Lean statement

One-sample matroid prophet inequality against an almighty adversary

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.

Theorem selected for verification: OAI.MatroidProphet.one_sample

Lean statement file: MatroidProphet.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 111 link it.

See the candidate theorem 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 one_sample.{u} : OneSampleChallenge.{u} := 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.MatroidProphet.Weights line 20
```
abbrev Weights (n : ℕ) := Fin n → ℝ
```

definition OAI.MatroidProphet.pairedCoordinates. line 66

def pairedCoordinates.{u} {n : ℕ} {Ω : Type u} (S V : Ω → Weights n)    (i : Fin n × Bool) : Ω → ℝ :=  if i.2 then fun ω => S ω i.1 else fun ω => V ω i.1

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: arrival, finite, independent, labeled, matroid, nonnegative, online, rule and five more.

Theorem 1.1 of One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary

Theorem

There is a measurable online selection rule, independent of the value distributions, with the following property. Let M=(E,\mathcal I) be any finite known labeled matroid, and let (S_e,V_e:e\in E) be mutually independent nonnegative real random variables, with S_e and V_e having the same arbitrary law D_e. Assume \Ex[\OPT_M(V)]<\infty. The rule receives exactly the sample vector S before arrivals and uses a seed R independent of S,V.

For every measurable permutation rule \pi depending on M, the laws (D_e), S, V, and R, the rule accepts an independent set A, irrevocably deciding at each arrival using only the available history, and

\Ex\left[\sum_{e\in A}V_e\right] \ge 2^{-310}\Ex[\OPT_M(V)].

Section 1, Introduction Read it in the paper TeX source, line 24

Show the TeX

There is a measurable online selection rule, independent of the value
distributions, with the following property. Let $M=(E,\mathcal I)$ be any
finite known labeled matroid, and let $(S_e,V_e:e\in E)$ be mutually
independent nonnegative real random variables, with $S_e$ and $V_e$ having
the same arbitrary law $D_e$. Assume
$\Ex[\OPT_M(V)]<\infty$. The rule receives exactly the sample vector $S$
before arrivals and uses a seed $R$ independent of $S,V$.

For every measurable permutation rule $\pi$ depending on $M$, the laws
$(D_e)$, $S$, $V$, and $R$, the rule accepts an independent set $A$,
irrevocably deciding at each arrival using only the available history, and
\begin{equation}
 \Ex\left[\sum_{e\in A}V_e\right]
       \ge 2^{-310}\Ex[\OPT_M(V)].
\end{equation}

Other possible paper matches, closest wording first

Corollary 7.1 of One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: order, choice, expectation, pairs, prefix, may and eleven more.
Proposition 2.1 of One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: feasible and rule.

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

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. 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

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, Quot.sound, Classical.choice

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 111’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: None; the hidden-vector selection result (constant 2^{-293}) is an extra supporting statement.
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

“The formalization proves a one-sample matroid prophet inequality with expected reward at least $2^{-310}$ times the expected offline optimum. Each element has one independent sample paired with an identically distributed nonnegative online value, and all coordinates are independent. The rule and finite seed law depend only on the labeled matroid; choices are irrevocable and feasible after every prefix. The arrival order may depend measurably on all samples, values, and the seed, and only the offline optimum must be integrable.”
Quoted from OpenAI’s Lean note for this family

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

Supporting material

The complete Lean statement file

94 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: 50 to its own declarations, 59 to Mathlib’s documentation. 17 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 one_sample 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 15 of the file’s 16 other declarations; with the theorem itself they span 57 lines. 1 of them is named in the statement itself.

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

abbrev Weightsline 20library
abbrev Seedline 21library
abbrev ArrivalOrderline 22library
abbrev Historyline 25library
structure OnlineRuleline 27library
def prefixIndexline 32library
def historyline 35library
def decisionAtline 38library
def acceptedThroughline 42library
def acceptedline 48library
def rewardline 52library
def optimumline 56library
def Feasibleline 61library
def pairedCoordinatesline 66library
def OneSampleChallengeline 70in the statementlibrary
theorem one_sampleline 89selected theoremlibrary

Other Lean rules that may affect the statement 1

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance arrivalOrderMeasurableSpace line 23

