| import Mathlib | |
| namespace OAI | |
| namespace Invariant233 | |
| theorem threshold_mem_Icc_of_abs_bounds {a x : ℝ} | |
| (ha : 0 < a) (hlo : a ≤ |x|) (hhi : |x| ≤ 1 / 2) : | |
| 1 - Int.fract x ∈ Set.Icc a (1 - a) := by | |
| rcases le_or_gt 0 x with hx | hx | |
| · have habs : |x| = x := abs_of_nonneg hx | |
| have hfract : Int.fract x = x := Int.fract_eq_self.mpr ⟨hx, by linarith⟩ | |
| rw [hfract] | |
| constructor <;> linarith | |
| · have habs : |x| = -x := abs_of_neg hx | |
| have hfract : Int.fract (-x) = -x := | |
| Int.fract_eq_self.mpr ⟨by linarith, by linarith⟩ | |
| have hnz : Int.fract (-x) ≠ 0 := by rw [hfract]; linarith | |
| have heq := Int.fract_neg hnz | |
| simp only [neg_neg] at heq | |
| rw [heq, hfract] | |
| constructor <;> linarith | |
| theorem phase_small_window {m K G α k : ℝ} | |
| (hm : 0 < m) (hK : 0 < K) (hG : 0 < G) | |
| (hαlo : 1 / (m * K) ≤ α) (hαhi : α ≤ 2 / (m * K)) | |
| (hklo : G * K ≤ |k|) (hkhi : |k| ≤ m * K / 4) : | |
| 1 - Int.fract (k * α) ∈ Set.Icc (G / m) (1 - G / m) := by | |
| have hmK : 0 < m * K := mul_pos hm hK | |
| have hα : 0 < α := lt_of_lt_of_le (one_div_pos.mpr hmK) hαlo | |
| have hlo' : G / m ≤ |k| * α := by | |
| calc | |
| G / m = (G * K) * (1 / (m * K)) := by field_simp | |
| _ ≤ |k| * α := mul_le_mul hklo hαlo (le_of_lt (one_div_pos.mpr hmK)) | |
| (abs_nonneg k) | |
| have hhi' : |k| * α ≤ 1 / 2 := by | |
| calc | |
| |k| * α ≤ (m * K / 4) * (2 / (m * K)) := | |
| mul_le_mul hkhi hαhi (le_of_lt hα) (by positivity) | |
| _ = 1 / 2 := by field_simp; ring | |
| apply threshold_mem_Icc_of_abs_bounds (div_pos hG hm) | |
| · simpa only [abs_mul, abs_of_pos hα] using hlo' | |
| · simpa only [abs_mul, abs_of_pos hα] using hhi' | |
| end Invariant233 | |
| end OAI | |