| import OAI.Probability.FreeIsing.Ising.CanonicalArtanhIdentification | |
| import OAI.Probability.FreeIsing.Ising.CommonStopBounds | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Filter Set | |
| open scoped NNReal | |
| namespace Problem367.IsingObservationFactor | |
| open IsingObservationInnovation IsingCopyEnergy | |
| variable {d : ℕ} {beta : ℝ} | |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] | |
| theorem ae_all_copyResponseDriftDifference_primitive | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (v : TreeVertex d) : | |
| ∀ᵐ z ∂copies beta μ, ∀ t : ℝ≥0, | |
| nnrealTimePrimitive (copyResponseDriftDifference beta v) t z = | |
| nnrealTimePrimitive (canonicalResponseDrift beta v) t z.2.1 - | |
| nnrealTimePrimitive (canonicalResponseDrift beta v) t z.2.2 := by | |
| have hi := ae_intervalIntegrable_horizons_of_joint | |
| (μ := observationLaw μ) (a := canonicalResponseDrift beta v) | |
| (fun N => by simpa only [NNReal.coe_natCast] using | |
| canonicalResponseDrift_integrable hμ hb v (N : ℝ≥0)) | |
| filter_upwards [(copyProjection_measurePreserving beta μ true).quasiMeasurePreserving.ae hi, | |
| (copyProjection_measurePreserving beta μ false).quasiMeasurePreserving.ae hi] | |
| with z hl hr | |
| intro t | |
| obtain ⟨N, hN⟩ := exists_nat_ge (t : ℝ) | |
| have hrestrict {f : ℝ → ℝ} (hf : IntervalIntegrable f volume 0 (N : ℝ)) : | |
| IntervalIntegrable f volume 0 (t : ℝ) := | |
| (intervalIntegrable_iff_integrableOn_Ioc_of_le t.coe_nonneg).mpr | |
| (((intervalIntegrable_iff_integrableOn_Ioc_of_le (Nat.cast_nonneg N)).mp hf).mono_set | |
| (Ioc_subset_Ioc le_rfl hN)) | |
| change (∫ s in (0 : ℝ)..(t : ℝ), | |
| canonicalResponseDrift beta v s.toNNReal z.2.1 - | |
| canonicalResponseDrift beta v s.toNNReal z.2.2) = _ | |
| exact intervalIntegral.integral_sub (hrestrict (hl N)) (hrestrict (hr N)) | |
| theorem copied_difference_decomposition | |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (v : TreeVertex d) | |
| {Y : ℝ≥0 → Paths d → ℝ} {Z : Bool → ℝ≥0 → Joint d → ℝ} | |
| (hfield : ∀ᵐ x ∂observationLaw μ, ∀ t, | |
| continuousField beta v t x = Y t x + | |
| nnrealTimePrimitive (canonicalResponseDrift beta v) t x) | |
| (hcopy : ∀ j, ∀ᵐ z ∂copies beta μ, ∀ t, | |
| Y t (projection j z) = Z j t z) : | |
| ∀ᵐ z ∂copies beta μ, ∀ t, | |
| continuousDifference beta v t z = (Z true t z - Z false t z) + | |
| nnrealTimePrimitive (copyResponseDriftDifference beta v) t z := by | |
| filter_upwards [(copyProjection_measurePreserving beta μ true).quasiMeasurePreserving.ae hfield, | |
| (copyProjection_measurePreserving beta μ false).quasiMeasurePreserving.ae hfield, | |
| hcopy true, hcopy false, ae_all_copyResponseDriftDifference_primitive hμ hb v] | |
| with z hl hr hzl hzr hprimitive | |
| intro t | |
| simp only [projection, Bool.false_eq_true, ite_true, ite_false] at hl hr hzl hzr | |
| change continuousField beta v t z.2.1 - continuousField beta v t z.2.2 = _ | |
| rw [hl t, hr t, hprimitive t] | |
| have hzl' := hzl t | |
| have hzr' := hzr t | |
| change Y t z.2.1 = Z true t z at hzl' | |
| change Y t z.2.2 = Z false t z at hzr' | |
| rw [hzl', hzr'] | |
| ring | |
| end Problem367.IsingObservationFactor | |
| end | |
| end OAI | |