task_id stringlengths 18 77 | harbor_task_name stringlengths 43 102 | harbor_task_path stringlengths 24 83 | theorem_name stringlengths 16 71 | theorem_header stringlengths 29 509 | source stringclasses 2
values | source_file stringlengths 39 54 | benchmark_bucket stringclasses 1
value | oracle_status stringclasses 1
value | licence_status stringclasses 4
values | redistribution_status stringclasses 1
value | formal_conjectures_commit stringclasses 1
value | lean_toolchain stringclasses 1
value | mathlib_commit stringclasses 1
value | allow_internet bool 1
class | agent_timeout_sec float64 86.4k 86.4k | verifier_timeout_sec float64 7.2k 7.2k | environment_build_timeout_sec float64 14.4k 14.4k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
arxiv-0911-2077-conjecture6-3-conjecture6-3 | formal-conjectures-bench/arxiv-0911-2077-conjecture6-3-conjecture6-3 | tasks/arxiv-0911-2077-conjecture6-3-conjecture6-3 | Arxiv.arxiv.id0911_2077.conjecture6_3 | theorem arxiv.id0911_2077.conjecture6_3
(p : ℝ) (h_p : p ∈ Set.Ioo 0 (1 / 2)) (k : ℕ) (hk : 0 < k)
(σ : ℝ) (h_σ : σ = (p * (1 - p)).sqrt) :
letI hp' : (⟨p, le_of_lt h_p.1⟩ : ℝ≥0) ≤ 1 := by
have : p ≤ 1 := le_trans (le_of_lt (Set.mem_Ioo.mp h_p).right) (by linarith)
exact this
1 - Φ ((1 / 2 ... | google-deepmind/formal-conjectures | FormalConjectures/Arxiv/0911.2077/Conjecture6_3.lean | gold_solution | bundled | MIT | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-100-erdos-100-piepmeyer | formal-conjectures-bench/erdosproblems-100-erdos-100-piepmeyer | tasks/erdosproblems-100-erdos-100-piepmeyer | Erdos100.erdos_100_piepmeyer | theorem erdos_100_piepmeyer :
∃ A : Finset ℝ², A.card = 9 ∧ DistancesSeparated A ∧
diam (A : Set ℝ²) < 5 | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/100.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-1043-erdos-1043 | formal-conjectures-bench/erdosproblems-1043-erdos-1043 | tasks/erdosproblems-1043-erdos-1043 | Erdos1043.erdos_1043 | theorem erdos_1043 :
answer(False) ↔ ∀ (f : ℂ[X]), f.Monic → f.degree ≥ 1 →
∃ (u : ℂ), ‖u‖ = 1 ∧
volume ((ℝ ∙ u).orthogonalProjection '' levelSet f) ≤ 2 | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/1043.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-1051-erdos-1051 | formal-conjectures-bench/erdosproblems-1051-erdos-1051 | tasks/erdosproblems-1051-erdos-1051 | Erdos1051.erdos_1051 | theorem erdos_1051 :
answer(True) ↔ ∀ (a : ℕ → ℤ), StrictMono a → GrowthCondition a →
Irrational (ErdosSeries a) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/1051.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-1052-even-of-isunitaryperfect | formal-conjectures-bench/erdosproblems-1052-even-of-isunitaryperfect | tasks/erdosproblems-1052-even-of-isunitaryperfect | Erdos1052.even_of_isUnitaryPerfect | theorem even_of_isUnitaryPerfect (n : ℕ) (hn : IsUnitaryPerfect n) : Even n | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/1052.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-1074-ehsnumbers-infinite | formal-conjectures-bench/erdosproblems-1074-ehsnumbers-infinite | tasks/erdosproblems-1074-ehsnumbers-infinite | Erdos1074.erdos_1074.variants.EHSNumbers_infinite | theorem erdos_1074.variants.EHSNumbers_infinite : EHSNumbers.Infinite | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/1074.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-1080-erdos-1080 | formal-conjectures-bench/erdosproblems-1080-erdos-1080 | tasks/erdosproblems-1080-erdos-1080 | Erdos1080.erdos_1080 | theorem erdos_1080 :
answer(False) ↔
∃ c > (0 : ℝ), ∀ (V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V) (X Y : Set V),
IsBipartition G X Y → X.ncard = ⌊(Fintype.card V : ℝ) ^ (2/3 : ℝ)⌋₊ →
G.edgeSet.ncard ≥ c * Fintype.card V →
∃ (v : V) (walk : G.Walk v v), walk.IsCycle ∧ walk.length ... | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/1080.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-1082-ii | formal-conjectures-bench/erdosproblems-1082-ii | tasks/erdosproblems-1082-ii | Erdos1082.erdos_1082.parts.ii | theorem erdos_1082.parts.ii : answer(False) ↔
∀ (A : Finset ℝ²) (hA : A.Nonempty) (hA_n3c : NonTrilinear (A : Set ℝ²)),
∃ (a : ℝ²) (ha : a ∈ A), A.card / 2 ≤ distinctDistancesFrom A a - 1 | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/1082.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-1151-nonempty-fixed-point | formal-conjectures-bench/erdosproblems-1151-nonempty-fixed-point | tasks/erdosproblems-1151-nonempty-fixed-point | Erdos1151NonemptyFixedPoint.erdos_1151_nonempty_fixed_point | theorem erdos_1151_nonempty_fixed_point
