| import OAI.Combinatorics.HalvingLines.Perturbation |
|
|
| namespace OAI |
|
|
| open Filter Topology |
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|
| namespace PlanarHalving |
|
|
| theorem halving_power : |
| ∃ ε : ℝ, 0 < ε ∧ ∃ C : ℝ, ∃ n₀ : ℕ, |
| ∀ n : ℕ, Even n → n₀ ≤ n → ∀ P : Fin n → Point, |
| GeneralPosition P → (halvingCount P : ℝ) ≤ C * (n : ℝ) ^ ((4 : ℝ) / 3 - ε) := by |
| obtain ⟨ε,hε,C,hbound⟩ := all_ranks_power |
| refine ⟨ε,hε,C,2,?_⟩ |
| intro n heven hn P hP |
| obtain ⟨m,hm⟩ := heven |
| have hn' : n=2*m := by omega |
| clear hm |
| subst n |
| obtain ⟨Q,hQ,hcount⟩ := Perturbation.generic_same_halvingCount hP |
| rw [hcount,halvingCount_eq_middle_switchCount (by omega : 0 < m) Q hQ] |
| exact hbound (2*m) (by omega) Q hQ m (by omega) |
| end PlanarHalving |
|
|
| namespace PlanarHalving |
|
|
| theorem power_bounds : |
| (∃ ε : ℝ, 0 < ε ∧ ∃ C : ℝ, ∃ n₀ : ℕ, |
| ∀ n : ℕ, Even n → n₀ ≤ n → ∀ P : Fin n → Point, |
| GeneralPosition P → (halvingCount P : ℝ) ≤ C * (n : ℝ) ^ ((4 : ℝ) / 3 - ε)) ∧ |
| (∃ ε : ℝ, 0 < ε ∧ ∃ C : ℝ, |
| ∀ n : ℕ, 1 ≤ n → ∀ P : Fin n → Point, Generic P → |
| ∀ k : ℕ, k ≤ n → (switchCount P k : ℝ) ≤ C * (n : ℝ) ^ ((4 : ℝ) / 3 - ε)) := |
| ⟨halving_power, all_ranks_power⟩ |
|
|
| end PlanarHalving |
|
|
| end OAI |
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|
