Lean statement

Randomized quasipolynomial mean-payoff algorithm

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.randomized_quasipolynomial_mean_payoff

Lean statement file: RandomizedMeanPayoff.leanShow on the mapFamily 104. 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.

noncomputable def MainClaim : Prop := by  classical  exact ∃ M : Machine, ∃ C : ℝ, 0 < C ∧ ∀ G : Game, ∃ T : ℕ,    (T : ℝ) ≤ (2 : ℝ) ^ (C * (Real.logb 2 ((input G).length + 2)) ^ 2) ∧    (∀ bits : Fin T → Bool, (result M G T bits).control = none) ∧    7 * 2 ^ T ≤ 8 * (Finset.univ.filter      (fun bits : Fin T → Bool => CorrectOutput G (result M G T bits))).card⋮theorem randomized_quasipolynomial_mean_payoff : MeanPayoff.MainClaim := 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.MeanPayoff.Game line 6

structure Game where  n : ℕ  positive : 0 < n  m : ℕ  tail : Fin m → Fin n  head : Fin m → Fin n  maxOwner : Fin n → Bool  weight : Fin m → ℤ  total : ∀ v : Fin n, ∃ e : Fin m, tail e = v

definition OAI.MeanPayoff.Strategy line 16

def Strategy (G : Game) :=  List (Fin G.m) → (v : Fin G.n) → {e : Fin G.m // G.tail e = v}

definition OAI.MeanPayoff.move line 19

def move (G : Game) (σ τ : Strategy G) (v : Fin G.n)    (history : List (Fin G.m)) : Fin G.m :=  if G.maxOwner v then (σ history v).val else (τ history v).val

definition OAI.MeanPayoff.position line 23

def position (G : Game) (σ τ : Strategy G) (start : Fin G.n) :    ℕ → Fin G.n × List (Fin G.m)  | 0 => (start, [])  | t + 1 =>    let p := position G σ τ start t    let e := move G σ τ p.1 p.2    (G.head e, p.2 ++ [e])

definition OAI.MeanPayoff.play line 31

def play (G : Game) (σ τ : Strategy G) (start : Fin G.n) (t : ℕ) : Fin G.m :=  let p := position G σ τ start t  move G σ τ p.1 p.2

definition OAI.MeanPayoff.meanPayoff line 35

noncomputable def meanPayoff (G : Game) (π : ℕ → Fin G.m) : ℝ :=  Filter.liminf (fun T : ℕ =>    (∑ j ∈ Finset.range T, (G.weight (π j) : ℝ)) / (T : ℝ)) Filter.atTop

definition OAI.MeanPayoff.Winning line 39

def Winning (G : Game) (v : Fin G.n) : Prop :=  ∃ σ : Strategy G, ∀ τ : Strategy G, 0 ≤ meanPayoff G (play G σ τ v)

definition OAI.MeanPayoff.encodeNat line 42

def encodeNat (n : ℕ) : List Bool :=  let bits := Nat.bits (n + 1)  List.replicate bits.length true ++ [false] ++ bits

definition OAI.MeanPayoff.encodeInt line 46

def encodeInt (z : ℤ) : List Bool :=  decide (z < 0) :: encodeNat z.natAbs

definition OAI.MeanPayoff.input line 49

def input (G : Game) : List Bool :=  encodeNat G.n ++ encodeNat G.m ++ List.ofFn G.maxOwner ++    (List.ofFn (fun e : Fin G.m =>      encodeNat ((G.tail e).val + 1) ++ encodeNat ((G.head e).val + 1) ++        encodeInt (G.weight e))).flatten

abbreviation OAI.MeanPayoff.Symbol line 55
```
abbrev Symbol := Option Bool
```

inductive type OAI.MeanPayoff.Direction line 57

inductive Direction where  | left | stay | right  deriving DecidableEq

definition OAI.MeanPayoff.Direction.offset line 61

def Direction.offset : Direction → ℤ  | .left => -1  | .stay => 0  | .right => 1

structure OAI.MeanPayoff.Action line 66

structure Action (q : ℕ) where  next : Fin q  write : Fin 3 → Symbol  shift : Fin 3 → Direction

