import OAI.Probability.FreeIsing.Energy.CoupledAttenuation import OAI.Probability.FreeIsing.Energy.MomentTail import OAI.Probability.FreeIsing.Estimates.LogInterpolation namespace OAI noncomputable section open MeasureTheory ProbabilityTheory namespace Problem367 def cavityLogVarianceCoefficient (theta : ℝ) : ℝ := 4 / (1 - theta) ^ 2 def cavityLogTailCoefficient (theta M : ℝ) : ℝ := 8 * (M / (1 - theta)) ^ 2 theorem cavityLogVarianceCoefficient_nonneg (theta : ℝ) : 0 ≤ cavityLogVarianceCoefficient theta := by unfold cavityLogVarianceCoefficient positivity theorem cavityLogTailCoefficient_nonneg (theta M : ℝ) : 0 ≤ cavityLogTailCoefficient theta M := by unfold cavityLogTailCoefficient positivity section Coupling variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ] variable {theta : ℝ} {X Y X' Y' H K H' K' : Ω → ℝ} theorem integral_cavity_log_discrepancy_sq_le_of_marginal_tails (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) (hX : Measurable X) (hX' : Measurable X') (hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ) (hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ) (heq : ∀ᵐ ω ∂μ, X ω + cavityMessage theta (Y ω) = H ω ∧ Y ω + cavityMessage theta (X ω) = K ω ∧ X' ω + cavityMessage theta (Y' ω) = H' ω ∧ Y' ω + cavityMessage theta (X' ω) = K' ω) {V M R B₁ B₂ : ℝ} (hM : 0 ≤ M) (hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M) (hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V) (hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V) (hTail : μ.real {ω | R < |X ω|} ≤ B₁) (hTail' : μ.real {ω | R < |X' ω|} ≤ B₂) : (∫ ω, (Real.log (responseAttenuation theta (X ω)) - Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ cavityLogVarianceCoefficient theta * R ^ 2 * V + min (cavityLogVarianceCoefficient theta * V) (4 * (M / (1 - theta)) ^ 2 * (B₁ + B₂)) := by have hθsq : theta ^ 2 ≤ 1 := by nlinarith have hInt := integrable_cavity_pair_sub_sq hθ0 hθ1 hX hX' hH hK heq have hVar := integral_cavity_pair_sub_sq_le hθ0 hθ1 hX hX' hH hK heq hHV hKV have hBound : ∀ᵐ ω ∂μ, |X ω - X' ω| ≤ M / (1 - theta) := by filter_upwards [heq, hbound] with ω he hb exact cavity_pair_sub_abs_le hθ0 hθ1 he.1 he.2.1 he.2.2.1 he.2.2.2 hb.1 hb.2 have h := integral_log_responseAttenuation_sub_sq_le_of_marginal_tails hθsq hX hX' hInt (div_nonneg hM (by linarith)) hBound hVar hTail hTail' have hfirst : 4 * R ^ 2 * (V / (1 - theta) ^ 2) = cavityLogVarianceCoefficient theta * R ^ 2 * V := by unfold cavityLogVarianceCoefficient ring have hsecond : 4 * (V / (1 - theta) ^ 2) = cavityLogVarianceCoefficient theta * V := by unfold cavityLogVarianceCoefficient ring simpa only [hfirst, hsecond] using h theorem integral_cavity_log_discrepancy_small_time (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) (hX : Measurable X) (hX' : Measurable X') (hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ) (hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ) (heq : ∀ᵐ ω ∂μ, X ω + cavityMessage theta (Y ω) = H ω ∧ Y ω + cavityMessage theta (X ω) = K ω ∧ X' ω + cavityMessage theta (Y' ω) = H' ω ∧ Y' ω + cavityMessage theta (X' ω) = K' ω) {V M t c : ℝ} (hM : 0 ≤ M) (ht : 0 < t) (hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M) (hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V) (hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V) (hTail : μ.real {ω | t ^ (1 / 8 : ℝ) < |X ω|} ≤ Real.exp (-c * (Real.log (1 / t)) ^ 2)) (hTail' : μ.real {ω | t ^ (1 / 8 : ℝ) < |X' ω|} ≤ Real.exp (-c * (Real.log (1 / t)) ^ 2)) : (∫ ω, (Real.log (responseAttenuation theta (X ω)) - Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V + min (cavityLogVarianceCoefficient theta * V) (cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2)) := by have h := integral_cavity_log_discrepancy_sq_le_of_marginal_tails hθ0 hθ1 hX hX' hH hK heq hM hbound hHV hKV hTail hTail' have hr : (t ^ (1 / 8 : ℝ)) ^ 2 = t ^ (1 / 4 : ℝ) := by rw [← Real.rpow_natCast, ← Real.rpow_mul ht.le] norm_num have htail : 4 * (M / (1 - theta)) ^ 2 * (Real.exp (-c * (Real.log (1 / t)) ^ 2) + Real.exp (-c * (Real.log (1 / t)) ^ 2)) = cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2) := by unfold cavityLogTailCoefficient ring simpa only [hr, htail] using h theorem