Lean statement

Partitional intersection, covering, and packing counterexamples

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorems selected for verification: OAI.InfiniteMatroidCounterexample.partitional_intersection_counterexample OAI.InfiniteMatroidCounterexample.separate_covering_packing_counterexamples

Lean statement file: InfiniteMatroidCorollaries.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 185 link it.

See the closest candidates Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem partitional_intersection_counterexample :    Countable E ∧ Infinite E ∧    ∃ Q : Matroid E0, Q.E = univ ∧ Q.dual = Q ∧ IsUniform Q ∧      let M₀ := rayMatroid Q false⋮theorem separate_covering_packing_counterexamples :    (∃ (F : Set E) (M : Bool → Matroid E),      (∀ i, (M i).E = F) ∧ ¬ CoveringConjectureFor F M) ∧    (∃ (F : Set E) (M : Bool → Matroid E),      (∀ i, (M i).E = F) ∧ ¬ PackingConjectureFor F M) := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.InfiniteMatroidCounterexample.D line 8

abbrev D := (m : ℕ) × ((Fin m → Bool) → Bool)

abbreviation OAI.InfiniteMatroidCounterexample.E0 line 9
```
abbrev E0 := D × Bool
```
abbreviation OAI.InfiniteMatroidCounterexample.E line 10
```
abbrev E := ℤ × D
```

definition OAI.InfiniteMatroidCounterexample.contractOnto line 12

def contractOnto {α : Type*} (M : Matroid α) (C : Set α) : Matroid α :=  (M.dual ↾ C).dual

definition OAI.InfiniteMatroidCounterexample.IsUniform line 81

def IsUniform {α : Type*} (M : Matroid α) : Prop :=  ∀ X ⊆ M.E, M.Indep X ∨ M.Spanning X

definition OAI.InfiniteMatroidCounterexample.FamilyCovering line 92

def FamilyCovering {α ι : Type*} (F : Set α) (M : ι → Matroid α) : Prop :=  ∃ I : ι → Set α, (∀ i, (M i).Indep (I i)) ∧ (⋃ i, I i) = F

definition OAI.InfiniteMatroidCounterexample.FamilyPacking line 95

def FamilyPacking {α ι : Type*} (M : ι → Matroid α) : Prop :=  ∃ S : ι → Set α, (Pairwise fun i j ↦ Disjoint (S i) (S j)) ∧    ∀ i, (M i).Spanning (S i)

definition OAI.InfiniteMatroidCounterexample.CoveringConjectureFor line 99

def CoveringConjectureFor {α ι : Type*} (F : Set α) (M : ι → Matroid α) : Prop :=  FamilyCovering F M ↔ ∀ Y ⊆ F,    FamilyPacking (fun i ↦ M i ↾ Y) → FamilyCovering Y (fun i ↦ M i ↾ Y)

definition OAI.InfiniteMatroidCounterexample.PackingConjectureFor line 103

def PackingConjectureFor {α ι : Type*} (F : Set α) (M : ι → Matroid α) : Prop :=  FamilyPacking M ↔ ∀ Y ⊆ F,    FamilyCovering Y (fun i ↦ contractOnto (M i) Y) →    FamilyPacking (fun i ↦ contractOnto (M i) Y)

No confident match with a paper statement

No paper theorem is paired with this Lean statement.

Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Other possible paper matches, closest wording first

Theorem 1.1 of A Counterexample to the Infinite Matroid Packing/Covering Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: covering, packing, matroid, infinite and conjecture. Found in the paper’s title, which its main theorem states: counterexample.
Corollary 7.1 of A Counterexample to the Infinite Matroid Packing/Covering Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: separate, covering, packing and conjectures.
Corollary 1.2 of A Counterexample to the Infinite Matroid Packing/Covering Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: partitional, intersection, countable, conjecture, infinite and matroids.
Theorem 5.5 of A Counterexample to the Infinite Matroid Packing/Covering Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: dual, family and matroid.
Proposition 5.4 of A Counterexample to the Infinite Matroid Packing/Covering Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: family and infinite.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 51 s; peak memory 6.1 GiB).
Reported 11 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Classical.choice, Quot.sound