{x0 : ℝ} (hx0 : x0 ∈ SpaceI)
{A : Set ℝ} (hA_closed : IsClosed A)
(hA_nonempty : A.Nonempty)
(hA_subset : A ⊆ SpaceI) :
∃ f : SpaceFun,
clusterSet (fun m : ℕ => chebLagEval x0 f (m.succ)) = A | formal-conjectures-bench/local | benchmark_sources/erdos1151/NonemptyFixedPoint.lean | gold_solution | bundled | project-local | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-12-i | formal-conjectures-bench/erdosproblems-12-i | tasks/erdosproblems-12-i | Erdos12.erdos_12.parts.i | theorem erdos_12.parts.i : answer(True) ↔ ∃ (A : Set ℕ), IsGood A ∧
(0 : ℝ) < Filter.atTop.liminf
(fun N => (A ∩ Icc 1 N).ncard / (N : ℝ).sqrt) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/12.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-12-ii | formal-conjectures-bench/erdosproblems-12-ii | tasks/erdosproblems-12-ii | Erdos12.erdos_12.parts.ii | theorem erdos_12.parts.ii : answer(False) ↔ ∃ c > (0 : ℝ), ∀ (A : Set ℕ), IsGood A →
{N : ℕ| (A ∩ Icc 1 N).ncard < (N : ℝ) ^ (1 - c)}.Infinite | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/12.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-12-isgood-example | formal-conjectures-bench/erdosproblems-12-isgood-example | tasks/erdosproblems-12-isgood-example | Erdos12.isGood_example | theorem isGood_example :
IsGood {p ^ 2 | (p : ℕ) (_ : p ≡ 3 [MOD 4]) (_ : p.Prime)} | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/12.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-125-erdos-125 | formal-conjectures-bench/erdosproblems-125-erdos-125 | tasks/erdosproblems-125-erdos-125 | Erdos125.erdos_125 | theorem erdos_125 :
answer(False) ↔ (A + B).HasPosDensity | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/125.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-125-positive-lower-density | formal-conjectures-bench/erdosproblems-125-positive-lower-density | tasks/erdosproblems-125-positive-lower-density | Erdos125.erdos_125.variants.positive_lower_density | theorem erdos_125.variants.positive_lower_density :
answer(False) ↔ 0 < (A + B).lowerDensity | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/125.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-138-difference | formal-conjectures-bench/erdosproblems-138-difference | tasks/erdosproblems-138-difference | Erdos138.erdos_138.variants.difference | theorem erdos_138.variants.difference :
answer(True) ↔ atTop.Tendsto (fun k => (W (k + 1) - W k)) atTop | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/138.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-138-monoapnumber-two-one | formal-conjectures-bench/erdosproblems-138-monoapnumber-two-one | tasks/erdosproblems-138-monoapnumber-two-one | Erdos138.monoAPNumber_two_one | theorem monoAPNumber_two_one : W 1 = 1 | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/138.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-138-monoapnumber-two-two | formal-conjectures-bench/erdosproblems-138-monoapnumber-two-two | tasks/erdosproblems-138-monoapnumber-two-two | Erdos138.monoAPNumber_two_two | theorem monoAPNumber_two_two : W 2 = 3 | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/138.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-152-erdos-152 | formal-conjectures-bench/erdosproblems-152-erdos-152 | tasks/erdosproblems-152-erdos-152 | Erdos152.erdos_152 | theorem erdos_152 : answer(True) ↔ Tendsto f atTop atTop | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/152.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-152-square | formal-conjectures-bench/erdosproblems-152-square | tasks/erdosproblems-152-square | Erdos152.erdos_152.variants.square | theorem erdos_152.variants.square : answer(True) ↔
(fun n => f n : ℕ → ℝ) ≫ (fun n => n ^ 2 : ℕ → ℝ) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/152.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-189-erdos-189 | formal-conjectures-bench/erdosproblems-189-erdos-189 | tasks/erdosproblems-189-erdos-189 | Erdos189.erdos_189 | theorem erdos_189 :
answer(False) ↔ Erdos189For
(fun a b c d ↦
line[ℝ, a, b].direction ⟂ line[ℝ, b, c].direction ∧
line[ℝ, b, c].direction ⟂ line[ℝ, c, d].direction ∧
line[ℝ, c, d].direction ⟂ line[ℝ, d, a].direction)
