Lean statement

Sharp integral fillings in proper $\mathrm{CAT}(0)$ spaces

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.CAT0Fillings.sharp_integral_filling

Lean statement file: SharpCAT0Filling.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 337 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 sharp_integral_filling    (X : Type u) [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X]    (hX : IsCAT0 X) (n : ℕ) (hn : 2 ≤ n) (T : IntegralCurrent X n)    (hTc : CompactlySupported T.val) (hTz : IsCycle T.val) :    ∃ S : IntegralCurrent X (n + 1), CompactlySupported S.val ∧      boundarySucc S.val = T.val ∧      mass S.val ≤ fillingCoefficient n * (mass T.val) ^ fillingPower n := 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.CAT0Fillings.Euc line 14

abbrev Euc (k : ℕ) := EuclideanSpace ℝ (Fin k)

abbreviation OAI.CAT0Fillings.Functional line 15

abbrev Functional (X : Type u) (k : ℕ) := (X → ℝ) → (Fin k → X → ℝ) → ℝ

definition OAI.CAT0Fillings.BoundedLip line 21

def BoundedLip (b : X → ℝ) : Prop :=  (∃ K : ℝ≥0, LipschitzWith K b) ∧ ∃ M : ℝ, ∀ x, |b x| ≤ M

definition OAI.CAT0Fillings.Admissible line 24

def Admissible {k : ℕ} (b : X → ℝ) (π : Fin k → X → ℝ) : Prop :=  BoundedLip b ∧ ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)

definition OAI.CAT0Fillings.Controls line 27

def Controls {k : ℕ} (T : Functional X k) (μ : Measure X) : Prop :=  ∀ (b : X → ℝ) (π : Fin k → X → ℝ), BoundedLip b →    (∀ i, LipschitzWith 1 (π i)) →    |T b π| ≤ ∫ x, |b x| ∂μ

definition OAI.CAT0Fillings.mass line 32

noncomputable def mass {k : ℕ} (T : Functional X k) : ℝ :=  sInf {r : ℝ | ∃ μ : Measure X,    IsFiniteMeasure μ ∧ Controls T μ ∧ r = μ.real univ}

structure OAI.CAT0Fillings.IsMetricCurrent line 36

structure IsMetricCurrent {k : ℕ} (T : Functional X k) : Prop where

Show the remaining 16 lines

  offDomain : ∀ b π, ¬Admissible b π → T b π = 0  linearFirst : ∀ b₁ b₂ π (a c : ℝ), BoundedLip b₁ → BoundedLip b₂ →    (∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) →    T (fun x => a * b₁ x + c * b₂ x) π = a * T b₁ π + c * T b₂ π  linearCoord : ∀ b π i f (a c : ℝ), Admissible b π →    (∃ K : ℝ≥0, LipschitzWith K f) →    T b (Function.update π i (fun x => a * π i x + c * f x)) =      a * T b π + c * T b (Function.update π i f)  sequentialContinuity : ∀ b π (πs : ℕ → Fin k → X → ℝ), BoundedLip b →    (∀ i, ∃ K : ℝ≥0, ∀ j, LipschitzWith K (πs j i)) →    (∀ i x, Tendsto (fun j => πs j i x) atTop (𝓝 (π i x))) →    Tendsto (fun j => T b (πs j)) atTop (𝓝 (T b π))  locality : ∀ b π, Admissible b π →    (∃ (i : Fin k) (U : Set X) (c : ℝ), IsOpen U ∧ Function.support b ⊆ U ∧      ∀ x ∈ U, π i x = c) → T b π = 0  finiteMass : ∃ μ : Measure X, IsFiniteMeasure μ ∧ Controls T μ

definition OAI.CAT0Fillings.boundarySucc line 54

noncomputable def boundarySucc {k : ℕ} (T : Functional X (k + 1)) : Functional X k :=  fun b π => if Admissible b π then T (fun _ => 1) (Matrix.vecCons b π) else 0

definition OAI.CAT0Fillings.boundary line 57

noncomputable def boundary : {k : ℕ} → Functional X k → Functional X (k - 1)  | 0, _ => fun _ _ => 0  | _ + 1, T => boundarySucc T

