| import OAI.Probability.FreeIsing.Gaussian.Moments | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace Problem367 | |
| theorem abs_le_cosh_mul_abs_tanh {y R : ℝ} (hR : 0 ≤ R) (hy : |y| ≤ R) : | |
| |y| ≤ Real.cosh R * |Real.tanh y| := by | |
| calc | |
| |y| ≤ Real.sinh |y| := Real.self_le_sinh_iff.mpr (abs_nonneg y) | |
| _ = |Real.sinh y| := (Real.abs_sinh y).symm | |
| _ = Real.cosh y * |Real.tanh y| := by | |
| rw [Real.tanh_eq_sinh_div_cosh, abs_div, abs_of_pos (Real.cosh_pos y)] | |
| field_simp | |
| _ ≤ Real.cosh R * |Real.tanh y| := by | |
| apply mul_le_mul_of_nonneg_right _ (abs_nonneg _) | |
| exact Real.cosh_le_cosh.mpr (by simpa only [abs_of_nonneg hR] using hy) | |
| theorem sq_le_tanh_sq_add_fourth {y x B : ℝ} (hB : 0 ≤ B) (hyx : |y - x| ≤ B) : | |
| y ^ 2 ≤ Real.cosh (B + 1) ^ 2 * Real.tanh y ^ 2 + | |
| 2 * (1 + B ^ 2) * |x| ^ 4 := by | |
| by_cases hx : |x| ≤ 1 | |
| · have hy : |y| ≤ B + 1 := by | |
| calc | |
| |y| = |y - x + x| := by congr 1; ring | |
| _ ≤ |y - x| + |x| := abs_add_le _ _ | |
| _ ≤ B + 1 := add_le_add hyx hx | |
| have hs := pow_le_pow_left₀ (abs_nonneg y) | |
| (abs_le_cosh_mul_abs_tanh (by linarith : 0 ≤ B + 1) hy) 2 | |
| simp only [mul_pow, sq_abs] at hs | |
| exact hs.trans (le_add_of_nonneg_right (by positivity)) | |
| · have hx1 : 1 < |x| := lt_of_not_ge hx | |
| have hx2 : 1 ≤ |x| ^ 2 := by nlinarith | |
| have hx4 : |x| ^ 2 ≤ |x| ^ 4 := by nlinarith [sq_nonneg (|x| ^ 2 - 1)] | |
| have hdiff := pow_le_pow_left₀ (abs_nonneg (y - x)) hyx 2 | |
| simp only [sq_abs] at hdiff | |
| have hY : y ^ 2 ≤ 2 * B ^ 2 + 2 * x ^ 2 := by nlinarith [sq_nonneg (y - 2 * x)] | |
| have hB4 : B ^ 2 ≤ B ^ 2 * |x| ^ 4 := by | |
| simpa using mul_le_mul_of_nonneg_left (hx2.trans hx4) (sq_nonneg B) | |
| have hmain : y ^ 2 ≤ 2 * (1 + B ^ 2) * |x| ^ 4 := by | |
| nlinarith [sq_abs x] | |
| exact hmain.trans (le_add_of_nonneg_left (by positivity)) | |
| theorem integrable_tanh_sq {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} | |
| [IsFiniteMeasure μ] {Y : Ω → ℝ} (hY : AEMeasurable Y μ) : | |
| Integrable (fun ω => Real.tanh (Y ω) ^ 2) μ := by | |
| have hm : AEMeasurable (fun ω => Real.tanh (Y ω)) μ := by | |
| simp_rw [Real.tanh_eq_sinh_div_cosh] | |
| exact (Real.continuous_sinh.measurable.comp_aemeasurable hY).div | |
| (Real.continuous_cosh.measurable.comp_aemeasurable hY) | |
| apply (integrable_const (1 : ℝ)).mono' (hm.pow_const 2).aestronglyMeasurable | |
| filter_upwards [] with ω | |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg (Real.tanh (Y ω)))] | |
| exact (Real.tanh_sq_lt_one (Y ω)).le | |
| theorem integral_sq_le_tanh_sq_add_fourth {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure Ω} [IsFiniteMeasure μ] {Y X : Ω → ℝ} {B : ℝ} | |
| (hB : 0 ≤ B) (hY : AEMeasurable Y μ) | |
| (hX4 : Integrable (fun ω => |X ω| ^ 4) μ) | |
| (hbound : ∀ᵐ ω ∂μ, |Y ω - X ω| ≤ B) : | |
| Integrable (fun ω => Y ω ^ 2) μ ∧ | |
| (∫ ω, Y ω ^ 2 ∂μ) ≤ Real.cosh (B + 1) ^ 2 * (∫ ω, Real.tanh (Y ω) ^ 2 ∂μ) + | |
