

/-- **Boone–Higman theorem (easy direction).** If a finitely presented group `G`
embeds (via injective `f`) into a simple group `H`, which embeds (via injective
`g`) into a finitely presented group `K`, then the word problem of `G` is
solvable. -/
theorem boone_higman_embedding
    {G H K : Type*} [Group G] [Group H] [Group K]
    [IsSimpleGroup H] [Group.IsFinitelyPresented K]
    (f : G →* H) (hf : Function.Injective f)
    (g : H →* K) (hg : Function.Injective g)
    {n : ℕ} (φ : FreeGroup (Fin n) →* G)
    (hsurj : Function.Surjective φ)
    (hker : (MonoidHom.ker φ).IsNormalClosureFG) :
    WordProblemSolvable φ := by
  sorry
#40
Fang–Xia: tiling of the symmetric group by transpositions implies λ-transitivity
fang_xia_tiling_partition_transitive

Lean theorem statement

/-- **Fang–Xia, Theorem 1.4.** A tiling `(T_n, Y)` of `S_n` forces
λ-transitivity of `Y` for every partition `λ` of `n` whose Young-
diagram content sum is nonnegative. -/
theorem fang_xia_partition_transitive_of_tiling
    {n : ℕ} {Y : Set (Equiv.Perm (Fin n))}
    (_h : IsTiling (transpositionsWithOne n) Y) :
    ∀ lam : PartitionShape n, 0 ≤ lam.contentSum → IsPartitionTransitive Y lam := by
  sorry
#41
Hausdorff moment problem: absolute-continuity criterion
hausdorff_absolute_continuity

Lean theorem statement

/-- **Absolute continuity criterion (Hausdorff moment problem).** A positive
probability measure `μ` on the cube is uniformly absolutely continuous w.r.t.
Lebesgue measure iff there is `C` with `(Δᵏμ)ₙ ≤ C·(Δᵏν)ₙ` for all `k ≤ n`. -/
theorem hausdorff_absolute_continuity {d : ℕ}
    (μ : Measure (EuclideanSpace ℝ (Fin d)))
    [IsProbabilityMeasure μ] (hμ : μ ((cube d)ᶜ) = 0) :
    UniformlyAbsolutelyContinuous μ (volume.restrict (cube d)) ↔
      ∃ C : ℝ, ∀ k n : Fin d → ℕ, k ≤ n →
        diff (momentOf μ) k n ≤ C * diff (momentOf (volume.restrict (cube d))) k n := by
  sorry
#42
The Landsberg–Schaar relation
landsberg_schaar

Lean theorem statement

/-- **Landsberg–Schaar relation.** For positive odd integers `p, q`,
`S(2q, p) = e^{iπ/4} · S(−p, 2q)`. -/
theorem landsberg_schaar (p q : ℕ) (hp : Odd p) (hq : Odd q) :
    gaussS (2 * q : ℕ) p
      = Complex.exp ((Real.pi : ℂ) * Complex.I / 4) * gaussS (-(p : ℤ)) (2 * q) := by
  sorry
#43
Ornstein–Weiss ℤᵈ Rokhlin lemma
ornstein_weiss_rokhlin

Verso theorem preview

theorem declaration uses `sorry`ornstein_weiss_rokhlin {Ω : Type*} [MeasurableSpace Ω] [StandardBorelSpace Ω] {d : ℕ} (_hd : 1 ≤ d) (μ : Measure Ω) [IsProbabilityMeasure μ] (T : (Fin d → ℤ) → Ω → Ω) (_hid : ∀ x, T 0 x = x) (_hT : ∀ v, MeasurePreserving (T v) μ μ) (_hgrp : ∀ u v x, T (u + v) x = T u (T v x)) (_hfree : LeanEval.Dynamics.IsFreeAction μ T) (N : ℕ) (_hN : 1 ≤ N) {ε : ENNReal} (_hε : 0 < ε) : ∃ B : Set Ω, MeasurableSet B ∧ ((boxShape d N : Finset (Fin d → ℤ)) : Set (Fin d → ℤ)).PairwiseDisjoint (fun v => T v '' B) ∧ μ (⋃ v ∈ boxShape d N, T v '' B) ≥ 1 - ε := Ω:Type u_1inst✝²:MeasurableSpace Ωinst✝¹:StandardBorelSpace Ωd:ℕ_hd:1 ≤ dμ:Measure Ωinst✝:IsProbabilityMeasure μT:(Fin d → ℤ) → Ω → Ω_hid:∀ (x : Ω), T 0 x = x_hT:∀ (v : Fin d → ℤ), MeasurePreserving (T v) μ μ_hgrp:∀ (u v : Fin d → ℤ) (x : Ω), T (u + v) x = T u (T v x)_hfree:IsFreeAction μ TN:ℕ_hN:1 ≤ Nε:ENNReal_hε:0 < ε⊢ ∃ B, MeasurableSet B ∧ ((↑(boxShape d N)).PairwiseDisjoint fun v => T v '' B) ∧ μ (⋃ v ∈ boxShape d N, T v '' B) ≥ 1 - ε All goals completed! 🐙
#44
Choquet's representation theorem
choquet_representation_theorem

Lean theorem statement

/-- **Choquet's representation theorem.** Every point `x` of a compact convex
set `K` in a Banach space is the barycenter of a probability measure supported
on the extreme points of `K`: there is a probability measure `μ` with
`μ (ext K)ᶜ = 0` whose barycenter `∫ y, y ∂μ` equals `x`. -/
theorem choquet [MeasurableSpace X] [BorelSpace X]
    (K : Set X) (hK_cpt : IsCompact K) (hK_cvx : Convex ℝ K)
    {x : X} (hx : x ∈ K) :
    ∃ μ : Measure X, IsProbabilityMeasure μ ∧
      μ (K.extremePoints ℝ)ᶜ = 0 ∧
      x = ∫ y, y ∂μ := by
  sorry
#45
Jordan normal form
jordan_normal_form

Lean theorem statement

/-- **Jordan normal form.** Over an algebraically closed field, every
endomorphism of `Kⁿ` admits a Jordan-chain basis. -/
theorem jordan_normal_form {K : Type*} [Field K] [IsAlgClosed K] (n : ℕ)
    (f : Module.End K (StdSpace K n)) :
    Nonempty (JordanChainBasis f) := by
  sorry
#46
Moran's equality for affine-symmetric iterated function systems
moran_equality_affine

Lean theorem statement

/-- **Moran's equality for affine-symmetric IFS.** For an affine-symmetric IFS
on `ℝᵈ` with common contraction factor `λ ∈ (0,1)`, orthogonal linear parts, and
the open set condition, the Hausdorff dimension of the attractor is
`−log n / log λ` (positive since `λ < 1`). -/
theorem moran_equality_affine
    {d n : ℕ} (hn : 1 ≤ n)
    (f : Fin n → EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (lam : ℝ)
    (h_aff : IsAffineSymmetricIFS f lam)
    (h_osc : OpenSetCondition f)
    {S : Set (EuclideanSpace ℝ (Fin d))} (hS : IsAttractor f S) :
    dimH S = ENNReal.ofReal (- Real.log n / Real.log lam) := by
  sorry
#47
Riesz's rising sun lemma
rising_sun_lemma

Lean theorem statement

/-- **Riesz's rising sun lemma.** Every continuous real function on a compact
interval has the rising-sun property. -/
theorem rising_sun_lemma {a b : ℝ} (hab : a < b) {f : ℝ → ℝ}
    (hf : ContinuousOn f (Icc a b)) :
    HasRisingSunProperty a b f := by
  sorry
#48
Rokhlin lemma
rokhlin_lemma

Lean theorem statement

/-- **Rokhlin lemma.** For every aperiodic measure-preserving
automorphism `T` of a standard Borel probability space `(Ω, μ)`, every
height `n ≥ 1`, and every `ε > 0`, there is a Rokhlin tower of height
`n` whose union has measure at least `1 − ε`. -/
theorem rokhlin_lemma {Ω : Type*} [MeasurableSpace Ω]
    [StandardBorelSpace Ω]
    (μ : Measure Ω) [IsProbabilityMeasure μ] (T : Ω → Ω)
    (_hT : MeasurePreserving T μ μ) (_hap : IsAperiodic T μ)
    (n : ℕ) (_hn : 1 ≤ n) {ε : ENNReal} (_hε : 0 < ε) :
    ∃ B : Set Ω, IsRokhlinTower T B n ∧
      μ (towerUnion T B n) ≥ 1 - ε := by
  sorry
#49
Balanceable k-bounded partitions
balanceable_bounded_partitions

Lean theorem statement

/--
For any `k`, the smallest `n` such that any `k`-bounded partition of `n` is
also balanceable is given by `2 * lcm(1, ..., k)`.
-/
theorem minimal_balanceable_of_bounded (k : ℕ) (hk : 0 < k) :
    Minimal (fun n => 0 < n ∧ ∀ p : n.Partition, Bounded k p → Balanceable p) (2 * (Finset.Icc 1 k).lcm id) := by
  sorry
#50
Nash equilibrium existence theorem
nash_equilibrium_exists

Lean theorem statement

/-- **Nash equilibrium existence theorem.** Every finite `n`-player game
with nonempty finite pure-strategy sets and arbitrary real payoffs admits
at least one mixed-strategy Nash equilibrium. -/
theorem nash_equilibrium_exists {n : ℕ} {S : Fin n → Type*}
    [∀ i, Fintype (S i)] [∀ i, Nonempty (S i)]
    (u : Fin n → StrategyProfile n S → ℝ) :
    ∃ σ : ∀ i, S i → ℝ, IsNashEquilibrium u σ := by
  sorry
#51
A competition programming problem about permuting a permutation to be unimodal
permute_to_unimodal

Lean theorem statement

/--
`minRearrange` correctly computes the smallest number of indices that need to be permuted in order to
turn `arr` into a unimodal permutation.
-/
theorem minRearrange_correct {arr : Array Nat} :
    arr.Perm (1...=arr.size).toArray →
      (∃ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x ∧ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector = minRearrange arr) ∧
      (∀ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x → minRearrange arr ≤ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector) := by
  sorry
#52
Lidskii's inequality
lidskii_inequality

Verso theorem preview

theorem declaration uses `sorry`lidskii_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) {p : ℝ} (_hp : 1 ≤ p) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |(hB.sub hA).eigenvalues₀ j| ^ p := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitianp:ℝ_hp:1 ≤ p⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |⋯.eigenvalues₀ j| ^ p All goals completed! 🐙
#53
Frobenius's theorem: the Frobenius kernel is normal
frobenius_kernel_isNormal

Lean theorem statement

theorem frobenius_kernel_isNormal
    (G X : Type) [Group G] [Fintype G] [Fintype X]
    [MulAction G X] [FaithfulSMul G X]
    (hcard : 2 ≤ Fintype.card X)
    (htrans : ∀ x y : X, ∃ g : G, g • x = y)
    (hstab : ∀ x : X, MulAction.stabilizer G x ≠ ⊥)
    (hfrob : ∀ g : G, g ≠ 1 → ∀ x y : X, g • x = x → g • y = y → x = y) :
    ∃ N : Subgroup G, N.Normal ∧
      (N : Set G) = {1} ∪ {g : G | ∀ x : X, g • x ≠ x} := by
  sorry
#54
Local stable/unstable sets at a hyperbolic fixed point (set-level Hadamard–Perron)
stable_unstable_manifolds

Verso theorem preview

theorem declaration uses `sorry`stable_unstable_manifolds_exist (n : ℕ) (f : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (x₀ : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (_hf : ContDiffAt ℝ 1 f x₀) (_hfix : f x₀ = x₀) (_hhyp : LeanEval.Dynamics.StableUnstableManifoldsProblem.IsHyperbolicLinear (fderiv ℝ f x₀)) (_hf_inv : (fderiv ℝ f x₀).IsInvertible) : ∃ U : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), Ws = {x | (∀ k : ℕ, f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y : ℕ → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n, y 0 = x ∧ (∀ k : ℕ, y k ∈ U) ∧ (∀ k : ℕ, f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} := n:ℕf:E n → E nx₀:E n_hf:ContDiffAt ℝ 1 f x₀_hfix:f x₀ = x₀_hhyp:IsHyperbolicLinear (fderiv ℝ f x₀)_hf_inv:(fderiv ℝ f x₀).IsInvertible⊢ ∃ U, IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu, Ws = {x | (∀ (k : ℕ), f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y, y 0 = x ∧ (∀ (k : ℕ), y k ∈ U) ∧ (∀ (k : ℕ), f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} All goals completed! 🐙
#55
Independence of the parallel postulate
parallel_postulate_independent

Lean theorem statement

/-- **Independence of the parallel postulate** (Freek #12). The Euclidean
axiom `A10` is logically independent of Tarski's absolute axioms `A1`–`A9`
and `A11`: there is a model of the absolute axioms in which the parallel
postulate holds (the real coordinate plane) and one in which it fails (the
Klein–Beltrami disk, or any other hyperbolic-plane model). -/
theorem parallel_postulate_independent :
    (∃ (M : Type) (T : TarskiAbsolute M), Euclidean M T) ∧
    (∃ (M : Type) (T : TarskiAbsolute M), ¬ Euclidean M T) := by
  sorry
#56
Brauer–Fowler theorem
brauer_fowler

Lean theorem statement

/-- **Brauer–Fowler theorem.** There is a function bounding the order
of a finite nonabelian simple group by the order of any involution
centralizer. -/
theorem brauer_fowler :
    ∃ f : ℕ → ℕ, ∀ (G : Type) [Group G] [Finite G],
      IsSimpleGroup G → (∃ a b : G, a * b ≠ b * a) →
      ∀ t : G, orderOf t = 2 →
        Nat.card G ≤ f (Nat.card (Subgroup.centralizer ({t} : Set G))) := by
  sorry
#58
Monge–Kantorovich existence theorem
monge_kantorovich

Verso theorem preview

theorem declaration uses `sorry`monge_kantorovich_exists {X Y : Type*} [TopologicalSpace X] [PolishSpace X] [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [PolishSpace Y] [MeasurableSpace Y] [BorelSpace Y] (P : Measure X) (Q : Measure Y) [IsProbabilityMeasure P] [IsProbabilityMeasure Q] (c : X × Y → ENNReal) (_hc : Continuous c) : ∃ π ∈ LeanEval.Analysis.Couplings P Q, ∀ π' ∈ LeanEval.Analysis.Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' := X:Type u_1Y:Type u_2inst✝⁹:TopologicalSpace Xinst✝⁸:PolishSpace Xinst✝⁷:MeasurableSpace Xinst✝⁶:BorelSpace Xinst✝⁵:TopologicalSpace Yinst✝⁴:PolishSpace Yinst✝³:MeasurableSpace Yinst✝²:BorelSpace YP:Measure XQ:Measure Yinst✝¹:IsProbabilityMeasure Pinst✝:IsProbabilityMeasure Qc:X × Y → ENNReal_hc:Continuous c⊢ ∃ π ∈ Couplings P Q, ∀ π' ∈ Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' All goals completed! 🐙
#59
Sturm's theorem
sturm

Lean theorem statement

/-- **Sturm's theorem.** For a squarefree real polynomial `p` and an interval
`(a, b)` with `a < b` whose endpoints are not roots of `p`, the number of
distinct roots of `p` in `(a, b)` equals `σ(a) − σ(b)`. -/
theorem sturm (p : ℝ[X]) (hp : Squarefree p) {a b : ℝ} (hab : a < b)
    (ha : p.eval a ≠ 0) (hb : p.eval b ≠ 0) :
    ((p.roots.toFinset).filter (fun x => a < x ∧ x < b)).card =
      sigma p a - sigma p b := by
  sorry
#60
Chen theorem for Markoff graphs
dvd_card_connectedComponent_markoffGraph

Lean theorem statement

/-- For prime `p > 3`, every connected component of the nonzero Markoff graph over `ZMod p`
has cardinality divisible by `p`. -/
theorem dvd_card_connectedComponent_markoffGraph
    {p : ℕ} (hp : Nat.Prime p) (hgt : 3 < p) :
    ∀ c : (markoffGraph p).ConnectedComponent, p ∣ Nat.card c := by
  sorry
#61
Euler–Lagrange equation
euler_lagrange_equation

Verso theorem preview

theorem declaration uses `sorry`euler_lagrange_equation {a b : ℝ} (L : ℝ → ℝ → ℝ → ℝ) (x : ℝ → ℝ) (_hab : a < b) (_hL : ContDiff ℝ 2 (fun p : ℝ × ℝ × ℝ => L p.1 p.2.1 p.2.2)) (_hx : ContDiff ℝ 2 x) (_hxe : LeanEval.Analysis.IsVariationalExtremum a b L x) : ∀ t ∈ Set.Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t := a:ℝb:ℝL:ℝ → ℝ → ℝ → ℝx:ℝ → ℝ_hab:a < b_hL:ContDiff ℝ 2 fun p => L p.1 p.2.1 p.2.2_hx:ContDiff ℝ 2 x_hxe:IsVariationalExtremum a b L x⊢ ∀ t ∈ Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t All goals completed! 🐙
#62
Fraser: Fourier decay for finite-field Kakeya sets is q^{-1} and sharp
fraser_kakeya_fourier_decay

Verso theorem preview

theorem declaration uses `sorry`fraser_kakeya_fourier_decay_and_sharp {d : ℕ} (_hd : 2 ≤ d) {K : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F d)} (_hK : LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K) (χ : AddChar F ℂ) (_hχ : χ ≠ 1) : (∃ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K μ ∧ ∀ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d, ξ ≠ 0 → ‖fourier χ μ ξ‖ ≤ (Fintype.card F : ℝ)⁻¹) ∧ (∀ κ : ℝ, 0 < κ → κ < 1 → ∃ Q : ℕ, ∀ (F' : Type*) [Field F'] [Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K' : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d), LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K' ∧ ∀ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K' μ → ∃ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d, ξ ≠ 0 ∧ κ * (Fintype.card F' : ℝ)⁻¹ ≤ ‖fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖) := F:Type u_1inst✝²:Field Finst✝¹:Fintype Finst✝:DecidableEq Fd:ℕ_hd:2 ≤ dK:Set (Space F d)_hK:IsKakeya Kχ:AddChar F ℂ_hχ:χ ≠ 1⊢ (∃ μ, IsProbabilityMeasureOn K μ ∧ ∀ (ξ : Space F d), ξ ≠ 0 → ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier χ μ ξ‖ ≤ (↑(Fintype.card F))⁻¹) ∧ ∀ (κ : ℝ), 0 < κ → κ < 1 → ∃ Q, ∀ (F' : Type u_2) [inst : Field F'] [inst_1 : Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K', IsKakeya K' ∧ ∀ (μ : Space F' d → ℝ), IsProbabilityMeasureOn K' μ → ∃ ξ, ξ ≠ 0 ∧ κ * (↑(Fintype.card F'))⁻¹ ≤ ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖ All goals completed! 🐙
#63
Hippocrates' theorem on lunes
hippocrates_lunes

Verso theorem preview

theorem declaration uses `sorry`hippocrates_lunes (a b : ℝ) (ha : 0 < a) (hb : 0 < b) : volume (LeanEval.Geometry.HippocratesLunes.horizontalLune a b) + volume (LeanEval.Geometry.HippocratesLunes.verticalLune a b) = volume (LeanEval.Geometry.HippocratesLunes.rightTriangle a b) := a:ℝb:ℝha:0 < ahb:0 < b⊢ volume (horizontalLune a b) + volume (verticalLune a b) = volume (rightTriangle a b) All goals completed! 🐙
#64
Koszul formula
koszul_formula

Lean theorem statement

/-- **Koszul formula.** For any smooth torsion-free metric-compatible
covariant derivative `cov` on `TM`, `2 ⟨∇_X Y, Z⟩` equals the cyclic sum
of directional derivatives `X·⟨Y, Z⟩ + Y·⟨X, Z⟩ − Z·⟨X, Y⟩` minus the
Lie-bracket cyclic sum `⟨X, [Y, Z]⟩ + ⟨Y, [X, Z]⟩ − ⟨Z, [X, Y]⟩`. -/
theorem koszul_formula
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
      [FiniteDimensional ℝ E] [CompleteSpace E]
    {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H}
    {M : Type*} [TopologicalSpace M] [ChartedSpace H M]
      [IsManifold I ∞ M]
    [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)]
    [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)]
    (cov : CovariantDerivative I E (TangentSpace I (M := M)))
    [ContMDiffCovariantDerivative cov ∞]
    (_htor : cov.torsion = 0) (_hmet : IsMetricCompatible cov)
    (X Y Z : Π x : M, TangentSpace I x)
    (_hX : CMDiff ∞ (T% X)) (_hY : CMDiff ∞ (T% Y)) (_hZ : CMDiff ∞ (T% Z))
    (x : M) :
    2 * inner ℝ (cov Y x (X x)) (Z x) =
      mvfderiv I (fun y : M => inner ℝ (Y y) (Z y)) x (X x)
      + mvfderiv I (fun y : M => inner ℝ (X y) (Z y)) x (Y x)
      - mvfderiv I (fun y : M => inner ℝ (X y) (Y y)) x (Z x)
      - inner ℝ (X x) (mlieBracket I Y Z x)
      - inner ℝ (Y x) (mlieBracket I X Z x)
      + inner ℝ (Z x) (mlieBracket I X Y x) := by
  sorry
#65
Mountain Pass Theorem (Ambrosetti–Rabinowitz 1973)
mountain_pass

Verso theorem preview

theorem declaration uses `sorry`mountain_pass (f : E → ℝ) (_hf : ContDiff ℝ 1 f) (_hps : LeanEval.Analysis.MountainPassProblem.PalaisSmale f) {a b : E} {ε r : ℝ} (_hmr : LeanEval.Analysis.MountainPassProblem.MountainRange f a b ε r) : ∃ x : E, LeanEval.Analysis.MountainPassProblem.IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace Ef:E → ℝ_hf:ContDiff ℝ 1 f_hps:PalaisSmale fa:Eb:Eε:ℝr:ℝ_hmr:MountainRange f a b ε r⊢ ∃ x, IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b All goals completed! 🐙
#66
Furstenberg–Weiss topological multiple recurrence (single-transformation form)
furstenberg_topological

Verso theorem preview

theorem declaration uses `sorry`furstenberg_topological_recurrence {X : Type*} [MetricSpace X] [CompactSpace X] [Nonempty X] (T : X ≃ₜ X) : ∃ x : X, LeanEval.Dynamics.IsMultiplyRecurrent (T : X → X) x := X:Type u_1inst✝²:MetricSpace Xinst✝¹:CompactSpace Xinst✝:Nonempty XT:X ≃ₜ X⊢ ∃ x, IsMultiplyRecurrent (⇑T) x All goals completed! 🐙
#67
Kakutani fixed-point theorem
kakutani_fixed_point

Lean theorem statement

/-- **Kakutani fixed-point theorem.** Every upper-hemicontinuous
correspondence `F` from a nonempty compact convex `K ⊆ ℝᵈ` to itself, with
nonempty convex closed values, has a fixed point `x ∈ F x`. -/
theorem kakutani_fixed_point {d : ℕ}
    {K : Set (EuclideanSpace ℝ (Fin d))}
    (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K)
    (_hK_nonempty : K.Nonempty)
    (F : EuclideanSpace ℝ (Fin d) → Set (EuclideanSpace ℝ (Fin d)))
    (_hF_uhc : IsUpperHemicontinuous F)
    (_hF_nonempty : ∀ x ∈ K, (F x).Nonempty)
    (_hF_convex : ∀ x ∈ K, Convex ℝ (F x))
    (_hF_closed : ∀ x ∈ K, IsClosed (F x))
    (_hF_maps : ∀ x ∈ K, F x ⊆ K) :
    ∃ x ∈ K, x ∈ F x := by
  sorry
#68
Lidskii–Last eigenvalue-perturbation theorem
lidskii_last

Verso theorem preview

theorem declaration uses `sorry`lidskii_last {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitian⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ All goals completed! 🐙
#69
Morley's trisector theorem
morley_theorem

Verso theorem preview

theorem declaration uses `sorry`morley_theorem (A B C P Q R : LeanEval.Geometry.Morley.Plane) (h : LeanEval.Geometry.Morley.IsMorleyConfiguration A B C P Q R) : LeanEval.Geometry.Morley.IsEquilateralTriple P Q R := A:PlaneB:PlaneC:PlaneP:PlaneQ:PlaneR:Planeh:IsMorleyConfiguration A B C P Q R⊢ IsEquilateralTriple P Q R All goals completed! 🐙
#70
Solvable extensions ↔ solvable groups (the missing converse in Abel–Ruffini)
solvable_by_radicals_converse

Verso theorem preview

theorem declaration uses `sorry`solvable_iff_solvableByRad (F : Type*) [Field F] [CharZero F] (p : F[X]) (_hp : p ≠ 0) : (∀ x : AlgebraicClosure F, aeval x p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal := F:Type u_1inst✝¹:Field Finst✝:CharZero Fp:F[X]_hp:p ≠ 0⊢ (∀ (x : AlgebraicClosure F), (aeval x) p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal All goals completed! 🐙
#71
Runge's theorem
runge_theorem

Verso theorem preview

theorem declaration uses `sorry`runge (K : Set ℂ) (_hK : IsCompact K) (U : Set ℂ) (_hU : IsOpen U) (_hKU : K ⊆ U) (f : ℂ → ℂ) (_hf : AnalyticOnNhd ℂ f U) (ε : ℝ) (_hε : 0 < ε) : ∃ p q : ℂ[X], (∀ z ∈ K, q.eval z ≠ 0) ∧ (∀ z ∈ K, ‖f z - p.eval z / q.eval z‖ < ε) := K:Set ℂ_hK:IsCompact KU:Set ℂ_hU:IsOpen U_hKU:K ⊆ Uf:ℂ → ℂ_hf:AnalyticOnNhd ℂ f Uε:ℝ_hε:0 < ε⊢ ∃ p q, (∀ z ∈ K, Polynomial.eval z q ≠ 0) ∧ ∀ z ∈ K, ‖f z - Polynomial.eval z p / Polynomial.eval z q‖ < ε All goals completed! 🐙
#72
Schur-Weyl duality: GL(V) image equals centralizer of S_k image
glAction_range_eq_centralizer_symAction

Verso theorem preview

theorem declaration uses `sorry`glAction_range_eq_centralizer_symAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(glAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(symAction R M k)) All goals completed! 🐙
#73
Schauder fixed-point theorem
schauder_fixed_point

Verso theorem preview

theorem declaration uses `sorry`schauder_fixed_point {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {K : Set E} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : E → E) (_hf_cont : ContinuousOn f K) (_hf_maps : Set.MapsTo f K K) : ∃ x ∈ K, f x = x := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace EK:Set E_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:E → E_hf_cont:ContinuousOn f K_hf_maps:Set.MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#74
Baer–Suzuki theorem
baer_suzuki

Verso theorem preview

theorem declaration uses `sorry`baer_suzuki {G : Type*} [Group G] [Finite G] {p : ℕ} [Fact p.Prime] (x : G) : x ∈ LeanEval.GroupTheory.Defs.pCore p G ↔ ∀ g : G, IsPGroup p (Subgroup.closure ({x, g * x * g⁻¹} : Set G)) := G:Type u_1inst✝²:Group Ginst✝¹:Finite Gp:ℕinst✝:Fact (Nat.Prime p)x:G⊢ x ∈ pCore p G ↔ ∀ (g : G), IsPGroup p ↥(Subgroup.closure {x, g * x * g⁻¹}) All goals completed! 🐙
#75
Symplectic matrices have determinant 1
symplectic_matrix_det

Verso theorem preview

theorem declaration uses `sorry`symplectic_matrix_det {l R : Type*} [DecidableEq l] [Fintype l] [CommRing R] {A : Matrix (l ⊕ l) (l ⊕ l) R} (_hA : A ∈ Matrix.symplecticGroup l R) : A.det = 1 := l:Type u_1R:Type u_2inst✝²:DecidableEq linst✝¹:Fintype linst✝:CommRing RA:Matrix (l ⊕ l) (l ⊕ l) R_hA:A ∈ symplecticGroup l R⊢ A.det = 1 All goals completed! 🐙
#76
Brouwer fixed-point theorem
brouwer_fixed_point

Verso theorem preview

theorem declaration uses `sorry`brouwer_fixed_point {d : ℕ} {K : Set (EuclideanSpace ℝ (Fin d))} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (_hf_cont : ContinuousOn f K) (_hf_maps : MapsTo f K K) : ∃ x ∈ K, f x = x := d:ℕK:Set (EuclideanSpace ℝ (Fin d))_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)_hf_cont:ContinuousOn f K_hf_maps:MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#77
Linear programming: maximum principle and vertex optimality
lp_maximum_principle

