| {"task_id": "arxiv-0911-2077-conjecture6-3-conjecture6-3", "harbor_task_name": "formal-conjectures-bench/arxiv-0911-2077-conjecture6-3-conjecture6-3", "harbor_task_path": "tasks/arxiv-0911-2077-conjecture6-3-conjecture6-3", "theorem_name": "Arxiv.arxiv.id0911_2077.conjecture6_3", "theorem_header": "theorem arxiv.id0911_2077.conjecture6_3\n (p : ℝ) (h_p : p ∈ Set.Ioo 0 (1 / 2)) (k : ℕ) (hk : 0 < k)\n (σ : ℝ) (h_σ : σ = (p * (1 - p)).sqrt) :\n letI hp' : (⟨p, le_of_lt h_p.1⟩ : ℝ≥0) ≤ 1 := by\n have : p ≤ 1 := le_trans (le_of_lt (Set.mem_Ioo.mp h_p).right) (by linarith)\n exact this\n 1 - Φ ((1 / 2 - p) * sqrt (2 * k : ℝ≥0) / σ)\n + (1 / 2) * ((2 * k).choose k) * σ ^ (2 * k)\n ≤ ((PMF.binomial (⟨p, le_of_lt h_p.1⟩) hp' (2 * k)).toMeasure\n (Set.Ici ⟨k, by omega⟩)).toReal", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Arxiv/0911.2077/Conjecture6_3.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "MIT", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-100-erdos-100-piepmeyer", "harbor_task_name": "formal-conjectures-bench/erdosproblems-100-erdos-100-piepmeyer", "harbor_task_path": "tasks/erdosproblems-100-erdos-100-piepmeyer", "theorem_name": "Erdos100.erdos_100_piepmeyer", "theorem_header": "theorem erdos_100_piepmeyer :\n ∃ A : Finset ℝ², A.card = 9 ∧ DistancesSeparated A ∧\n diam (A : Set ℝ²) < 5", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/100.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-1043-erdos-1043", "harbor_task_name": "formal-conjectures-bench/erdosproblems-1043-erdos-1043", "harbor_task_path": "tasks/erdosproblems-1043-erdos-1043", "theorem_name": "Erdos1043.erdos_1043", "theorem_header": "theorem erdos_1043 :\n answer(False) ↔ ∀ (f : ℂ[X]), f.Monic → f.degree ≥ 1 →\n ∃ (u : ℂ), ‖u‖ = 1 ∧\n volume ((ℝ ∙ u).orthogonalProjection '' levelSet f) ≤ 2", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/1043.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-1051-erdos-1051", "harbor_task_name": "formal-conjectures-bench/erdosproblems-1051-erdos-1051", "harbor_task_path": "tasks/erdosproblems-1051-erdos-1051", "theorem_name": "Erdos1051.erdos_1051", "theorem_header": "theorem erdos_1051 :\n answer(True) ↔ ∀ (a : ℕ → ℤ), StrictMono a → GrowthCondition a →\n Irrational (ErdosSeries a)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/1051.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-1052-even-of-isunitaryperfect", "harbor_task_name": "formal-conjectures-bench/erdosproblems-1052-even-of-isunitaryperfect", "harbor_task_path": "tasks/erdosproblems-1052-even-of-isunitaryperfect", "theorem_name": "Erdos1052.even_of_isUnitaryPerfect", "theorem_header": "theorem even_of_isUnitaryPerfect (n : ℕ) (hn : IsUnitaryPerfect n) : Even n", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/1052.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-1074-ehsnumbers-infinite", "harbor_task_name": "formal-conjectures-bench/erdosproblems-1074-ehsnumbers-infinite", "harbor_task_path": "tasks/erdosproblems-1074-ehsnumbers-infinite", "theorem_name": "Erdos1074.erdos_1074.variants.EHSNumbers_infinite", "theorem_header": "theorem erdos_1074.variants.EHSNumbers_infinite : EHSNumbers.Infinite", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/1074.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-1080-erdos-1080", "harbor_task_name": "formal-conjectures-bench/erdosproblems-1080-erdos-1080", "harbor_task_path": "tasks/erdosproblems-1080-erdos-1080", "theorem_name": "Erdos1080.erdos_1080", "theorem_header": "theorem erdos_1080 :\n answer(False) ↔\n ∃ c > (0 : ℝ), ∀ (V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V) (X Y : Set V),\n IsBipartition G X Y → X.ncard = ⌊(Fintype.card V : ℝ) ^ (2/3 : ℝ)⌋₊ →\n G.edgeSet.ncard ≥ c * Fintype.card V →\n ∃ (v : V) (walk : G.Walk v v), walk.IsCycle ∧ walk.length = 6", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/1080.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-1082-ii", "harbor_task_name": "formal-conjectures-bench/erdosproblems-1082-ii", "harbor_task_path": "tasks/erdosproblems-1082-ii", "theorem_name": "Erdos1082.erdos_1082.parts.ii", "theorem_header": "theorem erdos_1082.parts.ii : answer(False) ↔\n ∀ (A : Finset ℝ²) (hA : A.Nonempty) (hA_n3c : NonTrilinear (A : Set ℝ²)),\n ∃ (a : ℝ²) (ha : a ∈ A), A.card / 2 ≤ distinctDistancesFrom A a - 1", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/1082.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-1151-nonempty-fixed-point", "harbor_task_name": "formal-conjectures-bench/erdosproblems-1151-nonempty-fixed-point", "harbor_task_path": "tasks/erdosproblems-1151-nonempty-fixed-point", "theorem_name": "Erdos1151NonemptyFixedPoint.erdos_1151_nonempty_fixed_point", "theorem_header": "theorem erdos_1151_nonempty_fixed_point\n {x0 : ℝ} (hx0 : x0 ∈ SpaceI)\n {A : Set ℝ} (hA_closed : IsClosed A)\n (hA_nonempty : A.Nonempty)\n (hA_subset : A ⊆ SpaceI) :\n ∃ f : SpaceFun,\n clusterSet (fun m : ℕ => chebLagEval x0 f (m.succ)) = A", "source": "formal-conjectures-bench/local", "source_file": "benchmark_sources/erdos1151/NonemptyFixedPoint.