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 185’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: InfiniteMatroid.main gives countable infinite E, two self-dual matroids with ∀ I₀ I₁ indep, I₀ ∪ I₁ ≠ univ and ¬ HasPackingCovering M₀ M₁; partitional_intersection_counterexample adds IsPartitional M₀ ∧ IsPartitional M₁ ∧ ¬ HasIntersection M₀ M₁ (answering Joó's partitional question), and separate_covering_packing_counterexamples refutes the covering and packing conjectures separately. Uses Mathlib's infinite Matroid; the conjecture predicates (HasIntersection, HasPackingCovering, contractOnto M C := (M.dual ↾ C).dual) are written out by hand and look standard.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“Separate formalized consequences give counterexamples to the individual covering and packing conjectures. The construction is in ordinary set theory with Choice and assumes no finitary restriction on the matroids.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a consequence of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

128 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 61 to its own declarations, 75 to Mathlib’s documentation. 21 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves partitional_intersection_counterexample exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches 9 of the file’s 19 other declarations; with the theorem itself they span 35 lines. 5 of them are named in the statement itself.

Names in the namespace OAI.InfiniteMatroidCounterexample are listed without it.

abbrev Dline 8library
abbrev E0line 9in the statementlibrary
abbrev Eline 10in the statementlibrary
def contractOntoline 12library
def IsUniformline 81in the statementlibrary
def FamilyCoveringline 92library
def FamilyPackingline 95library
def CoveringConjectureForline 99in the statementlibrary
def PackingConjectureForline 103in the statementlibrary
theorem partitional_intersection_counterexampleline 108selected theoremlibrary
theorem separate_covering_packing_counterexamplesline 119selected theoremlibrary

10 declarations the statement does not reach by name

These definitions and lemmas were not found by following names in the statement. Lean may still use them through rules such as instances or macros.

def HasPackingCovering line 15
def bundleAt line 24
def parityBit line 27
abbrev Vertex line 28
def componentMap line 30
def globalComponentMap line 33
def componentEquiv line 36
def rayMatroid line 78
def IsPartitional line 84
def HasIntersection line 88