structure OAI.CAT0Fillings.IntegerChart line 61

structure IntegerChart (X : Type u) [MetricSpace X] (k : ℕ) where  domain : Set (Euc k)  borel : MeasurableSet domain  bounded : Bornology.IsBounded domain  param : domain → X  bilipschitz : ∃ L U : ℝ≥0, LipschitzWith L param ∧ AntilipschitzWith U param  multiplicity : Euc k → ℤ  integrable : Integrable (fun z => (multiplicity z : ℝ)) (volume.restrict domain)

definition OAI.CAT0Fillings.IntegerChart.scalar line 74

noncomputable def scalar (f : X → ℝ) (z : Euc k) : ℝ :=  if hz : z ∈ C.domain then f (C.param ⟨z, hz⟩) else 0

definition OAI.CAT0Fillings.IntegerChart.jacobian line 77

noncomputable def jacobian (π : Fin k → X → ℝ) (z : Euc k) : ℝ :=  Matrix.det (fun i j =>    (fderivWithin ℝ (C.scalar (π i)) C.domain z) (EuclideanSpace.single j 1))

definition OAI.CAT0Fillings.IntegerChart.action line 81

noncomputable def action : Functional X k :=  fun b π => if Admissible b π then    ∫ z in C.domain, (C.multiplicity z : ℝ) * C.scalar b z * C.jacobian π z  else 0

definition OAI.CAT0Fillings.IntegerChart.image line 86
```
def image : Set X := Set.range C.param
```

definition OAI.CAT0Fillings.IntegerRectifiable line 90

def IntegerRectifiable {k : ℕ} (T : Functional X k) : Prop :=  ∃ C : ℕ → IntegerChart X k,    Pairwise (fun i j => Disjoint (C i).image (C j).image) ∧    (∀ i, IsMetricCurrent (C i).action) ∧    Summable (fun i => mass (C i).action) ∧    ∀ b π, T b π = ∑' i, (C i).action b π

definition OAI.CAT0Fillings.IsIntegral line 97

def IsIntegral : (k : ℕ) → Functional X k → Prop  | 0, T => IsMetricCurrent T ∧ IntegerRectifiable T  | _ + 1, T => IsMetricCurrent T ∧ IntegerRectifiable T ∧      IsMetricCurrent (boundarySucc T) ∧ IntegerRectifiable (boundarySucc T)

abbreviation OAI.CAT0Fillings.IntegralCurrent line 102

abbrev IntegralCurrent (X : Type u) [MetricSpace X] [MeasurableSpace X] [BorelSpace X]    (k : ℕ) := { T : Functional X k // IsIntegral k T }

definition OAI.CAT0Fillings.CompactlySupported line 105

def CompactlySupported {k : ℕ} (T : Functional X k) : Prop :=  ∃ K : Set X, IsCompact K ∧    ∀ b π, Admissible b π → (∀ x ∈ K, b x = 0) → T b π = 0

definition OAI.CAT0Fillings.IsCycle line 109

def IsCycle {k : ℕ} (T : Functional X k) : Prop := boundary T = fun _ _ => 0

definition OAI.CAT0Fillings.IsCAT0 line 116

def IsCAT0 : Prop :=  ∃ segment : X → X → ℝ → X,    (∀ x y, segment x y 0 = x ∧ segment x y 1 = y) ∧    (∀ x y s t, s ∈ Icc (0 : ℝ) 1 → t ∈ Icc (0 : ℝ) 1 →      dist (segment x y s) (segment x y t) = |s - t| * dist x y) ∧    (∀ o x y s t, s ∈ Icc (0 : ℝ) 1 → t ∈ Icc (0 : ℝ) 1 →      dist (segment o x s) (segment o y t) ^ 2 ≤        (s * dist o x - t * dist o y) ^ 2 +          s * t * ((dist x y) ^ 2 - (dist o x - dist o y) ^ 2))

definition OAI.CAT0Fillings.omega line 128

noncomputable def omega (k : ℕ) : ℝ :=  (volume (Metric.ball (0 : Euc k) 1)).toReal

definition OAI.CAT0Fillings.sphereArea line 130

noncomputable def sphereArea (n : ℕ) : ℝ := (n + 1 : ℝ) * omega (n + 1)

definition OAI.CAT0Fillings.fillingCoefficient line 131

noncomputable def fillingCoefficient (n : ℕ) : ℝ :=  1 / ((n + 1 : ℝ) * (sphereArea n) ^ (1 / (n : ℝ)))

definition OAI.CAT0Fillings.fillingPower line 133

noncomputable def fillingPower (n : ℕ) : ℝ := (n + 1 : ℝ) / (n : ℝ)

