| import OAI.MeasureTheory.Falconer.Recursion.ProfileEndpointStability |
| import Mathlib.Algebra.Order.Floor.Ring |
|
|
| namespace OAI |
|
|
| /-! Inward rounding of profile intervals to a fixed finite scale grid. -/ |
| namespace Falconer |
|
|
| noncomputable def profileGridLeft (n : ℕ) (a : ℝ) : ℝ := (⌈(n : ℝ)*a⌉ : ℝ)/(n : ℝ) |
| noncomputable def profileGridRight (n : ℕ) (b : ℝ) : ℝ := (⌊(n : ℝ)*b⌋ : ℝ)/(n : ℝ) |
|
|
| theorem profileGridLeft_bounds {n : ℕ} (hn : 0 < n) (a : ℝ) : |
| a ≤ profileGridLeft n a ∧ profileGridLeft n a ≤ a+1/(n : ℝ) := by |
| have hnR : (0 : ℝ) < n := by exact_mod_cast hn |
| constructor |
| · apply (le_div_iff₀ hnR).2 |
| simpa only [mul_comm] using Int.le_ceil ((n : ℝ)*a) |
| · apply (div_le_iff₀ hnR).2 |
| have hb := (Int.ceil_lt_add_one ((n : ℝ)*a)).le |
| have he : (a+1/(n : ℝ))*(n : ℝ) = (n : ℝ)*a+1 := by field_simp |
| rw [he] |
| exact hb |
|
|
| theorem profileGridRight_bounds {n : ℕ} (hn : 0 < n) (b : ℝ) : |
| b-1/(n : ℝ) ≤ profileGridRight n b ∧ profileGridRight n b ≤ b := by |
| have hnR : (0 : ℝ) < n := by exact_mod_cast hn |
| constructor |
| · apply (le_div_iff₀ hnR).2 |
| have hb := (Int.lt_floor_add_one ((n : ℝ)*b)).le |
| have he : (b-1/(n : ℝ))*(n : ℝ) = (n : ℝ)*b-1 := by field_simp |
| rw [he] |
| linarith |
| · apply (div_le_iff₀ hnR).2 |
| simpa only [mul_comm] using Int.floor_le ((n : ℝ)*b) |
|
|
| theorem profileGrid_interval_bounds {n : ℕ} (hn : 0 < n) (a h : ℝ) : |
| profileGridRight n (a+h)-profileGridLeft n a ≤ h ∧ |
| h-2/(n : ℝ) ≤ profileGridRight n (a+h)-profileGridLeft n a ∧ |
| |profileGridLeft n a-a| ≤ 1/(n : ℝ) ∧ |
| |profileGridRight n (a+h)-(a+h)| ≤ 1/(n : ℝ) := by |
| have hl := profileGridLeft_bounds hn a |
| have hr := profileGridRight_bounds hn (a+h) |
| have he : 2/(n : ℝ) = 2*(1/(n : ℝ)) := by ring |
| refine ⟨by linarith,by rw [he]; linarith,?_,?_⟩ |
| · rw [abs_of_nonneg (sub_nonneg.mpr hl.1)] |
| linarith |
| · rw [abs_of_nonpos (sub_nonpos.mpr hr.2)] |
| linarith |
|
|
| end Falconer |
|
|
| end OAI |
|
|
