| import OAI.NumberTheory.TwoPoint.Circuits.CircuitBernoulli |
| import Mathlib.Data.Nat.Log |
|
|
| namespace OAI |
|
|
| /-! Only logarithmically many Bernoulli parameters are needed to isolate a |
| zero among the inputs of an AND gate. The success constant 1/8 is sufficient |
| for the application's unspecified absolute circuit constants. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| lemma exists_dyadic_single_hit (k z : ℕ) (hz : 0 < z) (hzk : z ≤ k) : |
| ∃ a : ℕ, a ≤ Nat.log 2 k + 2 ∧ |
| 1 / 8 ≤ (z : ℝ) * (1 / (2 : ℝ) ^ a) * |
| (1 - 1 / (2 : ℝ) ^ a) ^ (z - 1) := by |
| let l := Nat.log 2 z |
| have hlog := Nat.log_mono_right (b := 2) hzk |
| refine ⟨l + 2, by dsimp only [l]; omega, ?_⟩ |
| have hlow : (2 : ℝ) ^ l ≤ z := by |
| exact_mod_cast Nat.pow_log_le_self 2 (Nat.ne_of_gt hz) |
| have hhigh : (z : ℝ) < (2 : ℝ) ^ (l + 1) := by |
| exact_mod_cast (Nat.log_lt_iff_lt_pow (by norm_num : 1 < 2) |
| (Nat.ne_of_gt hz)).mp (Nat.lt_succ_self l) |
| have hpow : (2 : ℝ) ^ (l + 2) = 4 * (2 : ℝ) ^ l := by |
| rw [pow_add] |
| norm_num |
| ring |
| have hpow' : (2 : ℝ) ^ (l + 2) = 2 * (2 : ℝ) ^ (l + 1) := by |
| rw [show l + 2 = (l + 1) + 1 by omega, pow_add] |
| ring |
| have hpos : (0 : ℝ) < (2 : ℝ) ^ (l + 2) := by positivity |
| apply single_hit_probability_lower z hz _ (by positivity) |
| · rw [mul_one_div, div_le_iff₀ hpos, hpow'] |
| linarith |
| · rw [mul_one_div, le_div_iff₀ hpos, hpow] |
| linarith |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
