| import OAI.Probability.FreeIsing.Estimates.ExponentialTaylor | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| namespace Problem367 | |
| theorem norm_integral_weighted_exp_remainder_le | |
| {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) | |
| (Z : Ω → ℂ) (D : Ω → ℝ) | |
| (hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1) | |
| (hD3 : Integrable (fun ω => |D ω| ^ 3) μ) : | |
| ‖∫ ω, Z ω * (Complex.exp ((D ω : ℂ) * Complex.I) - 1 - | |
| (D ω : ℂ) * Complex.I + (D ω : ℂ) ^ 2 / 2) ∂μ‖ ≤ | |
| 4 * ∫ ω, |D ω| ^ 3 ∂μ := by | |
| rw [← integral_const_mul] | |
| apply norm_integral_le_of_norm_le (hD3.const_mul 4) | |
| filter_upwards [hZ] with ω hω | |
| rw [norm_mul] | |
| calc | |
| _ ≤ 1 * ‖Complex.exp ((D ω : ℂ) * Complex.I) - 1 - | |
| (D ω : ℂ) * Complex.I + (D ω : ℂ) ^ 2 / 2‖ := | |
| mul_le_mul_of_nonneg_right hω (norm_nonneg _) | |
| _ ≤ 4 * |D ω| ^ 3 := by | |
| simpa only [one_mul] using ExponentialTaylor.norm_exp_I_taylor_two_le (D ω) | |
| theorem norm_integral_weighted_characteristic_step | |
| {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ] | |
| (Z : Ω → ℂ) (D : Ω → ℝ) (h : ℝ) | |
| (hZm : AEStronglyMeasurable Z μ) (hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1) | |
| (hDm : Measurable D) (hD : Integrable D μ) | |
| (hD2 : Integrable (fun ω => D ω ^ 2) μ) | |
| (hD3 : Integrable (fun ω => |D ω| ^ 3) μ) | |
| (hzero : ∫ ω, Z ω * (D ω : ℂ) ∂μ = 0) : | |
| ‖(∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) - | |
| (1 - (h : ℂ) / 2) * ∫ ω, Z ω ∂μ‖ ≤ | |
| 4 * ∫ ω, |D ω| ^ 3 ∂μ + | |
| (1 / 2 : ℝ) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ‖ := by | |
| let R : Ω → ℂ := fun ω => Complex.exp ((D ω : ℂ) * Complex.I) - 1 - | |
| (D ω : ℂ) * Complex.I + (D ω : ℂ) ^ 2 / 2 | |
| have hZint : Integrable Z μ := | |
| (integrable_const (1 : ℝ)).mono' hZm hZ | |
| have hRm : Measurable R := by dsimp [R]; fun_prop | |
| have hRint : Integrable R μ := (hD3.const_mul 4).mono' hRm.aestronglyMeasurable | |
| (Filter.Eventually.of_forall fun ω => ExponentialTaylor.norm_exp_I_taylor_two_le (D ω)) | |
| have hZR : Integrable (fun ω => Z ω * R ω) μ := hRint.bdd_mul hZm hZ | |
| have hZD : Integrable (fun ω => Z ω * (D ω : ℂ)) μ := | |
| hD.ofReal.bdd_mul hZm hZ | |
| have hZQ : Integrable (fun ω => Z ω * ((D ω : ℂ) ^ 2 - h)) μ := by | |
| have hq : Integrable (fun ω => (D ω : ℂ) ^ 2 - h) μ := by | |
| have hd2 : Integrable (fun ω => (D ω : ℂ) ^ 2) μ := by | |
| convert hD2.ofReal using 1 | |
| funext ω | |
| exact (Complex.ofReal_pow (D ω) 2).symm | |
| exact hd2.sub (integrable_const (h : ℂ)) | |