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "project-local", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-12-i", "harbor_task_name": "formal-conjectures-bench/erdosproblems-12-i", "harbor_task_path": "tasks/erdosproblems-12-i", "theorem_name": "Erdos12.erdos_12.parts.i", "theorem_header": "theorem erdos_12.parts.i : answer(True) ↔ ∃ (A : Set ℕ), IsGood A ∧\n (0 : ℝ) < Filter.atTop.liminf\n (fun N => (A ∩ Icc 1 N).ncard / (N : ℝ).sqrt)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/12.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-12-ii", "harbor_task_name": "formal-conjectures-bench/erdosproblems-12-ii", "harbor_task_path": "tasks/erdosproblems-12-ii", "theorem_name": "Erdos12.erdos_12.parts.ii", "theorem_header": "theorem erdos_12.parts.ii : answer(False) ↔ ∃ c > (0 : ℝ), ∀ (A : Set ℕ), IsGood A →\n {N : ℕ| (A ∩ Icc 1 N).ncard < (N : ℝ) ^ (1 - c)}.Infinite", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/12.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-12-isgood-example", "harbor_task_name": "formal-conjectures-bench/erdosproblems-12-isgood-example", "harbor_task_path": "tasks/erdosproblems-12-isgood-example", "theorem_name": "Erdos12.isGood_example", "theorem_header": "theorem isGood_example :\n IsGood {p ^ 2 | (p : ℕ) (_ : p ≡ 3 [MOD 4]) (_ : p.Prime)}", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/12.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-125-erdos-125", "harbor_task_name": "formal-conjectures-bench/erdosproblems-125-erdos-125", "harbor_task_path": "tasks/erdosproblems-125-erdos-125", "theorem_name": "Erdos125.erdos_125", "theorem_header": "theorem erdos_125 :\n answer(False) ↔ (A + B).HasPosDensity", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/125.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-125-positive-lower-density", "harbor_task_name": "formal-conjectures-bench/erdosproblems-125-positive-lower-density", "harbor_task_path": "tasks/erdosproblems-125-positive-lower-density", "theorem_name": "Erdos125.erdos_125.variants.positive_lower_density", "theorem_header": "theorem erdos_125.variants.positive_lower_density :\n answer(False) ↔ 0 < (A + B).lowerDensity", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/125.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-138-difference", "harbor_task_name": "formal-conjectures-bench/erdosproblems-138-difference", "harbor_task_path": "tasks/erdosproblems-138-difference", "theorem_name": "Erdos138.erdos_138.variants.difference", "theorem_header": "theorem erdos_138.variants.difference :\n answer(True) ↔ atTop.Tendsto (fun k => (W (k + 1) - W k)) atTop", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/138.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-138-monoapnumber-two-one", "harbor_task_name": "formal-conjectures-bench/erdosproblems-138-monoapnumber-two-one", "harbor_task_path": "tasks/erdosproblems-138-monoapnumber-two-one", "theorem_name": "Erdos138.monoAPNumber_two_one", "theorem_header": "theorem monoAPNumber_two_one : W 1 = 1", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/138.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-138-monoapnumber-two-two", "harbor_task_name": "formal-conjectures-bench/erdosproblems-138-monoapnumber-two-two", "harbor_task_path": "tasks/erdosproblems-138-monoapnumber-two-two", "theorem_name": "Erdos138.monoAPNumber_two_two", "theorem_header": "theorem monoAPNumber_two_two : W 2 = 3", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/138.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-152-erdos-152", "harbor_task_name": "formal-conjectures-bench/erdosproblems-152-erdos-152", "harbor_task_path": "tasks/erdosproblems-152-erdos-152", "theorem_name": "Erdos152.erdos_152", "theorem_header": "theorem erdos_152 : answer(True) ↔ Tendsto f atTop atTop", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/152.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-152-square", "harbor_task_name": "formal-conjectures-bench/erdosproblems-152-square", "harbor_task_path": "tasks/erdosproblems-152-square", "theorem_name": "Erdos152.erdos_152.variants.square", "theorem_header": "theorem erdos_152.variants.square : answer(True) ↔\n (fun n => f n : ℕ → ℝ) ≫ (fun n => n ^ 2 : ℕ → ℝ)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/152.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-189-erdos-189", "harbor_task_name": "formal-conjectures-bench/erdosproblems-189-erdos-189", "harbor_task_path": "tasks/erdosproblems-189-erdos-189", "theorem_name": "Erdos189.erdos_189", "theorem_header": "theorem erdos_189 :\n answer(False) ↔ Erdos189For\n (fun a b c d ↦\n line[ℝ, a, b].direction ⟂ line[ℝ, b, c].direction ∧\n line[ℝ, b, c].direction ⟂ line[ℝ, c, d].direction ∧\n line[ℝ, c, d].direction ⟂ line[ℝ, d, a].direction)\n (fun a b c d ↦ dist a b * dist b c)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/189.