| import Mathlib.Data.Nat.Choose.Sum |
| import Mathlib.Analysis.SpecialFunctions.Log.Basic |
| import Mathlib.Analysis.SpecialFunctions.Exp |
|
|
| namespace OAI |
|
|
| namespace MatroidProphet.Pivots |
|
|
| open Finset |
|
|
| noncomputable def binomialSum (n d : ℕ) : ℕ := ∑ k ∈ Iic d, n.choose k |
|
|
| lemma binomialSum_le_weighted (n d : ℕ) (κ : ℝ) (hκ : 1 ≤ κ) : |
| (binomialSum n d : ℝ) ≤ κ ^ d * (1 + κ⁻¹) ^ n := by |
| have hkpos : 0 < κ := lt_of_lt_of_le zero_lt_one hκ |
| have htrunc : (binomialSum n d : ℝ) = ∑ k ∈ Iic d ∩ Iic n, (n.choose k : ℝ) := by |
| rw [binomialSum, Nat.cast_sum] |
| apply (sum_subset inter_subset_left ?_).symm |
| intro k hkd hnot |
| have hnk : n < k := by simpa only [mem_inter, hkd, true_and, mem_Iic, not_le] using hnot |
| simp [Nat.choose_eq_zero_of_lt hnk] |
| have hp (k : ℕ) (hkd : k ≤ d) : 1 ≤ κ ^ d * (κ⁻¹) ^ k := by |
| rw [inv_pow, ← div_eq_mul_inv] |
| exact (one_le_div₀ (pow_pos hkpos k)).2 (pow_le_pow_right₀ hκ hkd) |
| calc |
| (binomialSum n d : ℝ) = ∑ k ∈ Iic d ∩ Iic n, (n.choose k : ℝ) := htrunc |
| _ ≤ ∑ k ∈ Iic d ∩ Iic n, κ ^ d * ((κ⁻¹) ^ k * (n.choose k : ℝ)) := by |
| apply sum_le_sum |
| intro k hk |
| have h := mul_le_mul_of_nonneg_right (hp k (mem_Iic.mp (mem_inter.mp hk).1)) |
| (Nat.cast_nonneg (n.choose k) : (0 : ℝ) ≤ n.choose k) |
| simpa only [one_mul, mul_assoc] using h |
| _ ≤ ∑ k ∈ Iic n, κ ^ d * ((κ⁻¹) ^ k * (n.choose k : ℝ)) := |
| sum_le_sum_of_subset_of_nonneg inter_subset_right (by intros; positivity) |
| _ = κ ^ d * (1 + κ⁻¹) ^ n := by |
| rw [← mul_sum, add_comm 1, add_pow, Nat.range_succ_eq_Iic] |
| simp |
|
|
| lemma binomialSum_le_exp (n d : ℕ) (κ : ℝ) (hκ : 1 ≤ κ) : |
| (binomialSum n d : ℝ) ≤ Real.exp ((d : ℝ) * Real.log κ + (n : ℝ) / κ) := by |
| have hkpos : 0 < κ := lt_of_lt_of_le zero_lt_one hκ |
| have hone : 1 + κ⁻¹ ≤ Real.exp κ⁻¹ := by simpa [add_comm] using Real.add_one_le_exp κ⁻¹ |
| have hpow := pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ 1 + κ⁻¹) hone n |
| calc |
| (binomialSum n d : ℝ) ≤ κ ^ d * (1 + κ⁻¹) ^ n := binomialSum_le_weighted n d κ hκ |
| _ ≤ κ ^ d * (Real.exp κ⁻¹) ^ n := mul_le_mul_of_nonneg_left hpow (pow_nonneg hkpos.le d) |
| _ = Real.exp ((d : ℝ) * Real.log κ + (n : ℝ) / κ) := by |
| rw [Real.exp_add, Real.exp_nat_mul, Real.exp_log hkpos, div_eq_mul_inv, Real.exp_nat_mul] |
|
|
| lemma residual_raw_count_le_exp (n q : ℕ) (κ : ℝ) (hκ : 2 ≤ κ) |
| (hq : (q : ℝ) ≤ (n : ℝ) / κ) : |
| (((binomialSum n q) ^ 2 * (4 ^ (2 * q) * binomialSum n (2 * q)) ^ 2 : ℕ) : ℝ) ≤ |
| Real.exp (1000 * Real.log κ / κ * n) := by |
| have hkone : 1 ≤ κ := by linarith |
| have hkpos : 0 < κ := by linarith |
| have hlog : 0 ≤ Real.log κ := Real.log_nonneg hkone |
| have hloghalf : (1 : ℝ) / 2 ≤ Real.log κ := by |
| have htwo : (1 : ℝ) / 2 ≤ Real.log 2 := by |
| have h := Real.one_sub_inv_le_log_of_pos (by norm_num : (0 : ℝ) < 2) |
| norm_num at h ⊢ |
| linarith |
| exact htwo.trans (Real.log_le_log (by norm_num) hκ) |
| have h4 : (4 : ℝ) ^ (2 * q) ≤ Real.exp (((4 * q : ℕ) : ℝ) * Real.log κ) := by |
| rw [Real.exp_nat_mul, Real.exp_log hkpos] |
| calc |
| (4 : ℝ) ^ (2 * q) = (2 : ℝ) ^ (4 * q) := by |
| rw [show (4 : ℝ) = 2 ^ 2 by norm_num, ← pow_mul] |
| congr 1 |
| omega |
| _ ≤ κ ^ (4 * q) := pow_le_pow_left₀ (by norm_num) hκ _ |
| have hs := binomialSum_le_exp n q κ hkone |
| have hs2 := binomialSum_le_exp n (2 * q) κ hkone |
| have hproduct := mul_le_mul h4 hs2 (by positivity) (Real.exp_nonneg _) |
| have hsquare := pow_le_pow_left₀ (by positivity) hs 2 |
| have hproductsquare := pow_le_pow_left₀ (by positivity) hproduct 2 |
| have hall := mul_le_mul hsquare hproductsquare (by positivity) (by positivity) |
| have hexp : (((binomialSum n q) ^ 2 * (4 ^ (2 * q) * binomialSum n (2 * q)) ^ 2 : ℕ) : ℝ) ≤ |
| Real.exp (14 * (q : ℝ) * Real.log κ + 4 * ((n : ℝ) / κ)) := by |
| push_cast |
| convert hall using 1 |
| rw [← Real.exp_add, ← Real.exp_nat_mul, ← Real.exp_nat_mul, ← Real.exp_add] |
| congr 1 |
| push_cast |
| ring |
| apply hexp.trans (Real.exp_le_exp.mpr ?_) |
| have hqlog := mul_le_mul_of_nonneg_right hq hlog |
| have hn : (0 : ℝ) ≤ (n : ℝ) / κ := div_nonneg (Nat.cast_nonneg n) hkpos.le |
| have hnlog : 4 * ((n : ℝ) / κ) ≤ 8 * ((n : ℝ) / κ) * Real.log κ := by |
| nlinarith [mul_nonneg hn (sub_nonneg.mpr hloghalf)] |
| have hnlognonneg : 0 ≤ ((n : ℝ) / κ) * Real.log κ := mul_nonneg hn hlog |
| calc |
| 14 * (q : ℝ) * Real.log κ + 4 * ((n : ℝ) / κ) |
| ≤ 1000 * ((n : ℝ) / κ) * Real.log κ := by nlinarith |
| _ = 1000 * Real.log κ / κ * n := by ring |
|
|
| end MatroidProphet.Pivots |
|
|
| end OAI |
|
|