| exact hq.bdd_mul hZm hZ | |
| have hE : Integrable (fun ω => Z ω * Complex.exp ((D ω : ℂ) * Complex.I)) μ := by | |
| apply hZint.mul_bdd (c := 1) | |
| · apply Measurable.aestronglyMeasurable | |
| fun_prop | |
| · exact Filter.Eventually.of_forall (fun ω => by simp) | |
| have halg : (fun ω => Z ω * Complex.exp ((D ω : ℂ) * Complex.I) - | |
| (1 - (h : ℂ) / 2) * Z ω) = | |
| (fun ω => Z ω * R ω + (Z ω * (D ω : ℂ)) * Complex.I - | |
| (Z ω * ((D ω : ℂ) ^ 2 - h)) * (1 / 2 : ℂ)) := by | |
| funext ω | |
| dsimp [R] | |
| ring | |
| have hid : (∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) - | |
| (1 - (h : ℂ) / 2) * ∫ ω, Z ω ∂μ = | |
| (∫ ω, Z ω * R ω ∂μ) - | |
| (∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ) * (1 / 2 : ℂ) := by | |
| rw [← integral_const_mul, ← integral_sub hE (hZint.const_mul _), halg] | |
| have hsum : Integrable (fun ω => Z ω * R ω + Z ω * (D ω : ℂ) * Complex.I) μ := | |
| hZR.add (hZD.mul_const Complex.I) | |
| rw [integral_sub hsum (hZQ.mul_const (1 / 2 : ℂ)), | |
| integral_add hZR (hZD.mul_const Complex.I), integral_mul_const, integral_mul_const, hzero] | |
| simp | |
| rw [hid] | |
| calc | |
| _ ≤ ‖∫ ω, Z ω * R ω ∂μ‖ + | |
| ‖(∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ) * (1 / 2 : ℂ)‖ := norm_sub_le _ _ | |
| _ ≤ (4 * ∫ ω, |D ω| ^ 3 ∂μ) + | |
| (1 / 2 : ℝ) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ‖ := by | |
| have hr := norm_integral_weighted_exp_remainder_le μ Z D hZ hD3 | |
| have hn : ‖(1 / 2 : ℂ)‖ = (1 / 2 : ℝ) := by norm_num | |
| rw [norm_mul, hn, mul_comm _ (1 / 2 : ℝ)] | |
| exact add_le_add hr le_rfl | |
| theorem norm_gaussian_coefficient_sub_euler_le {h : ℝ} | |
| (h0 : 0 ≤ h) (h2 : h ≤ 2) : | |
| ‖Complex.exp (-(h : ℂ) / 2) - (1 - (h : ℂ) / 2)‖ ≤ h ^ 2 / 4 := by | |
| have hn : ‖-(h : ℂ) / 2‖ = h / 2 := by | |
| simp [Real.norm_eq_abs, abs_of_nonneg h0] | |
| have hb := Complex.norm_exp_sub_one_sub_id_le | |
| (x := -(h : ℂ) / 2) (by rw [hn]; linarith) | |
| rw [hn] at hb | |
| convert hb using 1 <;> congr 1 <;> ring | |
| theorem norm_gaussian_coefficient_sub_euler_le_global {h : ℝ} (h0 : 0 ≤ h) : | |
| ‖Complex.exp (-(h : ℂ) / 2) - (1 - (h : ℂ) / 2)‖ ≤ h ^ 2 / 4 := by | |
| have hb := ExponentialTaylor.norm_exp_neg_sub_one_add_le_sq | |
| (show 0 ≤ h / 2 by positivity) | |
| convert hb using 1 <;> push_cast <;> congr 1 <;> ring_nf | |
| theorem norm_integral_weighted_characteristic_gaussian_step_all | |
| {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] | |
| (Z : Ω → ℂ) (D : Ω → ℝ) {h : ℝ} (h0 : 0 ≤ h) | |
| (hZm : AEStronglyMeasurable Z μ) (hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1) | |
