| import OAI.Combinatorics.GraphThreshold.Basic |
| import Mathlib.Analysis.Complex.ExponentialBounds |
| import Mathlib.Data.Nat.Choose.Sum |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Finset |
|
|
| namespace LeanBlast.SecondKahnKalai |
|
|
| theorem two_lt_exp_one : (2 : ℝ) < Real.exp 1 := Real.exp_one_gt_two |
|
|
| theorem exp_one_lt_three : Real.exp 1 < (3 : ℝ) := Real.exp_one_lt_three |
|
|
| / |
| predecessors in the graph hierarchy. -/ |
| theorem predecessor_power_lt_one {r : ℕ} (hr : 1 ≤ r) : |
| 2 * (17 * Real.exp 1 / 128) ^ r < 1 := by |
| have hbase0 : 0 ≤ 17 * Real.exp 1 / 128 := by positivity |
| have hbase : 17 * Real.exp 1 / 128 < (1 / 2 : ℝ) := by |
| nlinarith [exp_one_lt_three] |
| have hp : (17 * Real.exp 1 / 128) ^ r ≤ 17 * Real.exp 1 / 128 := |
| pow_le_of_le_one hbase0 (by linarith) (by omega) |
| linarith |
|
|
| theorem exp_neg_fifty_lt_one : Real.exp (-50) < 1 := by |
| exact Real.exp_lt_one_iff.mpr (by norm_num) |
|
|
| theorem one_add_le_exp_one_of_le_exp_neg_fifty {q : ℝ} |
| (hq : q ≤ Real.exp (-50)) : 1 + q ≤ Real.exp 1 := by |
| linarith [exp_neg_fifty_lt_one, two_lt_exp_one] |
|
|
| theorem one_add_pow_le_exp_nat {q : ℝ} (hq0 : 0 ≤ q) |
| (hq : q ≤ Real.exp (-50)) (m : ℕ) : |
| (1 + q) ^ m ≤ Real.exp (m : ℝ) := by |
| calc |
| (1 + q) ^ m ≤ Real.exp 1 ^ m := |
| pow_le_pow_left₀ (by positivity) (one_add_le_exp_one_of_le_exp_neg_fifty hq) m |
| _ = Real.exp (m : ℝ) := by rw [← Real.exp_nat_mul]; simp |
|
|
| theorem two_pow_le_exp_nat (m : ℕ) : (2 : ℝ) ^ m ≤ Real.exp (m : ℝ) := by |
| calc |
| (2 : ℝ) ^ m ≤ Real.exp 1 ^ m := pow_le_pow_left₀ (by norm_num) two_lt_exp_one.le m |
| _ = Real.exp (m : ℝ) := by rw [← Real.exp_nat_mul]; simp |
|
|
| / |
| theorem large_intersection_power_le {m r : ℕ} {q : ℝ} |
| (hq0 : 0 ≤ q) (hq : q ≤ Real.exp (-50)) (hr : m < 2 * r) : |
| q ^ r ≤ Real.exp (-25 * (m : ℝ)) := by |
| have hr' : (m : ℝ) < 2 * (r : ℝ) := by exact_mod_cast hr |
| calc |
| q ^ r ≤ Real.exp (-50) ^ r := pow_le_pow_left₀ hq0 hq r |
| _ = Real.exp ((r : ℝ) * (-50)) := (Real.exp_nat_mul _ _).symm |
| _ ≤ Real.exp (-25 * (m : ℝ)) := Real.exp_le_exp.mpr (by linarith) |
| |
| /-- The finite binomial upper tail used by the resampling argument. -/ |
| def largeIntersectionTail (m : ℕ) (q : ℝ) : ℝ := |
| ∑ r ∈ range (m + 1), if m < 2 * r then (m.choose r : ℝ) * q ^ r else 0 |
| |
| theorem largeIntersectionTail_le {m : ℕ} {q : ℝ} |
| (hq0 : 0 ≤ q) (hq : q ≤ Real.exp (-50)) : |
| largeIntersectionTail m q ≤ (2 : ℝ) ^ m * Real.exp (-25 * (m : ℝ)) := by |
| have hsum : (∑ r ∈ range (m + 1), (m.choose r : ℝ)) = (2 : ℝ) ^ m := by |
| exact_mod_cast Nat.sum_range_choose m |
| unfold largeIntersectionTail |
| calc |
| (∑ r ∈ range (m + 1), if m < 2 * r then (m.choose r : ℝ) * q ^ r else 0) ≤ |
| ∑ r ∈ range (m + 1), (m.choose r : ℝ) * Real.exp (-25 * (m : ℝ)) := by |
| apply sum_le_sum |
| intro r hr |
| split_ifs with h |
| · exact mul_le_mul_of_nonneg_left (large_intersection_power_le hq0 hq h) (by positivity) |
| · positivity |
| _ = (2 : ℝ) ^ m * Real.exp (-25 * (m : ℝ)) := by rw [← sum_mul, hsum] |
| |
| theorem local_failure_exponential_combine {m : ℕ} (hm : 1 ≤ m) : |
| Real.exp (-10 * (m : ℝ)) + |
| Real.exp (10 * (m : ℝ)) * (2 : ℝ) ^ m * Real.exp (-25 * (m : ℝ)) ≤ |
| Real.exp (-9 * (m : ℝ)) := by |
| have hm0 : 0 ≤ (m : ℝ) := Nat.cast_nonneg m |
| have hm1 : 1 ≤ (m : ℝ) := by exact_mod_cast hm |
| have hterm : Real.exp (10 * (m : ℝ)) * (2 : ℝ) ^ m * |
| Real.exp (-25 * (m : ℝ)) ≤ Real.exp (-10 * (m : ℝ)) := by |
| calc |
| Real.exp (10 * (m : ℝ)) * (2 : ℝ) ^ m * Real.exp (-25 * (m : ℝ)) ≤ |
| Real.exp (10 * (m : ℝ)) * Real.exp (m : ℝ) * Real.exp (-25 * (m : ℝ)) := by |
| gcongr |
| exact two_pow_le_exp_nat m |
| _ = Real.exp (-14 * (m : ℝ)) := by rw [← Real.exp_add, ← Real.exp_add]; congr 1; ring |
| _ ≤ Real.exp (-10 * (m : ℝ)) := Real.exp_le_exp.mpr (by linarith) |
| have htwo : (2 : ℝ) ≤ Real.exp (m : ℝ) := |
| two_lt_exp_one.le.trans (Real.exp_le_exp.mpr hm1) |
| calc |
| Real.exp (-10 * (m : ℝ)) + |
| Real.exp (10 * (m : ℝ)) * (2 : ℝ) ^ m * Real.exp (-25 * (m : ℝ)) ≤ |
| 2 * Real.exp (-10 * (m : ℝ)) := by linarith |
| _ ≤ Real.exp (m : ℝ) * Real.exp (-10 * (m : ℝ)) := by gcongr |
| _ = Real.exp (-9 * (m : ℝ)) := by rw [← Real.exp_add]; congr 1; ring |
| |
| end LeanBlast.SecondKahnKalai |
| |
| end |
| |
| end OAI |
| |