(fun a b c d ↦ dist a b * dist b c) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/189.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-198-concrete | formal-conjectures-bench/erdosproblems-198-concrete | tasks/erdosproblems-198-concrete | Erdos198.erdos_198.variants.concrete | theorem erdos_198.variants.concrete : ∃ (A : Set ℕ), A = {n ! + n | n} ∧
IsSidon A ∧ (∀ Y, IsAPOfLength Y ⊤ → (A ∩ Y).Nonempty) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/198.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-198-erdos-198 | formal-conjectures-bench/erdosproblems-198-erdos-198 | tasks/erdosproblems-198-erdos-198 | Erdos198.erdos_198 | theorem erdos_198 : (∀ A : Set ℕ, IsSidon A → (∃ Y, IsAPOfLength Y ⊤ ∧ Y ⊆ Aᶜ)) ↔
answer(False) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/198.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-229-erdos-229 | formal-conjectures-bench/erdosproblems-229-erdos-229 | tasks/erdosproblems-229-erdos-229 | Erdos229.erdos_229 | theorem erdos_229 :
letI := Polynomial.algebraPi ℂ ℂ ℂ
answer(True) ↔ ∀ (S : ℕ → Set ℂ), (∀ n, derivedSet (S n) = ∅) →
∃ (f : ℂ → ℂ), Transcendental (Polynomial ℂ) f ∧ Differentiable ℂ f ∧ ∀ n ≥ 1,
∃ k, ∀ z ∈ S n, iteratedDeriv k f z = 0 | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/229.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-233-lower-bound | formal-conjectures-bench/erdosproblems-233-lower-bound | tasks/erdosproblems-233-lower-bound | Erdos233.erdos_233.variants.lower_bound | theorem erdos_233.variants.lower_bound :
(fun (N : ℕ) => N * (log N)^2) =O[atTop]
(fun N => ((∑ n ∈ Finset.range N, (primeGap n) ^ 2) : ℝ)) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/233.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-259-erdos-259 | formal-conjectures-bench/erdosproblems-259-erdos-259 | tasks/erdosproblems-259-erdos-259 | Erdos259.erdos_259 | theorem erdos_259 : Irrational (∑' n : ℕ, (μ n) ^ 2 * n / (2 ^ n)) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/259.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-26-erdos-26 | formal-conjectures-bench/erdosproblems-26-erdos-26 | tasks/erdosproblems-26-erdos-26 | Erdos26.erdos_26 | theorem erdos_26 : answer(False) ↔ ∀ A : ℕ → ℕ, StrictMono A → IsThick A →
∃ k, IsBehrend (A · + k) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/26.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-26-tenenbaum | formal-conjectures-bench/erdosproblems-26-tenenbaum | tasks/erdosproblems-26-tenenbaum | Erdos26.erdos_26.variants.tenenbaum | theorem erdos_26.variants.tenenbaum : answer(False) ↔ ∀ᵉ (A : ℕ → ℕ), StrictMono A → IsThick A →
(∀ ε > (0 : ℝ), ∃ k, IsWeaklyBehrend (A · + k) ε) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/26.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-263-ii | formal-conjectures-bench/erdosproblems-263-ii | tasks/erdosproblems-263-ii | Erdos263.erdos_263.parts.ii | theorem erdos_263.parts.ii : answer(False) ↔
∀ a : ℕ → ℕ,
IsIrrationalitySequence a →
atTop.Tendsto (fun n : ℕ => (a n : ℝ) ^ (1 / (n : ℝ))) atTop | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/263.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-267-specialization-pow-two | formal-conjectures-bench/erdosproblems-267-specialization-pow-two | tasks/erdosproblems-267-specialization-pow-two | Erdos267.erdos_267.variants.specialization_pow_two | theorem erdos_267.variants.specialization_pow_two :
Irrational <| ∑' k, 1 / (Nat.fib <| 2^k) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/267.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-275-erdos-275 | formal-conjectures-bench/erdosproblems-275-erdos-275 | tasks/erdosproblems-275-erdos-275 | Erdos275.erdos_275 | theorem erdos_275 (r : ℕ) (a : Fin r → ℤ) (n : Fin r → ℕ)
(H : ∃ k : ℤ, ∀ x ∈ Ico k (k + 2 ^ r), ∃ i, x ≡ a i [ZMOD n i]) (x : ℤ) :
∃ i, x ≡ a i [ZMOD n i] | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/275.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-316-erdos-316 | formal-conjectures-bench/erdosproblems-316-erdos-316 | tasks/erdosproblems-316-erdos-316 | Erdos316.erdos_316 | theorem erdos_316 : answer(False) ↔ ∀ A : Finset ℕ, 0 ∉ A → 1 ∉ A →
∑ n ∈ A, (1 / n : ℚ) < 2 → ∃ (A₁ A₂ : Finset ℕ),
Disjoint A₁ A₂ ∧ A = A₁ ∪ A₂ ∧
∑ n ∈ A₁, (1 / n : ℚ) < 1 ∧ ∑ n ∈ A₂, (1 / n : ℚ) < 1 | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/316.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-330-upper-density | formal-conjectures-bench/erdosproblems-330-upper-density | tasks/erdosproblems-330-upper-density | Erdos330UpperDensity.erdos_330_upper_density | theorem erdos_330_upper_density :