Verso theorem preview

/-- **Maximum principle for linear programming** (§101). A local maximiser of the LP objective on the feasible region is automatically a global maximiser; and whenever the objective is non-constant (`c ≠ 0`), the maximiser lies on the topological frontier of the feasible region. -/ theorem declaration uses `sorry`lp_maximum_principle {m n : ℕ} (lp : LinearProgram m n) (x : Fin m → ℝ) (_hx : x ∈ lp.feasible) (_hlocal : IsLocalMaxOn lp.objective lp.feasible x) : IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) := m:ℕn:ℕlp:LinearProgram m nx:Fin m → ℝ_hx:x ∈ lp.feasible_hlocal:IsLocalMaxOn lp.objective lp.feasible x⊢ IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) All goals completed! 🐙
/-- **Vertex optimality** (§101; the existence content of Dantzig's 1947 simplex algorithm). Every linear program with a nonempty bounded feasible region admits a global maximiser that is an extreme point (vertex) of the feasible region. -/ theorem declaration uses `sorry`simplex_algorithm {m n : ℕ} (lp : LinearProgram m n) (_hfeas : lp.feasible.Nonempty) (_hbdd : Bornology.IsBounded lp.feasible) : ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible := m:ℕn:ℕlp:LinearProgram m n_hfeas:lp.feasible.Nonempty_hbdd:Bornology.IsBounded lp.feasible⊢ ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible All goals completed! 🐙
#78
Wiener's atom-detection formula
wiener_atom_detection

Verso theorem preview

theorem declaration uses `sorry`wiener_atom_detection (μ : Measure (AddCircle (2 * Real.pi))) [IsProbabilityMeasure μ] : Tendsto (fun N : ℕ => (1 / (N : ℝ)) * ∑ k ∈ Finset.Icc (1 : ℤ) N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' x : AddCircle (2 * Real.pi), ((μ {x}).toReal) ^ 2)) := μ:Measure (AddCircle (2 * π))inst✝:IsProbabilityMeasure μ⊢ Tendsto (fun N => 1 / ↑N * ∑ k ∈ Finset.Icc 1 ↑N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' (x : AddCircle (2 * π)), (μ {x}).toReal ^ 2)) All goals completed! 🐙
#79
Bourbaki's locally convex extension of Banach–Alaoglu
banach_alaoglu_bourbaki

Verso theorem preview

theorem declaration uses `sorry`banach_alaoglu_bourbaki (E : Type*) [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] (U : Set E) (_hU : U ∈ 𝓝 (0 : E)) : IsCompact (LeanEval.Analysis.weakStarPolar E U) := E:Type u_1inst✝⁵:AddCommGroup Einst✝⁴:Module ℝ Einst✝³:TopologicalSpace Einst✝²:ContinuousAdd Einst✝¹:ContinuousSMul ℝ Einst✝:LocallyConvexSpace ℝ EU:Set E_hU:U ∈ 𝓝 0⊢ IsCompact (weakStarPolar E U) All goals completed! 🐙
#80
Kuznetsov's theorem: finitely presented simple groups have solvable word problem
boone_higman_simple

Verso theorem preview

theorem declaration uses `sorry`boone_higman_simple {G : Type*} [Group G] [IsSimpleGroup G] {n : ℕ} (φ : FreeGroup (Fin n) →* G) (_hsurj : Function.Surjective φ) (_hker : (MonoidHom.ker φ).IsNormalClosureFG) : LeanEval.GroupTheory.BooneHigmanSimpleProblem.WordProblemSolvable φ := G:Type u_1inst✝¹:Group Ginst✝:IsSimpleGroup Gn:ℕφ:FreeGroup (Fin n) →* G_hsurj:Function.Surjective ⇑φ_hker:φ.ker.IsNormalClosureFG⊢ WordProblemSolvable φ All goals completed! 🐙
#81
Schur-Weyl duality: S_k image equals centralizer of GL(V) image
symAction_range_eq_centralizer_glAction

Verso theorem preview

theorem declaration uses `sorry`symAction_range_eq_centralizer_glAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(symAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(glAction R M k)) All goals completed! 🐙
#82
Existence of a non-isotopic pair of oriented two-component links
exists_nonisotopic_link

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_link : ∃ L₁ L₂ : LeanEval.KnotTheory.TwoLink, ¬ L₁.Isotopic L₂ := ⊢ ∃ L₁ L₂, ¬L₁.Isotopic L₂ All goals completed! 🐙
#83
Pointwise and Cesàro convergence of Fourier series (Dirichlet, Fejér)
fourier_dirichlet_fejer

Verso theorem preview

/-- **Dirichlet's pointwise convergence theorem** (§46). For every `C¹` 2π-periodic complex function `f`, the symmetric Fourier partial sums `S_N(f)(x)` converge to `f(x)` at every point `x ∈ ℝ`. -/ theorem declaration uses `sorry`dirichlet_pointwise {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hC1 : ContDiff ℝ 1 f) (x : ℝ) : Tendsto (fun N : ℕ => fourierPartialSum f N x) atTop (𝓝 (f x)) := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hC1:ContDiff ℝ 1 fx:ℝ⊢ Tendsto (fun N => fourierPartialSum f N x) atTop (𝓝 (f x)) All goals completed! 🐙
/-- **Fejér's theorem** (§46). For every *continuous* 2π-periodic complex function `f` — without the `C¹` hypothesis of Dirichlet's theorem — the Cesàro means `σ_N(f)` of the symmetric Fourier partial sums converge to `f` uniformly on `ℝ`. -/ theorem declaration uses `sorry`fejer {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hcont : Continuous f) : TendstoUniformly (fun N : ℕ => fourierCesaroMean f N) f atTop := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hcont:Continuous f⊢ TendstoUniformly (fun N => fourierCesaroMean f N) f atTop All goals completed! 🐙
#84
Burnside p^a q^b theorem
finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow

Verso theorem preview

theorem declaration uses `sorry`finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow {G : Type*} [Group G] [Fintype G] {p q a b : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q) (hpq : p ≠ q) (hcard : Fintype.card G = p ^ a * q ^ b) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Fintype Gp:ℕq:ℕa:ℕb:ℕhp:Nat.Prime phq:Nat.Prime qhpq:p ≠ qhcard:Fintype.card G = p ^ a * q ^ b⊢ IsSolvable G All goals completed! 🐙
#85
Bing's house with two rooms is contractible
contractibleSpace_houseWithTwoRooms

Verso theorem preview

theorem declaration uses `sorry`contractibleSpace_houseWithTwoRooms : ContractibleSpace LeanEval.Topology.HouseWithTwoRooms := ⊢ ContractibleSpace ↑HouseWithTwoRooms All goals completed! 🐙
#86
Real cyclotomic integer with house at most 2
cyclotomic_integer_house_le_two

Verso theorem preview

theorem declaration uses `sorry`cyclotomic_integer_house_le_two {K : Type*} [Field K] [NumberField K] [Algebra ℚ K] (n : ℕ) [NeZero n] [IsCyclotomicExtension {n} ℚ K] {β : K} (hβ_int : IsIntegral ℤ β) (hβ_real : β ∈ NumberField.maximalRealSubfield K) : house β ≤ 2 → house β = 2 ∨ ∃ m : ℕ, 0 < m ∧ house β = 2 * Real.cos (Real.pi / m) := K:Type u_1inst✝⁴:Field Kinst✝³:NumberField Kinst✝²:Algebra ℚ Kn:ℕinst✝¹:NeZero ninst✝:IsCyclotomicExtension {n} ℚ Kβ:Khβ_int:IsIntegral ℤ βhβ_real:β ∈ maximalRealSubfield K⊢ house β ≤ 2 → house β = 2 ∨ ∃ m, 0 < m ∧ house β = 2 * Real.cos (Real.pi / ↑m) All goals completed! 🐙
#87
von Neumann double commutant theorem
vonNeumann_doubleCommutant_tfae

Verso theorem preview

theorem declaration uses `sorry`vonNeumann_doubleCommutant_tfae {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : StarSubalgebra ℂ (H →L[ℂ] H)) : List.TFAE [ Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S , IsClosed (ContinuousLinearMapWOT.ofCLM '' (S : Set (H →L[ℂ] H))) , IsClosed (ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H '' (S : Set (H →L[ℂ] H))) ] := H:Type u_1inst✝²:NormedAddCommGroup Hinst✝¹:InnerProductSpace ℂ Hinst✝:CompleteSpace HS:StarSubalgebra ℂ (H →L[ℂ] H)⊢ [(↑S).centralizer.centralizer = ↑S, IsClosed (ContinuousLinearMapWOT.ofCLM '' ↑S), IsClosed (⇑(ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H) '' ↑S)].TFAE All goals completed! 🐙
#88
Abel–Ruffini theorem
abel_ruffini

Verso theorem preview

theorem declaration uses `sorry`abel_ruffini (n : ℕ) (_hn : 1 ≤ n) : (∀ p : ℚ[X], p.natDegree = n → ∀ x : ℂ, aeval x p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 := n:ℕ_hn:1 ≤ n⊢ (∀ (p : ℚ[X]), p.natDegree = n → ∀ (x : ℂ), (aeval x) p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 All goals completed! 🐙
#89
Pell solutions are convergents of √d
pell_solution_convergent

Verso theorem preview

theorem declaration uses `sorry`pell_solution_is_convergent (d : ℤ) (_hd : Squarefree d) (_hd0 : 0 < d) (x y : ℤ) (_hx : 0 < x) (_hy : 0 < y) (_hsol : x ^ 2 - d * y ^ 2 = 1) : ∃ n : ℕ, (GenContFract.of (Real.sqrt (d : ℝ))).convs n = (x : ℝ) / (y : ℝ) := d:ℤ_hd:Squarefree d_hd0:0 < dx:ℤy:ℤ_hx:0 < x_hy:0 < y_hsol:x ^ 2 - d * y ^ 2 = 1⊢ ∃ n, (GenContFract.of √↑d).convs n = ↑x / ↑y All goals completed! 🐙
#90
Linear ODE with negative-real-part eigenvalues is asymptotically stable
linear_ode_asymptotic_stability

Verso theorem preview

theorem declaration uses `sorry`linear_ode_asymptotic_stability (n : ℕ) (A : Matrix (Fin n) (Fin n) ℝ) (hA : ∀ μ : ℂ, Module.End.HasEigenvalue (Matrix.toLin' (A.map (algebraMap ℝ ℂ))) μ → μ.re < 0) (x : ℝ → (Fin n → ℝ)) (hx : ∀ t : ℝ, 0 < t → HasDerivAt x (A.mulVec (x t)) t) : Filter.Tendsto (fun t : ℝ => ‖x t‖) Filter.atTop (nhds 0) := n:ℕA:Matrix (Fin n) (Fin n) ℝhA:∀ (μ : ℂ), Module.End.HasEigenvalue (Matrix.toLin' (A.map ⇑(algebraMap ℝ ℂ))) μ → μ.re < 0x:ℝ → Fin n → ℝhx:∀ (t : ℝ), 0 < t → HasDerivAt x (A *ᵥ x t) t⊢ Filter.Tendsto (fun t => ‖x t‖) Filter.atTop (nhds 0) All goals completed! 🐙
#91
Perron-Frobenius for irreducible nonnegative matrices
irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius

Verso theorem preview

theorem declaration uses `sorry`irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius {n : Type*} [Fintype n] [DecidableEq n] [Nonempty n] (A : Matrix n n ℝ) (hA : A.IsIrreducible) : ∃ v : n → ℝ, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ (∀ i, 0 < v i) := n:Type u_1inst✝²:Fintype ninst✝¹:DecidableEq ninst✝:Nonempty nA:Matrix n n ℝhA:A.IsIrreducible⊢ ∃ v, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ ∀ (i : n), 0 < v i All goals completed! 🐙
#92
Complementary polynomial on the unit circle
exists_complementary_polynomial_on_unit_circle

Verso theorem preview

theorem declaration uses `sorry`exists_complementary_polynomial_on_unit_circle (P : ℂ[X]) (hP : ∀ z : Circle, ‖P.eval (z : ℂ)‖ ≤ 1) : ∃ Q : ℂ[X], Q.natDegree ≤ P.natDegree ∧ ∀ z : Circle, ‖P.eval (z : ℂ)‖ ^ 2 + ‖Q.eval (z : ℂ)‖ ^ 2 = 1 := P:ℂ[X]hP:∀ (z : Circle), ‖eval (↑z) P‖ ≤ 1⊢ ∃ Q, Q.natDegree ≤ P.natDegree ∧ ∀ (z : Circle), ‖eval (↑z) P‖ ^ 2 + ‖eval (↑z) Q‖ ^ 2 = 1 All goals completed! 🐙
#93
Rouche theorem via zero counting
rouche_zero_count_eq

Verso theorem preview

theorem declaration uses `sorry`rouche_zero_count_eq {f g : ℂ → ℂ} {R : ℝ} (hR : 0 < R) (hf : MeromorphicNFOn f Set.univ) (hg : AnalyticOn ℂ g Set.univ) (hbound : ∀ z : ℂ, ‖z‖ = R → ‖g z‖ < ‖f z‖) : (∑ᶠ z, ((divisor (f + g) (Metric.closedBall 0 R))⁺) z) = (∑ᶠ z, ((divisor f (Metric.closedBall 0 R))⁺) z) := f:ℂ → ℂg:ℂ → ℂR:ℝhR:0 < Rhf:MeromorphicNFOn f Set.univhg:AnalyticOn ℂ g Set.univhbound:∀ (z : ℂ), ‖z‖ = R → ‖g z‖ < ‖f z‖⊢ ∑ᶠ (z : ℂ), (divisor (f + g) (Metric.closedBall 0 R))⁺ z = ∑ᶠ (z : ℂ), (divisor f (Metric.closedBall 0 R))⁺ z All goals completed! 🐙
#94
Character values of finite groups lie in cyclotomic fields
brauer_character_in_cyclotomic

Verso theorem preview

theorem declaration uses `sorry`brauer_character_in_cyclotomic (G : Type) [Group G] [Fintype G] : ∃ φ : CyclotomicField (Monoid.exponent G) ℚ →+* ℂ, ∀ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (_ : FiniteDimensional ℂ V) (ρ : Representation ℂ G V) (g : G), LinearMap.trace ℂ V (ρ g) ∈ φ.range := G:Typeinst✝¹:Group Ginst✝:Fintype G⊢ ∃ φ, ∀ (V : Type) (x : AddCommGroup V) (x_1 : Module ℂ V), FiniteDimensional ℂ V → ∀ (ρ : Representation ℂ G V) (g : G), (LinearMap.trace ℂ V) (ρ g) ∈ φ.range All goals completed! 🐙
#95
Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2
dirichlet_eigenvalues_eq_nat_sq

Verso theorem preview

theorem declaration uses `sorry`dirichlet_eigenvalues_eq_nat_sq (lam : ℝ) : (∃ (y : ℝ → ℝ) (J : Set ℝ), IsOpen J ∧ Set.Icc (0 : ℝ) Real.pi ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y Real.pi = 0 ∧ ∃ x ∈ Set.Ioo (0 : ℝ) Real.pi, y x ≠ 0) ↔ ∃ n : ℕ, 0 < n ∧ lam = (n : ℝ) ^ 2 := lam:ℝ⊢ (∃ y J, IsOpen J ∧ Set.Icc 0 π ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y π = 0 ∧ ∃ x ∈ Set.Ioo 0 π, y x ≠ 0) ↔ ∃ n, 0 < n ∧ lam = ↑n ^ 2 All goals completed! 🐙
#96
Minkowski-Caratheodory theorem
mem_convexHull_finset_extremePoints_of_mem_compact_convex

Verso theorem preview

theorem declaration uses `sorry`mem_convexHull_finset_extremePoints_of_mem_compact_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hscomp : IsCompact s) (hsconv : Convex ℝ s) (hx : x ∈ s) : ∃ t : Finset E, (↑t : Set E) ⊆ s.extremePoints ℝ ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ convexHull ℝ (↑t : Set E) := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:FiniteDimensional ℝ Es:Set Ex:Ehscomp:IsCompact shsconv:Convex ℝ shx:x ∈ s⊢ ∃ t, ↑t ⊆ extremePoints ℝ s ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ (convexHull ℝ) ↑t All goals completed! 🐙
#97
Entrywise exponential of a PSD matrix is PSD
posSemidef_map_exp

Verso theorem preview

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n] {A : Matrix n n ℝ} (hA : A.PosSemidef) : (A.map Real.exp).PosSemidef := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef All goals completed! 🐙
#98
Gaussian heat kernel solves the 1D heat equation
heat_kernel_solves_heat_equation

Verso theorem preview

theorem declaration uses `sorry`heat_kernel_solves_heat_equation (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_bdd : ∃ M : ℝ, ∀ x, |f x| ≤ M) : -- The PDE on (0, ∞) × ℝ. (∀ t : ℝ, 0 < t → ∀ x : ℝ, ∃ ux : ℝ → ℝ, ∃ uxx : ℝ, (∀ y : ℝ, HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ -- Initial condition recovered as a one-sided limit at t = 0. (∀ x : ℝ, Filter.Tendsto (fun t : ℝ => heatSolution f t x) (nhdsWithin (0 : ℝ) (Set.Ioi 0)) (nhds (f x))) := f:ℝ → ℝhf_cont:Continuous fhf_bdd:∃ M, ∀ (x : ℝ), |f x| ≤ M⊢ (∀ (t : ℝ), 0 < t → ∀ (x : ℝ), ∃ ux uxx, (∀ (y : ℝ), HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ ∀ (x : ℝ), Filter.Tendsto (fun t => heatSolution f t x) (nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) All goals completed! 🐙
#99
Oppenheim's inequality for Hadamard products
oppenheim_inequality

Verso theorem preview

theorem declaration uses `sorry`oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : A.det * ∏ i, B i i ≤ (A ⊙ B).det := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝB:Matrix n n ℝhA:A.PosSemidefhB:B.PosSemidef⊢ A.det * ∏ i, B i i ≤ (A ⊙ B).det All goals completed! 🐙
#100
pi_1 of the circle is Z
pi1_circle_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi1_circle_mulEquiv_int : Nonempty (HomotopyGroup.Pi 1 Circle (1 : Circle) ≃* Multiplicative ℤ) := ⊢ Nonempty (HomotopyGroup.Pi 1 Circle 1 ≃* Multiplicative ℤ) All goals completed! 🐙
#101
Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#104
Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#105
Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#108
Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#109
Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#110
Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#111

Test problems: multi_hole_helpers_example, noncomputable_hole_example, variable_binder_example, def_hole_example, instance_hole_example, ci_regenerate_main_check, list_append_singleton_length, two_plus_two (8 / 8 solved)

Submission history

First submissionJun 18, 2026
Last submissionJul 16, 2026

Contributors

lukerj00112
6Aleph Prover(logicalintelligence.com)56 solved

Other solved problems

Liouville–Arnold theorem on integrable systems
liouville_arnold

Verso theorem preview

theorem declaration uses `sorry`liouville_arnold {n : ℕ} (F : Fin n → LeanEval.Geometry.LiouvilleArnold.E n → ℝ) (U : Set (LeanEval.Geometry.LiouvilleArnold.E n)) (_hU : IsOpen U) (_hLI : LeanEval.Geometry.LiouvilleArnold.IsLiouvilleIntegrable F U) (c : Fin n → ℝ) (_hMc_sub : LeanEval.Geometry.LiouvilleArnold.levelSet F c ⊆ U) (_hMc_compact : IsCompact (LeanEval.Geometry.LiouvilleArnold.levelSet F c)) (_hMc_connected : IsConnected (LeanEval.Geometry.LiouvilleArnold.levelSet F c)) : Nonempty ((LeanEval.Geometry.LiouvilleArnold.levelSet F c) ≃ₜ (Fin n → AddCircle (1 : ℝ))) := n:ℕF:Fin n → E n → ℝU:Set (E n)_hU:IsOpen U_hLI:IsLiouvilleIntegrable F Uc:Fin n → ℝ_hMc_sub:levelSet F c ⊆ U_hMc_compact:IsCompact (levelSet F c)_hMc_connected:IsConnected (levelSet F c)⊢ Nonempty (↑(levelSet F c) ≃ₜ (Fin n → AddCircle 1)) All goals completed! 🐙
#1
How produced

Two attempts were made, for the second one we added guidance to only focus on homeomorphism(asked by the challenge) and not diffeomorphism(which is what the actual theorem claims). The first attempt got stuck in the part that's not well developed in Mathlib.

Local stable/unstable sets at a hyperbolic fixed point (set-level Hadamard–Perron)
stable_unstable_manifolds

Verso theorem preview

theorem declaration uses `sorry`stable_unstable_manifolds_exist (n : ℕ) (f : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (x₀ : LeanEval.Dynamics.StableUnstableManifoldsProblem.E n) (_hf : ContDiffAt ℝ 1 f x₀) (_hfix : f x₀ = x₀) (_hhyp : LeanEval.Dynamics.StableUnstableManifoldsProblem.IsHyperbolicLinear (fderiv ℝ f x₀)) (_hf_inv : (fderiv ℝ f x₀).IsInvertible) : ∃ U : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu : Set (LeanEval.Dynamics.StableUnstableManifoldsProblem.E n), Ws = {x | (∀ k : ℕ, f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y : ℕ → LeanEval.Dynamics.StableUnstableManifoldsProblem.E n, y 0 = x ∧ (∀ k : ℕ, y k ∈ U) ∧ (∀ k : ℕ, f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} := n:ℕf:E n → E nx₀:E n_hf:ContDiffAt ℝ 1 f x₀_hfix:f x₀ = x₀_hhyp:IsHyperbolicLinear (fderiv ℝ f x₀)_hf_inv:(fderiv ℝ f x₀).IsInvertible⊢ ∃ U, IsOpen U ∧ x₀ ∈ U ∧ ∃ Ws Wu, Ws = {x | (∀ (k : ℕ), f^[k] x ∈ U) ∧ Tendsto (fun k => f^[k] x) atTop (𝓝 x₀)} ∧ Wu = {x | ∃ y, y 0 = x ∧ (∀ (k : ℕ), y k ∈ U) ∧ (∀ (k : ℕ), f (y (k + 1)) = y k) ∧ Tendsto y atTop (𝓝 x₀)} ∧ Ws ∩ Wu = {x₀} All goals completed! 🐙
#2
How produced

Solved completely autonomously without human intervention

Mountain Pass Theorem (Ambrosetti–Rabinowitz 1973)
mountain_pass

Verso theorem preview

theorem declaration uses `sorry`mountain_pass (f : E → ℝ) (_hf : ContDiff ℝ 1 f) (_hps : LeanEval.Analysis.MountainPassProblem.PalaisSmale f) {a b : E} {ε r : ℝ} (_hmr : LeanEval.Analysis.MountainPassProblem.MountainRange f a b ε r) : ∃ x : E, LeanEval.Analysis.MountainPassProblem.IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace Ef:E → ℝ_hf:ContDiff ℝ 1 f_hps:PalaisSmale fa:Eb:Eε:ℝr:ℝ_hmr:MountainRange f a b ε r⊢ ∃ x, IsCriticalPoint f x ∧ f x = mountainPassLevel f a b ∧ ε ≤ mountainPassLevel f a b All goals completed! 🐙
#3
How produced

Solved completely autonomously without human intervention

Fraser: Fourier decay for finite-field Kakeya sets is q^{-1} and sharp
fraser_kakeya_fourier_decay

Verso theorem preview

theorem declaration uses `sorry`fraser_kakeya_fourier_decay_and_sharp {d : ℕ} (_hd : 2 ≤ d) {K : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F d)} (_hK : LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K) (χ : AddChar F ℂ) (_hχ : χ ≠ 1) : (∃ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K μ ∧ ∀ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F d, ξ ≠ 0 → ‖fourier χ μ ξ‖ ≤ (Fintype.card F : ℝ)⁻¹) ∧ (∀ κ : ℝ, 0 < κ → κ < 1 → ∃ Q : ℕ, ∀ (F' : Type*) [Field F'] [Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K' : Set (LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d), LeanEval.Combinatorics.FraserKakeyaProblem.IsKakeya K' ∧ ∀ μ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d → ℝ, LeanEval.Combinatorics.FraserKakeyaProblem.IsProbabilityMeasureOn K' μ → ∃ ξ : LeanEval.Combinatorics.FraserKakeyaProblem.Space F' d, ξ ≠ 0 ∧ κ * (Fintype.card F' : ℝ)⁻¹ ≤ ‖fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖) := F:Type u_1inst✝²:Field Finst✝¹:Fintype Finst✝:DecidableEq Fd:ℕ_hd:2 ≤ dK:Set (Space F d)_hK:IsKakeya Kχ:AddChar F ℂ_hχ:χ ≠ 1⊢ (∃ μ, IsProbabilityMeasureOn K μ ∧ ∀ (ξ : Space F d), ξ ≠ 0 → ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier χ μ ξ‖ ≤ (↑(Fintype.card F))⁻¹) ∧ ∀ (κ : ℝ), 0 < κ → κ < 1 → ∃ Q, ∀ (F' : Type u_2) [inst : Field F'] [inst_1 : Fintype F'] [DecidableEq F'], Q ≤ Fintype.card F' → ∃ K', IsKakeya K' ∧ ∀ (μ : Space F' d → ℝ), IsProbabilityMeasureOn K' μ → ∃ ξ, ξ ≠ 0 ∧ κ * (↑(Fintype.card F'))⁻¹ ≤ ‖LeanEval.Combinatorics.FraserKakeyaProblem.fourier (AddChar.FiniteField.primitiveChar_to_Complex F') μ ξ‖ All goals completed! 🐙
#4
How produced

Solved completely autonomously without human intervention

Euler–Lagrange equation
euler_lagrange_equation

Verso theorem preview

theorem declaration uses `sorry`euler_lagrange_equation {a b : ℝ} (L : ℝ → ℝ → ℝ → ℝ) (x : ℝ → ℝ) (_hab : a < b) (_hL : ContDiff ℝ 2 (fun p : ℝ × ℝ × ℝ => L p.1 p.2.1 p.2.2)) (_hx : ContDiff ℝ 2 x) (_hxe : LeanEval.Analysis.IsVariationalExtremum a b L x) : ∀ t ∈ Set.Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t := a:ℝb:ℝL:ℝ → ℝ → ℝ → ℝx:ℝ → ℝ_hab:a < b_hL:ContDiff ℝ 2 fun p => L p.1 p.2.1 p.2.2_hx:ContDiff ℝ 2 x_hxe:IsVariationalExtremum a b L x⊢ ∀ t ∈ Ioo a b, lagrangianPartialX L x t = deriv (lagrangianPartialV L x) t All goals completed! 🐙
#5
How produced

Solved completely autonomously without human intervention

Monge–Kantorovich existence theorem
monge_kantorovich

Verso theorem preview

theorem declaration uses `sorry`monge_kantorovich_exists {X Y : Type*} [TopologicalSpace X] [PolishSpace X] [MeasurableSpace X] [BorelSpace X] [TopologicalSpace Y] [PolishSpace Y] [MeasurableSpace Y] [BorelSpace Y] (P : Measure X) (Q : Measure Y) [IsProbabilityMeasure P] [IsProbabilityMeasure Q] (c : X × Y → ENNReal) (_hc : Continuous c) : ∃ π ∈ LeanEval.Analysis.Couplings P Q, ∀ π' ∈ LeanEval.Analysis.Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' := X:Type u_1Y:Type u_2inst✝⁹:TopologicalSpace Xinst✝⁸:PolishSpace Xinst✝⁷:MeasurableSpace Xinst✝⁶:BorelSpace Xinst✝⁵:TopologicalSpace Yinst✝⁴:PolishSpace Yinst✝³:MeasurableSpace Yinst✝²:BorelSpace YP:Measure XQ:Measure Yinst✝¹:IsProbabilityMeasure Pinst✝:IsProbabilityMeasure Qc:X × Y → ENNReal_hc:Continuous c⊢ ∃ π ∈ Couplings P Q, ∀ π' ∈ Couplings P Q, kantorovichCost c π ≤ kantorovichCost c π' All goals completed! 🐙
#6
How produced

Solved completely autonomously without human intervention

Lidskii–Last eigenvalue-perturbation theorem
lidskii_last

Verso theorem preview

theorem declaration uses `sorry`lidskii_last {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitian⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ≤ ∑ i, ∑ j, ‖A i j - B i j‖ All goals completed! 🐙
#7
How produced

The proof required `lidskii_inequality` solution, it was previously solved by Aleph, then for this run it had access to that whole proof. At the end, only the `lidskii_inequality` theorem itself was needed. No other human intervention other wise.

