import OAI.Probability.FreeIsing.Energy.DiscrepancyRowEnergy import OAI.Probability.FreeIsing.Estimates.RadialSummability import OAI.Probability.FreeIsing.Trees.CoupledPathDrift import OAI.Probability.FreeIsing.Trees.SubcriticalPathSums namespace OAI noncomputable section open MeasureTheory open scoped ENNReal namespace Problem367 theorem abs_tanh_weight_drift_le (h h' p p' : ℝ) : |(h - h') * (Real.tanh h * p ^ 2 - Real.tanh h' * p' ^ 2)| ≤ (h - h') ^ 2 * p ^ 2 + |h - h'| * |p ^ 2 - p' ^ 2| := by have hdiff : |Real.tanh h - Real.tanh h'| ≤ |h - h'| := by simpa only [Real.dist_eq, NNReal.coe_one, one_mul] using lipschitzWith_tanh_response.dist_le_mul h h' have hm : |Real.tanh h'| ≤ 1 := by have hsq := Real.tanh_sq_lt_one h' nlinarith [sq_abs (Real.tanh h'), abs_nonneg (Real.tanh h')] have hsplit : Real.tanh h * p ^ 2 - Real.tanh h' * p' ^ 2 = (Real.tanh h - Real.tanh h') * p ^ 2 + Real.tanh h' * (p ^ 2 - p' ^ 2) := by ring rw [abs_mul, hsplit] calc |h - h'| * |(Real.tanh h - Real.tanh h') * p ^ 2 + Real.tanh h' * (p ^ 2 - p' ^ 2)| ≤ |h - h'| * (|(Real.tanh h - Real.tanh h') * p ^ 2| + |Real.tanh h' * (p ^ 2 - p' ^ 2)|) := mul_le_mul_of_nonneg_left (abs_add_le _ _) (abs_nonneg _) _ = |h - h'| * (|Real.tanh h - Real.tanh h'| * p ^ 2 + |Real.tanh h'| * |p ^ 2 - p' ^ 2|) := by rw [abs_mul, abs_mul, abs_of_nonneg (sq_nonneg p)] _ ≤ |h - h'| * (|h - h'| * p ^ 2 + |p ^ 2 - p' ^ 2|) := by apply mul_le_mul_of_nonneg_left _ (abs_nonneg _) apply add_le_add (mul_le_mul_of_nonneg_right hdiff (sq_nonneg p)) exact mul_le_of_le_one_left (abs_nonneg _) hm _ = _ := by rw [mul_add, ← mul_assoc, ← sq, sq_abs] section Probability variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} theorem integral_abs_tanh_path_drift_le {ι : Type*} (s : Finset ι) {theta : ℝ} (hθ : theta ^ 2 ≤ 1) {X Y : ι → Ω → ℝ} {H H' : Ω → ℝ} (hX : ∀ i ∈ s, Measurable (X i)) (hY : ∀ i ∈ s, Measurable (Y i)) (hInt : ∀ i ∈ s, Integrable (fun ω => (X i ω - Y i ω) ^ 2) μ) (hH : Measurable H) (hH' : Measurable H') (hU : MemLp (fun ω => H ω - H' ω) 2 μ) : let P := fun ω => ∏ i ∈ s, responseAttenuation theta (X i ω) let P' := fun ω => ∏ i ∈ s, responseAttenuation theta (Y i ω) let Z := fun ω => ∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) - Real.log (responseAttenuation theta (Y i ω))) Integrable (fun ω => (H ω - H' ω) * (Real.tanh (H ω) * P ω ^ 2 - Real.tanh (H' ω) * P' ω ^ 2)) μ ∧ (∫ ω, |(H ω - H' ω) * (Real.tanh (H ω) * P ω ^ 2 - Real.tanh (H' ω) * P' ω ^ 2)| ∂μ) ≤ (∫ ω, (H ω - H' ω) ^ 2 * P ω ^ 2 ∂μ) + 2 * Real.sqrt (2 * ∫ ω, (H ω - H' ω) ^ 2 ∂μ) * Real.sqrt (∫ ω, (P ω ^ 2 + P' ω ^ 2) * Z ω ^ 2 ∂μ) := by dsimp only let U := fun ω => H ω - H' ω let P := fun ω => ∏ i ∈ s, responseAttenuation