import Mathlibnamespace OAInamespace InfiniteMatroidCounterexampleopen Set Matroidabbrev D := (m : ℕ) × ((Fin m → Bool) → Bool)abbrev E0 := D × Boolabbrev E := ℤ × Ddef contractOnto {α : Type*} (M : Matroid α) (C : Set α) : Matroid α :=  (M.dual ↾ C).dualdef HasPackingCovering {α : Type*} (M₀ M₁ : Matroid α) : Prop :=  ∃ P C S₀ S₁ I₀ I₁ : Set α,    Disjoint P C ∧ P ∪ C = M₀.E ∧ M₁.E = M₀.E ∧    S₀ ⊆ P ∧ S₁ ⊆ P ∧ Disjoint S₀ S₁ ∧    (M₀ ↾ P).Spanning S₀ ∧ (M₁ ↾ P).Spanning S₁ ∧    I₀ ⊆ C ∧ I₁ ⊆ C ∧    (contractOnto M₀ C).Indep I₀ ∧    (contractOnto M₁ C).Indep I₁ ∧ I₀ ∪ I₁ = Cdef bundleAt (n : ℤ) (b : Bool) : ℤ :=  if n < 0 then (if b then n - 1 else n) else (if b then n else n - 1)def parityBit (p : Bool) : ℤ := if p then 1 else 0abbrev Vertex (p : Bool) := {n : ℤ // n % 2 = parityBit p}def componentMap {α : Type*} {p : Bool} (v : Vertex p) (x : α × Bool) : ℤ × α :=  (bundleAt v x.2,x.1)def globalComponentMap (α : Type*) (p : Bool) (x : Vertex p × (α × Bool)) : ℤ × α :=  componentMap x.1 x.2noncomputable def componentEquiv (α : Type*) (p : Bool) : (Vertex p × (α × Bool)) ≃ (ℤ × α) :=  Equiv.ofBijective (globalComponentMap α p) (by    have bundleAt_bounds (n : ℤ) (b : Bool) : n - 1 ≤ bundleAt n b ∧ bundleAt n b ≤ n := by      unfold bundleAt      split_ifs <;> omega    have bundleAt_injective (n : ℤ) : Function.Injective (bundleAt n) := by      intro b c h      cases b <;> cases c <;> simp_all [bundleAt] <;> split_ifs at h <;> omega    have bundleAt_right (n : ℤ) : ∃ b, bundleAt n b = n := by      by_cases h : n < 0      · exact ⟨false,by simp [bundleAt,h]⟩      · exact ⟨true,by simp [bundleAt,h]⟩    have bundleAt_left (n : ℤ) : ∃ b, bundleAt n b = n - 1 := by      by_cases h : n < 0      · exact ⟨true,by simp [bundleAt,h]⟩      · exact ⟨false,by simp [bundleAt,h]⟩    have globalComponentMap_injective :        Function.Injective (globalComponentMap α p) := by      rintro ⟨⟨n,hn⟩,d,b⟩ ⟨⟨m,hm⟩,e,c⟩ h      have hnm : bundleAt n b = bundleAt m c := congrArg Prod.fst h      have hde : d = e := congrArg Prod.snd h      have hb := bundleAt_bounds n b      have hc := bundleAt_bounds m c      have hnm' : n = m := by omega      subst m      have hbc := bundleAt_injective n hnm      subst e; subst c      rfl    have globalComponentMap_surjective :        Function.Surjective (globalComponentMap α p) := by      rintro ⟨n,d⟩      by_cases hn : n % 2 = parityBit p      · obtain ⟨b,hb⟩ := bundleAt_right n        exact ⟨(⟨n,hn⟩,d,b),by simp [globalComponentMap,componentMap,hb]⟩      · have hn' : (n+1) % 2 = parityBit p := by          cases p <;> simp only [parityBit,Bool.false_eq_true,ite_false,ite_true] at hn ⊢ <;> omega        obtain ⟨b,hb⟩ := bundleAt_left (n+1)        exact ⟨(⟨n+1,hn'⟩,d,b),by simp [globalComponentMap,componentMap,hb]⟩    exact ⟨globalComponentMap_injective, globalComponentMap_surjective⟩)noncomputable def rayMatroid {α : Type*} (Q : Matroid (α × Bool)) (p : Bool) : Matroid (ℤ × α) :=  (Matroid.sum' (fun _ : Vertex p ↦ Q)).mapEquiv (componentEquiv α p)def IsUniform {α : Type*} (M : Matroid α) : Prop :=  ∀ X ⊆ M.E, M.Indep X ∨ M.Spanning Xdef IsPartitional {α : Type} (M : Matroid α) : Prop :=  ∃ (ι : Type) (β : ι → Type) (N : ∀ i, Matroid (β i)) (e : (Σ i, β i) ≃ α),    (∀ i, IsUniform (N i)) ∧ M = (Matroid.sigma N).mapEquiv edef HasIntersection {α : Type*} (M N : Matroid α) : Prop :=  ∃ J J₀ J₁ : Set α, M.Indep J ∧ N.Indep J ∧    Disjoint J₀ J₁ ∧ J₀ ∪ J₁ = J ∧ M.closure J₀ ∪ N.closure J₁ = M.Edef FamilyCovering {α ι : Type*} (F : Set α) (M : ι → Matroid α) : Prop :=  ∃ I : ι → Set α, (∀ i, (M i).Indep (I i)) ∧ (⋃ i, I i) = Fdef FamilyPacking {α ι : Type*} (M : ι → Matroid α) : Prop :=  ∃ S : ι → Set α, (Pairwise fun i j ↦ Disjoint (S i) (S j)) ∧    ∀ i, (M i).Spanning (S i)def CoveringConjectureFor {α ι : Type*} (F : Set α) (M : ι → Matroid α) : Prop :=  FamilyCovering F M ↔ ∀ Y ⊆ F,    FamilyPacking (fun i ↦ M i ↾ Y) → FamilyCovering Y (fun i ↦ M i ↾ Y)def PackingConjectureFor {α ι : Type*} (F : Set α) (M : ι → Matroid α) : Prop :=  FamilyPacking M ↔ ∀ Y ⊆ F,    FamilyCovering Y (fun i ↦ contractOnto (M i) Y) →    FamilyPacking (fun i ↦ contractOnto (M i) Y)theorem partitional_intersection_counterexample :    Countable E ∧ Infinite E ∧    ∃ Q : Matroid E0, Q.E = univ ∧ Q.dual = Q ∧ IsUniform Q ∧      let M₀ := rayMatroid Q false      let M₁ := rayMatroid Q true      M₀.E = univ ∧ M₁.E = univ ∧ M₀.dual = M₀ ∧ M₁.dual = M₁ ∧      IsPartitional M₀ ∧ IsPartitional M₁ ∧      (∀ I₀ I₁ : Set E, M₀.Indep I₀ → M₁.Indep I₁ → I₀ ∪ I₁ ≠ univ) ∧      ¬ HasPackingCovering M₀ M₁ ∧ ¬ HasIntersection M₀ M₁ := by  sorrytheorem separate_covering_packing_counterexamples :    (∃ (F : Set E) (M : Bool → Matroid E),      (∀ i, (M i).E = F) ∧ ¬ CoveringConjectureFor F M) ∧    (∃ (F : Set E) (M : Bool → Matroid E),      (∀ i, (M i).E = F) ∧ ¬ PackingConjectureFor F M) := by  sorryend InfiniteMatroidCounterexampleend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