structure OAI.MeanPayoff.Machine line 71

structure Machine where  states : ℕ  start : Fin states  transition : Fin states → (Fin 3 → Symbol) → Bool → Option (Action states)

structure OAI.MeanPayoff.Configuration line 76

structure Configuration (M : Machine) where  control : Option (Fin M.states)  head : Fin 3 → ℤ  tape : Fin 3 → ℤ → Symbol

definition OAI.MeanPayoff.initial line 81

def initial (M : Machine) (code : List Bool) : Configuration M where  control := some M.start  head := fun _ => 0  tape := fun k j => if k = 0 ∧ 0 ≤ j then code[j.toNat]? else none

definition OAI.MeanPayoff.step line 86

def step (M : Machine) (c : Configuration M) (bit : Bool) : Configuration M :=  match c.control with  | none => c  | some s =>    match M.transition s (fun k => c.tape k (c.head k)) bit with    | none => { c with control := none }    | some a =>      { control := some a.next        head := fun k => c.head k + (a.shift k).offset        tape := fun k => Function.update (c.tape k) (c.head k) (a.write k) }

definition OAI.MeanPayoff.run line 97

def run (M : Machine) (code : List Bool) (random : ℕ → Bool) : ℕ → Configuration M  | 0 => initial M code  | t + 1 => step M (run M code random t) (random t)

definition OAI.MeanPayoff.padBits line 101

def padBits {T : ℕ} (bits : Fin T → Bool) (j : ℕ) : Bool :=  if h : j < T then bits ⟨j, h⟩ else false

definition OAI.MeanPayoff.result line 104

def result (M : Machine) (G : Game) (T : ℕ) (bits : Fin T → Bool) : Configuration M :=  run M (input G) (padBits bits) T

definition OAI.MeanPayoff.CorrectOutput line 107

def CorrectOutput (G : Game) {M : Machine} (c : Configuration M) : Prop :=  c.control = none ∧  (∀ v : Fin G.n, c.tape 2 (v.val : ℤ) = some true ↔ Winning G v) ∧  (∀ v : Fin G.n, c.tape 2 (v.val : ℤ) = some false ↔ ¬ Winning G v) ∧  ∀ j : ℤ, (j < 0 ∨ (G.n : ℤ) ≤ j) → c.tape 2 j = none

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: pi, randomized, total, machine, random, tape, game, strategy and two more. Found in the paper’s title, which its main theorem states: quasipolynomial, payoff and mean. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: 7/8, probability, least and algorithm.

Statements in Randomized quasipolynomial-time mean-payoff games come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Randomized quasipolynomial-time mean-payoff games

Theorem

There is a uniform randomized algorithm that, for every such game, returns the entire set

U=\{v\in V:\text{there is a Max strategy }\sigma \text{ such that for every Min strategy }\tau,\ \operatorname{MP}_w(\pi_{v,\sigma,\tau})\ge0\}

correctly with probability at least 7/8. For an absolute constant C>0, on every random tape its total cost is at most

2^{C(\log_2(L+2))^2}

bit operations in a standard model polynomially equivalent to a uniform probabilistic Turing machine. The bound includes preprocessing, recursive calls, intermediate arithmetic, and random-bit generation.

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

Show the TeX

There is a uniform randomized algorithm that, for every such game,
returns the entire set
\[
 U=\{v\in V:\text{there is a Max strategy }\sigma
       \text{ such that for every Min strategy }\tau,\ 
       \operatorname{MP}_w(\pi_{v,\sigma,\tau})\ge0\}
\]
correctly with probability at least $7/8$.
For an absolute constant $C>0$, on every random tape its total cost is at
most
\[
 2^{C(\log_2(L+2))^2}
\]
bit operations in a standard model polynomially equivalent to a uniform
probabilistic Turing machine. The bound includes preprocessing, recursive
calls, intermediate arithmetic, and random-bit generation.