theta (X i ω) let P' := fun ω => ∏ i ∈ s, responseAttenuation theta (Y i ω) let Z := fun ω => ∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) - Real.log (responseAttenuation theta (Y i ω))) let D := fun ω => U ω * (Real.tanh (H ω) * P ω ^ 2 - Real.tanh (H' ω) * P' ω ^ 2) have hP : Measurable P := measurable_prod_responseAttenuation s theta hX have hP' : Measurable P' := measurable_prod_responseAttenuation s theta hY have hL := (integral_mul_prod_responseAttenuation_sq_le s hθ hX hU).1 have hB := integral_abs_mul_prod_responseAttenuation_sq_sub_sq_le s hθ hX hY hInt hU have htanh := lipschitzWith_tanh_response.continuous.measurable have hDm : Measurable D := (hH.sub hH').mul (((htanh.comp hH).mul (hP.pow_const 2)).sub ((htanh.comp hH').mul (hP'.pow_const 2))) have hpoint (ω : Ω) : |D ω| ≤ U ω ^ 2 * P ω ^ 2 + |U ω| * |P ω ^ 2 - P' ω ^ 2| := abs_tanh_weight_drift_le (H ω) (H' ω) (P ω) (P' ω) have hD : Integrable D μ := by apply (hL.add hB.1).mono' hDm.aestronglyMeasurable filter_upwards [] with ω exact (show ‖D ω‖ ≤ _ by simpa only [Real.norm_eq_abs, Pi.add_apply, U, P, P'] using hpoint ω) refine ⟨hD, ?_⟩ calc (∫ ω, |D ω| ∂μ) ≤ ∫ ω, U ω ^ 2 * P ω ^ 2 + |U ω| * |P ω ^ 2 - P' ω ^ 2| ∂μ := integral_mono hD.abs (hL.add hB.1) hpoint _ = (∫ ω, U ω ^ 2 * P ω ^ 2 ∂μ) + ∫ ω, |U ω| * |P ω ^ 2 - P' ω ^ 2| ∂μ := integral_add hL hB.1 _ ≤ _ := add_le_add le_rfl hB.2 theorem integrable_series_of_summable_integral_abs {ι : Type*} [Countable ι] {F : ι → Ω → ℝ} (hF : ∀ i, Integrable (F i) μ) (hs : Summable (fun i => ∫ ω, |F i ω| ∂μ)) : (∀ᵐ ω ∂μ, Summable (fun i => |F i ω|)) ∧ Integrable (fun ω => ∑' i, F i ω) μ ∧ (∫ ω, (∑' i, F i ω) ∂μ) = ∑' i, ∫ ω, F i ω ∂μ := by have hm (i : ι) : AEMeasurable (fun ω => ENNReal.ofReal |F i ω|) μ := (hF i).abs.aestronglyMeasurable.aemeasurable.ennreal_ofReal have hfin : (∫⁻ ω, ∑' i, ENNReal.ofReal |F i ω| ∂μ) < ∞ := by rw [lintegral_tsum hm] calc (∑' i, ∫⁻ ω, ENNReal.ofReal |F i ω| ∂μ) = ∑' i, ENNReal.ofReal (∫ ω, |F i ω| ∂μ) := by apply tsum_congr intro i exact (ofReal_integral_eq_lintegral_ofReal (hF i).abs (Filter.Eventually.of_forall (fun ω => abs_nonneg (F i ω)))).symm _ < ∞ := hs.tsum_ofReal_lt_top have hAe : ∀ᵐ ω ∂μ, Summable (fun i => |F i ω|) := by filter_upwards [ae_lt_top' (AEMeasurable.tsum hm) hfin.ne] with ω hω simpa only [ENNReal.toReal_ofReal (abs_nonneg _)] using ENNReal.summable_toReal hω.ne have hIabs : Integrable (fun ω => ∑' i, |F i ω|) μ := by have h := integrable_toReal_of_lintegral_ne_top (AEMeasurable.tsum hm) hfin.ne simpa only [ENNReal.tsum_toReal_eq (fun i => ENNReal.ofReal_ne_top), ENNReal.toReal_ofReal (abs_nonneg _)] using h refine ⟨hAe, ?_, ?