A Counterexample to the Infinite Matroid Packing/Covering Conjecture

Theorem

In ZFC, there exist a countably infinite set E and two infinite matroids M_0,M_1 on E such that M_i^*=M_i on the same labelled ground set for i=0,1, and

I_0\cup I_1\ne E \quad\text{for every independent }I_0\text{ in }M_0 \text{ and independent }I_1\text{ in }M_1.

The pair (M_0,M_1) has no packing/covering partition. Consequently, the infinite matroid packing/covering conjecture is false.

Section 1, Introduction Read it in the paper TeX source, line 32

Show the TeX

In ZFC, there exist a countably infinite set $E$ and two infinite matroids
$M_0,M_1$ on $E$ such that $M_i^*=M_i$ on the same labelled ground set for
$i=0,1$, and
\[
 I_0\cup I_1\ne E
 \quad\text{for every independent }I_0\text{ in }M_0
 \text{ and independent }I_1\text{ in }M_1.
\]
The pair $(M_0,M_1)$ has no packing/covering partition. Consequently, the
infinite matroid packing/covering conjecture is false.

Corollary

The same pair (M_0,M_1) constructed in Theorem thm:main consists of partitional matroids and does not satisfy intersection. More explicitly, for every set J independent in both M_0 and M_1 and every partition J=J_0\dot\cup J_1,

\cl_{M_0}(J_0)\cup\cl_{M_1}(J_1)\ne E.

Consequently, the unrestricted infinite Matroid Intersection Conjecture is false in ZFC. The pair also gives a negative ZFC answer to Joó’s Question 1.5 for two partitional matroids on a common countable ground set.

Section 1, Introduction Read it in the paper TeX source, line 103

Show the TeX

The same pair $(M_0,M_1)$ constructed in Theorem~\ref{thm:main} consists of
partitional matroids and does not satisfy intersection. More explicitly,
for every set $J$ independent in both $M_0$ and $M_1$ and every partition
$J=J_0\dot\cup J_1$,
\[
 \cl_{M_0}(J_0)\cup\cl_{M_1}(J_1)\ne E.
\]
Consequently, the unrestricted infinite Matroid Intersection Conjecture is
false in ZFC. The pair also gives a negative ZFC answer to Jo\'o's
Question~1.5 for two partitional matroids on a common countable ground set.

Configuration and file facts

File: lean/ComparatorChallenges/InfiniteMatroidCorollaries.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 185 link it.
Solution module: OAI.Combinatorics.InfiniteMatroid.Corollaries
Allowed axioms: propext, Classical.choice, Quot.soundThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 19 local vocabulary declarations
Lines of Lean, without comments or blank lines: 98
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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