MainTarget | formal-conjectures-bench/local | benchmark_sources/erdos330/UpperDensity.lean | gold_solution | bundled | project-local | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-331-erdos-331 | formal-conjectures-bench/erdosproblems-331-erdos-331 | tasks/erdosproblems-331-erdos-331 | Erdos331.erdos_331 | theorem erdos_331 :
answer(False) ↔
∀ A B : Set ℕ,
(fun (n : ℕ) ↦ (n : ℝ) ^ (1 / 2 : ℝ)) =O[atTop] (fun (n : ℕ) ↦ (count A n : ℝ)) →
(fun (n : ℕ) ↦ (n : ℝ) ^ (1 / 2 : ℝ)) =O[atTop] (fun (n : ℕ) ↦ (count B n : ℝ)) →
{ s : ℕ × ℕ × ℕ × ℕ | let ⟨a₁, a₂, b₁, b₂⟩ := s
a₁ ∈ A ∧ a₂ ∈ A ∧ b₁ ... | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/331.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-350-erdos-350 | formal-conjectures-bench/erdosproblems-350-erdos-350 | tasks/erdosproblems-350-erdos-350 | Erdos350.erdos_350 | theorem erdos_350 (A : Finset ℕ) (hA : DecidableDistinctSubsetSums A) :
∑ n ∈ A, (1 / n : ℝ) < 2 | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/350.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-355-erdos-355 | formal-conjectures-bench/erdosproblems-355-erdos-355 | tasks/erdosproblems-355-erdos-355 | Erdos355.erdos_355 | theorem erdos_355 :
answer(True) ↔ ∃ A : ℕ → ℕ, IsLacunary A ∧ ∃ u v : ℝ, u < v ∧ ∀ q : ℚ, ↑q ∈ Set.Ioo u v →
q ∈ {∑ a ∈ A', (1 / a : ℚ) | (A' : Finset ℕ) (_ : ↑A' ⊆ Set.range A)} | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/355.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-370-erdos-370 | formal-conjectures-bench/erdosproblems-370-erdos-370 | tasks/erdosproblems-370-erdos-370 | Erdos370.erdos_370 | theorem erdos_370 : answer(True) ↔
{ n | Nat.maxPrimeFac n < √n ∧ Nat.maxPrimeFac (n + 1) < √(n + 1) }.Infinite | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/370.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-379-erdos-379 | formal-conjectures-bench/erdosproblems-379-erdos-379 | tasks/erdosproblems-379-erdos-379 | Erdos379.erdos_379 | theorem erdos_379 : atTop.limsup (fun n => (S n : ℕ∞)) = ⊤ | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/379.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-397-erdos-397 | formal-conjectures-bench/erdosproblems-397-erdos-397 | tasks/erdosproblems-397-erdos-397 | Erdos397.erdos_397 | theorem erdos_397 :
answer(False) ↔
{(M, N) : Finset ℕ × Finset ℕ | Disjoint M N ∧
∏ i ∈ M, centralBinom i = ∏ j ∈ N, centralBinom j}.Finite | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/397.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-399-erdos-399 | formal-conjectures-bench/erdosproblems-399-erdos-399 | tasks/erdosproblems-399-erdos-399 | Erdos399.erdos_399 | theorem erdos_399 : answer(False) ↔
¬ ∃ (n x y k : ℕ), 1 < x * y ∧ 2 < k ∧ (n ! = x ^ k + y ^ k ∨ n ! + y ^ k = x ^ k) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/399.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-645-erdos-645 | formal-conjectures-bench/erdosproblems-645-erdos-645 | tasks/erdosproblems-645-erdos-645 | Erdos645.erdos_645 | theorem erdos_645 (c : ℕ → Bool) : ∃ x d, 0 < x ∧ x < d ∧
(∃ C, c x = C ∧ c (x + d) = C ∧ c (x + 2 * d) = C) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/645.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-707-erdos-707 | formal-conjectures-bench/erdosproblems-707-erdos-707 | tasks/erdosproblems-707-erdos-707 | Erdos707.erdos_707 | theorem erdos_707 : (∀ (A : Set ℕ) (h : A.Finite), IsSidon A →
∃ᵉ (B : Set ℕ) (n > 0), A ⊆ B ∧ IsPerfectDifferenceSet B n) ↔ False | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/707.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-741-i | formal-conjectures-bench/erdosproblems-741-i | tasks/erdosproblems-741-i | Erdos741.erdos_741.parts.i | theorem erdos_741.parts.i : answer(False) ↔ ∀ A : Set ℕ, HasPosDensity (A + A) → ∃ A₁ A₂,
A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ HasPosDensity (A₁ + A₁)
∧ HasPosDensity (A₂ + A₂) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/741.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-741-ii | formal-conjectures-bench/erdosproblems-741-ii | tasks/erdosproblems-741-ii | Erdos741.erdos_741.parts.ii | theorem erdos_741.parts.ii : answer(True) ↔ ∃ A : Set ℕ, IsAddBasisOfOrder (A ∪ {0}) 2 ∧ ∀ A₁ A₂,