_⟩ · apply hIabs.mono' (AEStronglyMeasurable.tsum (fun i => (hF i).aestronglyMeasurable)) filter_upwards [hAe] with ω hω simpa only [Real.norm_eq_abs] using norm_tsum_le_tsum_norm (show Summable (fun i => ‖F i ω‖) by simpa only [Real.norm_eq_abs] using hω) · exact (integral_tsum_of_summable_integral_norm hF (by simpa only [Real.norm_eq_abs] using hs)).symm theorem integral_series_le_of_abs_majorant {ι : Type*} [Countable ι] {F : ι → Ω → ℝ} {B : ι → ℝ} (hF : ∀ i, Integrable (F i) μ) (hB : Summable B) (hbound : ∀ i, (∫ ω, |F i ω| ∂μ) ≤ B i) : Integrable (fun ω => ∑' i, F i ω) μ ∧ |∫ ω, (∑' i, F i ω) ∂μ| ≤ ∑' i, B i := by have hs : Summable (fun i => ∫ ω, |F i ω| ∂μ) := Summable.of_nonneg_of_le (fun i => integral_nonneg (fun ω => abs_nonneg _)) hbound hB have hseries := integrable_series_of_summable_integral_abs hF hs refine ⟨hseries.2.1, ?_⟩ rw [hseries.2.2] have hnorm : Summable (fun i => |∫ ω, F i ω ∂μ|) := Summable.of_nonneg_of_le (fun i => abs_nonneg _) (fun i => abs_integral_le_integral_abs) hs calc |∑' i, ∫ ω, F i ω ∂μ| ≤ ∑' i, |∫ ω, F i ω ∂μ| := by simpa only [Real.norm_eq_abs] using norm_tsum_le_tsum_norm (show Summable (fun i => ‖∫ ω, F i ω ∂μ‖) by simpa only [Real.norm_eq_abs] using hnorm) _ ≤ ∑' i, B i := hnorm.tsum_le_tsum (fun i => abs_integral_le_integral_abs.trans (hbound i)) hB theorem integral_response_row_density_le {ι : Type*} [Countable ι] {c c' : ι → Ω → ℝ} {U m m' : Ω → ℝ} {N D : ι → ℝ} (hnoise : ∀ i, Integrable (fun ω => (c i ω - c' i ω) ^ 2) μ) (hdrift : ∀ i, Integrable (fun ω => U ω * (m ω * (c i ω) ^ 2 - m' ω * (c' i ω) ^ 2)) μ) (hN : Summable N) (hD : Summable D) (hNb : ∀ i, (∫ ω, (c i ω - c' i ω) ^ 2 ∂μ) ≤ N i) (hDb : ∀ i, (∫ ω, |U ω * (m ω * (c i ω) ^ 2 - m' ω * (c' i ω) ^ 2)| ∂μ) ≤ D i) (hrows : ∀ᵐ ω ∂μ, Summable (fun i => (c i ω) ^ 2) ∧ Summable (fun i => (c' i ω) ^ 2)) : Integrable (DiscrepancyEnergy.density c c' U m m') μ ∧ (∫ ω, DiscrepancyEnergy.density c c' U m m' ω ∂μ) ≤ (∑' i, N i) + 2 * ∑' i, D i := by let A := fun ω => ∑' i, (c i ω - c' i ω) ^ 2 let B := fun ω => ∑' i, U ω * (m ω * (c i ω) ^ 2 - m' ω * (c' i ω) ^ 2) have hAb : ∀ i, (∫ ω, |(c i ω - c' i ω) ^ 2| ∂μ) ≤ N i := by simpa only [abs_sq] using hNb have hA := integral_series_le_of_abs_majorant hnoise hN hAb have hB := integral_series_le_of_abs_majorant hdrift hD hDb have heq : DiscrepancyEnergy.density c c' U m m' =ᵐ[μ] (fun ω => A ω + 2 * B ω) := by filter_upwards [hrows] with ω hω have hs : Summable (fun i => (U ω * m ω) * (c i ω) ^ 2) := hω.1.mul_left (U ω * m ω) have hs' : Summable (fun i => (U ω * m' ω) * (c' i ω) ^ 2) := hω.2.mul_left (U ω * m' ω) have hBω : B ω = U ω * (m ω * (∑' i, (c i ω) ^ 2) - m' ω * (∑' i, (c' i ω) ^ 2)) := by calc B ω = ∑' i, ((U ω * m ω) * (c i ω) ^ 2 - (U ω * m' ω) * (c' i ω) ^ 2) := by apply tsum_congr intro i ring _ = _ := by rw [hs.tsum_sub hs', tsum_mul_left, tsum_mul_left]; ring dsimp only [DiscrepancyEnergy.density, A] rw [hBω] ring have hint : Integrable (DiscrepancyEnergy.density c c' U m m') μ := (hA.1.add (hB.1.const_mul 2)).congr heq.symm refine ⟨hint, ?