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: compactly, proper, coefficient, multiplicities, sharp, integer, currents, filling and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dimension.

Statements in Sharp integral fillings in CAT(0) spaces come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Sharp integral fillings in CAT(0) spaces

Theorem (Sharp integral filling)

Let X be a proper CAT(0) space and n\ge2 an integer. Every compactly supported T\in\I_n(X) with \partial T=0 admits a compactly supported S\in\I_{n+1}(X) such that

\partial S=T,\qquad \M(S)\le c_n\M(T)^{(n+1)/n} =\frac{\M(T)^{(n+1)/n}} {(n+1)^{(n+1)/n}\omega_{n+1}^{1/n}}.

There is no restriction on the dimension of X or on the integer multiplicities of the currents. The coefficient is optimal.

Section 1, Introduction Read it in the paper TeX source, line 26

Show the TeX

Let $X$ be a proper CAT(0) space and $n\ge2$ an integer.
Every compactly supported $T\in\I_n(X)$ with $\partial T=0$ admits a
compactly supported $S\in\I_{n+1}(X)$ such that
\begin{equation}
 \partial S=T,\qquad
 \M(S)\le c_n\M(T)^{(n+1)/n}
 =\frac{\M(T)^{(n+1)/n}}
 {(n+1)^{(n+1)/n}\omega_{n+1}^{1/n}}.
\end{equation}
There is no restriction on the dimension of $X$ or on the integer
multiplicities of the currents. The coefficient is optimal.

Other possible paper matches, closest wording first

Theorem 2.2 of Sharp integral fillings in CAT(0) spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: restricted, summable, det, pairwise, images, pi and 26 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: times.
Proposition 3.2 of Sharp integral fillings in CAT(0) spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: pi, sharp, cycle, filling, cat, integral and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dimension.
Proposition 2.6 of Sharp integral fillings in CAT(0) spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: action, integration, mass, current, charts, disjoint and one more.
Corollary 1.2 of Sharp integral fillings in CAT(0) spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: domain, area, omega, finite, boundary, compact and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dimension, inequality and ambient.
Proposition 5.1 of Sharp integral fillings in CAT(0) spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: real, scalar, integrable, first, chart, space and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: inequality.
Proposition 4.1 of Sharp integral fillings in CAT(0) spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: integral, boundary, omega, lipschitz 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.

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 337’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: sharp_integral_filling (every compactly supported integral n-cycle, n ≥ 2, in a proper CAT(0) space has a compactly supported integral filling of mass ≤ C_n·mass^((n+1)/n)) and coefficient_optimal (Euclidean sharpness) cover the summary's second claim only. The headline, the sharp Cartan-Hadamard isoperimetric inequality for finite-perimeter sets in complete simply connected manifolds with curvature ≤ κ ≤ 0 (and the Euclidean-ball equality case for κ = 0), has no statement; the docs concede only 'the selected integral-current form' is proved, and the Lean form does not imply the set-based inequality.
Definitions the group asks readers to compare with the paper: Hand-built metric currents: Functional X k := (X → ℝ) → (Fin k → X → ℝ) → ℝ, IsMetricCurrent (off-domain zero, multilinearity, sequential continuity, locality, finite mass), mass T := sInf {μ.real univ | Controls T μ}, IntegerRectifiable via countably many bi-Lipschitz IntegerCharts with integer multiplicity, boundarySucc T b π := if Admissible b π then T 1 (b, π) else 0; a Lipschitz-functional model of Ambrosio-Kirchheim integral currents, not in Mathlib ; IsCAT0 X := ∃ segment, geodesic-parametrization ∧ CN-comparison inequality (hand-built)
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: No Riemannian statement: neither the κ ≤ 0 isoperimetric comparison for finite-perimeter sets nor the κ = 0 equality rigidity is stated. (The κ = 0 inequality in dimension ≥ 3 would follow from the filling theorem via Hadamard manifolds being CAT(0) and the currents of finite-perimeter sets, but that bridge is not stated; κ < 0 and rigidity do not follow.)