| 2 * (1 + B ^ 2) * (∫ ω, |X ω| ^ 4 ∂μ) := by | |
| have hT := integrable_tanh_sq hY | |
| have hdom := (hT.const_mul (Real.cosh (B + 1) ^ 2)).add | |
| (hX4.const_mul (2 * (1 + B ^ 2))) | |
| have hle : ∀ᵐ ω ∂μ, Y ω ^ 2 ≤ | |
| Real.cosh (B + 1) ^ 2 * Real.tanh (Y ω) ^ 2 + | |
| 2 * (1 + B ^ 2) * |X ω| ^ 4 := by | |
| filter_upwards [hbound] with ω hω | |
| exact sq_le_tanh_sq_add_fourth hB hω | |
| have hint : Integrable (fun ω => Y ω ^ 2) μ := by | |
| apply hdom.mono' (hY.pow_const 2).aestronglyMeasurable | |
| filter_upwards [hle] with ω hω | |
| simpa only [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg (Y ω)), Pi.add_apply] using hω | |
| refine ⟨hint, ?_⟩ | |
| calc | |
| (∫ ω, Y ω ^ 2 ∂μ) ≤ ∫ ω, | |
| Real.cosh (B + 1) ^ 2 * Real.tanh (Y ω) ^ 2 + | |
| 2 * (1 + B ^ 2) * |X ω| ^ 4 ∂μ := integral_mono_ae hint hdom hle | |
| _ = _ := by rw [integral_add (hT.const_mul _) (hX4.const_mul _), integral_const_mul, | |
| integral_const_mul] | |
| theorem integral_abs_le_sqrt_integral_sq {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure Ω} [IsProbabilityMeasure μ] {Y : Ω → ℝ} | |
| (hY : AEMeasurable Y μ) (hY2 : Integrable (fun ω => Y ω ^ 2) μ) : | |
| (∫ ω, |Y ω| ∂μ) ≤ Real.sqrt (∫ ω, Y ω ^ 2 ∂μ) := by | |
| have hm : MemLp Y 2 μ := (memLp_two_iff_integrable_sq hY.aestronglyMeasurable).mpr hY2 | |
| have hma : MemLp (fun ω => |Y ω|) 2 μ := by simpa only [Real.norm_eq_abs] using hm.norm | |
| have hv := variance_nonneg (fun ω => |Y ω|) μ | |
| rw [variance_eq_sub hma] at hv | |
| simp only [Pi.pow_apply, sq_abs] at hv | |
| exact Real.le_sqrt_of_sq_le (by linarith) | |
| theorem integrable_abs_pow_of_bounded_perturbation {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure Ω} [IsFiniteMeasure μ] {Y X : Ω → ℝ} {B : ℝ} (p : ℕ) | |
| (hB : 0 ≤ B) (hY : AEMeasurable Y μ) | |
| (hX : Integrable (fun ω => |X ω| ^ p) μ) | |
| (hbound : ∀ᵐ ω ∂μ, |Y ω - X ω| ≤ B) : | |
| Integrable (fun ω => |Y ω| ^ p) μ := by | |
| apply ((hX.add (integrable_const (B ^ p))).const_mul (2 ^ (p - 1) : ℝ)).mono' | |
| (hY.abs.pow_const p).aestronglyMeasurable | |
| filter_upwards [hbound] with ω hω | |
| change ‖|Y ω| ^ p‖ ≤ 2 ^ (p - 1) * (|X ω| ^ p + B ^ p) | |
| rw [Real.norm_eq_abs, abs_of_nonneg (pow_nonneg (abs_nonneg _) p)] | |
| have hsum : |Y ω| ≤ |X ω| + B := by | |
| have htriangle := abs_add_le (Y ω - X ω) (X ω) | |
| have heq : Y ω - X ω + X ω = Y ω := by ring | |
| rw [heq] at htriangle | |
| linarith | |
| exact (pow_le_pow_left₀ (abs_nonneg _) hsum p).trans (add_pow_le (abs_nonneg _) hB p) | |
| theorem integrable_cavity_abs_pow_of_gaussian_bound {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure Ω} [IsProbabilityMeasure μ] {G Y : Ω → ℝ} {B t : ℝ} | |
| (hG : HasLaw G (gaussianReal 0 1) μ) (hY : AEMeasurable Y μ) | |
| (hB : 0 ≤ B) (ht : 0 ≤ t) (ht1 : t ≤ 1) | |
| (hbound : ∀ᵐ ω ∂μ, |Y ω - (t + Real.sqrt t * G ω)| ≤ B) (p : ℕ) : | |
| Integrable (fun ω => |Y ω| ^ p) μ := | |
| integrable_abs_pow_of_bounded_perturbation p hB hY | |
| (integrable_gaussian_observation_abs_pow_of_hasLaw hG ht ht1 p) hbound | |