Other possible paper matches, closest wording first

Proposition 2.5 of Randomized quasipolynomial-time mean-payoff games. Words of the Lean statement, its names split into words, that also appear in the theorem: play, history, mean, payoff, winning, eq and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: threshold, average, zero, dependent and least.
Corollary 8.1 of Randomized quasipolynomial-time mean-payoff games. Words of the Lean statement, its names split into words, that also appear in the theorem: symbol, randomized, tape, integer, winning, strategy and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: average, 7/8, probability, least and algorithm.
Proposition 2.1 of Mean-payoff parity games in quasipolynomial time. Words of the Lean statement, its names split into words, that also appear in the theorem: encoding, total, weights, length, history, integer and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: signed, finite, binary, dependent and least.
Proposition 2.2 of Deterministic quasipolynomial-time mean-payoff games. Words of the Lean statement, its names split into words, that also appear in the theorem: play, starting, history, winning, mean, payoff and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: average, allow, zero, dependent and least.
Corollary 6.4 of Randomized quasipolynomial-time mean-payoff games. Words of the Lean statement, its names split into words, that also appear in the theorem: top, weights, output and winning. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: 7/8, complete, probability and least.
Theorem 1.1 of Deterministic quasipolynomial-time mean-payoff games. Words of the Lean statement, its names split into words, that also appear in the theorem: pi, machine, output, game, strategy, max and one more. Found in the paper’s title, which its main theorem states: quasipolynomial, payoff and mean. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: complete and algorithm.

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 104’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “weaker-statement”.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record
Their note: Only a randomized (error 1/8) quasipolynomial algorithm for the winning set; the deterministic algorithm, exact values/optimal positional strategies, turn-based stochastic games and mean-payoff-parity games (two of four index papers) are not stated. TruffetCounterexample is a side negative result.
Definitions the group asks readers to compare with the paper: Custom 3-tape randomized Turing machine (structure Machine, step, run M code random, one random bit per step, alphabet Option Bool, heads in ℤ) with T existentially quantified per input and time = number of steps; standard in form but NOT Mathlib's TM model, so 'quasipolynomial bit time' rests on this hand-rolled definition. Mean payoff is liminf of running averages and strategies are history-dependent (Strategy G := List (Fin G.m) → ...), both standard
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: Headline: deterministic quasipolynomial algorithms that compute values and optimal positional strategies (ordinary games), the nonnegative-value set of stochastic games, and mean-payoff-parity winning sets. Lean: one randomized (≥ 7/8 success) quasipolynomial algorithm for the zero-threshold winning set of ordinary games, plus a finite counterexample to Truffet's procedure. Of the four manuscripts, only the randomized one is formalized.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result gives one randomized algorithm for the complete zero-threshold winning set of every finite mean-payoff game with signed binary weights. It succeeds with probability at least $7/8$ and runs in quasipolynomial bit time on every random tape. Self-loops, parallel edges, and history-dependent strategies are allowed, and winning means a nonnegative liminf average payoff.”
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

126 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: 59 to its own declarations, 90 to Mathlib’s documentation. 23 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 randomized_quasipolynomial_mean_payoff 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 23 other declarations of the file; with the theorem itself they span 94 lines. 1 of them is named in the statement itself.

