Lean statement

One-third stability of Brenier maps and sharpness

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.Problem358.one_third_stability OAI.Problem358.one_third_exponent_is_sharp

Lean statement file: Brenier.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 374 link it.

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

Problems reported

Problem reported Reported as broken in the same way: its definitions are listed as holes, so a trivial replacement makes the checked statement vacuous. Alex Meiburg, Several of the comparator challenges in the OpenAI release are broken and hackable, 2026-10-08 Source

The Lean statement

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

theorem one_third_stability :    ∀ (dimension : ℕ) (source container : Set (E dimension)),      2 ≤ dimension → IsCompact source → Convex ℝ source → (interior source).Nonempty →      IsCompact container → container.Nonempty →      ∃ constant : ℝ, 0 ≤ constant ∧        ∀ (first second : Measure (E dimension)),          IsProbabilityMeasure first → IsProbabilityMeasure second →          IsSupported first container → IsSupported second container →          ∃ firstMap secondMap : E dimension → E dimension,            IsUniqueQuadraticOptimalMap (uniformMeasure source) first firstMap ∧            IsUniqueQuadraticOptimalMap (uniformMeasure source) second secondMap ∧            mapL2Dist (uniformMeasure source) firstMap secondMap ≤              constant * Real.rpow (wasserstein2 first second) (1 / 3 : ℝ) := by  sorry⋮theorem one_third_exponent_is_sharp :    ∀ (dimension : ℕ), 2 ≤ dimension → ∀ exponent : ℝ, (1 / 3 : ℝ) < exponent →      ∀ constant : ℝ, 0 < constant →        ∃ (first second : Measure (E dimension)) (firstMap secondMap : E dimension → E dimension),          IsProbabilityMeasure first ∧ IsProbabilityMeasure second ∧          IsSupported first (cube dimension) ∧ IsSupported second (cube dimension) ∧          HasExactlyThreeAtoms first (cube dimension) ∧          HasExactlyThreeAtoms second (cube dimension) ∧          first ≠ second ∧          IsUniqueQuadraticOptimalMap (uniformMeasure (cube dimension)) first firstMap ∧          IsUniqueQuadraticOptimalMap (uniformMeasure (cube dimension)) second secondMap ∧          0 < wasserstein2 first second ∧          constant * Real.rpow (wasserstein2 first second) exponent <            mapL2Dist (uniformMeasure (cube dimension)) firstMap secondMap := 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.Problem358.E line 12

abbrev E (dimension : ℕ) := EuclideanSpace ℝ (Fin dimension)

definition OAI.Problem358.uniformMeasure line 14

def uniformMeasure {dimension : ℕ} (source : Set (E dimension)) : Measure (E dimension) :=  (volume source)⁻¹ • volume.restrict source

definition OAI.Problem358.IsSupported line 17

def IsSupported {dimension : ℕ} (measure : Measure (E dimension))    (container : Set (E dimension)) : Prop :=  measure (containerᶜ) = 0

definition OAI.Problem358.IsCoupling line 21

def IsCoupling {dimension : ℕ} (source target : Measure (E dimension))    (plan : Measure (E dimension × E dimension)) : Prop :=  IsProbabilityMeasure plan ∧    Measure.map Prod.fst plan = source ∧    Measure.map Prod.snd plan = target

definition OAI.Problem358.quadraticPlanCost line 27

def quadraticPlanCost {dimension : ℕ} (plan : Measure (E dimension × E dimension)) :    ℝ≥0∞ :=  ∫⁻ pair, ENNReal.ofReal (‖pair.1 - pair.2‖ ^ 2) ∂plan

definition OAI.Problem358.wasserstein2 line 31

def wasserstein2 {dimension : ℕ} (source target : Measure (E dimension)) : ℝ :=  Real.sqrt    (ENNReal.toReal      (sInf {cost : ℝ≥0∞ |        ∃ plan : Measure (E dimension × E dimension),          IsCoupling source target plan ∧ quadraticPlanCost plan = cost}))

