Lean statement

Injectivity of Bogomolov–Pop reconstruction modulo the stated ambiguities

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.BogomolovPop.main_graph_injective

Lean statement file: BogomolovPopInjectivity.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 009 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem main_graph_injective (ell : ℕ) [Fact ell.Prime]    (k K l L : Type*) [Field k] [Field K] [Field l] [Field L]    [IsAlgClosed k] [IsAlgClosed l] [Algebra k K] [Algebra l L]    [Algebra.EssFiniteType k K] [Algebra.EssFiniteType l L]    (hk : (ell : k) ≠ 0) (hl : (ell : l) ≠ 0)    (hK : 2 ≤ Algebra.trdeg k K) (hL : 2 ≤ Algebra.trdeg l L) :    ∀ (a a' : IsomF k K l L) (b : IsomCUnits ell K L),      PhiGraph a b → PhiGraph a' b → a = a'  := 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.BogomolovPop.proEllKernel line 8

def proEllKernel (ell : ℕ) (G : Type*) [Group G] [TopologicalSpace G] : Subgroup G :=  ⨅ (H : OpenNormalSubgroup G) (_ : IsPGroup ell (G ⧸ H.toSubgroup)), H.toSubgroup

abbreviation OAI.BogomolovPop.ProEll line 17

abbrev ProEll (ell : ℕ) (F : Type*) [Field F] :=  Field.absoluteGaloisGroup F ⧸ proEllKernel ell (Field.absoluteGaloisGroup F)

definition OAI.BogomolovPop.closedGammaThree line 20

def closedGammaThree (G : Type*) [Group G] [TopologicalSpace G] [IsTopologicalGroup G] :    Subgroup G :=  (⁅(commutator G).topologicalClosure, (⊤ : Subgroup G)⁆).topologicalClosure

abbreviation OAI.BogomolovPop.PiA line 29

abbrev PiA (ell : ℕ) (F : Type*) [Field F] := TopologicalAbelianization (ProEll ell F)

abbreviation OAI.BogomolovPop.PiC line 31

abbrev PiC (ell : ℕ) (F : Type*) [Field F] := ProEll ell F ⧸ closedGammaThree (ProEll ell F)

definition OAI.BogomolovPop.centralProjection line 38

def centralProjection (ell : ℕ) (F : Type*) [Field F] : PiC ell F →* PiA ell F :=  QuotientGroup.map _ _ (MonoidHom.id _) (closedGammaThree_le _)

abbreviation OAI.BogomolovPop.Delta line 41

abbrev Delta (ell : ℕ) (F : Type*) [Field F] := (centralProjection ell F).ker

definition OAI.BogomolovPop.sourceCommutator line 43

def sourceCommutator {G : Type*} [Group G] (x y : G) : G := x⁻¹ * y⁻¹ * x * y

definition OAI.BogomolovPop.abelianLift line 45

def abelianLift (ell : ℕ) (F : Type*) [Field F] (x : PiA ell F) : ProEll ell F :=  Quotient.out x

definition OAI.BogomolovPop.bracket line 51

def bracket (ell : ℕ) (F : Type*) [Field F] (x y : PiA ell F) : Delta ell F :=  ⟨QuotientGroup.mk' _ (sourceCommutator (abelianLift ell F x) (abelianLift ell F y)), by    change centralProjection ell F _ = 1    simp only [centralProjection, sourceCommutator, map_mul, map_inv]    simp [mul_comm]⟩

definition OAI.BogomolovPop.BracketCompatible line 57

def BracketCompatible {ell : ℕ} {K L : Type*} [Field K] [Field L]    (φ : PiA ell L ≃ₜ* PiA ell K) : Prop :=  ∃ ψ : Delta ell L ≃ₜ* Delta ell K,    ∀ σ τ, ψ (bracket ell L σ τ) = bracket ell K (φ σ) (φ τ)

abbreviation OAI.BogomolovPop.PerfectFieldClosure line 62

abbrev PerfectFieldClosure (F : Type*) [Field F] := perfectClosure F (AlgebraicClosure F)

definition OAI.BogomolovPop.PerfectIsom line 64

def PerfectIsom (k K l L : Type*) [Field k] [Field K] [Field l] [Field L]    [Algebra k K] [Algebra l L] :=  { α : PerfectFieldClosure K ≃+* PerfectFieldClosure L //    α '' Set.range (algebraMap k (PerfectFieldClosure K)) =      Set.range (algebraMap l (PerfectFieldClosure L)) }

