| import Mathlib |
| import OAI.Probability.SKBarriers.Dynamics.Uniformization |
|
|
| namespace OAI |
|
|
| section |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem clockWeight_succ_mul (a : ℝ) (k : ℕ) : |
| ((k+1:ℕ):ℝ)*clockWeight a (k+1) = a*clockWeight a k := by |
| unfold clockWeight |
| rw [Nat.factorial_succ,Nat.cast_mul,Nat.cast_add,Nat.cast_one,pow_succ] |
| have h1 : (k:ℝ)+1 ≠ 0 := by positivity |
| have h2 : (k.factorial:ℝ) ≠ 0 := by exact_mod_cast Nat.factorial_ne_zero k |
| field_simp |
|
|
| theorem clockWeight_mean_hasSum (a : ℝ) : HasSum (fun k : ℕ => (k:ℝ)*clockWeight a k) a := by |
| have H := (clockWeight_hasSum a).mul_left a |
| rw [mul_one] at H |
| have HH : HasSum (fun k : ℕ => ((k+1:ℕ):ℝ)*clockWeight a (k+1)) a := by |
| simpa only [clockWeight_succ_mul] using H |
| have HHH := (hasSum_nat_add_iff (f := fun k : ℕ => (k:ℝ)*clockWeight a k) (g := a) 1).mp HH |
| simpa only [Finset.sum_range_one,Nat.cast_zero,zero_mul,add_zero] using HHH |
|
|
| theorem clockWeight_tail_summable {a : ℝ} (ha : 0 ≤ a) (m : ℕ) : |
| Summable (fun k : ℕ => if m < k then clockWeight a k else 0) := by |
| apply Summable.of_nonneg_of_le _ _ (clockWeight_hasSum a).summable |
| · intro k; split_ifs |
| · exact clockWeight_nonneg ha k |
| · exact le_refl 0 |
| · intro k; split_ifs |
| · exact le_refl _ |
| · exact clockWeight_nonneg ha k |
| |
| theorem clockWeight_tail_le {a : ℝ} (ha : 0 ≤ a) {m : ℕ} (hm : 0 < m) : |
| (∑' k : ℕ, if m < k then clockWeight a k else 0) ≤ a/(m:ℝ) := by |
| have H := ((clockWeight_tail_summable ha m).mul_left (m:ℝ)).tsum_le_tsum |
| (fun k => show (m:ℝ)*(if m < k then clockWeight a k else 0) ≤ (k:ℝ)*clockWeight a k from by |
| split_ifs with hk |
| · exact mul_le_mul_of_nonneg_right (by exact_mod_cast le_of_lt hk) (clockWeight_nonneg ha k) |
| · simp only [mul_zero]; exact mul_nonneg (by positivity) (clockWeight_nonneg ha k)) |
| (clockWeight_mean_hasSum a).summable |
| rw [tsum_mul_left,(clockWeight_mean_hasSum a).tsum_eq] at H |
| exact (le_div_iff₀ (show (0:ℝ) < m by exact_mod_cast hm)).mpr (by simpa only [mul_comm] using H) |
|
|
| theorem attempt_horizon_tail (n : ℕ) (L : ℝ) : |
| (∑' k : ℕ, if ⌈Real.exp (2*L)⌉₊ < k then clockWeight ((n:ℝ)*Real.exp L) k else 0) ≤ |
| (n:ℝ)*Real.exp (-L) := by |
| have hExp := Real.exp_pos (2*L) |
| have hm : 0 < ⌈Real.exp (2*L)⌉₊ := Nat.ceil_pos.mpr hExp |
| have hceil : Real.exp (2*L) ≤ (⌈Real.exp (2*L)⌉₊:ℝ) := Nat.le_ceil _ |
| calc |
| _ ≤ ((n:ℝ)*Real.exp L)/(⌈Real.exp (2*L)⌉₊:ℝ) := clockWeight_tail_le (by positivity) hm |
| _ ≤ ((n:ℝ)*Real.exp L)/Real.exp (2*L) := div_le_div_of_nonneg_left (by positivity) hExp hceil |
| _ = (n:ℝ)*Real.exp (-L) := by |
| rw [mul_div_assoc,← Real.exp_sub] |
| congr 2 |
| ring |
| |
| end SK.Analytic |
| |
| end |
| end |
| |
| end |
| |
| end OAI |
| |
