Lean statement

Uniformly discrete Lipschitz-free space with AP but no BAP

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.DiscreteFree.source_endpoints

Lean statement file: DiscreteLipschitzFree.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 330 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 source_endpoints : BlockStatement ∧ MainStatement := 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.

definition OAI.DiscreteFree.lipSubmodule line 16

def lipSubmodule (o : M) : Submodule ℝ (M → ℝ) where

Show the remaining 26 lines

  carrier := {f | f o = 0 ∧ ∃ C : ℝ, 0 ≤ C ∧    ∀ x y, |f x - f y| ≤ C * dist x y}  zero_mem' := by    refine ⟨rfl, 0, le_rfl, ?_⟩    intro x y    simp  add_mem' := by    rintro f g ⟨hfo, C, hC, hf⟩ ⟨hgo, D, hD, hg⟩    refine ⟨by simpa using congrArg₂ (· + ·) hfo hgo,      C + D, add_nonneg hC hD, ?_⟩    intro x y    change |f x + g x - (f y + g y)| ≤ (C + D) * dist x y    calc      _ = |(f x - f y) + (g x - g y)| := by ring_nf      _ ≤ |f x - f y| + |g x - g y| := abs_add_le _ _      _ ≤ C * dist x y + D * dist x y := add_le_add (hf x y) (hg x y)      _ = _ := by ring  smul_mem' := by    rintro c f ⟨hfo, C, hC, hf⟩    refine ⟨by simp [hfo], |c| * C, mul_nonneg (abs_nonneg c) hC, ?_⟩    intro x y    change |c * f x - c * f y| ≤ (|c| * C) * dist x y    calc      _ = |c| * |f x - f y| := by rw [← mul_sub, abs_mul]      _ ≤ |c| * (C * dist x y) := mul_le_mul_of_nonneg_left (hf x y) (abs_nonneg c)      _ = _ := by ring

definition OAI.DiscreteFree.Lip0 line 44
```
def Lip0 (o : M) := ↥(lipSubmodule o)
```

definition OAI.DiscreteFree.Lip0.slope line 55

def slope (o : M) : Lip0 o →ₗ[ℝ] lp (fun _ : M × M => ℝ) ∞ where

Show the remaining 22 lines

  toFun f := ⟨fun z : M × M => (f z.1 - f z.2) / dist z.1 z.2, by    obtain ⟨C, hC, hf⟩ := f.property.2    apply memℓp_infty    refine ⟨C, ?_⟩    rintro r ⟨⟨x, y⟩, rfl⟩    change ‖(f x - f y) / dist x y‖ ≤ C    rw [Real.norm_eq_abs, abs_div, abs_of_nonneg dist_nonneg]    by_cases hxy : x = y    · simp [hxy, hC]    · exact (div_le_iff₀ (dist_pos.mpr hxy)).2 (hf x y)⟩  map_add' f g := by    apply lp.ext    funext z    change ((f z.1 + g z.1) - (f z.2 + g z.2)) / dist z.1 z.2 =      (f z.1 - f z.2) / dist z.1 z.2 + (g z.1 - g z.2) / dist z.1 z.2    ring  map_smul' c f := by    apply lp.ext    funext z    change (c * f z.1 - c * f z.2) / dist z.1 z.2 =      c * ((f z.1 - f z.2) / dist z.1 z.2)    ring

definition OAI.DiscreteFree.evaluation line 96

def evaluation (o x : M) : (Lip0 o →L[ℝ] ℝ) :=

Show the remaining 14 lines

  LinearMap.mkContinuous    ({ toFun := fun f : Lip0 o => f x       map_add' := fun _ _ => rfl       map_smul' := fun _ _ => rfl } : Lip0 o →ₗ[ℝ] ℝ)    (dist x o) (fun f => by      change ‖f x‖ ≤ dist x o * ‖f‖      rw [mul_comm]      have hx := lp.norm_apply_le_norm (by simp : (∞ : ℝ≥0∞) ≠ 0)        (Lip0.slope o f) (x, o)      change ‖(f x - f o) / dist x o‖ ≤ ‖f‖ at hx      rw [f.property.1, sub_zero, norm_div, Real.norm_of_nonneg dist_nonneg] at hx      by_cases hxo : x = o      · rw [hxo, f.property.1, norm_zero, dist_self, mul_zero]      · exact (div_le_iff₀ (dist_pos.mpr hxo)).mp hx)