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

structure Gameline 6library
def Strategyline 16library
def moveline 19library
def positionline 23library
def playline 31library
def meanPayoffline 35library
def Winningline 39library
def encodeNatline 42library
def encodeIntline 46library
def inputline 49library
abbrev Symbolline 55library
inductive Directionline 57library
def Direction.offsetline 61library
structure Actionline 66library
structure Machineline 71library
structure Configurationline 76library
def initialline 81library
def stepline 86library
def runline 97library
def padBitsline 101library
def resultline 104library
def CorrectOutputline 107library
def MainClaimline 113in the statementlibrary
theorem OAI.randomized_quasipolynomial_mean_payoffline 123selected theorem

import Mathlibnamespace OAInamespace MeanPayoffstructure Game where  n : ℕ  positive : 0 < n  m : ℕ  tail : Fin m → Fin n  head : Fin m → Fin n  maxOwner : Fin n → Bool  weight : Fin m → ℤ  total : ∀ v : Fin n, ∃ e : Fin m, tail e = vdef Strategy (G : Game) :=  List (Fin G.m) → (v : Fin G.n) → {e : Fin G.m // G.tail e = v}def move (G : Game) (σ τ : Strategy G) (v : Fin G.n)    (history : List (Fin G.m)) : Fin G.m :=  if G.maxOwner v then (σ history v).val else (τ history v).valdef position (G : Game) (σ τ : Strategy G) (start : Fin G.n) :    ℕ → Fin G.n × List (Fin G.m)  | 0 => (start, [])  | t + 1 =>    let p := position G σ τ start t    let e := move G σ τ p.1 p.2    (G.head e, p.2 ++ [e])def play (G : Game) (σ τ : Strategy G) (start : Fin G.n) (t : ℕ) : Fin G.m :=  let p := position G σ τ start t  move G σ τ p.1 p.2noncomputable def meanPayoff (G : Game) (π : ℕ → Fin G.m) : ℝ :=  Filter.liminf (fun T : ℕ =>    (∑ j ∈ Finset.range T, (G.weight (π j) : ℝ)) / (T : ℝ)) Filter.atTopdef Winning (G : Game) (v : Fin G.n) : Prop :=  ∃ σ : Strategy G, ∀ τ : Strategy G, 0 ≤ meanPayoff G (play G σ τ v)def encodeNat (n : ℕ) : List Bool :=  let bits := Nat.bits (n + 1)  List.replicate bits.length true ++ [false] ++ bitsdef encodeInt (z : ℤ) : List Bool :=  decide (z < 0) :: encodeNat z.natAbsdef input (G : Game) : List Bool :=  encodeNat G.n ++ encodeNat G.m ++ List.ofFn G.maxOwner ++    (List.ofFn (fun e : Fin G.m =>      encodeNat ((G.tail e).val + 1) ++ encodeNat ((G.head e).val + 1) ++        encodeInt (G.weight e))).flattenabbrev Symbol := Option Boolinductive Direction where  | left | stay | right  deriving DecidableEqdef Direction.offset : Direction → ℤ  | .left => -1  | .stay => 0  | .right => 1structure Action (q : ℕ) where  next : Fin q  write : Fin 3 → Symbol  shift : Fin 3 → Directionstructure Machine where  states : ℕ  start : Fin states  transition : Fin states → (Fin 3 → Symbol) → Bool → Option (Action states)structure Configuration (M : Machine) where  control : Option (Fin M.states)  head : Fin 3 → ℤ  tape : Fin 3 → ℤ → Symboldef initial (M : Machine) (code : List Bool) : Configuration M where  control := some M.start  head := fun _ => 0  tape := fun k j => if k = 0 ∧ 0 ≤ j then code[j.toNat]? else nonedef step (M : Machine) (c : Configuration M) (bit : Bool) : Configuration M :=  match c.control with  | none => c  | some s =>    match M.transition s (fun k => c.tape k (c.head k)) bit with    | none => { c with control := none }    | some a =>      { control := some a.next        head := fun k => c.head k + (a.shift k).offset        tape := fun k => Function.update (c.tape k) (c.head k) (a.write k) }def run (M : Machine) (code : List Bool) (random : ℕ → Bool) : ℕ → Configuration M  | 0 => initial M code  | t + 1 => step M (run M code random t) (random t)def padBits {T : ℕ} (bits : Fin T → Bool) (j : ℕ) : Bool :=  if h : j < T then bits ⟨j, h⟩ else falsedef result (M : Machine) (G : Game) (T : ℕ) (bits : Fin T → Bool) : Configuration M :=  run M (input G) (padBits bits) Tdef CorrectOutput (G : Game) {M : Machine} (c : Configuration M) : Prop :=  c.control = none ∧  (∀ v : Fin G.n, c.tape 2 (v.val : ℤ) = some true ↔ Winning G v) ∧  (∀ v : Fin G.n, c.tape 2 (v.val : ℤ) = some false ↔ ¬ Winning G v) ∧  ∀ j : ℤ, (j < 0 ∨ (G.n : ℤ) ≤ j) → c.tape 2 j = nonenoncomputable def MainClaim : Prop := by  classical  exact ∃ M : Machine, ∃ C : ℝ, 0 < C ∧ ∀ G : Game, ∃ T : ℕ,    (T : ℝ) ≤ (2 : ℝ) ^ (C * (Real.logb 2 ((input G).length + 2)) ^ 2) ∧    (∀ bits : Fin T → Bool, (result M G T bits).control = none) ∧    7 * 2 ^ T ≤ 8 * (Finset.univ.filter      (fun bits : Fin T → Bool => CorrectOutput G (result M G T bits))).cardend MeanPayofftheorem randomized_quasipolynomial_mean_payoff : MeanPayoff.MainClaim := by  sorryend 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.

