openAI-math-dataset / lean /OAI /Analysis /Conductivity /Variational /CompactParametricIntegral.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Model | |
| namespace OAI | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set MeasureTheory Filter Topology Metric | |
| variable {E P G : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [ProperSpace E] | |
| [NormedAddCommGroup P] [NormedSpace ℝ P] [SecondCountableTopology P] | |
| [MeasurableSpace P] [BorelSpace P] [NormedAddCommGroup G] [NormedSpace ℝ G] | |
| [CompleteSpace G] {μ : Measure P} [IsLocallyFiniteMeasure μ] | |
| omit [NormedSpace ℝ P] [CompleteSpace G] in | |
| lemma hasFDerivAt_compact_integral {F : E × P → G} {D : E × P → E →L[ℝ] G} | |
| (hF : Continuous F) (hD : Continuous D) | |
| (hderiv : ∀ x p, HasFDerivAt (fun y => F (y,p)) (D (x,p)) x) | |
| {K : Set P} (hK : IsCompact K) (x : E) : | |
| HasFDerivAt (fun y => ∫ p in K, F (y,p) ∂μ) (∫ p in K, D (x,p) ∂μ) x := by | |
| obtain ⟨C,hC⟩ := ((isCompact_closedBall x 1).prod hK).exists_bound_of_continuousOn | |
| hD.continuousOn | |
| have hFd (y : E) : Continuous (fun p => F (y,p)) := | |
| hF.comp (continuous_const.prodMk continuous_id) | |
| have hDd (y : E) : Continuous (fun p => D (y,p)) := | |
| hD.comp (continuous_const.prodMk continuous_id) | |
| have hi : Integrable (fun _ : P => C) (μ.restrict K) := | |
| ContinuousOn.integrableOn_compact hK continuous_const.continuousOn | |
| apply hasFDerivAt_integral_of_dominated_of_fderiv_le | |
| (μ := μ.restrict K) (s := closedBall x 1) (bound := fun _ => C) | |
| (closedBall_mem_nhds x (by norm_num)) | |
| (Filter.Eventually.of_forall fun y => (hFd y).aestronglyMeasurable) | |
| (ContinuousOn.integrableOn_compact hK (hFd x).continuousOn) | |
| (hDd x).aestronglyMeasurable _ hi | |
| · exact Filter.Eventually.of_forall (fun p y _ => hderiv y p) | |
| · filter_upwards [ae_restrict_mem hK.measurableSet] with p hp | |
| intro y hy | |
| exact hC (y,p) ⟨hy,hp⟩ | |
| lemma contDiff_compact_integral_nat (n : ℕ) {F : E × P → G} | |
| (hF : ContDiff ℝ n F) {K : Set P} (hK : IsCompact K) : | |
| ContDiff ℝ n (fun x => ∫ p in K, F (x,p) ∂μ) := by | |
| induction n generalizing G with | |
| | zero => | |
| exact contDiff_zero.mpr (continuous_parametric_integral_of_continuous (f := fun x p => F (x,p)) hF.continuous hK) | |
| | succ n ih => | |
| let D : E × P → E →L[ℝ] G := fun z => | |
| (fderiv ℝ F z).comp (ContinuousLinearMap.inl ℝ E P) | |
| have hD : ContDiff ℝ n D := by | |
| exact (hF.fderiv_right (by norm_cast)).clm_comp contDiff_const | |
| have hderiv : ∀ x p, HasFDerivAt (fun y => F (y,p)) (D (x,p)) x := by | |
| intro x p | |
| have hf := (hF.differentiable (by norm_cast)).differentiableAt.hasFDerivAt | |
| (x := (x,p)) | |
| exact hf.comp x ((hasFDerivAt_id x).prodMk (hasFDerivAt_const p x)) | |
| rw [Nat.cast_add, Nat.cast_one] | |
| apply contDiff_succ_iff_hasFDerivAt.mpr | |
| exact ⟨fun x => ∫ p in K, D (x,p) ∂μ, ih hD, | |
| hasFDerivAt_compact_integral hF.continuous hD.continuous hderiv hK⟩ | |
| lemma contDiff_compact_integral {F : E × P → G} | |
| (hF : ContDiff ℝ (↑(⊤ : ℕ∞)) F) {K : Set P} (hK : IsCompact K) : | |
| ContDiff ℝ (↑(⊤ : ℕ∞)) (fun x => ∫ p in K, F (x,p) ∂μ) := by | |
| apply contDiff_infty.mpr | |
| intro n | |
| exact contDiff_compact_integral_nat n (contDiff_infty.mp hF n) hK | |
| end ScalarConductivity | |
| end | |
| end OAI | |