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-198-concrete", "harbor_task_name": "formal-conjectures-bench/erdosproblems-198-concrete", "harbor_task_path": "tasks/erdosproblems-198-concrete", "theorem_name": "Erdos198.erdos_198.variants.concrete", "theorem_header": "theorem erdos_198.variants.concrete : ∃ (A : Set ℕ), A = {n ! + n | n} ∧\n IsSidon A ∧ (∀ Y, IsAPOfLength Y ⊤ → (A ∩ Y).Nonempty)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/198.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-198-erdos-198", "harbor_task_name": "formal-conjectures-bench/erdosproblems-198-erdos-198", "harbor_task_path": "tasks/erdosproblems-198-erdos-198", "theorem_name": "Erdos198.erdos_198", "theorem_header": "theorem erdos_198 : (∀ A : Set ℕ, IsSidon A → (∃ Y, IsAPOfLength Y ⊤ ∧ Y ⊆ Aᶜ)) ↔\n answer(False)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/198.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-229-erdos-229", "harbor_task_name": "formal-conjectures-bench/erdosproblems-229-erdos-229", "harbor_task_path": "tasks/erdosproblems-229-erdos-229", "theorem_name": "Erdos229.erdos_229", "theorem_header": "theorem erdos_229 :\n letI := Polynomial.algebraPi ℂ ℂ ℂ\n answer(True) ↔ ∀ (S : ℕ → Set ℂ), (∀ n, derivedSet (S n) = ∅) →\n ∃ (f : ℂ → ℂ), Transcendental (Polynomial ℂ) f ∧ Differentiable ℂ f ∧ ∀ n ≥ 1,\n ∃ k, ∀ z ∈ S n, iteratedDeriv k f z = 0", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/229.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-233-lower-bound", "harbor_task_name": "formal-conjectures-bench/erdosproblems-233-lower-bound", "harbor_task_path": "tasks/erdosproblems-233-lower-bound", "theorem_name": "Erdos233.erdos_233.variants.lower_bound", "theorem_header": "theorem erdos_233.variants.lower_bound :\n (fun (N : ℕ) => N * (log N)^2) =O[atTop]\n (fun N => ((∑ n ∈ Finset.range N, (primeGap n) ^ 2) : ℝ))", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/233.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-259-erdos-259", "harbor_task_name": "formal-conjectures-bench/erdosproblems-259-erdos-259", "harbor_task_path": "tasks/erdosproblems-259-erdos-259", "theorem_name": "Erdos259.erdos_259", "theorem_header": "theorem erdos_259 : Irrational (∑' n : ℕ, (μ n) ^ 2 * n / (2 ^ n))", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/259.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-26-erdos-26", "harbor_task_name": "formal-conjectures-bench/erdosproblems-26-erdos-26", "harbor_task_path": "tasks/erdosproblems-26-erdos-26", "theorem_name": "Erdos26.erdos_26", "theorem_header": "theorem erdos_26 : answer(False) ↔ ∀ A : ℕ → ℕ, StrictMono A → IsThick A →\n ∃ k, IsBehrend (A · + k)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/26.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-26-tenenbaum", "harbor_task_name": "formal-conjectures-bench/erdosproblems-26-tenenbaum", "harbor_task_path": "tasks/erdosproblems-26-tenenbaum", "theorem_name": "Erdos26.erdos_26.variants.tenenbaum", "theorem_header": "theorem erdos_26.variants.tenenbaum : answer(False) ↔ ∀ᵉ (A : ℕ → ℕ), StrictMono A → IsThick A →\n (∀ ε > (0 : ℝ), ∃ k, IsWeaklyBehrend (A · + k) ε)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/26.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-263-ii", "harbor_task_name": "formal-conjectures-bench/erdosproblems-263-ii", "harbor_task_path": "tasks/erdosproblems-263-ii", "theorem_name": "Erdos263.erdos_263.parts.ii", "theorem_header": "theorem erdos_263.parts.ii : answer(False) ↔\n ∀ a : ℕ → ℕ,\n IsIrrationalitySequence a →\n atTop.Tendsto (fun n : ℕ => (a n : ℝ) ^ (1 / (n : ℝ))) atTop", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/263.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-267-specialization-pow-two", "harbor_task_name": "formal-conjectures-bench/erdosproblems-267-specialization-pow-two", "harbor_task_path": "tasks/erdosproblems-267-specialization-pow-two", "theorem_name": "Erdos267.erdos_267.variants.specialization_pow_two", "theorem_header": "theorem erdos_267.variants.specialization_pow_two :\n Irrational <| ∑' k, 1 / (Nat.fib <| 2^k)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/267.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-275-erdos-275", "harbor_task_name": "formal-conjectures-bench/erdosproblems-275-erdos-275", "harbor_task_path": "tasks/erdosproblems-275-erdos-275", "theorem_name": "Erdos275.erdos_275", "theorem_header": "theorem erdos_275 (r : ℕ) (a : Fin r → ℤ) (n : Fin r → ℕ)\n (H : ∃ k : ℤ, ∀ x ∈ Ico k (k + 2 ^ r), ∃ i, x ≡ a i [ZMOD n i]) (x : ℤ) :\n ∃ i, x ≡ a i [ZMOD n i]", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/275.