| (hDm : Measurable D) (hD : Integrable D μ) | |
| (hD2 : Integrable (fun ω => D ω ^ 2) μ) | |
| (hD3 : Integrable (fun ω => |D ω| ^ 3) μ) | |
| (hzero : ∫ ω, Z ω * (D ω : ℂ) ∂μ = 0) : | |
| ‖(∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) - | |
| Complex.exp (-(h : ℂ) / 2) * ∫ ω, Z ω ∂μ‖ ≤ | |
| 4 * ∫ ω, |D ω| ^ 3 ∂μ + | |
| (1 / 2 : ℝ) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ‖ + h ^ 2 / 4 := by | |
| have he := norm_integral_weighted_characteristic_step μ Z D h | |
| hZm hZ hDm hD hD2 hD3 hzero | |
| have hZi : ‖∫ ω, Z ω ∂μ‖ ≤ 1 := by | |
| simpa using norm_integral_le_of_norm_le (integrable_const (1 : ℝ)) hZ | |
| have hc : ‖(1 - (h : ℂ) / 2) - Complex.exp (-(h : ℂ) / 2)‖ ≤ h ^ 2 / 4 := by | |
| rw [norm_sub_rev] | |
| exact norm_gaussian_coefficient_sub_euler_le_global h0 | |
| have halg : (∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) - | |
| Complex.exp (-(h : ℂ) / 2) * ∫ ω, Z ω ∂μ = | |
| ((∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) - | |
| (1 - (h : ℂ) / 2) * ∫ ω, Z ω ∂μ) + | |
| ((1 - (h : ℂ) / 2) - Complex.exp (-(h : ℂ) / 2)) * ∫ ω, Z ω ∂μ := by ring | |
| rw [halg] | |
| apply (norm_add_le _ _).trans | |
| apply add_le_add he | |
| rw [norm_mul] | |
| calc | |
| _ ≤ ‖(1 - (h : ℂ) / 2) - Complex.exp (-(h : ℂ) / 2)‖ * 1 := | |
| mul_le_mul_of_nonneg_left hZi (norm_nonneg _) | |
| _ ≤ h ^ 2 / 4 := by simpa only [mul_one] using hc | |
| theorem norm_integral_weighted_characteristic_gaussian_step | |
| {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] | |
| (Z : Ω → ℂ) (D : Ω → ℝ) {h : ℝ} (h0 : 0 ≤ h) (_h2 : h ≤ 2) | |
| (hZm : AEStronglyMeasurable Z μ) (hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1) | |
| (hDm : Measurable D) (hD : Integrable D μ) | |
| (hD2 : Integrable (fun ω => D ω ^ 2) μ) | |
| (hD3 : Integrable (fun ω => |D ω| ^ 3) μ) | |
| (hzero : ∫ ω, Z ω * (D ω : ℂ) ∂μ = 0) : | |
| ‖(∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) - | |
| Complex.exp (-(h : ℂ) / 2) * ∫ ω, Z ω ∂μ‖ ≤ | |
| 4 * ∫ ω, |D ω| ^ 3 ∂μ + | |
| (1 / 2 : ℝ) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ‖ + h ^ 2 / 4 := | |
| norm_integral_weighted_characteristic_gaussian_step_all μ Z D h0 | |
| hZm hZ hDm hD hD2 hD3 hzero | |
| theorem norm_integral_weighted_characteristic_gaussian_step_all_mul | |
| {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] | |
| (Z : Ω → ℂ) (D : Ω → ℝ) (u : ℝ) {v : ℝ} | |
| (hv : 0 ≤ v) | |
| (hZm : AEStronglyMeasurable Z μ) (hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1) | |
| (hDm : Measurable D) (hD : Integrable D μ) | |