Deterministic quasipolynomial-time mean-payoff games

Theorem

A uniform deterministic algorithm returns exactly the set

U=\bigl\{v\in V:\exists\text{ Max strategy }\sigma\ \forall\text{ Min strategies }\tau, \ \MP_w(\pi_{v,\sigma,\tau})\ge0\bigr\}

for every game specified above. For an absolute constant C>0, the complete computation uses at most

2^{C(\log_2(L+2))^2}

bit operations in a standard model polynomially equivalent to a uniform deterministic Turing machine. The bound includes preprocessing, exact arithmetic, recursion, and output of the entire set U.

Section 1, Introduction Read it in the paper TeX source, line 34

Show the TeX

A uniform deterministic algorithm returns exactly the set
\[
 U=\bigl\{v\in V:\exists\text{ Max strategy }\sigma\ 
       \forall\text{ Min strategies }\tau,
       \ \MP_w(\pi_{v,\sigma,\tau})\ge0\bigr\}
\]
for every game specified above. For an absolute constant $C>0$, the
complete computation uses at most
\[
                      2^{C(\log_2(L+2))^2}
\]
bit operations in a standard model polynomially equivalent to a
uniform deterministic Turing machine. The bound includes
preprocessing, exact arithmetic, recursion, and output of the entire
set $U$.

Randomized quasipolynomial-time mean-payoff games

Theorem

There is a uniform randomized algorithm that, for every such game, returns the entire set

U=\{v\in V:\text{there is a Max strategy }\sigma \text{ such that for every Min strategy }\tau,\ \operatorname{MP}_w(\pi_{v,\sigma,\tau})\ge0\}

correctly with probability at least 7/8. For an absolute constant C>0, on every random tape its total cost is at most

2^{C(\log_2(L+2))^2}

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

Show the TeX

There is a uniform randomized algorithm that, for every such game,
returns the entire set
\[
 U=\{v\in V:\text{there is a Max strategy }\sigma
       \text{ such that for every Min strategy }\tau,\ 
       \operatorname{MP}_w(\pi_{v,\sigma,\tau})\ge0\}
\]
correctly with probability at least $7/8$.
For an absolute constant $C>0$, on every random tape its total cost is at
most
\[
 2^{C(\log_2(L+2))^2}
\]
bit operations in a standard model polynomially equivalent to a uniform
probabilistic Turing machine. The bound includes preprocessing, recursive
calls, intermediate arithmetic, and random-bit generation.

Configuration and file facts

File: lean/ComparatorChallenges/RandomizedMeanPayoff.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Computability/RandomMean/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: 23 local vocabulary declarations
Lines of Lean, without comments or blank lines: 94
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