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves the sharp Euclidean filling inequality for every compactly supported integral $n$-cycle in a proper $\mathrm{CAT}(0)$ space, for $n\ge2$. It constructs a compactly supported integral filling whose mass is at most $C_n$ times the boundary mass to the power $(n+1)/n$, where $C_n$ is the Euclidean isoperimetric coefficient. Integer multiplicities and the ambient dimension are unrestricted.”
Quoted from OpenAI’s Lean note for this family

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

Supporting material

The complete Lean statement file

146 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: 59 to its own declarations, 75 to Mathlib’s documentation. 25 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. 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 sharp_integral_filling 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 all 24 other declarations of the file; with the theorem itself they span 92 lines. 8 of them are named in the statement itself.

Names in the namespace OAI.CAT0Fillings are listed without it.

abbrev Eucline 14library
abbrev Functionalline 15library
def BoundedLipline 21library
def Admissibleline 24library
def Controlsline 27library
def massline 32in the statementlibrary
structure IsMetricCurrentline 36library
def boundarySuccline 54in the statementlibrary
def boundaryline 57library
structure IntegerChartline 61library
def IntegerChart.scalarline 74library
def IntegerChart.jacobianline 77library
def IntegerChart.actionline 81library
def IntegerChart.imageline 86library
def IntegerRectifiableline 90library
def IsIntegralline 97library
abbrev IntegralCurrentline 102in the statementlibrary
def CompactlySupportedline 105in the statementlibrary
def IsCycleline 109in the statementlibrary
def IsCAT0line 116in the statementlibrary
def omegaline 128library
def sphereArealine 130library
def fillingCoefficientline 131in the statementlibrary
def fillingPowerline 133in the statementlibrary
theorem sharp_integral_fillingline 135selected 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.