A = A₁ ∪ A₂ → Disjoint A₁ A₂ → ¬ (IsSyndetic (A₁ + A₁) ∧ IsSyndetic (A₂ + A₂)) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/741.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-741-upper | formal-conjectures-bench/erdosproblems-741-upper | tasks/erdosproblems-741-upper | Erdos741.erdos_741.variants.upper | theorem erdos_741.variants.upper : answer(True) ↔ ∀ A : Set ℕ, 0 < upperDensity (A + A) → ∃ A₁ A₂,
A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ 0 < upperDensity (A₁ + A₁)
∧ 0 < upperDensity (A₂ + A₂) | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/741.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-828-phi-dvd-self-iff-pow2-pow3 | formal-conjectures-bench/erdosproblems-828-phi-dvd-self-iff-pow2-pow3 | tasks/erdosproblems-828-phi-dvd-self-iff-pow2-pow3 | Erdos828.erdos_828.variants.phi_dvd_self_iff_pow2_pow3 | theorem erdos_828.variants.phi_dvd_self_iff_pow2_pow3 {n : ℕ} :
φ n ∣ n ↔ n ≤ 1 ∨ ∃ᵉ (a > 0) (b), n = 2 ^ a * 3 ^ b | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/828.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-846-erdos-846 | formal-conjectures-bench/erdosproblems-846-erdos-846 | tasks/erdosproblems-846-erdos-846 | Erdos846.erdos_846 | theorem erdos_846 : answer(False) ↔ ∀ᵉ (A : Set ℝ²) (ε > 0), A.Infinite → NonTrilinearFor A ε →
WeaklyNonTrilinear A | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/846.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-897-i | formal-conjectures-bench/erdosproblems-897-i | tasks/erdosproblems-897-i | Erdos897.erdos_897.parts.i | theorem erdos_897.parts.i : answer(False) ↔ ∀ (f : ℕ → ℝ),
(∀ᵉ (a > 0) (b > 0), a.Coprime b → f (a * b) = f a + f b) →
((Filter.atTop ⊓ Filter.principal {(p, k) : ℕ × ℕ | p.Prime}).limsup
(fun (p, k) => (f (p^k) / (p^k : ℝ).log : EReal)) = ⊤) →
Filter.atTop.limsup (fun (n : ℕ) => ((f (n+1) - f n) / (n... | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/897.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-897-ii | formal-conjectures-bench/erdosproblems-897-ii | tasks/erdosproblems-897-ii | Erdos897.erdos_897.parts.ii | theorem erdos_897.parts.ii : answer(False) ↔ ∀ (f : ℕ → ℝ),
(∀ᵉ (a > 0) (b > 0), a.Coprime b → f (a * b) = f a + f b) →
((Filter.atTop ⊓ Filter.principal {(p, k) : ℕ × ℕ | p.Prime}).limsup
(fun (p, k) => (f (p^k) / (p^k : ℝ).log : EReal)) = ⊤) →
Filter.atTop.limsup (fun (n : ℕ) => (f (n+1) / f n : ERe... | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/897.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
erdosproblems-978-allow-fixed-divisors | formal-conjectures-bench/erdosproblems-978-allow-fixed-divisors | tasks/erdosproblems-978-allow-fixed-divisors | Erdos978.erdos_978.variants.allow_fixed_divisors | theorem erdos_978.variants.allow_fixed_divisors : answer(False) ↔
∀ {f : ℤ[X]}, Irreducible f → f.natDegree > 3 →
(¬ ∃ l : ℕ, f.natDegree = 2 ^ l) → 0 < f.leadingCoeff →
(¬ ∃ p : ℕ, p.Prime ∧ ∀ n : ℕ, (p : ℤ) ^ (f.natDegree - 1) ∣ f.eval (n : ℤ)) →
{n : ℕ | Powerfree (f.natDegree - 2) (f.eval (n : ℤ))}.... | google-deepmind/formal-conjectures | FormalConjectures/ErdosProblems/978.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
greensopenproblems-57-green-57 | formal-conjectures-bench/greensopenproblems-57-green-57 | tasks/greensopenproblems-57-green-57 | Green57.green_57 | theorem green_57 :
answer(False) ↔
∀ (G : Type) [AddCommGroup G] [Fintype G] [DecidableEq G],
Φ G = Φ' G | google-deepmind/formal-conjectures | FormalConjectures/GreensOpenProblems/57.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
greensopenproblems-57-z3 | formal-conjectures-bench/greensopenproblems-57-z3 | tasks/greensopenproblems-57-z3 | Green57.green_57.variants.z3 | theorem green_57.variants.z3 :
answer(False) ↔ (Φ (ZMod 3) = Φ' (ZMod 3)) | google-deepmind/formal-conjectures | FormalConjectures/GreensOpenProblems/57.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
greensopenproblems-57-z3-functional | formal-conjectures-bench/greensopenproblems-57-z3-functional | tasks/greensopenproblems-57-z3-functional | Green57.green_57.variants.z3_functional | theorem green_57.variants.z3_functional :
let a : ZMod 3 → ℂ := ![(-1 : ℂ), -3, 3]
answer(True) ↔
supportFn (ZMod 3) a (Φ' (ZMod 3)) < supportFn (ZMod 3) a (Φ (ZMod 3)) | google-deepmind/formal-conjectures | FormalConjectures/GreensOpenProblems/57.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
