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8769036 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | 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
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