Lean statement

EXPTIME-completeness of variable-dimension Weisfeiler–Leman 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.VariableWL.main

Lean statement file: VariableWL.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 133 link it.

See the closest candidates 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 : EXPTIMEComplete WL ∧ EXPTIMEComplete SubWL := 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.VariableWL.Word line 7
```
abbrev Word := List Bool
```

inductive type OAI.VariableWL.Motion line 9

inductive Motion where  | left | stay | right  deriving DecidableEq

definition OAI.VariableWL.Motion.displacement line 13

def Motion.displacement : Motion → ℤ  | .left => -1  | .stay => 0  | .right => 1

structure OAI.VariableWL.Machine line 18

structure Machine where  states : ℕ  workSymbols : ℕ  extraTapes : ℕ  transition : Fin (states + 1) →    (Fin (extraTapes + 1) → Fin (workSymbols + 3)) →    Option (Fin (states + 1) ×      (Fin (extraTapes + 1) → Fin (workSymbols + 3) × Motion))

structure OAI.VariableWL.Configuration line 27

structure Configuration (M : Machine) where  control : Fin (M.states + 1)  heads : Fin (M.extraTapes + 1) → ℤ  cells : Fin (M.extraTapes + 1) → ℤ → Fin (M.workSymbols + 3)

definition OAI.VariableWL.wordCells line 32

def wordCells (a : ℕ) (w : Word) (i : ℤ) : Fin (a + 3) :=  if 0 ≤ i then    match w[i.toNat]? with    | none => 0    | some false => 1    | some true => 2  else 0

definition OAI.VariableWL.initial line 40

def initial (M : Machine) (w : Word) : Configuration M where  control := 0  heads := fun _ => 0  cells := fun j => if j = 0 then wordCells M.workSymbols w else fun _ => 0

definition OAI.VariableWL.scanned line 45

def scanned {M : Machine} (c : Configuration M) :    Fin (M.extraTapes + 1) → Fin (M.workSymbols + 3) :=  fun j => c.cells j (c.heads j)

definition OAI.VariableWL.step line 49

def step (M : Machine) (c : Configuration M) : Option (Configuration M) :=  (M.transition c.control (scanned c)).map fun data =>    { control := data.1      heads := fun j => c.heads j + (data.2 j).2.displacement      cells := fun j i => if i = c.heads j then (data.2 j).1 else c.cells j i }

definition OAI.VariableWL.run line 55

def run (M : Machine) (w : Word) : ℕ → Configuration M  | 0 => initial M w  | n + 1 => (step M (run M w n)).getD (run M w n)

definition OAI.VariableWL.OutputsWithin line 59

def OutputsWithin (M : Machine) (w out : Word) (bound : ℕ) : Prop :=  ∃ t ≤ bound, step M (run M w t) = none ∧    (run M w t).cells 0 = wordCells M.workSymbols out

abbreviation OAI.VariableWL.Language line 63
```
abbrev Language := Set Word
```

definition OAI.VariableWL.InEXPTIME line 65

def InEXPTIME (L : Language) : Prop :=  ∃ M : Machine, ∃ c d : ℕ, ∀ w : Word,    ∃ b : Bool, (b = true ↔ w ∈ L) ∧      OutputsWithin M w [b] (2 ^ (c * (w.length + 1) ^ d))

definition OAI.VariableWL.PolytimeReduces line 70

def PolytimeReduces (L R : Language) : Prop :=  ∃ M : Machine, ∃ c d : ℕ, ∀ w : Word,    ∃ out : Word, OutputsWithin M w out (c * (w.length + 1) ^ d) ∧      (w ∈ L ↔ out ∈ R)

definition OAI.VariableWL.EXPTIMEComplete line 75

def EXPTIMEComplete (L : Language) : Prop :=  InEXPTIME L ∧ ∀ R : Language, InEXPTIME R → PolytimeReduces R L

structure OAI.VariableWL.Graph line 78

structure Graph where  order : ℕ  graph : SimpleGraph (Fin order)