Furstenberg–Weiss topological multiple recurrence (single-transformation form)
furstenberg_topological

Verso theorem preview

theorem declaration uses `sorry`furstenberg_topological_recurrence {X : Type*} [MetricSpace X] [CompactSpace X] [Nonempty X] (T : X ≃ₜ X) : ∃ x : X, LeanEval.Dynamics.IsMultiplyRecurrent (T : X → X) x := X:Type u_1inst✝²:MetricSpace Xinst✝¹:CompactSpace Xinst✝:Nonempty XT:X ≃ₜ X⊢ ∃ x, IsMultiplyRecurrent (⇑T) x All goals completed! 🐙
#8
How produced

Solved completely autonomously without human intervention

Lidskii's inequality
lidskii_inequality

Verso theorem preview

theorem declaration uses `sorry`lidskii_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℂ} (hA : A.IsHermitian) (hB : B.IsHermitian) {p : ℝ} (_hp : 1 ≤ p) : ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |(hB.sub hA).eigenvalues₀ j| ^ p := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℂB:Matrix n n ℂhA:A.IsHermitianhB:B.IsHermitianp:ℝ_hp:1 ≤ p⊢ ∑ j, |hA.eigenvalues₀ j - hB.eigenvalues₀ j| ^ p ≤ ∑ j, |⋯.eigenvalues₀ j| ^ p All goals completed! 🐙
#9
How produced

Solved completely autonomously without human intervention

Solvable extensions ↔ solvable groups (the missing converse in Abel–Ruffini)
solvable_by_radicals_converse

Verso theorem preview

theorem declaration uses `sorry`solvable_iff_solvableByRad (F : Type*) [Field F] [CharZero F] (p : F[X]) (_hp : p ≠ 0) : (∀ x : AlgebraicClosure F, aeval x p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal := F:Type u_1inst✝¹:Field Finst✝:CharZero Fp:F[X]_hp:p ≠ 0⊢ (∀ (x : AlgebraicClosure F), (aeval x) p = 0 → x ∈ solvableByRad F (AlgebraicClosure F)) ↔ IsSolvable p.Gal All goals completed! 🐙
#10
How produced

Solved completely autonomously without human intervention

Frobenius's theorem: the Frobenius kernel is normal
frobenius_kernel_isNormal

Lean theorem statement

theorem frobenius_kernel_isNormal
    (G X : Type) [Group G] [Fintype G] [Fintype X]
    [MulAction G X] [FaithfulSMul G X]
    (hcard : 2 ≤ Fintype.card X)
    (htrans : ∀ x y : X, ∃ g : G, g • x = y)
    (hstab : ∀ x : X, MulAction.stabilizer G x ≠ ⊥)
    (hfrob : ∀ g : G, g ≠ 1 → ∀ x y : X, g • x = x → g • y = y → x = y) :
    ∃ N : Subgroup G, N.Normal ∧
      (N : Set G) = {1} ∪ {g : G | ∀ x : X, g • x ≠ x} := by
  sorry
#11
How produced

Solved completely autonomously without human intervention

Nash equilibrium existence theorem
nash_equilibrium_exists

Lean theorem statement

/-- **Nash equilibrium existence theorem.** Every finite `n`-player game
with nonempty finite pure-strategy sets and arbitrary real payoffs admits
at least one mixed-strategy Nash equilibrium. -/
theorem nash_equilibrium_exists {n : ℕ} {S : Fin n → Type*}
    [∀ i, Fintype (S i)] [∀ i, Nonempty (S i)]
    (u : Fin n → StrategyProfile n S → ℝ) :
    ∃ σ : ∀ i, S i → ℝ, IsNashEquilibrium u σ := by
  sorry
#12
How produced

The solution used schauder_fixed_point. Schauder fixed-point theorem was first proven by Aleph and then manually inlined. Other than that, the solution was produced autonomously.

Kakutani fixed-point theorem
kakutani_fixed_point

Lean theorem statement

/-- **Kakutani fixed-point theorem.** Every upper-hemicontinuous
correspondence `F` from a nonempty compact convex `K ⊆ ℝᵈ` to itself, with
nonempty convex closed values, has a fixed point `x ∈ F x`. -/
theorem kakutani_fixed_point {d : ℕ}
    {K : Set (EuclideanSpace ℝ (Fin d))}
    (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K)
    (_hK_nonempty : K.Nonempty)
    (F : EuclideanSpace ℝ (Fin d) → Set (EuclideanSpace ℝ (Fin d)))
    (_hF_uhc : IsUpperHemicontinuous F)
    (_hF_nonempty : ∀ x ∈ K, (F x).Nonempty)
    (_hF_convex : ∀ x ∈ K, Convex ℝ (F x))
    (_hF_closed : ∀ x ∈ K, IsClosed (F x))
    (_hF_maps : ∀ x ∈ K, F x ⊆ K) :
    ∃ x ∈ K, x ∈ F x := by
  sorry
#13
How produced

The solution used brouwer_fixed_point. Brouwer fixed-point theorem was first proved by Aleph and then manually inlined. Other than that, the solutions were produced autonomously.

Sturm's theorem
sturm

Lean theorem statement

/-- **Sturm's theorem.** For a squarefree real polynomial `p` and an interval
`(a, b)` with `a < b` whose endpoints are not roots of `p`, the number of
distinct roots of `p` in `(a, b)` equals `σ(a) − σ(b)`. -/
theorem sturm (p : ℝ[X]) (hp : Squarefree p) {a b : ℝ} (hab : a < b)
    (ha : p.eval a ≠ 0) (hb : p.eval b ≠ 0) :
    ((p.roots.toFinset).filter (fun x => a < x ∧ x < b)).card =
      sigma p a - sigma p b := by
  sorry
#14
How produced

Solved completely autonomously without human intervention

Koszul formula
koszul_formula

Lean theorem statement

/-- **Koszul formula.** For any smooth torsion-free metric-compatible
covariant derivative `cov` on `TM`, `2 ⟨∇_X Y, Z⟩` equals the cyclic sum
of directional derivatives `X·⟨Y, Z⟩ + Y·⟨X, Z⟩ − Z·⟨X, Y⟩` minus the
Lie-bracket cyclic sum `⟨X, [Y, Z]⟩ + ⟨Y, [X, Z]⟩ − ⟨Z, [X, Y]⟩`. -/
theorem koszul_formula
    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
      [FiniteDimensional ℝ E] [CompleteSpace E]
    {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℝ E H}
    {M : Type*} [TopologicalSpace M] [ChartedSpace H M]
      [IsManifold I ∞ M]
    [RiemannianBundle (fun (x : M) ↦ TangentSpace I x)]
    [IsContMDiffRiemannianBundle I ∞ E (fun (x : M) ↦ TangentSpace I x)]
    (cov : CovariantDerivative I E (TangentSpace I (M := M)))
    [ContMDiffCovariantDerivative cov ∞]
    (_htor : cov.torsion = 0) (_hmet : IsMetricCompatible cov)
    (X Y Z : Π x : M, TangentSpace I x)
    (_hX : CMDiff ∞ (T% X)) (_hY : CMDiff ∞ (T% Y)) (_hZ : CMDiff ∞ (T% Z))
    (x : M) :
    2 * inner ℝ (cov Y x (X x)) (Z x) =
      mvfderiv I (fun y : M => inner ℝ (Y y) (Z y)) x (X x)
      + mvfderiv I (fun y : M => inner ℝ (X y) (Z y)) x (Y x)
      - mvfderiv I (fun y : M => inner ℝ (X y) (Y y)) x (Z x)
      - inner ℝ (X x) (mlieBracket I Y Z x)
      - inner ℝ (Y x) (mlieBracket I X Z x)
      + inner ℝ (Z x) (mlieBracket I X Y x) := by
  sorry
#15
How produced

Solved completely autonomously without human intervention

Independence of the parallel postulate
parallel_postulate_independent

Lean theorem statement

/-- **Independence of the parallel postulate** (Freek #12). The Euclidean
axiom `A10` is logically independent of Tarski's absolute axioms `A1`–`A9`
and `A11`: there is a model of the absolute axioms in which the parallel
postulate holds (the real coordinate plane) and one in which it fails (the
Klein–Beltrami disk, or any other hyperbolic-plane model). -/
theorem parallel_postulate_independent :
    (∃ (M : Type) (T : TarskiAbsolute M), Euclidean M T) ∧
    (∃ (M : Type) (T : TarskiAbsolute M), ¬ Euclidean M T) := by
  sorry
#16
How produced

Solved completely autonomously without human intervention

Brauer–Fowler theorem
brauer_fowler

Lean theorem statement

/-- **Brauer–Fowler theorem.** There is a function bounding the order
of a finite nonabelian simple group by the order of any involution
centralizer. -/
theorem brauer_fowler :
    ∃ f : ℕ → ℕ, ∀ (G : Type) [Group G] [Finite G],
      IsSimpleGroup G → (∃ a b : G, a * b ≠ b * a) →
      ∀ t : G, orderOf t = 2 →
        Nat.card G ≤ f (Nat.card (Subgroup.centralizer ({t} : Set G))) := by
  sorry
#17
How produced

Solved completely autonomously without human intervention

Balanceable k-bounded partitions
balanceable_bounded_partitions

Lean theorem statement

/--
For any `k`, the smallest `n` such that any `k`-bounded partition of `n` is
also balanceable is given by `2 * lcm(1, ..., k)`.
-/
theorem minimal_balanceable_of_bounded (k : ℕ) (hk : 0 < k) :
    Minimal (fun n => 0 < n ∧ ∀ p : n.Partition, Bounded k p → Balanceable p) (2 * (Finset.Icc 1 k).lcm id) := by
  sorry
#18
How produced

Solved without human intervention. Two `native_decide`'s were used in the AI-generated proof(one as `interval_cases k <;> native_decide`). Those were replaced with `decide` manually. The goals that were solved by `native_decide`->`decide` were: ``` ⊢ ∑ i ∈ Finset.Icc 1 0, i * (i - 2) ≤ 0 * (0 - 1) * (0 - 2) / 2 case pos.«0» k : ℕ ih : ∑ i ∈ Finset.Icc 1 0, i * (i - 2) ≤ 0 * (0 - 1) * (0 - 2) / 2 hk : 0 < 3 ⊢ ∑ i ∈ Finset.Icc 1 (0 + 1), i * (i - 2) ≤ (0 + 1) * (0 + 1 - 1) * (0 + 1 - 2) / 2 case pos.«1» k : ℕ ih : ∑ i ∈ Finset.Icc 1 1, i * (i - 2) ≤ 1 * (1 - 1) * (1 - 2) / 2 hk : 1 < 3 ⊢ ∑ i ∈ Finset.Icc 1 (1 + 1), i * (i - 2) ≤ (1 + 1) * (1 + 1 - 1) * (1 + 1 - 2) / 2 case pos.«2» k : ℕ ih : ∑ i ∈ Finset.Icc 1 2, i * (i - 2) ≤ 2 * (2 - 1) * (2 - 2) / 2 hk : 2 < 3 ⊢ ∑ i ∈ Finset.Icc 1 (2 + 1), i * (i - 2) ≤ (2 + 1) * (2 + 1 - 1) * (2 + 1 - 2) / 2 ``` (Aleph wasn't rerun in strict mode just to save time)

A competition programming problem about permuting a permutation to be unimodal
permute_to_unimodal

Lean theorem statement

/--
`minRearrange` correctly computes the smallest number of indices that need to be permuted in order to
turn `arr` into a unimodal permutation.
-/
theorem minRearrange_correct {arr : Array Nat} :
    arr.Perm (1...=arr.size).toArray →
      (∃ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x ∧ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector = minRearrange arr) ∧
      (∀ (x : Array Nat) (hx : x.Perm (1...=arr.size).toArray), Unimodal x → minRearrange arr ≤ differences (Vector.mk x (by simpa using hx.size_eq)) arr.toVector) := by
  sorry
#19
How produced

Solved completely autonomously without human intervention

Chen theorem for Markoff graphs
dvd_card_connectedComponent_markoffGraph

Lean theorem statement

/-- For prime `p > 3`, every connected component of the nonzero Markoff graph over `ZMod p`
has cardinality divisible by `p`. -/
theorem dvd_card_connectedComponent_markoffGraph
    {p : ℕ} (hp : Nat.Prime p) (hgt : 3 < p) :
    ∀ c : (markoffGraph p).ConnectedComponent, p ∣ Nat.card c := by
  sorry
#20
How produced

Solved completely autonomously without human intervention

Linear programming: maximum principle and vertex optimality
lp_maximum_principle

Verso theorem preview

/-- **Maximum principle for linear programming** (§101). A local maximiser of the LP objective on the feasible region is automatically a global maximiser; and whenever the objective is non-constant (`c ≠ 0`), the maximiser lies on the topological frontier of the feasible region. -/ theorem declaration uses `sorry`lp_maximum_principle {m n : ℕ} (lp : LinearProgram m n) (x : Fin m → ℝ) (_hx : x ∈ lp.feasible) (_hlocal : IsLocalMaxOn lp.objective lp.feasible x) : IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) := m:ℕn:ℕlp:LinearProgram m nx:Fin m → ℝ_hx:x ∈ lp.feasible_hlocal:IsLocalMaxOn lp.objective lp.feasible x⊢ IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) All goals completed! 🐙
/-- **Vertex optimality** (§101; the existence content of Dantzig's 1947 simplex algorithm). Every linear program with a nonempty bounded feasible region admits a global maximiser that is an extreme point (vertex) of the feasible region. -/ theorem declaration uses `sorry`simplex_algorithm {m n : ℕ} (lp : LinearProgram m n) (_hfeas : lp.feasible.Nonempty) (_hbdd : Bornology.IsBounded lp.feasible) : ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible := m:ℕn:ℕlp:LinearProgram m n_hfeas:lp.feasible.Nonempty_hbdd:Bornology.IsBounded lp.feasible⊢ ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible All goals completed! 🐙
#21
How produced

Solved completely autonomously without human intervention

Kuznetsov's theorem: finitely presented simple groups have solvable word problem
boone_higman_simple

Verso theorem preview

theorem declaration uses `sorry`boone_higman_simple {G : Type*} [Group G] [IsSimpleGroup G] {n : ℕ} (φ : FreeGroup (Fin n) →* G) (_hsurj : Function.Surjective φ) (_hker : (MonoidHom.ker φ).IsNormalClosureFG) : LeanEval.GroupTheory.BooneHigmanSimpleProblem.WordProblemSolvable φ := G:Type u_1inst✝¹:Group Ginst✝:IsSimpleGroup Gn:ℕφ:FreeGroup (Fin n) →* G_hsurj:Function.Surjective ⇑φ_hker:φ.ker.IsNormalClosureFG⊢ WordProblemSolvable φ All goals completed! 🐙
#22
How produced

Solved completely autonomously without human intervention

Bourbaki's locally convex extension of Banach–Alaoglu
banach_alaoglu_bourbaki

Verso theorem preview

theorem declaration uses `sorry`banach_alaoglu_bourbaki (E : Type*) [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] (U : Set E) (_hU : U ∈ 𝓝 (0 : E)) : IsCompact (LeanEval.Analysis.weakStarPolar E U) := E:Type u_1inst✝⁵:AddCommGroup Einst✝⁴:Module ℝ Einst✝³:TopologicalSpace Einst✝²:ContinuousAdd Einst✝¹:ContinuousSMul ℝ Einst✝:LocallyConvexSpace ℝ EU:Set E_hU:U ∈ 𝓝 0⊢ IsCompact (weakStarPolar E U) All goals completed! 🐙
#23
How produced

Solved completely autonomously without human intervention

Wiener's atom-detection formula
wiener_atom_detection

Verso theorem preview

theorem declaration uses `sorry`wiener_atom_detection (μ : Measure (AddCircle (2 * Real.pi))) [IsProbabilityMeasure μ] : Tendsto (fun N : ℕ => (1 / (N : ℝ)) * ∑ k ∈ Finset.Icc (1 : ℤ) N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' x : AddCircle (2 * Real.pi), ((μ {x}).toReal) ^ 2)) := μ:Measure (AddCircle (2 * π))inst✝:IsProbabilityMeasure μ⊢ Tendsto (fun N => 1 / ↑N * ∑ k ∈ Finset.Icc 1 ↑N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' (x : AddCircle (2 * π)), (μ {x}).toReal ^ 2)) All goals completed! 🐙
#24
How produced

Solved completely autonomously without human intervention

Runge's theorem
runge_theorem

Verso theorem preview

theorem declaration uses `sorry`runge (K : Set ℂ) (_hK : IsCompact K) (U : Set ℂ) (_hU : IsOpen U) (_hKU : K ⊆ U) (f : ℂ → ℂ) (_hf : AnalyticOnNhd ℂ f U) (ε : ℝ) (_hε : 0 < ε) : ∃ p q : ℂ[X], (∀ z ∈ K, q.eval z ≠ 0) ∧ (∀ z ∈ K, ‖f z - p.eval z / q.eval z‖ < ε) := K:Set ℂ_hK:IsCompact KU:Set ℂ_hU:IsOpen U_hKU:K ⊆ Uf:ℂ → ℂ_hf:AnalyticOnNhd ℂ f Uε:ℝ_hε:0 < ε⊢ ∃ p q, (∀ z ∈ K, Polynomial.eval z q ≠ 0) ∧ ∀ z ∈ K, ‖f z - Polynomial.eval z p / Polynomial.eval z q‖ < ε All goals completed! 🐙
#25
How produced

Solved completely autonomously without human intervention

Baer–Suzuki theorem
baer_suzuki

Verso theorem preview

theorem declaration uses `sorry`baer_suzuki {G : Type*} [Group G] [Finite G] {p : ℕ} [Fact p.Prime] (x : G) : x ∈ LeanEval.GroupTheory.Defs.pCore p G ↔ ∀ g : G, IsPGroup p (Subgroup.closure ({x, g * x * g⁻¹} : Set G)) := G:Type u_1inst✝²:Group Ginst✝¹:Finite Gp:ℕinst✝:Fact (Nat.Prime p)x:G⊢ x ∈ pCore p G ↔ ∀ (g : G), IsPGroup p ↥(Subgroup.closure {x, g * x * g⁻¹}) All goals completed! 🐙
#26
How produced

Solved completely autonomously without human intervention

Schauder fixed-point theorem
schauder_fixed_point

Verso theorem preview

theorem declaration uses `sorry`schauder_fixed_point {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {K : Set E} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : E → E) (_hf_cont : ContinuousOn f K) (_hf_maps : Set.MapsTo f K K) : ∃ x ∈ K, f x = x := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace EK:Set E_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:E → E_hf_cont:ContinuousOn f K_hf_maps:Set.MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#27
How produced

The solution used brouwer_fixed_point. Brouwer fixed-point theorem was first proved by Aleph and then manually inlined. Other than that, the solutions were produced autonomously.

Brouwer fixed-point theorem
brouwer_fixed_point

Verso theorem preview

theorem declaration uses `sorry`brouwer_fixed_point {d : ℕ} {K : Set (EuclideanSpace ℝ (Fin d))} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (_hf_cont : ContinuousOn f K) (_hf_maps : MapsTo f K K) : ∃ x ∈ K, f x = x := d:ℕK:Set (EuclideanSpace ℝ (Fin d))_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)_hf_cont:ContinuousOn f K_hf_maps:MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#28
How produced

First attempt used `pin_sphere_n_mulEquiv_int` which Aleph can't prove yet. For the second attempt we asked Aleph to use Milnor's proof instead and to avoid "`HomotopyGroup.Pi`, `pin_sphere_n_mulEquiv_int`, `pi1_circle_mulEquiv_int`, singular/cellular homology, or fundamental groups. Mathlib's algebraic-topology infrastructure is too thin to prove "S^(d-1) is not contractible"(that was human judgement).

Symplectic matrices have determinant 1
symplectic_matrix_det

Verso theorem preview

theorem declaration uses `sorry`symplectic_matrix_det {l R : Type*} [DecidableEq l] [Fintype l] [CommRing R] {A : Matrix (l ⊕ l) (l ⊕ l) R} (_hA : A ∈ Matrix.symplecticGroup l R) : A.det = 1 := l:Type u_1R:Type u_2inst✝²:DecidableEq linst✝¹:Fintype linst✝:CommRing RA:Matrix (l ⊕ l) (l ⊕ l) R_hA:A ∈ symplecticGroup l R⊢ A.det = 1 All goals completed! 🐙
#29
How produced

Solved completely autonomously without human intervention

Schur-Weyl duality: S_k image equals centralizer of GL(V) image
symAction_range_eq_centralizer_glAction

Verso theorem preview

theorem declaration uses `sorry`symAction_range_eq_centralizer_glAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(symAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(glAction R M k)) All goals completed! 🐙
#30
How produced

Solved without human intervention Solution to glAction_range_eq_centralizer_symAction was used for this proof as internally all the benchmark problems were put into a single Lean project. We manually inlined that solution.

Schur-Weyl duality: GL(V) image equals centralizer of S_k image
glAction_range_eq_centralizer_symAction

Verso theorem preview

theorem declaration uses `sorry`glAction_range_eq_centralizer_symAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(glAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(symAction R M k)) All goals completed! 🐙
#31
How produced

Solved completely autonomously without human intervention

Pointwise and Cesàro convergence of Fourier series (Dirichlet, Fejér)
fourier_dirichlet_fejer

Verso theorem preview

/-- **Dirichlet's pointwise convergence theorem** (§46). For every `C¹` 2π-periodic complex function `f`, the symmetric Fourier partial sums `S_N(f)(x)` converge to `f(x)` at every point `x ∈ ℝ`. -/ theorem declaration uses `sorry`dirichlet_pointwise {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hC1 : ContDiff ℝ 1 f) (x : ℝ) : Tendsto (fun N : ℕ => fourierPartialSum f N x) atTop (𝓝 (f x)) := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hC1:ContDiff ℝ 1 fx:ℝ⊢ Tendsto (fun N => fourierPartialSum f N x) atTop (𝓝 (f x)) All goals completed! 🐙
/-- **Fejér's theorem** (§46). For every *continuous* 2π-periodic complex function `f` — without the `C¹` hypothesis of Dirichlet's theorem — the Cesàro means `σ_N(f)` of the symmetric Fourier partial sums converge to `f` uniformly on `ℝ`. -/ theorem declaration uses `sorry`fejer {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hcont : Continuous f) : TendstoUniformly (fun N : ℕ => fourierCesaroMean f N) f atTop := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hcont:Continuous f⊢ TendstoUniformly (fun N => fourierCesaroMean f N) f atTop All goals completed! 🐙
#32
How produced

Solved completely autonomously without human intervention

Abel–Ruffini theorem
abel_ruffini

Verso theorem preview

theorem declaration uses `sorry`abel_ruffini (n : ℕ) (_hn : 1 ≤ n) : (∀ p : ℚ[X], p.natDegree = n → ∀ x : ℂ, aeval x p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 := n:ℕ_hn:1 ≤ n⊢ (∀ (p : ℚ[X]), p.natDegree = n → ∀ (x : ℂ), (aeval x) p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 All goals completed! 🐙
#33
How produced

Solved completely autonomously without human intervention

Pell solutions are convergents of √d
pell_solution_convergent

Verso theorem preview

theorem declaration uses `sorry`pell_solution_is_convergent (d : ℤ) (_hd : Squarefree d) (_hd0 : 0 < d) (x y : ℤ) (_hx : 0 < x) (_hy : 0 < y) (_hsol : x ^ 2 - d * y ^ 2 = 1) : ∃ n : ℕ, (GenContFract.of (Real.sqrt (d : ℝ))).convs n = (x : ℝ) / (y : ℝ) := d:ℤ_hd:Squarefree d_hd0:0 < dx:ℤy:ℤ_hx:0 < x_hy:0 < y_hsol:x ^ 2 - d * y ^ 2 = 1⊢ ∃ n, (GenContFract.of √↑d).convs n = ↑x / ↑y All goals completed! 🐙
#34
How produced

Solved completely autonomously without human intervention

Bing's house with two rooms is contractible
contractibleSpace_houseWithTwoRooms

Verso theorem preview

theorem declaration uses `sorry`contractibleSpace_houseWithTwoRooms : ContractibleSpace LeanEval.Topology.HouseWithTwoRooms := ⊢ ContractibleSpace ↑HouseWithTwoRooms All goals completed! 🐙
#35
How produced

Solved completely autonomously without human intervention

Existence of a non-isotopic pair of oriented two-component links
exists_nonisotopic_link

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_link : ∃ L₁ L₂ : LeanEval.KnotTheory.TwoLink, ¬ L₁.Isotopic L₂ := ⊢ ∃ L₁ L₂, ¬L₁.Isotopic L₂ All goals completed! 🐙
#36
How produced

Solved completely autonomously without human intervention

von Neumann double commutant theorem
vonNeumann_doubleCommutant_tfae

Verso theorem preview

theorem declaration uses `sorry`vonNeumann_doubleCommutant_tfae {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : StarSubalgebra ℂ (H →L[ℂ] H)) : List.TFAE [ Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S , IsClosed (ContinuousLinearMapWOT.ofCLM '' (S : Set (H →L[ℂ] H))) , IsClosed (ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H '' (S : Set (H →L[ℂ] H))) ] := H:Type u_1inst✝²:NormedAddCommGroup Hinst✝¹:InnerProductSpace ℂ Hinst✝:CompleteSpace HS:StarSubalgebra ℂ (H →L[ℂ] H)⊢ [(↑S).centralizer.centralizer = ↑S, IsClosed (ContinuousLinearMapWOT.ofCLM '' ↑S), IsClosed (⇑(ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H) '' ↑S)].TFAE All goals completed! 🐙
#37
How produced

Solved completely autonomously without human intervention

Real cyclotomic integer with house at most 2
cyclotomic_integer_house_le_two

Verso theorem preview

theorem declaration uses `sorry`cyclotomic_integer_house_le_two {K : Type*} [Field K] [NumberField K] [Algebra ℚ K] (n : ℕ) [NeZero n] [IsCyclotomicExtension {n} ℚ K] {β : K} (hβ_int : IsIntegral ℤ β) (hβ_real : β ∈ NumberField.maximalRealSubfield K) : house β ≤ 2 → house β = 2 ∨ ∃ m : ℕ, 0 < m ∧ house β = 2 * Real.cos (Real.pi / m) := K:Type u_1inst✝⁴:Field Kinst✝³:NumberField Kinst✝²:Algebra ℚ Kn:ℕinst✝¹:NeZero ninst✝:IsCyclotomicExtension {n} ℚ Kβ:Khβ_int:IsIntegral ℤ βhβ_real:β ∈ maximalRealSubfield K⊢ house β ≤ 2 → house β = 2 ∨ ∃ m, 0 < m ∧ house β = 2 * Real.cos (Real.pi / ↑m) All goals completed! 🐙
#38
How produced

Solved completely autonomously without human intervention

Burnside p^a q^b theorem
finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow

Verso theorem preview

theorem declaration uses `sorry`finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow {G : Type*} [Group G] [Fintype G] {p q a b : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q) (hpq : p ≠ q) (hcard : Fintype.card G = p ^ a * q ^ b) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Fintype Gp:ℕq:ℕa:ℕb:ℕhp:Nat.Prime phq:Nat.Prime qhpq:p ≠ qhcard:Fintype.card G = p ^ a * q ^ b⊢ IsSolvable G All goals completed! 🐙
#39
How produced

Solved completely autonomously without human intervention

Perron-Frobenius for irreducible nonnegative matrices
irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius

Verso theorem preview

theorem declaration uses `sorry`irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius {n : Type*} [Fintype n] [DecidableEq n] [Nonempty n] (A : Matrix n n ℝ) (hA : A.IsIrreducible) : ∃ v : n → ℝ, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ (∀ i, 0 < v i) := n:Type u_1inst✝²:Fintype ninst✝¹:DecidableEq ninst✝:Nonempty nA:Matrix n n ℝhA:A.IsIrreducible⊢ ∃ v, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ ∀ (i : n), 0 < v i All goals completed! 🐙
#40
How produced

Solved completely autonomously without human intervention

Linear ODE with negative-real-part eigenvalues is asymptotically stable
linear_ode_asymptotic_stability

