Lean statement

Exponential deterministic state lower bound for liveness

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.OneWayLiveness.main_theorem

Lean statement file: OneWayLiveness.leanShow on the mapFamily 129. OpenAI lists this as a main result in its Lean catalogue.

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_theorem (h : ℕ) (hh : 2 ≤ h) :    (∃ N : NMachine (Alphabet h) (h + 3), N.NoLeft ∧      ∀ positive : Bool, N.Recognizes positive (OWL h)) ∧    (∀ (positive : Bool) (s : ℕ) (D : DMachine (Alphabet h) s),      D.Recognizes positive (OWL h) →      2 ^ ((h - 2) / 31) ≤ 4 * (s + if positive then 2 else 1) ^ 2) := 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.

structure OAI.OneWayLiveness.BRel line 7

@[ext] structure BRel (H : Type*) where  holds : H → H → Prop

definition OAI.OneWayLiveness.BRel.live line 24

def live (w : List (BRel H)) : Prop := ∃ x y, w.prod.holds x y

abbreviation OAI.OneWayLiveness.Alphabet line 27
```
abbrev Alphabet (h : ℕ) := BRel (Fin h)
```

definition OAI.OneWayLiveness.OWL line 28

def OWL (h : ℕ) : Set (List (Alphabet h)) := {w | BRel.live w}

inductive type OAI.OneWayLiveness.Symbol line 30

inductive Symbol (Alpha : Type*) where  | left : Symbol Alpha  | letter : Alpha → Symbol Alpha  | right : Symbol Alpha

inductive type OAI.OneWayLiveness.Move line 35

inductive Move where  | left | stay | right  deriving DecidableEq

definition OAI.OneWayLiveness.Move.Rel line 39

def Move.Rel : Move → ℕ → ℕ → Prop  | .left, i, j => j + 1 = i  | .stay, i, j => j = i  | .right, i, j => j = i + 1

definition OAI.OneWayLiveness.scanned line 45

def scanned {Alpha : Type*} (w : List Alpha) (i : Fin (w.length + 2)) : Symbol Alpha :=  if h0 : i.val = 0 then .left  else if hi : i.val < w.length + 1 then .letter (w[i.val - 1]'(by omega))  else .right

abbreviation OAI.OneWayLiveness.Configuration line 50

abbrev Configuration {Alpha : Type*} (w : List Alpha) (n : ℕ) := Fin (w.length + 2) × Fin n

structure OAI.OneWayLiveness.DMachine line 52

structure DMachine (Alpha : Type*) (s : ℕ) where  initial : Fin s  accepting : Set (Fin s)  transition : Fin s → Symbol Alpha → Option (Fin s × Move)  left_boundary : ∀ q q', transition q .left ≠ some (q', .left)  right_boundary : ∀ q q', transition q .right ≠ some (q', .right)

structure OAI.OneWayLiveness.NMachine line 59

structure NMachine (Alpha : Type*) (n : ℕ) where  initial : Fin n  accepting : Set (Fin n)  transition : Fin n → Symbol Alpha → Set (Fin n × Move)  left_boundary : ∀ q q', (q', .left) ∉ transition q .left  right_boundary : ∀ q q', (q', .right) ∉ transition q .right

definition OAI.OneWayLiveness.DMachine.Step line 66

def DMachine.Step {Alpha : Type*} {s : ℕ} (M : DMachine Alpha s) (w : List Alpha)    (c c' : Configuration w s) : Prop :=  ∃ d, M.transition c.2 (scanned w c.1) = some (c'.2, d) ∧ d.Rel c.1.val c'.1.val

definition OAI.OneWayLiveness.FiniteRun line 75

def FiniteRun {C : Type*} (positive : Bool) (step : C → C → Prop) (c c' : C) : Prop :=  if positive then Relation.TransGen step c c' else Relation.ReflTransGen step c c'

definition OAI.OneWayLiveness.DMachine.Accepts line 78

def DMachine.Accepts {Alpha : Type*} {s : ℕ} (M : DMachine Alpha s)    (positive : Bool) (w : List Alpha) : Prop :=  ∃ c, FiniteRun positive (M.Step w) (⟨0, by omega⟩, M.initial) c ∧ c.2 ∈ M.accepting

definition OAI.OneWayLiveness.NMachine.NoLeft line 86

def NMachine.NoLeft {Alpha : Type*} {n : ℕ} (M : NMachine Alpha n) : Prop :=  ∀ q a q', (q', Move.left) ∉ M.transition q a

definition OAI.OneWayLiveness.NMachine.Recognizes line 89

def NMachine.Recognizes {Alpha : Type*} {n : ℕ} (M : NMachine Alpha n)    (positive : Bool) (L : Set (List Alpha)) : Prop := ∀ w, M.Accepts positive w ↔ w ∈ L

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: eq, configuration, owl, accepting, integer and recognized. Found in the paper’s title, which its main theorem states: one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: acceptance, language, way, deterministic, nondeterministic, two, bound and states.

