openAI-math-dataset / lean /OAI /MathematicalPhysics /Boltzmann /Compactness /LocalizedBlockComparison.lean
| import OAI.MathematicalPhysics.Boltzmann.Branching.BoundedEntryPortion | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.DampedBoundedEndpointTransport | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.DampedLateBoxPhysicalBlock | |
| import OAI.MathematicalPhysics.Boltzmann.Branching.LateBoxAccessMeasure | |
| import OAI.MathematicalPhysics.Boltzmann.Compactness.LocalizedDampedBalance | |
| namespace OAI | |
| section | |
| /-! # Retaining the actual open-left collision-time window in physical histories -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal NNReal ProbabilityTheory | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| def physicalCollisionWindow (s T : ℝ) : Set PhysicalFlightState := {p | T-p.1 ∈ Ioc s T} | |
| theorem physicalCollisionWindow_measurable (s T : ℝ) : MeasurableSet (physicalCollisionWindow s T) := | |
| (measurable_const.sub measurable_fst) measurableSet_Ioc | |
| def localizedNormalWindowEvent (b : BeamCoefficients) (ρ C T a η l s : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) : | |
| Kernel PhysicalFlightState PhysicalFlightState := | |
| (localizedNormalPhysicalEvent b ρ C T a η l hΩ).restrict (physicalCollisionWindow_measurable s T) | |
| instance localizedNormalWindowEvent_sFinite (b : BeamCoefficients) (ρ C T a η l s : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) : | |
| IsSFiniteKernel (localizedNormalWindowEvent b ρ C T a η l s hΩ) := by | |
| unfold localizedNormalWindowEvent | |
| infer_instance | |
| theorem localizedNormalWindowEvent_transport (b : BeamCoefficients) (ρ : ℝ) | |
| (C : ℝ≥0) (T a η l s : ℝ) {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) : | |
| localizedNormalWindowEvent b ρ C T a η l s hΩ = | |
| transportedKernel (flightCharacteristicEquiv T) | |
| (((localizedNormalBeamKernel b ρ C a η l hΩ).restrict | |
| (measurable_fst measurableSet_Ioc : MeasurableSet (Prod.fst ⁻¹' Ioc s T))) ∘ₖ | |
| WeakTime.residenceKernel C T) := by | |
| rw [transportedKernel_compKernel,transportedKernel_restrict, | |
| ← physicalWaitingKernel_eq_transport,Kernel.comp_restrict] | |
| simp only [localizedNormalWindowEvent,localizedNormalPhysicalEvent,localizedNormalCollision, | |
| Real.toNNReal_coe] | |
| rfl | |
| theorem boundedHistory_le_localized_window {C : ℝ≥0} (hC : referenceClock ≤ C) | |
| (b : BeamCoefficients) (ρ : ℝ≥0) (T a η l : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) | |
| (p : PhysicalFlightState) (ht : 0 ≤ p.1) (hv : ‖p.2.2‖ ≤ 6) | |
| (hΩcone : ∀ q, boundedFlightCone p q → flightCharacteristicEquiv T q ∈ Ω) | |
| (hlocal : ∀ q, boundedFlightCone p q → | |
| flightCharacteristicEquiv T q ∈ coldCharacteristicDomain a η l ∧ | |
| 0 ≤ b.plusDensity (characteristicBeamPoint (flightCharacteristicEquiv T q)) ∧ | |
| 0 ≤ b.minusDensity (characteristicBeamPoint (flightCharacteristicEquiv T q)) ∧ | |
| characteristicBeamFrequency b (flightCharacteristicEquiv T q) ≤ (referenceClock:ℝ)) | |
| (n : ℕ) : | |
| ((dampedBoundedDensityScaledEvent C b T ρ)^n) p ≤ | |
| ((localizedNormalWindowEvent b ρ (ρ*C) T a η l (T-p.1) hΩ)^n) p := by | |
