| import OAI.Probability.FreeIsing.Energy.TimeRowEnergyHasSum |
| import OAI.Probability.FreeIsing.Ising.ContinuousRootResponse |
| import OAI.Probability.FreeIsing.Ising.CopyEnergyIntegrability |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped NNReal Topology |
|
|
| namespace Problem367.IsingObservationFactor |
| open IsingObservationInnovation |
|
|
| variable {d : ℕ} {beta : ℝ} |
|
|
| private theorem measurable_real_of_progressive |
| {Ω : Type*} [mΩ : MeasurableSpace Ω] {F : Filtration ℝ≥0 mΩ} |
| {M : ℝ≥0 → Ω → ℝ} (hM : IsStronglyProgressive F M) : |
| Measurable (fun p : ℝ × Ω => M p.1.toNNReal p.2) := by |
| have hj : Measurable (fun p : ℝ≥0 × Ω => M p.1 p.2) := |
| (stronglyMeasurable_joint_of_isStronglyProgressive hM).measurable |
| have ht : Measurable (fun p : ℝ × Ω => (p.1.toNNReal, p.2)) := |
| (measurable_real_toNNReal.comp measurable_fst).prodMk measurable_snd |
| exact hj.comp ht |
|
|
| private theorem measurable_response_real (hb : 0 ≤ beta) (v : TreeVertex d) : |
| Measurable (fun p : ℝ × Paths d => continuousRootResponse hb v p.1.toNNReal p.2) := |
| measurable_real_of_progressive (continuousRootResponse_progressive hb v) |
|
|
| private theorem measurable_covariance_real (hb : 0 ≤ beta) (v : TreeVertex d) : |
| Measurable (fun p : ℝ × Paths d => continuousRootCovariance hb v p.1.toNNReal p.2) := |
| measurable_real_of_progressive (continuousRootCovariance_progressive hb v) |
|
|
| theorem continuousRootResponse_row_eq_ae |
| (hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (T : ℝ) : |
| ∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ), |
| ∀ v, continuousRootResponse hb v p.1.toNNReal p.2 = |
| IsingCopyEnergy.responseRow beta (ConnectedHull.root d) v p := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| apply ae_all_iff.mpr |
| intro v |
| apply (Measure.ae_prod_iff_ae_ae (measurableSet_eq_fun |
| (measurable_response_real hb v) |
| (IsingCopyEnergy.measurable_responseRow beta (ConnectedHull.root d) v))).mpr |
| exact Eventually.of_forall fun t => |
| continuousRootResponse_eq_response_ae hd hμ hb hcrit v t.toNNReal |
|
|
| theorem continuousRootResponse_row_energy |
| (hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| {T : ℝ} (hT : 0 ≤ T) : |
| (∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ), |
| Summable (fun v => continuousRootResponse hb v p.1.toNNReal p.2 ^ 2)) ∧ |
| Integrable (fun p : ℝ × Paths d => ∑' v, |
| continuousRootResponse hb v p.1.toNNReal p.2 ^ 2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) ∧ |
| (∫ p : ℝ × Paths d, ∑' v, continuousRootResponse hb v p.1.toNNReal p.2 ^ 2 |
| ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) ≤ |
| 2 * ((d : ℝ) * beta + T + Real.sqrt T) := by |
| have hraw := IsingCopyEnergy.observation_row_energy_integrable hμ hb |
| (ConnectedHull.root d) hT |
| have he := continuousRootResponse_row_eq_ae hd hμ hb hcrit T |
| have hsum : (fun p : ℝ × Paths d => ∑' v, |
| continuousRootResponse hb v p.1.toNNReal p.2 ^ 2) =ᵐ[ |
| (volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)] |
| (fun p => ∑' v, IsingCopyEnergy.responseRow beta (ConnectedHull.root d) v p ^ 2) := by |
| filter_upwards [he] with p hp |
| exact tsum_congr (fun v => congrArg (fun r : ℝ => r ^ 2) (hp v)) |
| refine ⟨?_, hraw.2.1.congr hsum.symm, (integral_congr_ae hsum).trans_le hraw.2.2⟩ |
| filter_upwards [he, hraw.1] with p hp hs |
| simpa only [hp] using hs |
|
|
| theorem continuousRootResponse_hasSum_timeEnergy |
| (hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (T : ℝ≥0) : |
| HasSum (fun v => ∫ t in (0 : ℝ)..(T : ℝ), |
| ∫ x, continuousRootResponse hb v t.toNNReal x ^ 2 ∂observationLaw μ) |
| (∫ p : ℝ × Paths d, ∑' v, continuousRootResponse hb v p.1.toNNReal p.2 ^ 2 |
| ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have h := continuousRootResponse_row_energy hd hμ hb hcrit T.coe_nonneg |
| exact DiscrepancyEnergy.hasSum_intervalIntegral_sq_row T |
| (fun v => (measurable_response_real hb v).aestronglyMeasurable) h.1 h.2.1 |
| |
| theorem continuousRootCovariance_sq_le_response_sq |
| (hb : 0 ≤ beta) (v : TreeVertex d) (t : ℝ≥0) (x : Paths d) : |
| continuousRootCovariance hb v t x ^ 2 ≤ continuousRootResponse hb v t x ^ 2 := by |
| have h0 : 0 ≤ 1 - Real.tanh (continuousField beta (ConnectedHull.root d) t x) ^ 2 := |
| (sub_pos.mpr (Real.tanh_sq_lt_one _)).le |
| have h1 : 1 - Real.tanh (continuousField beta (ConnectedHull.root d) t x) ^ 2 ≤ 1 := |
| sub_le_self _ (sq_nonneg _) |
| have hs : (1 - Real.tanh (continuousField beta (ConnectedHull.root d) t x) ^ 2) ^ 2 ≤ 1 := by |
| nlinarith |
| rw [continuousRootCovariance, mul_pow] |
| exact (mul_le_mul_of_nonneg_right hs (sq_nonneg _)).trans_eq (one_mul _) |
| |
| theorem continuousRootCovariance_row_energy |
| (hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) |
| {T : ℝ} (hT : 0 ≤ T) : |
| (∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ), |
| Summable (fun v => continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2)) ∧ |
| Integrable (fun p : ℝ × Paths d => ∑' v, |
| continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) ∧ |
| (∫ p : ℝ × Paths d, ∑' v, continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2 |
| ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) ≤ |
| 2 * ((d : ℝ) * beta + T + Real.sqrt T) := by |
| have hR := continuousRootResponse_row_energy hd hμ hb hcrit hT |
| have hs : ∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ), |
| Summable (fun v => continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2) := by |
| filter_upwards [hR.1] with p hp |
| exact Summable.of_nonneg_of_le (fun v => sq_nonneg _) |
| (fun v => continuousRootCovariance_sq_le_response_sq hb v _ _) hp |
| have hle : ∀ᵐ p ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ), |
| (∑' v, continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2) ≤ |
| ∑' v, continuousRootResponse hb v p.1.toNNReal p.2 ^ 2 := by |
| filter_upwards [hs, hR.1] with p hcp hrp |
| exact hcp.tsum_le_tsum |
| (fun v => continuousRootCovariance_sq_le_response_sq hb v _ _) hrp |
| have hi : Integrable (fun p : ℝ × Paths d => ∑' v, |
| continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2) |
| ((volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) := by |
| apply hR.2.1.mono' |
| (Measurable.tsum (fun v => (measurable_covariance_real hb v).pow_const 2)).aestronglyMeasurable |
| filter_upwards [hle] with p hp |
| rw [Real.norm_eq_abs, abs_of_nonneg (tsum_nonneg |
| (fun v => sq_nonneg (continuousRootCovariance hb v p.1.toNNReal p.2)))] |
| exact hp |
| exact ⟨hs, hi, (integral_mono_ae hi hR.2.1 hle).trans hR.2.2⟩ |
| |
| theorem continuousRootCovariance_hasSum_timeEnergy |
| (hd : 3 ≤ d) {μ : Measure (TreeVertex d → IsingSpin)} |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) |
| (hcrit : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 ≤ 1) (T : ℝ≥0) : |
| HasSum (fun v => ∫ t in (0 : ℝ)..(T : ℝ), |
| ∫ x, continuousRootCovariance hb v t.toNNReal x ^ 2 ∂observationLaw μ) |
| (∫ p : ℝ × Paths d, ∑' v, continuousRootCovariance hb v p.1.toNNReal p.2 ^ 2 |
| ∂(volume.restrict (Ioc (0 : ℝ) T)).prod (observationLaw μ)) := by |
| let : IsProbabilityMeasure μ := hμ.isProbabilityMeasure |
| have h := continuousRootCovariance_row_energy hd hμ hb hcrit T.coe_nonneg |
| exact DiscrepancyEnergy.hasSum_intervalIntegral_sq_row T |
| (fun v => (measurable_covariance_real hb v).aestronglyMeasurable) h.1 h.2.1 |
|
|
| end Problem367.IsingObservationFactor |
|
|
| end |
|
|
| end OAI |
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|