definition OAI.DiscreteFree.freeSubmodule line 112

def freeSubmodule (o : M) : Submodule ℝ ((Lip0 o →L[ℝ] ℝ)) :=  (Submodule.span ℝ (Set.range (evaluation o))).topologicalClosure

definition OAI.DiscreteFree.FreeSpace line 115

def FreeSpace (o : M) := ↥(freeSubmodule o)

definition OAI.DiscreteFree.delta line 123

def delta (o x : M) : FreeSpace o :=  ⟨evaluation o x, Submodule.le_topologicalClosure _    (Submodule.subset_span ⟨x, rfl⟩)⟩

definition OAI.DiscreteFree.HasAP line 127

def HasAP (X : Type*) [NormedAddCommGroup X] [NormedSpace ℝ X] : Prop :=  ∀ C : Set X, IsCompact C → ∀ ε : ℝ, 0 < ε →    ∃ T : X →L[ℝ] X, Module.Finite ℝ (LinearMap.range T.toLinearMap) ∧      ∃ η : ℝ, η < ε ∧ ∀ u ∈ C, ‖T u - u‖ ≤ η

definition OAI.DiscreteFree.HasBAP line 132

def HasBAP (X : Type*) [NormedAddCommGroup X] [NormedSpace ℝ X] (Λ : ℝ) : Prop :=  ∀ C : Set X, IsCompact C → ∀ ε : ℝ, 0 < ε →    ∃ T : X →L[ℝ] X, Module.Finite ℝ (LinearMap.range T.toLinearMap) ∧      ‖T‖ ≤ Λ ∧ ∃ η : ℝ, η < ε ∧ ∀ u ∈ C, ‖T u - u‖ ≤ η

definition OAI.DiscreteFree.blockBound line 137

def blockBound (p : ℕ) : ℝ := 300 * (p : ℝ)^2 + 150 * (p : ℝ) + 5

definition OAI.DiscreteFree.BlockStatement line 139

def BlockStatement : Prop :=  ∀ p : ℕ, 1 ≤ p → ∃ (M : Type) (d : MetricSpace M) (o : M),    letI := d    Countable M ∧ (∀ x y : M, x ≠ y → 1 ≤ dist x y) ∧      (∀ x y : M, dist x y ≤ blockBound p) ∧      ∃ A : Finset M, o ∈ A ∧ ∀ T : FreeSpace o →L[ℝ] FreeSpace o,        Module.Finite ℝ (LinearMap.range T.toLinearMap) → ‖T‖ ≤ (p : ℝ) →          ∃ a ∈ A, (1 / 2 : ℝ) ≤ ‖T (delta o a) - delta o a‖

definition OAI.DiscreteFree.MainStatement line 148

def MainStatement : Prop :=  ∃ (M : Type) (d : MetricSpace M) (o : M),    letI := d    Countable M ∧ (∀ x y : M, x ≠ y → 1 ≤ dist x y) ∧      (¬ Bornology.IsBounded (Set.univ : Set M)) ∧      (¬ ProperSpace M) ∧ HasAP (FreeSpace o) ∧      ∀ Λ : ℝ, 1 ≤ Λ → ¬ HasBAP (FreeSpace o) Λ

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: ap, finite, countable, metric, bap and space. Found in the paper’s title, which its main theorem states: free and bounded. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unbounded, distinct, fails and pointed.

Theorem 1.1 of A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces

Theorem

There is a countable, unbounded, nonproper pointed metric space (M,d,o) such that d(x,y)\ge1 for distinct points and \F(M) has AP but fails \lambda-BAP for every finite \lambda\ge1.

Section 1, Introduction Read it in the paper TeX source, line 39

Show the TeX

There is a countable, unbounded, nonproper pointed metric space $(M,d,o)$
such that $d(x,y)\ge1$ for distinct points and $\F(M)$ has AP but fails
$\lambda$-BAP for every finite $\lambda\ge1$.

Other possible paper matches, closest wording first

Proposition 2.1 of A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: zero, map, delta, countable, linear, 1/2 and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: distinct and pointed.
Theorem 1.2 of A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, delta, bounded, integer, linear, 1/2 and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: pointed.
Corollary 1.3 of A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: real, map, integer, 1/2, norm and bap. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: fails.

