| import OAI.Probability.FreeIsing.Energy.DiscrepancyEnergy | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory Filter Set | |
| open scoped Topology | |
| namespace Problem367.DiscrepancyEnergy | |
| theorem intervalIntegrable_primitive_mul | |
| {b : ℝ → ℝ} {a t : ℝ} (hb : IntervalIntegrable b volume a t) : | |
| IntervalIntegrable (fun s => (∫ u in a..s, b u) * b s) volume a t := by | |
| exact hb.continuousOn_mul | |
| (hb.absolutelyContinuousOnInterval_intervalIntegral (c := a) (by simp)).continuousOn | |
| theorem primitive_sq_eq_two_integral | |
| {b : ℝ → ℝ} {a t : ℝ} (hb : IntervalIntegrable b volume a t) : | |
| (∫ s in a..t, b s) ^ 2 = | |
| 2 * ∫ s in a..t, (∫ u in a..s, b u) * b s := by | |
| let A : ℝ → ℝ := fun s => ∫ u in a..s, b u | |
| have hA : AbsolutelyContinuousOnInterval A a t := | |
| hb.absolutelyContinuousOnInterval_intervalIntegral (c := a) (by simp) | |
| have h := hA.integral_deriv_mul_eq_sub hA | |
| have heq : (∫ s in a..t, deriv A s * A s + A s * deriv A s) = | |
| ∫ s in a..t, 2 * (A s * b s) := by | |
| apply intervalIntegral.integral_congr_ae | |
| filter_upwards [hb.ae_hasDerivAt_integral] with s hs hst | |
| have hd : deriv A s = b s := | |
| (hs (uIoc_subset_uIcc hst) a (by simp)).deriv | |
| rw [hd] | |
| ring | |
| rw [heq, intervalIntegral.integral_const_mul] at h | |
| simpa only [A, intervalIntegral.integral_same, mul_zero, sub_zero, pow_two] using h.symm | |
| theorem ae_timePrimitive_sq | |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [SFinite μ] | |
| {b : ℝ → Ω → ℝ} {T t : ℝ} (ht : t ∈ Icc 0 T) | |
| (hb : Integrable (fun p : ℝ × Ω => b p.1 p.2) | |
| ((volume.restrict (Ioc 0 T)).prod μ)) : | |
| ∀ᵐ ω ∂μ, (∫ s in (0 : ℝ)..t, b s ω) ^ 2 = | |
| 2 * ∫ s in (0 : ℝ)..t, (∫ u in (0 : ℝ)..s, b u ω) * b s ω := by | |
| have hbt : Integrable (fun p : ℝ × Ω => b p.1 p.2) | |
| ((volume.restrict (Ioc 0 t)).prod μ) := | |
| hb.mono_measure (Measure.prod_mono | |
| (Measure.restrict_mono_set _ (Ioc_subset_Ioc_right ht.2)) le_rfl) | |
| filter_upwards [hbt.prod_left_ae] with ω hω | |
| exact primitive_sq_eq_two_integral | |
| ((intervalIntegrable_iff_integrableOn_Ioc_of_le ht.1).mpr hω) | |
| theorem integral_timePrimitive_sq | |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [SFinite μ] | |
| {b : ℝ → Ω → ℝ} {T t : ℝ} (ht : t ∈ Icc 0 T) | |
| (hb : Integrable (fun p : ℝ × Ω => b p.1 p.2) | |
| ((volume.restrict (Ioc 0 T)).prod μ)) | |
| (hAb : Integrable (fun p : ℝ × Ω => | |
| (∫ u in (0 : ℝ)..p.1, b u p.2) * b p.1 p.2) | |
| ((volume.restrict (Ioc 0 T)).prod μ)) : | |
| (∫ ω, (∫ s in (0 : ℝ)..t, b s ω) ^ 2 ∂μ) = | |
| 2 * ∫ s in (0 : ℝ)..t, ∫ ω, (∫ u in (0 : ℝ)..s, b u ω) * b s ω ∂μ := by | |
| rw [integral_congr_ae (ae_timePrimitive_sq ht hb), integral_const_mul] | |
| congr 1 | |
| exact integral_timePrimitive ht hAb | |
| theorem integrable_timePrimitive_sq | |
| {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [SFinite μ] | |
| {b : ℝ → Ω → ℝ} {T t : ℝ} (ht : t ∈ Icc 0 T) | |
| (hb : Integrable (fun p : ℝ × Ω => b p.1 p.2) | |
| ((volume.restrict (Ioc 0 T)).prod μ)) | |
| (hAb : Integrable (fun p : ℝ × Ω => | |
| (∫ u in (0 : ℝ)..p.1, b u p.2) * b p.1 p.2) | |
| ((volume.restrict (Ioc 0 T)).prod μ)) : | |
| Integrable (fun ω => (∫ s in (0 : ℝ)..t, b s ω) ^ 2) μ := by | |
| have hAbt : Integrable (fun p : ℝ × Ω => | |
| (∫ u in (0 : ℝ)..p.1, b u p.2) * b p.1 p.2) | |
| ((volume.restrict (Ioc 0 t)).prod μ) := | |
| hAb.mono_measure (Measure.prod_mono | |
| (Measure.restrict_mono_set _ (Ioc_subset_Ioc_right ht.2)) le_rfl) | |
| have hI : Integrable (fun ω => ∫ s in (0 : ℝ)..t, | |
| (∫ u in (0 : ℝ)..s, b u ω) * b s ω) μ := by | |
| simpa only [intervalIntegral.integral_of_le ht.1] using hAbt.integral_prod_right | |
| exact (hI.const_mul 2).congr ((ae_timePrimitive_sq ht hb).mono (fun ω hω => hω.symm)) | |
| end Problem367.DiscrepancyEnergy | |
| end | |
| end OAI | |