openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Transfer /FiniteRealChartIntersections.lean
| import OAI.Analysis.CircleDomains.Topology.MeanSigmaCompactPhysicalCount | |
| import OAI.Analysis.CircleDomains.Modulus.SphericalChartCost | |
| import OAI.Analysis.CircleDomains.Topology.SphereSliceChart | |
| namespace OAI | |
| /-! Finite generic intersections on a bounded physical chart follow | |
| from the actual L2 occupation density and the coordinate's differential | |
| energy. No simplicity of the sampled path is needed. | |
| -/ | |
| open Set Function Filter MeasureTheory | |
| open scoped Topology ENNReal NNReal | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| attribute [local instance] sphereMetric sphereBorel | |
| local instance : BorelSpace Sphere := ⟨rfl⟩ | |
| variable [MeasurableSpace C(PathTime, Sphere)] [BorelSpace C(PathTime, Sphere)] | |
| theorem mean_real_chart_circle_count_finite | |
| (hcoarea : PlanarLipschitzFiberCoarea) | |
| {G U : Set Sphere} (e : G ≃ₜ U) (hU : IsOpen U) (y₀ : Sphere) | |
| (π : Sphere → Sphere) (hπ : Continuous π) (he : ∀ y : G, (e y).val = π y.val) | |
| {A D K : Set ℂ} (hA : IsSigmaCompact A) (hK : IsCompact K) (hAK : A ⊆ K) | |
| (hAG : ∀ z ∈ A, (z : Sphere) ∈ G) (hAD : A ⊆ D) (hD : IsOpen D) | |
| {f : ℂ → ℝ} (hf : ContDiffOn ℝ 1 f D) | |
| (henergy : (∫⁻ z in D, ENNReal.ofReal ‖fderiv ℝ f z‖ ^ 2) < ∞) | |
| (μ : ProbabilityMeasure C(PathTime, Sphere)) | |
| (hfinite : ∀ᵐ γ ∂(μ : Measure _), physicalDomainLength e y₀ γ < ∞) | |
| (ρ : Sphere → ℝ≥0∞) (hρ : Measurable ρ) | |
| (hocc : meanPhysicalOccupation e y₀ μ = sphericalArea.withDensity ρ) | |
| (hρ2 : (∫⁻ y, ρ y ^ 2 ∂sphericalArea) < ∞) : | |
| (∫⁻ t : UnitAddCircle, ∫⁻ γ, | |
| pathImageCount ((fun z : ℂ => π (z : Sphere)) '' (A ∩ {z | (f z : UnitAddCircle) = t})) | |
| γ ∂(μ : Measure _)) < ∞ := by | |
| obtain ⟨B, hB⟩ := finiteCoord_lipschitz_on_compact_chart hK | |
| have hAm : MeasurableSet A := by | |
| obtain ⟨Q, hQ, hcov⟩ := hA | |
| rw [← hcov] | |
| exact MeasurableSet.iUnion fun n => (hQ n).measurableSet | |
| have hcoeA : MeasurableSet (((↑) : ℂ → Sphere) '' A) := | |
| OnePoint.isOpenEmbedding_coe.measurableEmbedding.measurableSet_image.mpr hAm | |
| let k : ℂ → ℝ≥0∞ := fun z => ENNReal.ofReal ‖fderiv ℝ f z‖ | |
| let g := localizedSphericalChartWeight B A k | |
| have hk : Measurable k := (measurable_fderiv ℝ f).norm.ennreal_ofReal | |
| have hg : Measurable g := measurable_localizedSphericalChartWeight B hAm hk | |
| have hg2 : (∫⁻ y, g y ^ 2 ∂sphericalArea) < ∞ := | |
| localizedSphericalChartWeight_square_finite B hAm hk | |
| ((lintegral_mono_set hAD).trans_lt henergy) | |
| have hcost : (∫⁻ y, g y ∂meanPhysicalOccupation e y₀ μ) < ∞ := by | |
| rw [← lintegral_physicalOccupationCode e hU y₀ μ g hg] | |
