| import OAI.Probability.FreeIsing.Ising.CanonicalContinuousFields |
| import OAI.Probability.FreeIsing.Ising.CopyStationarity |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open MeasureTheory |
| open scoped NNReal |
|
|
| namespace Problem367.ProgressiveVersionPushforward |
|
|
| theorem ae_all_version_comp_eq |
| {Ω Ξ : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| {M : ℝ≥0 → Ξ → ℝ} {N : ℝ≥0 → Ω → ℝ} {φ : Ω → Ξ} |
| (h : ∀ t, (fun ω => M t (φ ω)) =ᵐ[μ] N t) : |
| ∀ᵐ ω ∂μ, ∀ t, version M t (φ ω) = version N t ω := by |
| have hgrid : ∀ᵐ ω ∂μ, ∀ p : ℤ × ℕ, |
| M (gridTime p) (φ ω) = N (gridTime p) ω := |
| ae_all_iff.mpr (fun p => h (gridTime p)) |
| filter_upwards [hgrid] with ω hω |
| intro t |
| change ContinuousModification.sampledLimit leftTime |
| (fun s ω => M s (φ ω)) t ω = |
| ContinuousModification.sampledLimit leftTime N t ω |
| apply ContinuousModification.sampledLimit_congr |
| intro n |
| obtain ⟨p, hp⟩ := leftTime_mem_range t n |
| rw [← hp] |
| exact hω p |
|
|
| end Problem367.ProgressiveVersionPushforward |
|
|
| namespace Problem367.IsingObservationFactor |
|
|
| open IsingObservationInnovation IsingCopyStationarity |
|
|
| variable {d : ℕ} {beta : ℝ} |
| variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ] |
|
|
| omit [IsProbabilityMeasure μ] in |
| theorem continuousField_array_relabel_ae [_hprob : IsProbabilityMeasure μ] |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (g : TreeAut d) : |
| ∀ᵐ x ∂observationLaw μ, |
| (fun v t => continuousField beta v t (relabel g x)) = |
| relabel g (fun v t => continuousField beta v t x) := by |
| have hcoord (v : TreeVertex d) : ∀ᵐ x ∂observationLaw μ, ∀ t, |
| continuousField beta v t (relabel g x) = |
| continuousField beta (g.toEquiv.symm v) t x := by |
| apply ProgressiveVersionPushforward.ae_all_version_comp_eq |
| intro t |
| filter_upwards [field_array_relabel_ae hμ g t] with x hx |
| exact congrFun hx v |
| filter_upwards [ae_all_iff.mpr hcoord] with x hx |
| exact funext (fun v => funext (hx v)) |
|
|
| theorem continuousDifference_array_relabel_ae |
| (hμ : HasFreeIsingLaw (Real.tanh beta) μ) (g : TreeAut d) : |
| ∀ᵐ q ∂copies beta μ, |
| (fun v t => continuousDifference beta v t (copyRelabel g q)) = |
| relabel g (fun v t => continuousDifference beta v t q) := by |
| obtain ⟨hl, hr⟩ := measurePreserving_observations (beta := beta) (μ := μ) |
| filter_upwards |
| [hl.quasiMeasurePreserving.ae (continuousField_array_relabel_ae hμ g), |
| hr.quasiMeasurePreserving.ae (continuousField_array_relabel_ae hμ g)] |
| with q hql hqr |
| funext v t |
| exact congrArg₂ (fun a b : ℝ => a - b) |
| (congrFun (congrFun hql v) t) (congrFun (congrFun hqr v) t) |
|
|
| end Problem367.IsingObservationFactor |
|
|
| end |
|
|
| end OAI |
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|
