Datasets:
id stringlengths 5 52 | splittable bool 2
classes | prompt stringlengths 161 1.23k ⌀ | correct stringlengths 137 1.56k | answer stringlengths 7 629 ⌀ | reason stringlengths 166 514 ⌀ | formal_statement stringlengths 389 291k | reference_proof stringlengths 1.11k 602k | header stringlengths 15 3.89k | theorem_names listlengths 1 35 | repo_url stringlengths 35 60 | repo_commit_sha stringlengths 40 40 | license stringclasses 2
values | source listlengths 0 7 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
BerkovichUncu | false | null | Four statistic-transfer lemmas about the file's four concrete maps between partition classes of n: φ : D(n) → A1(n), Φ : A1(n) → A2(n), ψ⁻¹ : A2(n) → O(n), and G : O(n) → D(n), where D(n) is strict partitions of n, O(n) is partitions of n into odd parts, and A1(n), A2(n) are the two Chen–Gao–Ji–Li classes. Assuming the... | null | The text combines four separate lemmas about four different maps, each with its own hypothesis and conclusion. There is no single claim to split; instead there are four distinct statements. | set_option backward.isDefEq.respectTransparency false
/-
# Problem Description
This file formalizes the explicit Berkovich--Uncu bijection and its statistic
transfer, following the paper's four-map construction
`F = G ∘ ψInv ∘ Φ ∘ φ : D n → D n`,
where
`D n --φ--> A1 n --Φ--> A2 n --ψInv--> O n --G--> D n`.
H... | section
set_option backward.isDefEq.respectTransparency false
/-
# Problem Description
This file formalizes the explicit Berkovich--Uncu bijection and its statistic
transfer, following the paper's four-map construction
`F = G ∘ ψInv ∘ Φ ∘ φ : D n → D n`,
where
`D n --φ--> A1 n --Φ--> A2 n --ψInv--> O n --G--> ... | import Mathlib
| [
"BerkovichUncu.phi_stat",
"BerkovichUncu.Phi_stat",
"BerkovichUncu.psiInv_oddMultiplicity",
"BerkovichUncu.G_oddPart"
] | https://github.com/AxiomMath/BerkovichUncu | d949f32c1d28f8d2b60e80a4f3219dd3b4d712d4 | MIT | [] |
Biswal:theorem1 | false | null | "Let p_0 = p_1 = 1 and p_{n+2} = p_{n+1} - x·p_n for n ≥ 0, and for a partition ξ put p_ξ = ∏(...TRUNCATED) | null | "The text is one indivisible statement: it first defines notation, then states a condition-dependent(...TRUNCATED) | "noncomputable def polyP (R : Type*) [CommRing R] : ℕ → Polynomial R\n | 0 => 1\n | 1 => 1\n (...TRUNCATED) | "section\nnoncomputable def polyP (R : Type*) [CommRing R] : ℕ → Polynomial R\n | 0 => 1\n | 1(...TRUNCATED) | "import Mathlib\n\n/-\n# Problem: Positivity of Coefficients of a Partition Generating Function\n\nW(...TRUNCATED) | [
"genFun_eq_one_of_m_eq_one",
"genFun_is_polynomial",
"genFun_coeff_eventually_pos"
] | https://github.com/AxiomMath/Biswal | 71bcc619a22c77d2e118b1f53dcf68165707f662 | MIT | [] |
Biswal:theorem2and3 | true | "Let p₀=p₁=1 and p_{n+2}=p_{n+1}−x·p_n, and let C(n,j) be the ordinary binomial coefficient, (...TRUNCATED) | "Let p₀=p₁=1, p_{n+2}=p_{n+1}−x·p_n, and let C(n,j) be the ordinary binomial coefficient (0 w(...TRUNCATED) | "Over any commutative ring, [x^j]p_n=(−1)^j C(n−j,j), which vanishes when 2j>n. Over any field K(...TRUNCATED) | null | "/-- Chebyshev-type polynomials defined by the recurrence\n$p_0 = 1$, $p_1 = 1$, $p_{n+2} = p_{n+1} (...TRUNCATED) | "section\nnoncomputable def polyP (R : Type*) [CommRing R] : ℕ → Polynomial R\n | 0 => 1\n | 1(...TRUNCATED) | "import Mathlib\n\n/-\n# Problem: Combinatorial formula and Dyck path identity for Chebyshev partiti(...TRUNCATED) | ["genFun_eq_reduced","D_coeff_mk","polyP_coeff","thm_comb_cauchy","thm_comb","thm_manifest_factoriza(...TRUNCATED) | https://github.com/AxiomMath/Biswal | 71bcc619a22c77d2e118b1f53dcf68165707f662 | MIT | [] |