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-316-erdos-316", "harbor_task_name": "formal-conjectures-bench/erdosproblems-316-erdos-316", "harbor_task_path": "tasks/erdosproblems-316-erdos-316", "theorem_name": "Erdos316.erdos_316", "theorem_header": "theorem erdos_316 : answer(False) ↔ ∀ A : Finset ℕ, 0 ∉ A → 1 ∉ A →\n ∑ n ∈ A, (1 / n : ℚ) < 2 → ∃ (A₁ A₂ : Finset ℕ),\n Disjoint A₁ A₂ ∧ A = A₁ ∪ A₂ ∧\n ∑ n ∈ A₁, (1 / n : ℚ) < 1 ∧ ∑ n ∈ A₂, (1 / n : ℚ) < 1", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/316.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-330-upper-density", "harbor_task_name": "formal-conjectures-bench/erdosproblems-330-upper-density", "harbor_task_path": "tasks/erdosproblems-330-upper-density", "theorem_name": "Erdos330UpperDensity.erdos_330_upper_density", "theorem_header": "theorem erdos_330_upper_density :\n MainTarget", "source": "formal-conjectures-bench/local", "source_file": "benchmark_sources/erdos330/UpperDensity.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "project-local", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-331-erdos-331", "harbor_task_name": "formal-conjectures-bench/erdosproblems-331-erdos-331", "harbor_task_path": "tasks/erdosproblems-331-erdos-331", "theorem_name": "Erdos331.erdos_331", "theorem_header": "theorem erdos_331 :\n answer(False) ↔\n ∀ A B : Set ℕ,\n (fun (n : ℕ) ↦ (n : ℝ) ^ (1 / 2 : ℝ)) =O[atTop] (fun (n : ℕ) ↦ (count A n : ℝ)) →\n (fun (n : ℕ) ↦ (n : ℝ) ^ (1 / 2 : ℝ)) =O[atTop] (fun (n : ℕ) ↦ (count B n : ℝ)) →\n { s : ℕ × ℕ × ℕ × ℕ | let ⟨a₁, a₂, b₁, b₂⟩ := s\n a₁ ∈ A ∧ a₂ ∈ A ∧ b₁ ∈ B ∧ b₂ ∈ B ∧\n a₁ ≠ a₂ ∧ a₁ + b₂ = a₂ + b₁ }.Infinite", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/331.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-350-erdos-350", "harbor_task_name": "formal-conjectures-bench/erdosproblems-350-erdos-350", "harbor_task_path": "tasks/erdosproblems-350-erdos-350", "theorem_name": "Erdos350.erdos_350", "theorem_header": "theorem erdos_350 (A : Finset ℕ) (hA : DecidableDistinctSubsetSums A) :\n ∑ n ∈ A, (1 / n : ℝ) < 2", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/350.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-355-erdos-355", "harbor_task_name": "formal-conjectures-bench/erdosproblems-355-erdos-355", "harbor_task_path": "tasks/erdosproblems-355-erdos-355", "theorem_name": "Erdos355.erdos_355", "theorem_header": "theorem erdos_355 :\n answer(True) ↔ ∃ A : ℕ → ℕ, IsLacunary A ∧ ∃ u v : ℝ, u < v ∧ ∀ q : ℚ, ↑q ∈ Set.Ioo u v →\n q ∈ {∑ a ∈ A', (1 / a : ℚ) | (A' : Finset ℕ) (_ : ↑A' ⊆ Set.range A)}", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/355.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-370-erdos-370", "harbor_task_name": "formal-conjectures-bench/erdosproblems-370-erdos-370", "harbor_task_path": "tasks/erdosproblems-370-erdos-370", "theorem_name": "Erdos370.erdos_370", "theorem_header": "theorem erdos_370 : answer(True) ↔\n { n | Nat.maxPrimeFac n < √n ∧ Nat.maxPrimeFac (n + 1) < √(n + 1) }.Infinite", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/370.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-379-erdos-379", "harbor_task_name": "formal-conjectures-bench/erdosproblems-379-erdos-379", "harbor_task_path": "tasks/erdosproblems-379-erdos-379", "theorem_name": "Erdos379.erdos_379", "theorem_header": "theorem erdos_379 : atTop.limsup (fun n => (S n : ℕ∞)) = ⊤", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/379.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-397-erdos-397", "harbor_task_name": "formal-conjectures-bench/erdosproblems-397-erdos-397", "harbor_task_path": "tasks/erdosproblems-397-erdos-397", "theorem_name": "Erdos397.erdos_397", "theorem_header": "theorem erdos_397 :\n answer(False) ↔\n {(M, N) : Finset ℕ × Finset ℕ | Disjoint M N ∧\n ∏ i ∈ M, centralBinom i = ∏ j ∈ N, centralBinom j}.Finite", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/397.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-399-erdos-399", "harbor_task_name": "formal-conjectures-bench/erdosproblems-399-erdos-399", "harbor_task_path": "tasks/erdosproblems-399-erdos-399", "theorem_name": "Erdos399.erdos_399", "theorem_header": "theorem erdos_399 : answer(False) ↔\n ¬ ∃ (n x y k : ℕ), 1 < x * y ∧ 2 < k ∧ (n ! = x ^ k + y ^ k ∨ n ! + y ^ k = x ^ k)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/399.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-645-erdos-645", "harbor_task_name": "formal-conjectures-bench/erdosproblems-645-erdos-645", "harbor_task_path": "tasks/erdosproblems-645-erdos-645", "theorem_name": "Erdos645.erdos_645", "theorem_header": "theorem erdos_645 (c : ℕ → Bool) : ∃ x d, 0 < x ∧ x < d ∧\n (∃ C, c x = C ∧ c (x + d) = C ∧ c (x + 2 * d) = C)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/645.