definition OAI.Problem358.graphPlan line 38

def graphPlan {dimension : ℕ} (source : Measure (E dimension))    (transport : E dimension → E dimension) : Measure (E dimension × E dimension) :=  Measure.map (fun point => (point, transport point)) source

definition OAI.Problem358.IsQuadraticOptimalMap line 42

def IsQuadraticOptimalMap {dimension : ℕ} (source target : Measure (E dimension))    (transport : E dimension → E dimension) : Prop :=  Measurable transport ∧    Measure.map transport source = target ∧    ∀ plan : Measure (E dimension × E dimension),      IsCoupling source target plan →        quadraticPlanCost (graphPlan source transport) ≤ quadraticPlanCost plan

definition OAI.Problem358.IsUniqueQuadraticOptimalMap line 50

def IsUniqueQuadraticOptimalMap {dimension : ℕ} (source target : Measure (E dimension))    (transport : E dimension → E dimension) : Prop :=  IsQuadraticOptimalMap source target transport ∧    ∀ plan : Measure (E dimension × E dimension),      IsCoupling source target plan →        quadraticPlanCost plan ≤ quadraticPlanCost (graphPlan source transport) →          plan = graphPlan source transport

definition OAI.Problem358.mapL2Dist line 58

def mapL2Dist {dimension : ℕ} (source : Measure (E dimension))    (first second : E dimension → E dimension) : ℝ :=  Real.sqrt (∫ point, ‖first point - second point‖ ^ 2 ∂source)

definition OAI.Problem358.cube line 62

def cube (dimension : ℕ) : Set (E dimension) :=  {point | ∀ index, |point index| ≤ 1}

definition OAI.Problem358.HasExactlyThreeAtoms line 65

def HasExactlyThreeAtoms {dimension : ℕ} (measure : Measure (E dimension))    (container : Set (E dimension)) : Prop :=  ∃ atoms : Fin 3 → E dimension, Function.Injective atoms ∧    (∀ index, atoms index ∈ container) ∧    ∃ weights : Fin 3 → ℝ≥0∞, (∀ index, 0 < weights index) ∧      (∑ index, weights index) = 1 ∧      measure = ∑ index, weights index • Measure.dirac (atoms index)

Definitions taken from OpenAI’s library 12

Comparator compares only the name and type of each definition listed here. When a theorem is configured, its proof uses the definition in OpenAI’s solution library. Read that definition as well as the one in this file when comparing the statement with the paper.

Comparator checks the name and type of E; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.
Comparator checks the name and type of uniformMeasure; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.
Comparator checks the name and type of IsSupported; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.
Comparator checks the name and type of IsCoupling; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.
Comparator checks the name and type of quadraticPlanCost; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.
Comparator checks the name and type of wasserstein2; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.
Comparator checks the name and type of graphPlan; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.
Comparator checks the name and type of IsQuadraticOptimalMap; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.
Comparator checks the name and type of IsUniqueQuadraticOptimalMap; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.
Comparator checks the name and type of mapL2Dist; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.
Comparator checks the name and type of cube; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.
Comparator checks the name and type of HasExactlyThreeAtoms; its definition is in OpenAI’s library, in the modules that OAI.Analysis.Brenier.Stability imports.

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: exactly, constant, atoms, three, exponent, 1/3, targets and prob. Found in the paper’s title, which its main theorem states: one, third, sharp, stability and maps.

Theorem 1.1 of Sharp One-Third Stability of Brenier Maps

Theorem

For every K,Y as above there is a finite constant C(K,Y) such that

\|T_\mu-T_\nu\|_{L^2(\rho)} \le C(K,Y)\,W_2(\mu,\nu)^{1/3} \qquad\bigl(\mu,\nu\in\Prob(Y)\bigr).