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 86 s; peak memory 6.2 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, 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 330’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: MainStatement (inside source_endpoints) gives a countable metric space M with all distinct distances ≥ 1, unbounded and not proper, such that the real Lipschitz-free space FreeSpace o has HasAP but ¬ HasBAP for every Λ ≥ 1, exactly the summary; BlockStatement (finite quantitative blocks) and the ℓ¹ renorming with 2B_p-BAP but not p-BAP are the retained earlier results.
Definitions the group asks readers to compare with the paper: FreeSpace o := closure of span {evaluation o x} ⊆ (Lip0 o →L[ℝ] ℝ) with Lip0 o normed by the slope map into ℓ^∞(M × M): hand-built Lipschitz-free space, standard; HasAP/HasBAP state approximation of the identity on every compact set by finite-rank operators (with ‖T‖ ≤ Λ for BAP)
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 constructs a countable metric space with all distinct points at distance at least one whose real Lipschitz-free space has the approximation property but fails every bounded approximation bound. The metric space is unbounded and not proper.”
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

163 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: 35 to its own declarations, 109 to Mathlib’s documentation. 13 declared names also link to the same declaration in OpenAI’s library. 4 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 source_endpoints 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 12 of the file’s 19 other declarations; with the theorem itself they span 98 lines. 2 of them are named in the statement itself.

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

def lipSubmoduleline 16library
def Lip0line 44library
def Lip0.slopeline 55library
def evaluationline 96library
def freeSubmoduleline 112library
def FreeSpaceline 115library
def deltaline 123library
def HasAPline 127library
def HasBAPline 132library
def blockBoundline 137library
def BlockStatementline 139in the statementlibrary
def MainStatementline 148in the statementlibrary
theorem source_endpointsline 156selected theoremlibrary

Other Lean rules that may affect the statement 7

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 AddCommGroup (Lip0 o) line 50
instance Module ℝ (Lip0 o) line 51
instance CoeFun (Lip0 o) (fun _ => M → ℝ) line 53
instance (o : M) : NormedAddCommGroup (Lip0 o) line 79
instance (o : M) : NormedSpace ℝ (Lip0 o) line 91
instance (o : M) : NormedAddCommGroup (FreeSpace o) line 117
instance (o : M) : NormedSpace ℝ (FreeSpace o) line 120