Verso theorem preview

theorem declaration uses `sorry`linear_ode_asymptotic_stability (n : ℕ) (A : Matrix (Fin n) (Fin n) ℝ) (hA : ∀ μ : ℂ, Module.End.HasEigenvalue (Matrix.toLin' (A.map (algebraMap ℝ ℂ))) μ → μ.re < 0) (x : ℝ → (Fin n → ℝ)) (hx : ∀ t : ℝ, 0 < t → HasDerivAt x (A.mulVec (x t)) t) : Filter.Tendsto (fun t : ℝ => ‖x t‖) Filter.atTop (nhds 0) := n:ℕA:Matrix (Fin n) (Fin n) ℝhA:∀ (μ : ℂ), Module.End.HasEigenvalue (Matrix.toLin' (A.map ⇑(algebraMap ℝ ℂ))) μ → μ.re < 0x:ℝ → Fin n → ℝhx:∀ (t : ℝ), 0 < t → HasDerivAt x (A *ᵥ x t) t⊢ Filter.Tendsto (fun t => ‖x t‖) Filter.atTop (nhds 0) All goals completed! 🐙
#41
How produced

Solved completely autonomously without human intervention

Rouche theorem via zero counting
rouche_zero_count_eq

Verso theorem preview

theorem declaration uses `sorry`rouche_zero_count_eq {f g : ℂ → ℂ} {R : ℝ} (hR : 0 < R) (hf : MeromorphicNFOn f Set.univ) (hg : AnalyticOn ℂ g Set.univ) (hbound : ∀ z : ℂ, ‖z‖ = R → ‖g z‖ < ‖f z‖) : (∑ᶠ z, ((divisor (f + g) (Metric.closedBall 0 R))⁺) z) = (∑ᶠ z, ((divisor f (Metric.closedBall 0 R))⁺) z) := f:ℂ → ℂg:ℂ → ℂR:ℝhR:0 < Rhf:MeromorphicNFOn f Set.univhg:AnalyticOn ℂ g Set.univhbound:∀ (z : ℂ), ‖z‖ = R → ‖g z‖ < ‖f z‖⊢ ∑ᶠ (z : ℂ), (divisor (f + g) (Metric.closedBall 0 R))⁺ z = ∑ᶠ (z : ℂ), (divisor f (Metric.closedBall 0 R))⁺ z All goals completed! 🐙
#42
How produced

Solved completely autonomously without human intervention

Character values of finite groups lie in cyclotomic fields
brauer_character_in_cyclotomic

Verso theorem preview

theorem declaration uses `sorry`brauer_character_in_cyclotomic (G : Type) [Group G] [Fintype G] : ∃ φ : CyclotomicField (Monoid.exponent G) ℚ →+* ℂ, ∀ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (_ : FiniteDimensional ℂ V) (ρ : Representation ℂ G V) (g : G), LinearMap.trace ℂ V (ρ g) ∈ φ.range := G:Typeinst✝¹:Group Ginst✝:Fintype G⊢ ∃ φ, ∀ (V : Type) (x : AddCommGroup V) (x_1 : Module ℂ V), FiniteDimensional ℂ V → ∀ (ρ : Representation ℂ G V) (g : G), (LinearMap.trace ℂ V) (ρ g) ∈ φ.range All goals completed! 🐙
#43
How produced

Solved completely autonomously without human intervention

Complementary polynomial on the unit circle
exists_complementary_polynomial_on_unit_circle

Verso theorem preview

theorem declaration uses `sorry`exists_complementary_polynomial_on_unit_circle (P : ℂ[X]) (hP : ∀ z : Circle, ‖P.eval (z : ℂ)‖ ≤ 1) : ∃ Q : ℂ[X], Q.natDegree ≤ P.natDegree ∧ ∀ z : Circle, ‖P.eval (z : ℂ)‖ ^ 2 + ‖Q.eval (z : ℂ)‖ ^ 2 = 1 := P:ℂ[X]hP:∀ (z : Circle), ‖eval (↑z) P‖ ≤ 1⊢ ∃ Q, Q.natDegree ≤ P.natDegree ∧ ∀ (z : Circle), ‖eval (↑z) P‖ ^ 2 + ‖eval (↑z) Q‖ ^ 2 = 1 All goals completed! 🐙
#44
How produced

Solved completely autonomously without human intervention

Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2
dirichlet_eigenvalues_eq_nat_sq

Verso theorem preview

theorem declaration uses `sorry`dirichlet_eigenvalues_eq_nat_sq (lam : ℝ) : (∃ (y : ℝ → ℝ) (J : Set ℝ), IsOpen J ∧ Set.Icc (0 : ℝ) Real.pi ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y Real.pi = 0 ∧ ∃ x ∈ Set.Ioo (0 : ℝ) Real.pi, y x ≠ 0) ↔ ∃ n : ℕ, 0 < n ∧ lam = (n : ℝ) ^ 2 := lam:ℝ⊢ (∃ y J, IsOpen J ∧ Set.Icc 0 π ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y π = 0 ∧ ∃ x ∈ Set.Ioo 0 π, y x ≠ 0) ↔ ∃ n, 0 < n ∧ lam = ↑n ^ 2 All goals completed! 🐙
#45
How produced

Solved completely autonomously without human intervention

Gaussian heat kernel solves the 1D heat equation
heat_kernel_solves_heat_equation

Verso theorem preview

theorem declaration uses `sorry`heat_kernel_solves_heat_equation (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_bdd : ∃ M : ℝ, ∀ x, |f x| ≤ M) : -- The PDE on (0, ∞) × ℝ. (∀ t : ℝ, 0 < t → ∀ x : ℝ, ∃ ux : ℝ → ℝ, ∃ uxx : ℝ, (∀ y : ℝ, HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ -- Initial condition recovered as a one-sided limit at t = 0. (∀ x : ℝ, Filter.Tendsto (fun t : ℝ => heatSolution f t x) (nhdsWithin (0 : ℝ) (Set.Ioi 0)) (nhds (f x))) := f:ℝ → ℝhf_cont:Continuous fhf_bdd:∃ M, ∀ (x : ℝ), |f x| ≤ M⊢ (∀ (t : ℝ), 0 < t → ∀ (x : ℝ), ∃ ux uxx, (∀ (y : ℝ), HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ ∀ (x : ℝ), Filter.Tendsto (fun t => heatSolution f t x) (nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) All goals completed! 🐙
#46
How produced

Solved completely autonomously without human intervention

Minkowski-Caratheodory theorem
mem_convexHull_finset_extremePoints_of_mem_compact_convex

Verso theorem preview

theorem declaration uses `sorry`mem_convexHull_finset_extremePoints_of_mem_compact_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hscomp : IsCompact s) (hsconv : Convex ℝ s) (hx : x ∈ s) : ∃ t : Finset E, (↑t : Set E) ⊆ s.extremePoints ℝ ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ convexHull ℝ (↑t : Set E) := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:FiniteDimensional ℝ Es:Set Ex:Ehscomp:IsCompact shsconv:Convex ℝ shx:x ∈ s⊢ ∃ t, ↑t ⊆ extremePoints ℝ s ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ (convexHull ℝ) ↑t All goals completed! 🐙
#47
How produced

Solved completely autonomously without human intervention

Oppenheim's inequality for Hadamard products
oppenheim_inequality

Verso theorem preview

theorem declaration uses `sorry`oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : A.det * ∏ i, B i i ≤ (A ⊙ B).det := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝB:Matrix n n ℝhA:A.PosSemidefhB:B.PosSemidef⊢ A.det * ∏ i, B i i ≤ (A ⊙ B).det All goals completed! 🐙
#48
How produced

Solved completely autonomously without human intervention

pi_1 of the circle is Z
pi1_circle_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi1_circle_mulEquiv_int : Nonempty (HomotopyGroup.Pi 1 Circle (1 : Circle) ≃* Multiplicative ℤ) := ⊢ Nonempty (HomotopyGroup.Pi 1 Circle 1 ≃* Multiplicative ℤ) All goals completed! 🐙
#49
How produced

Solved completely autonomously without human intervention

Entrywise exponential of a PSD matrix is PSD
posSemidef_map_exp

Verso theorem preview

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n] {A : Matrix n n ℝ} (hA : A.PosSemidef) : (A.map Real.exp).PosSemidef := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef All goals completed! 🐙
#50
How produced

Solved completely autonomously without human intervention

Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#51
How produced

Solved completely autonomously without human intervention

Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#52
How produced

Solved completely autonomously without human intervention

Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#55
How produced

Solved completely autonomously without human intervention

Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#58
How produced

Solved completely autonomously without human intervention

Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#59
How produced

Solved without human intervention. Then manual one-line fix applied for 4.30 because, as of now, Aleph only supports lean/mathlib versions up to 4.29. ``` 34 - exact ⟨(SimpleGraph.Embedding.induce S).comp f⟩ 34 + exact ⟨(SimpleGraph.Copy.induce G S).comp f⟩ ```

Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#60
How produced

Solved completely autonomously without human intervention

Test problems: def_hole_example, instance_hole_example, ci_regenerate_main_check, list_append_singleton_length, two_plus_two (5 / 8 solved)

Submission history

First submissionMay 7, 2026
Last submissionJun 5, 2026

Contributors

mayorov-m-a57antpavzhi4
7MerLean-Prover30 solved

Other solved problems

Hippocrates' theorem on lunes
hippocrates_lunes

Verso theorem preview

theorem declaration uses `sorry`hippocrates_lunes (a b : ℝ) (ha : 0 < a) (hb : 0 < b) : volume (LeanEval.Geometry.HippocratesLunes.horizontalLune a b) + volume (LeanEval.Geometry.HippocratesLunes.verticalLune a b) = volume (LeanEval.Geometry.HippocratesLunes.rightTriangle a b) := a:ℝb:ℝha:0 < ahb:0 < b⊢ volume (horizontalLune a b) + volume (verticalLune a b) = volume (rightTriangle a b) All goals completed! 🐙
#1
Schauder fixed-point theorem
schauder_fixed_point

Verso theorem preview

theorem declaration uses `sorry`schauder_fixed_point {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {K : Set E} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : E → E) (_hf_cont : ContinuousOn f K) (_hf_maps : Set.MapsTo f K K) : ∃ x ∈ K, f x = x := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:CompleteSpace EK:Set E_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:E → E_hf_cont:ContinuousOn f K_hf_maps:Set.MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#2
Kuznetsov's theorem: finitely presented simple groups have solvable word problem
boone_higman_simple

Verso theorem preview

theorem declaration uses `sorry`boone_higman_simple {G : Type*} [Group G] [IsSimpleGroup G] {n : ℕ} (φ : FreeGroup (Fin n) →* G) (_hsurj : Function.Surjective φ) (_hker : (MonoidHom.ker φ).IsNormalClosureFG) : LeanEval.GroupTheory.BooneHigmanSimpleProblem.WordProblemSolvable φ := G:Type u_1inst✝¹:Group Ginst✝:IsSimpleGroup Gn:ℕφ:FreeGroup (Fin n) →* G_hsurj:Function.Surjective ⇑φ_hker:φ.ker.IsNormalClosureFG⊢ WordProblemSolvable φ All goals completed! 🐙
#3
Linear programming: maximum principle and vertex optimality
lp_maximum_principle

Verso theorem preview

/-- **Maximum principle for linear programming** (§101). A local maximiser of the LP objective on the feasible region is automatically a global maximiser; and whenever the objective is non-constant (`c ≠ 0`), the maximiser lies on the topological frontier of the feasible region. -/ theorem declaration uses `sorry`lp_maximum_principle {m n : ℕ} (lp : LinearProgram m n) (x : Fin m → ℝ) (_hx : x ∈ lp.feasible) (_hlocal : IsLocalMaxOn lp.objective lp.feasible x) : IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) := m:ℕn:ℕlp:LinearProgram m nx:Fin m → ℝ_hx:x ∈ lp.feasible_hlocal:IsLocalMaxOn lp.objective lp.feasible x⊢ IsMaxOn lp.objective lp.feasible x ∧ (lp.c ≠ 0 → x ∈ frontier lp.feasible) All goals completed! 🐙
/-- **Vertex optimality** (§101; the existence content of Dantzig's 1947 simplex algorithm). Every linear program with a nonempty bounded feasible region admits a global maximiser that is an extreme point (vertex) of the feasible region. -/ theorem declaration uses `sorry`simplex_algorithm {m n : ℕ} (lp : LinearProgram m n) (_hfeas : lp.feasible.Nonempty) (_hbdd : Bornology.IsBounded lp.feasible) : ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible := m:ℕn:ℕlp:LinearProgram m n_hfeas:lp.feasible.Nonempty_hbdd:Bornology.IsBounded lp.feasible⊢ ∃ x ∈ lp.feasible, IsMaxOn lp.objective lp.feasible x ∧ x ∈ Set.extremePoints ℝ lp.feasible All goals completed! 🐙
#4
Wiener's atom-detection formula
wiener_atom_detection

Verso theorem preview

theorem declaration uses `sorry`wiener_atom_detection (μ : Measure (AddCircle (2 * Real.pi))) [IsProbabilityMeasure μ] : Tendsto (fun N : ℕ => (1 / (N : ℝ)) * ∑ k ∈ Finset.Icc (1 : ℤ) N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' x : AddCircle (2 * Real.pi), ((μ {x}).toReal) ^ 2)) := μ:Measure (AddCircle (2 * π))inst✝:IsProbabilityMeasure μ⊢ Tendsto (fun N => 1 / ↑N * ∑ k ∈ Finset.Icc 1 ↑N, ‖fourierCoeffMeasure μ k‖ ^ 2) atTop (𝓝 (∑' (x : AddCircle (2 * π)), (μ {x}).toReal ^ 2)) All goals completed! 🐙
#5
Bourbaki's locally convex extension of Banach–Alaoglu
banach_alaoglu_bourbaki

Verso theorem preview

theorem declaration uses `sorry`banach_alaoglu_bourbaki (E : Type*) [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] (U : Set E) (_hU : U ∈ 𝓝 (0 : E)) : IsCompact (LeanEval.Analysis.weakStarPolar E U) := E:Type u_1inst✝⁵:AddCommGroup Einst✝⁴:Module ℝ Einst✝³:TopologicalSpace Einst✝²:ContinuousAdd Einst✝¹:ContinuousSMul ℝ Einst✝:LocallyConvexSpace ℝ EU:Set E_hU:U ∈ 𝓝 0⊢ IsCompact (weakStarPolar E U) All goals completed! 🐙
#6
Brouwer fixed-point theorem
brouwer_fixed_point

Verso theorem preview

theorem declaration uses `sorry`brouwer_fixed_point {d : ℕ} {K : Set (EuclideanSpace ℝ (Fin d))} (_hK_compact : IsCompact K) (_hK_convex : Convex ℝ K) (_hK_nonempty : K.Nonempty) (f : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)) (_hf_cont : ContinuousOn f K) (_hf_maps : MapsTo f K K) : ∃ x ∈ K, f x = x := d:ℕK:Set (EuclideanSpace ℝ (Fin d))_hK_compact:IsCompact K_hK_convex:Convex ℝ K_hK_nonempty:K.Nonemptyf:EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d)_hf_cont:ContinuousOn f K_hf_maps:MapsTo f K K⊢ ∃ x ∈ K, f x = x All goals completed! 🐙
#7
Runge's theorem
runge_theorem

Verso theorem preview

theorem declaration uses `sorry`runge (K : Set ℂ) (_hK : IsCompact K) (U : Set ℂ) (_hU : IsOpen U) (_hKU : K ⊆ U) (f : ℂ → ℂ) (_hf : AnalyticOnNhd ℂ f U) (ε : ℝ) (_hε : 0 < ε) : ∃ p q : ℂ[X], (∀ z ∈ K, q.eval z ≠ 0) ∧ (∀ z ∈ K, ‖f z - p.eval z / q.eval z‖ < ε) := K:Set ℂ_hK:IsCompact KU:Set ℂ_hU:IsOpen U_hKU:K ⊆ Uf:ℂ → ℂ_hf:AnalyticOnNhd ℂ f Uε:ℝ_hε:0 < ε⊢ ∃ p q, (∀ z ∈ K, Polynomial.eval z q ≠ 0) ∧ ∀ z ∈ K, ‖f z - Polynomial.eval z p / Polynomial.eval z q‖ < ε All goals completed! 🐙
#8
Abel–Ruffini theorem
abel_ruffini

Verso theorem preview

theorem declaration uses `sorry`abel_ruffini (n : ℕ) (_hn : 1 ≤ n) : (∀ p : ℚ[X], p.natDegree = n → ∀ x : ℂ, aeval x p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 := n:ℕ_hn:1 ≤ n⊢ (∀ (p : ℚ[X]), p.natDegree = n → ∀ (x : ℂ), (aeval x) p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 All goals completed! 🐙
#9
Pointwise and Cesàro convergence of Fourier series (Dirichlet, Fejér)
fourier_dirichlet_fejer

Verso theorem preview

/-- **Dirichlet's pointwise convergence theorem** (§46). For every `C¹` 2π-periodic complex function `f`, the symmetric Fourier partial sums `S_N(f)(x)` converge to `f(x)` at every point `x ∈ ℝ`. -/ theorem declaration uses `sorry`dirichlet_pointwise {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hC1 : ContDiff ℝ 1 f) (x : ℝ) : Tendsto (fun N : ℕ => fourierPartialSum f N x) atTop (𝓝 (f x)) := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hC1:ContDiff ℝ 1 fx:ℝ⊢ Tendsto (fun N => fourierPartialSum f N x) atTop (𝓝 (f x)) All goals completed! 🐙
/-- **Fejér's theorem** (§46). For every *continuous* 2π-periodic complex function `f` — without the `C¹` hypothesis of Dirichlet's theorem — the Cesàro means `σ_N(f)` of the symmetric Fourier partial sums converge to `f` uniformly on `ℝ`. -/ theorem declaration uses `sorry`fejer {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hcont : Continuous f) : TendstoUniformly (fun N : ℕ => fourierCesaroMean f N) f atTop := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hcont:Continuous f⊢ TendstoUniformly (fun N => fourierCesaroMean f N) f atTop All goals completed! 🐙
#10
Pell solutions are convergents of √d
pell_solution_convergent

Verso theorem preview

theorem declaration uses `sorry`pell_solution_is_convergent (d : ℤ) (_hd : Squarefree d) (_hd0 : 0 < d) (x y : ℤ) (_hx : 0 < x) (_hy : 0 < y) (_hsol : x ^ 2 - d * y ^ 2 = 1) : ∃ n : ℕ, (GenContFract.of (Real.sqrt (d : ℝ))).convs n = (x : ℝ) / (y : ℝ) := d:ℤ_hd:Squarefree d_hd0:0 < dx:ℤy:ℤ_hx:0 < x_hy:0 < y_hsol:x ^ 2 - d * y ^ 2 = 1⊢ ∃ n, (GenContFract.of √↑d).convs n = ↑x / ↑y All goals completed! 🐙
#11
Burnside p^a q^b theorem
finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow

Verso theorem preview

theorem declaration uses `sorry`finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow {G : Type*} [Group G] [Fintype G] {p q a b : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q) (hpq : p ≠ q) (hcard : Fintype.card G = p ^ a * q ^ b) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Fintype Gp:ℕq:ℕa:ℕb:ℕhp:Nat.Prime phq:Nat.Prime qhpq:p ≠ qhcard:Fintype.card G = p ^ a * q ^ b⊢ IsSolvable G All goals completed! 🐙
#12
Existence of a non-isotopic pair of oriented two-component links
exists_nonisotopic_link

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_link : ∃ L₁ L₂ : LeanEval.KnotTheory.TwoLink, ¬ L₁.Isotopic L₂ := ⊢ ∃ L₁ L₂, ¬L₁.Isotopic L₂ All goals completed! 🐙
#13
von Neumann double commutant theorem
vonNeumann_doubleCommutant_tfae

Verso theorem preview

theorem declaration uses `sorry`vonNeumann_doubleCommutant_tfae {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : StarSubalgebra ℂ (H →L[ℂ] H)) : List.TFAE [ Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S , IsClosed (ContinuousLinearMapWOT.ofCLM '' (S : Set (H →L[ℂ] H))) , IsClosed (ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H '' (S : Set (H →L[ℂ] H))) ] := H:Type u_1inst✝²:NormedAddCommGroup Hinst✝¹:InnerProductSpace ℂ Hinst✝:CompleteSpace HS:StarSubalgebra ℂ (H →L[ℂ] H)⊢ [(↑S).centralizer.centralizer = ↑S, IsClosed (ContinuousLinearMapWOT.ofCLM '' ↑S), IsClosed (⇑(ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H) '' ↑S)].TFAE All goals completed! 🐙
#14
Linear ODE with negative-real-part eigenvalues is asymptotically stable
linear_ode_asymptotic_stability

Verso theorem preview

theorem declaration uses `sorry`linear_ode_asymptotic_stability (n : ℕ) (A : Matrix (Fin n) (Fin n) ℝ) (hA : ∀ μ : ℂ, Module.End.HasEigenvalue (Matrix.toLin' (A.map (algebraMap ℝ ℂ))) μ → μ.re < 0) (x : ℝ → (Fin n → ℝ)) (hx : ∀ t : ℝ, 0 < t → HasDerivAt x (A.mulVec (x t)) t) : Filter.Tendsto (fun t : ℝ => ‖x t‖) Filter.atTop (nhds 0) := n:ℕA:Matrix (Fin n) (Fin n) ℝhA:∀ (μ : ℂ), Module.End.HasEigenvalue (Matrix.toLin' (A.map ⇑(algebraMap ℝ ℂ))) μ → μ.re < 0x:ℝ → Fin n → ℝhx:∀ (t : ℝ), 0 < t → HasDerivAt x (A *ᵥ x t) t⊢ Filter.Tendsto (fun t => ‖x t‖) Filter.atTop (nhds 0) All goals completed! 🐙
#15
Complementary polynomial on the unit circle
exists_complementary_polynomial_on_unit_circle

Verso theorem preview

theorem declaration uses `sorry`exists_complementary_polynomial_on_unit_circle (P : ℂ[X]) (hP : ∀ z : Circle, ‖P.eval (z : ℂ)‖ ≤ 1) : ∃ Q : ℂ[X], Q.natDegree ≤ P.natDegree ∧ ∀ z : Circle, ‖P.eval (z : ℂ)‖ ^ 2 + ‖Q.eval (z : ℂ)‖ ^ 2 = 1 := P:ℂ[X]hP:∀ (z : Circle), ‖eval (↑z) P‖ ≤ 1⊢ ∃ Q, Q.natDegree ≤ P.natDegree ∧ ∀ (z : Circle), ‖eval (↑z) P‖ ^ 2 + ‖eval (↑z) Q‖ ^ 2 = 1 All goals completed! 🐙
#16
Rouche theorem via zero counting
rouche_zero_count_eq

Verso theorem preview

theorem declaration uses `sorry`rouche_zero_count_eq {f g : ℂ → ℂ} {R : ℝ} (hR : 0 < R) (hf : MeromorphicNFOn f Set.univ) (hg : AnalyticOn ℂ g Set.univ) (hbound : ∀ z : ℂ, ‖z‖ = R → ‖g z‖ < ‖f z‖) : (∑ᶠ z, ((divisor (f + g) (Metric.closedBall 0 R))⁺) z) = (∑ᶠ z, ((divisor f (Metric.closedBall 0 R))⁺) z) := f:ℂ → ℂg:ℂ → ℂR:ℝhR:0 < Rhf:MeromorphicNFOn f Set.univhg:AnalyticOn ℂ g Set.univhbound:∀ (z : ℂ), ‖z‖ = R → ‖g z‖ < ‖f z‖⊢ ∑ᶠ (z : ℂ), (divisor (f + g) (Metric.closedBall 0 R))⁺ z = ∑ᶠ (z : ℂ), (divisor f (Metric.closedBall 0 R))⁺ z All goals completed! 🐙
#17
Character values of finite groups lie in cyclotomic fields
brauer_character_in_cyclotomic

Verso theorem preview

theorem declaration uses `sorry`brauer_character_in_cyclotomic (G : Type) [Group G] [Fintype G] : ∃ φ : CyclotomicField (Monoid.exponent G) ℚ →+* ℂ, ∀ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (_ : FiniteDimensional ℂ V) (ρ : Representation ℂ G V) (g : G), LinearMap.trace ℂ V (ρ g) ∈ φ.range := G:Typeinst✝¹:Group Ginst✝:Fintype G⊢ ∃ φ, ∀ (V : Type) (x : AddCommGroup V) (x_1 : Module ℂ V), FiniteDimensional ℂ V → ∀ (ρ : Representation ℂ G V) (g : G), (LinearMap.trace ℂ V) (ρ g) ∈ φ.range All goals completed! 🐙
#18
Gaussian heat kernel solves the 1D heat equation
heat_kernel_solves_heat_equation

Verso theorem preview

theorem declaration uses `sorry`heat_kernel_solves_heat_equation (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_bdd : ∃ M : ℝ, ∀ x, |f x| ≤ M) : -- The PDE on (0, ∞) × ℝ. (∀ t : ℝ, 0 < t → ∀ x : ℝ, ∃ ux : ℝ → ℝ, ∃ uxx : ℝ, (∀ y : ℝ, HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ -- Initial condition recovered as a one-sided limit at t = 0. (∀ x : ℝ, Filter.Tendsto (fun t : ℝ => heatSolution f t x) (nhdsWithin (0 : ℝ) (Set.Ioi 0)) (nhds (f x))) := f:ℝ → ℝhf_cont:Continuous fhf_bdd:∃ M, ∀ (x : ℝ), |f x| ≤ M⊢ (∀ (t : ℝ), 0 < t → ∀ (x : ℝ), ∃ ux uxx, (∀ (y : ℝ), HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ ∀ (x : ℝ), Filter.Tendsto (fun t => heatSolution f t x) (nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) All goals completed! 🐙
#19
Minkowski-Caratheodory theorem
mem_convexHull_finset_extremePoints_of_mem_compact_convex

Verso theorem preview

theorem declaration uses `sorry`mem_convexHull_finset_extremePoints_of_mem_compact_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hscomp : IsCompact s) (hsconv : Convex ℝ s) (hx : x ∈ s) : ∃ t : Finset E, (↑t : Set E) ⊆ s.extremePoints ℝ ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ convexHull ℝ (↑t : Set E) := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:FiniteDimensional ℝ Es:Set Ex:Ehscomp:IsCompact shsconv:Convex ℝ shx:x ∈ s⊢ ∃ t, ↑t ⊆ extremePoints ℝ s ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ (convexHull ℝ) ↑t All goals completed! 🐙
#20
Oppenheim's inequality for Hadamard products
oppenheim_inequality

Verso theorem preview

theorem declaration uses `sorry`oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : A.det * ∏ i, B i i ≤ (A ⊙ B).det := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝB:Matrix n n ℝhA:A.PosSemidefhB:B.PosSemidef⊢ A.det * ∏ i, B i i ≤ (A ⊙ B).det All goals completed! 🐙
#21
pi_1 of the circle is Z
pi1_circle_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi1_circle_mulEquiv_int : Nonempty (HomotopyGroup.Pi 1 Circle (1 : Circle) ≃* Multiplicative ℤ) := ⊢ Nonempty (HomotopyGroup.Pi 1 Circle 1 ≃* Multiplicative ℤ) All goals completed! 🐙
#22
Entrywise exponential of a PSD matrix is PSD
posSemidef_map_exp

Verso theorem preview

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n] {A : Matrix n n ℝ} (hA : A.PosSemidef) : (A.map Real.exp).PosSemidef := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef All goals completed! 🐙
#23
Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2
dirichlet_eigenvalues_eq_nat_sq

Verso theorem preview

theorem declaration uses `sorry`dirichlet_eigenvalues_eq_nat_sq (lam : ℝ) : (∃ (y : ℝ → ℝ) (J : Set ℝ), IsOpen J ∧ Set.Icc (0 : ℝ) Real.pi ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y Real.pi = 0 ∧ ∃ x ∈ Set.Ioo (0 : ℝ) Real.pi, y x ≠ 0) ↔ ∃ n : ℕ, 0 < n ∧ lam = (n : ℝ) ^ 2 := lam:ℝ⊢ (∃ y J, IsOpen J ∧ Set.Icc 0 π ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y π = 0 ∧ ∃ x ∈ Set.Ioo 0 π, y x ≠ 0) ↔ ∃ n, 0 < n ∧ lam = ↑n ^ 2 All goals completed! 🐙
#24
Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#25
Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#26
Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#27
Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#28
Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#29
Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#30

Submission history

First submissionMay 31, 2026
Last submissionJun 25, 2026

Contributors

doxtor630
8Antigravity (Multi-Model Ensemble: Gemini 3.1 Pro, Gemini 3 Flash, Claude 4.6 Sonnet/Opus)21 solved