_⟩ rw [integral_congr_ae heq, integral_add hA.1 (hB.1.const_mul 2), integral_const_mul] exact add_le_add ((le_abs_self _).trans hA.2) (mul_le_mul_of_nonneg_left ((le_abs_self _).trans hB.2) (by norm_num)) namespace ResponseRows def pathProduct {ι : Type*} (theta : ℝ) (s : Finset ι) (X : ι → Ω → ℝ) (ω : Ω) : ℝ := ∏ i ∈ s, responseAttenuation theta (X i ω) def pathLogDiff {ι : Type*} (theta : ℝ) (s : Finset ι) (X Y : ι → Ω → ℝ) (ω : Ω) : ℝ := ∑ i ∈ s, (Real.log (responseAttenuation theta (X i ω)) - Real.log (responseAttenuation theta (Y i ω))) def pathNoiseEnergy {ι : Type*} (μ : Measure Ω) (theta : ℝ) (s : Finset ι) (X Y : ι → Ω → ℝ) : ℝ := ∫ ω, (pathProduct theta s X ω ^ 2 + pathProduct theta s Y ω ^ 2) * pathLogDiff theta s X Y ω ^ 2 ∂μ def pathLinearEnergy {ι : Type*} (μ : Measure Ω) (theta : ℝ) (s : Finset ι) (X : ι → Ω → ℝ) (U : Ω → ℝ) : ℝ := ∫ ω, U ω ^ 2 * pathProduct theta s X ω ^ 2 ∂μ def coefficient {J ι : Type*} (theta : ℝ) (a : J → ℝ) (s : J → Finset ι) (X : J → ι → Ω → ℝ) (j : J) (ω : Ω) : ℝ := a j * pathProduct theta (s j) (X j) ω theorem integral_density_le_path_energies {J ι : Type*} [Countable J] (s : J → Finset ι) (a : J → ℝ) {theta : ℝ} (hθ : theta ^ 2 ≤ 1) {X Y : J → ι → Ω → ℝ} {H H' : Ω → ℝ} (hX : ∀ j i, i ∈ s j → Measurable (X j i)) (hY : ∀ j i, i ∈ s j → Measurable (Y j i)) (hInt : ∀ j i, i ∈ s j → Integrable (fun ω => (X j i ω - Y j i ω) ^ 2) μ) (hH : Measurable H) (hH' : Measurable H') (hU : MemLp (fun ω => H ω - H' ω) 2 μ) (hQsum : Summable (fun j => (a j) ^ 2 * pathNoiseEnergy μ theta (s j) (X j) (Y j))) (hLsum : Summable (fun j => (a j) ^ 2 * pathLinearEnergy μ theta (s j) (X j) (fun ω => H ω - H' ω))) (hRsum : Summable (fun j => (a j) ^ 2 * Real.sqrt (pathNoiseEnergy μ theta (s j) (X j) (Y j)))) (hrows : ∀ᵐ ω ∂μ, Summable (fun j => (coefficient theta a s X j ω) ^ 2) ∧ Summable (fun j => (coefficient theta a s Y j ω) ^ 2)) : Integrable (DiscrepancyEnergy.density (coefficient theta a s X) (coefficient theta a s Y) (fun ω => H ω - H' ω) (fun ω => Real.tanh (H ω)) (fun ω => Real.tanh (H' ω))) μ ∧ (∫ ω, DiscrepancyEnergy.density (coefficient theta a s X) (coefficient theta a s Y) (fun ω => H ω - H' ω) (fun ω => Real.tanh (H ω)) (fun ω => Real.tanh (H' ω)) ω ∂μ) ≤ (∑' j, (a j) ^ 2 * pathNoiseEnergy μ theta (s j) (X j) (Y j)) + 2 * (∑' j, (a j) ^ 2 * pathLinearEnergy μ theta (s j) (X j) (fun ω => H ω - H' ω)) + 4 * Real.sqrt (2 * ∫ ω, (H ω - H' ω) ^ 2 ∂μ) * (∑' j, (a j) ^ 2 * Real.sqrt (pathNoiseEnergy μ theta (s j) (X j) (Y j))) := by let U := fun ω => H ω - H' ω let P := fun j => pathProduct theta (s j) (X j) let P' := fun j => pathProduct theta (s j) (Y j) let Q := fun j => pathNoiseEnergy μ theta (s j) (X j) (Y j) let L := fun j => pathLinearEnergy μ theta (s j) (X j) U let V := ∫ ω, U ω ^ 2 ∂μ let K := 2 * Real.sqrt (2 * V) let F := fun j ω => U ω * (Real.tanh (H ω) * P j ω ^ 2 - Real.tanh (H' ω) * P' j ω ^ 2) have hfiniteN (j : J) := integral_prod_responseAttenuation_sub_sq_le (s j) hθ (hX j) (hY j) (hInt j) have hfiniteD (j : J) := integral_abs_tanh_path_drift_le (s j) hθ (hX j) (hY j) (hInt j) hH hH' hU have hnoiseEq (j : J) (ω : Ω) : (coefficient theta a s X j ω - coefficient theta a s Y j ω) ^ 2 = (a j) ^ 2 * (P j ω - P' j ω) ^ 2 := by dsimp only [coefficient, P, P'] ring have hdriftEq (j : J) (ω : Ω) : U ω * (Real.tanh (H ω) * coefficient theta a s X j ω ^ 2 - Real.tanh (H' ω) * coefficient theta a s Y j ω ^ 2) = (a j) ^ 2 * F j ω := by dsimp only [coefficient, F, P, P'] ring have hnoise (j : J) : Integrable (fun ω => (coefficient theta a s X j ω - coefficient theta a s Y j ω) ^ 2) μ := by simp_rw [hnoiseEq] exact (hfiniteN j).2.1.const_mul _ have hdrift (j : J) : Integrable (fun ω => U ω * (Real.tanh (H ω) * coefficient theta a s X j ω ^ 2 - Real.tanh (H' ω) * coefficient theta a s Y j ω ^ 2)) μ := by simp_rw [hdriftEq] exact (hfiniteD j).1.const_mul _ have hNb (j : J) : (∫ ω, (coefficient theta a s X j ω - coefficient theta a s Y j ω) ^ 2 ∂μ) ≤ (a j) ^ 2 * Q j := by simp_rw [hnoiseEq] rw [integral_const_mul] exact mul_le_mul_of_nonneg_left (hfiniteN j).2.2 (sq_nonneg _) have hDb (j : J) : (∫ ω, |U ω * (Real.tanh (H ω) * coefficient theta a s X j ω ^ 2 - Real.tanh (H' ω) * coefficient theta a s Y j ω ^ 2)| ∂μ) ≤ (a j) ^ 2 * (L j + K * Real.sqrt (Q j)) := by simp_rw [hdriftEq, abs_mul, abs_sq] rw [integral_const_mul] exact mul_le_mul_of_nonneg_left (hfiniteD j).2 (sq_nonneg _) have hDsum : Summable (fun j => (a j) ^ 2 * (L j + K * Real.sqrt (Q j))) := by convert hLsum.add (hRsum.mul_left K) using 1 funext j dsimp only [L, Q, U] ring have hb := integral_response_row_density_le hnoise hdrift hQsum hDsum hNb hDb hrows refine ⟨hb.1, hb.2.trans_eq ?_⟩ have hsumD : (∑' j, (a j) ^ 2 * (L j + K * Real.sqrt (Q j))) = (∑' j, (a j) ^ 2 * L j) + K * ∑' j, (a j) ^ 2 * Real.sqrt (Q j) := by calc _ = ∑' j, ((a j) ^ 2 * L j + K * ((a j) ^ 2 * Real.sqrt (Q j))) := by apply tsum_congr intro j ring _ = _ := by rw [hLsum.tsum_add (hRsum.mul_left K), tsum_mul_left] rw [hsumD] dsimp only [K, V, Q, L, U] ring end ResponseRows end Probability end Problem367 end end OAI