| import OAI.Probability.FreeIsing.Estimates.ThresholdArithmetic |
| import OAI.Probability.FreeIsing.Gaussian.ComponentSignalAttenuation |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace Problem367 |
|
|
| theorem half_path_cut_admissible {n m : ℕ} (hn : 8 ≤ n) (hm : n / 2 ≤ m) : |
| let k := m / 2 - 1 |
| let L := m - m / 2 |
| k + 1 + L = m ∧ |
| (n : ℝ) / 8 - 1 ≤ (k + 1 : ℕ) ∧ (n : ℝ) / 8 ≤ (L : ℝ) := by |
| dsimp only |
| have hk : m / 2 - 1 + 1 = m / 2 := by omega |
| rw [hk] |
| refine ⟨by omega, ?_, ?_⟩ |
| · have hround : n ≤ 8 * (m / 2) + 8 := by omega |
| have hr : (n : ℝ) ≤ 8 * (m / 2 : ℕ) + 8 := by exact_mod_cast hround |
| linarith |
| · have hround : n ≤ 8 * (m - m / 2) := by omega |
| have hr : (n : ℝ) ≤ 8 * (m - m / 2 : ℕ) := by exact_mod_cast hround |
| linarith |
|
|
| def criticalNoiseEta (d : ℕ) (beta : ℝ) : ℝ := |
| min (ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) / 4) |
| (Real.log ((d : ℝ) - 1) / 8) |
|
|
| def criticalNoiseLengthConstant (beta : ℝ) : ℝ := |
| max 8 (32 * Real.log 2 / ComponentSignalAttenuation.lossCoefficient (Real.tanh beta)) |
|
|
| def criticalNoiseTailBase (d : ℕ) : ℝ := ((d : ℝ) - 1) ^ (1 / 16 : ℝ) |
|
|
| theorem criticalNoise_constants_positive {d : ℕ} {beta : ℝ} |
| (hd : 3 ≤ d) (hb : 0 ≤ beta) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) : |
| 0 < criticalNoiseEta d beta ∧ 0 < criticalNoiseLengthConstant beta ∧ |
| 1 < criticalNoiseTailBase d ∧ 0 < 1 / (2 * ((d : ℝ) - 1)) := by |
| have hb1 : 1 < (d : ℝ) - 1 := by |
| have hdR : (3 : ℝ) ≤ d := by exact_mod_cast hd |
| linarith |
| have hθ : 0 < Real.tanh beta := by |
| by_contra h |
| have hz := le_antisymm (not_lt.mp h) (tanh_nonneg_of_beta_nonneg hb) |
| simp [hz] at hcritical |
| have hγ := ComponentSignalAttenuation.lossCoefficient_pos hθ (Real.tanh_lt_one beta) |
| refine ⟨lt_min (div_pos hγ (by norm_num)) |
| (div_pos (Real.log_pos hb1) (by norm_num)), ?_, ?_, ?_⟩ |
| · exact lt_of_lt_of_le (by norm_num) (le_max_left _ _) |
| · exact Real.one_lt_rpow hb1 (by norm_num) |
| · have : 0 < (d : ℝ) - 1 := zero_lt_one.trans hb1 |
| positivity |
|
|
| theorem criticalNoise_length_admissible {d n : ℕ} {beta ω : ℝ} |
| (_hd : 3 ≤ d) (hb : 0 ≤ beta) |
| (hcritical : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1) |
| (hω : 0 < ω) (hω1 : ω ≤ 1) |
| (hn : criticalNoiseLengthConstant beta / ω ≤ (n : ℝ)) : |
| 8 ≤ n ∧ 8 * Real.log 2 ≤ |
| ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) * (ω * ((n : ℝ) / 4)) := by |
| have hθ : 0 < Real.tanh beta := by |
| by_contra h |
| have hz := le_antisymm (not_lt.mp h) (tanh_nonneg_of_beta_nonneg hb) |
| simp [hz] at hcritical |
| have hγ := ComponentSignalAttenuation.lossCoefficient_pos hθ (Real.tanh_lt_one beta) |
| have hlarge := (div_le_iff₀ hω).mp hn |
| have hlow : (8 : ℝ) ≤ criticalNoiseLengthConstant beta := le_max_left _ _ |
| have hmul := mul_le_mul_of_nonneg_left hω1 (Nat.cast_nonneg n : (0 : ℝ) ≤ _) |
| have hnR : (8 : ℝ) ≤ n := by nlinarith |
| refine ⟨by exact_mod_cast hnR, ?_⟩ |
| have hfrac : 32 * Real.log 2 / ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) ≤ |
| (n : ℝ) * ω := (le_max_right _ _).trans hlarge |
| have h := (div_le_iff₀ hγ).mp hfrac |
| nlinarith |
|
|
| theorem criticalNoise_threshold_eq (d : ℕ) (beta ω : ℝ) (n : ℕ) : |
| Real.exp (-(criticalNoiseEta d beta * ω) * ((n : ℝ) / 4)) = |
| Real.exp (-((criticalNoiseEta d beta / 4) * ω) * n) := by congr 1; ring |
|
|
| theorem criticalNoise_tail_eq {d : ℕ} (hd : 3 ≤ d) (n : ℕ) : |
| Real.exp (-(((d : ℝ) - 1) ^ (((n : ℝ) / 4) / 4)) / |
| (2 * ((d : ℝ) - 1))) = |
| Real.exp (-(1 / (2 * ((d : ℝ) - 1))) * criticalNoiseTailBase d ^ n) := by |
| have hb : 0 ≤ (d : ℝ) - 1 := by |
| have hdR : (3 : ℝ) ≤ d := by exact_mod_cast hd |
| linarith |
| have hp : ((d : ℝ) - 1) ^ (((n : ℝ) / 4) / 4) = criticalNoiseTailBase d ^ n := by |
| rw [show ((n : ℝ) / 4) / 4 = (1 / 16 : ℝ) * n by ring, |
| Real.rpow_mul hb, Real.rpow_natCast] |
| rfl |
| rw [hp] |
| congr 1 |
| ring |
|
|
| end Problem367 |
|
|
| end |
|
|
| end OAI |
|
|