import Mathlibnamespace OAI/-!# One-sample matroid selectionThe rule and its finite seed law depend only on the known labeled matroid.A decision sees the initial sample vector, the entire seed, and the arrivinglabel/value prefix. Every prefix is feasible, and decisions are irrevocable.The arrival permutation may depend on all samples, values, and seed bits.Only the actual offline optimum is required to be integrable.-/open MeasureTheory ProbabilityTheoryopen scoped BigOperatorsnamespace MatroidProphetabbrev Weights (n : ℕ) := Fin n → ℝabbrev Seed (bits : ℕ) := Fin bits → Boolabbrev ArrivalOrder (n : ℕ) := Equiv.Perm (Fin n)instance arrivalOrderMeasurableSpace (n : ℕ) : MeasurableSpace (ArrivalOrder n) := ⊤abbrev History (n : ℕ) (k : Fin n) := Fin (k.val + 1) → Fin n × ℝstructure OnlineRule (n bits : ℕ) where  decide : (k : Fin n) → Seed bits → Weights n → History n k → Bool  measurable_decide : ∀ k, Measurable    (fun x : Seed bits × (Weights n × History n k) => decide k x.1 x.2.1 x.2.2)def prefixIndex {n : ℕ} (k : Fin n) (j : Fin (k.val + 1)) : Fin n :=  ⟨j.val, Nat.lt_of_lt_of_le j.isLt (Nat.succ_le_of_lt k.isLt)⟩def history {n : ℕ} (v : Weights n) (π : ArrivalOrder n) (k : Fin n) : History n k :=  fun j => let e := π (prefixIndex k j); (e, v e)def decisionAt {n bits : ℕ} (A : OnlineRule n bits) (r : Seed bits)    (s v : Weights n) (π : ArrivalOrder n) (k : Fin n) : Bool :=  A.decide k r s (history v π k)noncomputable def acceptedThrough {n bits : ℕ} (A : OnlineRule n bits) (r : Seed bits)    (s v : Weights n) (π : ArrivalOrder n) (t : ℕ) : Finset (Fin n) := by  classical  exact Finset.univ.filter fun e =>    (π.symm e).val < t ∧ decisionAt A r s v π (π.symm e) = truenoncomputable def accepted {n bits : ℕ} (A : OnlineRule n bits) (r : Seed bits)    (s v : Weights n) (π : ArrivalOrder n) : Finset (Fin n) :=  acceptedThrough A r s v π nnoncomputable def reward {n bits : ℕ} (A : OnlineRule n bits) (r : Seed bits)    (s v : Weights n) (π : ArrivalOrder n) : ℝ :=  ∑ e ∈ accepted A r s v π, v enoncomputable def optimum {n : ℕ} (M : Matroid (Fin n)) (v : Weights n) : ℝ := by  classical  exact Finset.univ.sup' Finset.univ_nonempty    (fun I : Finset (Fin n) => if M.Indep (I : Set (Fin n)) then ∑ e ∈ I, v e else 0)def Feasible {n bits : ℕ} (M : Matroid (Fin n)) (A : OnlineRule n bits) : Prop :=  ∀ (r : Seed bits) (s v : Weights n) (π : ArrivalOrder n) (t : ℕ),    (∀ e, 0 ≤ s e) → (∀ e, 0 ≤ v e) →    M.Indep (acceptedThrough A r s v π t : Set (Fin n))def pairedCoordinates.{u} {n : ℕ} {Ω : Type u} (S V : Ω → Weights n)    (i : Fin n × Bool) : Ω → ℝ :=  if i.2 then fun ω => S ω i.1 else fun ω => V ω i.1def OneSampleChallenge.{u} : Prop :=  ∀ (n : ℕ) (M : Matroid (Fin n)), M.E = Set.univ →    ∃ (bits : ℕ) (ν : Measure (Seed bits)), IsProbabilityMeasure ν ∧    ∃ (A : OnlineRule n bits), Feasible M A ∧      ∀ {Ω : Type u} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]        (S V : Ω → Weights n) (R : Ω → Seed bits),        Measurable S → Measurable V → Measurable R →        (∀ᵐ ω ∂μ, ∀ e, 0 ≤ S ω e) →        (∀ᵐ ω ∂μ, ∀ e, 0 ≤ V ω e) →        iIndepFun (pairedCoordinates S V) μ →        (∀ e, Measure.map (fun ω => S ω e) μ = Measure.map (fun ω => V ω e) μ) →        IndepFun (fun ω => (S ω, V ω)) R μ →        Measure.map R μ = ν →        Integrable (fun ω => optimum M (V ω)) μ →        ∀ (π : Ω → ArrivalOrder n), Measurable π →          Integrable (fun ω => reward A (R ω) (S ω) (V ω) (π ω)) μ ∧          ((2 : ℝ) ^ 310)⁻¹ * (∫ ω, optimum M (V ω) ∂μ) ≤            ∫ ω, reward A (R ω) (S ω) (V ω) (π ω) ∂μtheorem one_sample.{u} : OneSampleChallenge.{u} := by  sorryend MatroidProphetend 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.

One Sample Suffices for Matroid Prophet Inequalities against an Almighty Adversary

Theorem

\Ex\left[\sum_{e\in A}V_e\right] \ge 2^{-310}\Ex[\OPT_M(V)].

Section 1, Introduction Read it in the paper TeX source, line 24

Show the TeX

There is a measurable online selection rule, independent of the value
distributions, with the following property. Let $M=(E,\mathcal I)$ be any
finite known labeled matroid, and let $(S_e,V_e:e\in E)$ be mutually
independent nonnegative real random variables, with $S_e$ and $V_e$ having
the same arbitrary law $D_e$. Assume
$\Ex[\OPT_M(V)]<\infty$. The rule receives exactly the sample vector $S$
before arrivals and uses a seed $R$ independent of $S,V$.

For every measurable permutation rule $\pi$ depending on $M$, the laws
$(D_e)$, $S$, $V$, and $R$, the rule accepts an independent set $A$,
irrevocably deciding at each arrival using only the available history, and
\begin{equation}
 \Ex\left[\sum_{e\in A}V_e\right]
       \ge 2^{-310}\Ex[\OPT_M(V)].
\end{equation}

Configuration and file facts

File: lean/ComparatorChallenges/MatroidProphet.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 111 link it.
Solution module: OAI.Probability.MatroidProphet.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe 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: 16 local vocabulary declarations
Lines of Lean, without comments or blank lines: 58
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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