The targets may be atomic, singular, or absolutely continuous. The exponent 1/3 cannot be replaced by any larger exponent in general, even for K=Y=[-1,1]^d and targets with exactly three atoms.

Section 1, Introduction Read it in the paper TeX source, line 31

Show the TeX

For every \(K,Y\) as above there is a finite constant \(C(K,Y)\) such that
\[
 \|T_\mu-T_\nu\|_{L^2(\rho)}
 \le C(K,Y)\,W_2(\mu,\nu)^{1/3}
 \qquad\bigl(\mu,\nu\in\Prob(Y)\bigr).
\]
The targets may be atomic, singular, or absolutely continuous. The
exponent \(1/3\) cannot be replaced by any larger exponent in general,
even for \(K=Y=[-1,1]^d\) and targets with exactly three atoms.

Other possible paper matches, closest wording first

Proposition 6.3 of Sharp One-Third Stability of Brenier Maps. Words of the Lean statement, its names split into words, that also appear in the theorem: quadratic, optimal, point, unique, map and prob.
Proposition 2.2 of Sharp One-Third Stability of Brenier Maps. Words of the Lean statement, its names split into words, that also appear in the theorem: exactly, atoms, three, exponent, 1/3, targets and one more.
Corollary 6.2 of Sharp One-Third Stability of Brenier Maps. Words of the Lean statement, its names split into words, that also appear in the theorem: sqrt, stability, supported, 1/3, target and probabilities.
Proposition 5.1 of Sharp One-Third Stability of Brenier Maps. Words of the Lean statement, its names split into words, that also appear in the theorem: convex, functions, sqrt, supported and probabilities.
Proposition 3.1 of Sharp One-Third Stability of Brenier Maps. Words of the Lean statement, its names split into words, that also appear in the theorem: graph, constant and maps.
Proposition 3.2 of Sharp One-Third Stability of Brenier Maps. Words of the Lean statement, its names split into words, that also appear in the theorem: weights, unique and function.

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.

Definitions compared by name and type only (12): E, uniformMeasure, IsSupported, IsCoupling, quadraticPlanCost, wasserstein2, graphPlan, IsQuadraticOptimalMap, IsUniqueQuadraticOptimalMap, mapL2Dist, cube, HasExactlyThreeAtoms. Their bodies come from OpenAI’s solution library.

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 374’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: All 12 definition_names were read: uniformMeasure ((vol S)⁻¹·vol|_S), IsCoupling, quadraticPlanCost (∫⁻ ofReal ‖x-y‖²), wasserstein2 (√ toReal sInf of coupling costs), graphPlan, IsQuadraticOptimalMap, IsUniqueQuadraticOptimalMap (optimal and every coupling of cost ≤ it equals the graph plan), mapL2Dist, cube, HasExactlyThreeAtoms, IsSupported, E; all standard. one_third_stability gives, for d ≥ 2, any compact convex source with interior and any compact nonempty container, one constant with ‖T_μ - T_ν‖_{L²(uniform)} ≤ C·W₂(μ,ν)^(1/3) for all probability targets in the container (unique optimal maps exist), and one_third_exponent_is_sharp shows for every exponent > 1/3 and constant, three-atom targets in the cube violate the bound (so the conjectured square root fails).
Definitions the group asks readers to compare with the paper: wasserstein2 and mapL2Dist use ENNReal.toReal/Bochner integrals (value 0 if infinite or non-integrable), harmless here because targets lie in a compact container so all quantities are finite
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 one-third Hölder stability of Brenier maps from the uniform measure on a compact convex body with nonempty interior in $\mathbb R^d$, for every $d\ge2$. For targets supported in one fixed nonempty compact set, the $L^2$ distance between the unique quadratic optimal maps is at most a constant times the one-third power of the targets' $2$-Wasserstein distance. The constant is uniform over those targets.”
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

108 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: 81 to its own declarations, 53 to Mathlib’s documentation. 14 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_third_stability 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. For a definition marked “name and type only”, Comparator does not compare the definition’s body with the one here. When a theorem is configured, it verifies the proof using the definition in OpenAI’s library. Follow the library link to compare the two definitions.