abbreviation OAI.VariableWL.Tuple line 82

abbrev Tuple (G : Graph) (k : ℕ) := Fin k → Fin G.order

definition OAI.VariableWL.Color line 84

def Color (k : ℕ) : ℕ → Type  | 0 => Fin k → Fin k → Bool × Bool  | s + 1 => Color k s × Multiset (Fin k → Color k s)

definition OAI.VariableWL.color line 88

noncomputable def color (G : Graph) (k : ℕ) : (s : ℕ) → Tuple G k → Color k s  | 0, v => by      classical      exact fun i j => (decide (v i = v j), decide (G.graph.Adj (v i) (v j)))  | s + 1, v => (color G k s v,      Finset.univ.val.map (fun z => fun i => color G k s (Function.update v i z)))

definition OAI.VariableWL.histogram line 95

noncomputable def histogram (G : Graph) (k s : ℕ) : Multiset (Color k s) :=  Finset.univ.val.map (color G k s)

definition OAI.VariableWL.Equivalent line 98

def Equivalent (G H : Graph) (k : ℕ) : Prop :=  ∀ s : ℕ, histogram G k s = histogram H k s

definition OAI.VariableWL.integerCode line 101

def integerCode (n : ℕ) : Word :=  List.replicate (Nat.bits n).length true ++ [false] ++ Nat.bits n

definition OAI.VariableWL.graphCode line 104

noncomputable def graphCode (G : Graph) : Word := by  classical  exact integerCode G.order ++    (List.ofFn fun i : Fin G.order =>      List.ofFn fun j : Fin G.order => decide (G.graph.Adj i j)).flatten

definition OAI.VariableWL.pairCode line 110

noncomputable def pairCode (G H : Graph) (k : ℕ) : Word :=  graphCode G ++ graphCode H ++ integerCode k

3 more definitions are reached the same way; the full file lists them.

No confident match with a paper statement

No paper theorem is paired with this Lean statement.

Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Statements in Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs come first because the release’s name for this Lean statement shares words with that paper’s title.

Other possible paper matches, closest wording first

Theorem 1.1 of Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: complete, wl and exptime. Found in the paper’s title, which its main theorem states: subcubic and graphs. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: many, reductions, polynomial, time and one.
Proposition 8.2 of Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: output, degree, word, connected, machine, order and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: maximum, three, input, binary, positive, polynomial and three more.
Corollary 3.5 of Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: language, word, exptime, integer, machine and length. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: reduction, binary, polynomial, dimensions and time.
Corollary 8.3 of Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: complete, wl and exptime. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: many, encoded, reductions, binary, polynomial, dimension and two more.
Theorem 1.1 of The complexity of identifying a graph by Weisfeiler–Leman refinement. Words of the Lean statement, its names split into words, that also appear in the theorem: color, update, complete, exptime, integer, simple and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: many, joint, adjacency, reductions, binary, positive and five more.

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, 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 133’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; all three headline claims are stated. The conditional running-time exclusions are not (docs).
Definitions the group asks readers to compare with the paper: Hand-rolled computation models: TM (multi-tape, fixed finite tables, halting states 1/2), a RAM with logarithmic word width (WordWidth.logarithmic), memory Word w → Word w, compute (op : WordOperation R) where a whole word operation (any map implemented by a TM within constant * (w+1)^exponent) counts as one instruction and one step runs a finite instruction tree; InEXPTIME (2^(c (n+1)^d) steps), PolytimeReduces/PolytimeManyOne, EXPTIMEComplete; WL colors are canonical nested multisets (Color c k t) with Equivalent := ∀ t, histogram G t = histogram H t
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 EXPTIME-completeness of deciding joint-update $k$-dimensional Weisfeiler–Leman equivalence when $k\ge2$ is part of the input in binary and the graphs are encoded by explicit adjacency matrices. It proves both the general result and the restriction to connected simple uncolored graphs of equal positive order and maximum degree at most three. Completeness uses polynomial-time many-one reductions.”
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

130 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: 80 to its own declarations, 51 to Mathlib’s documentation. 27 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 27 other declarations of the file; with the theorem itself they span 93 lines. 3 of them are named in the statement itself.