definition OAI.BogomolovPop.FrobeniusStep line 70

def FrobeniusStep {k K l L : Type*} [Field k] [Field K] [Field l] [Field L]    [Algebra k K] [Algebra l L] (α β : PerfectIsom k K l L) : Prop :=  ∃ p : ℕ, p.Prime ∧ (p : L) = 0 ∧    ∀ x : PerfectFieldClosure K, β.1 x = (α.1 x) ^ p

abbreviation OAI.BogomolovPop.IsomF line 75

abbrev IsomF (k K l L : Type*) [Field k] [Field K] [Field l] [Field L]    [Algebra k K] [Algebra l L] :=  Quotient (Relation.EqvGen.setoid (FrobeniusStep (k := k) (K := K) (l := l) (L := L)))

abbreviation OAI.BogomolovPop.IsomC line 79

abbrev IsomC (ell : ℕ) (K L : Type*) [Field K] [Field L] :=  {φ : PiA ell L ≃ₜ* PiA ell K // BracketCompatible φ}

definition OAI.BogomolovPop.ScalarStep line 82

def ScalarStep {ell : ℕ} [Fact ell.Prime] {K L : Type*} [Field K] [Field L]    (φ χ : IsomC ell K L) : Prop :=  ∃ u : (PadicInt ell)ˣ, ∀ x : PiA ell L,    Filter.Tendsto (fun n : ℕ => (φ.1 x) ^ PadicInt.appr (u : PadicInt ell) n)      Filter.atTop (nhds (χ.1 x))

abbreviation OAI.BogomolovPop.IsomCUnits line 88

abbrev IsomCUnits (ell : ℕ) [Fact ell.Prime] (K L : Type*) [Field K] [Field L] :=  Quotient (Relation.EqvGen.setoid (ScalarStep (ell := ell) (K := K) (L := L)))

definition OAI.BogomolovPop.absoluteToPiA line 91

def absoluteToPiA (ell : ℕ) (F : Type*) [Field F] :    Field.absoluteGaloisGroup F →* PiA ell F :=  (QuotientGroup.mk' _).comp (QuotientGroup.mk' _)

definition OAI.BogomolovPop.absoluteAction line 95

def absoluteAction {F : Type*} [Field F] (σ : Field.absoluteGaloisGroup F)    (x : AlgebraicClosure F) : AlgebraicClosure F := by  change AlgebraicClosure F ≃ₐ[F] AlgebraicClosure F at σ  exact σ x

definition OAI.BogomolovPop.PullbackGraph line 100

def PullbackGraph {ell : ℕ} {k K l L : Type*}    [Field k] [Field K] [Field l] [Field L] [Algebra k K] [Algebra l L]    (α : PerfectIsom k K l L) (φ : IsomC ell K L) : Prop :=  ∃ β : AlgebraicClosure K ≃+* AlgebraicClosure L,    (∀ x : PerfectFieldClosure K, β x = ((α.1 x : PerfectFieldClosure L) :      AlgebraicClosure L)) ∧    ∀ σ : Field.absoluteGaloisGroup L, ∃ τ : Field.absoluteGaloisGroup K,      (∀ x : AlgebraicClosure K,        β (absoluteAction τ x) = absoluteAction σ (β x)) ∧      φ.1 (absoluteToPiA ell L σ) = absoluteToPiA ell K τ

definition OAI.BogomolovPop.PhiGraph line 111

def PhiGraph {ell : ℕ} [Fact ell.Prime] {k K l L : Type*}    [Field k] [Field K] [Field l] [Field L] [Algebra k K] [Algebra l L]    (a : IsomF k K l L) (b : IsomCUnits ell K L) : Prop :=  ∃ α : PerfectIsom k K l L, ∃ φ : IsomC ell K L,    Quotient.mk _ α = a ∧ Quotient.mk _ φ = b ∧ PullbackGraph α φ

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: algebraically, bogomolov, closed, different, fields, function and reconstruction.

Statements in The Bogomolov-Pop reconstruction theorem come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of The Bogomolov-Pop reconstruction theorem

Theorem (Bogomolov–Pop reconstruction)

Let k and l be arbitrary algebraically closed fields of characteristics different from \ell, and let K/k and L/l be function fields with \trdeg(K/k),\trdeg(L/l)\ge2. Then

\Phi_{K,L}:\Isom_F^i(K,L)\longrightarrow \Isom^c(\Pi_L^a,\Pi_K^a)/\Zell^\times, \qquad [\alpha]\longmapsto[\alpha^*],

is a bijection.

