| import Mathlib | |
| namespace OAI | |
| namespace Problem367 | |
| theorem critical_estimator_cutoff {t : ℝ} (ht : 0 < t) (ht1 : t ≤ 1) : | |
| ∃ n : ℕ, Real.sqrt t / 5 ≤ | |
| t * ((n : ℝ) + 1) / (1 + t * ((n : ℝ) + 1) ^ 2) := by | |
| let n : ℕ := ⌊1 / Real.sqrt t⌋₊ | |
| have hs : 0 < Real.sqrt t := Real.sqrt_pos.mpr ht | |
| have hs1 : Real.sqrt t ≤ 1 := Real.sqrt_le_one.mpr ht1 | |
| have hsq : Real.sqrt t ^ 2 = t := Real.sq_sqrt ht.le | |
| have hinv : 1 ≤ 1 / Real.sqrt t := (le_div_iff₀ hs).mpr (by simpa using hs1) | |
| have hnlow : 1 / Real.sqrt t ≤ (n : ℝ) + 1 := | |
| (Nat.lt_floor_add_one (1 / Real.sqrt t)).le | |
| have hnhigh : (n : ℝ) + 1 ≤ 2 / Real.sqrt t := by | |
| have hfloor := Nat.floor_le (by positivity : 0 ≤ 1 / Real.sqrt t) | |
| change (n : ℝ) ≤ 1 / Real.sqrt t at hfloor | |
| calc | |
| (n : ℝ) + 1 ≤ 1 / Real.sqrt t + 1 := by linarith | |
| _ ≤ 1 / Real.sqrt t + 1 / Real.sqrt t := by linarith | |
| _ = 2 / Real.sqrt t := by ring | |
| have hmul_low : 1 ≤ Real.sqrt t * ((n : ℝ) + 1) := by | |
| have h := (div_le_iff₀ hs).mp hnlow | |
| nlinarith | |
| have hmul_high : Real.sqrt t * ((n : ℝ) + 1) ≤ 2 := by | |
| have h := (le_div_iff₀ hs).mp hnhigh | |
| nlinarith | |
| have hnum : Real.sqrt t ≤ t * ((n : ℝ) + 1) := by | |
| have h := mul_le_mul_of_nonneg_left hmul_low hs.le | |
| nlinarith [hsq] | |
| have hprod_sq : (Real.sqrt t * ((n : ℝ) + 1)) ^ 2 = | |
| t * ((n : ℝ) + 1) ^ 2 := by rw [mul_pow, hsq] | |
| have hden : 1 + t * ((n : ℝ) + 1) ^ 2 ≤ 5 := by | |
| have h := mul_self_le_mul_self (by positivity : | |
| 0 ≤ Real.sqrt t * ((n : ℝ) + 1)) hmul_high | |
| nlinarith [hprod_sq] | |
| have hdenpos : 0 < 1 + t * ((n : ℝ) + 1) ^ 2 := by positivity | |
| refine ⟨n, (le_div_iff₀ hdenpos).mpr ?_⟩ | |
| calc | |
| Real.sqrt t / 5 * (1 + t * ((n : ℝ) + 1) ^ 2) ≤ | |
| Real.sqrt t / 5 * 5 := | |
| mul_le_mul_of_nonneg_left hden (by positivity) | |
| _ ≤ t * ((n : ℝ) + 1) := by nlinarith | |
| theorem critical_information_lower_of_estimators {t M : ℝ} (ht : 0 < t) | |
| (h : ∀ n : ℕ, | |
| t * ((n : ℝ) + 1) / (1 + t * ((n : ℝ) + 1) ^ 2) ≤ M) : | |
| min (Real.sqrt t) 1 / 5 ≤ M := by | |
| by_cases ht1 : t ≤ 1 | |
| · obtain ⟨n, hn⟩ := critical_estimator_cutoff ht ht1 | |
| rw [min_eq_left (Real.sqrt_le_one.mpr ht1)] | |
| exact hn.trans (h n) | |
| · have ht1' : 1 ≤ t := le_of_lt (lt_of_not_ge ht1) | |
| rw [min_eq_right (Real.one_le_sqrt.mpr ht1')] | |
| have h0 : t / (1 + t) ≤ M := by simpa using h 0 | |
| have hhalf : (1 / 2 : ℝ) ≤ t / (1 + t) := by | |
| apply (le_div_iff₀ (by positivity : 0 < 1 + t)).mpr | |
| linarith | |
| linarith | |
| theorem sqrt_bound_of_quadratic_information_loss {M c K t : ℝ} | |
| (hc : 0 < c) (hK : 0 ≤ K) | |
| (h : M ≤ M - c * M ^ 2 + K * t) : | |
| M ≤ Real.sqrt (K / c) * Real.sqrt t := by | |
| have hquad : M ^ 2 ≤ K * t / c := by | |
| apply (le_div_iff₀ hc).mpr | |
| nlinarith | |
| calc | |
| M ≤ Real.sqrt (K * t / c) := Real.le_sqrt_of_sq_le hquad | |
| _ = Real.sqrt (K / c) * Real.sqrt t := by | |
| rw [show K * t / c = (K / c) * t by ring, | |
| Real.sqrt_mul (div_nonneg hK hc.le)] | |
| theorem critical_information_upper_of_fusion {b theta M t : ℝ} | |
| (hθ0 : 0 < theta) (hθ1 : theta < 1) | |
| (hcritical : b * theta ^ 2 = 1) (ht : 0 ≤ t) (ht1 : t ≤ 1) | |
| (h : M ≤ b * theta ^ 2 * M - | |
| 2 * (1 - theta ^ 2) * theta ^ 4 * M ^ 2 + (t + t ^ 2)) : | |
| M ≤ Real.sqrt (1 / ((1 - theta ^ 2) * theta ^ 4)) * Real.sqrt t := by | |
| have hθsq : theta ^ 2 < 1 := | |
| (sq_lt_one_iff_abs_lt_one theta).mpr (abs_lt.mpr ⟨by linarith, hθ1⟩) | |
| have hc : 0 < (1 - theta ^ 2) * theta ^ 4 := | |
| mul_pos (sub_pos.mpr hθsq) (pow_pos hθ0 4) | |
| have htsq : t ^ 2 ≤ t := by nlinarith | |
| rw [hcritical, one_mul] at h | |
| have hrec : M ≤ M - (2 * ((1 - theta ^ 2) * theta ^ 4)) * M ^ 2 + 2 * t := by | |
| nlinarith | |
| have hbound := sqrt_bound_of_quadratic_information_loss | |
| (mul_pos (by norm_num : (0 : ℝ) < 2) hc) (by norm_num : (0 : ℝ) ≤ 2) hrec | |
| have hratio : (2 : ℝ) / (2 * ((1 - theta ^ 2) * theta ^ 4)) = | |
| 1 / ((1 - theta ^ 2) * theta ^ 4) := by field_simp | |
| simpa only [hratio] using hbound | |
| end Problem367 | |
| end OAI | |