Definitions bundle OAIQ031A188 of the split of OpenAI, An explicit counterexample to the Auslander-Reiten conjecture (result family 199)

Definition

OAIQ031A188

by dbenbenn · Oct 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

A definitions bundle of the split of OpenAI's proof of OAI.ArExplicit.Statement.main, carved from OpenAI's openai/math development (OpenAI, An explicit counterexample to the Auslander-Reiten conjecture (result family 199), OpenAI Math Release, September 23, 2026, result family 199, github.com/openai/math at commit adc7f12, Apache License 2.0) and ported to Lean 4.33.1 and Mathlib 0df444a. The declarations and their proofs are OpenAI's.

Contents. It holds 2 declarations of OpenAI's Lean: 2 theorems.

Sources. They come from these modules of lean/OAI/Algebra, in whole or in part: AuslanderReiten_Fibers_FiberBranchConsequences, AuslanderReiten_Fibers_FiberProductCoordinates.

Imports. It imports the bundles OAIQ031A185 and OAIQ031A186 of this split.

Imported by (directly, in this split): the bundle OAIQ031A190.

Definition code

import Mathlib
import Definitions.Def_OAIQ031A185
import Definitions.Def_OAIQ031A186

section
-- module Solutions.OAIQ031_AR.AuslanderReiten_Fibers_FiberProductCoordinates
attribute [-instance] _root_.OAI.ArExplicit.Resolution.aInverseRestriction_isEquivalence _root_.OAI.ArExplicit.Resolution.cResolution_finite_terms _root_.OAI.ArExplicit.Resolution.complex_scalarTower _root_.OAI.ArExplicit.Resolution.instIsScalarTowerMulOppositeBaseAlgebraSimple _root_.OAI.ArExplicit.Resolution.simpleRightAction _root_.OAI.ArExplicit.Resolution.simpleRightModule _root_.OAI.ArExplicit.Resolution.tInducedCochain_kprojective _root_.OAI.ArExplicit.T.explicitBimodule_finite_left _root_.OAI.ArExplicit.T.explicitBimodule_finite_right _root_.OAI.ArExplicit.T.explicitBimodule_projective_left _root_.OAI.ArExplicit.T.explicitBimodule_projective_right _root_.OAI.ArExplicit.TrivialInduction.dualModule _root_.OAI.ArExplicit.TrivialInduction.induced_term_projective _root_.OAI.ArExplicit.TrivialInduction.instAdditiveModuleCatTrivialExtensionInflate _root_.OAI.ArExplicit.TrivialInduction.instIsScalarTowerTrivialExtensionCoinduced _root_.OAI.ArExplicit.TrivialInduction.instIsScalarTowerTrivialExtensionQuotient _root_.OAI.ArExplicit.TrivialInduction.instPreservesHomologyModuleCatTrivialExtensionInflate _root_.OAI.ArExplicit.TrivialInduction.instSMulCommClassTrivialExtensionCoinduced _root_.OAI.ArExplicit.TrivialInduction.instSMulCommClassTrivialExtensionMulOppositeCoinduced _root_.OAI.ArExplicit.TrivialInduction.instSMulCommClassTrivialExtensionMulOppositeQuotient _root_.OAI.ArExplicit.TrivialInduction.instSMulCommClassTrivialExtensionQuotient _root_.OAI.ArExplicit.TrivialInduction.quotientModule _root_.OAI.ArExplicit.VectorCochain.vectorSpaceInjective _root_.OAI.ArExplicit.cochainNatTensorSigns _root_.OAI.ArExplicit.Resolution.homETower _root_.OAI.ArExplicit.Resolution.homFTower _root_.OAI.ArExplicit.BalancedTensor.finitePiProjective _root_.OAI.ArExplicit.TriangularInduction.finitePiProjective
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section
namespace ArExplicit.T
open CategoryTheory CategoryTheory.Limits
open scoped TensorProduct ModuleCat.Algebra
universe u
variable {k : Type u} [Field k] [CharP k 2] (q : k) (H₁ H₂ : kˣ)
local notation "X₀" => simpleHomotopyObject q
local notation "T₀" => fiberProductTriangle q H₁ H₂

theorem fiberRhoPair_bijective_of_productFirst (n : ℤ)
    (hf : Function.Bijective (TriangleGradedSequence.first k T₀ X₀ n)) :
    Function.Bijective (fiberHomRhoPair q H₁ H₂ n) := by
  have he : (fiberHomRhoPair q H₁ H₂ n : FiberHom q H₁ H₂ n → _) =
      (fiberProductMiddleEquiv q H₁ H₂ n) ∘ (TriangleGradedSequence.first k T₀ X₀ n) := by
    funext v
    exact (fiberProductFirst_coordinates q H₁ H₂ n v).symm
  rw [he]
  exact (fiberProductMiddleEquiv q H₁ H₂ n).bijective.comp hf

end ArExplicit.T

end

end OAI
end

section
-- module Solutions.OAIQ031_AR.AuslanderReiten_Fibers_FiberBranchConsequences
attribute [-instance] _root_.OAI.ArExplicit.Resolution.aInverseRestriction_isEquivalence _root_.OAI.ArExplicit.Resolution.cResolution_finite_terms _root_.OAI.ArExplicit.Resolution.complex_scalarTower _root_.OAI.ArExplicit.Resolution.instIsScalarTowerMulOppositeBaseAlgebraSimple _root_.OAI.ArExplicit.Resolution.simpleRightAction _root_.OAI.ArExplicit.Resolution.simpleRightModule _root_.OAI.ArExplicit.Resolution.tInducedCochain_kprojective _root_.OAI.ArExplicit.T.explicitBimodule_finite_left _root_.OAI.ArExplicit.T.explicitBimodule_finite_right _root_.OAI.ArExplicit.T.explicitBimodule_projective_left _root_.OAI.ArExplicit.T.explicitBimodule_projective_right _root_.OAI.ArExplicit.TrivialInduction.dualModule _root_.OAI.ArExplicit.TrivialInduction.induced_term_projective _root_.OAI.ArExplicit.TrivialInduction.instAdditiveModuleCatTrivialExtensionInflate _root_.OAI.ArExplicit.TrivialInduction.instIsScalarTowerTrivialExtensionCoinduced _root_.OAI.ArExplicit.TrivialInduction.instIsScalarTowerTrivialExtensionQuotient _root_.OAI.ArExplicit.TrivialInduction.instPreservesHomologyModuleCatTrivialExtensionInflate _root_.OAI.ArExplicit.TrivialInduction.instSMulCommClassTrivialExtensionCoinduced _root_.OAI.ArExplicit.TrivialInduction.instSMulCommClassTrivialExtensionMulOppositeCoinduced _root_.OAI.ArExplicit.TrivialInduction.instSMulCommClassTrivialExtensionMulOppositeQuotient _root_.OAI.ArExplicit.TrivialInduction.instSMulCommClassTrivialExtensionQuotient _root_.OAI.ArExplicit.TrivialInduction.quotientModule _root_.OAI.ArExplicit.VectorCochain.vectorSpaceInjective _root_.OAI.ArExplicit.cochainNatTensorSigns _root_.OAI.ArExplicit.Resolution.homETower _root_.OAI.ArExplicit.Resolution.homFTower _root_.OAI.ArExplicit.BalancedTensor.finitePiProjective _root_.OAI.ArExplicit.TriangularInduction.finitePiProjective
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section
namespace ArExplicit.T
open CategoryTheory
open CochainComplex.HomComplex CompleteComparison
open scoped TensorProduct ModuleCat.Algebra
universe u
variable {k : Type u} [Field k] [CharP k 2] (q : k) (H₁ H₂ : kˣ)
local notation "T₀" => fiberProductTriangle q H₁ H₂
local notation "X₀" => simpleHomotopyObject q

variable (hq1 : 1+q ≠ 0) (e : BranchHom q 0 ≃ₗ[k] k)
variable (hg : branchHom q H₁ = branchHom q H₂) (hn : branchHom q H₁ ≠ 0)

include hq1 e hg hn

theorem fiberFullRhoPair_positive_bijective
    (hpos : ∀ n : ℤ, 0 < n → Subsingleton (BranchHom q n))
    (n : ℤ) (hnpos : 0 < n) :
    Function.Bijective (fiberFullRhoPair q H₁ H₂ n) := by
  apply fiberFullRhoPair_bijective_of_rho
  apply fiberRhoPair_bijective_of_productFirst
  exact TriangleGradedSequence.first_positive_bijective k T₀
    (fiberProductTriangle_distinguished q H₁ H₂) X₀ hpos
    (fiberSecondZero_surjective q H₁ H₂ hq1 e hg hn) n hnpos

end ArExplicit.T

end

end OAI
end

Source

OpenAI, An explicit counterexample to the Auslander-Reiten conjecture (result family 199), OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 199; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026/paper.pdf; Apache License 2.0); definitions carved from lean/OAI/Algebra at commit adc7f12

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