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-707-erdos-707", "harbor_task_name": "formal-conjectures-bench/erdosproblems-707-erdos-707", "harbor_task_path": "tasks/erdosproblems-707-erdos-707", "theorem_name": "Erdos707.erdos_707", "theorem_header": "theorem erdos_707 : (∀ (A : Set ℕ) (h : A.Finite), IsSidon A →\n ∃ᵉ (B : Set ℕ) (n > 0), A ⊆ B ∧ IsPerfectDifferenceSet B n) ↔ False", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/707.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-741-i", "harbor_task_name": "formal-conjectures-bench/erdosproblems-741-i", "harbor_task_path": "tasks/erdosproblems-741-i", "theorem_name": "Erdos741.erdos_741.parts.i", "theorem_header": "theorem erdos_741.parts.i : answer(False) ↔ ∀ A : Set ℕ, HasPosDensity (A + A) → ∃ A₁ A₂,\n A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ HasPosDensity (A₁ + A₁)\n ∧ HasPosDensity (A₂ + A₂)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/741.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-741-ii", "harbor_task_name": "formal-conjectures-bench/erdosproblems-741-ii", "harbor_task_path": "tasks/erdosproblems-741-ii", "theorem_name": "Erdos741.erdos_741.parts.ii", "theorem_header": "theorem erdos_741.parts.ii : answer(True) ↔ ∃ A : Set ℕ, IsAddBasisOfOrder (A ∪ {0}) 2 ∧ ∀ A₁ A₂,\n A = A₁ ∪ A₂ → Disjoint A₁ A₂ → ¬ (IsSyndetic (A₁ + A₁) ∧ IsSyndetic (A₂ + A₂))", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/741.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-741-upper", "harbor_task_name": "formal-conjectures-bench/erdosproblems-741-upper", "harbor_task_path": "tasks/erdosproblems-741-upper", "theorem_name": "Erdos741.erdos_741.variants.upper", "theorem_header": "theorem erdos_741.variants.upper : answer(True) ↔ ∀ A : Set ℕ, 0 < upperDensity (A + A) → ∃ A₁ A₂,\n A = A₁ ∪ A₂ ∧ Disjoint A₁ A₂ ∧ 0 < upperDensity (A₁ + A₁)\n ∧ 0 < upperDensity (A₂ + A₂)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/741.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-828-phi-dvd-self-iff-pow2-pow3", "harbor_task_name": "formal-conjectures-bench/erdosproblems-828-phi-dvd-self-iff-pow2-pow3", "harbor_task_path": "tasks/erdosproblems-828-phi-dvd-self-iff-pow2-pow3", "theorem_name": "Erdos828.erdos_828.variants.phi_dvd_self_iff_pow2_pow3", "theorem_header": "theorem erdos_828.variants.phi_dvd_self_iff_pow2_pow3 {n : ℕ} :\n φ n ∣ n ↔ n ≤ 1 ∨ ∃ᵉ (a > 0) (b), n = 2 ^ a * 3 ^ b", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/828.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-846-erdos-846", "harbor_task_name": "formal-conjectures-bench/erdosproblems-846-erdos-846", "harbor_task_path": "tasks/erdosproblems-846-erdos-846", "theorem_name": "Erdos846.erdos_846", "theorem_header": "theorem erdos_846 : answer(False) ↔ ∀ᵉ (A : Set ℝ²) (ε > 0), A.Infinite → NonTrilinearFor A ε →\n WeaklyNonTrilinear A", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/846.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-897-i", "harbor_task_name": "formal-conjectures-bench/erdosproblems-897-i", "harbor_task_path": "tasks/erdosproblems-897-i", "theorem_name": "Erdos897.erdos_897.parts.i", "theorem_header": "theorem erdos_897.parts.i : answer(False) ↔ ∀ (f : ℕ → ℝ),\n (∀ᵉ (a > 0) (b > 0), a.Coprime b → f (a * b) = f a + f b) →\n ((Filter.atTop ⊓ Filter.principal {(p, k) : ℕ × ℕ | p.Prime}).limsup\n (fun (p, k) => (f (p^k) / (p^k : ℝ).log : EReal)) = ⊤) →\n Filter.atTop.limsup (fun (n : ℕ) => ((f (n+1) - f n) / (n : ℝ).log : EReal)) = ⊤", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/897.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-897-ii", "harbor_task_name": "formal-conjectures-bench/erdosproblems-897-ii", "harbor_task_path": "tasks/erdosproblems-897-ii", "theorem_name": "Erdos897.erdos_897.parts.ii", "theorem_header": "theorem erdos_897.parts.ii : answer(False) ↔ ∀ (f : ℕ → ℝ),\n (∀ᵉ (a > 0) (b > 0), a.Coprime b → f (a * b) = f a + f b) →\n ((Filter.atTop ⊓ Filter.principal {(p, k) : ℕ × ℕ | p.Prime}).limsup\n (fun (p, k) => (f (p^k) / (p^k : ℝ).log : EReal)) = ⊤) →\n Filter.atTop.limsup (fun (n : ℕ) => (f (n+1) / f n : EReal)) = ⊤", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/897.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "erdosproblems-978-allow-fixed-divisors", "harbor_task_name": "formal-conjectures-bench/erdosproblems-978-allow-fixed-divisors", "harbor_task_path": "tasks/erdosproblems-978-allow-fixed-divisors", "theorem_name": "Erdos978.erdos_978.variants.allow_fixed_divisors", "theorem_header": "theorem erdos_978.variants.allow_fixed_divisors : answer(False) ↔\n ∀ {f : ℤ[X]}, Irreducible f → f.natDegree > 3 →\n (¬ ∃ l : ℕ, f.natDegree = 2 ^ l) → 0 < f.leadingCoeff →\n (¬ ∃ p : ℕ, p.Prime ∧ ∀ n : ℕ, (p : ℤ) ^ (f.natDegree - 1) ∣ f.eval (n : ℤ)) →\n {n : ℕ | Powerfree (f.natDegree - 2) (f.eval (n : ℤ))}.Infinite", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/ErdosProblems/978.