File size: 4,271 Bytes
16a4018 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | # openai/math challenge `BackwardIntertwiners` (family 293)
Prove the following result from OpenAI's [openai/math](https://github.com/openai/math) release in
Lean 4, with a proof the Lean kernel accepts.
Context: this statement belongs to family 293 of the release, *A counterexample to the hyperinvariant-subspace problem*
(Operator algebras). The family as a whole: Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type $\mathrm{II}_1$ factor.
The challenge is `BackwardIntertwiners`, also at `/opt/openai-math/challenges/BackwardIntertwiners.lean`:
```lean
import Mathlib
namespace OAI
namespace Hyperinvariant233
open Filter
open scoped Topology
universe u
variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
def commutant (T : H →L[ℂ] H) : Subalgebra ℂ (H →L[ℂ] H) :=
Subalgebra.centralizer ℂ ({T} : Set (H →L[ℂ] H))
def TransitiveCommutant (T : H →L[ℂ] H) : Prop :=
∀ K : Submodule ℂ H, IsClosed (K : Set H) →
(∀ A : H →L[ℂ] H, A * T = T * A → ∀ x ∈ K, A x ∈ K) →
K = ⊥ ∨ K = ⊤
@[instance_reducible]
def strongOperatorTopology : TopologicalSpace (H →L[ℂ] H) :=
TopologicalSpace.induced (fun A : H →L[ℂ] H => (fun x : H => A x)) inferInstance
def FullClaim : Prop :=
∃ T : H →L[ℂ] H, T ≠ 0 ∧
Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0) ∧
TransitiveCommutant T ∧ commutant T ≠ ⊤ ∧
@IsClosed (H →L[ℂ] H) strongOperatorTopology (commutant T : Set (H →L[ℂ] H))
end Hyperinvariant233
namespace BackwardIntertwiners
open Filter Topology MeasureTheory Set
noncomputable section
open scoped ENNReal
theorem direct_algebra_corollary (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H]
[CompleteSpace H] [TopologicalSpace.SeparableSpace H] (hInf : ¬FiniteDimensional ℂ H) :
Hyperinvariant233.FullClaim (H:=H) := by
sorry
end
end BackwardIntertwiners
end OAI
```
## What to submit
Write `/workspace/Submission.lean`. Start from a copy of the challenge:
```bash
cp /opt/openai-math/challenges/BackwardIntertwiners.lean /workspace/Submission.lean
```
then replace every `sorry` with a proof. The file is graded on three things:
- **Same statements.** The theorem `OAI.BackwardIntertwiners.direct_algebra_corollary` must keep exactly the statement shown above: same names,
namespaces, binders and types. Every definition the statements use must stay exactly as written.
Change nothing except the proofs.
- **Standard axioms only.** Proofs may use only `propext`, `Quot.sound` and `Classical.choice`.
`sorry`, `admit`, new `axiom`s and `native_decide` (it introduces an axiom of its own) are rejected.
- **Keep the challenge's declarations as they are.** Put new lemmas and instances *after* the
definitions the statements use, or in a separate `Submission/*.lean` module. A declaration added
before them can change how they elaborate, and then they no longer match the challenge.
- **Kernel-checked.** The proofs are re-checked by the Lean kernel, not just the elaborator.
Long proofs can be split into modules under `/workspace/Submission/` (module names
`Submission.Foo`, `Submission.Foo.Bar`) imported from `Submission.lean`. Only `.lean` files at
those two paths are graded.
## Environment
- Lean `v4.34.1` and Mathlib at commit `d13f23b` are installed and prebuilt; `/workspace` is a
Lake project.
- The sandbox has 4 CPUs and 8 GB of memory; `LEAN_NUM_THREADS=3` keeps `lake build` to three
parallel jobs. Check your work with `cd /workspace && lake build Submission`. Add `#print axioms <name>` to see
which axioms a proof uses.
- There is no internet access. OpenAI's own proofs are not installed.
## Grading
When you finish, `Submission.lean` and `Submission/**.lean` are copied to a fresh machine and
checked with [Comparator](https://github.com/leanprover/comparator), the Lean FRO's proof checker.
The reward is 1 if Comparator accepts the proof and 0 otherwise. A partial proof scores 0.
|