greensopenproblems-94-green-94-outer-measure | formal-conjectures-bench/greensopenproblems-94-green-94-outer-measure | tasks/greensopenproblems-94-green-94-outer-measure | Green94.green_94_outer_measure | theorem green_94_outer_measure :
answer(False) ↔ ∀ A : Set ℝ,
volume A > 0 →
∃ a b : ℝ, a ≠ 0 ∧ ∀ n : ℕ, a * (1 / 2^n) + b ∈ A | google-deepmind/formal-conjectures | FormalConjectures/GreensOpenProblems/94.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
mathoverflow-10799-mathoverflow-10799 | formal-conjectures-bench/mathoverflow-10799-mathoverflow-10799 | tasks/mathoverflow-10799-mathoverflow-10799 | Mathoverflow10799.mathoverflow_10799 | theorem mathoverflow_10799 : answer(False) ↔
∀ (n : ℕ) (_ : 2 ≤ n)
(F : Finset (Finset (Fin n))) (_ : IsMonotoneIncreasing F)
(s t : ℝ) (_ : 0 < s) (_ : s ≤ t) (_ : t < 1)
(_ : t / s > 1000 * Real.log n),
∃ p, s ≤ p ∧ p ≤ t ∧ IsOptimal p F | google-deepmind/formal-conjectures | FormalConjectures/Mathoverflow/10799.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
mathoverflow-486451-exists-semiring-unique-left-right-maximal-ne | formal-conjectures-bench/mathoverflow-486451-exists-semiring-unique-left-right-maximal-ne | tasks/mathoverflow-486451-exists-semiring-unique-left-right-maximal-ne | Mathoverflow486451.exists_semiring_unique_left_right_maximal_ne | theorem exists_semiring_unique_left_right_maximal_ne :
answer(True) ↔ ∃ (R : Type) (_ : Semiring R) (hI : ∃! I : Ideal R, I.IsMaximal)
(hJ : ∃! J : Ideal Rᵐᵒᵖ, J.IsMaximal),
(hI.choose : Set R) ≠ MulOpposite.op ⁻¹' hJ.choose | google-deepmind/formal-conjectures | FormalConjectures/Mathoverflow/486451.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
openquantumproblems-13-mutuallyunbiasedbases-dim6-bounds | formal-conjectures-bench/openquantumproblems-13-mutuallyunbiasedbases-dim6-bounds | tasks/openquantumproblems-13-mutuallyunbiasedbases-dim6-bounds | OpenQuantumProblem13.mutuallyUnbiasedBases_dim6_bounds | theorem mutuallyUnbiasedBases_dim6_bounds :
HasMUBs 6 3 ∧ ∀ m : ℕ, HasMUBs 6 m → m ≤ 7 | google-deepmind/formal-conjectures | FormalConjectures/OpenQuantumProblems/13.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
openquantumproblems-35-ame-11-5-open | formal-conjectures-bench/openquantumproblems-35-ame-11-5-open | tasks/openquantumproblems-35-ame-11-5-open | OpenQuantumProblem35.ame_11_5_open | theorem ame_11_5_open :
answer(True) ↔ ExistsAME 11 5 | google-deepmind/formal-conjectures | FormalConjectures/OpenQuantumProblems/35.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
paper-casasalvero-positive-char-counterexample | formal-conjectures-bench/paper-casasalvero-positive-char-counterexample | tasks/paper-casasalvero-positive-char-counterexample | CasasAlvero.casas_alvero.positive_char_counterexample | theorem casas_alvero.positive_char_counterexample {p : ℕ} (hp : p.Prime) :
∃ (K : Type*) (_ : Field K) (_ : CharP K p),
let P := X ^ (p + 1) - X ^ p
Monic P ∧ HasCasasAlveroProp P ∧
¬∃ α : K, P = (X - C α) ^ P.natDegree | google-deepmind/formal-conjectures | FormalConjectures/Paper/CasasAlvero.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
paper-claudescycles-cube-hamiltonian-arc-decomposition | formal-conjectures-bench/paper-claudescycles-cube-hamiltonian-arc-decomposition | tasks/paper-claudescycles-cube-hamiltonian-arc-decomposition | ClaudesCycles.cube_hamiltonian_arc_decomposition | theorem cube_hamiltonian_arc_decomposition {m : ℕ} [NeZero m] (hm : Odd m) (hm' : 1 < m) :
HasHamiltonianArcDecomposition m | google-deepmind/formal-conjectures | FormalConjectures/Paper/ClaudesCycles.lean | gold_solution | bundled | no-explicit-license | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
paper-monochromaticquantumgraph-eqsystem-no-solution-even-ge4-d-eq-n-explicit | formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem-no-solution-even-ge4-d-eq-n-explicit | tasks/paper-monochromaticquantumgraph-eqsystem-no-solution-even-ge4-d-eq-n-explicit | MonochromaticQuantumGraph.eqSystem_no_solution_even_ge4_d_eq_n_explicit | theorem eqSystem_no_solution_even_ge4_d_eq_n_explicit :
answer(True) ↔
∀ N : Nat, N ≥ 4 → Even N →
¬ ∃ W : WeightsN N N ℂ, EqSystemN N N W | google-deepmind/formal-conjectures | FormalConjectures/Paper/MonochromaticQuantumGraph.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
