openAI-math-dataset / lean /OAI /Analysis /Conductivity /Variational /CompactDepthReductionRegular.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Variational.TsupportFiniteSumSubset | |
| import OAI.Analysis.Conductivity.Variational.Unchanged | |
| namespace OAI | |
| noncomputable section | |
| open MeasureTheory | |
| open scoped ENNReal | |
| open Matrix Filter Topology | |
| open Set MeasureTheory Filter Topology | |
| open scoped BigOperators | |
| open Set MeasureTheory Filter Topology | |
| open scoped Manifold | |
| open Set Filter | |
| open scoped Topology | |
| open Set Filter MeasureTheory | |
| open scoped Topology Manifold ENNReal | |
| open Set | |
| namespace ScalarConductivity | |
| open Matrix Set MeasureTheory Filter Topology | |
| open scoped Matrix.Norms.Elementwise | |
| theorem exists_compact_depth_reduction_regular | |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] | |
| (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) | |
| {U : Set Coord3} (hU : IsOpen U) (hUb : Bornology.IsBounded U) | |
| (hu : ContDiffOn ℝ (↑(⊤ : ℕ∞)) u U) | |
| (hA : ContDiffOn ℝ (↑(⊤ : ℕ∞)) (fun x => (A x).val) U) | |
| (hdiv : ∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → HasCompactSupport ψ → | |
| tsupport ψ ⊆ U → (∫ x, fderiv ℝ ψ x ((conductivityFlux u A x).col j) ∂μ) = 0) | |
| (hr : TwoFieldRankRegular u U) | |
| {a b : ℝ} (ha : 0 < a) {T : Set DiagonalTriple} | |
| (hT : IsOpen T) (hsub : ∀ d ∈ T, IsFiniteLaminate a b d) | |
| (hperm : ∀ d ∈ T, ∀ e : Equiv.Perm (Fin 3), d ∘ e ∈ T) | |
| {p : Coord3} (hpU : p ∈ U) {n : ℕ} | |
| (hp : A p ∈ spectralExtension (depthClass T (n+1))) | |
| (hsc : (spectralExtension (depthClass T n)).Nonempty) : | |
| ∃ W : Set Coord3, IsOpen W ∧ p ∈ W ∧ closure W ⊆ U ∧ IsCompact (closure W) ∧ | |
| ∀ O : Set Coord3, IsOpen O → O ⊆ W → ∀ ε : ℝ, 0 < ε → | |
| ∃ R : CompactCompatibleReplacement μ O u A, | |
| TwoFieldRankRegular (fun x => u x+R.du x) O ∧ | |
| (∀ x ∈ O, R.tensor x ∈ matrixFiniteLaminate a b) ∧ | |
| μ {x | x ∈ O ∧ R.tensor x ∉ spectralExtension (depthClass T n)} ≤ ENNReal.ofReal ε := by | |
| classical | |
| by_cases hn : A p ∈ spectralExtension (depthClass T n) | |
| · have hTo := isOpen_spectralExtension (isOpen_depthClass ha hT hsub n) (depthClass_permute hperm n) | |
| have hAc : ContinuousAt A p := Topology.IsInducing.subtypeVal.continuousAt_iff.mpr | |
| (hA.continuousOn.continuousAt (hU.mem_nhds hpU)) | |
| have hev : ∀ᶠ x in 𝓝 p, x ∈ U ∧ A x ∈ spectralExtension (depthClass T n) := | |
| Filter.Eventually.and (hU.mem_nhds hpU) (hAc.preimage_mem_nhds (hTo.mem_nhds hn)) | |
| obtain ⟨V, hVs, hV, hpV⟩ := mem_nhds_iff.mp hev | |
| obtain ⟨W, hW, hpW, hWV, hWc⟩ := exists_precompact_open_inside hV hpV | |
| have hVU : V ⊆ U := fun x hx => (hVs hx).1 | |
| refine ⟨W, hW, hpW, hWV.trans hVU, hWc, ?_⟩ | |
| intro O hO hOW ε _ | |
| have hOV := hOW.trans (subset_closure.trans hWV) | |
| have hOU := hOV.trans hVU | |
| let R := CompactCompatibleReplacement.unchanged μ O u A (hA.mono hOU) | |
| refine ⟨R, ?_, fun x hx => spectral_depth_subset hsub n (hVs (hOV hx)).2, ?_⟩ | |
| · simpa only [R, CompactCompatibleReplacement.unchanged, Pi.zero_apply, add_zero] using hr.mono hOU | |