Section 1, The reconstruction problem Read it in the paper TeX source, line 55

Show the TeX

Let $k$ and $l$ be arbitrary algebraically closed fields of characteristics
different from $\ell$, and let $K/k$ and $L/l$ be function fields with
$\trdeg(K/k),\trdeg(L/l)\ge2$.  Then
\[
 \Phi_{K,L}:\Isom_F^i(K,L)\longrightarrow
 \Isom^c(\Pi_L^a,\Pi_K^a)/\Zell^\times,
 \qquad [\alpha]\longmapsto[\alpha^*],
\]
is a bijection.

Other possible paper matches, closest wording first

Proposition 7.1 of The Bogomolov-Pop reconstruction theorem. Words of the Lean statement, its names split into words, that also appear in the theorem: perfect, graph, top, abelian, closures, abelianizations and ten more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: modulo, data, pops, reconstruction and isomorphism.
Proposition 3.5 of The Bogomolov-Pop reconstruction theorem. Words of the Lean statement, its names split into words, that also appear in the theorem: bracket, map, compatible, ell, prime, closed and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: isomorphism.
Proposition 8.4 of Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory. Words of the Lean statement, its names split into words, that also appear in the theorem: action, scalar, frobenius and field.
Proposition 4.3 of Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, units, algebraic, ell, finite, prime and two more.
Theorem 1.1 of Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory. Words of the Lean statement, its names split into words, that also appear in the theorem: trdeg, map, compatible, ell, prime, closed and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: isomorphism.
Proposition 3.7 of The Bogomolov-Pop reconstruction theorem. Words of the Lean statement, its names split into words, that also appear in the theorem: units, algebraic, subgroups, finite, prime and field. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: isomorphism.

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 110 s; peak memory 6.5 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 009’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: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: Only injectivity half of Bogomolov-Pop reconstruction (Galois data side); no existence/surjectivity and no Milnor K-theoretic reconstruction or perfect-closure/constants recovery statement.
Definitions the group asks readers to compare with the paper: def ScalarStep ... ∃ u : (PadicInt ell)ˣ, ∀ x, Tendsto (fun n => (φ.1 x) ^ PadicInt.appr (u : PadicInt ell) n) atTop (nhds (χ.1 x)) (ℓ-adic unit scaling encoded via appr approximants; custom but plausible); abelianLift := Quotient.out x (choice-based section used in bracket, harmless)
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: Both Milnor K-theory reconstruction theorems (mod ℓ with ℓ ≠ char; mod p in equal characteristic) are not stated at all. Of Bogomolov–Pop only injectivity is stated (existence omitted, declared).

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The selected statement proves this uniqueness; existence of a field isomorphism for every admissible Galois-data isomorphism is outside it.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states part of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

131 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: 63 to its own declarations, 136 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_graph_injective 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 23 of the file’s 26 other declarations; with the theorem itself they span 81 lines. 3 of them are named in the statement itself.

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

def proEllKernelline 8library
abbrev ProEllline 17library
def closedGammaThreeline 20library
abbrev PiAline 29library
abbrev PiCline 31library
theorem closedGammaThree_leline 33library
def centralProjectionline 38library
abbrev Deltaline 41library
def sourceCommutatorline 43library
def abelianLiftline 45library
def bracketline 51library
def BracketCompatibleline 57library
abbrev PerfectFieldClosureline 62library
def PerfectIsomline 64library
def FrobeniusStepline 70library
abbrev IsomFline 75in the statementlibrary
abbrev IsomCline 79library
def ScalarStepline 82library
abbrev IsomCUnitsline 88in the statementlibrary
def absoluteToPiAline 91library
def absoluteActionline 95library
def PullbackGraphline 100library
def PhiGraphline 111in the statementlibrary
theorem main_graph_injectiveline 117selected theoremlibrary

Other Lean rules that may affect the statement 2

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 proEllKernel_normal line 11
instance closedGammaThree_normal line 24

1 declaration the statement does not reach by name

These definitions and lemmas were not found by following names in the statement. Lean may still use them through rules such as instances or macros.