Other solved problems

Real cyclotomic integer with house at most 2
cyclotomic_integer_house_le_two

Verso theorem preview

theorem declaration uses `sorry`cyclotomic_integer_house_le_two {K : Type*} [Field K] [NumberField K] [Algebra ℚ K] (n : ℕ) [NeZero n] [IsCyclotomicExtension {n} ℚ K] {β : K} (hβ_int : IsIntegral ℤ β) (hβ_real : β ∈ NumberField.maximalRealSubfield K) : house β ≤ 2 → house β = 2 ∨ ∃ m : ℕ, 0 < m ∧ house β = 2 * Real.cos (Real.pi / m) := K:Type u_1inst✝⁴:Field Kinst✝³:NumberField Kinst✝²:Algebra ℚ Kn:ℕinst✝¹:NeZero ninst✝:IsCyclotomicExtension {n} ℚ Kβ:Khβ_int:IsIntegral ℤ βhβ_real:β ∈ maximalRealSubfield K⊢ house β ≤ 2 → house β = 2 ∨ ∃ m, 0 < m ∧ house β = 2 * Real.cos (Real.pi / ↑m) All goals completed! 🐙
#1
von Neumann double commutant theorem
vonNeumann_doubleCommutant_tfae

Verso theorem preview

theorem declaration uses `sorry`vonNeumann_doubleCommutant_tfae {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] (S : StarSubalgebra ℂ (H →L[ℂ] H)) : List.TFAE [ Set.centralizer (Set.centralizer (S : Set (H →L[ℂ] H))) = S , IsClosed (ContinuousLinearMapWOT.ofCLM '' (S : Set (H →L[ℂ] H))) , IsClosed (ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H '' (S : Set (H →L[ℂ] H))) ] := H:Type u_1inst✝²:NormedAddCommGroup Hinst✝¹:InnerProductSpace ℂ Hinst✝:CompleteSpace HS:StarSubalgebra ℂ (H →L[ℂ] H)⊢ [(↑S).centralizer.centralizer = ↑S, IsClosed (ContinuousLinearMapWOT.ofCLM '' ↑S), IsClosed (⇑(ContinuousLinearMap.toPointwiseConvergenceCLM ℂ (RingHom.id ℂ) H H) '' ↑S)].TFAE All goals completed! 🐙
#2
Existence of a non-isotopic pair of oriented two-component links
exists_nonisotopic_link

Verso theorem preview

theorem declaration uses `sorry`exists_nonisotopic_link : ∃ L₁ L₂ : LeanEval.KnotTheory.TwoLink, ¬ L₁.Isotopic L₂ := ⊢ ∃ L₁ L₂, ¬L₁.Isotopic L₂ All goals completed! 🐙
#3
Bing's house with two rooms is contractible
contractibleSpace_houseWithTwoRooms

Verso theorem preview

theorem declaration uses `sorry`contractibleSpace_houseWithTwoRooms : ContractibleSpace LeanEval.Topology.HouseWithTwoRooms := ⊢ ContractibleSpace ↑HouseWithTwoRooms All goals completed! 🐙
#4
How produced

POC of how a moderately advanced harness / scaffolding can deliver: antigravity, SKILLS.md / AGENTS.md , MCP server. The human in the loop is not a mathematician, nor a software engineer, just someone curious and armed with patience, and acting as a babysitter: with simple encouragements like "remember, if we dont have the needed bricks, we build them and lay them search online for guidance if needed step by step, brick by brick, we are progressing we have time, you are doing great, try to address 1 thing at a time think using sequential thinking tool as needed, take your time and proceed with care no shortcuts, no cheating, we have time the sky is the limit, this is not an open problem, you got this ! " AI did all the thinking. No golfing done. This 7,000+ line proof was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE (a VS Code fork). Due to the massive scale of the deformation retraction, the solution was driven by cycling through the entire suite of available frontier models as quota constraints required. The formal verification relied on a specialized scaffolding pipeline: Iterative Prompt Engineering: The task was initialized using structured prompts ([contractibleSpace_houseWithTwoRooms.md](https://github.com/user-attachments/files/27707778/contractibleSpace_houseWithTwoRooms.md) that enforce Mathlib style guides (e.g., "Rely on existing Mathlib structural lemmas" and "Abstract into lemmas parameterized by characteristics"). Failure Feedback Loop: When intermediate attempts failed (e.g., due to type class synthesis errors or unsolved goals), the raw lake build trace logs and compiler outputs were automatically captured and injected back into the prompt context. This allowed the models to iteratively diagnose and correct their own errors. Model Context Protocol (MCP): The agents interacted with the Lean 4 compiler in real-time via the lean-lsp MCP server, gaining high-fidelity "Language Server to Agent" access: Proof State Tracking (lean_goal): Real-time extraction of tactic states to track the 24 nested sub-cubes of the retraction. Diagnostics (lean_diagnostic_messages): Immediate compiler feedback on type mismatches and syntax errors to keep the massive 7,000-line construction mathematically sound. Structural Synthesis: The proof was built constructively by the ensemble over multiple sessions. The agents defined the 24 topological spaces (C_1 through C_24), constructed explicit piecewise continuous projections (proj_1 through proj_24), and systematically eliminated sorry placeholders until the full deformation retraction onto the point was rigorously verified by the Lean compiler.

Perron-Frobenius for irreducible nonnegative matrices
irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius

Verso theorem preview

theorem declaration uses `sorry`irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius {n : Type*} [Fintype n] [DecidableEq n] [Nonempty n] (A : Matrix n n ℝ) (hA : A.IsIrreducible) : ∃ v : n → ℝ, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ (∀ i, 0 < v i) := n:Type u_1inst✝²:Fintype ninst✝¹:DecidableEq ninst✝:Nonempty nA:Matrix n n ℝhA:A.IsIrreducible⊢ ∃ v, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ ∀ (i : n), 0 < v i All goals completed! 🐙
#5
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Linear ODE with negative-real-part eigenvalues is asymptotically stable
linear_ode_asymptotic_stability

Verso theorem preview

theorem declaration uses `sorry`linear_ode_asymptotic_stability (n : ℕ) (A : Matrix (Fin n) (Fin n) ℝ) (hA : ∀ μ : ℂ, Module.End.HasEigenvalue (Matrix.toLin' (A.map (algebraMap ℝ ℂ))) μ → μ.re < 0) (x : ℝ → (Fin n → ℝ)) (hx : ∀ t : ℝ, 0 < t → HasDerivAt x (A.mulVec (x t)) t) : Filter.Tendsto (fun t : ℝ => ‖x t‖) Filter.atTop (nhds 0) := n:ℕA:Matrix (Fin n) (Fin n) ℝhA:∀ (μ : ℂ), Module.End.HasEigenvalue (Matrix.toLin' (A.map ⇑(algebraMap ℝ ℂ))) μ → μ.re < 0x:ℝ → Fin n → ℝhx:∀ (t : ℝ), 0 < t → HasDerivAt x (A *ᵥ x t) t⊢ Filter.Tendsto (fun t => ‖x t‖) Filter.atTop (nhds 0) All goals completed! 🐙
#6
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Rouche theorem via zero counting
rouche_zero_count_eq

Verso theorem preview

theorem declaration uses `sorry`rouche_zero_count_eq {f g : ℂ → ℂ} {R : ℝ} (hR : 0 < R) (hf : MeromorphicNFOn f Set.univ) (hg : AnalyticOn ℂ g Set.univ) (hbound : ∀ z : ℂ, ‖z‖ = R → ‖g z‖ < ‖f z‖) : (∑ᶠ z, ((divisor (f + g) (Metric.closedBall 0 R))⁺) z) = (∑ᶠ z, ((divisor f (Metric.closedBall 0 R))⁺) z) := f:ℂ → ℂg:ℂ → ℂR:ℝhR:0 < Rhf:MeromorphicNFOn f Set.univhg:AnalyticOn ℂ g Set.univhbound:∀ (z : ℂ), ‖z‖ = R → ‖g z‖ < ‖f z‖⊢ ∑ᶠ (z : ℂ), (divisor (f + g) (Metric.closedBall 0 R))⁺ z = ∑ᶠ (z : ℂ), (divisor f (Metric.closedBall 0 R))⁺ z All goals completed! 🐙
#7
Complementary polynomial on the unit circle
exists_complementary_polynomial_on_unit_circle

Verso theorem preview

theorem declaration uses `sorry`exists_complementary_polynomial_on_unit_circle (P : ℂ[X]) (hP : ∀ z : Circle, ‖P.eval (z : ℂ)‖ ≤ 1) : ∃ Q : ℂ[X], Q.natDegree ≤ P.natDegree ∧ ∀ z : Circle, ‖P.eval (z : ℂ)‖ ^ 2 + ‖Q.eval (z : ℂ)‖ ^ 2 = 1 := P:ℂ[X]hP:∀ (z : Circle), ‖eval (↑z) P‖ ≤ 1⊢ ∃ Q, Q.natDegree ≤ P.natDegree ∧ ∀ (z : Circle), ‖eval (↑z) P‖ ^ 2 + ‖eval (↑z) Q‖ ^ 2 = 1 All goals completed! 🐙
#8
Character values of finite groups lie in cyclotomic fields
brauer_character_in_cyclotomic

Verso theorem preview

theorem declaration uses `sorry`brauer_character_in_cyclotomic (G : Type) [Group G] [Fintype G] : ∃ φ : CyclotomicField (Monoid.exponent G) ℚ →+* ℂ, ∀ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (_ : FiniteDimensional ℂ V) (ρ : Representation ℂ G V) (g : G), LinearMap.trace ℂ V (ρ g) ∈ φ.range := G:Typeinst✝¹:Group Ginst✝:Fintype G⊢ ∃ φ, ∀ (V : Type) (x : AddCommGroup V) (x_1 : Module ℂ V), FiniteDimensional ℂ V → ∀ (ρ : Representation ℂ G V) (g : G), (LinearMap.trace ℂ V) (ρ g) ∈ φ.range All goals completed! 🐙
#9
How produced

Antigravity orchestrated the solving using [Jules](https://jules.google.com/) and guided the agent during the task. Direct Spectral Decomposition Approach Strategy: Instead of moving to the global virtual character ring, we perform a local eigenvalue decomposition at the element level. 1. We state that the trace of the linear operator $\rho(g)$ is the sum of its eigenvalues (roots of the characteristic polynomial) using the Mathlib lemma: Module.End.trace_eq_sum_roots_charpoly_of_splits 2. Because $g^{\exp(G)} = 1$ in the group, we have $\rho(g)^{\exp(G)} = 1$. 3. By the Spectral Mapping Theorem (spectrum.pow_mem_pow), if $x$ is an eigenvalue of $\rho(g)$, then $x^{\exp(G)}$ must be an eigenvalue of the identity operator, forcing $x^{\exp(G)} = 1$. 4. Thus, every individual eigenvalue is an $\exp(G)$-th root of unity. 5. The range of our cyclotomic embedding $\varphi$ contains the image of the primitive root $\zeta_{\exp(G)}$, which algebraically generates all $\exp(G)$-th roots of unity in $\mathbb{C}$. Hence, each individual eigenvalue lies in $\varphi\text{.range}$. 6. Since $\varphi\text{.range}$ is a subring (subsemiring in Mathlib), it is closed under addition, and the trace (the sum of the eigenvalues) automatically lies in $\varphi\text{.range}$.

Oppenheim's inequality for Hadamard products
oppenheim_inequality

Verso theorem preview

theorem declaration uses `sorry`oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : A.det * ∏ i, B i i ≤ (A ⊙ B).det := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝB:Matrix n n ℝhA:A.PosSemidefhB:B.PosSemidef⊢ A.det * ∏ i, B i i ≤ (A ⊙ B).det All goals completed! 🐙
#10
Minkowski-Caratheodory theorem
mem_convexHull_finset_extremePoints_of_mem_compact_convex

Verso theorem preview

theorem declaration uses `sorry`mem_convexHull_finset_extremePoints_of_mem_compact_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hscomp : IsCompact s) (hsconv : Convex ℝ s) (hx : x ∈ s) : ∃ t : Finset E, (↑t : Set E) ⊆ s.extremePoints ℝ ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ convexHull ℝ (↑t : Set E) := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:FiniteDimensional ℝ Es:Set Ex:Ehscomp:IsCompact shsconv:Convex ℝ shx:x ∈ s⊢ ∃ t, ↑t ⊆ extremePoints ℝ s ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ (convexHull ℝ) ↑t All goals completed! 🐙
#11
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2
dirichlet_eigenvalues_eq_nat_sq

Verso theorem preview

theorem declaration uses `sorry`dirichlet_eigenvalues_eq_nat_sq (lam : ℝ) : (∃ (y : ℝ → ℝ) (J : Set ℝ), IsOpen J ∧ Set.Icc (0 : ℝ) Real.pi ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y Real.pi = 0 ∧ ∃ x ∈ Set.Ioo (0 : ℝ) Real.pi, y x ≠ 0) ↔ ∃ n : ℕ, 0 < n ∧ lam = (n : ℝ) ^ 2 := lam:ℝ⊢ (∃ y J, IsOpen J ∧ Set.Icc 0 π ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y π = 0 ∧ ∃ x ∈ Set.Ioo 0 π, y x ≠ 0) ↔ ∃ n, 0 < n ∧ lam = ↑n ^ 2 All goals completed! 🐙
#12
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Gaussian heat kernel solves the 1D heat equation
heat_kernel_solves_heat_equation

Verso theorem preview

theorem declaration uses `sorry`heat_kernel_solves_heat_equation (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_bdd : ∃ M : ℝ, ∀ x, |f x| ≤ M) : -- The PDE on (0, ∞) × ℝ. (∀ t : ℝ, 0 < t → ∀ x : ℝ, ∃ ux : ℝ → ℝ, ∃ uxx : ℝ, (∀ y : ℝ, HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ -- Initial condition recovered as a one-sided limit at t = 0. (∀ x : ℝ, Filter.Tendsto (fun t : ℝ => heatSolution f t x) (nhdsWithin (0 : ℝ) (Set.Ioi 0)) (nhds (f x))) := f:ℝ → ℝhf_cont:Continuous fhf_bdd:∃ M, ∀ (x : ℝ), |f x| ≤ M⊢ (∀ (t : ℝ), 0 < t → ∀ (x : ℝ), ∃ ux uxx, (∀ (y : ℝ), HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ ∀ (x : ℝ), Filter.Tendsto (fun t => heatSolution f t x) (nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) All goals completed! 🐙
#13
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

pi_1 of the circle is Z
pi1_circle_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi1_circle_mulEquiv_int : Nonempty (HomotopyGroup.Pi 1 Circle (1 : Circle) ≃* Multiplicative ℤ) := ⊢ Nonempty (HomotopyGroup.Pi 1 Circle 1 ≃* Multiplicative ℤ) All goals completed! 🐙
#14
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Entrywise exponential of a PSD matrix is PSD
posSemidef_map_exp

Verso theorem preview

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n] {A : Matrix n n ℝ} (hA : A.PosSemidef) : (A.map Real.exp).PosSemidef := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef All goals completed! 🐙
#15
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#16
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#17
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#20
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#23
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#24
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#25
How produced

This solution was produced interactively via an AI-driven pair-programming workflow using Antigravity, Google's agentic IDE. The proof was built structurally by cycling through the available ensemble of frontier models over multiple sessions. The formal verification relied on a specialized scaffolding pipeline: (1) Iterative prompt engineering utilizing Mathlib style constraints. (2) A failure feedback loop that injected raw compiler diagnostics back into the prompt context to diagnose errors. (3) Real-time Model Context Protocol (MCP) integration with the lean-lsp server, providing high-fidelity "Language Server to Agent" access to lean_goal (tactic state tracking) and lean_diagnostic_messages (immediate compiler feedback).

Test problems: def_hole_example, instance_hole_example, ci_regenerate_main_check, list_append_singleton_length, two_plus_two (5 / 8 solved)

Submission history

First submissionMay 13, 2026
Last submissionMay 22, 2026

Contributors

daouid26
9Claude Opus 4.7 (1M context)17 solved

Other solved problems

Schur-Weyl duality: GL(V) image equals centralizer of S_k image
glAction_range_eq_centralizer_symAction

Verso theorem preview

theorem declaration uses `sorry`glAction_range_eq_centralizer_symAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(glAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(symAction R M k)) All goals completed! 🐙
#1
How produced

Claude Opus 4.7 (1M context) with human (Jeroen Zuiddam) direction.

Schur-Weyl duality: S_k image equals centralizer of GL(V) image
symAction_range_eq_centralizer_glAction

Verso theorem preview

theorem declaration uses `sorry`symAction_range_eq_centralizer_glAction {R : Type*} [Field R] {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M] {k : ℕ} [Invertible (k.factorial : R)] : Algebra.adjoin R (Set.range (LeanEval.RepresentationTheory.symAction R M k)) = Subalgebra.centralizer R (Set.range (LeanEval.RepresentationTheory.glAction R M k)) := R:Type u_1inst✝⁴:Field RM:Type u_2inst✝³:AddCommGroup Minst✝²:Module R Minst✝¹:FiniteDimensional R Mk:ℕinst✝:Invertible ↑k.factorial⊢ Algebra.adjoin R (Set.range ⇑(symAction R M k)) = Subalgebra.centralizer R (Set.range ⇑(glAction R M k)) All goals completed! 🐙
#2
How produced

Claude Opus 4.7 (1M context) with human (Jeroen Zuiddam) direction.

Complementary polynomial on the unit circle
exists_complementary_polynomial_on_unit_circle

Verso theorem preview

theorem declaration uses `sorry`exists_complementary_polynomial_on_unit_circle (P : ℂ[X]) (hP : ∀ z : Circle, ‖P.eval (z : ℂ)‖ ≤ 1) : ∃ Q : ℂ[X], Q.natDegree ≤ P.natDegree ∧ ∀ z : Circle, ‖P.eval (z : ℂ)‖ ^ 2 + ‖Q.eval (z : ℂ)‖ ^ 2 = 1 := P:ℂ[X]hP:∀ (z : Circle), ‖eval (↑z) P‖ ≤ 1⊢ ∃ Q, Q.natDegree ≤ P.natDegree ∧ ∀ (z : Circle), ‖eval (↑z) P‖ ^ 2 + ‖eval (↑z) Q‖ ^ 2 = 1 All goals completed! 🐙
#3
Rouche theorem via zero counting
rouche_zero_count_eq

Verso theorem preview

theorem declaration uses `sorry`rouche_zero_count_eq {f g : ℂ → ℂ} {R : ℝ} (hR : 0 < R) (hf : MeromorphicNFOn f Set.univ) (hg : AnalyticOn ℂ g Set.univ) (hbound : ∀ z : ℂ, ‖z‖ = R → ‖g z‖ < ‖f z‖) : (∑ᶠ z, ((divisor (f + g) (Metric.closedBall 0 R))⁺) z) = (∑ᶠ z, ((divisor f (Metric.closedBall 0 R))⁺) z) := f:ℂ → ℂg:ℂ → ℂR:ℝhR:0 < Rhf:MeromorphicNFOn f Set.univhg:AnalyticOn ℂ g Set.univhbound:∀ (z : ℂ), ‖z‖ = R → ‖g z‖ < ‖f z‖⊢ ∑ᶠ (z : ℂ), (divisor (f + g) (Metric.closedBall 0 R))⁺ z = ∑ᶠ (z : ℂ), (divisor f (Metric.closedBall 0 R))⁺ z All goals completed! 🐙
#4
Character values of finite groups lie in cyclotomic fields
brauer_character_in_cyclotomic

Verso theorem preview

theorem declaration uses `sorry`brauer_character_in_cyclotomic (G : Type) [Group G] [Fintype G] : ∃ φ : CyclotomicField (Monoid.exponent G) ℚ →+* ℂ, ∀ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (_ : FiniteDimensional ℂ V) (ρ : Representation ℂ G V) (g : G), LinearMap.trace ℂ V (ρ g) ∈ φ.range := G:Typeinst✝¹:Group Ginst✝:Fintype G⊢ ∃ φ, ∀ (V : Type) (x : AddCommGroup V) (x_1 : Module ℂ V), FiniteDimensional ℂ V → ∀ (ρ : Representation ℂ G V) (g : G), (LinearMap.trace ℂ V) (ρ g) ∈ φ.range All goals completed! 🐙
#5
Gaussian heat kernel solves the 1D heat equation
heat_kernel_solves_heat_equation

Verso theorem preview

theorem declaration uses `sorry`heat_kernel_solves_heat_equation (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_bdd : ∃ M : ℝ, ∀ x, |f x| ≤ M) : -- The PDE on (0, ∞) × ℝ. (∀ t : ℝ, 0 < t → ∀ x : ℝ, ∃ ux : ℝ → ℝ, ∃ uxx : ℝ, (∀ y : ℝ, HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ -- Initial condition recovered as a one-sided limit at t = 0. (∀ x : ℝ, Filter.Tendsto (fun t : ℝ => heatSolution f t x) (nhdsWithin (0 : ℝ) (Set.Ioi 0)) (nhds (f x))) := f:ℝ → ℝhf_cont:Continuous fhf_bdd:∃ M, ∀ (x : ℝ), |f x| ≤ M⊢ (∀ (t : ℝ), 0 < t → ∀ (x : ℝ), ∃ ux uxx, (∀ (y : ℝ), HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ ∀ (x : ℝ), Filter.Tendsto (fun t => heatSolution f t x) (nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) All goals completed! 🐙
#6
Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2
dirichlet_eigenvalues_eq_nat_sq

Verso theorem preview

theorem declaration uses `sorry`dirichlet_eigenvalues_eq_nat_sq (lam : ℝ) : (∃ (y : ℝ → ℝ) (J : Set ℝ), IsOpen J ∧ Set.Icc (0 : ℝ) Real.pi ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y Real.pi = 0 ∧ ∃ x ∈ Set.Ioo (0 : ℝ) Real.pi, y x ≠ 0) ↔ ∃ n : ℕ, 0 < n ∧ lam = (n : ℝ) ^ 2 := lam:ℝ⊢ (∃ y J, IsOpen J ∧ Set.Icc 0 π ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y π = 0 ∧ ∃ x ∈ Set.Ioo 0 π, y x ≠ 0) ↔ ∃ n, 0 < n ∧ lam = ↑n ^ 2 All goals completed! 🐙
#7
Minkowski-Caratheodory theorem
mem_convexHull_finset_extremePoints_of_mem_compact_convex

Verso theorem preview

theorem declaration uses `sorry`mem_convexHull_finset_extremePoints_of_mem_compact_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hscomp : IsCompact s) (hsconv : Convex ℝ s) (hx : x ∈ s) : ∃ t : Finset E, (↑t : Set E) ⊆ s.extremePoints ℝ ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ convexHull ℝ (↑t : Set E) := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:FiniteDimensional ℝ Es:Set Ex:Ehscomp:IsCompact shsconv:Convex ℝ shx:x ∈ s⊢ ∃ t, ↑t ⊆ extremePoints ℝ s ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ (convexHull ℝ) ↑t All goals completed! 🐙
#8
pi_1 of the circle is Z
pi1_circle_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi1_circle_mulEquiv_int : Nonempty (HomotopyGroup.Pi 1 Circle (1 : Circle) ≃* Multiplicative ℤ) := ⊢ Nonempty (HomotopyGroup.Pi 1 Circle 1 ≃* Multiplicative ℤ) All goals completed! 🐙
#9
Oppenheim's inequality for Hadamard products
oppenheim_inequality

Verso theorem preview

theorem declaration uses `sorry`oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : A.det * ∏ i, B i i ≤ (A ⊙ B).det := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝB:Matrix n n ℝhA:A.PosSemidefhB:B.PosSemidef⊢ A.det * ∏ i, B i i ≤ (A ⊙ B).det All goals completed! 🐙
#10
Entrywise exponential of a PSD matrix is PSD
posSemidef_map_exp

Verso theorem preview

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n] {A : Matrix n n ℝ} (hA : A.PosSemidef) : (A.map Real.exp).PosSemidef := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef All goals completed! 🐙
#11
Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#12
Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#13
How produced

Just asking to solve the problems in Claude Code.

Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#16
Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#19
Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#20
How produced

Just asking to solve the problems in Claude Code.

Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#21
How produced

Just asking to solve the problems in Claude Code.

Test problems: def_hole_example, instance_hole_example, ci_regenerate_main_check, list_append_singleton_length, two_plus_two (5 / 8 solved)

Submission history

First submissionMay 2, 2026
Last submissionMay 24, 2026

Contributors

rkirov20jzuiddam2
10GPT-5.516 solved

Other solved problems

Bing's house with two rooms is contractible
contractibleSpace_houseWithTwoRooms

Verso theorem preview

theorem declaration uses `sorry`contractibleSpace_houseWithTwoRooms : ContractibleSpace LeanEval.Topology.HouseWithTwoRooms := ⊢ ContractibleSpace ↑HouseWithTwoRooms All goals completed! 🐙
#1
How produced

autonomous lmp

Linear ODE with negative-real-part eigenvalues is asymptotically stable
linear_ode_asymptotic_stability

Verso theorem preview

theorem declaration uses `sorry`linear_ode_asymptotic_stability (n : ℕ) (A : Matrix (Fin n) (Fin n) ℝ) (hA : ∀ μ : ℂ, Module.End.HasEigenvalue (Matrix.toLin' (A.map (algebraMap ℝ ℂ))) μ → μ.re < 0) (x : ℝ → (Fin n → ℝ)) (hx : ∀ t : ℝ, 0 < t → HasDerivAt x (A.mulVec (x t)) t) : Filter.Tendsto (fun t : ℝ => ‖x t‖) Filter.atTop (nhds 0) := n:ℕA:Matrix (Fin n) (Fin n) ℝhA:∀ (μ : ℂ), Module.End.HasEigenvalue (Matrix.toLin' (A.map ⇑(algebraMap ℝ ℂ))) μ → μ.re < 0x:ℝ → Fin n → ℝhx:∀ (t : ℝ), 0 < t → HasDerivAt x (A *ᵥ x t) t⊢ Filter.Tendsto (fun t => ‖x t‖) Filter.atTop (nhds 0) All goals completed! 🐙
#2
How produced

autonomous lmp

Perron-Frobenius for irreducible nonnegative matrices
irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius

Verso theorem preview

theorem declaration uses `sorry`irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius {n : Type*} [Fintype n] [DecidableEq n] [Nonempty n] (A : Matrix n n ℝ) (hA : A.IsIrreducible) : ∃ v : n → ℝ, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ (∀ i, 0 < v i) := n:Type u_1inst✝²:Fintype ninst✝¹:DecidableEq ninst✝:Nonempty nA:Matrix n n ℝhA:A.IsIrreducible⊢ ∃ v, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ ∀ (i : n), 0 < v i All goals completed! 🐙
#3
Character values of finite groups lie in cyclotomic fields
brauer_character_in_cyclotomic

Verso theorem preview

theorem declaration uses `sorry`brauer_character_in_cyclotomic (G : Type) [Group G] [Fintype G] : ∃ φ : CyclotomicField (Monoid.exponent G) ℚ →+* ℂ, ∀ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (_ : FiniteDimensional ℂ V) (ρ : Representation ℂ G V) (g : G), LinearMap.trace ℂ V (ρ g) ∈ φ.range := G:Typeinst✝¹:Group Ginst✝:Fintype G⊢ ∃ φ, ∀ (V : Type) (x : AddCommGroup V) (x_1 : Module ℂ V), FiniteDimensional ℂ V → ∀ (ρ : Representation ℂ G V) (g : G), (LinearMap.trace ℂ V) (ρ g) ∈ φ.range All goals completed! 🐙
#4
Gaussian heat kernel solves the 1D heat equation
heat_kernel_solves_heat_equation

Verso theorem preview

theorem declaration uses `sorry`heat_kernel_solves_heat_equation (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_bdd : ∃ M : ℝ, ∀ x, |f x| ≤ M) : -- The PDE on (0, ∞) × ℝ. (∀ t : ℝ, 0 < t → ∀ x : ℝ, ∃ ux : ℝ → ℝ, ∃ uxx : ℝ, (∀ y : ℝ, HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ -- Initial condition recovered as a one-sided limit at t = 0. (∀ x : ℝ, Filter.Tendsto (fun t : ℝ => heatSolution f t x) (nhdsWithin (0 : ℝ) (Set.Ioi 0)) (nhds (f x))) := f:ℝ → ℝhf_cont:Continuous fhf_bdd:∃ M, ∀ (x : ℝ), |f x| ≤ M⊢ (∀ (t : ℝ), 0 < t → ∀ (x : ℝ), ∃ ux uxx, (∀ (y : ℝ), HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ ∀ (x : ℝ), Filter.Tendsto (fun t => heatSolution f t x) (nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) All goals completed! 🐙
#5
How produced

autonomous lmp

Oppenheim's inequality for Hadamard products
oppenheim_inequality

Verso theorem preview

theorem declaration uses `sorry`oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : A.det * ∏ i, B i i ≤ (A ⊙ B).det := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝB:Matrix n n ℝhA:A.PosSemidefhB:B.PosSemidef⊢ A.det * ∏ i, B i i ≤ (A ⊙ B).det All goals completed! 🐙
#6
How produced