Statements in An exponential two-way deterministic state lower bound for one-way liveness come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of An exponential two-way deterministic state lower bound for one-way liveness

Theorem

For every integer h\ge2, the language \OWL_h is recognized by a nondeterministic automaton with h+3 states using no left moves. Every equivalent deterministic two-way automaton with s states satisfies

4(s+2)^2\ge 2^{\lfloor(h-2)/31\rfloor}.

If an initially accepting configuration counts as acceptance, s+2 in (eq:main-bound) may be replaced by s+1.

Section 1, Introduction Read it in the paper TeX source, line 99

Show the TeX

For every integer $h\ge2$, the language $\OWL_h$ is recognized by a
nondeterministic automaton with $h+3$ states using no left moves. Every
equivalent deterministic two-way automaton with $s$ states satisfies
\begin{equation}

 4(s+2)^2\ge 2^{\lfloor(h-2)/31\rfloor}.
\end{equation}
If an initially accepting configuration counts as acceptance, $s+2$ in
\eqref{eq:main-bound} may be replaced by $s+1$.

Other possible paper matches, closest wording first

Theorem 4.1 of An exponential two-way deterministic state lower bound for one-way liveness. Words of the Lean statement, its names split into words, that also appear in the theorem: length, transitions, symbols, word, positive, machine and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: acceptance, requires, deterministic, two and state.
Corollary 1.2 of An exponential two-way deterministic state lower bound for one-way liveness. Words of the Lean statement, its names split into words, that also appear in the theorem: alphabet and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: way, deterministic, nondeterministic, two and state.
Theorem 1.1 of An exponential state lower bound for two-way nondeterministic complementation. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, alphabet, recognizing and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bound and state.
Proposition 5.1 of An exponential two-way deterministic state lower bound for one-way liveness. Words of the Lean statement, its names split into words, that also appear in the theorem: owl. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: language, nondeterministic and states.
Corollary 3.2 of An exponential two-way deterministic state lower bound for one-way liveness. Words of the Lean statement, its names split into words, that also appear in the theorem: monoid, relation and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: required and bound.
Corollary 5.3 of An exponential state lower bound for two-way nondeterministic complementation. Words of the Lean statement, its names split into words, that also appear in the theorem: alphabet, recognizing and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: language, bound and state.

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 76 s; peak memory 6.1 GiB).
Reported 10 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 129’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: None; alphabet is SetRel (Fin (n-2)) (Fin (n-2)) (resp. BRel (Fin h)), so no alphabet-independent polynomial bound, exactly as the summary says.
Definitions the group asks readers to compare with the paper: Hand-rolled 2-way automata (TwoNFA, DMachine, NMachine) with endmarkers, stay moves and acceptance by an existential finite run (FiniteRun positive), a convention that differs from textbook halting-acceptance (documented in the docs and handled via the positive flag); ((n - 4) / 127) is natural-number floor division in the exponent
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 gives a family of one-way nondeterministic automata with $h+3$ states whose languages require exponentially many states for two-way deterministic recognition. For every $h\ge2$, a deterministic recognizer with $s$ states satisfies $2^{\lfloor(h-2)/31\rfloor}\le4(s+2)^2$ under both acceptance conventions considered in the paper; one convention has the sharper factor $4(s+1)^2$.”
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

105 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: 36 to its own declarations, 46 to Mathlib’s documentation. 20 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_theorem 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 19 of the file’s 22 other declarations; with the theorem itself they span 60 lines. 7 of them are named in the statement itself.