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

abbrev Wordline 7library
inductive Motionline 9library
def Motion.displacementline 13library
structure Machineline 18library
structure Configurationline 27library
def wordCellsline 32library
def initialline 40library
def scannedline 45library
def stepline 49library
def runline 55library
def OutputsWithinline 59library
abbrev Languageline 63library
def InEXPTIMEline 65library
def PolytimeReducesline 70library
def EXPTIMECompleteline 75in the statementlibrary
structure Graphline 78library
abbrev Tupleline 82library
def Colorline 84library
def colorline 88library
def histogramline 95library
def Equivalentline 98library
def integerCodeline 101library
def graphCodeline 104library
def pairCodeline 110library
def WLline 113in the statementlibrary
def Subcubicline 116library
def SubWLline 120in the statementlibrary
theorem mainline 125selected theorem

import Mathlibnamespace OAInamespace VariableWLabbrev Word := List Boolinductive Motion where  | left | stay | right  deriving DecidableEqdef Motion.displacement : Motion → ℤ  | .left => -1  | .stay => 0  | .right => 1structure Machine where  states : ℕ  workSymbols : ℕ  extraTapes : ℕ  transition : Fin (states + 1) →    (Fin (extraTapes + 1) → Fin (workSymbols + 3)) →    Option (Fin (states + 1) ×      (Fin (extraTapes + 1) → Fin (workSymbols + 3) × Motion))structure Configuration (M : Machine) where  control : Fin (M.states + 1)  heads : Fin (M.extraTapes + 1) → ℤ  cells : Fin (M.extraTapes + 1) → ℤ → Fin (M.workSymbols + 3)def wordCells (a : ℕ) (w : Word) (i : ℤ) : Fin (a + 3) :=  if 0 ≤ i then    match w[i.toNat]? with    | none => 0    | some false => 1    | some true => 2  else 0def initial (M : Machine) (w : Word) : Configuration M where  control := 0  heads := fun _ => 0  cells := fun j => if j = 0 then wordCells M.workSymbols w else fun _ => 0def scanned {M : Machine} (c : Configuration M) :    Fin (M.extraTapes + 1) → Fin (M.workSymbols + 3) :=  fun j => c.cells j (c.heads j)def step (M : Machine) (c : Configuration M) : Option (Configuration M) :=  (M.transition c.control (scanned c)).map fun data =>    { control := data.1      heads := fun j => c.heads j + (data.2 j).2.displacement      cells := fun j i => if i = c.heads j then (data.2 j).1 else c.cells j i }def run (M : Machine) (w : Word) : ℕ → Configuration M  | 0 => initial M w  | n + 1 => (step M (run M w n)).getD (run M w n)def OutputsWithin (M : Machine) (w out : Word) (bound : ℕ) : Prop :=  ∃ t ≤ bound, step M (run M w t) = none ∧    (run M w t).cells 0 = wordCells M.workSymbols outabbrev Language := Set Worddef InEXPTIME (L : Language) : Prop :=  ∃ M : Machine, ∃ c d : ℕ, ∀ w : Word,    ∃ b : Bool, (b = true ↔ w ∈ L) ∧      OutputsWithin M w [b] (2 ^ (c * (w.length + 1) ^ d))def PolytimeReduces (L R : Language) : Prop :=  ∃ M : Machine, ∃ c d : ℕ, ∀ w : Word,    ∃ out : Word, OutputsWithin M w out (c * (w.length + 1) ^ d) ∧      (w ∈ L ↔ out ∈ R)def EXPTIMEComplete (L : Language) : Prop :=  InEXPTIME L ∧ ∀ R : Language, InEXPTIME R → PolytimeReduces R Lstructure Graph where  order : ℕ  graph : SimpleGraph (Fin order)abbrev Tuple (G : Graph) (k : ℕ) := Fin k → Fin G.orderdef Color (k : ℕ) : ℕ → Type  | 0 => Fin k → Fin k → Bool × Bool  | s + 1 => Color k s × Multiset (Fin k → Color k s)noncomputable def color (G : Graph) (k : ℕ) : (s : ℕ) → Tuple G k → Color k s  | 0, v => by      classical      exact fun i j => (decide (v i = v j), decide (G.graph.Adj (v i) (v j)))  | s + 1, v => (color G k s v,      Finset.univ.val.map (fun z => fun i => color G k s (Function.update v i z)))noncomputable def histogram (G : Graph) (k s : ℕ) : Multiset (Color k s) :=  Finset.univ.val.map (color G k s)def Equivalent (G H : Graph) (k : ℕ) : Prop :=  ∀ s : ℕ, histogram G k s = histogram H k sdef integerCode (n : ℕ) : Word :=  List.replicate (Nat.bits n).length true ++ [false] ++ Nat.bits nnoncomputable def graphCode (G : Graph) : Word := by  classical  exact integerCode G.order ++    (List.ofFn fun i : Fin G.order =>      List.ofFn fun j : Fin G.order => decide (G.graph.Adj i j)).flattennoncomputable def pairCode (G H : Graph) (k : ℕ) : Word :=  graphCode G ++ graphCode H ++ integerCode kdef WL : Language :=  { w | ∃ G H : Graph, ∃ k : ℕ, w = pairCode G H k ∧ 2 ≤ k ∧ Equivalent G H k }noncomputable def Subcubic (G : Graph) : Prop := by  classical  exact ∀ v : Fin G.order, G.graph.degree v ≤ 3def SubWL : Language :=  { w | ∃ G H : Graph, ∃ k : ℕ, w = pairCode G H k ∧ 2 ≤ k ∧    0 < G.order ∧ G.order = H.order ∧ G.graph.Connected ∧ H.graph.Connected ∧    Subcubic G ∧ Subcubic H ∧ Equivalent G H k }theorem main : EXPTIMEComplete WL ∧ EXPTIMEComplete SubWL := by  sorryend VariableWLend 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.

Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence

Theorem (Parity reduction)

Fix k\ge4, either refinement convention, and the corresponding t in (eq:bag-parameter). From every choice system on t domains one can explicitly construct two uncolored simple graphs X_0,X_\star of equal order such that

X_0\equiv_k X_\star \quad\Longleftrightarrow\quad \text{the choice system has no successful choice}.

The assertion holds both with common color names on separate graphs and with pure-tuple comparison in their disjoint union.

If every domain and label set has size at most A, where A\ge1, their common order is at most A M_t, where

M_t=t\,2^{\,t-2+3\binom{t-1}{2}}+12\binom t3.

When a successful choice exists, the two graphs already have different histograms after at most two joint rounds, or one separate round.

Section 1, Introduction Read it in the paper TeX source, line 76

Show the TeX

Fix \(k\ge4\), either refinement convention, and the corresponding
\(t\) in \eqref{eq:bag-parameter}. From every choice system on \(t\)
domains one can explicitly construct two uncolored simple graphs
\(X_0,X_\star\) of equal order such that
\[
 X_0\equiv_k X_\star
 \quad\Longleftrightarrow\quad
 \text{the choice system has no successful choice}.
\]
The assertion holds both with common color names on separate graphs
and with pure-tuple comparison in their disjoint union.

If every domain and label set has size at most \(A\), where \(A\ge1\),
their common order is at most \(A M_t\), where
\[
 M_t=t\,2^{\,t-2+3\binom{t-1}{2}}+12\binom t3.
\]
When a successful choice exists, the two graphs already have different
histograms after at most two joint rounds, or one separate round.