attribute [local instance] Classical.propDecidable line 10

import Mathlibnamespace OAIopen Set Filter MeasureTheoryopen scoped Topology ENNReal NNRealnamespace CAT0Fillingsattribute [local instance] Classical.propDecidableuniverse uabbrev Euc (k : ℕ) := EuclideanSpace ℝ (Fin k)abbrev Functional (X : Type u) (k : ℕ) := (X → ℝ) → (Fin k → X → ℝ) → ℝsection MetricCurrentsvariable {X : Type u} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]def BoundedLip (b : X → ℝ) : Prop :=  (∃ K : ℝ≥0, LipschitzWith K b) ∧ ∃ M : ℝ, ∀ x, |b x| ≤ Mdef Admissible {k : ℕ} (b : X → ℝ) (π : Fin k → X → ℝ) : Prop :=  BoundedLip b ∧ ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)def Controls {k : ℕ} (T : Functional X k) (μ : Measure X) : Prop :=  ∀ (b : X → ℝ) (π : Fin k → X → ℝ), BoundedLip b →    (∀ i, LipschitzWith 1 (π i)) →    |T b π| ≤ ∫ x, |b x| ∂μnoncomputable def mass {k : ℕ} (T : Functional X k) : ℝ :=  sInf {r : ℝ | ∃ μ : Measure X,    IsFiniteMeasure μ ∧ Controls T μ ∧ r = μ.real univ}structure IsMetricCurrent {k : ℕ} (T : Functional X k) : Prop where  offDomain : ∀ b π, ¬Admissible b π → T b π = 0  linearFirst : ∀ b₁ b₂ π (a c : ℝ), BoundedLip b₁ → BoundedLip b₂ →    (∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) →    T (fun x => a * b₁ x + c * b₂ x) π = a * T b₁ π + c * T b₂ π  linearCoord : ∀ b π i f (a c : ℝ), Admissible b π →    (∃ K : ℝ≥0, LipschitzWith K f) →    T b (Function.update π i (fun x => a * π i x + c * f x)) =      a * T b π + c * T b (Function.update π i f)  sequentialContinuity : ∀ b π (πs : ℕ → Fin k → X → ℝ), BoundedLip b →    (∀ i, ∃ K : ℝ≥0, ∀ j, LipschitzWith K (πs j i)) →    (∀ i x, Tendsto (fun j => πs j i x) atTop (𝓝 (π i x))) →    Tendsto (fun j => T b (πs j)) atTop (𝓝 (T b π))  locality : ∀ b π, Admissible b π →    (∃ (i : Fin k) (U : Set X) (c : ℝ), IsOpen U ∧ Function.support b ⊆ U ∧      ∀ x ∈ U, π i x = c) → T b π = 0  finiteMass : ∃ μ : Measure X, IsFiniteMeasure μ ∧ Controls T μnoncomputable def boundarySucc {k : ℕ} (T : Functional X (k + 1)) : Functional X k :=  fun b π => if Admissible b π then T (fun _ => 1) (Matrix.vecCons b π) else 0noncomputable def boundary : {k : ℕ} → Functional X k → Functional X (k - 1)  | 0, _ => fun _ _ => 0  | _ + 1, T => boundarySucc Tstructure IntegerChart (X : Type u) [MetricSpace X] (k : ℕ) where  domain : Set (Euc k)  borel : MeasurableSet domain  bounded : Bornology.IsBounded domain  param : domain → X  bilipschitz : ∃ L U : ℝ≥0, LipschitzWith L param ∧ AntilipschitzWith U param  multiplicity : Euc k → ℤ  integrable : Integrable (fun z => (multiplicity z : ℝ)) (volume.restrict domain)namespace IntegerChartvariable {k : ℕ} (C : IntegerChart X k)noncomputable def scalar (f : X → ℝ) (z : Euc k) : ℝ :=  if hz : z ∈ C.domain then f (C.param ⟨z, hz⟩) else 0noncomputable def jacobian (π : Fin k → X → ℝ) (z : Euc k) : ℝ :=  Matrix.det (fun i j =>    (fderivWithin ℝ (C.scalar (π i)) C.domain z) (EuclideanSpace.single j 1))noncomputable def action : Functional X k :=  fun b π => if Admissible b π then    ∫ z in C.domain, (C.multiplicity z : ℝ) * C.scalar b z * C.jacobian π z  else 0def image : Set X := Set.range C.paramend IntegerChartdef IntegerRectifiable {k : ℕ} (T : Functional X k) : Prop :=  ∃ C : ℕ → IntegerChart X k,    Pairwise (fun i j => Disjoint (C i).image (C j).image) ∧    (∀ i, IsMetricCurrent (C i).action) ∧    Summable (fun i => mass (C i).action) ∧    ∀ b π, T b π = ∑' i, (C i).action b πdef IsIntegral : (k : ℕ) → Functional X k → Prop  | 0, T => IsMetricCurrent T ∧ IntegerRectifiable T  | _ + 1, T => IsMetricCurrent T ∧ IntegerRectifiable T ∧      IsMetricCurrent (boundarySucc T) ∧ IntegerRectifiable (boundarySucc T)abbrev IntegralCurrent (X : Type u) [MetricSpace X] [MeasurableSpace X] [BorelSpace X]    (k : ℕ) := { T : Functional X k // IsIntegral k T }def CompactlySupported {k : ℕ} (T : Functional X k) : Prop :=  ∃ K : Set X, IsCompact K ∧    ∀ b π, Admissible b π → (∀ x ∈ K, b x = 0) → T b π = 0def IsCycle {k : ℕ} (T : Functional X k) : Prop := boundary T = fun _ _ => 0end MetricCurrentssection Geometryvariable (X : Type u) [MetricSpace X]def IsCAT0 : Prop :=  ∃ segment : X → X → ℝ → X,    (∀ x y, segment x y 0 = x ∧ segment x y 1 = y) ∧    (∀ x y s t, s ∈ Icc (0 : ℝ) 1 → t ∈ Icc (0 : ℝ) 1 →      dist (segment x y s) (segment x y t) = |s - t| * dist x y) ∧    (∀ o x y s t, s ∈ Icc (0 : ℝ) 1 → t ∈ Icc (0 : ℝ) 1 →      dist (segment o x s) (segment o y t) ^ 2 ≤        (s * dist o x - t * dist o y) ^ 2 +          s * t * ((dist x y) ^ 2 - (dist o x - dist o y) ^ 2))end Geometrynoncomputable def omega (k : ℕ) : ℝ :=  (volume (Metric.ball (0 : Euc k) 1)).toRealnoncomputable def sphereArea (n : ℕ) : ℝ := (n + 1 : ℝ) * omega (n + 1)noncomputable def fillingCoefficient (n : ℕ) : ℝ :=  1 / ((n + 1 : ℝ) * (sphereArea n) ^ (1 / (n : ℝ)))noncomputable def fillingPower (n : ℕ) : ℝ := (n + 1 : ℝ) / (n : ℝ)theorem sharp_integral_filling    (X : Type u) [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X]    (hX : IsCAT0 X) (n : ℕ) (hn : 2 ≤ n) (T : IntegralCurrent X n)    (hTc : CompactlySupported T.val) (hTz : IsCycle T.val) :    ∃ S : IntegralCurrent X (n + 1), CompactlySupported S.val ∧      boundarySucc S.val = T.val ∧      mass S.val ≤ fillingCoefficient n * (mass T.val) ^ fillingPower n := by  sorryend CAT0Fillingsend 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 integral fillings in CAT(0) spaces

