GapConjecture.lean`

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

1 declarations, 1 contain sorry, 0 declared as axioms

Gap conjecture

References:

Wikipedia
On the Gap Conjecture concerning group growth by Rostislav Grigorchuk

Imports (1)

FormalConjecturesUtil

theorem GapConjecture.gap_conjecture If a finitely generated group has superpolynomial growth, then with respect to any finite generating set its growth function is at least sorry line 41 source
Source documentation
If a finitely generated group has superpolynomial growth, then with respect to any finite generating set its growth function is at least $e^{\sqrt n}$ in Grigorchuk's preorder on growth functions, where the comparison is witnessed by linearly rescaling the radius.

depends on sorry what this means
Statement uses: Group, Set, Set.Finite, Eq, Subgroup, Subgroup.closure, Top.top, GromovPolynomialGrowth.HasSuperPolynomialGrowth, Exists, Nat, And, LT.lt, OfNat.ofNat, Filter.Eventually, LE.le, Real, Real.exp, Real.sqrt, GromovPolynomialGrowth.GrowthFunction, HMul.hMul and 1 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem gap_conjecture :
    ∀ (G : Type) [Group G] (S : Set G), S.Finite → Subgroup.closure S = ⊤ →
      HasSuperPolynomialGrowth G →
      ∃ C : ℕ, 0 < C ∧
        ∀ᶠ n : ℕ in atTop, Real.exp (Real.sqrt (n : ℝ)) ≤
          (GrowthFunction S (C * n) : ℝ) := by
…
```