It reaches all 12 other declarations of the file; with the theorem itself they span 78 lines. 8 of them are named in the statement itself.

Names in the namespace OAI.Problem358 are listed without it.

abbrev Eline 12in the statementname and type onlylibrary
def uniformMeasureline 14in the statementname and type onlylibrary
def IsSupportedline 17in the statementname and type onlylibrary
def IsCouplingline 21name and type onlylibrary
def quadraticPlanCostline 27name and type onlylibrary
def wasserstein2line 31in the statementname and type onlylibrary
def graphPlanline 38name and type onlylibrary
def IsQuadraticOptimalMapline 42name and type onlylibrary
def IsUniqueQuadraticOptimalMapline 50in the statementname and type onlylibrary
def mapL2Distline 58in the statementname and type onlylibrary
def cubeline 62in the statementname and type onlylibrary
def HasExactlyThreeAtomsline 65in the statementname and type onlylibrary
theorem one_third_stabilityline 73selected theoremlibrary
theorem one_third_exponent_is_sharpline 88selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen MeasureTheoryopen scoped BigOperators ENNRealnamespace Problem358abbrev E (dimension : ℕ) := EuclideanSpace ℝ (Fin dimension)def uniformMeasure {dimension : ℕ} (source : Set (E dimension)) : Measure (E dimension) :=  (volume source)⁻¹ • volume.restrict sourcedef IsSupported {dimension : ℕ} (measure : Measure (E dimension))    (container : Set (E dimension)) : Prop :=  measure (containerᶜ) = 0def IsCoupling {dimension : ℕ} (source target : Measure (E dimension))    (plan : Measure (E dimension × E dimension)) : Prop :=  IsProbabilityMeasure plan ∧    Measure.map Prod.fst plan = source ∧    Measure.map Prod.snd plan = targetdef quadraticPlanCost {dimension : ℕ} (plan : Measure (E dimension × E dimension)) :    ℝ≥0∞ :=  ∫⁻ pair, ENNReal.ofReal (‖pair.1 - pair.2‖ ^ 2) ∂plandef wasserstein2 {dimension : ℕ} (source target : Measure (E dimension)) : ℝ :=  Real.sqrt    (ENNReal.toReal      (sInf {cost : ℝ≥0∞ |        ∃ plan : Measure (E dimension × E dimension),          IsCoupling source target plan ∧ quadraticPlanCost plan = cost}))def graphPlan {dimension : ℕ} (source : Measure (E dimension))    (transport : E dimension → E dimension) : Measure (E dimension × E dimension) :=  Measure.map (fun point => (point, transport point)) sourcedef IsQuadraticOptimalMap {dimension : ℕ} (source target : Measure (E dimension))    (transport : E dimension → E dimension) : Prop :=  Measurable transport ∧    Measure.map transport source = target ∧    ∀ plan : Measure (E dimension × E dimension),      IsCoupling source target plan →        quadraticPlanCost (graphPlan source transport) ≤ quadraticPlanCost plandef IsUniqueQuadraticOptimalMap {dimension : ℕ} (source target : Measure (E dimension))    (transport : E dimension → E dimension) : Prop :=  IsQuadraticOptimalMap source target transport ∧    ∀ plan : Measure (E dimension × E dimension),      IsCoupling source target plan →        quadraticPlanCost plan ≤ quadraticPlanCost (graphPlan source transport) →          plan = graphPlan source transportdef mapL2Dist {dimension : ℕ} (source : Measure (E dimension))    (first second : E dimension → E dimension) : ℝ :=  Real.sqrt (∫ point, ‖first point - second point‖ ^ 2 ∂source)def cube (dimension : ℕ) : Set (E dimension) :=  {point | ∀ index, |point index| ≤ 1}def HasExactlyThreeAtoms {dimension : ℕ} (measure : Measure (E dimension))    (container : Set (E dimension)) : Prop :=  ∃ atoms : Fin 3 → E dimension, Function.Injective atoms ∧    (∀ index, atoms index ∈ container) ∧    ∃ weights : Fin 3 → ℝ≥0∞, (∀ index, 0 < weights index) ∧      (∑ index, weights index) = 1 ∧      measure = ∑ index, weights index • Measure.dirac (atoms index)theorem one_third_stability :    ∀ (dimension : ℕ) (source container : Set (E dimension)),      2 ≤ dimension → IsCompact source → Convex ℝ source → (interior source).Nonempty →      IsCompact container → container.Nonempty →      ∃ constant : ℝ, 0 ≤ constant ∧        ∀ (first second : Measure (E dimension)),          IsProbabilityMeasure first → IsProbabilityMeasure second →          IsSupported first container → IsSupported second container →          ∃ firstMap secondMap : E dimension → E dimension,            IsUniqueQuadraticOptimalMap (uniformMeasure source) first firstMap ∧            IsUniqueQuadraticOptimalMap (uniformMeasure source) second secondMap ∧            mapL2Dist (uniformMeasure source) firstMap secondMap ≤              constant * Real.rpow (wasserstein2 first second) (1 / 3 : ℝ) := by  sorrytheorem one_third_exponent_is_sharp :    ∀ (dimension : ℕ), 2 ≤ dimension → ∀ exponent : ℝ, (1 / 3 : ℝ) < exponent →      ∀ constant : ℝ, 0 < constant →        ∃ (first second : Measure (E dimension)) (firstMap secondMap : E dimension → E dimension),          IsProbabilityMeasure first ∧ IsProbabilityMeasure second ∧          IsSupported first (cube dimension) ∧ IsSupported second (cube dimension) ∧          HasExactlyThreeAtoms first (cube dimension) ∧          HasExactlyThreeAtoms second (cube dimension) ∧          first ≠ second ∧          IsUniqueQuadraticOptimalMap (uniformMeasure (cube dimension)) first firstMap ∧          IsUniqueQuadraticOptimalMap (uniformMeasure (cube dimension)) second secondMap ∧          0 < wasserstein2 first second ∧          constant * Real.rpow (wasserstein2 first second) exponent <            mapL2Dist (uniformMeasure (cube dimension)) firstMap secondMap := by  sorryend Problem358endend 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.

