| import OAI.MeasureTheory.Falconer.Shells.HighShellCoefficientBounds | |
| namespace OAI | |
| namespace PlanarFalconer | |
| theorem high_shell_radial_power_bound {v r : ℝ} (hr : (2:ℝ)^(v-1) ≤ r) (j : ℕ) : | |
| 0 ≤ ((r/(2:ℝ)^v)⁻¹)^j ∧ ((r/(2:ℝ)^v)⁻¹)^j ≤ (2:ℝ)^j := by | |
| have hrpos : 0 < r := (Real.rpow_pos_of_pos (by norm_num) _).trans_le hr | |
| have hd : (1/2:ℝ) ≤ r/(2:ℝ)^v := by | |
| apply (le_div_iff₀ (by positivity)).mpr | |
| have he : (1/2:ℝ)*(2:ℝ)^v = (2:ℝ)^(v-1) := by | |
| rw [Real.rpow_sub (by norm_num : (0:ℝ) < 2),Real.rpow_one] | |
| ring | |
| rw [he] | |
| exact hr | |
| have hi : (r/(2:ℝ)^v)⁻¹ ≤ 2 := by | |
| simpa only [one_div, inv_inv] using inv_anti₀ (by norm_num : (0:ℝ) < 1/2) hd | |
| exact ⟨pow_nonneg (by positivity) _,pow_le_pow_left₀ (by positivity) hi j⟩ | |
| theorem uniform_minimal_inverseLeibniz_subpower {eps : ℝ} (heps : 0 < eps) (j : ℕ) : | |
| ∃ N₀ : ℕ, ∀ N ≥ N₀, ∀ (A : ℕ → ℝ) (q a t : ℕ), 0 < q → | |
| ((inverseLeibnizTerms j (pairMaskIndices (policyDescriptorList A N q a t))).map | |
| (fun b => ‖b.1‖)).sum * (2:ℝ)^j ≤ ((2:ℝ)^N)^eps := by | |
| let C := (inverseStationaryExpr j).cost * (2:ℝ)^(2*j) * (2:ℝ)^j | |
| obtain ⟨N₁,h₁⟩ := fixed_constant_absorbed (half_pos heps) C | |
| obtain ⟨N₂,h₂⟩ := fixed_polynomial_subpower (6*j) (half_pos heps) | |
| refine ⟨max N₁ N₂,?_⟩ | |
| intro N hN A q a t hq | |
| have hpoly : (((N:ℝ)+1)^3)^(2*j) = ((N:ℝ)+1)^(6*j) := by | |
| rw [← pow_mul] | |
| congr 1 | |
| omega | |
| have hbound := mul_le_mul_of_nonneg_right (minimal_inverseLeibniz_cost A (N := N) (a := a) (t := t) hq j) | |
| (by positivity : 0 ≤ (2:ℝ)^j) | |
| apply hbound.trans | |
| calc | |
| _ = C*((N:ℝ)+1)^(6*j) := by rw [mul_pow,hpoly]; dsimp [C]; ring | |
| _ ≤ C*(4*(N:ℝ)+1)^(6*j) := mul_le_mul_of_nonneg_left | |
| (pow_le_pow_left₀ (by positivity) (by have := Nat.cast_nonneg (α := ℝ) N; linarith) _) | |
| (by dsimp [C]; exact mul_nonneg (mul_nonneg (inverseStationaryExpr j).cost_nonneg (by positivity)) (by positivity)) | |
| _ ≤ ((2:ℝ)^N)^(eps/2)*((2:ℝ)^N)^(eps/2) := | |
| mul_le_mul (h₁ N ((le_max_left _ _).trans hN)) (h₂ N ((le_max_right _ _).trans hN)) | |
| (by positivity) (by positivity) | |
| _ = _ := by rw [← Real.rpow_add (by positivity)]; congr 1; ring | |
| end PlanarFalconer | |
| end OAI | |