53.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Green's Open Problem 53

References:

[Gr24] Green's Open Problems

Imports (1)

FormalConjecturesUtil

theorem Green53.green_53 Suppose that $\mathbb{F}_2^n$ is partitioned in to sets $A_1, ..., A_K$ sorry line 40 source
Source documentation
Suppose that $\mathbb{F}_2^n$ is partitioned in to sets $A_1, ..., A_K$. Does $2A_i$ contain a coset of codimension $O_K(1)$ for some $i$?

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Fin, Set, 𝔽₂, Eq, Set.iUnion, Set.univ, Pairwise, Function.onFun, Disjoint, AffineSubspace, ZMod, OfNat.ofNat, And, LE.le, HAdd.hAdd, Module.finrank and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
80 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem green_53 :
    answer(sorry) ↔ ∃ (c : ℕ → ℕ), ∀ (n K : ℕ) (A : Fin K → Set (𝔽₂ n)),
      (⋃ i, A i) = Set.univ →
      Pairwise (Disjoint on A) →
      ∃ (i : Fin K) (S : AffineSubspace (ZMod 2) (𝔽₂ n)),
        (S : Set (𝔽₂ n)) ⊆ A i + A i ∧
        n ≤ Module.finrank (ZMod 2) S.direction + c K := by
…
```
Mentions: c, 𝔽₂