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

structure BRelline 7library
def BRel.liveline 24library
abbrev Alphabetline 27in the statementlibrary
def OWLline 28in the statementlibrary
inductive Symbolline 30library
inductive Moveline 35library
def Move.Relline 39library
def scannedline 45library
abbrev Configurationline 50library
structure DMachineline 52in the statementlibrary
structure NMachineline 59in the statementlibrary
def DMachine.Stepline 66library
def NMachine.Stepline 70library
def FiniteRunline 75library
def DMachine.Acceptsline 78library
def NMachine.Acceptsline 82library
def NMachine.NoLeftline 86in the statementlibrary
def NMachine.Recognizesline 89in the statementlibrary
def DMachine.Recognizesline 92in the statementlibrary
theorem main_theoremline 95selected theoremlibrary

Other Lean rules that may affect the statement 3

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 One (BRel H) line 12
instance Mul (BRel H) line 13
instance Monoid (BRel H) line 14

import Mathlibnamespace OAInamespace OneWayLiveness@[ext] structure BRel (H : Type*) where  holds : H → H → Propnamespace BRelvariable {H : Type*}instance : One (BRel H) := ⟨⟨Eq⟩⟩instance : Mul (BRel H) := ⟨fun a b => ⟨fun x z => ∃ y, a.holds x y ∧ b.holds y z⟩⟩instance : Monoid (BRel H) where  mul_assoc a b c := by    ext x z    change (∃ y, (∃ w, a.holds x w ∧ b.holds w y) ∧ c.holds y z) ↔      ∃ w, a.holds x w ∧ ∃ y, b.holds w y ∧ c.holds y z    aesop  one_mul a := by ext x z; change (∃ y, x = y ∧ a.holds y z) ↔ a.holds x z; simp  mul_one a := by ext x z; change (∃ y, a.holds x y ∧ y = z) ↔ a.holds x z; simp/-- Nonemptiness is the existence of a path through the word's ordered product. -/def live (w : List (BRel H)) : Prop := ∃ x y, w.prod.holds x yend BRelabbrev Alphabet (h : ℕ) := BRel (Fin h)def OWL (h : ℕ) : Set (List (Alphabet h)) := {w | BRel.live w}inductive Symbol (Alpha : Type*) where  | left : Symbol Alpha  | letter : Alpha → Symbol Alpha  | right : Symbol Alphainductive Move where  | left | stay | right  deriving DecidableEqdef Move.Rel : Move → ℕ → ℕ → Prop  | .left, i, j => j + 1 = i  | .stay, i, j => j = i  | .right, i, j => j = i + 1/-- The finite word including both endmarkers. -/def scanned {Alpha : Type*} (w : List Alpha) (i : Fin (w.length + 2)) : Symbol Alpha :=  if h0 : i.val = 0 then .left  else if hi : i.val < w.length + 1 then .letter (w[i.val - 1]'(by omega))  else .rightabbrev Configuration {Alpha : Type*} (w : List Alpha) (n : ℕ) := Fin (w.length + 2) × Fin nstructure DMachine (Alpha : Type*) (s : ℕ) where  initial : Fin s  accepting : Set (Fin s)  transition : Fin s → Symbol Alpha → Option (Fin s × Move)  left_boundary : ∀ q q', transition q .left ≠ some (q', .left)  right_boundary : ∀ q q', transition q .right ≠ some (q', .right)structure NMachine (Alpha : Type*) (n : ℕ) where  initial : Fin n  accepting : Set (Fin n)  transition : Fin n → Symbol Alpha → Set (Fin n × Move)  left_boundary : ∀ q q', (q', .left) ∉ transition q .left  right_boundary : ∀ q q', (q', .right) ∉ transition q .rightdef DMachine.Step {Alpha : Type*} {s : ℕ} (M : DMachine Alpha s) (w : List Alpha)    (c c' : Configuration w s) : Prop :=  ∃ d, M.transition c.2 (scanned w c.1) = some (c'.2, d) ∧ d.Rel c.1.val c'.1.valdef NMachine.Step {Alpha : Type*} {n : ℕ} (M : NMachine Alpha n) (w : List Alpha)    (c c' : Configuration w n) : Prop :=  ∃ d, (c'.2, d) ∈ M.transition c.2 (scanned w c.1) ∧ d.Rel c.1.val c'.1.val/-- The positive convention explicitly excludes a length-zero computation. -/def FiniteRun {C : Type*} (positive : Bool) (step : C → C → Prop) (c c' : C) : Prop :=  if positive then Relation.TransGen step c c' else Relation.ReflTransGen step c c'def DMachine.Accepts {Alpha : Type*} {s : ℕ} (M : DMachine Alpha s)    (positive : Bool) (w : List Alpha) : Prop :=  ∃ c, FiniteRun positive (M.Step w) (⟨0, by omega⟩, M.initial) c ∧ c.2 ∈ M.acceptingdef NMachine.Accepts {Alpha : Type*} {n : ℕ} (M : NMachine Alpha n)    (positive : Bool) (w : List Alpha) : Prop :=  ∃ c, FiniteRun positive (M.Step w) (⟨0, by omega⟩, M.initial) c ∧ c.2 ∈ M.acceptingdef NMachine.NoLeft {Alpha : Type*} {n : ℕ} (M : NMachine Alpha n) : Prop :=  ∀ q a q', (q', Move.left) ∉ M.transition q adef NMachine.Recognizes {Alpha : Type*} {n : ℕ} (M : NMachine Alpha n)    (positive : Bool) (L : Set (List Alpha)) : Prop := ∀ w, M.Accepts positive w ↔ w ∈ Ldef DMachine.Recognizes {Alpha : Type*} {s : ℕ} (M : DMachine Alpha s)    (positive : Bool) (L : Set (List Alpha)) : Prop := ∀ w, M.Accepts positive w ↔ w ∈ Ltheorem main_theorem (h : ℕ) (hh : 2 ≤ h) :    (∃ N : NMachine (Alphabet h) (h + 3), N.NoLeft ∧      ∀ positive : Bool, N.Recognizes positive (OWL h)) ∧    (∀ (positive : Bool) (s : ℕ) (D : DMachine (Alphabet h) s),      D.Recognizes positive (OWL h) →      2 ^ ((h - 2) / 31) ≤ 4 * (s + if positive then 2 else 1) ^ 2) := by  sorryend OneWayLivenessend 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.