Theorem (Sharp integral filling)

Let X be a proper CAT(0) space and n\ge2 an integer. Every compactly supported T\in\I_n(X) with \partial T=0 admits a compactly supported S\in\I_{n+1}(X) such that

\partial S=T,\qquad \M(S)\le c_n\M(T)^{(n+1)/n} =\frac{\M(T)^{(n+1)/n}} {(n+1)^{(n+1)/n}\omega_{n+1}^{1/n}}.

There is no restriction on the dimension of X or on the integer multiplicities of the currents. The coefficient is optimal.

Section 1, Introduction Read it in the paper TeX source, line 26

Show the TeX

Let $X$ be a proper CAT(0) space and $n\ge2$ an integer.
Every compactly supported $T\in\I_n(X)$ with $\partial T=0$ admits a
compactly supported $S\in\I_{n+1}(X)$ such that
\begin{equation}
 \partial S=T,\qquad
 \M(S)\le c_n\M(T)^{(n+1)/n}
 =\frac{\M(T)^{(n+1)/n}}
 {(n+1)^{(n+1)/n}\omega_{n+1}^{1/n}}.
\end{equation}
There is no restriction on the dimension of $X$ or on the integer
multiplicities of the currents. The coefficient is optimal.

Corollary

Let M be a Cartan–Hadamard manifold of dimension d\ge3. For every bounded domain \Omega\subset M with smooth boundary,

\operatorname{area}(\partial\Omega) \ge d\omega_d^{1/d}\vol(\Omega)^{(d-1)/d}.

The same inequality holds for relatively compact sets of finite perimeter, with ambient perimeter in place of boundary area.

Section 1, Introduction Read it in the paper TeX source, line 57

Show the TeX

Let $M$ be a Cartan--Hadamard manifold of dimension $d\ge3$.
For every bounded domain $\Omega\subset M$ with smooth boundary,
\begin{equation}
 \operatorname{area}(\partial\Omega)
 \ge d\omega_d^{1/d}\vol(\Omega)^{(d-1)/d}.
\end{equation}
The same inequality holds for relatively compact sets of finite
perimeter, with ambient perimeter in place of boundary area.

Configuration and file facts

File: lean/ComparatorChallenges/SharpCAT0Filling.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 337 link it.
Solution module: OAI.Geometry.CAT0Fillings.Main
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: 24 local vocabulary declarations
Lines of Lean, without comments or blank lines: 93
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