Lean statement

Two-way nondeterministic complementation lower bound

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.TwoWayComplementation.complementation_lower_bound_finite

Lean statement file: TwoWayComplementation.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 complementation_lower_bound_finite (n : ℕ) (hn : 4 ≤ n) :    ∃ A : TwoNFA (SetRel (Fin (n - 2)) (Fin (n - 2))) (Fin n),      ∀ (Q : Type*) [Fintype Q], ∀ B : TwoNFA (SetRel (Fin (n - 2)) (Fin (n - 2))) Q,        B.language = A.languageᶜ →          (1 / 2 : ℝ) * 2 ^ ((n - 4) / 127) - 1 ≤ (Fintype.card Q : ℝ) := 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.TwoWayComplementation.relStar line 9

def relStar {α : Type*} (R : SetRel α α) : SetRel α α :=  {(x, y) | Relation.ReflTransGen (fun a b => (a, b) ∈ R) x y}

structure OAI.TwoWayComplementation.TwoNFA line 12

structure TwoNFA (Sigma Q : Type*) where  initial : Q  accepting : Set Q  transition : Q → (Fin 2 ⊕ Sigma) → Set (Q × Fin 3)

definition OAI.TwoWayComplementation.tapeSymbol line 17

def tapeSymbol {Sigma : Type*} (w : List Sigma) (p : Fin (w.length + 2)) : Fin 2 ⊕ Sigma :=  if hleft : p.val = 0 then .inl 0  else if hright : p.val = w.length + 1 then .inl 1  else .inr (w[p.val - 1]'(by omega))

definition OAI.TwoWayComplementation.TwoNFA.step line 24

def step {Sigma Q : Type*} (A : TwoNFA Sigma Q) (w : List Sigma) :    SetRel (Q × Fin (w.length + 2)) (Q × Fin (w.length + 2)) :=  {(c, d) | ∃ move : Fin 3,    (d.1, move) ∈ A.transition c.1 (tapeSymbol w c.2) ∧    (d.2.val : ℤ) = (c.2.val : ℤ) + (move.val : ℤ) - 1}

definition OAI.TwoWayComplementation.TwoNFA.language line 30

def language {Sigma Q : Type*} (A : TwoNFA Sigma Q) : Set (List Sigma) :=  {w | ∃ c, c.1 ∈ A.accepting ∧    ((A.initial, ⟨0, by omega⟩), c) ∈ relStar (A.step w)}

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: 1/2, complementation, integer, nfa, sigma, bound and finite. Found in the paper’s title, which its main theorem states: lower and two. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: least and state.

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

Theorem 1.1 of 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.

Other possible paper matches, closest wording first

Theorem 1.1 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: accepting, integer, language, two and bound. Found in the paper’s title, which its main theorem states: lower. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: way, automaton, nondeterministic and states.
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: nfa, language, bound and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: least and state.
Proposition 2.3 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: nfa, sigma, two and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: least and state.
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, accepts, sigma and two. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: requires and state.
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: relation, bound and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: required.
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: two and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: way, automaton, nondeterministic 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 52 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, for every $n\ge4$, an explicit $n$-state automaton whose complement requires at least $\tfrac12 2^{\lfloor(n-4)/127\rfloor}-1$ states.”
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

45 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: 6 to its own declarations, 26 to Mathlib’s documentation. 6 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 complementation_lower_bound_finite 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 5 other declarations of the file; with the theorem itself they span 24 lines. 2 of them are named in the statement itself.

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

def relStarline 9library
structure TwoNFAline 12in the statementlibrary
def tapeSymbolline 17library
def TwoNFA.stepline 24library
def TwoNFA.languageline 30in the statementlibrary
theorem complementation_lower_bound_finiteline 36selected theoremlibrary

import Mathlibnamespace OAInamespace TwoWayComplementationopen scoped SetReldef relStar {α : Type*} (R : SetRel α α) : SetRel α α :=  {(x, y) | Relation.ReflTransGen (fun a b => (a, b) ∈ R) x y}structure TwoNFA (Sigma Q : Type*) where  initial : Q  accepting : Set Q  transition : Q → (Fin 2 ⊕ Sigma) → Set (Q × Fin 3)def tapeSymbol {Sigma : Type*} (w : List Sigma) (p : Fin (w.length + 2)) : Fin 2 ⊕ Sigma :=  if hleft : p.val = 0 then .inl 0  else if hright : p.val = w.length + 1 then .inl 1  else .inr (w[p.val - 1]'(by omega))namespace TwoNFAdef step {Sigma Q : Type*} (A : TwoNFA Sigma Q) (w : List Sigma) :    SetRel (Q × Fin (w.length + 2)) (Q × Fin (w.length + 2)) :=  {(c, d) | ∃ move : Fin 3,    (d.1, move) ∈ A.transition c.1 (tapeSymbol w c.2) ∧    (d.2.val : ℤ) = (c.2.val : ℤ) + (move.val : ℤ) - 1}def language {Sigma Q : Type*} (A : TwoNFA Sigma Q) : Set (List Sigma) :=  {w | ∃ c, c.1 ∈ A.accepting ∧    ((A.initial, ⟨0, by omega⟩), c) ∈ relStar (A.step w)}end TwoNFAtheorem complementation_lower_bound_finite (n : ℕ) (hn : 4 ≤ n) :    ∃ A : TwoNFA (SetRel (Fin (n - 2)) (Fin (n - 2))) (Fin n),      ∀ (Q : Type*) [Fintype Q], ∀ B : TwoNFA (SetRel (Fin (n - 2)) (Fin (n - 2))) Q,        B.language = A.languageᶜ →          (1 / 2 : ℝ) * 2 ^ ((n - 4) / 127) - 1 ≤ (Fintype.card Q : ℝ) := by  sorryend TwoWayComplementationend 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

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 $s+1$.

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/TwoWayComplementation.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Combinatorics/TwoWayAutomata/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: 5 local vocabulary declarations
Lines of Lean, without comments or blank lines: 24
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