Lean statement

Boone–Higman equivalence

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.FiniteAlgebraicEnvelopes.main

Lean statement file: BooneHigman.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 250 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 main (GroupType : Type u) [Group GroupType] [Group.FG GroupType] :    HasDecidableWordProblem GroupType ↔ EmbedsInFinitelyPresentedSimpleGroup GroupType := 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.FiniteAlgebraicEnvelopes.Word line 9

abbrev Word (generatorCount : ℕ) := List (Fin generatorCount × Bool)

definition OAI.FiniteAlgebraicEnvelopes.evalWord line 11

def evalWord {GroupType : Type u} [Group GroupType] {generatorCount : ℕ}    (generators : Fin generatorCount → GroupType) (word : Word generatorCount) : GroupType :=  (word.map fun letter =>    if letter.2 then generators letter.1 else (generators letter.1)⁻¹).prod

definition OAI.FiniteAlgebraicEnvelopes.HasDecidableWordProblem line 16

def HasDecidableWordProblem (GroupType : Type u) [Group GroupType] : Prop :=  ∃ (generatorCount : ℕ) (generators : Fin generatorCount → GroupType),    Subgroup.closure (Set.range generators) = ⊤ ∧    ComputablePred (fun word : Word generatorCount => evalWord generators word = 1)

definition OAI.FiniteAlgebraicEnvelopes.EmbedsInFinitelyPresentedSimpleGroup line 21

def EmbedsInFinitelyPresentedSimpleGroup (GroupType : Type u) [Group GroupType] : Prop :=  ∃ (TargetGroup : Type u) (_ : Group TargetGroup),    Group.IsFinitelyPresented TargetGroup ∧ IsSimpleGroup TargetGroup ∧    ∃ embedding : GroupType →* TargetGroup, Function.Injective embedding

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: embeds, simple, decidable, word, presented, group and finitely. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: generated.

Statements in Finite algebraic envelopes and the Boone–Higman conjecture come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Finite algebraic envelopes and the Boone–Higman conjecture

Theorem

For every finitely generated group G, the following are equivalent:

(i)
G has decidable word problem.
(ii)
G embeds in a finitely presented simple group.

Section 1, Introduction Read it in the paper TeX source, line 9

Show the TeX

For every finitely generated group \(G\), the following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(G\) has decidable word problem.
\item \(G\) embeds in a finitely presented simple group.
\end{enumerate}

Other possible paper matches, closest wording first

Theorem 1.2 of Finite algebraic envelopes and the Boone–Higman conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: embeds, decidable, word, presented, group and finitely. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: generated.
Theorem 2.3 of Finite algebraic envelopes and the Boone–Higman conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: countable, embeds, simple, presented, group and finitely. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: generated.
Theorem 2.2 of Finite algebraic envelopes and the Boone–Higman conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: embedding, simple, word, decidable, injective, presented and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: homomorphism and generated.
Theorem 1.1 of Simple F∞ overgroups of groups with decidable word problem. Words of the Lean statement, its names split into words, that also appear in the theorem: simple, decidable, word, injective, group and finitely. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: generated and homomorphism.
Theorem 6.2 of Finite algebraic envelopes and the Boone–Higman conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: map, word, decidable, injective, presented, finitely and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite, homomorphism and generated.
Corollary 1.2 of A universal group of type F∞. Words of the Lean statement, its names split into words, that also appear in the theorem: embeds, presented, group and finitely. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: generated.

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 113 s; peak memory 6.3 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 250’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: BooneHigman.main (G) [Group.FG G] : HasDecidableWordProblem G ↔ EmbedsInFinitelyPresentedSimpleGroup G (ComputablePred of evalWord g w = 1 for some finite generating tuple; Mathlib Group.IsFinitelyPresented, IsSimpleGroup), SimpleOvergroups.main (G f.g. with decidable word problem ⇒ ∃ nontrivial simple H of type F∞ and injective G →* H) and UniversalFInfinity.universal_group_of_type_FInfinity (∃ H of type F∞ into which every Group.IsFinitelyPresented group embeds). All three claims are stated; type F∞ = aspherical Hausdorff connected CW complex with FiniteType (finitely many cells per dimension).
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 the equivalence in full: a finitely generated group has a computable word problem for a finite generating set exactly when it embeds by an injective homomorphism into a finitely presented simple group.”
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

32 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: 5 to its own declarations, 17 to Mathlib’s documentation. 5 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 main 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 4 other declarations of the file; with the theorem itself they span 16 lines. 2 of them are named in the statement itself.