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "greensopenproblems-57-green-57", "harbor_task_name": "formal-conjectures-bench/greensopenproblems-57-green-57", "harbor_task_path": "tasks/greensopenproblems-57-green-57", "theorem_name": "Green57.green_57", "theorem_header": "theorem green_57 :\n answer(False) ↔\n ∀ (G : Type) [AddCommGroup G] [Fintype G] [DecidableEq G],\n Φ G = Φ' G", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/GreensOpenProblems/57.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "greensopenproblems-57-z3", "harbor_task_name": "formal-conjectures-bench/greensopenproblems-57-z3", "harbor_task_path": "tasks/greensopenproblems-57-z3", "theorem_name": "Green57.green_57.variants.z3", "theorem_header": "theorem green_57.variants.z3 :\n answer(False) ↔ (Φ (ZMod 3) = Φ' (ZMod 3))", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/GreensOpenProblems/57.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "greensopenproblems-57-z3-functional", "harbor_task_name": "formal-conjectures-bench/greensopenproblems-57-z3-functional", "harbor_task_path": "tasks/greensopenproblems-57-z3-functional", "theorem_name": "Green57.green_57.variants.z3_functional", "theorem_header": "theorem green_57.variants.z3_functional :\n let a : ZMod 3 → ℂ := ![(-1 : ℂ), -3, 3]\n answer(True) ↔\n supportFn (ZMod 3) a (Φ' (ZMod 3)) < supportFn (ZMod 3) a (Φ (ZMod 3))", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/GreensOpenProblems/57.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "greensopenproblems-94-green-94-outer-measure", "harbor_task_name": "formal-conjectures-bench/greensopenproblems-94-green-94-outer-measure", "harbor_task_path": "tasks/greensopenproblems-94-green-94-outer-measure", "theorem_name": "Green94.green_94_outer_measure", "theorem_header": "theorem green_94_outer_measure :\n answer(False) ↔ ∀ A : Set ℝ,\n volume A > 0 →\n ∃ a b : ℝ, a ≠ 0 ∧ ∀ n : ℕ, a * (1 / 2^n) + b ∈ A", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/GreensOpenProblems/94.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "mathoverflow-10799-mathoverflow-10799", "harbor_task_name": "formal-conjectures-bench/mathoverflow-10799-mathoverflow-10799", "harbor_task_path": "tasks/mathoverflow-10799-mathoverflow-10799", "theorem_name": "Mathoverflow10799.mathoverflow_10799", "theorem_header": "theorem mathoverflow_10799 : answer(False) ↔\n ∀ (n : ℕ) (_ : 2 ≤ n)\n (F : Finset (Finset (Fin n))) (_ : IsMonotoneIncreasing F)\n (s t : ℝ) (_ : 0 < s) (_ : s ≤ t) (_ : t < 1)\n (_ : t / s > 1000 * Real.log n),\n ∃ p, s ≤ p ∧ p ≤ t ∧ IsOptimal p F", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Mathoverflow/10799.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "mathoverflow-486451-exists-semiring-unique-left-right-maximal-ne", "harbor_task_name": "formal-conjectures-bench/mathoverflow-486451-exists-semiring-unique-left-right-maximal-ne", "harbor_task_path": "tasks/mathoverflow-486451-exists-semiring-unique-left-right-maximal-ne", "theorem_name": "Mathoverflow486451.exists_semiring_unique_left_right_maximal_ne", "theorem_header": "theorem exists_semiring_unique_left_right_maximal_ne :\n answer(True) ↔ ∃ (R : Type) (_ : Semiring R) (hI : ∃! I : Ideal R, I.IsMaximal)\n (hJ : ∃! J : Ideal Rᵐᵒᵖ, J.IsMaximal),\n (hI.choose : Set R) ≠ MulOpposite.op ⁻¹' hJ.choose", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Mathoverflow/486451.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "openquantumproblems-13-mutuallyunbiasedbases-dim6-bounds", "harbor_task_name": "formal-conjectures-bench/openquantumproblems-13-mutuallyunbiasedbases-dim6-bounds", "harbor_task_path": "tasks/openquantumproblems-13-mutuallyunbiasedbases-dim6-bounds", "theorem_name": "OpenQuantumProblem13.mutuallyUnbiasedBases_dim6_bounds", "theorem_header": "theorem mutuallyUnbiasedBases_dim6_bounds :\n HasMUBs 6 3 ∧ ∀ m : ℕ, HasMUBs 6 m → m ≤ 7", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/OpenQuantumProblems/13.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "openquantumproblems-35-ame-11-5-open", "harbor_task_name": "formal-conjectures-bench/openquantumproblems-35-ame-11-5-open", "harbor_task_path": "tasks/openquantumproblems-35-ame-11-5-open", "theorem_name": "OpenQuantumProblem35.ame_11_5_open", "theorem_header": "theorem ame_11_5_open :\n answer(True) ↔ ExistsAME 11 5", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/OpenQuantumProblems/35.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "paper-casasalvero-positive-char-counterexample", "harbor_task_name": "formal-conjectures-bench/paper-casasalvero-positive-char-counterexample", "harbor_task_path": "tasks/paper-casasalvero-positive-char-counterexample", "theorem_name": "CasasAlvero.casas_alvero.positive_char_counterexample", "theorem_header": "theorem casas_alvero.positive_char_counterexample {p : ℕ} (hp : p.Prime) :\n ∃ (K : Type*) (_ : Field K) (_ : CharP K p),\n let P := X ^ (p + 1) - X ^ p\n Monic P ∧ HasCasasAlveroProp P ∧\n ¬∃ α : K, P = (X - C α) ^ P.natDegree", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Paper/CasasAlvero.