theorem abelianLift_spec line 48

import Mathlibnamespace OAInoncomputable sectionnamespace BogomolovPopdef proEllKernel (ell : ℕ) (G : Type*) [Group G] [TopologicalSpace G] : Subgroup G :=  ⨅ (H : OpenNormalSubgroup G) (_ : IsPGroup ell (G ⧸ H.toSubgroup)), H.toSubgroupinstance proEllKernel_normal (ell : ℕ) (G : Type*) [Group G] [TopologicalSpace G] :    (proEllKernel ell G).Normal := by  unfold proEllKernel  exact Subgroup.normal_iInf_normal fun H =>    Subgroup.normal_iInf_normal fun _ => H.isNormal'abbrev ProEll (ell : ℕ) (F : Type*) [Field F] :=  Field.absoluteGaloisGroup F ⧸ proEllKernel ell (Field.absoluteGaloisGroup F)def closedGammaThree (G : Type*) [Group G] [TopologicalSpace G] [IsTopologicalGroup G] :    Subgroup G :=  (⁅(commutator G).topologicalClosure, (⊤ : Subgroup G)⁆).topologicalClosureinstance closedGammaThree_normal (G : Type*) [Group G] [TopologicalSpace G]    [IsTopologicalGroup G] : (closedGammaThree G).Normal := by  unfold closedGammaThree  exact Subgroup.is_normal_topologicalClosure _abbrev PiA (ell : ℕ) (F : Type*) [Field F] := TopologicalAbelianization (ProEll ell F)abbrev PiC (ell : ℕ) (F : Type*) [Field F] := ProEll ell F ⧸ closedGammaThree (ProEll ell F)theorem closedGammaThree_le (G : Type*) [Group G] [TopologicalSpace G]    [IsTopologicalGroup G] : closedGammaThree G ≤ (commutator G).topologicalClosure := by  exact Subgroup.topologicalClosure_minimal _ (Subgroup.commutator_le_left _ _)    (Subgroup.isClosed_topologicalClosure _)def centralProjection (ell : ℕ) (F : Type*) [Field F] : PiC ell F →* PiA ell F :=  QuotientGroup.map _ _ (MonoidHom.id _) (closedGammaThree_le _)abbrev Delta (ell : ℕ) (F : Type*) [Field F] := (centralProjection ell F).kerdef sourceCommutator {G : Type*} [Group G] (x y : G) : G := x⁻¹ * y⁻¹ * x * ydef abelianLift (ell : ℕ) (F : Type*) [Field F] (x : PiA ell F) : ProEll ell F :=  Quotient.out xtheorem abelianLift_spec (ell : ℕ) (F : Type*) [Field F] (x : PiA ell F) :    QuotientGroup.mk' _ (abelianLift ell F x) = x := Quotient.out_eq xdef bracket (ell : ℕ) (F : Type*) [Field F] (x y : PiA ell F) : Delta ell F :=  ⟨QuotientGroup.mk' _ (sourceCommutator (abelianLift ell F x) (abelianLift ell F y)), by    change centralProjection ell F _ = 1    simp only [centralProjection, sourceCommutator, map_mul, map_inv]    simp [mul_comm]⟩def BracketCompatible {ell : ℕ} {K L : Type*} [Field K] [Field L]    (φ : PiA ell L ≃ₜ* PiA ell K) : Prop :=  ∃ ψ : Delta ell L ≃ₜ* Delta ell K,    ∀ σ τ, ψ (bracket ell L σ τ) = bracket ell K (φ σ) (φ τ)abbrev PerfectFieldClosure (F : Type*) [Field F] := perfectClosure F (AlgebraicClosure F)def PerfectIsom (k K l L : Type*) [Field k] [Field K] [Field l] [Field L]    [Algebra k K] [Algebra l L] :=  { α : PerfectFieldClosure K ≃+* PerfectFieldClosure L //    α '' Set.range (algebraMap k (PerfectFieldClosure K)) =      Set.range (algebraMap l (PerfectFieldClosure L)) }def FrobeniusStep {k K l L : Type*} [Field k] [Field K] [Field l] [Field L]    [Algebra k K] [Algebra l L] (α β : PerfectIsom k K l L) : Prop :=  ∃ p : ℕ, p.Prime ∧ (p : L) = 0 ∧    ∀ x : PerfectFieldClosure K, β.1 x = (α.1 x) ^ pabbrev IsomF (k K l L : Type*) [Field k] [Field K] [Field l] [Field L]    [Algebra k K] [Algebra l L] :=  Quotient (Relation.EqvGen.setoid (FrobeniusStep (k := k) (K := K) (l := l) (L := L)))abbrev IsomC (ell : ℕ) (K L : Type*) [Field K] [Field L] :=  {φ : PiA ell L ≃ₜ* PiA ell K // BracketCompatible φ}def ScalarStep {ell : ℕ} [Fact ell.Prime] {K L : Type*} [Field K] [Field L]    (φ χ : IsomC ell K L) : Prop :=  ∃ u : (PadicInt ell)ˣ, ∀ x : PiA ell L,    Filter.Tendsto (fun n : ℕ => (φ.1 x) ^ PadicInt.appr (u : PadicInt ell) n)      Filter.atTop (nhds (χ.1 x))abbrev IsomCUnits (ell : ℕ) [Fact ell.Prime] (K L : Type*) [Field K] [Field L] :=  Quotient (Relation.EqvGen.setoid (ScalarStep (ell := ell) (K := K) (L := L)))def absoluteToPiA (ell : ℕ) (F : Type*) [Field F] :    Field.absoluteGaloisGroup F →* PiA ell F :=  (QuotientGroup.mk' _).comp (QuotientGroup.mk' _)def absoluteAction {F : Type*} [Field F] (σ : Field.absoluteGaloisGroup F)    (x : AlgebraicClosure F) : AlgebraicClosure F := by  change AlgebraicClosure F ≃ₐ[F] AlgebraicClosure F at σ  exact σ xdef PullbackGraph {ell : ℕ} {k K l L : Type*}    [Field k] [Field K] [Field l] [Field L] [Algebra k K] [Algebra l L]    (α : PerfectIsom k K l L) (φ : IsomC ell K L) : Prop :=  ∃ β : AlgebraicClosure K ≃+* AlgebraicClosure L,    (∀ x : PerfectFieldClosure K, β x = ((α.1 x : PerfectFieldClosure L) :      AlgebraicClosure L)) ∧    ∀ σ : Field.absoluteGaloisGroup L, ∃ τ : Field.absoluteGaloisGroup K,      (∀ x : AlgebraicClosure K,        β (absoluteAction τ x) = absoluteAction σ (β x)) ∧      φ.1 (absoluteToPiA ell L σ) = absoluteToPiA ell K τdef PhiGraph {ell : ℕ} [Fact ell.Prime] {k K l L : Type*}    [Field k] [Field K] [Field l] [Field L] [Algebra k K] [Algebra l L]    (a : IsomF k K l L) (b : IsomCUnits ell K L) : Prop :=  ∃ α : PerfectIsom k K l L, ∃ φ : IsomC ell K L,    Quotient.mk _ α = a ∧ Quotient.mk _ φ = b ∧ PullbackGraph α φtheorem main_graph_injective (ell : ℕ) [Fact ell.Prime]    (k K l L : Type*) [Field k] [Field K] [Field l] [Field L]    [IsAlgClosed k] [IsAlgClosed l] [Algebra k K] [Algebra l L]    [Algebra.EssFiniteType k K] [Algebra.EssFiniteType l L]    (hk : (ell : k) ≠ 0) (hl : (ell : l) ≠ 0)    (hK : 2 ≤ Algebra.trdeg k K) (hL : 2 ≤ Algebra.trdeg l L) :    ∀ (a a' : IsomF k K l L) (b : IsomCUnits ell K L),      PhiGraph a b → PhiGraph a' b → a = a'  := by  sorryend BogomolovPopendend 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.