integral_cavity_log_discrepancy_small_time_of_moments (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) (hX : Measurable X) (hX' : Measurable X') (hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ) (hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ) (heq : ∀ᵐ ω ∂μ, X ω + cavityMessage theta (Y ω) = H ω ∧ Y ω + cavityMessage theta (X ω) = K ω ∧ X' ω + cavityMessage theta (Y' ω) = H' ω ∧ Y' ω + cavityMessage theta (X' ω) = K' ω) {V M t A : ℝ} (hM : 0 ≤ M) (ht : 0 < t) (hA : 0 < A) (hsmall : 32 * A ≤ Real.log (1 / t)) (hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M) (hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V) (hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V) (hInt : ∀ p : ℕ, 1 ≤ p → Integrable (fun ω => |X ω| ^ p) μ) (hInt' : ∀ p : ℕ, 1 ≤ p → Integrable (fun ω => |X' ω| ^ p) μ) (hmom : ∀ p : ℕ, 1 ≤ p → (∫ ω, |X ω| ^ p ∂μ) ≤ Real.exp (A * (p : ℝ) ^ 2) * t ^ ((p : ℝ) / 4)) (hmom' : ∀ p : ℕ, 1 ≤ p → (∫ ω, |X' ω| ^ p ∂μ) ≤ Real.exp (A * (p : ℝ) ^ 2) * t ^ ((p : ℝ) / 4)) : (∫ ω, (Real.log (responseAttenuation theta (X ω)) - Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V + min (cavityLogVarianceCoefficient theta * V) (cavityLogTailCoefficient theta M * Real.exp (-(1 / (512 * A)) * (Real.log (1 / t)) ^ 2)) := by have he : -(1 / (512 * A)) * (Real.log (1 / t)) ^ 2 = -((Real.log (1 / t)) ^ 2) / (512 * A) := by ring apply integral_cavity_log_discrepancy_small_time hθ0 hθ1 hX hX' hH hK heq hM ht hbound hHV hKV · rw [he] exact measureReal_abs_gt_rpow_le_of_moments hA ht hsmall hInt hmom · rw [he] exact measureReal_abs_gt_rpow_le_of_moments hA ht hsmall hInt' hmom' theorem integral_cavity_log_discrepancy_small_time_of_common_law {Ξ : Type*} [MeasurableSpace Ξ] {ν : Measure Ξ} {Z : Ξ → ℝ} (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) (hX : Measurable X) (hX' : Measurable X') (hH : Integrable (fun ω => (H ω - H' ω) ^ 2) μ) (hK : Integrable (fun ω => (K ω - K' ω) ^ 2) μ) (heq : ∀ᵐ ω ∂μ, X ω + cavityMessage theta (Y ω) = H ω ∧ Y ω + cavityMessage theta (X ω) = K ω ∧ X' ω + cavityMessage theta (Y' ω) = H' ω ∧ Y' ω + cavityMessage theta (X' ω) = K' ω) {V M t c : ℝ} (hM : 0 ≤ M) (ht : 0 < t) (hbound : ∀ᵐ ω ∂μ, |H ω - H' ω| ≤ M ∧ |K ω - K' ω| ≤ M) (hHV : (∫ ω, (H ω - H' ω) ^ 2 ∂μ) ≤ V) (hKV : (∫ ω, (K ω - K' ω) ^ 2 ∂μ) ≤ V) (hLaw : IdentDistrib X Z μ ν) (hLaw' : IdentDistrib X' Z μ ν) (hTail : ν {z | t ^ (1 / 8 : ℝ) < |Z z|} ≤ ENNReal.ofReal (Real.exp (-c * (Real.log (1 / t)) ^ 2))) : (∫ ω, (Real.log (responseAttenuation theta (X ω)) - Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V + min (cavityLogVarianceCoefficient theta * V) (cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2)) := by have transfer {W : Ω → ℝ} (hW : IdentDistrib W Z μ ν) : μ.real {ω | t ^ (1 / 8 : ℝ) < |W ω|} ≤ Real.exp (-c * (Real.log (1 / t)) ^ 2) := by apply ENNReal.toReal_le_of_le_ofReal (Real.exp_nonneg _) have hm := (hW.comp measurable_abs).measure_mem_eq (measurableSet_Ioi : MeasurableSet (Set.Ioi (t ^ (1 / 8 : ℝ)))) change μ {ω | t ^ (1 / 8 : ℝ) < |W ω|} = ν {z | t ^ (1 / 8 : ℝ) < |Z z|} at hm rw [hm] exact hTail exact integral_cavity_log_discrepancy_small_time hθ0 hθ1 hX hX' hH hK heq hM ht hbound hHV hKV (transfer hLaw) (transfer hLaw') omit [IsProbabilityMeasure μ] in theorem integral_cavity_log_discrepancy_osgood_of_small_time {V M t c L : ℝ} (hc : 0 < c) (hL : 1 ≤ L) (hV : 0 < V) (hVL : V ≤ L) (ht : 0 < t) (ht1 : t < 1) (hsmall : (∫ ω, (Real.log (responseAttenuation theta (X ω)) - Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ cavityLogVarianceCoefficient theta * t ^ (1 / 4 : ℝ) * V + min (cavityLogVarianceCoefficient theta * V) (cavityLogTailCoefficient theta M * Real.exp (-c * (Real.log (1 / t)) ^ 2))) : (∫ ω, (Real.log (responseAttenuation theta (X ω)) - Real.log (responseAttenuation theta (X' ω))) ^ 2 ∂μ) ≤ (cavityLogVarianceCoefficient theta + 2 * (cavityLogVarianceCoefficient theta + cavityLogTailCoefficient theta M) / c) * V * (t ^ (1 / 4 : ℝ) + (1 + Real.log (L / V)) / (Real.log (1 / t)) ^ 2) := by exact small_time_log_loss_of_minimum (cavityLogVarianceCoefficient_nonneg theta) (cavityLogTailCoefficient_nonneg theta M) hc hL hV hVL ht ht1 hsmall end Coupling end Problem367 end end OAI