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "paper-claudescycles-cube-hamiltonian-arc-decomposition", "harbor_task_name": "formal-conjectures-bench/paper-claudescycles-cube-hamiltonian-arc-decomposition", "harbor_task_path": "tasks/paper-claudescycles-cube-hamiltonian-arc-decomposition", "theorem_name": "ClaudesCycles.cube_hamiltonian_arc_decomposition", "theorem_header": "theorem cube_hamiltonian_arc_decomposition {m : ℕ} [NeZero m] (hm : Odd m) (hm' : 1 < m) :\n HasHamiltonianArcDecomposition m", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Paper/ClaudesCycles.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "no-explicit-license", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "paper-monochromaticquantumgraph-eqsystem-no-solution-even-ge4-d-eq-n-explicit", "harbor_task_name": "formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem-no-solution-even-ge4-d-eq-n-explicit", "harbor_task_path": "tasks/paper-monochromaticquantumgraph-eqsystem-no-solution-even-ge4-d-eq-n-explicit", "theorem_name": "MonochromaticQuantumGraph.eqSystem_no_solution_even_ge4_d_eq_n_explicit", "theorem_header": "theorem eqSystem_no_solution_even_ge4_d_eq_n_explicit :\n answer(True) ↔\n ∀ N : Nat, N ≥ 4 → Even N →\n ¬ ∃ W : WeightsN N N ℂ, EqSystemN N N W", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Paper/MonochromaticQuantumGraph.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "paper-monochromaticquantumgraph-eqsystem10-no-solution-d10", "harbor_task_name": "formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem10-no-solution-d10", "harbor_task_path": "tasks/paper-monochromaticquantumgraph-eqsystem10-no-solution-d10", "theorem_name": "MonochromaticQuantumGraph.eqSystem10_no_solution_d10", "theorem_header": "theorem eqSystem10_no_solution_d10 :\n answer(True) ↔\n ¬ ∃ W : WeightsN 10 10 ℂ, EqSystemN 10 10 W", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Paper/MonochromaticQuantumGraph.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "paper-monochromaticquantumgraph-eqsystem4-no-solution-d4", "harbor_task_name": "formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem4-no-solution-d4", "harbor_task_path": "tasks/paper-monochromaticquantumgraph-eqsystem4-no-solution-d4", "theorem_name": "MonochromaticQuantumGraph.eqSystem4_no_solution_d4", "theorem_header": "theorem eqSystem4_no_solution_d4 :\n answer(True) ↔\n ¬ ∃ W : WeightsN 4 4 ℂ, EqSystemN 4 4 W", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Paper/MonochromaticQuantumGraph.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4", "harbor_task_name": "formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4", "harbor_task_path": "tasks/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4", "theorem_name": "MonochromaticQuantumGraph.eqSystem4_no_solution_ge4", "theorem_header": "theorem eqSystem4_no_solution_ge4 :\n answer(True) ↔\n ∀ D : Nat, D ≥ 4 →\n ¬ ∃ W : WeightsN 4 D ℂ, EqSystemN 4 D W", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Paper/MonochromaticQuantumGraph.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-int", "harbor_task_name": "formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-int", "harbor_task_path": "tasks/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-int", "theorem_name": "MonochromaticQuantumGraph.eqSystem4_no_solution_ge4_int", "theorem_header": "theorem eqSystem4_no_solution_ge4_int :\n answer(True) ↔\n ∀ D : Nat, D ≥ 4 →\n ¬ ∃ W : WeightsN 4 D ℤ, EqSystemN 4 D W", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Paper/MonochromaticQuantumGraph.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-real", "harbor_task_name": "formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-real", "harbor_task_path": "tasks/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-real", "theorem_name": "MonochromaticQuantumGraph.eqSystem4_no_solution_ge4_real", "theorem_header": "theorem eqSystem4_no_solution_ge4_real :\n answer(True) ↔\n ∀ D : Nat, D ≥ 4 →\n ¬ ∃ W : WeightsN 4 D ℝ, EqSystemN 4 D W", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Paper/MonochromaticQuantumGraph.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-trinary-int", "harbor_task_name": "formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-trinary-int", "harbor_task_path": "tasks/paper-monochromaticquantumgraph-eqsystem4-no-solution-ge4-trinary-int", "theorem_name": "MonochromaticQuantumGraph.eqSystem4_no_solution_ge4_trinary_int", "theorem_header": "theorem eqSystem4_no_solution_ge4_trinary_int :\n answer(True) ↔\n ∀ D : Nat, D ≥ 4 →\n ¬ ∃ W : WeightsN 4 D ℤ,\n (∀ e, W e = (-1 : ℤ) ∨ W e = 0 ∨ W e = 1) ∧\n EqSystemN 4 D W", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Paper/MonochromaticQuantumGraph.