| · have he : {x | x ∈ O ∧ R.tensor x ∉ spectralExtension (depthClass T n)} = ∅ := by | |
| ext x | |
| simp only [mem_ofPred_eq, mem_empty_iff_false, iff_false, not_and, not_not] | |
| intro hx | |
| exact (hVs (hOV hx)).2 | |
| rw [he, measure_empty] | |
| exact bot_le | |
| · exact exists_compact_depth_reduction_regular_nontrivial μ u A hU hUb hu hA hdiv hr | |
| ha hT hsub hperm hpU hp hn hsc | |
| end ScalarConductivity | |
| namespace ScalarConductivity | |
| open Matrix Set MeasureTheory Filter Topology | |
| open scoped Matrix.Norms.Elementwise | |
| theorem assemble_compatible_replacements | |
| (μ : Measure Coord3) [μ.IsAddHaarMeasure] | |
| {I : Type*} [Fintype I] [DecidableEq I] | |
| (O : I → Set Coord3) (hO : ∀ i, IsOpen (O i)) | |
| (hd : Pairwise (fun i j => Disjoint (O i) (O j))) | |
| (u : Coord3 → Fin 2 → ℝ) (A : Coord3 → Symmetric3) | |
| (R : ∀ i, CompactCompatibleReplacement μ (O i) u A) : | |
| ∃ (du : Coord3 → Fin 2 → ℝ) (dF : Coord3 → Matrix (Fin 3) (Fin 2) ℝ), | |
| ContDiff ℝ (↑(⊤ : ℕ∞)) du ∧ HasCompactSupport du ∧ tsupport du ⊆ ⋃ i, O i ∧ | |
| ContDiff ℝ (↑(⊤ : ℕ∞)) dF ∧ HasCompactSupport dF ∧ tsupport dF ⊆ ⋃ i, O i ∧ | |
| (∀ j (ψ : Coord3 → ℝ), ContDiff ℝ (↑(⊤ : ℕ∞)) ψ → | |
| (∫ x, fderiv ℝ ψ x ((dF x).col j) ∂μ) = 0) ∧ | |
| (∀ i, EqOn du (R i).du (O i) ∧ EqOn dF (R i).dF (O i) ∧ | |
| ∀ x ∈ O i, | |
| fderiv ℝ (fun y => u y+du y) x = fderiv ℝ (fun y => u y+(R i).du y) x) ∧ | |
| (∀ x ∉ ⋃ i, O i, du x = 0 ∧ dF x = 0) := by | |
| let du : Coord3 → Fin 2 → ℝ := fun x => ∑ i, (R i).du x | |
| let dF : Coord3 → Matrix (Fin 3) (Fin 2) ℝ := fun x => ∑ i, (R i).dF x | |
| have heu : ∀ i, EqOn du (R i).du (O i) := | |
| supported_sum_eq_on O hd (fun i => (R i).du) (fun i => (R i).support_du) | |
| have heF : ∀ i, EqOn dF (R i).dF (O i) := | |
| supported_sum_eq_on O hd (fun i => (R i).dF) (fun i => (R i).support_dF) | |
| have hcdu : HasCompactSupport du := by | |
| simpa only [du, ← Finset.sum_apply] using | |
| (HasCompactSupport.finset_sum (s := Finset.univ) (fun i _ => (R i).compact_du)) | |
| have hcdF : HasCompactSupport dF := by | |
| simpa only [dF, ← Finset.sum_apply] using | |
| (HasCompactSupport.finset_sum (s := Finset.univ) (fun i _ => (R i).compact_dF)) | |
| refine ⟨du, dF, ContDiff.sum (fun i _ => (R i).smooth_du), | |
| hcdu, | |
| (tsupport_finite_sum_subset _).trans (iUnion_mono (fun i => (R i).support_du)), | |
| ContDiff.sum (fun i _ => (R i).smooth_dF), | |
| hcdF, | |
| (tsupport_finite_sum_subset _).trans (iUnion_mono (fun i => (R i).support_dF)), ?_, ?_, ?_⟩ | |
| · exact finite_sum_cauchy_preserving μ (fun i => (R i).dF) | |
| (fun i => (R i).smooth_dF) (fun i => (R i).compact_dF) (fun i => (R i).cauchy_dF) | |
| · intro i | |
| refine ⟨heu i, heF i, ?_⟩ | |
| intro x hx | |
| apply Filter.EventuallyEq.fderiv_eq | |
| filter_upwards [(hO i).mem_nhds hx] with y hy | |
| rw [heu i hy] | |
| · intro x hx | |
| exact ⟨supported_sum_eq_zero_off O _ (fun i => (R i).support_du) hx, | |
| supported_sum_eq_zero_off O _ (fun i => (R i).support_dF) hx⟩ | |
| end ScalarConductivity | |
| end | |
| end OAI | |