| import Mathlib |
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| namespace OAI |
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| section |
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| section |
| noncomputable section |
| namespace SK.Analytic |
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| theorem overlap_error_arithmetic {k v : ℝ} (hk : 1 ≤ k) (hv : 0 < v) |
| (hkv : k/Real.sqrt v ≤ 1) : |
| let E := 1/(k+1)+2*(k+1)/v |
| k*Real.sqrt (2/v)+3*Real.sqrt k/(k+1)+E+2*Real.sqrt E+4/(k+1) ≤ |
| 20*(k/Real.sqrt v+1/Real.sqrt k) := by |
| have hk0 : 0 < k := by linarith |
| have hrk : 0 < Real.sqrt k := Real.sqrt_pos.2 hk0 |
| have hrv : 0 < Real.sqrt v := Real.sqrt_pos.2 hv |
| have hksq : k^2 ≤ v := by |
| have hh := (div_le_one hrv).1 hkv |
| nlinarith [Real.sq_sqrt hv.le] |
| have hroot : Real.sqrt k ≤ k := by |
| nlinarith [Real.sq_sqrt hk0.le] |
| have hK : 0 < k+1 := by linarith |
| have hE : 1/(k+1)+2*(k+1)/v ≤ 5/k := by |
| have h1 := one_div_le_one_div_of_le hk0 (show k ≤ k+1 by linarith) |
| have h2 : 2*(k+1)/v ≤ 4/k := by |
| calc |
| _ ≤ 2*(2*k)/(k^2) := div_le_div₀ (by positivity) (by nlinarith) (sq_pos_of_pos hk0) hksq |
| _ = _ := by field_simp; ring |
| calc |
| _ ≤ 1/k+4/k := add_le_add h1 h2 |
| _ = _ := by ring |
| have hEsqrt : Real.sqrt (1/(k+1)+2*(k+1)/v) ≤ 3/Real.sqrt k := by |
| calc |
| _ ≤ Real.sqrt (9/k) := Real.sqrt_le_sqrt (hE.trans (div_le_div_of_nonneg_right (by norm_num) hk0.le)) |
| _ = _ := by rw [Real.sqrt_div (by norm_num)]; norm_num |
| have hfirst : k*Real.sqrt (2/v) ≤ 2*(k/Real.sqrt v) := by |
| rw [Real.sqrt_div (by norm_num)] |
| have htwo : Real.sqrt (2 : ℝ) ≤ 2 := by nlinarith [Real.sq_sqrt (show (0 : ℝ) ≤ 2 by norm_num),Real.sqrt_nonneg (2 : ℝ)] |
| have H := mul_le_mul_of_nonneg_left (div_le_div_of_nonneg_right htwo hrv.le) hk0.le |
| calc |
| _ ≤ k*(2/Real.sqrt v) := H |
| _ = _ := by ring |
| have hmean : 3*Real.sqrt k/(k+1) ≤ 3/Real.sqrt k := by |
| apply (div_le_div_iff₀ hK hrk).2 |
| nlinarith [Real.sq_sqrt hk0.le] |
| have hEfinal : 1/(k+1)+2*(k+1)/v ≤ 5/Real.sqrt k := |
| hE.trans (div_le_div_of_nonneg_left (by norm_num) hrk hroot) |
| have hlast : 4/(k+1) ≤ 4/Real.sqrt k := |
| div_le_div_of_nonneg_left (by norm_num) hrk (hroot.trans (by linarith)) |
| dsimp only |
| have ha : 0 ≤ k/Real.sqrt v := by positivity |
| have hb : 0 ≤ 1/Real.sqrt k := by positivity |
| simp only [div_eq_mul_inv] at hEsqrt hfirst hmean hEfinal hlast ha hb ⊢ |
| linarith |
| end SK.Analytic |
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| end |
| end |
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| end |
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| end OAI |
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