Feige | true | "For any positive integer n and any independent, measurable, integrable, pointwise nonnegative rando(...TRUNCATED) | "For any positive integer n and any independent, measurable, integrable, pointwise nonnegative rando(...TRUNCATED) | "P(S < E[S] + 1) ≥ (n/(n+1))ⁿ, and for each fixed n this constant is optimal: it is the largest (...TRUNCATED) | null | "open scoped BigOperators\nopen MeasureTheory ProbabilityTheory\n\nnamespace PalomarFeige\n\n/-- The(...TRUNCATED) | "section\nnamespace Feige\n\nopen Filter Topology\n\n/-- The unit-slack sharp constant `bₙ,₁` in(...TRUNCATED) | "import Mathlib.MeasureTheory.Integral.Bochner.Basic\nimport Mathlib.MeasureTheory.Measure.Real\nimp(...TRUNCATED) | [
"PalomarFeige.sharp_unit_slack_feige_complete"
] | https://github.com/pengzhang91/Feige | 19c61d6cb556d1e967dbb0ada9f293c290837867 | Apache-2.0 | [{"authors":["Weibo Fu","Yanjun Han","Guanyang Wang","Jun Yan","Peng Zhang","Zhengqing Zhou"],"id":"(...TRUNCATED) |
HigherDyson:Batch1 | false | null | "With a fixed nome q ∈ ℂˣ, the project formalizes: character-sum collapse when x^d = 1 and x (...TRUNCATED) | null | "The text is not a single mathematical claim with a separable hypothesis/conclusion structure; it is(...TRUNCATED) | "open Finset\n\n-- Main Definition(s)\n\n/-- The character sum `C_x = ∑_{r=0}^{d-1} x^r`. -/\nnonc(...TRUNCATED) | "section\nset_option maxHeartbeats 1000000\n\n/-\n# Problem Description\n\nThis corpus comes from Al(...TRUNCATED) | "import Mathlib\n\n/-\n# Problem Description\n\nThis corpus comes from Alfes-Ono, *Higher Dyson rank(...TRUNCATED) | ["KF_collapse","exponent_identity","KF_recurrence","KF_boundary","rational_reduction_one","rational_(...TRUNCATED) | https://github.com/AxiomMath/HigherDyson | fed112b469b55bebb5bbcd1823110cd66bf8ee34 | MIT | [] |
HigherDyson:Batch2 | true | "Let m ≥ 2, d = 2m+1, x,q ∈ ℂˣ, ω a primitive d-th root of unity, C_x := Σ_{r<d} x^r, G, X(...TRUNCATED) | "Let m ≥ 2, d = 2m+1, x, q ∈ ℂˣ, ω a primitive d-th root of unity, C_x := Σ_{r<d} x^r, G, X(...TRUNCATED) | "Y := X₀Ψ − Σ_{r<d} D_{r,x}Ω_r − E_xΩ_ext (torsion: D_r for D_{r,x}, no E_x term) satisfie(...TRUNCATED) | null | "open Polynomial\n\nnoncomputable section\n\n/-! ## Abbreviations for the field and the indeterminat(...TRUNCATED) | "section\nopen Polynomial\n\nnoncomputable section\n\n/-! ## Abbreviations for the field and the ind(...TRUNCATED) | "import Mathlib\n\n/-\n# Problem Description\n\nWe formalize three \"Key Formulas\" from the Dyson s(...TRUNCATED) | ["Tpoly_natDegree_le","Tpoly_coeff_zero","Cx_one","Cx_torsion_ne","Tpoly_factor_one","Tpoly_factor_t(...TRUNCATED) | https://github.com/AxiomMath/HigherDyson | fed112b469b55bebb5bbcd1823110cd66bf8ee34 | MIT | [] |