Theorem (Fixed-dimension ETH consequence)

Assume positive exponential-rate ETH. There are constants c>0 and K\ge4 such that, for every fixed integer k\ge K, no deterministic algorithm decides k-WL equivalence of two n-vertex graphs in time O(n^{ck}). This holds for both joint and separate refinement. The inputs are explicit adjacency matrices of uncolored simple undirected graphs, and time is measured on a multitape Turing machine.

Section 1, Introduction Read it in the paper TeX source, line 197

Show the TeX

Assume positive exponential-rate ETH. There are constants \(c>0\)
and \(K\ge4\) such that, for every fixed integer \(k\ge K\), no
deterministic algorithm decides \(k\)-WL equivalence of two
\(n\)-vertex graphs in time \(O(n^{ck})\).
This holds for both joint and separate refinement. The inputs are
explicit adjacency matrices of uncolored simple undirected graphs,
and time is measured on a multitape Turing machine.

The complexity of identifying a graph by Weisfeiler–Leman refinement

Theorem

The following problem is EXPTIME-complete under deterministic polynomial-time many-one reductions: given a nonempty finite simple undirected uncolored graph G by its adjacency matrix and a positive integer k in binary, decide whether \WLdim(G)\le k. Here dimension one uses ordinary neighbor color refinement, and dimensions at least two use the joint-update rule described above.

Section 1, Introduction Read it in the paper TeX source, line 36

Show the TeX

The following problem is EXPTIME-complete under deterministic
polynomial-time many-one reductions: given a nonempty finite simple
undirected uncolored graph $G$ by its adjacency matrix and a positive
integer $k$ in binary, decide whether $\WLdim(G)\le k$.
Here dimension one uses ordinary neighbor color refinement, and
dimensions at least two use the joint-update rule described above.

Unconditional time lower bounds for Weisfeiler–Leman equivalence

Theorem

There are absolute constants c>0 and k_0 with the following property. For every fixed integer k\ge k_0 and every correct deterministic decider A for k-Weisfeiler–Leman equivalence, there is an integer n_0=n_0(A,k) such that

T_A(n)\ge n^{ck} \qquad\text{for every integer }n\ge n_0.

This holds for multitape Turing machines and for sequential random-access machines with O(\log(n+2))-bit addresses and words, boundedly many memory accesses per instruction, and fixed instructions computable in time polynomial in the word length. The same bound holds if A is required to decide correctly only on pairs of simple connected uncolored graphs of diameter at most two, with T_A(n) maximized over that class. It applies both to the joint replacement-multiset convention for k-dimensional refinement and to the separate-multiset convention defined below.

Section 1, Introduction Read it in the paper TeX source, line 82

Show the TeX

There are absolute constants $c>0$ and $k_0$ with the following
property. For every fixed integer $k\ge k_0$ and every correct
deterministic decider $A$ for $k$-Weisfeiler--Leman equivalence, there
is an integer $n_0=n_0(A,k)$ such that
\[
                     T_A(n)\ge n^{ck}
             \qquad\text{for every integer }n\ge n_0.
\]
This holds for multitape Turing machines and for sequential
random-access machines with $O(\log(n+2))$-bit addresses and words,
boundedly many memory accesses per instruction, and fixed instructions
computable in time polynomial in the word length.
The same bound holds if $A$ is required to decide correctly only on
pairs of simple connected uncolored graphs of diameter at most two,
with $T_A(n)$ maximized over that class. It applies both to the
joint replacement-multiset convention for $k$-dimensional refinement
and to the separate-multiset convention defined below.

Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs

Theorem

Both \WL and \SubWL are \EXPTIME-complete under deterministic polynomial-time many-one reductions.

Section 1, Introduction Read it in the paper TeX source, line 28

Show the TeX

Both \(\WL\) and \(\SubWL\) are \(\EXPTIME\)-complete under deterministic
polynomial-time many-one reductions.

Configuration and file facts

File: lean/ComparatorChallenges/VariableWL.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 133 link it.
Solution module: OAI.Combinatorics.VariableWL.Completeness
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: 27 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