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

abbrev Wordline 9library
def evalWordline 11library
def HasDecidableWordProblemline 16in the statementlibrary
def EmbedsInFinitelyPresentedSimpleGroupline 21in the statementlibrary
theorem mainline 26selected theoremlibrary

import Mathlibnamespace OAInamespace FiniteAlgebraicEnvelopesuniverse uabbrev Word (generatorCount : ℕ) := List (Fin generatorCount × Bool)def evalWord {GroupType : Type u} [Group GroupType] {generatorCount : ℕ}    (generators : Fin generatorCount → GroupType) (word : Word generatorCount) : GroupType :=  (word.map fun letter =>    if letter.2 then generators letter.1 else (generators letter.1)⁻¹).proddef HasDecidableWordProblem (GroupType : Type u) [Group GroupType] : Prop :=  ∃ (generatorCount : ℕ) (generators : Fin generatorCount → GroupType),    Subgroup.closure (Set.range generators) = ⊤ ∧    ComputablePred (fun word : Word generatorCount => evalWord generators word = 1)def EmbedsInFinitelyPresentedSimpleGroup (GroupType : Type u) [Group GroupType] : Prop :=  ∃ (TargetGroup : Type u) (_ : Group TargetGroup),    Group.IsFinitelyPresented TargetGroup ∧ IsSimpleGroup TargetGroup ∧    ∃ embedding : GroupType →* TargetGroup, Function.Injective embeddingtheorem main (GroupType : Type u) [Group GroupType] [Group.FG GroupType] :    HasDecidableWordProblem GroupType ↔ EmbedsInFinitelyPresentedSimpleGroup GroupType := by  sorryend FiniteAlgebraicEnvelopesend 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.

Finite algebraic envelopes and the Boone–Higman conjecture

Theorem

For every finitely generated group G, the following are equivalent:

(i)
G has decidable word problem.
(ii)
G embeds in a finitely presented simple group.

Section 1, Introduction Read it in the paper TeX source, line 9

Show the TeX

For every finitely generated group \(G\), the following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(G\) has decidable word problem.
\item \(G\) embeds in a finitely presented simple group.
\end{enumerate}

Theorem

Every finitely generated group with decidable word problem embeds in a finitely presented group \Gamma admitting a faithful two-transitive action on a countably infinite set. Every point stabilizer of this action is finitely generated.

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

Show the TeX

Every finitely generated group with decidable word problem embeds in a
finitely presented group \(\Gamma\) admitting a faithful two-transitive
action on a countably infinite set. Every point stabilizer of this action
is finitely generated.

Simple F∞ overgroups of groups with decidable word problem

Theorem

Every finitely generated group G with decidable word problem admits an injective homomorphism into a nontrivial simple group H of type \Finfty.

Section 1, Introduction Read it in the paper TeX source, line 33

Show the TeX

Every finitely generated group \(G\) with decidable word problem admits
an injective homomorphism into a nontrivial simple group \(H\) of type
\(\Finfty\).

A universal group of type F∞

Theorem

There exists a group H of type \Finfty such that every finitely presented group admits an injective homomorphism into H.

Section 1, Introduction Read it in the paper TeX source, line 10

Show the TeX

There exists a group $H$ of type $\Finfty$ such that every finitely
presented group admits an injective homomorphism into $H$.

Corollary

A finitely generated group embeds in H if and only if it is recursively presented.

Section 1, Introduction Read it in the paper TeX source, line 20

Show the TeX

A finitely generated group embeds in $H$ if and only if it is
recursively presented.

Configuration and file facts

File: lean/ComparatorChallenges/BooneHigman.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 250 link it.
Solution module: OAI.GroupTheory.BooneHigman.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: 4 local vocabulary declarations
Lines of Lean, without comments or blank lines: 16
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