| (hD2 : Integrable (fun ω => D ω ^ 2) μ) | |
| (hD3 : Integrable (fun ω => |D ω| ^ 3) μ) | |
| (hzero : ∫ ω, Z ω * (D ω : ℂ) ∂μ = 0) : | |
| ‖(∫ ω, Z ω * Complex.exp (((u * D ω : ℝ) : ℂ) * Complex.I) ∂μ) - | |
| Complex.exp (-((u ^ 2 * v : ℝ) : ℂ) / 2) * ∫ ω, Z ω ∂μ‖ ≤ | |
| 4 * |u| ^ 3 * (∫ ω, |D ω| ^ 3 ∂μ) + | |
| (u ^ 2 / 2) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - v) ∂μ‖ + | |
| u ^ 4 * v ^ 2 / 4 := by | |
| have hs2 : Integrable (fun ω => (u * D ω) ^ 2) μ := by | |
| simpa only [mul_pow] using hD2.const_mul (u ^ 2) | |
| have hs3 : Integrable (fun ω => |u * D ω| ^ 3) μ := by | |
| simpa only [abs_mul, mul_pow] using hD3.const_mul (|u| ^ 3) | |
| have hsz : ∫ ω, Z ω * ((u * D ω : ℝ) : ℂ) ∂μ = 0 := by | |
| calc | |
| _ = (u : ℂ) * ∫ ω, Z ω * (D ω : ℂ) ∂μ := by | |
| rw [← integral_const_mul] | |
| congr 1 | |
| funext ω | |
| push_cast | |
| ring | |
| _ = 0 := by rw [hzero, mul_zero] | |
| have hs := norm_integral_weighted_characteristic_gaussian_step_all μ Z | |
| (fun ω => u * D ω) (mul_nonneg (sq_nonneg u) hv) | |
| hZm hZ (hDm.const_mul u) (hD.const_mul u) hs2 hs3 hsz | |
| have hthird : (∫ ω, |u * D ω| ^ 3 ∂μ) = | |
| |u| ^ 3 * ∫ ω, |D ω| ^ 3 ∂μ := by | |
| simp only [abs_mul, mul_pow, integral_const_mul] | |
| have hquad : (∫ ω, Z ω * (((u * D ω : ℝ) : ℂ) ^ 2 - (u ^ 2 * v : ℝ)) ∂μ) = | |
| (u : ℂ) ^ 2 * ∫ ω, Z ω * ((D ω : ℂ) ^ 2 - v) ∂μ := by | |
| rw [← integral_const_mul] | |
| congr 1 | |
| funext ω | |
| push_cast | |
| ring | |
| rw [hthird, hquad, norm_mul] at hs | |
| have hn : ‖(u : ℂ) ^ 2‖ = u ^ 2 := by simp [norm_pow, sq_abs] | |
| rw [hn] at hs | |
| convert hs using 1 | |
| ring | |
| theorem norm_integral_weighted_characteristic_gaussian_step_mul | |
| {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ] | |
| (Z : Ω → ℂ) (D : Ω → ℝ) (u : ℝ) {v : ℝ} | |
| (hv : 0 ≤ v) (_huv : u ^ 2 * v ≤ 2) | |
| (hZm : AEStronglyMeasurable Z μ) (hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1) | |
| (hDm : Measurable D) (hD : Integrable D μ) | |
| (hD2 : Integrable (fun ω => D ω ^ 2) μ) | |
| (hD3 : Integrable (fun ω => |D ω| ^ 3) μ) | |
| (hzero : ∫ ω, Z ω * (D ω : ℂ) ∂μ = 0) : | |
| ‖(∫ ω, Z ω * Complex.exp (((u * D ω : ℝ) : ℂ) * Complex.I) ∂μ) - | |
| Complex.exp (-((u ^ 2 * v : ℝ) : ℂ) / 2) * ∫ ω, Z ω ∂μ‖ ≤ | |
| 4 * |u| ^ 3 * (∫ ω, |D ω| ^ 3 ∂μ) + | |
| (u ^ 2 / 2) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - v) ∂μ‖ + | |
| u ^ 4 * v ^ 2 / 4 := | |
| norm_integral_weighted_characteristic_gaussian_step_all_mul μ Z D u hv | |
| hZm hZ hDm hD hD2 hD3 hzero | |
| end Problem367 | |
| end | |
| end OAI | |