paper-monochromaticquantumgraph-eqsystem10-no-solution-d10 | formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem10-no-solution-d10 | tasks/paper-monochromaticquantumgraph-eqsystem10-no-solution-d10 | MonochromaticQuantumGraph.eqSystem10_no_solution_d10 | theorem eqSystem10_no_solution_d10 :
answer(True) ↔
¬ ∃ W : WeightsN 10 10 ℂ, EqSystemN 10 10 W | google-deepmind/formal-conjectures | FormalConjectures/Paper/MonochromaticQuantumGraph.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
paper-monochromaticquantumgraph-eqsystem4-no-solution-d4 | formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem4-no-solution-d4 | tasks/paper-monochromaticquantumgraph-eqsystem4-no-solution-d4 | MonochromaticQuantumGraph.eqSystem4_no_solution_d4 | theorem eqSystem4_no_solution_d4 :
answer(True) ↔
¬ ∃ W : WeightsN 4 4 ℂ, EqSystemN 4 4 W | google-deepmind/formal-conjectures | FormalConjectures/Paper/MonochromaticQuantumGraph.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4 | formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4 | tasks/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4 | MonochromaticQuantumGraph.eqSystem4_no_solution_ge4 | theorem eqSystem4_no_solution_ge4 :
answer(True) ↔
∀ D : Nat, D ≥ 4 →
¬ ∃ W : WeightsN 4 D ℂ, EqSystemN 4 D W | google-deepmind/formal-conjectures | FormalConjectures/Paper/MonochromaticQuantumGraph.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-int | formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-int | tasks/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-int | MonochromaticQuantumGraph.eqSystem4_no_solution_ge4_int | theorem eqSystem4_no_solution_ge4_int :
answer(True) ↔
∀ D : Nat, D ≥ 4 →
¬ ∃ W : WeightsN 4 D ℤ, EqSystemN 4 D W | google-deepmind/formal-conjectures | FormalConjectures/Paper/MonochromaticQuantumGraph.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-real | formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-real | tasks/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-real | MonochromaticQuantumGraph.eqSystem4_no_solution_ge4_real | theorem eqSystem4_no_solution_ge4_real :
answer(True) ↔
∀ D : Nat, D ≥ 4 →
¬ ∃ W : WeightsN 4 D ℝ, EqSystemN 4 D W | google-deepmind/formal-conjectures | FormalConjectures/Paper/MonochromaticQuantumGraph.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-trinary-int | formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-trinary-int | tasks/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-trinary-int | MonochromaticQuantumGraph.eqSystem4_no_solution_ge4_trinary_int | theorem eqSystem4_no_solution_ge4_trinary_int :
answer(True) ↔
∀ D : Nat, D ≥ 4 →
¬ ∃ W : WeightsN 4 D ℤ,
(∀ e, W e = (-1 : ℤ) ∨ W e = 0 ∨ W e = 1) ∧
EqSystemN 4 D W | google-deepmind/formal-conjectures | FormalConjectures/Paper/MonochromaticQuantumGraph.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
paper-monochromaticquantumgraph-eqsystem6-no-solution-d6 | formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem6-no-solution-d6 | tasks/paper-monochromaticquantumgraph-eqsystem6-no-solution-d6 | MonochromaticQuantumGraph.eqSystem6_no_solution_d6 | theorem eqSystem6_no_solution_d6 :
answer(True) ↔
¬ ∃ W : WeightsN 6 6 ℂ, EqSystemN 6 6 W | google-deepmind/formal-conjectures | FormalConjectures/Paper/MonochromaticQuantumGraph.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
paper-monochromaticquantumgraph-eqsystem8-no-solution-d10 | formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem8-no-solution-d10 | tasks/paper-monochromaticquantumgraph-eqsystem8-no-solution-d10 | MonochromaticQuantumGraph.eqSystem8_no_solution_d10 | theorem eqSystem8_no_solution_d10 :
answer(True) ↔
¬ ∃ W : WeightsN 8 10 ℂ, EqSystemN 8 10 W | google-deepmind/formal-conjectures | FormalConjectures/Paper/MonochromaticQuantumGraph.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
wikipedia-agohgiuga-isweakgiuga-iff-prime-dvd | formal-conjectures-bench/wikipedia-agohgiuga-isweakgiuga-iff-prime-dvd | tasks/wikipedia-agohgiuga-isweakgiuga-iff-prime-dvd | AgohGiuga.isWeakGiuga_iff_prime_dvd | theorem isWeakGiuga_iff_prime_dvd {n : ℕ} (hn : n.Composite) :
IsWeakGiuga n ↔ ∀ p ∈ n.primeFactors, p ∣ (n / p - 1) | google-deepmind/formal-conjectures | FormalConjectures/Wikipedia/AgohGiuga.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
wikipedia-eulerbrick-cuboidone | formal-conjectures-bench/wikipedia-eulerbrick-cuboidone | tasks/wikipedia-eulerbrick-cuboidone | EulerBrick.cuboidOne | theorem cuboidOne : CuboidOne | google-deepmind/formal-conjectures | FormalConjectures/Wikipedia/EulerBrick.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
