File size: 1,713 Bytes
f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 | 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
|