| apply kernel_pow_le_on_invariant _ _ {q | boundedFlightCone p q} _ _ n p | |
| (boundedFlightCone_refl ht hv) | |
| · intro q hq | |
| filter_upwards [dampedBoundedDensityScaledEvent_ae_cone C b T ρ q hq.2.2.1] with r hr | |
| exact boundedFlightCone_trans hq hr | |
| · intro q hq | |
| have he : dampedBoundedDensityScaledEvent C b T ρ q ≤ | |
| localizedNormalPhysicalEvent b ρ (ρ*C) T a η l hΩ q := by | |
| simpa only [pow_one] using boundedHistory_le_localized hC b ρ T a η l hΩ q hq.1 hq.2.2.1 | |
| (fun r hr => hΩcone r (boundedFlightCone_trans hq hr)) | |
| (fun r hr => hlocal r (boundedFlightCone_trans hq hr)) 1 | |
| have htwin : ∀ᵐ r ∂dampedBoundedDensityScaledEvent C b T ρ q, | |
| r ∈ physicalCollisionWindow (T-p.1) T := | |
| dampedBoundedDensityScaledEvent_ae_time_window C b T ρ hq | |
| calc | |
| _ = (dampedBoundedDensityScaledEvent C b T ρ q).restrict | |
| (physicalCollisionWindow (T-p.1) T) := (Measure.restrict_eq_self_of_ae_mem htwin).symm | |
| _ ≤ _ := Measure.restrict_mono_measure he _ | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| section | |
| /-! # Physical normal-angle histories end at the hot trace's characteristic endpoint -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| open scoped ENNReal NNReal ProbabilityTheory | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| theorem localizedNormalWindowEvent_pow_transport (b : BeamCoefficients) (ρ : ℝ) | |
| (C : ℝ≥0) (T a η l s : ℝ) {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) (n : ℕ) : | |
| (localizedNormalWindowEvent b ρ C T a η l s hΩ)^n = | |
| transportedKernel (flightCharacteristicEquiv T) | |
| ((((localizedNormalBeamKernel b ρ C a η l hΩ).restrict | |
| (measurable_fst measurableSet_Ioc : MeasurableSet (Prod.fst ⁻¹' Set.Ioc s T))) ∘ₖ | |
| WeakTime.residenceKernel C T)^n) := by | |
| rw [transportedKernel_pow,← localizedNormalWindowEvent_transport] | |
| theorem localizedNormal_endpoint_characteristic (b : BeamCoefficients) (ρ : ℝ) | |
| (C : ℝ≥0) (T a η l s : ℝ) {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) | |
| (n : ℕ) (p : PhysicalFlightState) {f : V × V → ℝ≥0∞} (hf : Measurable f) : | |
| (∫⁻ q, clockSurvival C q.1*f (finishMediumFlight q) | |
| ∂((localizedNormalWindowEvent b ρ C T a η l s hΩ)^n) p) = | |
| ∫⁻ q, f (EuclideanFlight.flight T q) | |
| ∂(WeakTime.phaseEndpointKernel C T ∘ₖ | |
| ((((localizedNormalBeamKernel b ρ C a η l hΩ).restrict | |
| (measurable_fst measurableSet_Ioc : MeasurableSet (Prod.fst ⁻¹' Set.Ioc s T))) ∘ₖ | |
| WeakTime.residenceKernel C T)^n)) (flightCharacteristicEquiv T p) := by | |
| have hg : Measurable (fun z => f (EuclideanFlight.flight T z)) := | |
| hf.comp (EuclideanFlight.flight_measurePreserving T).measurable | |
| have hw : Measurable (fun q : PhysicalFlightState => clockSurvival C q.1*f (finishMediumFlight q)) := | |
| (show Measurable (fun q : PhysicalFlightState => clockSurvival C q.1) by | |
| unfold clockSurvival; fun_prop).mul (hf.comp finishMediumFlight_measurable) | |
| rw [localizedNormalWindowEvent_pow_transport,transportedKernel_lintegral _ _ _ hw, | |
| Kernel.lintegral_comp _ _ _ hg] | |
| apply lintegral_congr | |
| intro z | |