| exact (lintegral_physical_cost_le_square_norms e hU y₀ μ sphericalArea ρ g hρ hg hocc).trans_lt | |
| (ENNReal.mul_lt_top | |
| (ENNReal.rpow_lt_top_of_nonneg (by norm_num) hρ2.ne) | |
| (ENNReal.rpow_lt_top_of_nonneg (by norm_num) hg2.ne)) | |
| have hbound := mean_sigmaCompact_physical_chart_circle_count_le hcoarea e hU y₀ π hπ he | |
| ((↑) : ℂ → Sphere) OnePoint.continuous_coe hA hAG hAD hD finiteCoord | |
| measurable_finiteCoord (fun z _ => finiteCoord_coe z) B (hB.mono (image_mono hAK)) | |
| μ hfinite hf | |
| have hi : (∫⁻ t in Ico (0 : ℝ) 1, ∫⁻ γ, | |
| pathImageCount ((fun z : ℂ => π (z : Sphere)) '' | |
| (A ∩ {z | (f z : UnitAddCircle) = (t : UnitAddCircle)})) γ ∂(μ : Measure _)) < ∞ := by | |
| apply hbound.trans_lt | |
| simpa only [g, localizedSphericalChartWeight, lintegral_const_mul' _ _ ENNReal.coe_ne_top, | |
| lintegral_indicator hcoeA] using hcost | |
| have hc := measurable_sigmaCompact_chart_circle_count | |
| (fun z : ℂ => π (z : Sphere)) (hπ.comp OnePoint.continuous_coe) hA f | |
| (hf.continuousOn.mono hAD) | |
| have hm := (hc.comp measurable_swap).lintegral_prod_right' (ν := (μ : Measure _)) | |
| have heq := (AddCircle.measurePreserving_mk (1 : ℝ) 0).lintegral_comp hm | |
| rw [restrict_Ico_eq_restrict_Ioc] at hi | |
| simp only [zero_add] at heq | |
| dsimp only [Function.comp_def, Prod.swap] at heq | |
| exact heq ▸ hi | |
| theorem ae_real_chart_circle_count_finite | |
| (π : Sphere → Sphere) (hπ : Continuous π) {A : Set ℂ} (hA : IsSigmaCompact A) | |
| (f : ℂ → ℝ) (hf : ContinuousOn f A) | |
| (μ : ProbabilityMeasure C(PathTime, Sphere)) | |
| (hmean : (∫⁻ t : UnitAddCircle, ∫⁻ γ, | |
| pathImageCount ((fun z : ℂ => π (z : Sphere)) '' (A ∩ {z | (f z : UnitAddCircle) = t})) | |
| γ ∂(μ : Measure _)) < ∞) : | |
| ∀ᵐ γ ∂(μ : Measure _), ∀ᵐ t : UnitAddCircle ∂volume, | |
| pathImageCount ((fun z : ℂ => π (z : Sphere)) '' (A ∩ {z | (f z : UnitAddCircle) = t})) γ < ∞ := by | |
| have hc := measurable_sigmaCompact_chart_circle_count | |
| (fun z : ℂ => π (z : Sphere)) (hπ.comp OnePoint.continuous_coe) hA f hf | |
| have hm := (hc.comp measurable_swap).lintegral_prod_right' (ν := (μ : Measure _)) | |
| have hg : ∀ᵐ t : UnitAddCircle ∂volume, ∀ᵐ γ ∂(μ : Measure _), | |
| pathImageCount ((fun z : ℂ => π (z : Sphere)) '' (A ∩ {z | (f z : UnitAddCircle) = t})) γ < ∞ := by | |
| filter_upwards [ae_lt_top hm hmean.ne] with t ht | |
| exact ae_lt_top (hc.comp (measurable_id.prodMk measurable_const)) ht.ne | |
| have hmeas : MeasurableSet {p : C(PathTime, Sphere) × UnitAddCircle | | |
| pathImageCount ((fun z : ℂ => π (z : Sphere)) '' | |
| (A ∩ {z | (f z : UnitAddCircle) = p.2})) p.1 < ∞} := | |
| measurableSet_lt hc measurable_const | |
| exact (Measure.ae_ae_comm hmeas).mpr hg | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |