Individual proper infiniteness implies strong pure infiniteness (Lean)

The Lean statement

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

theorem individual_to_strong_main (hP : PropertyP A) :    (∀ a b : A, 0 ≤ a → 0 ≤ b → ∀ c : A, ∀ ε : ℝ, 0 < ε →      ∃ s t : A,        ‖star s * a * s - a‖ < ε ∧        ‖star t * b * t - b‖ < ε ∧        ‖star s * c * t‖ < ε) ∧    MainGap.StronglyPurelyInfinite 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.MainGap.scalarCorner line 14

def scalarCorner (h : A) : CStarMatrix (Fin 1) (Fin 1) A :=  CStarMatrix.ofMatrix (fun _ _ => h)

definition OAI.MainGap.doubleCorner line 17

def doubleCorner (h : A) : CStarMatrix (Fin 2) (Fin 2) A :=  CStarMatrix.ofMatrix (Matrix.diagonal (fun _ => h))

definition OAI.MainGap.Diagonalization line 20

def Diagonalization (a b c : A) (ε : ℝ) : Prop :=  ∃ s t : A,    ‖star s * a * s - a‖ < ε ∧    ‖star t * b * t - b‖ < ε ∧    ‖star s * c * t‖ < ε

definition OAI.MainGap.StronglyPurelyInfinite line 30

def StronglyPurelyInfinite (A : Type*) [NonUnitalCStarAlgebra A] : Prop :=  ∀ x y : A, 0 ≤ x → 0 ≤ y → ∀ ε : ℝ, 0 < ε →    Diagonalization (x * x) (y * y) (x * y) ε

abbreviation OAI.MainGap.Stable.Mat line 44

abbrev Mat (A : Type*) (n : ℕ) := CStarMatrix (Fin n) (Fin n) A

definition OAI.MainGap.Stable.pad line 46

def pad {n m : ℕ} (_ : n ≤ m) (x : Mat A n) : Mat A m :=  CStarMatrix.ofMatrix fun i j =>    if hi : i.val < n then if hj : j.val < n then x ⟨i,hi⟩ ⟨j,hj⟩ else 0 else 0

definition OAI.MainGap.Stable.padHom line 86

def padHom {n m : ℕ} (h : n ≤ m) : Mat A n →⋆ₙₐ[ℂ] Mat A m where  toFun := pad h  map_zero' := by ext i j; simp [pad_entry]  map_add' x y := by ext i j; by_cases hi : i.val < n <;> by_cases hj : j.val < n <;> simp [pad_entry, hi, hj]  map_mul' := pad_mul h  map_smul' c x := by ext i j; by_cases hi : i.val < n <;> by_cases hj : j.val < n <;> simp [pad_entry, hi, hj]  map_star' x := by ext i j; by_cases hi : i.val < n <;> by_cases hj : j.val < n <;> simp [pad_entry, hi, hj, CStarMatrix.star_apply]

abbreviation OAI.MainGap.Stable.Algebraic line 122

abbrev Algebraic (A : Type*) [NonUnitalCStarAlgebra A] :=  DirectLimit (Mat A) (fun _ _ h => padHom h)

abbreviation OAI.MainGap.Stable.Stabilization line 177

abbrev Stabilization (A : Type*) [NonUnitalCStarAlgebra A] := UniformSpace.Completion (Algebraic A)

definition OAI.MainGap.Manuscript.CuntzLE line 308

def CuntzLE (d v : Stable.Stabilization A) : Prop :=  ∃ r : ℕ → Stable.Stabilization A,    Tendsto (fun j => star (r j) * v * r j) atTop (𝓝 d)

definition OAI.MainGap.Manuscript.ProperlyInfinite line 312

def ProperlyInfinite (h : A) : Prop :=  CuntzLE ((⟦⟨2, MainGap.doubleCorner h⟩⟧ : Stable.Algebraic A) : Stable.Stabilization A)    ((⟦⟨1, MainGap.scalarCorner h⟩⟧ : Stable.Algebraic A) : Stable.Stabilization A)

definition OAI.MainGap.Manuscript.PropertyP line 316

def PropertyP (A : Type*) [NonUnitalCStarAlgebra A] : Prop :=  ∀ h : A, 0 ≤ h → ProperlyInfinite h

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: algebra, complex, element, exact, infinite, positive, properly, purely and one more.

Theorem 1.2 of Weak pure infiniteness and O-infinity absorption

Theorem

Let C be an exact complex C^*-algebra. Suppose that there is one integer n\ge1 such that h^{\oplus n} is properly infinite for every h\in C_+. Then every positive element of C is properly infinite. Consequently C is strongly purely infinite.

Section 1, Introduction Read it in the paper TeX source, line 98

Show the TeX

Let \(C\) be an exact complex \(C^*\)-algebra.  Suppose that there is
one integer \(n\ge1\) such that \(h^{\oplus n}\) is properly infinite
for every \(h\in C_+\).  Then every positive element of \(C\) is
properly infinite.  Consequently \(C\) is strongly purely infinite.

Other possible paper matches, closest wording first

Theorem 1.1 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: purely, strongly, complex, infinite and algebra.
Proposition 9.3 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: direct, purely, strongly, infinite and algebra.
Proposition 7.1 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: gap, unital, property, uniform, integer and algebra.
Theorem 6.1 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: properly, complex, infinite, integer and algebra. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nuclearity, element and positive.
Proposition 4.3 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: individual. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: proper, infiniteness and positive.
Proposition 3.1 of Weak pure infiniteness and O-infinity absorption. Words of the Lean statement, its names split into words, that also appear in the theorem: eq, property, integer and algebra. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: implies, elements and positive.

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 364 s; peak memory 6.4 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 303’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 disagree 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: individual_to_strong_main (hP : PropertyP A) : unrestricted diagonalization ∧ StronglyPurelyInfinite A holds for every non-unital C*-algebra A, where PropertyP = every positive element properly infinite (Kirchberg-Rørdam's characterization of ordinary pure infiniteness is the one routine identification step), and ExactInfiniteness gives the fixed-amplification version for exact A (WeaklyPurelyInfinite with some n ≥ 1 ⇒ ProperlyInfinite (diagonalCopies 1 a) and strongly purely infinite). The summary's final consequence (separable nuclear algebras with this property absorb O_∞) is not stated.
Definitions the group asks readers to compare with the paper: def Exact (A : Type u) ... := ∀ (C D : Type u) (q : C →⋆ₙₐ[ℂ] D), Surjective q → ∀ z : MinimalTensor A C, (∀ π : SpatialRepresentation A D, z (π.precompRight q) = 0) → z.val ∈ tensorKernelIdeal q: hand-built exactness via the spatial (min) tensor product as an ℓ^∞ sum over all pairs of Hilbert-space representations; matches the standard exactness of A ⊗_min - but tests only C, D in the universe of A ; Two different hand-built notions of proper infiniteness: matrix-level CuntzSubequiv (blockSum a a) a (ε-approximation by v : CStarMatrix k m A) in ExactInfiniteness, and CuntzLE in the completed stabilization (MainGap.Stable.Stabilization, direct limit of CStarMatrix algebras) in IndividualStrongInfiniteness; both read as standard, the Lean theorem uses the weaker (stable) form as hypothesis in the ordinary-to-strong direction
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: The O_∞-absorption clause (the family title's 'Cuntz-algebra absorption') is not stated; it is a consequence of the stated result via the (unformalized) Kirchberg–Rørdam absorption theorem.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves that a complex $C^*$-algebra in which every positive element is properly infinite is strongly purely infinite, with the full positive-element diagonalization property. No exactness, nuclearity, unitality, or simplicity assumption is needed for this implication.”
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

333 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: 198 to its own declarations, 243 to Mathlib’s documentation. 44 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAInoncomputable sectionopen Filteropen scoped Topology CStarAlgebra ComplexOrder InnerProductSpacenamespace MainGapvariable {A : Type*} [NonUnitalCStarAlgebra A]local instance positivityOrder : PartialOrder A := CStarAlgebra.spectralOrder Alocal instance positivityStarOrder : StarOrderedRing A := CStarAlgebra.spectralOrderedRing Adef scalarCorner (h : A) : CStarMatrix (Fin 1) (Fin 1) A :=  CStarMatrix.ofMatrix (fun _ _ => h)def doubleCorner (h : A) : CStarMatrix (Fin 2) (Fin 2) A :=  CStarMatrix.ofMatrix (Matrix.diagonal (fun _ => h))def Diagonalization (a b c : A) (ε : ℝ) : Prop :=  ∃ s t : A,    ‖star s * a * s - a‖ < ε ∧    ‖star t * b * t - b‖ < ε ∧    ‖star s * c * t‖ < εdef UnrestrictedDiagonalization (A : Type*) [NonUnitalCStarAlgebra A] : Prop :=  ∀ a b : A, 0 ≤ a → 0 ≤ b → ∀ c : A, ∀ ε : ℝ, 0 < ε →    Diagonalization a b c εdef StronglyPurelyInfinite (A : Type*) [NonUnitalCStarAlgebra A] : Prop :=  ∀ x y : A, 0 ≤ x → 0 ≤ y → ∀ ε : ℝ, 0 < ε →    Diagonalization (x * x) (y * y) (x * y) εend MainGapnoncomputable sectionopen Filteropen scoped Topologynamespace MainGap.Stablevariable {A : Type*} [NonUnitalCStarAlgebra A]local instance positivityOrder : PartialOrder A := CStarAlgebra.spectralOrder Alocal instance positivityStarOrder : StarOrderedRing A := CStarAlgebra.spectralOrderedRing Aabbrev Mat (A : Type*) (n : ℕ) := CStarMatrix (Fin n) (Fin n) Adef pad {n m : ℕ} (_ : n ≤ m) (x : Mat A n) : Mat A m :=  CStarMatrix.ofMatrix fun i j =>    if hi : i.val < n then if hj : j.val < n then x ⟨i,hi⟩ ⟨j,hj⟩ else 0 else 0@[simp] lemma pad_entry {n m : ℕ} (h : n ≤ m) (x : Mat A n) (i j : Fin m) :    pad h x i j = (if hi : i.val < n then if hj : j.val < n then x ⟨i,hi⟩ ⟨j,hj⟩ else 0 else 0) := rfl@[simp] lemma pad_apply {n m : ℕ} (h : n ≤ m) (x : Mat A n) (i j : Fin n) :    pad h x (Fin.castLE h i) (Fin.castLE h j) = x i j := by simp [pad_entry]lemma sum_pad {n m : ℕ} (h : n ≤ m) {B : Type*} [AddCommMonoid B] (f : Fin n → B) :    ∑ k : Fin m, (if hk : k.val < n then f ⟨k,hk⟩ else 0) = ∑ k : Fin n, f k := by  classical  symm  apply Finset.sum_bij_ne_zero (fun i _ _ => Fin.castLE h i)  · simp  · intro i _ _ j _ _ hij    exact Fin.ext (congrArg (fun k : Fin m => k.val) hij)  · intro j _ hj0    have hj : j.val < n := by      by_contra hn      simp [hn] at hj0    exact ⟨⟨j,hj⟩, Finset.mem_univ _, by simpa [hj] using hj0, rfl⟩  · intro i _ _    simplemma pad_mul {n m : ℕ} (h : n ≤ m) (x y : Mat A n) :    pad h (x*y) = pad h x * pad h y := by  ext i j  by_cases hi : i.val < n <;> by_cases hj : j.val < n  · simp only [pad_entry, dite_eq_left hi, dite_eq_left hj,      CStarMatrix.mul_apply]    rw [← sum_pad h (fun k => x ⟨i,hi⟩ k * y k ⟨j,hj⟩)]    apply Finset.sum_congr rfl    intro k _    by_cases hk : k.val < n <;> simp [hk]  · simp [pad_entry, hi, hj, CStarMatrix.mul_apply]  · simp [pad_entry, hi, CStarMatrix.mul_apply]  · simp [pad_entry, hi, CStarMatrix.mul_apply]def padHom {n m : ℕ} (h : n ≤ m) : Mat A n →⋆ₙₐ[ℂ] Mat A m where  toFun := pad h  map_zero' := by ext i j; simp [pad_entry]  map_add' x y := by ext i j; by_cases hi : i.val < n <;> by_cases hj : j.val < n <;> simp [pad_entry, hi, hj]  map_mul' := pad_mul h  map_smul' c x := by ext i j; by_cases hi : i.val < n <;> by_cases hj : j.val < n <;> simp [pad_entry, hi, hj]  map_star' x := by ext i j; by_cases hi : i.val < n <;> by_cases hj : j.val < n <;> simp [pad_entry, hi, hj, CStarMatrix.star_apply]lemma pad_injective {n m : ℕ} (h : n ≤ m) : Function.Injective (pad (A := A) h) := by  intro x y hxy  ext i j  have hh := congrArg (fun z : Mat A m => z (Fin.castLE h i) (Fin.castLE h j)) hxy  simpa using hhlemma norm_pad {n m : ℕ} (h : n ≤ m) (x : Mat A n) : ‖pad h x‖ = ‖x‖ :=  NonUnitalStarAlgHom.norm_map (padHom h) (pad_injective h) xlemma pad_self (n : ℕ) (x : Mat A n) : pad le_rfl x = x := by  ext i j  simp [pad_entry]lemma pad_pad {n m k : ℕ} (h : n ≤ m) (h' : m ≤ k) (x : Mat A n) :    pad h' (pad h x) = pad (h.trans h') x := by  ext i j  by_cases hi : i.val < n <;> by_cases hj : j.val < n  · have hi' : i.val < m := lt_of_lt_of_le hi h    have hj' : j.val < m := lt_of_lt_of_le hj h    simp [pad_entry, hi, hj, hi', hj']  · by_cases hi' : i.val < m <;> by_cases hj' : j.val < m <;> simp [pad_entry, hi, hj, hi', hj']  · by_cases hi' : i.val < m <;> by_cases hj' : j.val < m <;> simp [pad_entry, hi, hi', hj']  · by_cases hi' : i.val < m <;> by_cases hj' : j.val < m <;> simp [pad_entry, hi, hi', hj']instance pad_directedSystem : DirectedSystem (Mat A) (fun _ _ h => padHom h) where  map_self := fun {_} x => pad_self _ x  map_map := fun {_ _ _} h h' x => pad_pad h h' xabbrev Algebraic (A : Type*) [NonUnitalCStarAlgebra A] :=  DirectLimit (Mat A) (fun _ _ h => padHom h)local instance matLinear (n m : ℕ) :    LinearMapClass (Mat A n →⋆ₙₐ[ℂ] Mat A m) ℂ (Mat A n) (Mat A m) where  map_add f x y := f.map_add x y  map_smulₛₗ f c x := f.map_smul c xinstance : Module ℂ (Algebraic A) := DirectLimit.instModuleinstance : Norm (Algebraic A) where  norm := DirectLimit.lift _ (fun _ x => ‖x‖) (fun _ _ h x => (norm_pad h x).symm)@[simp] lemma norm_mk (n : ℕ) (x : Mat A n) :    ‖(⟦⟨n,x⟩⟧ : Algebraic A)‖ = ‖x‖ := rfllemma normedCore : NormedSpace.Core ℂ (Algebraic A) where  norm_nonneg := DirectLimit.induction _ fun n x => norm_nonneg x  norm_smul c := DirectLimit.induction _ fun n x => by    rw [DirectLimit.smul_def, norm_mk, norm_mk, norm_smul]  norm_triangle := DirectLimit.induction₂ _ fun n x y => by    rw [DirectLimit.add_def, norm_mk, norm_mk, norm_mk]    exact norm_add_le x y  norm_eq_zero_iff := DirectLimit.induction _ fun n x => by    rw [norm_mk, norm_eq_zero]    constructor    · rintro rfl      exact (DirectLimit.zero_def (f := fun _ _ h => padHom (A := A) h) n).symm    · intro h      apply pad_injective (le_rfl : n ≤ n)      have hi := DirectLimit.mk_injective (fun _ _ h => padHom (A := A) h) (fun _ _ h => pad_injective h) n      exact congrArg (pad le_rfl) (hi (h.trans (DirectLimit.zero_def (f := fun _ _ h => padHom (A := A) h) n)))instance : NormedAddCommGroup (Algebraic A) := .ofCore (𝕜 := ℂ) (normedCore (A := A))instance : NormedSpace ℂ (Algebraic A) := .ofCore (normedCore (A := A))instance : NonUnitalNormedRing (Algebraic A) where  dist_eq _ _ := rfl  norm_mul_le := DirectLimit.induction₂ _ fun n x y => by    rw [DirectLimit.mul_def, norm_mk, norm_mk, norm_mk]    exact norm_mul_le x yinstance : CStarRing (Algebraic A) where  norm_mul_self_le := DirectLimit.induction _ fun n x => by    rw [DirectLimit.star_def, DirectLimit.mul_def, norm_mk, norm_mk]    exact CStarRing.norm_mul_self_le xdef cornerAlg (n : ℕ) : Mat A n →⋆ₙₐ[ℂ] Algebraic A where  __ := DirectLimit.NonUnitalAlgebra.of (Mat A) (fun _ _ h => padHom h) n  map_star' _ := rfl@[simp] lemma cornerAlg_apply (n : ℕ) (x : Mat A n) :    cornerAlg n x = (⟦⟨n,x⟩⟧ : Algebraic A) := rfllemma cornerAlg_isometry (n : ℕ) : Isometry (cornerAlg (A := A) n) :=  AddMonoidHomClass.isometry_of_norm (cornerAlg n) (fun _ => rfl)abbrev Stabilization (A : Type*) [NonUnitalCStarAlgebra A] := UniformSpace.Completion (Algebraic A)local instance algebraicTopology : TopologicalSpace (Algebraic A) :=  (inferInstance : PseudoMetricSpace (Algebraic A)).toUniformSpace.toTopologicalSpaceopen UniformSpaceopen UniformSpace.Completioninstance : Mul (Stabilization A) :=  ⟨Function.curry <| (isDenseInducing_coe.prodMap isDenseInducing_coe).extend    ((↑) ∘ Function.uncurry (· * · : Algebraic A → Algebraic A → Algebraic A))⟩@[simp, norm_cast] lemma coe_mul_stable (x y : Algebraic A) :    ((x*y : Algebraic A) : Stabilization A) = (x : Stabilization A) * (y : Stabilization A) :=  ((isDenseInducing_coe.prodMap isDenseInducing_coe).extend_eq    ((continuous_coe _).comp continuous_mul) (x,y)).symminstance : ContinuousMul (Stabilization A) where  continuous_mul := by    let m := (AddMonoidHom.mul : Algebraic A →+ Algebraic A →+ Algebraic A).compr₂ toCompl    have hh : Continuous fun p : Algebraic A × Algebraic A => m p.1 p.2 :=      (continuous_coe _).comp continuous_mul    have di : IsDenseInducing (toCompl : Algebraic A → Stabilization A) := isDenseInducing_coe    exact di.extend_Z_bilin di hhinstance : NonUnitalRing (Stabilization A) where  __ : AddCommGroup (Stabilization A) := inferInstance  zero_mul x := Completion.induction_on x (isClosed_eq (by fun_prop) continuous_const)    fun x => by rw [← coe_zero, ← coe_mul_stable, zero_mul]  mul_zero x := Completion.induction_on x (isClosed_eq (by fun_prop) continuous_const)    fun x => by rw [← coe_zero, ← coe_mul_stable, mul_zero]  mul_assoc x y z := Completion.induction_on₃ x y z    (isClosed_eq (by fun_prop) (by fun_prop)) fun x y z => by      rw [← coe_mul_stable, ← coe_mul_stable, ← coe_mul_stable, ← coe_mul_stable, mul_assoc]  left_distrib x y z := Completion.induction_on₃ x y z    (isClosed_eq (by fun_prop) (by fun_prop)) fun x y z => by      rw [← coe_add, ← coe_mul_stable, ← coe_mul_stable, ← coe_mul_stable, ← coe_add, mul_add]  right_distrib x y z := Completion.induction_on₃ x y z    (isClosed_eq (by fun_prop) (by fun_prop)) fun x y z => by      rw [← coe_add, ← coe_mul_stable, ← coe_mul_stable, ← coe_mul_stable, ← coe_add, add_mul]instance : NonUnitalNormedRing (Stabilization A) where  dist_eq := dist_eq_norm_neg_add  norm_mul_le x y := Completion.induction_on₂ x y    (isClosed_le (by fun_prop) (by fun_prop)) fun x y => by      simpa only [← coe_mul_stable, norm_coe] using norm_mul_le x yinstance : Star (Stabilization A) := ⟨Completion.map (star : Algebraic A → Algebraic A)⟩@[simp, norm_cast] lemma coe_star_stable (x : Algebraic A) :    ((star x : Algebraic A) : Stabilization A) = star (x : Stabilization A) :=  (Completion.map_coe star_isometry.uniformContinuous x).symminstance : ContinuousStar (Stabilization A) where  continuous_star := Completion.continuous_mapinstance : StarRing (Stabilization A) where  star_involutive x := Completion.induction_on x    (isClosed_eq (by fun_prop) continuous_id) fun x => by      rw [← coe_star_stable, ← coe_star_stable, star_star]  star_add x y := Completion.induction_on₂ x y    (isClosed_eq (by fun_prop) (by fun_prop)) fun x y => by      rw [← coe_add, ← coe_star_stable, ← coe_star_stable, ← coe_star_stable, ← coe_add, star_add]  star_mul x y := Completion.induction_on₂ x y    (isClosed_eq (by fun_prop) (by fun_prop)) fun x y => by      rw [← coe_mul_stable, ← coe_star_stable, ← coe_star_stable, ← coe_star_stable,        ← coe_mul_stable, star_mul]instance : CStarRing (Stabilization A) where  norm_mul_self_le x := Completion.induction_on x    (isClosed_le (by fun_prop) (by fun_prop)) fun x => by      simpa only [← coe_star_stable, ← coe_mul_stable, norm_coe] using CStarRing.norm_mul_self_le xinstance : IsScalarTower ℂ (Stabilization A) (Stabilization A) where  smul_assoc c x y := by    change (c • x) * y = c • (x*y)    refine Completion.induction_on₂ x y ?_ ?_    · exact isClosed_eq ((continuous_fst.const_smul c).mul continuous_snd)        ((continuous_fst.mul continuous_snd).const_smul c)    · intro x y      rw [← coe_smul, ← coe_mul_stable, ← coe_mul_stable, ← coe_smul, smul_mul_assoc]instance : SMulCommClass ℂ (Stabilization A) (Stabilization A) where  smul_comm c x y := by    change c • (x*y) = x * (c • y)    refine Completion.induction_on₂ x y ?_ ?_    · exact isClosed_eq ((continuous_fst.mul continuous_snd).const_smul c)        (continuous_fst.mul (continuous_snd.const_smul c))    · intro x y      rw [← coe_mul_stable, ← coe_smul, ← coe_smul, ← coe_mul_stable, mul_smul_comm]instance : StarModule ℂ (Stabilization A) where  star_smul c x := by    refine Completion.induction_on x ?_ ?_    · exact isClosed_eq (continuous_star.comp (continuous_const_smul c))        (continuous_star.const_smul (star c))    · intro x      rw [← coe_smul, ← coe_star_stable, ← coe_star_stable, ← coe_smul, star_smul]@[instance_reducible] def stabilizationCStarAlgebra : NonUnitalCStarAlgebra (Stabilization A) wheredef corner (n : ℕ) : Mat A n →⋆ₙₐ[ℂ] Stabilization A where  toFun x := (cornerAlg n x : Stabilization A)  map_zero' := by rw [map_zero, coe_zero]  map_add' x y := by simp [map_add, coe_add]  map_mul' x y := by simp [map_mul]  map_smul' c x := by simp [map_smul, coe_smul]  map_star' x := by simp [map_star]@[simp] lemma corner_apply (n : ℕ) (x : Mat A n) :    corner n x = (cornerAlg n x : Stabilization A) := rfllemma norm_corner (n : ℕ) (x : Mat A n) : ‖corner n x‖ = ‖x‖ := by  rw [corner_apply, norm_coe, cornerAlg_apply, norm_mk]lemma corner_isometry (n : ℕ) : Isometry (corner (A := A) n) :=  AddMonoidHomClass.isometry_of_norm (corner n) (norm_corner n)lemma corner_pad {n m : ℕ} (h : n ≤ m) (x : Mat A n) :    corner m (pad h x) = corner n x := by  change ((⟦⟨m,padHom h x⟩⟧ : Algebraic A) : Stabilization A) =    ((⟦⟨n,x⟩⟧ : Algebraic A) : Stabilization A)  rw [DirectLimit.mk_apply]end MainGap.Stablenamespace MainGap.Manuscriptvariable {A : Type*} [NonUnitalCStarAlgebra A]local instance positivityOrder : PartialOrder A := CStarAlgebra.spectralOrder Alocal instance positivityStarOrder : StarOrderedRing A := CStarAlgebra.spectralOrderedRing Adef CuntzLE (d v : Stable.Stabilization A) : Prop :=  ∃ r : ℕ → Stable.Stabilization A,    Tendsto (fun j => star (r j) * v * r j) atTop (𝓝 d)def ProperlyInfinite (h : A) : Prop :=  CuntzLE ((⟦⟨2, MainGap.doubleCorner h⟩⟧ : Stable.Algebraic A) : Stable.Stabilization A)    ((⟦⟨1, MainGap.scalarCorner h⟩⟧ : Stable.Algebraic A) : Stable.Stabilization A)def PropertyP (A : Type*) [NonUnitalCStarAlgebra A] : Prop :=  ∀ h : A, 0 ≤ h → ProperlyInfinite htheorem individual_to_strong_main (hP : PropertyP A) :    (∀ a b : A, 0 ≤ a → 0 ≤ b → ∀ c : A, ∀ ε : ℝ, 0 < ε →      ∃ s t : A,        ‖star s * a * s - a‖ < ε ∧        ‖star t * b * t - b‖ < ε ∧        ‖star s * c * t‖ < ε) ∧    MainGap.StronglyPurelyInfinite A := by  sorryend MainGap.Manuscriptendendend 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.

Weak pure infiniteness and O-infinity absorption

Theorem

Let C be a complex C^*-algebra satisfying \propP. For all a,b\in C_+, c\in C, and \varepsilon>0, there are s,t\in C such that

Consequently C is strongly purely infinite.

Section 1, Introduction Read it in the paper TeX source, line 73

Show the TeX

Let \(C\) be a complex \(C^*\)-algebra satisfying \(\propP\).
For all \(a,b\in C_+\), \(c\in C\), and \(\varepsilon>0\), there are
\(s,t\in C\) such that
\begin{equation}
 \|s^*as-a\|<\varepsilon,\qquad
 \|t^*bt-b\|<\varepsilon,\qquad
 \|s^*ct\|<\varepsilon.
\end{equation}
Consequently \(C\) is strongly purely infinite.

Theorem

Let C be an exact complex C^*-algebra. Suppose that there is one integer n\ge1 such that h^{\oplus n} is properly infinite for every h\in C_+. Then every positive element of C is properly infinite. Consequently C is strongly purely infinite.

Section 1, Introduction Read it in the paper TeX source, line 98

Show the TeX

Let \(C\) be an exact complex \(C^*\)-algebra.  Suppose that there is
one integer \(n\ge1\) such that \(h^{\oplus n}\) is properly infinite
for every \(h\in C_+\).  Then every positive element of \(C\) is
properly infinite.  Consequently \(C\) is strongly purely infinite.

Corollary

Let A be a separable nuclear complex C^*-algebra. If there is one integer n\ge1 such that h^{\oplus n} is properly infinite for every h\in A_+, then

A\cong A\otimes_{\min}\Oinf.

Section 1, Introduction Read it in the paper TeX source, line 129

Show the TeX

Let \(A\) be a separable nuclear complex \(C^*\)-algebra.  If there is
one integer \(n\ge1\) such that \(h^{\oplus n}\) is properly infinite
for every \(h\in A_+\), then
\[
 A\cong A\otimes_{\min}\Oinf.
\]

Configuration and file facts

File: lean/ComparatorChallenges/IndividualStrongInfiniteness.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 303 link it.
Solution module: OAI.Analysis.CStarAlgebra.PureInfiniteness.Main
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: 42 local vocabulary declarations
Lines of Lean, without comments or blank lines: 248
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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