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "paper-monochromaticquantumgraph-eqsystem6-no-solution-d6", "harbor_task_name": "formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem6-no-solution-d6", "harbor_task_path": "tasks/paper-monochromaticquantumgraph-eqsystem6-no-solution-d6", "theorem_name": "MonochromaticQuantumGraph.eqSystem6_no_solution_d6", "theorem_header": "theorem eqSystem6_no_solution_d6 :\n answer(True) ↔\n ¬ ∃ W : WeightsN 6 6 ℂ, EqSystemN 6 6 W", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Paper/MonochromaticQuantumGraph.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "paper-monochromaticquantumgraph-eqsystem8-no-solution-d10", "harbor_task_name": "formal-conjectures-bench/paper-monochromaticquantumgraph-eqsystem8-no-solution-d10", "harbor_task_path": "tasks/paper-monochromaticquantumgraph-eqsystem8-no-solution-d10", "theorem_name": "MonochromaticQuantumGraph.eqSystem8_no_solution_d10", "theorem_header": "theorem eqSystem8_no_solution_d10 :\n answer(True) ↔\n ¬ ∃ W : WeightsN 8 10 ℂ, EqSystemN 8 10 W", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Paper/MonochromaticQuantumGraph.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "wikipedia-agohgiuga-isweakgiuga-iff-prime-dvd", "harbor_task_name": "formal-conjectures-bench/wikipedia-agohgiuga-isweakgiuga-iff-prime-dvd", "harbor_task_path": "tasks/wikipedia-agohgiuga-isweakgiuga-iff-prime-dvd", "theorem_name": "AgohGiuga.isWeakGiuga_iff_prime_dvd", "theorem_header": "theorem isWeakGiuga_iff_prime_dvd {n : ℕ} (hn : n.Composite) :\n IsWeakGiuga n ↔ ∀ p ∈ n.primeFactors, p ∣ (n / p - 1)", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Wikipedia/AgohGiuga.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "wikipedia-eulerbrick-cuboidone", "harbor_task_name": "formal-conjectures-bench/wikipedia-eulerbrick-cuboidone", "harbor_task_path": "tasks/wikipedia-eulerbrick-cuboidone", "theorem_name": "EulerBrick.cuboidOne", "theorem_header": "theorem cuboidOne : CuboidOne", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Wikipedia/EulerBrick.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "wikipedia-legendreconjecture-bounded-gap-legendre", "harbor_task_name": "formal-conjectures-bench/wikipedia-legendreconjecture-bounded-gap-legendre", "harbor_task_path": "tasks/wikipedia-legendreconjecture-bounded-gap-legendre", "theorem_name": "LegendreConjecture.bounded_gap_legendre", "theorem_header": "theorem bounded_gap_legendre\n (H : ∃ c > 0, ∀ᶠ n in atTop, (n + 1).nth Nat.Prime - n.nth Nat.Prime <\n (n.nth Nat.Prime : ℝ) ^ (1 / (2 : ℝ) - c)) :\n ∀ᶠ n in atTop, ∃ p ∈ Set.Ioo (n ^ 2) ((n + 1) ^ 2), Nat.Prime p", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Wikipedia/LegendreConjecture.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "wikipedia-perfectnumbers-euler-form", "harbor_task_name": "formal-conjectures-bench/wikipedia-perfectnumbers-euler-form", "harbor_task_path": "tasks/wikipedia-perfectnumbers-euler-form", "theorem_name": "PerfectNumbers.odd_perfect_number.euler_form", "theorem_header": "theorem odd_perfect_number.euler_form (n : ℕ) (hn : Odd n) (hp : Perfect n) :\n ∃ (p m α : ℕ),\n p.Prime ∧\n p ≡ 1 [ZMOD 4] ∧\n α ≡ 1 [ZMOD 4] ∧\n ¬ p ∣ m ∧\n n = p^α * m^2", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Wikipedia/PerfectNumbers.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "wikipedia-wolstenholmeprime-wolstenholme-prime-16483", "harbor_task_name": "formal-conjectures-bench/wikipedia-wolstenholmeprime-wolstenholme-prime-16483", "harbor_task_path": "tasks/wikipedia-wolstenholmeprime-wolstenholme-prime-16483", "theorem_name": "WolstenholmePrime.wolstenholme_prime_16483", "theorem_header": "theorem wolstenholme_prime_16483 : IsWolstenholmePrime 16843", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Wikipedia/WolstenholmePrime.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
| {"task_id": "wikipedia-wolstenholmeprime-wolstenholme-theorem", "harbor_task_name": "formal-conjectures-bench/wikipedia-wolstenholmeprime-wolstenholme-theorem", "harbor_task_path": "tasks/wikipedia-wolstenholmeprime-wolstenholme-theorem", "theorem_name": "WolstenholmePrime.wolstenholme_theorem", "theorem_header": "theorem wolstenholme_theorem (p : ℕ) (h : p > 3) (hp : Nat.Prime p) :\n (2 * p - 1).choose (p - 1) ≡ 1 [MOD p ^ 3]", "source": "google-deepmind/formal-conjectures", "source_file": "FormalConjectures/Wikipedia/WolstenholmePrime.lean", "benchmark_bucket": "gold_solution", "oracle_status": "bundled", "licence_status": "Apache-2.0", "redistribution_status": "bundled", "formal_conjectures_commit": "233a10e857ef78e79fd9fe661d37db724089170a", "lean_toolchain": "leanprover/lean4:v4.27.0", "mathlib_commit": "a3a10db0e9d66acbebf76c5e6a135066525ac900", "allow_internet": false, "agent_timeout_sec": 86400.0, "verifier_timeout_sec": 7200.0, "environment_build_timeout_sec": 14400.0} |
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