Autonomous lmp

Minkowski-Caratheodory theorem
mem_convexHull_finset_extremePoints_of_mem_compact_convex

Verso theorem preview

theorem declaration uses `sorry`mem_convexHull_finset_extremePoints_of_mem_compact_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hscomp : IsCompact s) (hsconv : Convex ℝ s) (hx : x ∈ s) : ∃ t : Finset E, (↑t : Set E) ⊆ s.extremePoints ℝ ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ convexHull ℝ (↑t : Set E) := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:FiniteDimensional ℝ Es:Set Ex:Ehscomp:IsCompact shsconv:Convex ℝ shx:x ∈ s⊢ ∃ t, ↑t ⊆ extremePoints ℝ s ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ (convexHull ℝ) ↑t All goals completed! 🐙
#7
Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2
dirichlet_eigenvalues_eq_nat_sq

Verso theorem preview

theorem declaration uses `sorry`dirichlet_eigenvalues_eq_nat_sq (lam : ℝ) : (∃ (y : ℝ → ℝ) (J : Set ℝ), IsOpen J ∧ Set.Icc (0 : ℝ) Real.pi ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y Real.pi = 0 ∧ ∃ x ∈ Set.Ioo (0 : ℝ) Real.pi, y x ≠ 0) ↔ ∃ n : ℕ, 0 < n ∧ lam = (n : ℝ) ^ 2 := lam:ℝ⊢ (∃ y J, IsOpen J ∧ Set.Icc 0 π ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y π = 0 ∧ ∃ x ∈ Set.Ioo 0 π, y x ≠ 0) ↔ ∃ n, 0 < n ∧ lam = ↑n ^ 2 All goals completed! 🐙
#8
pi_1 of the circle is Z
pi1_circle_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi1_circle_mulEquiv_int : Nonempty (HomotopyGroup.Pi 1 Circle (1 : Circle) ≃* Multiplicative ℤ) := ⊢ Nonempty (HomotopyGroup.Pi 1 Circle 1 ≃* Multiplicative ℤ) All goals completed! 🐙
#9
Entrywise exponential of a PSD matrix is PSD
posSemidef_map_exp

Verso theorem preview

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n] {A : Matrix n n ℝ} (hA : A.PosSemidef) : (A.map Real.exp).PosSemidef := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef All goals completed! 🐙
#10
Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#13
Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#14
Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#17
Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#18
Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#19
Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#20

Test problems: instance_hole_example, def_hole_example, ci_regenerate_main_check, list_append_singleton_length, two_plus_two (5 / 8 solved)

Submission history

First submissionMay 1, 2026
Last submissionJun 8, 2026

Contributors

sqrt-of-214Morgan-Griffiths10A-M-Berns4kim-em1
11EVO (deepthought.com.au)8 solved

Other solved problems

Pell solutions are convergents of √d
pell_solution_convergent

Verso theorem preview

theorem declaration uses `sorry`pell_solution_is_convergent (d : ℤ) (_hd : Squarefree d) (_hd0 : 0 < d) (x y : ℤ) (_hx : 0 < x) (_hy : 0 < y) (_hsol : x ^ 2 - d * y ^ 2 = 1) : ∃ n : ℕ, (GenContFract.of (Real.sqrt (d : ℝ))).convs n = (x : ℝ) / (y : ℝ) := d:ℤ_hd:Squarefree d_hd0:0 < dx:ℤy:ℤ_hx:0 < x_hy:0 < y_hsol:x ^ 2 - d * y ^ 2 = 1⊢ ∃ n, (GenContFract.of √↑d).convs n = ↑x / ↑y All goals completed! 🐙
#3
Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2
dirichlet_eigenvalues_eq_nat_sq

Verso theorem preview

theorem declaration uses `sorry`dirichlet_eigenvalues_eq_nat_sq (lam : ℝ) : (∃ (y : ℝ → ℝ) (J : Set ℝ), IsOpen J ∧ Set.Icc (0 : ℝ) Real.pi ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y Real.pi = 0 ∧ ∃ x ∈ Set.Ioo (0 : ℝ) Real.pi, y x ≠ 0) ↔ ∃ n : ℕ, 0 < n ∧ lam = (n : ℝ) ^ 2 := lam:ℝ⊢ (∃ y J, IsOpen J ∧ Set.Icc 0 π ⊆ J ∧ (∀ x ∈ J, HasDerivAt y (deriv y x) x) ∧ (∀ x ∈ J, HasDerivAt (deriv y) (-(lam * y x)) x) ∧ y 0 = 0 ∧ y π = 0 ∧ ∃ x ∈ Set.Ioo 0 π, y x ≠ 0) ↔ ∃ n, 0 < n ∧ lam = ↑n ^ 2 All goals completed! 🐙
#4
Entrywise exponential of a PSD matrix is PSD
posSemidef_map_exp

Verso theorem preview

theorem declaration uses `sorry`posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n] {A : Matrix n n ℝ} (hA : A.PosSemidef) : (A.map Real.exp).PosSemidef := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝhA:A.PosSemidef⊢ (A.map Real.exp).PosSemidef All goals completed! 🐙
#5
Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#6
Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#7
Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#12
Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#13
Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#14

Test problems: multi_hole_helpers_example, variable_binder_example, def_hole_example, instance_hole_example, ci_regenerate_main_check, list_append_singleton_length, two_plus_two (7 / 8 solved)

Submission history

First submissionJun 6, 2026
Last submissionJun 23, 2026

Contributors

test1-deepthought15
12Public accepted source + Codex packaging7 solved

Other solved problems

Perron-Frobenius for irreducible nonnegative matrices
irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius

Verso theorem preview

theorem declaration uses `sorry`irreducible_nonnegative_matrix_has_positive_eigenvector_at_spectralRadius {n : Type*} [Fintype n] [DecidableEq n] [Nonempty n] (A : Matrix n n ℝ) (hA : A.IsIrreducible) : ∃ v : n → ℝ, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ (∀ i, 0 < v i) := n:Type u_1inst✝²:Fintype ninst✝¹:DecidableEq ninst✝:Nonempty nA:Matrix n n ℝhA:A.IsIrreducible⊢ ∃ v, Module.End.HasEigenvector (Matrix.toLin' A) (spectralRadius ℝ A).toReal v ∧ ∀ (i : n), 0 < v i All goals completed! 🐙
#1
How produced

Copied from a public accepted Lean Eval submission, repackaged into a commit-pinned workspace, then checked locally with `lake build Submission`, `lake env lean Submission.lean`, and a solver-owned forbidden-token scan.

Rouche theorem via zero counting
rouche_zero_count_eq

Verso theorem preview

theorem declaration uses `sorry`rouche_zero_count_eq {f g : ℂ → ℂ} {R : ℝ} (hR : 0 < R) (hf : MeromorphicNFOn f Set.univ) (hg : AnalyticOn ℂ g Set.univ) (hbound : ∀ z : ℂ, ‖z‖ = R → ‖g z‖ < ‖f z‖) : (∑ᶠ z, ((divisor (f + g) (Metric.closedBall 0 R))⁺) z) = (∑ᶠ z, ((divisor f (Metric.closedBall 0 R))⁺) z) := f:ℂ → ℂg:ℂ → ℂR:ℝhR:0 < Rhf:MeromorphicNFOn f Set.univhg:AnalyticOn ℂ g Set.univhbound:∀ (z : ℂ), ‖z‖ = R → ‖g z‖ < ‖f z‖⊢ ∑ᶠ (z : ℂ), (divisor (f + g) (Metric.closedBall 0 R))⁺ z = ∑ᶠ (z : ℂ), (divisor f (Metric.closedBall 0 R))⁺ z All goals completed! 🐙
#2
How produced

Copied from a public accepted Lean Eval submission, repackaged into a commit-pinned workspace, then checked locally with `lake build Submission`, `lake env lean Submission.lean`, and a solver-owned forbidden-token scan.

Complementary polynomial on the unit circle
exists_complementary_polynomial_on_unit_circle

Verso theorem preview

theorem declaration uses `sorry`exists_complementary_polynomial_on_unit_circle (P : ℂ[X]) (hP : ∀ z : Circle, ‖P.eval (z : ℂ)‖ ≤ 1) : ∃ Q : ℂ[X], Q.natDegree ≤ P.natDegree ∧ ∀ z : Circle, ‖P.eval (z : ℂ)‖ ^ 2 + ‖Q.eval (z : ℂ)‖ ^ 2 = 1 := P:ℂ[X]hP:∀ (z : Circle), ‖eval (↑z) P‖ ≤ 1⊢ ∃ Q, Q.natDegree ≤ P.natDegree ∧ ∀ (z : Circle), ‖eval (↑z) P‖ ^ 2 + ‖eval (↑z) Q‖ ^ 2 = 1 All goals completed! 🐙
#3
How produced

Copied from a public accepted Lean Eval submission, repackaged into a commit-pinned workspace, then checked locally with `lake build Submission`, `lake env lean Submission.lean`, and a solver-owned forbidden-token scan.

Gaussian heat kernel solves the 1D heat equation
heat_kernel_solves_heat_equation

Verso theorem preview

theorem declaration uses `sorry`heat_kernel_solves_heat_equation (f : ℝ → ℝ) (hf_cont : Continuous f) (hf_bdd : ∃ M : ℝ, ∀ x, |f x| ≤ M) : -- The PDE on (0, ∞) × ℝ. (∀ t : ℝ, 0 < t → ∀ x : ℝ, ∃ ux : ℝ → ℝ, ∃ uxx : ℝ, (∀ y : ℝ, HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ -- Initial condition recovered as a one-sided limit at t = 0. (∀ x : ℝ, Filter.Tendsto (fun t : ℝ => heatSolution f t x) (nhdsWithin (0 : ℝ) (Set.Ioi 0)) (nhds (f x))) := f:ℝ → ℝhf_cont:Continuous fhf_bdd:∃ M, ∀ (x : ℝ), |f x| ≤ M⊢ (∀ (t : ℝ), 0 < t → ∀ (x : ℝ), ∃ ux uxx, (∀ (y : ℝ), HasDerivAt (fun z => heatSolution f t z) (ux y) y) ∧ HasDerivAt ux uxx x ∧ HasDerivAt (fun s => heatSolution f s x) uxx t) ∧ ∀ (x : ℝ), Filter.Tendsto (fun t => heatSolution f t x) (nhdsWithin 0 (Set.Ioi 0)) (nhds (f x)) All goals completed! 🐙
#4
How produced

Copied from a public accepted Lean Eval submission, repackaged into a commit-pinned workspace, then checked locally with `lake build Submission`, `lake env lean Submission.lean`, and a solver-owned forbidden-token scan.

Oppenheim's inequality for Hadamard products
oppenheim_inequality

Verso theorem preview

theorem declaration uses `sorry`oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n] {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) : A.det * ∏ i, B i i ≤ (A ⊙ B).det := n:Type u_1inst✝¹:Fintype ninst✝:DecidableEq nA:Matrix n n ℝB:Matrix n n ℝhA:A.PosSemidefhB:B.PosSemidef⊢ A.det * ∏ i, B i i ≤ (A ⊙ B).det All goals completed! 🐙
#5
How produced

Copied from a public accepted Lean Eval submission, repackaged into a commit-pinned workspace, then checked locally with `lake build Submission`, `lake env lean Submission.lean`, and a solver-owned forbidden-token scan.

Minkowski-Caratheodory theorem
mem_convexHull_finset_extremePoints_of_mem_compact_convex

Verso theorem preview

theorem declaration uses `sorry`mem_convexHull_finset_extremePoints_of_mem_compact_convex {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E} (hscomp : IsCompact s) (hsconv : Convex ℝ s) (hx : x ∈ s) : ∃ t : Finset E, (↑t : Set E) ⊆ s.extremePoints ℝ ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ convexHull ℝ (↑t : Set E) := E:Type u_1inst✝²:NormedAddCommGroup Einst✝¹:NormedSpace ℝ Einst✝:FiniteDimensional ℝ Es:Set Ex:Ehscomp:IsCompact shsconv:Convex ℝ shx:x ∈ s⊢ ∃ t, ↑t ⊆ extremePoints ℝ s ∧ t.card ≤ Module.finrank ℝ E + 1 ∧ x ∈ (convexHull ℝ) ↑t All goals completed! 🐙
#6
How produced

Copied from a public accepted Lean Eval submission, repackaged into a commit-pinned workspace, then checked locally with `lake build Submission`, `lake env lean Submission.lean`, and a solver-owned forbidden-token scan.

Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#7
How produced

Copied from a public accepted Lean Eval submission, repackaged into a commit-pinned workspace, then checked locally with `lake build Submission`, `lake env lean Submission.lean`, and a solver-owned forbidden-token scan.

Submission history

First submissionJun 19, 2026
Last submissionJun 19, 2026

Contributors

rishistyping7
13Claude Fable 54 solved

Other solved problems

Jordan curve theorem
jordan_curve

Verso theorem preview

theorem declaration uses `sorry`jordan_curve (r : Metric.sphere (0 : EuclideanSpace ℝ (Fin 2)) 1 → EuclideanSpace ℝ (Fin 2)) (_hcont : Continuous r) (_hinj : Function.Injective r) : Nat.card (ConnectedComponents ((Set.range r)ᶜ : Set (EuclideanSpace ℝ (Fin 2)))) = 2 := r:↑(Metric.sphere 0 1) → EuclideanSpace ℝ (Fin 2)_hcont:Continuous r_hinj:Function.Injective r⊢ Nat.card (ConnectedComponents ↑(Set.range r)ᶜ) = 2 All goals completed! 🐙
#1
Radó's theorem on Riemann surfaces
rado_riemannSurface

Verso theorem preview

theorem declaration uses `sorry`rado_riemannSurface {X : Type*} [TopologicalSpace X] [T2Space X] [ConnectedSpace X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) 1 X] : SecondCountableTopology X := X:Type u_1inst✝⁴:TopologicalSpace Xinst✝³:T2Space Xinst✝²:ConnectedSpace Xinst✝¹:ChartedSpace ℂ Xinst✝:IsManifold (modelWithCornersSelf ℂ ℂ) 1 X⊢ SecondCountableTopology X All goals completed! 🐙
#2
Jacobian of a compact Riemann surface (Buzzard challenge)
jacobian_challenge_diffgeo

Verso theorem preview

def declaration uses `sorry`genus (X : Type u) [TopologicalSpace X] [T2Space X] [CompactSpace X] [ConnectedSpace X] [Nonempty X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) ω X] : ℕ := sorry
theorem declaration uses `sorry`genus_eq_zero_iff_homeo : genus X = 0 ↔ Nonempty (X ≃ₜ (Metric.sphere (0 : EuclideanSpace ℝ (Fin 3)) 1)) := sorry
def declaration uses `sorry`Jacobian (X : Type u) [TopologicalSpace X] [T2Space X] [CompactSpace X] [ConnectedSpace X] [Nonempty X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) ω X] : Type u := sorry
instance declaration uses `sorry`instAddCommGroup : AddCommGroup (Jacobian X) := sorry
instance declaration uses `sorry`instTopologicalSpace : TopologicalSpace (Jacobian X) := sorry
instance declaration uses `sorry`instT2Space : T2Space (Jacobian X) := sorry
instance declaration uses `sorry`instCompactSpace : CompactSpace (Jacobian X) := sorry
instance declaration uses `sorry`instChartedSpace : ChartedSpace (Fin (genus X) → ℂ) (Jacobian X) := sorry
instance declaration uses `sorry`instIsManifold : IsManifold (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (Jacobian X) := sorry
instance declaration uses `sorry`instLieAddGroup : LieAddGroup (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (Jacobian X) := sorry
def declaration uses `sorry`ofCurve (P : X) : X → Jacobian X := sorry
theorem declaration uses `sorry`ofCurve_contMDiff (P : X) : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (ofCurve P) := sorry
theorem declaration uses `sorry`ofCurve_self (P : X) : ofCurve P P = 0 := sorry
theorem declaration uses `sorry`ofCurve_inj (P : X) (h : 0 < genus X) : Function.Injective (ofCurve P) := sorry
def declaration uses `sorry`pushforward (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : Jacobian X →ₜ+ Jacobian Y := sorry
theorem declaration uses `sorry`pushforward_contMDiff (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ContMDiff (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) (modelWithCornersSelf ℂ (Fin (genus Y) → ℂ)) ω (pushforward f hf) := sorry
theorem declaration uses `sorry`pushforward_id_apply (P : Jacobian X) : pushforward id contMDiff_id P = P := sorry
theorem declaration uses `sorry`pushforward_comp_apply (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (g : Y → Z) (hg : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω g) (P : Jacobian X) : pushforward (g ∘ f) (hg.comp hf) P = pushforward g hg (pushforward f hf P) := sorry
def declaration uses `sorry`pullback (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : Jacobian Y →ₜ+ Jacobian X := sorry
theorem declaration uses `sorry`pullback_contMDiff (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ContMDiff (modelWithCornersSelf ℂ (Fin (genus Y) → ℂ)) (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (pullback f hf) := sorry
theorem declaration uses `sorry`pullback_id_apply (P : Jacobian X) : pullback id contMDiff_id P = P := sorry
theorem declaration uses `sorry`pullback_comp_apply (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (g : Y → Z) (hg : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω g) (P : Jacobian Z) : pullback (g.comp f) (hg.comp hf) P = pullback f hf (pullback g hg P) := sorry
def declaration uses `sorry`degree (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ℕ := sorry -- 0 for constant case
theorem declaration uses `sorry`pushforward_pullback (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (P : Jacobian Y) : pushforward f hf (pullback f hf P) = (degree f hf) • P := sorry
#3
Pick's theorem
pick

Verso theorem preview

theorem declaration uses `sorry`pick {n : ℕ} (hn : 3 ≤ n) (v : Fin n → ℤ × ℤ) (hsimple : LeanEval.Geometry.PicksTheorem.IsSimple (LeanEval.Geometry.PicksTheorem.latPoly v)) : area ((LeanEval.Geometry.PicksTheorem.latPoly v).boundary (R := ℝ)) = (interiorPts v : ℝ) + (boundaryPts v : ℝ) / 2 - 1 := n:ℕhn:3 ≤ nv:Fin n → ℤ × ℤhsimple:IsSimple (latPoly v)⊢ area (Polygon.boundary ℝ (latPoly v)) = ↑(interiorPts v) + ↑(boundaryPts v) / 2 - 1 All goals completed! 🐙
#4

Submission history

First submissionJul 7, 2026
Last submissionJul 10, 2026

Contributors

rkirov4
14DeepSeek V4 Flash3 solved

Other solved problems

Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#1
How produced

Setup: Opencode with lean-lsp MCP installed, model "DeepSeek V4 Flash Free" (under "Opencode Zen"). Prompt: "prove Submission.lean". Solve time: 40 minutes. Cost: $0.00.

Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#2
How produced

Setup: Opencode with lean-lsp MCP installed, model "DeepSeek V4 Flash Free" (under "Opencode Zen"). Prompt: "prove Submission.lean", adding the informal solution. Solve time: 45 min. Cost: $0.00.

Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#3
How produced

Setup: Opencode with lean-lsp MCP installed, model "DeepSeek V4 Flash Free" (under "Opencode Zen"). Prompt: "prove Submission.lean". Solve time: 1 hours. Cost: $0.00.

Submission history

First submissionJul 12, 2026
Last submissionJul 12, 2026

Contributors

rwst3
15Grok 4.53 solved

Other solved problems

Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#1
How produced

Setup: Grok CLI, picking up lean4-skills and lean-lsp MCP. Prompt: "/lean4:prove @Submission.lean --commit=never --golf=yes". Time: 40 min. Cost: 0$ (currently free).

Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#2
How produced

Setup: Grok CLI, picking up lean4-skills and lean-lsp MCP. Prompt: "/lean4:prove @Submission.lean --commit=never --golf=yes". Time: 15 min. Cost: 0$ (currently free).

Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#3
How produced

Setup: Grok CLI, picking up lean4-skills and lean-lsp MCP. Prompt: "/lean4:prove @Submission.lean --commit=never --golf=yes". Time: 15 min. Cost: 0$ (currently free).

Submission history

First submissionJul 22, 2026
Last submissionJul 23, 2026

Contributors

rwst3
16GPT-5.5 Codex2 solved

Other solved problems

Sturm separation theorem
sturm_separation

Verso theorem preview

theorem declaration uses `sorry`sturm_separation (p q y₁ y₂ : ℝ → ℝ) (a b : ℝ) (hab : a < b) (J : Set ℝ) (hJ_open : IsOpen J) (hJ_conn : IsPreconnected J) (hJ_sub : Set.Icc a b ⊆ J) (hp : ContinuousOn p J) (hq : ContinuousOn q J) (hy₁ : ∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) x) (hy₁' : ∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) x) (hy₂ : ∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) x) (hy₂' : ∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) x) (hW : ∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0) (hza : y₁ a = 0) (hzb : y₁ b = 0) (hne : ∀ x ∈ Set.Ioo a b, y₁ x ≠ 0) : ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 := p:ℝ → ℝq:ℝ → ℝy₁:ℝ → ℝy₂:ℝ → ℝa:ℝb:ℝhab:a < bJ:Set ℝhJ_open:IsOpen JhJ_conn:IsPreconnected JhJ_sub:Set.Icc a b ⊆ Jhp:ContinuousOn p Jhq:ContinuousOn q Jhy₁:∀ x ∈ J, HasDerivAt y₁ (deriv y₁ x) xhy₁':∀ x ∈ J, HasDerivAt (deriv y₁) (-(p x * deriv y₁ x + q x * y₁ x)) xhy₂:∀ x ∈ J, HasDerivAt y₂ (deriv y₂ x) xhy₂':∀ x ∈ J, HasDerivAt (deriv y₂) (-(p x * deriv y₂ x + q x * y₂ x)) xhW:∃ x₀ ∈ J, y₁ x₀ * deriv y₂ x₀ - y₂ x₀ * deriv y₁ x₀ ≠ 0hza:y₁ a = 0hzb:y₁ b = 0hne:∀ x ∈ Set.Ioo a b, y₁ x ≠ 0⊢ ∃! c, c ∈ Set.Ioo a b ∧ y₂ c = 0 All goals completed! 🐙
#1
Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#2

Test problems: two_plus_two (1 / 8 solved)

Submission history

First submissionMay 6, 2026
Last submissionMay 7, 2026

Contributors

A-M-Berns3
17Gemini 3.1 Pro2 solved

Other solved problems

Cayley graph connected iff generators generate the group
mulCayley_connected_iff_closure_eq_top

Verso theorem preview

theorem declaration uses `sorry`mulCayley_connected_iff_closure_eq_top {G : Type*} [Group G] (S : Set G) : (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ := G:Type u_1inst✝:Group GS:Set G⊢ (SimpleGraph.mulCayley S).Connected ↔ Subgroup.closure S = ⊤ All goals completed! 🐙
#1
Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#6

Test problems: instance_hole_example, def_hole_example, list_append_singleton_length, ci_regenerate_main_check, two_plus_two (5 / 8 solved)

Submission history

First submissionMay 1, 2026
Last submissionMay 10, 2026

Contributors

sqrt-of-27kim-em1
18[submission] aegis-of-the-unit-circle-logos2 solved

Other solved problems

Real cyclotomic integer with house at most 2
cyclotomic_integer_house_le_two

Verso theorem preview

theorem declaration uses `sorry`cyclotomic_integer_house_le_two {K : Type*} [Field K] [NumberField K] [Algebra ℚ K] (n : ℕ) [NeZero n] [IsCyclotomicExtension {n} ℚ K] {β : K} (hβ_int : IsIntegral ℤ β) (hβ_real : β ∈ NumberField.maximalRealSubfield K) : house β ≤ 2 → house β = 2 ∨ ∃ m : ℕ, 0 < m ∧ house β = 2 * Real.cos (Real.pi / m) := K:Type u_1inst✝⁴:Field Kinst✝³:NumberField Kinst✝²:Algebra ℚ Kn:ℕinst✝¹:NeZero ninst✝:IsCyclotomicExtension {n} ℚ Kβ:Khβ_int:IsIntegral ℤ βhβ_real:β ∈ maximalRealSubfield K⊢ house β ≤ 2 → house β = 2 ∨ ∃ m, 0 < m ∧ house β = 2 * Real.cos (Real.pi / ↑m) All goals completed! 🐙
#1
How produced

Comparator-accepted Lean Eval solution for cyclotomic_integer_house_le_two. Developed and verified in a private repository. Local checks included direct Lean Eval comparator and CI-equivalent evaluate_submission.py.

pi_1 of the circle is Z
pi1_circle_mulEquiv_int

Verso theorem preview

theorem declaration uses `sorry`pi1_circle_mulEquiv_int : Nonempty (HomotopyGroup.Pi 1 Circle (1 : Circle) ≃* Multiplicative ℤ) := ⊢ Nonempty (HomotopyGroup.Pi 1 Circle 1 ≃* Multiplicative ℤ) All goals completed! 🐙
#2
How produced

Comparator-accepted Lean Eval solution for cyclotomic_integer_house_le_two. Developed and verified in a private repository. Local checks included direct Lean Eval comparator and CI-equivalent evaluate_submission.py.

Test problems: ci_regenerate_main_check, two_plus_two (2 / 8 solved)

Submission history

First submissionMay 12, 2026
Last submissionMay 12, 2026

Contributors

rishistyping4
19EVO2 solved

Other solved problems

Comparison principle for the Dirichlet BVP
bvp_comparison

Verso theorem preview

theorem declaration uses `sorry`bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J) (u v : ℝ → ℝ) (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x) (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x) (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x) (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x) (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x) (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) : ∀ x ∈ Set.Icc (0 : ℝ) 1, u x ≤ v x := J:Set ℝhJ_open:IsOpen JhJ_sub:Set.Icc 0 1 ⊆ Ju:ℝ → ℝv:ℝ → ℝhu:∀ x ∈ J, HasDerivAt u (deriv u x) xhu':∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) xhv:∀ x ∈ J, HasDerivAt v (deriv v x) xhv':∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) xhineq:∀ x ∈ Set.Ioo 0 1, -deriv (deriv u) x ≤ -deriv (deriv v) xhu0:u 0 ≤ v 0hu1:u 1 ≤ v 1⊢ ∀ x ∈ Set.Icc 0 1, u x ≤ v x All goals completed! 🐙
#6
Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#7

Test problems: variable_binder_example, instance_hole_example, def_hole_example, list_append_singleton_length, ci_regenerate_main_check, two_plus_two (6 / 8 solved)

Submission history

First submissionMay 18, 2026
Last submissionJun 4, 2026

Contributors

test1-deepthought8machinelearning20143
20Claude Opus 4.8 (1M context)2 solved

Other solved problems

Jordan curve theorem
jordan_curve

Verso theorem preview

theorem declaration uses `sorry`jordan_curve (r : Metric.sphere (0 : EuclideanSpace ℝ (Fin 2)) 1 → EuclideanSpace ℝ (Fin 2)) (_hcont : Continuous r) (_hinj : Function.Injective r) : Nat.card (ConnectedComponents ((Set.range r)ᶜ : Set (EuclideanSpace ℝ (Fin 2)))) = 2 := r:↑(Metric.sphere 0 1) → EuclideanSpace ℝ (Fin 2)_hcont:Continuous r_hinj:Function.Injective r⊢ Nat.card (ConnectedComponents ↑(Set.range r)ᶜ) = 2 All goals completed! 🐙
#1
How produced

~1 week of grinding on $100/month subscription. Minimal human intervention.

Pick's theorem
pick

Verso theorem preview

theorem declaration uses `sorry`pick {n : ℕ} (hn : 3 ≤ n) (v : Fin n → ℤ × ℤ) (hsimple : LeanEval.Geometry.PicksTheorem.IsSimple (LeanEval.Geometry.PicksTheorem.latPoly v)) : area ((LeanEval.Geometry.PicksTheorem.latPoly v).boundary (R := ℝ)) = (interiorPts v : ℝ) + (boundaryPts v : ℝ) / 2 - 1 := n:ℕhn:3 ≤ nv:Fin n → ℤ × ℤhsimple:IsSimple (latPoly v)⊢ area (Polygon.boundary ℝ (latPoly v)) = ↑(interiorPts v) + ↑(boundaryPts v) / 2 - 1 All goals completed! 🐙
#2
How produced

~1 week of grinding on $100/month subscription. Minimal human intervention.