| rw [WeakTime.phaseEndpointKernel_lintegral _ _ _ hg,physical_terminal_characteristic T C f] | |
| simp only [MeasurableEquiv.apply_symm_apply] | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| section | |
| /-! # The finite physical branching block lies below the actual angular endpoint histories -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| open scoped ENNReal NNReal ProbabilityTheory | |
| namespace BoltzmannNonuniqueness.ReferenceGeometry | |
| def localizedHistoryBlockMass (b : BeamCoefficients) (ρ C T a η l s : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) (A : Set (V × V)) | |
| (N : ℕ) (z : ℝ × WeakTime.Phase) : ℝ≥0∞ := | |
| ∑ n ∈ Finset.range N, ∫⁻ q, A.indicator 1 (EuclideanFlight.flight T q) | |
| ∂(WeakTime.phaseEndpointKernel C T ∘ₖ | |
| ((((localizedNormalBeamKernel b ρ C a η l hΩ).restrict | |
| (measurable_fst measurableSet_Ioc : MeasurableSet (Prod.fst ⁻¹' Ioc s T))) ∘ₖ | |
| WeakTime.residenceKernel C T)^n)) z | |
| theorem physicalBlock_le_localized {C : ℝ≥0} (hC : referenceClock ≤ C) | |
| (b : BeamCoefficients) (ρ : ℝ≥0) (s Δ a η l κ d : ℝ) | |
| {Ω : Set (ℝ × WeakTime.Phase)} (hΩ : MeasurableSet Ω) (x : V) (v : RetainedState) | |
| (hΔ : 0 ≤ Δ) (hv : ‖(v:V)‖ ≤ 6) (hp : (x,(v:V)) ∈ lateBox κ d s) | |
| (hΩcone : ∀ q, boundedFlightCone (Δ,(x,(v:V))) q → flightCharacteristicEquiv (s+Δ) q ∈ Ω) | |
| (hlocal : ∀ q, boundedFlightCone (Δ,(x,(v:V))) q → | |
| flightCharacteristicEquiv (s+Δ) q ∈ coldCharacteristicDomain a η l ∧ | |
| 0 ≤ b.plusDensity (characteristicBeamPoint (flightCharacteristicEquiv (s+Δ) q)) ∧ | |
| 0 ≤ b.minusDensity (characteristicBeamPoint (flightCharacteristicEquiv (s+Δ) q)) ∧ | |
| characteristicBeamFrequency b (flightCharacteristicEquiv (s+Δ) q) ≤ (referenceClock:ℝ)) | |
| (N : ℕ) : | |
| dampedDensityScaledBlockMass C b (s+Δ) ρ x Δ v κ d N ≤ | |
| localizedHistoryBlockMass b ρ (ρ*C) (s+Δ) a η l s hΩ (lateBox κ d (s+Δ)) N | |
| (flightCharacteristicEquiv (s+Δ) (Δ,(x,(v:V)))) := by | |
| let A := lateBox κ d (s+Δ) | |
| have hA : MeasurableSet A := lateBox_measurable _ _ _ | |
| have hf : Measurable (A.indicator (1 : V × V → ℝ≥0∞)) := measurable_const.indicator hA | |
| unfold dampedDensityScaledBlockMass localizedHistoryBlockMass | |
| simp only [Real.toNNReal_coe] | |
| apply Finset.sum_le_sum | |
| intro n _hn | |
| have hle := boundedHistory_le_localized_window hC b ρ (s+Δ) a η l hΩ | |
| (Δ,(x,(v:V))) hΔ hv hΩcone hlocal n | |
| have htime : (s+Δ)-Δ = s := by ring | |
| rw [htime] at hle | |
| calc | |
| _ ≤ ∫⁻ q, clockSurvival (ρ*C) q.1*A.indicator 1 (finishMediumFlight q) | |
| ∂((dampedBoundedDensityScaledEvent C b (s+Δ) ρ)^n) (Δ,(x,(v:V))) := by | |
| apply lintegral_mono | |
| intro q | |
| dsimp only | |
| gcongr | |
| exact fun _ hy => physicalBlockTarget_subset_lateBox hp hy | |
| _ ≤ ∫⁻ q, clockSurvival (ρ*C) q.1*A.indicator 1 (finishMediumFlight q) | |
| ∂((localizedNormalWindowEvent b ρ (ρ*C) (s+Δ) a η l s hΩ)^n) (Δ,(x,(v:V))) := | |
| lintegral_mono' hle le_rfl | |
| _ = _ := by | |
| simpa only [NNReal.coe_mul] using | |
| localizedNormal_endpoint_characteristic b ρ (ρ*C) (s+Δ) a η l s hΩ n (Δ,(x,(v:V))) hf | |
| end BoltzmannNonuniqueness.ReferenceGeometry | |
| end | |
| end | |
| end OAI | |