An exponential state lower bound for two-way nondeterministic complementation

Theorem

For every integer n\ge4, set H=\{1,\ldots,n-2\} and \Sigma_n=\mathcal R_H. There is an n-state 2NFA A_n recognizing L_H such that every 2NFA recognizing \Sigma_n^*\setminus L(A_n) has at least

\frac12\,2^{\lfloor(n-4)/127\rfloor}-1

states. In particular, there is no polynomial state bound for complementation that is independent of the finite alphabet.

Section 1, Introduction Read it in the paper TeX source, line 38

Show the TeX

For every integer $n\ge4$, set $H=\{1,\ldots,n-2\}$ and
$\Sigma_n=\mathcal R_H$. There is an $n$-state 2NFA $A_n$ recognizing
$L_H$ such that every 2NFA recognizing
$\Sigma_n^*\setminus L(A_n)$ has at least
\[
 \frac12\,2^{\lfloor(n-4)/127\rfloor}-1
\]
states. In particular, there is no polynomial state bound for
complementation that is independent of the finite alphabet.

An exponential two-way deterministic state lower bound for one-way liveness

Theorem

4(s+2)^2\ge 2^{\lfloor(h-2)/31\rfloor}.

If an initially accepting configuration counts as acceptance, s+2 in (eq:main-bound) may be replaced by s+1.

Section 1, Introduction Read it in the paper TeX source, line 99

Show the TeX

For every integer $h\ge2$, the language $\OWL_h$ is recognized by a
nondeterministic automaton with $h+3$ states using no left moves. Every
equivalent deterministic two-way automaton with $s$ states satisfies
\begin{equation}

 4(s+2)^2\ge 2^{\lfloor(h-2)/31\rfloor}.
\end{equation}
If an initially accepting configuration counts as acceptance, $s+2$ in
\eqref{eq:main-bound} may be replaced by $s+1$.

Corollary

There are no absolute constants C,c>0 such that every n-state two-way nondeterministic automaton over every finite alphabet has an equivalent two-way deterministic automaton with at most Cn^c states.

Section 1, Introduction Read it in the paper TeX source, line 112

Show the TeX

There are no absolute constants $C,c>0$ such that every $n$-state two-way
nondeterministic automaton over every finite alphabet has an equivalent
two-way deterministic automaton with at most $Cn^c$ states.

Configuration and file facts

File: lean/ComparatorChallenges/OneWayLiveness.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Combinatorics/Automata/Main.lean
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: 22 local vocabulary declarations
Lines of Lean, without comments or blank lines: 70
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