| import Mathlib |
| import OAI.Probability.BinarySweep.SparseBounds.FewSparseBases |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
|
|
| namespace BinaryCoordinateSweeps |
| open GridSplit |
|
|
| lemma balanced_counts (b : ℕ) (hb : 2 ≤ b) : ∃m n : ℕ, |
| b=m+n ∧ 1 ≤ m ∧ 1 ≤ n ∧ m < b ∧ n < b ∧ m ≤ 2*n ∧ n ≤ 2*m := by |
| refine ⟨b/2,b-b/2,by omega,by omega,by omega,by omega,by omega,by omega,by omega⟩ |
|
|
| lemma balanced_grid_bounds {m n r : ℕ} (bits : Fin (m+n) → ℕ) |
| (hm : 1 ≤ m) (hn : 1 ≤ n) (hmn : m ≤ 2*n) (hnm : n ≤ 2*m) |
| (hd : ∀j,r ≤ bits j ∧ bits j ≤ 2*r) (hr : 400000000 ≤ (r:ℝ)*Real.log 2) : |
| 1 ≤ Real.log (gridSize (leftBits bits)) ∧ 1 ≤ Real.log (gridSize (rightBits bits)) ∧ |
| Real.log (gridSize (leftBits bits)) ≤ (4/5:ℝ)*Real.log (gridSize bits) ∧ |
| Real.log (gridSize (rightBits bits)) ≤ (4/5:ℝ)*Real.log (gridSize bits) ∧ |
| Real.log 4*(m+n) ≤ (1/1000:ℝ)*Real.log (gridSize bits) := by |
| have hl := grid_log_lower (fun j : Fin m => (hd (j.castAdd n)).1) |
| have hr' := grid_log_lower (fun j : Fin n => (hd (j.natAdd m)).1) |
| have hlu := grid_log_upper (fun j : Fin m => (hd (j.castAdd n)).2) |
| have hru := grid_log_upper (fun j : Fin n => (hd (j.natAdd m)).2) |
| change (m:ℝ)*((r:ℝ)*Real.log 2) ≤ Real.log (gridSize (leftBits bits)) at hl |
| change (n:ℝ)*((r:ℝ)*Real.log 2) ≤ Real.log (gridSize (rightBits bits)) at hr' |
| change Real.log (gridSize (leftBits bits)) ≤ (m:ℝ)*(2*r*Real.log 2) at hlu |
| change Real.log (gridSize (rightBits bits)) ≤ (n:ℝ)*(2*r*Real.log 2) at hru |
| have hsum : Real.log (gridSize bits)=Real.log (gridSize (leftBits bits))+Real.log (gridSize (rightBits bits)) := by |
| rw [size_split,Nat.cast_mul] |
| exact Real.log_mul (by exact_mod_cast (gridSize_pos (leftBits bits)).ne') |
| (by exact_mod_cast (gridSize_pos (rightBits bits)).ne') |
| have hmR : (1:ℝ) ≤ m := by exact_mod_cast hm |
| have hnR : (1:ℝ) ≤ n := by exact_mod_cast hn |
| have hmnR : (m:ℝ) ≤ 2*n := by exact_mod_cast hmn |
| have hnmR : (n:ℝ) ≤ 2*m := by exact_mod_cast hnm |
| have hscale : 0 ≤ (r:ℝ)*Real.log 2 := by linarith |
| have hmnl := mul_le_mul_of_nonneg_right hmnR hscale |
| have hnml := mul_le_mul_of_nonneg_right hnmR hscale |
| have hm1 := mul_le_mul_of_nonneg_right hmR hscale |
| have hn1 := mul_le_mul_of_nonneg_right hnR hscale |
| have haxes := grid_log_lower (fun j => (hd j).1) |
| have hlog4 : Real.log 4 ≤ 4 := (Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<4)).trans (by norm_num) |
| have hbR : (0:ℝ) ≤ m+n := by positivity |
| have hax := mul_le_mul_of_nonneg_right hlog4 hbR |
| have hsc := mul_le_mul_of_nonneg_left hr hbR |
| push_cast at haxes |
| refine ⟨by nlinarith,by nlinarith,by nlinarith,by nlinarith,by nlinarith⟩ |
|
|
| end BinaryCoordinateSweeps |
|
|
| end |
|
|
| end OAI |
|
|