The Bogomolov-Pop reconstruction theorem

Theorem (Bogomolov–Pop reconstruction)

Let k and l be arbitrary algebraically closed fields of characteristics different from \ell, and let K/k and L/l be function fields with \trdeg(K/k),\trdeg(L/l)\ge2. Then

\Phi_{K,L}:\Isom_F^i(K,L)\longrightarrow \Isom^c(\Pi_L^a,\Pi_K^a)/\Zell^\times, \qquad [\alpha]\longmapsto[\alpha^*],

is a bijection.

Section 1, The reconstruction problem Read it in the paper TeX source, line 55

Show the TeX

Let $k$ and $l$ be arbitrary algebraically closed fields of characteristics
different from $\ell$, and let $K/k$ and $L/l$ be function fields with
$\trdeg(K/k),\trdeg(L/l)\ge2$.  Then
\[
 \Phi_{K,L}:\Isom_F^i(K,L)\longrightarrow
 \Isom^c(\Pi_L^a,\Pi_K^a)/\Zell^\times,
 \qquad [\alpha]\longmapsto[\alpha^*],
\]
is a bijection.

Configuration and file facts

File: lean/ComparatorChallenges/BogomolovPopInjectivity.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 009 link it.
Solution module: OAI.Algebra.BogomolovPop.Injectivity
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: 24 local vocabulary declarations
Lines of Lean, without comments or blank lines: 92
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.

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