import Mathlibnamespace OAI/-!# Real Lipschitz-free spaces and approximation properties-/noncomputable sectionopen scoped BigOperators ENNRealnamespace DiscreteFreevariable {M : Type*} [MetricSpace M]def lipSubmodule (o : M) : Submodule ℝ (M → ℝ) where  carrier := {f | f o = 0 ∧ ∃ C : ℝ, 0 ≤ C ∧    ∀ x y, |f x - f y| ≤ C * dist x y}  zero_mem' := by    refine ⟨rfl, 0, le_rfl, ?_⟩    intro x y    simp  add_mem' := by    rintro f g ⟨hfo, C, hC, hf⟩ ⟨hgo, D, hD, hg⟩    refine ⟨by simpa using congrArg₂ (· + ·) hfo hgo,      C + D, add_nonneg hC hD, ?_⟩    intro x y    change |f x + g x - (f y + g y)| ≤ (C + D) * dist x y    calc      _ = |(f x - f y) + (g x - g y)| := by ring_nf      _ ≤ |f x - f y| + |g x - g y| := abs_add_le _ _      _ ≤ C * dist x y + D * dist x y := add_le_add (hf x y) (hg x y)      _ = _ := by ring  smul_mem' := by    rintro c f ⟨hfo, C, hC, hf⟩    refine ⟨by simp [hfo], |c| * C, mul_nonneg (abs_nonneg c) hC, ?_⟩    intro x y    change |c * f x - c * f y| ≤ (|c| * C) * dist x y    calc      _ = |c| * |f x - f y| := by rw [← mul_sub, abs_mul]      _ ≤ |c| * (C * dist x y) := mul_le_mul_of_nonneg_left (hf x y) (abs_nonneg c)      _ = _ := by ringdef Lip0 (o : M) := ↥(lipSubmodule o)namespace Lip0variable {o : M}instance : AddCommGroup (Lip0 o) := inferInstanceAs (AddCommGroup ↥(lipSubmodule o))instance : Module ℝ (Lip0 o) := inferInstanceAs (Module ℝ ↥(lipSubmodule o))instance : CoeFun (Lip0 o) (fun _ => M → ℝ) := ⟨fun f => f.val⟩def slope (o : M) : Lip0 o →ₗ[ℝ] lp (fun _ : M × M => ℝ) ∞ where  toFun f := ⟨fun z : M × M => (f z.1 - f z.2) / dist z.1 z.2, by    obtain ⟨C, hC, hf⟩ := f.property.2    apply memℓp_infty    refine ⟨C, ?_⟩    rintro r ⟨⟨x, y⟩, rfl⟩    change ‖(f x - f y) / dist x y‖ ≤ C    rw [Real.norm_eq_abs, abs_div, abs_of_nonneg dist_nonneg]    by_cases hxy : x = y    · simp [hxy, hC]    · exact (div_le_iff₀ (dist_pos.mpr hxy)).2 (hf x y)⟩  map_add' f g := by    apply lp.ext    funext z    change ((f z.1 + g z.1) - (f z.2 + g z.2)) / dist z.1 z.2 =      (f z.1 - f z.2) / dist z.1 z.2 + (g z.1 - g z.2) / dist z.1 z.2    ring  map_smul' c f := by    apply lp.ext    funext z    change (c * f z.1 - c * f z.2) / dist z.1 z.2 =      c * ((f z.1 - f z.2) / dist z.1 z.2)    ringinstance (o : M) : NormedAddCommGroup (Lip0 o) :=  NormedAddCommGroup.induced _ _ (slope o).toAddMonoidHom (by    intro f g h    apply Subtype.ext    funext x    have hx := congrArg (fun q : lp (fun _ : M × M => ℝ) ∞ => q (x, o)) h    change (f x - f o) / dist x o = (g x - g o) / dist x o at hx    rw [f.property.1, g.property.1, sub_zero, sub_zero] at hx    by_cases hxo : x = o    · rw [hxo, f.property.1, g.property.1]    · exact (div_left_inj' (ne_of_gt (dist_pos.mpr hxo))).mp hx)instance (o : M) : NormedSpace ℝ (Lip0 o) :=  NormedSpace.induced ℝ _ _ (slope o)end Lip0def evaluation (o x : M) : (Lip0 o →L[ℝ] ℝ) :=  LinearMap.mkContinuous    ({ toFun := fun f : Lip0 o => f x       map_add' := fun _ _ => rfl       map_smul' := fun _ _ => rfl } : Lip0 o →ₗ[ℝ] ℝ)    (dist x o) (fun f => by      change ‖f x‖ ≤ dist x o * ‖f‖      rw [mul_comm]      have hx := lp.norm_apply_le_norm (by simp : (∞ : ℝ≥0∞) ≠ 0)        (Lip0.slope o f) (x, o)      change ‖(f x - f o) / dist x o‖ ≤ ‖f‖ at hx      rw [f.property.1, sub_zero, norm_div, Real.norm_of_nonneg dist_nonneg] at hx      by_cases hxo : x = o      · rw [hxo, f.property.1, norm_zero, dist_self, mul_zero]      · exact (div_le_iff₀ (dist_pos.mpr hxo)).mp hx)def freeSubmodule (o : M) : Submodule ℝ ((Lip0 o →L[ℝ] ℝ)) :=  (Submodule.span ℝ (Set.range (evaluation o))).topologicalClosuredef FreeSpace (o : M) := ↥(freeSubmodule o)instance (o : M) : NormedAddCommGroup (FreeSpace o) :=  inferInstanceAs (NormedAddCommGroup ↥(freeSubmodule o))instance (o : M) : NormedSpace ℝ (FreeSpace o) :=  inferInstanceAs (NormedSpace ℝ ↥(freeSubmodule o))def delta (o x : M) : FreeSpace o :=  ⟨evaluation o x, Submodule.le_topologicalClosure _    (Submodule.subset_span ⟨x, rfl⟩)⟩def HasAP (X : Type*) [NormedAddCommGroup X] [NormedSpace ℝ X] : Prop :=  ∀ C : Set X, IsCompact C → ∀ ε : ℝ, 0 < ε →    ∃ T : X →L[ℝ] X, Module.Finite ℝ (LinearMap.range T.toLinearMap) ∧      ∃ η : ℝ, η < ε ∧ ∀ u ∈ C, ‖T u - u‖ ≤ ηdef HasBAP (X : Type*) [NormedAddCommGroup X] [NormedSpace ℝ X] (Λ : ℝ) : Prop :=  ∀ C : Set X, IsCompact C → ∀ ε : ℝ, 0 < ε →    ∃ T : X →L[ℝ] X, Module.Finite ℝ (LinearMap.range T.toLinearMap) ∧      ‖T‖ ≤ Λ ∧ ∃ η : ℝ, η < ε ∧ ∀ u ∈ C, ‖T u - u‖ ≤ ηdef blockBound (p : ℕ) : ℝ := 300 * (p : ℝ)^2 + 150 * (p : ℝ) + 5def BlockStatement : Prop :=  ∀ p : ℕ, 1 ≤ p → ∃ (M : Type) (d : MetricSpace M) (o : M),    letI := d    Countable M ∧ (∀ x y : M, x ≠ y → 1 ≤ dist x y) ∧      (∀ x y : M, dist x y ≤ blockBound p) ∧      ∃ A : Finset M, o ∈ A ∧ ∀ T : FreeSpace o →L[ℝ] FreeSpace o,        Module.Finite ℝ (LinearMap.range T.toLinearMap) → ‖T‖ ≤ (p : ℝ) →          ∃ a ∈ A, (1 / 2 : ℝ) ≤ ‖T (delta o a) - delta o a‖def MainStatement : Prop :=  ∃ (M : Type) (d : MetricSpace M) (o : M),    letI := d    Countable M ∧ (∀ x y : M, x ≠ y → 1 ≤ dist x y) ∧      (¬ Bornology.IsBounded (Set.univ : Set M)) ∧      (¬ ProperSpace M) ∧ HasAP (FreeSpace o) ∧      ∀ Λ : ℝ, 1 ≤ Λ → ¬ HasBAP (FreeSpace o) Λtheorem source_endpoints : BlockStatement ∧ MainStatement := by  sorryend DiscreteFreeendend 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 uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces

Theorem

There is a countable, unbounded, nonproper pointed metric space (M,d,o) such that d(x,y)\ge1 for distinct points and \F(M) has AP but fails \lambda-BAP for every finite \lambda\ge1.

Section 1, Introduction Read it in the paper TeX source, line 39

Show the TeX

There is a countable, unbounded, nonproper pointed metric space $(M,d,o)$
such that $d(x,y)\ge1$ for distinct points and $\F(M)$ has AP but fails
$\lambda$-BAP for every finite $\lambda\ge1$.

Theorem

For each integer p\ge1, there are a bounded countably infinite pointed metric space (M_p,d_p,o) and a finite set A_p\subseteq M_p containing o such that

d_p(x,y)\ge1\quad(x\ne y),\qquad \operatorname{diam}M_p\le B_p:=300p^2+150p+5,

and every bounded finite-rank linear operator T_0:\F(M_p)\to\F(M_p) with \norm{T_0}\le p satisfies

\max_{a\in A_p}\norm{T_0\delta_a-\delta_a}\ge\frac12.

Section 1, Introduction Read it in the paper TeX source, line 92

Show the TeX

For each integer $p\ge1$, there are a bounded countably infinite pointed
metric space $(M_p,d_p,o)$ and a finite set $A_p\subseteq M_p$ containing
$o$ such that
\[
 d_p(x,y)\ge1\quad(x\ne y),\qquad
 \operatorname{diam}M_p\le B_p:=300p^2+150p+5,
\]
and every bounded finite-rank linear operator
$T_0:\F(M_p)\to\F(M_p)$ with $\norm{T_0}\le p$ satisfies
\[
 \max_{a\in A_p}\norm{T_0\delta_a-\delta_a}\ge\frac12.
\]

Corollary

For each integer p\ge1, real \ell_1 has an equivalent norm \norm{\cdot}_p such that

\tfrac12\norm{c}_1\le\norm{c}_p\le B_p\norm{c}_1,

and (\ell_1,\norm{\cdot}_p) has 2B_p-BAP but fails p-BAP. For p=1, this norm fails MAP.

Section 1, Introduction Read it in the paper TeX source, line 119

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For each integer $p\ge1$, real $\ell_1$ has an equivalent norm
$\norm{\cdot}_p$ such that
\[
 \tfrac12\norm{c}_1\le\norm{c}_p\le B_p\norm{c}_1,
\]
and $(\ell_1,\norm{\cdot}_p)$ has $2B_p$-BAP but fails $p$-BAP.
For $p=1$, this norm fails MAP.

Configuration and file facts

File: lean/ComparatorChallenges/DiscreteLipschitzFree.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 330 link it.
Solution module: OAI.Analysis.LipschitzFree.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: 19 local vocabulary declarations
Lines of Lean, without comments or blank lines: 118
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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