| import OAI.Combinatorics.ContingencyTables.Dense.DenseScales |
| import Mathlib.Analysis.SpecialFunctions.Pow.Real |
|
|
| namespace OAI |
|
|
| /-! # Numerical interpolation parameters of the dense bin chain -/ |
|
|
| namespace ContingencyTables.GridBoundary |
|
|
| noncomputable def denseInterpolation (d K : ℕ) : ℝ := 1 / (4 * (d : ℝ) * K) |
|
|
| theorem denseInterpolation_bounds (d K q : ℕ) (hd : 14 ≤ d) (hK : 1 ≤ K) (hq : q ≤ d) : |
| 0 < denseInterpolation d K ∧ denseInterpolation d K < 1 ∧ |
| denseInterpolation d K * K ≤ 1 / 4 ∧ |
| (q - 1 : ℕ) * (2 * (denseInterpolation d K * K)) ≤ (1 / 2 : ℝ) := by |
| have hd₀ : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega) |
| have hd₁ : (1 : ℝ) ≤ d := by exact_mod_cast (show 1 ≤ d by omega) |
| have hK₀ : (0 : ℝ) < K := by exact_mod_cast (show 0 < K by omega) |
| have hK₁ : (1 : ℝ) ≤ K := by exact_mod_cast hK |
| have heq : denseInterpolation d K * K = 1 / (4 * (d : ℝ)) := by |
| unfold denseInterpolation |
| field_simp |
| refine ⟨by unfold denseInterpolation; positivity, ?_, ?_, ?_⟩ |
| · unfold denseInterpolation |
| apply (div_lt_one (by positivity)).mpr |
| nlinarith |
| · rw [heq] |
| apply (div_le_iff₀ (by positivity)).mpr |
| linarith |
| · rw [heq] |
| have hqd : ((q - 1 : ℕ) : ℝ) ≤ d := by exact_mod_cast (Nat.sub_le q 1).trans hq |
| calc |
| _ ≤ (d : ℝ) * (2 * (1 / (4 * (d : ℝ)))) := by gcongr |
| _ = 1 / 2 := by field_simp; ring |
|
|
| theorem two_rpow_gain (t : ℝ) (ht : 0 ≤ t) : t / 2 ≤ (2 : ℝ) ^ t - 1 := by |
| have hl := Real.one_sub_inv_le_log_of_pos (show (0 : ℝ) < 2 by norm_num) |
| norm_num at hl |
| have he := Real.add_one_le_exp (Real.log 2 * t) |
| rw [Real.rpow_def_of_pos (by norm_num : (0 : ℝ) < 2)] |
| nlinarith [mul_le_mul_of_nonneg_right hl ht] |
|
|
| end ContingencyTables.GridBoundary |
|
|
| end OAI |
|
|
