openAI-math-dataset / lean /OAI /MathematicalPhysics /NavierStokes /ForcedComputation /Detector /CylinderCompactCalculus.lean
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Detector.CylinderLocalEnergyInput | |
| import OAI.MathematicalPhysics.NavierStokes.ForcedComputation.Scalar.PlaneTestIntegral | |
| namespace OAI | |
| /-! Compact horizontal localization of integrals over one vertical period. | |
| These lemmas are the measure-theoretic part of the localized energy identity. -/ | |
| noncomputable section | |
| namespace ForcedComputation.VelocityDetector.CylinderLocalCalculus | |
| open ShearFlows Set MeasureTheory Filter | |
| open scoped Topology | |
| local instance : IsLocallyFiniteMeasure cylinderMeasure := by | |
| unfold cylinderMeasure verticalPeriodMeasure | |
| infer_instance | |
| theorem ae_vertical_period : ∀ᵐ y ∂cylinderMeasure, y.2 ∈ Icc (0 : ℝ) 1 := by | |
| apply (Measure.ae_prod_iff_ae_ae | |
| (measurableSet_Icc.preimage measurable_snd)).2 | |
| exact Eventually.of_forall (fun _ => ae_restrict_mem measurableSet_Icc) | |
| theorem integrable_horizontal_compact {f : Plane × ℝ → ℝ} | |
| (hf : Continuous f) {K : Set Plane} (hK : IsCompact K) | |
| (hs : ∀ y, y.1 ∉ K → f y = 0) : Integrable f cylinderMeasure := by | |
| have hi : IntegrableOn f (K ×ˢ Icc (0 : ℝ) 1) cylinderMeasure := | |
| hf.continuousOn.integrableOn_compact (hK.prod isCompact_Icc) | |
| apply hi.integrable_of_ae_notMem_eq_zero | |
| filter_upwards [ae_vertical_period] with y hy | |
| intro hn | |
| exact hs y (fun hx => hn ⟨hx, hy⟩) | |
| private theorem slice_continuous {g : ℝ × (Plane × ℝ) → ℝ} {a b t : ℝ} | |
| (hg : ContinuousOn g (Icc a b ×ˢ univ)) (ht : t ∈ Icc a b) : | |
| Continuous (fun y => g (t, y)) := | |
| hg.comp_continuous (continuous_const.prodMk continuous_id) | |
| (fun _ => ⟨ht, mem_univ _⟩) | |
| theorem compact_integral_hasDerivAt {g d : ℝ × (Plane × ℝ) → ℝ} {a b t : ℝ} | |
| (hg : ContinuousOn g (Icc a b ×ˢ univ)) | |
| (hd : ContinuousOn d (Icc a b ×ˢ univ)) | |
| (he : ∀ s ∈ Ioo a b, ∀ y, HasDerivAt (fun r => g (r, y)) (d (s, y)) s) | |
| {K : Set Plane} (hK : IsCompact K) (ht : t ∈ Ioo a b) : | |
| HasDerivAt (fun r => ∫ y in K ×ˢ Icc (0 : ℝ) 1, g (r, y) ∂cylinderMeasure) | |
| (∫ y in K ×ˢ Icc (0 : ℝ) 1, d (t, y) ∂cylinderMeasure) t := by | |
| let S := K ×ˢ Icc (0 : ℝ) 1 | |
| have hS : IsCompact S := hK.prod isCompact_Icc | |
| obtain ⟨B, hB⟩ := (isCompact_Icc.prod hS).bddAbove_image | |
| (hd.norm.mono (fun _ hp => ⟨hp.1, mem_univ _⟩)) | |
| have hb : ∀ᵐ y ∂(cylinderMeasure.restrict S), | |
| ∀ s ∈ Ioo a b, ‖d (s, y)‖ ≤ B := by | |
| filter_upwards [ae_restrict_mem hS.measurableSet] with y hy | |
| intro s hs | |
| exact hB (mem_image_of_mem _ ⟨Ioo_subset_Icc_self hs, hy⟩) | |
| have hm : ∀ᶠ s in 𝓝 t, | |
| AEStronglyMeasurable (fun y => g (s, y)) (cylinderMeasure.restrict S) := by | |
| filter_upwards [Ioo_mem_nhds ht.1 ht.2] with s hs | |
| exact (slice_continuous hg (Ioo_subset_Icc_self hs)).aestronglyMeasurable | |
| have hi : IntegrableOn (fun y => g (t, y)) S cylinderMeasure := | |
| (slice_continuous hg (Ioo_subset_Icc_self ht)).continuousOn.integrableOn_compact hS | |
| have hdm : AEStronglyMeasurable (fun y => d (t, y)) (cylinderMeasure.restrict S) := | |
| (slice_continuous hd (Ioo_subset_Icc_self ht)).aestronglyMeasurable | |
| exact (hasDerivAt_integral_of_dominated_loc_of_deriv_le (Ioo_mem_nhds ht.1 ht.2) | |
| hm hi hdm hb (integrableOn_const hS.measure_ne_top) | |
| (Eventually.of_forall (fun y s hs => he s hs y))).2 | |
| theorem integral_restrict_eq_of_horizontal_support {f : Plane × ℝ → ℝ} {K : Set Plane} | |
| (hs : ∀ y, y.1 ∉ K → f y = 0) : | |
| (∫ y in K ×ˢ Icc (0 : ℝ) 1, f y ∂cylinderMeasure) = ∫ y, f y ∂cylinderMeasure := by | |
| apply setIntegral_eq_integral_of_ae_compl_eq_zero | |
| filter_upwards [ae_vertical_period] with y hy | |
| intro hn | |
| exact hs y (fun hx => hn ⟨hx, hy⟩) | |
| theorem supported_integral_hasDerivAt {g d : ℝ × (Plane × ℝ) → ℝ} {a b t : ℝ} | |
| (hg : ContinuousOn g (Icc a b ×ˢ univ)) | |
| (hd : ContinuousOn d (Icc a b ×ˢ univ)) | |
| (he : ∀ s ∈ Ioo a b, ∀ y, HasDerivAt (fun r => g (r, y)) (d (s, y)) s) | |
| {K : Set Plane} (hK : IsCompact K) | |
| (hs : ∀ r y, y.1 ∉ K → g (r, y) = 0) | |
| (hds : ∀ y, y.1 ∉ K → d (t, y) = 0) (ht : t ∈ Ioo a b) : | |
| HasDerivAt (fun r => ∫ y, g (r, y) ∂cylinderMeasure) | |
| (∫ y, d (t, y) ∂cylinderMeasure) t := by | |
| have h := compact_integral_hasDerivAt hg hd he hK ht | |
| have hi : (fun r => ∫ y in K ×ˢ Icc (0 : ℝ) 1, g (r, y) ∂cylinderMeasure) = | |
| (fun r => ∫ y, g (r, y) ∂cylinderMeasure) := | |
| funext (fun r => integral_restrict_eq_of_horizontal_support (hs r)) | |
| rw [hi, integral_restrict_eq_of_horizontal_support hds] at h | |
| exact h | |
| end ForcedComputation.VelocityDetector.CylinderLocalCalculus | |
| end | |
| end OAI | |