Submission history

First submissionJul 1, 2026
Last submissionJul 1, 2026

Contributors

rkirov2
21Kimi K2.72 solved

Other solved problems

Polynomial decay rate of y' = -y^3
cubic_decay_asymptotic

Verso theorem preview

theorem declaration uses `sorry`cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t) (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0) (hy0 : y 0 = 1) : Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := y:ℝ → ℝhy_diff:∀ (t : ℝ), 0 < t → HasDerivAt y (-y t ^ 3) thy_cont:ContinuousWithinAt y (Set.Ici 0) 0hy0:y 0 = 1⊢ Tendsto (fun t => y t * √t) atTop (𝓝 (1 / √2)) All goals completed! 🐙
#6
How produced

Freshly started Kimi Code with a Kimi version of the cameronfreer/lean4-skills plugin. Model K2.7, default settings. Command: /lean4:prove Submission.lean . Then only giving permissions that were asked for. When the context reached 60% of the window the commands "Prepare for compaction", "/compact", "continue" were given. Used one hour of a $20/month subscription.

Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#9
How produced

Freshly started Kimi Code with a Kimi version of the cameronfreer/lean4-skills plugin. Model K2.7, default settings. Command: `/lean4:prove Submission.lean` . Then only giving permissions that were asked for. No input during proof process. The question of golfing was answered positively. Uses small part of a $20/month subscription.

Test problems: multi_hole_helpers_example, noncomputable_hole_example, variable_binder_example, def_hole_example, instance_hole_example, ci_regenerate_main_check, list_append_singleton_length, two_plus_two (8 / 8 solved)

Submission history

First submissionJul 6, 2026
Last submissionJul 10, 2026

Contributors

rwst10
22deepseek-v4 and gpt-5.x2 solved

Problems uniquely solved by this model

Brauer–Suzuki theorem (quaternion Sylow 2-subgroup)
brauer_suzuki

Verso theorem preview

theorem declaration uses `sorry`brauer_suzuki {G : Type*} [Group G] [Finite G] (n : ℕ) (hn : 3 ≤ n) (P : Sylow 2 G) (hquat : Nonempty ((P : Subgroup G) ≃* QuaternionGroup (2 ^ (n - 2)))) (t : G) (ht_mem : t ∈ (P : Subgroup G)) (ht_ord : orderOf t = 2) : (QuotientGroup.mk t : G ⧸ LeanEval.GroupTheory.Defs.oddCore G) ∈ Subgroup.center (G ⧸ LeanEval.GroupTheory.Defs.oddCore G) := G:Type u_1inst✝¹:Group Ginst✝:Finite Gn:ℕhn:3 ≤ nP:Sylow 2 Ghquat:Nonempty (↥↑P ≃* QuaternionGroup (2 ^ (n - 2)))t:Ght_mem:t ∈ ↑Pht_ord:orderOf t = 2⊢ ↑t ∈ Subgroup.center (G ⧸ oddCore G) All goals completed! 🐙
#1

Other solved problems

Feit–Thompson odd-order theorem
feit_thompson

Verso theorem preview

theorem declaration uses `sorry`feit_thompson {G : Type*} [Group G] [Finite G] (_h : Odd (Nat.card G)) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Finite G_h:Odd (Nat.card G)⊢ IsSolvable G All goals completed! 🐙
#2

Submission history

First submissionJul 24, 2026
Last submissionJul 24, 2026

Contributors

ivwumupy2
23Claude Opus 4.71 solved

Other solved problems

Finite Ramsey theorem for graphs
finite_graph_ramsey_theorem

Verso theorem preview

theorem declaration uses `sorry`finite_graph_ramsey_theorem : ∀ r s : ℕ, 2 ≤ r → 2 ≤ s → ∃ n : ℕ, ∀ G : SimpleGraph (Fin n), ¬ G.CliqueFree r ∨ ¬ Gᶜ.CliqueFree s := ⊢ ∀ (r s : ℕ), 2 ≤ r → 2 ≤ s → ∃ n, ∀ (G : SimpleGraph (Fin n)), ¬G.CliqueFree r ∨ ¬Gᶜ.CliqueFree s All goals completed! 🐙
#2

Test problems: list_append_singleton_length, two_plus_two (2 / 8 solved)

Submission history

First submissionApr 30, 2026
Last submissionApr 30, 2026

Contributors

rkirov3kim-em1
24Claude Opus 4.7 + GPT-5.5 (human-in-the-loop)1 solved

Other solved problems

Baer–Suzuki theorem
baer_suzuki

Verso theorem preview

theorem declaration uses `sorry`baer_suzuki {G : Type*} [Group G] [Finite G] {p : ℕ} [Fact p.Prime] (x : G) : x ∈ LeanEval.GroupTheory.Defs.pCore p G ↔ ∀ g : G, IsPGroup p (Subgroup.closure ({x, g * x * g⁻¹} : Set G)) := G:Type u_1inst✝²:Group Ginst✝¹:Finite Gp:ℕinst✝:Fact (Nat.Prime p)x:G⊢ x ∈ pCore p G ↔ ∀ (g : G), IsPGroup p ↥(Subgroup.closure {x, g * x * g⁻¹}) All goals completed! 🐙
#1
How produced

Claude Opus 4.7 + GPT-5.5, human-in-the-loop direction and review. Self-contained: inlines a port of Isaacs, Finite Group Theory Ch. 1-2 (p-core / Fitting subgroup / minimal-normal machinery) under Submission/, since Mathlib has no pCore. Builds on Mathlib rev 5450b53e5ddc; proof routes through Aschbacher §31 (normal closure of x is a p-group)

Submission history

First submissionMay 29, 2026
Last submissionMay 29, 2026

Contributors

yawara1
25GPT-5 Codex + Aristotle1 solved

Other solved problems

Symplectic matrices have determinant 1
symplectic_matrix_det

Verso theorem preview

theorem declaration uses `sorry`symplectic_matrix_det {l R : Type*} [DecidableEq l] [Fintype l] [CommRing R] {A : Matrix (l ⊕ l) (l ⊕ l) R} (_hA : A ∈ Matrix.symplecticGroup l R) : A.det = 1 := l:Type u_1R:Type u_2inst✝²:DecidableEq linst✝¹:Fintype linst✝:CommRing RA:Matrix (l ⊕ l) (l ⊕ l) R_hA:A ∈ symplecticGroup l R⊢ A.det = 1 All goals completed! 🐙
#1
How produced

Produced by GPT-5 Codex in a Lean Eval workspace with many focused Aristotle API runs and external research prompts. The final proof uses a block/paired-minor route for symplectic_matrix_det, with local Lean validation, forbidden-token scan, and axiom audit before submission. Local evidence before submission: lake env lean Submission/Helpers.lean, lake build Submission.Helpers, lake env lean Submission.lean, lake build Submission, and lake build all passed; forbidden constructs were absent; axiom audit reported only [propext, Classical.choice, Quot.sound].

Submission history

First submissionJun 1, 2026
Last submissionJun 1, 2026

Contributors

rishistyping1
26Autoform-Bot1 solved

Other solved problems

Pointwise and Cesàro convergence of Fourier series (Dirichlet, Fejér)
fourier_dirichlet_fejer

Verso theorem preview

/-- **Dirichlet's pointwise convergence theorem** (§46). For every `C¹` 2π-periodic complex function `f`, the symmetric Fourier partial sums `S_N(f)(x)` converge to `f(x)` at every point `x ∈ ℝ`. -/ theorem declaration uses `sorry`dirichlet_pointwise {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hC1 : ContDiff ℝ 1 f) (x : ℝ) : Tendsto (fun N : ℕ => fourierPartialSum f N x) atTop (𝓝 (f x)) := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hC1:ContDiff ℝ 1 fx:ℝ⊢ Tendsto (fun N => fourierPartialSum f N x) atTop (𝓝 (f x)) All goals completed! 🐙
/-- **Fejér's theorem** (§46). For every *continuous* 2π-periodic complex function `f` — without the `C¹` hypothesis of Dirichlet's theorem — the Cesàro means `σ_N(f)` of the symmetric Fourier partial sums converge to `f` uniformly on `ℝ`. -/ theorem declaration uses `sorry`fejer {f : ℝ → ℂ} (_hperiod : Function.Periodic f (2 * Real.pi)) (_hcont : Continuous f) : TendstoUniformly (fun N : ℕ => fourierCesaroMean f N) f atTop := f:ℝ → ℂ_hperiod:Function.Periodic f (2 * Real.pi)_hcont:Continuous f⊢ TendstoUniformly (fun N => fourierCesaroMean f N) f atTop All goals completed! 🐙
#1
How produced

This was created by building on the Atlas-Lean repository.

Submission history

First submissionJun 2, 2026
Last submissionJun 2, 2026

Contributors

niketp031
27Leanstral 1.41 solved

Other solved problems

Abel–Ruffini theorem
abel_ruffini

Verso theorem preview

theorem declaration uses `sorry`abel_ruffini (n : ℕ) (_hn : 1 ≤ n) : (∀ p : ℚ[X], p.natDegree = n → ∀ x : ℂ, aeval x p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 := n:ℕ_hn:1 ≤ n⊢ (∀ (p : ℚ[X]), p.natDegree = n → ∀ (x : ℂ), (aeval x) p = 0 → x ∈ solvableByRad ℚ ℂ) ↔ n ≤ 4 All goals completed! 🐙
#1
How produced

Leanstral 1.4 produced it with the prompt "prove this theorem: {theorem statement}". Cost is 1.95 dollars. It used 4.31 so there was a bit of post-hoc modification to suit the submission. Second submission since the first one has some package import issue

Submission history

First submissionJun 6, 2026
Last submissionJun 6, 2026

Contributors

albertqjiang1
28Claude Opus 4.7, 4.8 and Fable 5 + OSS contributions1 solved

Other solved problems

Jacobian of a compact Riemann surface (Buzzard challenge)
jacobian_challenge_diffgeo

Verso theorem preview

def declaration uses `sorry`genus (X : Type u) [TopologicalSpace X] [T2Space X] [CompactSpace X] [ConnectedSpace X] [Nonempty X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) ω X] : ℕ := sorry
theorem declaration uses `sorry`genus_eq_zero_iff_homeo : genus X = 0 ↔ Nonempty (X ≃ₜ (Metric.sphere (0 : EuclideanSpace ℝ (Fin 3)) 1)) := sorry
def declaration uses `sorry`Jacobian (X : Type u) [TopologicalSpace X] [T2Space X] [CompactSpace X] [ConnectedSpace X] [Nonempty X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) ω X] : Type u := sorry
instance declaration uses `sorry`instAddCommGroup : AddCommGroup (Jacobian X) := sorry
instance declaration uses `sorry`instTopologicalSpace : TopologicalSpace (Jacobian X) := sorry
instance declaration uses `sorry`instT2Space : T2Space (Jacobian X) := sorry
instance declaration uses `sorry`instCompactSpace : CompactSpace (Jacobian X) := sorry
instance declaration uses `sorry`instChartedSpace : ChartedSpace (Fin (genus X) → ℂ) (Jacobian X) := sorry
instance declaration uses `sorry`instIsManifold : IsManifold (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (Jacobian X) := sorry
instance declaration uses `sorry`instLieAddGroup : LieAddGroup (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (Jacobian X) := sorry
def declaration uses `sorry`ofCurve (P : X) : X → Jacobian X := sorry
theorem declaration uses `sorry`ofCurve_contMDiff (P : X) : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (ofCurve P) := sorry
theorem declaration uses `sorry`ofCurve_self (P : X) : ofCurve P P = 0 := sorry
theorem declaration uses `sorry`ofCurve_inj (P : X) (h : 0 < genus X) : Function.Injective (ofCurve P) := sorry
def declaration uses `sorry`pushforward (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : Jacobian X →ₜ+ Jacobian Y := sorry
theorem declaration uses `sorry`pushforward_contMDiff (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ContMDiff (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) (modelWithCornersSelf ℂ (Fin (genus Y) → ℂ)) ω (pushforward f hf) := sorry
theorem declaration uses `sorry`pushforward_id_apply (P : Jacobian X) : pushforward id contMDiff_id P = P := sorry
theorem declaration uses `sorry`pushforward_comp_apply (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (g : Y → Z) (hg : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω g) (P : Jacobian X) : pushforward (g ∘ f) (hg.comp hf) P = pushforward g hg (pushforward f hf P) := sorry
def declaration uses `sorry`pullback (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : Jacobian Y →ₜ+ Jacobian X := sorry
theorem declaration uses `sorry`pullback_contMDiff (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ContMDiff (modelWithCornersSelf ℂ (Fin (genus Y) → ℂ)) (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (pullback f hf) := sorry
theorem declaration uses `sorry`pullback_id_apply (P : Jacobian X) : pullback id contMDiff_id P = P := sorry
theorem declaration uses `sorry`pullback_comp_apply (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (g : Y → Z) (hg : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω g) (P : Jacobian Z) : pullback (g.comp f) (hg.comp hf) P = pullback f hf (pullback g hg P) := sorry
def declaration uses `sorry`degree (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ℕ := sorry -- 0 for constant case
theorem declaration uses `sorry`pushforward_pullback (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (P : Jacobian Y) : pushforward f hf (pullback f hf P) = (degree f hf) • P := sorry
#1
How produced

vanilla Claude Code using Opus 4.7, 4.8 and Fable 5. Partial ports from https://github.com/Brsanch/jacobian-lean-challenge https://github.com/tangentstorm/JacobianChallenge and https://github.com/mrdouglasny/jacobian-challenge.

Submission history

First submissionJun 11, 2026
Last submissionJun 11, 2026

Contributors

rkirov1
29Community multi-model project1 solved

Other solved problems

Jacobian of a compact Riemann surface (Buzzard challenge)
jacobian_challenge_diffgeo

Verso theorem preview

def declaration uses `sorry`genus (X : Type u) [TopologicalSpace X] [T2Space X] [CompactSpace X] [ConnectedSpace X] [Nonempty X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) ω X] : ℕ := sorry
theorem declaration uses `sorry`genus_eq_zero_iff_homeo : genus X = 0 ↔ Nonempty (X ≃ₜ (Metric.sphere (0 : EuclideanSpace ℝ (Fin 3)) 1)) := sorry
def declaration uses `sorry`Jacobian (X : Type u) [TopologicalSpace X] [T2Space X] [CompactSpace X] [ConnectedSpace X] [Nonempty X] [ChartedSpace ℂ X] [IsManifold (modelWithCornersSelf ℂ ℂ) ω X] : Type u := sorry
instance declaration uses `sorry`instAddCommGroup : AddCommGroup (Jacobian X) := sorry
instance declaration uses `sorry`instTopologicalSpace : TopologicalSpace (Jacobian X) := sorry
instance declaration uses `sorry`instT2Space : T2Space (Jacobian X) := sorry
instance declaration uses `sorry`instCompactSpace : CompactSpace (Jacobian X) := sorry
instance declaration uses `sorry`instChartedSpace : ChartedSpace (Fin (genus X) → ℂ) (Jacobian X) := sorry
instance declaration uses `sorry`instIsManifold : IsManifold (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (Jacobian X) := sorry
instance declaration uses `sorry`instLieAddGroup : LieAddGroup (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (Jacobian X) := sorry
def declaration uses `sorry`ofCurve (P : X) : X → Jacobian X := sorry
theorem declaration uses `sorry`ofCurve_contMDiff (P : X) : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (ofCurve P) := sorry
theorem declaration uses `sorry`ofCurve_self (P : X) : ofCurve P P = 0 := sorry
theorem declaration uses `sorry`ofCurve_inj (P : X) (h : 0 < genus X) : Function.Injective (ofCurve P) := sorry
def declaration uses `sorry`pushforward (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : Jacobian X →ₜ+ Jacobian Y := sorry
theorem declaration uses `sorry`pushforward_contMDiff (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ContMDiff (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) (modelWithCornersSelf ℂ (Fin (genus Y) → ℂ)) ω (pushforward f hf) := sorry
theorem declaration uses `sorry`pushforward_id_apply (P : Jacobian X) : pushforward id contMDiff_id P = P := sorry
theorem declaration uses `sorry`pushforward_comp_apply (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (g : Y → Z) (hg : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω g) (P : Jacobian X) : pushforward (g ∘ f) (hg.comp hf) P = pushforward g hg (pushforward f hf P) := sorry
def declaration uses `sorry`pullback (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : Jacobian Y →ₜ+ Jacobian X := sorry
theorem declaration uses `sorry`pullback_contMDiff (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ContMDiff (modelWithCornersSelf ℂ (Fin (genus Y) → ℂ)) (modelWithCornersSelf ℂ (Fin (genus X) → ℂ)) ω (pullback f hf) := sorry
theorem declaration uses `sorry`pullback_id_apply (P : Jacobian X) : pullback id contMDiff_id P = P := sorry
theorem declaration uses `sorry`pullback_comp_apply (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (g : Y → Z) (hg : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω g) (P : Jacobian Z) : pullback (g.comp f) (hg.comp hf) P = pullback f hf (pullback g hg P) := sorry
def declaration uses `sorry`degree (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) : ℕ := sorry -- 0 for constant case
theorem declaration uses `sorry`pushforward_pullback (f : X → Y) (hf : ContMDiff (modelWithCornersSelf ℂ ℂ) (modelWithCornersSelf ℂ ℂ) ω f) (P : Jacobian Y) : pushforward f hf (pullback f hf P) = (degree f hf) • P := sorry
#1
How produced

Multi-agent community project (mrdouglasny/jacobian-challenge) under light human steering; zero human-written Lean. Claude Code with Opus 4.8 and Sonnet 4.6 as the primary models (Claude Fable 5 only in the final ~2 days), Codex/GPT-5.4 rescue passes, and Gemini deep-think axiom vetting; ~8 weeks wall-clock. An independent, complementary solution to Rado Kirov's (the first lean-eval pass): a different construction (period-lattice / H1 route), with explicit positive-genus curve instances (elliptic, hyperelliptic, plane) and a machine-checked finding that Buzzard's 24 requirements are non-categorical, plus the Albanese universal-property repair. All 24 obligations sorry-free and axiom-free [propext, Classical.choice, Quot.sound], confirmed by a local Lean FRO comparator run on main. Builds on Rado Kirov's Dolbeault library (rkirov/jacobian-claude, Apache 2.0, vendored) and Michal Wallace's modules (tangentstorm/JacobianChallenge, MIT).

Submission history

First submissionJun 15, 2026
Last submissionJun 15, 2026

Contributors

mrdouglasny1
30Codex (with human in the loop)1 solved

Problems uniquely solved by this model

Erdős's unit-distance conjecture is false
erdos_unit_distance_conjecture_false

Verso theorem preview

theorem declaration uses `sorry`erdos_unit_distance_conjecture_false : ∃ δ : ℝ, 0 < δ ∧ ∀ N : ℕ, ∃ (n : ℕ) (P : Finset (EuclideanSpace ℝ (Fin 2))), N ≤ n ∧ P.card = n ∧ (n : ℝ) ^ (1 + δ) ≤ (unitDist P : ℝ) := ⊢ ∃ δ, 0 < δ ∧ ∀ (N : ℕ), ∃ n P, N ≤ n ∧ P.card = n ∧ ↑n ^ (1 + δ) ≤ ↑(unitDist P) All goals completed! 🐙
#1

Submission history

First submissionJun 26, 2026
Last submissionJun 26, 2026

Contributors

plby1
31Claude Opus 4.8 + Fable 5 (multi-agent)1 solved

Other solved problems

Chudnovsky formula for pi inverse
chudnovsky_formula_for_pi_inv

Verso theorem preview

theorem declaration uses `sorry`chudnovsky_formula_for_pi_inv : chudnovskySum = π⁻¹ := ⊢ chudnovskySum = π⁻¹ All goals completed! 🐙
#1
How produced

Formalization of Mathlib's chudnovskySum = π⁻¹ following Milla's complex-analytic proof (arXiv:1809.00533v6). Produced by Claude Opus 4.8 and Fable 5 (Claude Code) via multi-agent, file-parallel orchestration across multiple sessions; human direction/review at the architecture level (statement pinning, wave planning, output review). ~20.8k lines across 30 modules under Submission/Pi/. Axiom-clean: #print axioms yields exactly [propext, Classical.choice, Quot.sound] (no sorryAx). Toolchain leanprover/lean4:v4.32.0-rc1, Mathlib rev 360da6f.

Submission history

First submissionJul 5, 2026
Last submissionJul 5, 2026

Contributors

ldct1
32Hy31 solved

Other solved problems

Catalan generating function via compositional inversion
substInv_X_sub_X_sq_eq_catalan

Verso theorem preview

theorem declaration uses `sorry`substInv_X_sub_X_sq_eq_catalan (n : ℕ) : haveI : Invertible (coeff 1 ((X : ℚ⟦X⟧) - X ^ 2)) := n:ℕ⊢ Invertible ((coeff 1) (X - X ^ 2)) n:ℕ⊢ Invertible 1; All goals completed! 🐙 coeff (n + 1) (substInv ((X : ℚ⟦X⟧) - X ^ 2)) = (Nat.choose (2 * n) n : ℚ) / (↑n + 1) := n:ℕ⊢ (coeff (n + 1)) (X - X ^ 2).substInv = ↑((2 * n).choose n) / (↑n + 1) All goals completed! 🐙
#1
How produced

Setup: Opencode with lean-lsp MCP installed, model "Hy3 Free" (under "Opencode Zen". Prompt: "prove Submission.lean". Solve time: 2 hours. Cost: $0.00.

Submission history

First submissionJul 12, 2026
Last submissionJul 12, 2026

Contributors

rwst1
33UNICO/NOUS pipeline - Claude (Anthropic)1 solved

Other solved problems

Morley's trisector theorem
morley_theorem

Verso theorem preview

theorem declaration uses `sorry`morley_theorem (A B C P Q R : LeanEval.Geometry.Morley.Plane) (h : LeanEval.Geometry.Morley.IsMorleyConfiguration A B C P Q R) : LeanEval.Geometry.Morley.IsEquilateralTriple P Q R := A:PlaneB:PlaneC:PlaneP:PlaneQ:PlaneR:Planeh:IsMorleyConfiguration A B C P Q R⊢ IsEquilateralTriple P Q R All goals completed! 🐙
#1
How produced

Produced by an autonomous theorem-proving and certification pipeline built with Claude Code, directed and operated by a non-mathematician developer. All mathematics is AI-generated (Claude by Anthropic); exact polynomial cofactors for the `linear_combination` steps were computed symbolically with sympy and verified before kernel certification. The pipeline accepts nothing unless the local Lean kernel certifies it; this workspace was additionally verified locally with the official comparator at the CI-pinned tool commits (landrun `5ed4a3db`, lean4export `3de59f10`, comparator `71b52ec2`) on the pinned toolchain (v4.32.0-rc1): "Your solution is okay!". The proof is independent of all previously recorded solutions (none were consulted). It transports the benchmark's unoriented-angle configuration, via barycentric coordinates that recover the orientation sign from the convex-hull hypothesis, onto an oriented-ray complex-plane Morley development (Connes-style algebraic core closed by `linear_combination` with exact cofactors; ray-intersection master lemma; `Complex.arg` toolkit). A standalone statement with existence/uniqueness and non-degeneracy companions lives in the same repository (`UnicoProofs/Morley.lean`). Axioms: `propext`, `Classical.choice`, `Quot.sound` only - no `sorry`, no `native_decide`. Estimated cost: one afternoon of wall-clock time end-to-end (bridge design + proof + hardening, ~25 local kernel verdicts), subscription-based model usage, consumer Apple Silicon hardware. Human role: direction, operation and review of process - not of the mathematics. Context: announced on the Lean Zulip ("Morley's theorem (#84): AI-generated proof, please check", 2026-07-13, #AI authored projects), where Jeremy Chen kindly pointed us to this benchmark.

Submission history

First submissionJul 13, 2026
Last submissionJul 13, 2026

Contributors

Solarys4311
34Claude Fable 5 (orchestrating Opus/Sonnet 4.8)1 solved

Other solved problems

Burnside p^a q^b theorem
finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow

Verso theorem preview

theorem declaration uses `sorry`finite_group_isSolvable_of_card_eq_prime_pow_mul_prime_pow {G : Type*} [Group G] [Fintype G] {p q a b : ℕ} (hp : Nat.Prime p) (hq : Nat.Prime q) (hpq : p ≠ q) (hcard : Fintype.card G = p ^ a * q ^ b) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Fintype Gp:ℕq:ℕa:ℕb:ℕhp:Nat.Prime phq:Nat.Prime qhpq:p ≠ qhcard:Fintype.card G = p ^ a * q ^ b⊢ IsSolvable G All goals completed! 🐙
#1
How produced

Claude Fable 5 orchestrating fresh subagents (Fable/Sonnet implementers, adversarial reviewers per task) under human direction (Ian Klatzco, Rado Kirov). Clean-room w.r.t. the seven prior leaderboard solutions.

Submission history

First submissionJul 14, 2026
Last submissionJul 14, 2026

Contributors

ianklatzco1
35GPT-5.6 (human-in-the-loop)1 solved

Other solved problems

Existence of a simple group of order 10200960 with a 22-dim irrep whose tensor square has 4 isotypic components
m23_irrep_tensor_square_decomp

Verso theorem preview

theorem declaration uses `sorry`m23_irrep_tensor_square_decomp : ∃ (G : Type) (_ : Group G) (_ : Fintype G), Fintype.card G = 10200960 ∧ IsSimpleGroup G ∧ ∃ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (ρ : Representation ℂ G V), Module.finrank ℂ V = 22 ∧ ρ.IsIrreducible ∧ (@isotypicComponents (MonoidAlgebra ℂ G) (V ⊗[ℂ] V) _ _ (Module.compHom (V ⊗[ℂ] V) (Representation.asAlgebraHom (ρ.tprod ρ)).toRingHom)).ncard = 4 := ⊢ ∃ G x x_1, Fintype.card G = 10200960 ∧ IsSimpleGroup G ∧ ∃ V x_2 x_3 ρ, Module.finrank ℂ V = 22 ∧ ρ.IsIrreducible ∧ (isotypicComponents (MonoidAlgebra ℂ G) (V ⊗[ℂ] V)).ncard = 4 All goals completed! 🐙
#1
How produced

Running GPT-5.6-sol in opencode with some minimal human guidance. Used most of a $20 Codex sub's weekly limit with estimated API cost of $106.

Submission history

First submissionJul 14, 2026
Last submissionJul 14, 2026

Contributors

matthewjasper1
36Various models ( with human in the loop )1 solved

Other solved problems

Feit–Thompson odd-order theorem
feit_thompson

Verso theorem preview

theorem declaration uses `sorry`feit_thompson {G : Type*} [Group G] [Finite G] (_h : Odd (Nat.card G)) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Finite G_h:Odd (Nat.card G)⊢ IsSolvable G All goals completed! 🐙
#1
How produced

This submission was produced with assistance from Codex 5.5 and 5.6, and Claude Code using Opus 4.7, Opus 4.8, and Fable 5, with human direction, review, and integration throughout.

Submission history

First submissionJul 16, 2026
Last submissionJul 16, 2026

Contributors

yawara1
37Trellis1 solved

Other solved problems

Spencer-Szemerédi-Trotter unit-distance upper bound
unit_distance_upper_bound

Verso theorem preview

theorem declaration uses `sorry`unit_distance_upper_bound : ∃ C : ℝ, 0 < C ∧ ∀ P : Finset (EuclideanSpace ℝ (Fin 2)), (unitDist P : ℝ) ≤ C * (P.card : ℝ) ^ ((4 : ℝ) / 3) := ⊢ ∃ C, 0 < C ∧ ∀ (P : Finset (EuclideanSpace ℝ (Fin 2))), ↑(unitDist P) ≤ C * ↑P.card ^ (4 / 3) All goals completed! 🐙
#1
How produced

Trellis ran autonomously for 3 weeks, from 6/29/2026 through 7/20/2026 on the usage budget of a single ChatGPT Pro account. The only input was a 16-page .tex manuscript based on Székely's paper, “Crossing Numbers and Hard Erdős Problems in Discrete Geometry," but including detailed plane graph preliminaries. The public Trellis repo for this formalization is at https://github.com/wpegden/crossing-consequences/ It includes a cycle-by-cycle view of every edit made over time to the formalization by the Trellis process. The reference paper (prepared by ChatGPT Pro) is also available there. This single Trellis run formalized the following theorems (all consequences of the Crossing Lemma, below): *) Szemerédi–Trotter / Point-line incidences: There is an absolute constant C such that for any n points and ℓ lines in the plane, the number of point-line incidences is at most C·((nℓ)^(2/3) + n + ℓ). *) Rich lines: There is an absolute constant C such that, for 2 ≤ k ≤ √n, the number of lines containing at least k points from an n-point set is at most C·n²/k³. *) Unit distances (the challenge target) There is an absolute constant C such that every set of n points in the plane determines at most C·n^(4/3) unit distances. Trellis formalized these by defining plane drawings of graphs with polygonal edges and formalizing the Crossing Lemma with respect to polygonal drawings: For a simple graph with n ≥ 1 vertices and e edges, if e ≥ 4n then the crossing number satisfies cr(G) ≥ e³/(100n²). From the time Trellis started the challenge to when it finished, the pinned mathlib version changed from v4.30.0-rc2 to v4.32.0-rc1. Thus, after the autonomous run completed, minor edits were necessary to make the project still build against the new challenge mathlib version. These were made by a single codex agent outside the Trellis formalization, given precisely that task. The version before these edits can be seen either at the public repo linked above or as the first commit of this challenge repo.

Submission history

First submissionJul 20, 2026
Last submissionJul 20, 2026

Contributors

wpegden1
38UNICO/NOUS: Claude (Fable 5 + Opus 4.8) + GPT-5.6-Sol1 solved

Problems uniquely solved by this model

Platonic classification
platonic_classification

Verso theorem preview

theorem declaration uses `sorry`platonic_classification : platonicCount 2 = ⊤ ∧ platonicCount 3 = 5 ∧ platonicCount 4 = 6 ∧ ∀ d, 5 ≤ d → platonicCount d = 3 := ⊢ platonicCount 2 = ⊤ ∧ platonicCount 3 = 5 ∧ platonicCount 4 = 6 ∧ ∀ (d : ℕ), 5 ≤ d → platonicCount d = 3 All goals completed! 🐙
#1
How produced

Produced by UNICO/NOUS, a multi-model pipeline: Claude Fable 5 as the orchestrator in the main loop, with GPT-5.6-Sol (via Codex CLI) and Claude Opus 4.8 as prover/builder agents. A Python numerical scout computed exact golden-ratio data (flags, exposing functionals, wall reflections of the 24-cell, 600-cell and 120-cell) that guided the Lean formalization; every step was then proved from scratch and kernel-checked. Human direction throughout (strategy, adversarial review gates, final audits). The final theorem depends only on [propext, Classical.choice, Quot.sound]; no sorry, axiom or native_decide anywhere in the submission. Solved 2026-07-18 to 2026-07-21. (Third attempt: the repo is now structured as a generated workspace importing ChallengeDeps, root theorem in the Submission namespace; Solution.lean builds in a local replica of the CI environment.)

Submission history

First submissionJul 21, 2026
Last submissionJul 21, 2026

Contributors

Solarys4311
39various model (deepseek-v4, gpt-5.*)1 solved

Other solved problems

Feit–Thompson odd-order theorem
feit_thompson

Verso theorem preview

theorem declaration uses `sorry`feit_thompson {G : Type*} [Group G] [Finite G] (_h : Odd (Nat.card G)) : IsSolvable G := G:Type u_1inst✝¹:Group Ginst✝:Finite G_h:Odd (Nat.card G)⊢ IsSolvable G All goals completed! 🐙
#1

Submission history

First submissionJul 24, 2026
Last submissionJul 24, 2026

Contributors

ivwumupy1
40Kimi K2.60 solved

Test problems: two_plus_two (1 / 8 solved)

Submission history

First submissionMay 1, 2026
Last submissionMay 1, 2026

Contributors

kim-em1
41Mistral Large 30 solved

Test problems: two_plus_two (1 / 8 solved)

Submission history

First submissionMay 1, 2026
Last submissionMay 1, 2026

Contributors

kim-em1
42DeepSeek V4 Pro0 solved

Test problems: two_plus_two (1 / 8 solved)

Submission history

First submissionMay 1, 2026
Last submissionMay 1, 2026

Contributors

kim-em1
43Qwen3.6 Max0 solved

Test problems: two_plus_two (1 / 8 solved)

Submission history

First submissionMay 1, 2026
Last submissionMay 1, 2026

Contributors

kim-em1
44Grok 4.30 solved

Test problems: two_plus_two (1 / 8 solved)

Submission history

First submissionMay 1, 2026
Last submissionMay 1, 2026

Contributors

kim-em1
45Claude Sonnet 4.60 solved

Test problems: two_plus_two (1 / 8 solved)

Submission history

First submissionMay 1, 2026
Last submissionMay 1, 2026

Contributors

kim-em1
46Leanstral-26030 solved

Test problems: instance_hole_example, def_hole_example, list_append_singleton_length, ci_regenerate_main_check, two_plus_two (5 / 8 solved)

Submission history

First submissionMay 11, 2026
Last submissionMay 11, 2026

Contributors

sqrt-of-25
Coverage

Per-problem coverage

Which problems each model has solved. Hidden on narrow screens.

ProblemSeed Prover (ByteDance)Aristotle (Harmonic)Stealth ModelGPT-5.6Tau (caj.al)Aleph Prover(logicalintelligence.com)MerLean-ProverAntigravity (Multi-Model Ensemble: Gemini 3.1 Pro, Gemini 3 Flash, Claude 4.6 Sonnet/Opus)Claude Opus 4.7 (1M context)GPT-5.5EVO (deepthought.com.au)Public accepted source + Codex packagingClaude Fable 5DeepSeek V4 FlashGrok 4.5GPT-5.5 CodexGemini 3.1 Pro[submission] aegis-of-the-unit-circle-logosEVOClaude Opus 4.8 (1M context)Kimi K2.7deepseek-v4 and gpt-5.xClaude Opus 4.7Claude Opus 4.7 + GPT-5.5 (human-in-the-loop)GPT-5 Codex + AristotleAutoform-BotLeanstral 1.4Claude Opus 4.7, 4.8 and Fable 5 + OSS contributionsCommunity multi-model projectCodex (with human in the loop)Claude Opus 4.8 + Fable 5 (multi-agent)Hy3UNICO/NOUS pipeline - Claude (Anthropic)Claude Fable 5 (orchestrating Opus/Sonnet 4.8)GPT-5.6 (human-in-the-loop)Various models ( with human in the loop )TrellisUNICO/NOUS: Claude (Fable 5 + Opus 4.8) + GPT-5.6-Solvarious model (deepseek-v4, gpt-5.*)Kimi K2.6Mistral Large 3DeepSeek V4 ProQwen3.6 MaxGrok 4.3Claude Sonnet 4.6Leanstral-2603
No bounded projection from L^1 onto H^1main✓✓✓✓✓—————————————————————————————————————————
Abel–Ruffini theoremmain✓✓✓✓✓✓✓———————————————————✓———————————————————
The alternating sign matrix theoremmain——————————————————————————————————————————————
The Annulus Theorem in dimension 4 (Quinn)main——————————————————————————————————————————————
The Annulus Theorem in dimension ≥ 5 (Kirby)main——————————————————————————————————————————————
Anosov–Bowen shadowing lemmamain✓✓✓✓✓—————————————————————————————————————————
Existence of an aspherical integer homology 4-spheremain——————————————————————————————————————————————
Baer–Suzuki theoremmain✓✓✓✓✓✓—————————————————✓——————————————————————
Baker-Wüstholz theorem on linear forms in logarithmsmain——————————————————————————————————————————————
Balanceable k-bounded partitionsmain✓✓✓✓✓✓————————————————————————————————————————
Bourbaki's locally convex extension of Banach–Alaoglumain✓✓✓✓✓✓✓———————————————————————————————————————
Bauer's uniqueness at extreme pointsmain✓✓✓✓✓—————————————————————————————————————————
Bézout's theorem (projective, with multiplicity)main✓✓————————————————————————————————————————————
Boone–Higman theorem (easy direction)main✓✓✓✓✓—————————————————————————————————————————
Kuznetsov's theorem: finitely presented simple groups have solvable word problemmain✓✓✓✓✓✓✓———————————————————————————————————————
Bourgain's polynomial ergodic theoremmain——————————————————————————————————————————————
Character values of finite groups lie in cyclotomic fieldsmain✓✓✓✓✓✓✓✓✓✓————————————————————————————————————
Brauer–Fowler theoremmain✓✓✓✓✓✓————————————————————————————————————————
Brauer's splitting field theoremmain——————————————————————————————————————————————
Brauer–Suzuki theorem (quaternion Sylow 2-subgroup)main—————————————————————✓————————————————————————
Brouwer fixed-point theoremmain✓✓✓✓✓✓✓———————————————————————————————————————
Brun's theorem (convergence of the twin-prime reciprocal sum)main✓✓✓✓✓—————————————————————————————————————————
Comparison principle for the Dirichlet BVPmain✓✓✓✓✓✓✓✓✓✓✓————✓✓—✓———————————————————————————
Cauchy–Kovalevskaya theoremmain✓✓✓✓——————————————————————————————————————————
Cerf's theorem: every self-diffeomorphism of S3 is smoothly isotopic to a linear isometrymain——————————————————————————————————————————————
Hardy–Littlewood sign-change for the prime race mod 4main——————————————————————————————————————————————
Chen's theoremmain——————————————————————————————————————————————
Choquet's representation theoremmain✓✓✓✓✓—————————————————————————————————————————
Chudnovsky formula for pi inversemain✓—✓———————————————————————————✓———————————————
Coherent cohomology of a proper scheme over ℚ is finite-dimensionalmain——————————————————————————————————————————————
Commuting probabilities are closedmain✓✓————————————————————————————————————————————
Complete reducibility for compact groupsmain✓✓✓✓✓—————————————————————————————————————————
Bing's house with two rooms is contractiblemain✓✓✓✓✓✓—✓—✓————————————————————————————————————
The Conway knot is not smoothly slicemain——————————————————————————————————————————————
The Conway knot is topologically slicemain——————————————————————————————————————————————
Conway–Schneeberger fifteen theoremmain——————————————————————————————————————————————
Polynomial decay rate of y' = -y^3main✓✓✓✓✓✓✓✓✓✓———✓✓—————✓—————————————————————————
Real cyclotomic integer with house in (2, 76/33)main——————————————————————————————————————————————
Real cyclotomic integer with house at most 2main✓✓✓✓✓✓—✓—————————✓————————————————————————————
Darboux's theorem (symplectic forms are locally standard)main✓✓——✓—————————————————————————————————————————
De Branges's theorem (Bieberbach conjecture)main——————————————————————————————————————————————
Dehn–Sommerville equations for simplicial spheresmain✓—————————————————————————————————————————————
Dirichlet eigenvalues of -y'' = lambda y on [0,pi] are n^2main✓✓✓✓✓✓✓✓✓✓✓———————————————————————————————————
Chen theorem for Markoff graphsmain✓✓✓✓✓✓————————————————————————————————————————
Existence of a 779247-dim irreducible e₈-representation with 40 tensor-square isotypic componentsmain——————————————————————————————————————————————
Lai-Sang Young entropy–dimension–Lyapunov theoremmain——————————————————————————————————————————————
Equichordal point theorem (convex curves have a unique equichordal point)main——————————————————————————————————————————————
Erdős's unit-distance conjecture is falsemain—————————————————————————————✓————————————————
Euler–Lagrange equationmain✓✓✓✓✓✓————————————————————————————————————————
Existence of a chiral oriented knotmain✓—————————————————————————————————————————————
Complementary polynomial on the unit circlemain✓✓✓✓✓✓✓✓✓——✓——————————————————————————————————
Existence of a non-isotopic pair of oriented knotsmain✓✓✓✓——————————————————————————————————————————
Existence of a non-isotopic pair of oriented two-component linksmain✓✓✓✓✓✓✓✓——————————————————————————————————————
Existence of a topologically slice, not smoothly slice knotmain——————————————————————————————————————————————
Morrison–Walker Lemma B.0.1: adapting families of maps to open coversmain✓✓✓✓——————————————————————————————————————————
Fang–Xia: tiling of the symmetric group by transpositions implies λ-transitivitymain✓✓✓✓✓—————————————————————————————————————————
Fáry–Milnor theorem (knot total curvature ≤ 4π implies unknotted)main✓—————————————————————————————————————————————
Fatou–Julia / Cantor dichotomymain✓—————————————————————————————————————————————
Feit–Thompson odd-order theoremmain——✓✓—————————————————✓—————————————✓——✓———————
Fermat's Last Theoremmain——————————————————————————————————————————————
Finite Ramsey theorem for graphsmain✓✓✓✓✓✓✓✓✓✓✓——✓✓———✓———✓———————————————————————
Burnside p^a q^b theoremmain✓✓✓✓✓✓✓——————————————————————————✓————————————
Possible orders of 5-transitive finite permutation groupsmain——————————————————————————————————————————————
Pointwise and Cesàro convergence of Fourier series (Dirichlet, Fejér)main✓✓✓✓✓✓✓——————————————————✓————————————————————
Fraser: Fourier decay for finite-field Kakeya sets is q^{-1} and sharpmain✓✓✓✓✓✓————————————————————————————————————————
Frobenius determinant theoremmain✓✓✓✓✓—————————————————————————————————————————
Frobenius's theorem: the Frobenius kernel is normalmain✓✓✓✓✓✓————————————————————————————————————————
Fundamental theorem of topos theorymain✓✓✓✓✓—————————————————————————————————————————
Furstenberg measure-preserving multiple recurrencemain——————————————————————————————————————————————
Furstenberg–Weiss topological multiple recurrence (single-transformation form)main✓✓✓✓✓✓————————————————————————————————————————
Existence of a 64-dim irreducible g₂-representation with 14 tensor-square isotypic componentsmain——————————————————————————————————————————————
Gauss-Wantzel constructible regular polygon theoremmain✓✓✓✓✓—————————————————————————————————————————
Schur-Weyl duality: GL(V) image equals centralizer of S_k imagemain✓✓✓✓✓✓——✓—————————————————————————————————————
Glauberman's Z* theorem for isolated involutionsmain——————————————————————————————————————————————
Gleason's theorem (finite-dimensional)main✓—————————————————————————————————————————————
Gleason's theorem (separable Hilbert space)main✓—————————————————————————————————————————————
The Golod–Shafarevich inequalitymain✓✓✓✓——————————————————————————————————————————
Gorenstein–Walter theorem (dihedral Sylow 2-subgroup)main——————————————————————————————————————————————
Green–Tao theoremmain——————————————————————————————————————————————
Hadwiger's theoremmain——————————————————————————————————————————————
Halmos's generic weak-mixing theoremmain✓✓✓✓——————————————————————————————————————————
Hausdorff moment problem: absolute-continuity criterionmain✓✓✓✓✓—————————————————————————————————————————
The Hausdorff–Hildebrandt–Schoenberg moment theoremmain✓✓✓✓✓—————————————————————————————————————————
The Hausdorff positivity (complete-monotonicity) criterionmain✓✓✓✓✓—————————————————————————————————————————
Gaussian heat kernel solves the 1D heat equationmain✓✓✓✓✓✓✓✓✓✓—✓——————————————————————————————————
Higman's infinite finitely-presented simple groupmain——————————————————————————————————————————————
Hippocrates' theorem on lunesmain✓✓✓✓✓—✓———————————————————————————————————————
Hopf–Rinow theoremmain——————————————————————————————————————————————
The Hopf Umlaufsatz (theorem of turning tangents)main✓—✓✓——————————————————————————————————————————
Hurewicz theorem in degree 1 (H₁ = abelianization of π₁)main✓✓✓✓✓—————————————————————————————————————————
Perron-Frobenius for irreducible nonnegative matricesmain✓✓✓✓✓✓—✓—✓—✓——————————————————————————————————
Onsager's 2D Ising phase transitionmain✓✓————————————————————————————————————————————
Isoperimetric inequality (n-dim, topological-frontier form)main——————————————————————————————————————————————
Jacobian of a smooth proper curve (Merten challenge)main——————————————————————————————————————————————
Jacobian of a compact Riemann surface (Buzzard challenge)main——✓—————————✓——————————————✓✓—————————————————
Jordan–Brouwer separation theoremmain✓✓————————————————————————————————————————————
Jordan curve theoremmain✓—✓—————————✓——————✓——————————————————————————
Jordan normal formmain✓✓✓✓✓—————————————————————————————————————————
Kakutani fixed-point theoremmain✓✓✓✓✓✓————————————————————————————————————————
KAM persistence of an invariant curvemain✓✓————————————————————————————————————————————
Kepler conjecture (optimal sphere packing in ℝ³)main——————————————————————————————————————————————
Kirk's normal-structure fixed point theoremmain✓✓✓✓✓—————————————————————————————————————————
Kolmogorov–Arnold superposition theorem (non-universal Lorentz form)main✓✓✓✓✓—————————————————————————————————————————
Koszul formulamain✓✓✓✓✓✓————————————————————————————————————————
The Landsberg–Schaar relationmain✓✓✓✓✓—————————————————————————————————————————
Lax's approximation theorem for toral homeomorphismsmain✓✓✓✓✓—————————————————————————————————————————
Fundamental theorem of Riemannian geometry (Levi-Civita)main✓✓✓✓✓—————————————————————————————————————————
Lidskii's inequalitymain✓✓✓✓✓✓————————————————————————————————————————
Lidskii–Last eigenvalue-perturbation theoremmain✓✓✓✓✓✓————————————————————————————————————————
Lindemann's theorem (e and π transcendental)main✓✓✓✓✓—————————————————————————————————————————
The Lindemann–Weierstrass theoremmain✓✓✓✓✓—————————————————————————————————————————
Linear ODE with negative-real-part eigenvalues is asymptotically stablemain✓✓✓✓✓✓✓✓—✓————————————————————————————————————
Liouville–Arnold theorem on integrable systemsmain✓✓✓✓—✓————————————————————————————————————————
Linear programming: maximum principle and vertex optimalitymain✓✓✓✓✓✓✓———————————————————————————————————————
Existence of a simple group of order 10200960 with a 22-dim irrep whose tensor square has 4 isotypic componentsmain✓—————————————————————————————————✓———————————
Mandelbar (tricorn) is not path-connected (Hubbard–Schleicher)main——————————————————————————————————————————————
Hausdorff dimension of the Mandelbrot boundary (Shishikura)main——————————————————————————————————————————————
Mandelbrot set is connected (Douady–Hubbard)main✓—————————————————————————————————————————————
Manolescu's disproof of the triangulation conjecturemain——————————————————————————————————————————————
Margulis–Ruelle inequalitymain✓✓————————————————————————————————————————————
Martinet's asymptotically-good totally real towersmain——————————————————————————————————————————————
Mazur's torsion theoremmain——————————————————————————————————————————————
Minkowski-Caratheodory theoremmain✓✓✓✓✓✓✓✓✓✓—✓——————————————————————————————————
Mergelyan's theoremmain✓✓————————————————————————————————————————————
Milnor's exotic 7-spheremain——————————————————————————————————————————————
Monge–Kantorovich existence theoremmain✓✓✓✓✓✓————————————————————————————————————————
Moran's equality for affine-symmetric iterated function systemsmain✓✓✓✓✓—————————————————————————————————————————
Morley's categoricity theoremmain——————————————————————————————————————————————
Morley's trisector theoremmain✓✓✓✓✓———————————————————————————✓—————————————
Morse inequalitiesmain✓—————————————————————————————————————————————
Mountain Pass Theorem (Ambrosetti–Rabinowitz 1973)main✓✓✓✓✓✓————————————————————————————————————————
Cayley graph connected iff generators generate the groupmain✓✓✓✓✓✓✓✓✓✓✓—————✓—————————————————————————————
Nash equilibrium existence theoremmain✓✓✓✓✓✓————————————————————————————————————————
Neukirch–Uchida theoremmain——————————————————————————————————————————————
A 3-manifold group with no faithful representation into GL(4, ℝ)main✓—✓✓——————————————————————————————————————————
Normal spectral theoremmain✓✓✓✓✓—————————————————————————————————————————
Novikov's theorem: the word problem is undecidable for finitely presented groupsmain✓—————————————————————————————————————————————
Nyquist–Shannon sampling theoremmain✓✓✓✓✓—————————————————————————————————————————
Oppenheim's inequality for Hadamard productsmain✓✓✓✓✓✓✓✓✓✓—✓——————————————————————————————————
Ornstein–Weiss ℤᵈ Rokhlin lemmamain✓✓✓✓✓—————————————————————————————————————————
Independence of the parallel postulatemain✓✓✓✓✓✓————————————————————————————————————————
Pascal's theoremmain✓✓✓✓✓—————————————————————————————————————————
Peano existence theorem for ODEsmain✓✓✓✓✓—————————————————————————————————————————
Pell solutions are convergents of √dmain✓✓✓✓✓✓✓———✓———————————————————————————————————
A competition programming problem about permuting a permutation to be unimodalmain✓✓✓✓✓✓————————————————————————————————————————
Pesin entropy formula (symplectic surface case)main——————————————————————————————————————————————
pi_1 of the circle is Zmain✓✓✓✓✓✓✓✓✓✓———————✓————————————————————————————
pi_3 of the 2-sphere is Zmain✓✓————————————————————————————————————————————
pi_(n+1) of S^n is Z/2 for n at least 3main——————————————————————————————————————————————
Pick's theoremmain✓✓✓—————————✓——————✓——————————————————————————
pi_n of the n-sphere is Zmain✓✓————————————————————————————————————————————
Platonic classificationmain—————————————————————————————————————✓————————
3D smooth Poincaré conjecture (Perelman)main——————————————————————————————————————————————
3D topological Poincaré conjecture (Perelman)main——————————————————————————————————————————————
4D topological Poincaré conjecture (Freedman)main——————————————————————————————————————————————
Poincaré–Bendixson theoremmain✓✓————————————————————————————————————————————
Generalized topological Poincaré conjecture in dimensions ≥ 5 (Smale)main——————————————————————————————————————————————
Poincaré–Siegel linearisation theoremmain✓✓✓✓✓—————————————————————————————————————————
Entrywise exponential of a PSD matrix is PSDmain✓✓✓✓✓✓✓✓✓✓✓———————————————————————————————————
Radó's theorem on Riemann surfacesmain✓✓✓—————————✓—————————————————————————————————
Radon transform: Fourier-slice diagonalization and pseudo-inversionmain✓✓✓✓✓—————————————————————————————————————————
Sard's regular-value corollarymain✓✓✓✓✓—————————————————————————————————————————
Lagarias criterion is equivalent to RHmain——————————————————————————————————————————————
Riesz brothers' theoremmain✓✓✓✓✓—————————————————————————————————————————
Riesz's rising sun lemmamain✓✓✓✓✓—————————————————————————————————————————
Rokhlin lemmamain✓✓✓✓✓—————————————————————————————————————————
Rouche theorem via zero countingmain✓✓✓✓✓✓✓✓✓——✓——————————————————————————————————
Runge's theoremmain✓✓✓✓✓✓✓———————————————————————————————————————
Sard's theorem (critical-set image has measure zero)main✓✓✓✓✓—————————————————————————————————————————
Schauder fixed-point theoremmain✓✓✓✓✓✓✓———————————————————————————————————————
Schläfli classification of regular polytopesmain——————————————————————————————————————————————
Schoenflies theoremmain——————————————————————————————————————————————
Schreier's conjecture: outer automorphism group of a finite simple group is solvablemain——————————————————————————————————————————————
Radial symmetry for positive semilinear Poisson solutionsmain✓✓✓✓✓—————————————————————————————————————————
Shafarevich's relation-rank boundmain——————————————————————————————————————————————
Shannon capacity of the pentagonmain✓✓✓✓✓—————————————————————————————————————————
Smale conjecture (Hatcher) in relative parameterized formmain——————————————————————————————————————————————
Pannwitz–Kuperberg quadrisecant theoremmain——————————————————————————————————————————————
Sobolev embedding theorem (Morrey regime)main✓✓✓✓✓—————————————————————————————————————————
Solvable extensions ↔ solvable groups (the missing converse in Abel–Ruffini)main✓✓✓✓✓✓————————————————————————————————————————
230 space groups (Fedorov 1891 / Schoenflies 1891)main——————————————————————————————————————————————
Differentiable sphere theorem (Brendle–Schoen)main——————————————————————————————————————————————
Topological sphere theorem (Berger–Klingenberg–Rauch)main——————————————————————————————————————————————
Local stable/unstable sets at a hyperbolic fixed point (set-level Hadamard–Perron)main✓✓✓✓✓✓————————————————————————————————————————
Strong Subadditivity of von Neumann Entropymain✓✓✓✓✓—————————————————————————————————————————
Sturm's theoremmain✓✓✓✓✓✓————————————————————————————————————————
Sturm separation theoremmain✓✓✓✓✓✓✓✓✓✓✓————✓——————————————————————————————
Catalan generating function via compositional inversionmain✓✓✓✓✓✓✓✓✓✓✓✓—✓✓—————✓——————————✓——————————————
Schur-Weyl duality: S_k image equals centralizer of GL(V) imagemain✓✓✓✓✓✓——✓—————————————————————————————————————
Symplectic matrices have determinant 1main✓✓✓✓✓✓——————————————————✓—————————————————————
Szemerédi's theoremmain——————————————————————————————————————————————
Thue–Siegel–Roth theorem (irrationality measure ≤ 2 for algebraic irrationals)main✓✓————————————————————————————————————————————
Topological classification of surfacesmain——————————————————————————————————————————————
Trace Cayley-Hamilton / Newton identitymain✓✓✓✓✓—————————————————————————————————————————
General recursive equals Turing computablemain✓✓✓✓✓—————————————————————————————————————————
Tverberg's theoremmain✓✓✓✓✓—————————————————————————————————————————
Uniformization theorem for Riemann surfacesmain——————————————————————————————————————————————
Spencer-Szemerédi-Trotter unit-distance upper boundmain——✓—————————————————————————————————✓—————————
Upper bound theorem for geometric simplicial spheres (Stanley 1975)main✓—————————————————————————————————————————————
von Neumann double commutant theoremmain✓✓✓✓✓✓✓✓——————————————————————————————————————
Seventeen wallpaper groups (Pólya–Niggli 1924)main✓✓————————————————————————————————————————————
Watanabe's disproof of the 4-dimensional Smale conjecturemain——————————————————————————————————————————————
Weak Morse inequalitiesmain✓—————————————————————————————————————————————
Weinstein conjecture in dimension three (Taubes 2007)main——————————————————————————————————————————————
Whitney embedding theorem (strong form, dimension 2n)main——————————————————————————————————————————————
Wieferich's theorem g(3) = 9main✓—————————————————————————————————————————————
Wiener's atom-detection formulamain✓✓✓✓✓✓✓———————————————————————————————————————
Wiener's 1/f theoremmain✓✓✓✓✓—————————————————————————————————————————
Wiener–Lévy theoremmain✓✓✓✓✓—————————————————————————————————————————
Wigner semicircle lawmain✓✓✓✓✓—————————————————————————————————————————
Bounded gaps between primesmain——————————————————————————————————————————————
CI regenerate-main checktest—✓✓—✓✓—✓✓✓✓—————✓✓✓—✓————————————————————————✓
def-hole minimal exampletest—✓✓—✓✓—✓✓✓✓—————✓—✓—✓————————————————————————✓
instance-hole minimal exampletest—✓✓—✓✓—✓✓✓✓—————✓—✓—✓————————————————————————✓
Appending a singleton increases the list lengthtest—✓✓—✓✓—✓✓✓✓—————✓—✓—✓—✓——————————————————————✓
multi-hole-with-helpers regression exampletest——✓—✓—————✓—————————✓—————————————————————————
noncomputable-hole minimal exampletest—✓✓—✓———————————————✓—————————————————————————
2 + 2 = 4test—✓✓—✓✓—✓✓✓✓————✓✓✓✓—✓—✓————————————————✓✓✓✓✓✓✓
variable-binder minimal exampletest—✓✓—✓—————✓———————✓—✓—————————————————————————

Welcome to lean-eval, a Lean formalization benchmark and public leaderboard.

You can submit new problems for review, and solutions for existing problems. New problems will be carefully reviewed and added to future benchmark releases if they are accepted. Solutions are automatically verified using comparator and added to the public leaderboard.

This benchmark intends to capture hard Lean formalization problems, consisting of mathematical problems that are currently stateable mostly using existing Mathlib definitions, perhaps with a page or so of additional setup. They should be hard, but usually not open problems: in fact, it's preferred if the problem has a known informal solution which is publicly available.

Our hope is that at launch, the problem set will be mostly, but not entirely, out of reach for current publicly available frontier models, or simple orchestration layers built on top of these. So some genuine mathematical subtlety is required!

It's also important to say what this benchmark is not: we are not trying to capture the ability to write readable or reusable code, or to follow best practices in Lean. In particular, the only requirement for a solution to be accepted is that it is correct and passes the comparator tests.

I'd like to acknowledge the use of Aristotle, Claude Code, and Codex in the preparation of many of the problems here. In particular I should point out that Aristotle has a handicap on the leaderboard: typically, if a single query to Aristotle could resolve a problem, I would deem it too easy and drop it from consideration for the eval set. I think it's a testament to the public service that Aristotle provides that this is both possible, and useful!