wikipedia-legendreconjecture-bounded-gap-legendre | formal-conjectures-bench/wikipedia-legendreconjecture-bounded-gap-legendre | tasks/wikipedia-legendreconjecture-bounded-gap-legendre | LegendreConjecture.bounded_gap_legendre | theorem bounded_gap_legendre
(H : ∃ c > 0, ∀ᶠ n in atTop, (n + 1).nth Nat.Prime - n.nth Nat.Prime <
(n.nth Nat.Prime : ℝ) ^ (1 / (2 : ℝ) - c)) :
∀ᶠ n in atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p | google-deepmind/formal-conjectures | FormalConjectures/Wikipedia/LegendreConjecture.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
wikipedia-perfectnumbers-euler-form | formal-conjectures-bench/wikipedia-perfectnumbers-euler-form | tasks/wikipedia-perfectnumbers-euler-form | PerfectNumbers.odd_perfect_number.euler_form | theorem odd_perfect_number.euler_form (n : ℕ) (hn : Odd n) (hp : Perfect n) :
∃ (p m α : ℕ),
p.Prime ∧
p ≡ 1 [ZMOD 4] ∧
α ≡ 1 [ZMOD 4] ∧
¬ p ∣ m ∧
n = p^α * m^2 | google-deepmind/formal-conjectures | FormalConjectures/Wikipedia/PerfectNumbers.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
wikipedia-wolstenholmeprime-wolstenholme-prime-16483 | formal-conjectures-bench/wikipedia-wolstenholmeprime-wolstenholme-prime-16483 | tasks/wikipedia-wolstenholmeprime-wolstenholme-prime-16483 | WolstenholmePrime.wolstenholme_prime_16483 | theorem wolstenholme_prime_16483 : IsWolstenholmePrime 16843 | google-deepmind/formal-conjectures | FormalConjectures/Wikipedia/WolstenholmePrime.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
wikipedia-wolstenholmeprime-wolstenholme-theorem | formal-conjectures-bench/wikipedia-wolstenholmeprime-wolstenholme-theorem | tasks/wikipedia-wolstenholmeprime-wolstenholme-theorem | WolstenholmePrime.wolstenholme_theorem | theorem wolstenholme_theorem (p : ℕ) (h : p > 3) (hp : Nat.Prime p) :
(2 * p - 1).choose (p - 1) ≡ 1 [MOD p ^ 3] | google-deepmind/formal-conjectures | FormalConjectures/Wikipedia/WolstenholmePrime.lean | gold_solution | bundled | Apache-2.0 | bundled | 233a10e857ef78e79fd9fe661d37db724089170a | leanprover/lean4:v4.27.0 | a3a10db0e9d66acbebf76c5e6a135066525ac900 | false | 86,400 | 7,200 | 14,400 |
FormalConjectures-Bench Gold
FormalConjectures-Bench Gold v1.1.0 is a set of 74 offline Lean 4 theorem-proving tasks with bundled oracle solutions. The executable benchmark is distributed as a Harbor dataset; this Hugging Face dataset is the metadata and discovery mirror.
The benchmark is intended for evaluation or RL. Benchmark target files and oracle solutions should not be included in training corpora.
Running
Install Harbor and run the registry-backed dataset:
uv tool install harbor
harbor run \
--dataset formal-conjectures-gold@1.1.0 \
--registry-url https://raw.githubusercontent.com/AllenGrahamHart/FormalConjectures-Bench/v1.1.0/registry.json \
--agent terminus-2 \
--model <provider/model>
Run a single task:
harbor run \
--dataset formal-conjectures-gold@1.1.0 \
--registry-url https://raw.githubusercontent.com/AllenGrahamHart/FormalConjectures-Bench/v1.1.0/registry.json \
--task-name erdosproblems-399-erdos-399 \
--agent terminus-2 \
--model <provider/model>
The release defaults are intentionally generous: 24 hours for agent execution, 2 hours for verification, and 4 hours for environment builds. Use Harbor's timeout multiplier to scale them:
harbor run \
--dataset formal-conjectures-gold@1.1.0 \
--registry-url https://raw.githubusercontent.com/AllenGrahamHart/FormalConjectures-Bench/v1.1.0/registry.json \
--timeout-multiplier 0.25
Files
tasks.jsonlcontains one metadata row per task.- The executable task directories live in the GitHub repository.
registry.jsonin the GitHub repository is the Harbor dataset registry.
Versions
- Dataset:
formal-conjectures-gold@1.1.0 - Task source commit:
a3674901510f2f8d8a8c1de0c28c568fcd48828e - Lean toolchain:
leanprover/lean4:v4.27.0 - Mathlib commit:
a3a10db0e9d66acbebf76c5e6a135066525ac900 - Formal Conjectures commit:
233a10e857ef78e79fd9fe661d37db724089170a
Licensing
The task statements derive from Google DeepMind's Formal Conjectures project.
Bundled oracle metadata is recorded per task in tasks.jsonl; the source
repository also includes manifest/licence_review.csv.
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