Sharp One-Third Stability of Brenier Maps

Theorem

For every K,Y as above there is a finite constant C(K,Y) such that

\|T_\mu-T_\nu\|_{L^2(\rho)} \le C(K,Y)\,W_2(\mu,\nu)^{1/3} \qquad\bigl(\mu,\nu\in\Prob(Y)\bigr).

The targets may be atomic, singular, or absolutely continuous. The exponent 1/3 cannot be replaced by any larger exponent in general, even for K=Y=[-1,1]^d and targets with exactly three atoms.

Section 1, Introduction Read it in the paper TeX source, line 31

Show the TeX

For every \(K,Y\) as above there is a finite constant \(C(K,Y)\) such that
\[
 \|T_\mu-T_\nu\|_{L^2(\rho)}
 \le C(K,Y)\,W_2(\mu,\nu)^{1/3}
 \qquad\bigl(\mu,\nu\in\Prob(Y)\bigr).
\]
The targets may be atomic, singular, or absolutely continuous. The
exponent \(1/3\) cannot be replaced by any larger exponent in general,
even for \(K=Y=[-1,1]^d\) and targets with exactly three atoms.

Configuration and file facts

File: lean/ComparatorChallenges/Brenier.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 374 link it.
Solution module: OAI.Analysis.Brenier.Stability
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: 12 local vocabulary declarations
Lines of Lean, without comments or blank lines: 78
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.

Plectis Atlas