HigherDyson:Batch3 | false | null | "Fix m≥2, d=2m+1, a unit x, τ, ω, q=e^{2πiτ}, u(w)=e^{2πiw}. The q-series, kernel and theta s(...TRUNCATED) | null | "The text is a dense, self-contained block of definitions and hypotheses followed by a list of concl(...TRUNCATED) | "namespace HigherDyson\n\n-- Fixed data (Definitions 1, 3, 7).\nvariable (m : ℕ) (x : ℂˣ) (τ :(...TRUNCATED) | "section\nnamespace HigherDyson\n\n-- Fixed data (Definitions 1, 3, 7).\nvariable (m : ℕ) (x : ℂ(...TRUNCATED) | "import Mathlib\n\n/-\n# Problem Description\n\nHigher Dyson ranks and meromorphic elliptic correcti(...TRUNCATED) | ["HigherDyson.stmt1_i","HigherDyson.stmt1_ii","HigherDyson.stmt2_a","HigherDyson.stmt2_b_left","High(...TRUNCATED) | https://github.com/AxiomMath/HigherDyson | fed112b469b55bebb5bbcd1823110cd66bf8ee34 | MIT | [] |
HigherDyson:Batch4 | true | "Let m≥2, d=2m+1, x,ω∈ℂˣ, q∈ℂ with q≠0, τ∈ℂ, and let w↦u(w)∈ℂˣ satisfy u(w(...TRUNCATED) | "Fix m≥2, d=2m+1, units x,ω∈ℂˣ, nonzero q∈ℂ, and τ∈ℂ, with an elliptic variable w(...TRUNCATED) | "For all w, with u=u(w): B(w+τ)=x²q^{-m}u^{-2m}B(w)−q^{-m}u^{-2m}P_d(u); for r<d, A_r(w+τ)=x²q(...TRUNCATED) | null | "namespace AppellKernelShift\n\n/- We record the unit `qᵤ : ℂˣ` corresponding to the nonzero sc(...TRUNCATED) | "section\nnamespace AppellKernelShift\n\n/- We record the unit `qᵤ : ℂˣ` corresponding to the n(...TRUNCATED) | "import Mathlib\n\n/-\n# Problem Description\n\nWe work over the complex numbers `ℂ`. Fix the foll(...TRUNCATED) | [
"AppellKernelShift.thm_Bshift",
"AppellKernelShift.thm_Arshift",
"AppellKernelShift.thm_Aextshift"
] | https://github.com/AxiomMath/HigherDyson | fed112b469b55bebb5bbcd1823110cd66bf8ee34 | MIT | [] |
IMO2026:Q1 | true | "A board is a multiset of natural numbers; call it initial if it has exactly 2026 entries, all > 1. (...TRUNCATED) | "A board is a multiset of natural numbers; call it initial if it has exactly 2026 entries, all > 1. (...TRUNCATED) | "(1) no infinite sequence of boards starts at B₀ with each consecutive pair related by a move; (2)(...TRUNCATED) | null | "set_option backward.isDefEq.respectTransparency false\n\n/-- A *board* is a finite multiset of natu(...TRUNCATED) | "section\nset_option backward.isDefEq.respectTransparency false\n\n/-- A *board* is a finite multise(...TRUNCATED) | import Mathlib
| ["Q1.statement_a_termination","Q1.statement_a_unique_large","Q1.statement_b_invariance","Q1.terminal(...TRUNCATED) | https://github.com/AxiomMath/IMO2026 | c5a6a089d06d3619afe7ff45c5ccab9e2a30d5d2 | MIT | [{"authors":["International Mathematical Olympiad"],"id":"IMO 2026","location":"","note":"","relatio(...TRUNCATED) |
IMO2026:Q2 | true | "Let ABC be a nondegenerate triangle, with M, N the midpoints of AB, AC. Let K lie strictly inside t(...TRUNCATED) | "Let ABC be a nondegenerate triangle, with M, N the midpoints of AB, AC. Let K lie strictly inside t(...TRUNCATED) | OM = ON | null | "set_option backward.isDefEq.respectTransparency false\n\nopen EuclideanGeometry\n\n-- We work in th(...TRUNCATED) | "section\nset_option backward.isDefEq.respectTransparency false\n\nopen EuclideanGeometry\n\n-- We w(...TRUNCATED) | import Mathlib
| [
"Q2.main_theorem"
] | https://github.com/AxiomMath/IMO2026 | c5a6a089d06d3619afe7ff45c5ccab9e2a30d5d2 | MIT | [{"authors":["International Mathematical Olympiad"],"id":"IMO 2026","location":"","note":"","relatio(...TRUNCATED) |
End of preview. Expand in Data Studio