| def cavityBaseConstant (B C : ℝ) : ℝ := | |
| Real.cosh (B + 1) ^ 2 * C + 2 * (1 + B ^ 2) * Real.exp 32 | |
| theorem cavityBaseConstant_pos {B C : ℝ} (hC : 0 ≤ C) : | |
| 0 < cavityBaseConstant B C := by | |
| unfold cavityBaseConstant | |
| positivity | |
| theorem cavity_base_second_moment {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure Ω} [IsProbabilityMeasure μ] {G Y : Ω → ℝ} {B C t : ℝ} | |
| (hG : HasLaw G (gaussianReal 0 1) μ) (hY : AEMeasurable Y μ) | |
| (hB : 0 ≤ B) (ht : 0 < t) (ht1 : t ≤ 1) | |
| (hbound : ∀ᵐ ω ∂μ, |Y ω - (t + Real.sqrt t * G ω)| ≤ B) | |
| (hinfo : (∫ ω, Real.tanh (Y ω) ^ 2 ∂μ) ≤ C * Real.sqrt t) : | |
| Integrable (fun ω => Y ω ^ 2) μ ∧ | |
| (∫ ω, Y ω ^ 2 ∂μ) ≤ cavityBaseConstant B C * Real.sqrt t := by | |
| have hX4 := integrable_gaussian_observation_abs_pow_of_hasLaw hG ht.le ht1 4 | |
| have hX4bound := integral_gaussian_observation_abs_pow_le_of_hasLaw | |
| hG (p := 4) ht ht1 (by decide) | |
| norm_num at hX4bound | |
| obtain ⟨hint, hsecond⟩ := integral_sq_le_tanh_sq_add_fourth hB hY hX4 hbound | |
| refine ⟨hint, hsecond.trans ?_⟩ | |
| calc | |
| Real.cosh (B + 1) ^ 2 * (∫ ω, Real.tanh (Y ω) ^ 2 ∂μ) + | |
| 2 * (1 + B ^ 2) * (∫ ω, |t + Real.sqrt t * G ω| ^ 4 ∂μ) ≤ | |
| Real.cosh (B + 1) ^ 2 * (C * Real.sqrt t) + | |
| 2 * (1 + B ^ 2) * (Real.exp 32 * t) := by | |
| exact add_le_add (mul_le_mul_of_nonneg_left hinfo (sq_nonneg _)) | |
| (mul_le_mul_of_nonneg_left hX4bound (by positivity)) | |
| _ = Real.cosh (B + 1) ^ 2 * C * Real.sqrt t + | |
| (2 * (1 + B ^ 2) * Real.exp 32) * t := by ring | |
| _ ≤ Real.cosh (B + 1) ^ 2 * C * Real.sqrt t + | |
| (2 * (1 + B ^ 2) * Real.exp 32) * Real.sqrt t := by | |
| exact add_le_add le_rfl (mul_le_mul_of_nonneg_left | |
| (Real.le_sqrt_self_iff.mpr ht1) (by positivity)) | |
| _ = cavityBaseConstant B C * Real.sqrt t := by unfold cavityBaseConstant; ring | |
| theorem cavity_base_first_moment {Ω : Type*} [MeasurableSpace Ω] | |
| {μ : Measure Ω} [IsProbabilityMeasure μ] {G Y : Ω → ℝ} {B C t : ℝ} | |
| (hG : HasLaw G (gaussianReal 0 1) μ) (hY : AEMeasurable Y μ) | |
| (hB : 0 ≤ B) (hC : 0 ≤ C) (ht : 0 < t) (ht1 : t ≤ 1) | |
| (hbound : ∀ᵐ ω ∂μ, |Y ω - (t + Real.sqrt t * G ω)| ≤ B) | |
| (hinfo : (∫ ω, Real.tanh (Y ω) ^ 2 ∂μ) ≤ C * Real.sqrt t) : | |
| (∫ ω, |Y ω| ∂μ) ≤ Real.sqrt (cavityBaseConstant B C) * t ^ (1 / 4 : ℝ) := by | |
| obtain ⟨hint, hsecond⟩ := cavity_base_second_moment hG hY hB ht ht1 hbound hinfo | |
| have hquarter : Real.sqrt (Real.sqrt t) = t ^ (1 / 4 : ℝ) := by | |
| rw [Real.sqrt_eq_rpow, Real.sqrt_eq_rpow, ← Real.rpow_mul ht.le] | |
| congr 1 | |
| norm_num | |
| calc | |
| (∫ ω, |Y ω| ∂μ) ≤ Real.sqrt (∫ ω, Y ω ^ 2 ∂μ) := | |
| integral_abs_le_sqrt_integral_sq hY hint | |
| _ ≤ Real.sqrt (cavityBaseConstant B C * Real.sqrt t) := | |
| Real.sqrt_le_sqrt hsecond | |
| _ = Real.sqrt (cavityBaseConstant B C) * t ^ (1 / 4 : ℝ) := by | |
| rw [Real.sqrt_mul (cavityBaseConstant_pos hC).le, hquarter] | |
| end Problem367 | |
| end | |
| end OAI | |