Strictly Speaking
Does a model's mathematical understanding hold up strictly speaking, at Lean-grade precision, or is it only right in the ordinary, looser sense that informal writing usually gets away with?
Each row is derived from a real, already-formalized Lean 4 theorem (drawn from
Pradheep1647/lean-verifier-formalizations).
The theorem's informal statement is split into two pieces: the hypotheses/setup (prompt), and
the conclusion that was elided from it (answer) - the actual mathematical content being tested
(a formula, bound, or characterization). A model is given only prompt and asked to derive or
state the conclusion itself, in plain English. It never sees or writes any Lean.
The intended use is to grade that answer two separate ways, against two different ground truths:
- Against
answer(the original informal conclusion) - ordinary correctness, the loose bar. - Against a backtranslation of
formal_statement/reference_proofinto plain English - the strict bar, capturing precision an informal statement often glosses over (open vs closed sets, exact hypothesis scope, edge cases) but a real formalization has to get exactly right.
The gap between those two scores is the point: whether a model's math reasoning is informally right but formally loose, or actually precise enough to survive formal-grade scrutiny, without ever touching a proof assistant.
Splittable rows
Not every theorem has a hypothesis/conclusion structure that can be split without either giving
away the answer in the prompt or leaving the answer meaningless on its own (a multi-part lemma
list, a tautological unpacking of a definition, or a "prove this holds" claim with no separable
result to state). splittable marks whether the split worked; reason explains why not, for the
21 rows where it didn't. Non-splittable rows still carry formal_statement/reference_proof for
reference, but prompt/answer are null.
Columns
| Column | Description |
|---|---|
id |
Row id, matching the parent dataset's id. |
splittable |
Whether a usable hypothesis/conclusion split was found. |
prompt |
The setup shown to the model (null if not splittable). |
correct |
The original, full informal statement, unsplit - for reference/audit. |
answer |
The elided conclusion, the grading target for the loose bar (null if not splittable). |
reason |
Why a row isn't splittable (null if it is). |
formal_statement |
The real Lean target declaration (the theorem, sorry in place of the proof). |
reference_proof |
The real, human-written Lean proof. |
header |
Fixed imports/opens the formal statement compiles against. |
theorem_names |
Which declaration(s) in formal_statement are graded. |
repo_url, repo_commit_sha |
Provenance: the source repo and pinned commit. |
license |
The source repo's own license (MIT or Apache-2.0), per row. |
source |
Paper/competition citation metadata for the row's origin. |
License
Two license scopes apply, same split as the parent dataset:
- Source code (
header,formal_statement,reference_proof): each row carries its source repository's original license, recorded per-row in thelicensecolumn. - Curated content (
prompt,correct,answer,reason, the hypothesis/conclusion split itself): released under CC-BY-4.0.
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