openAI-math-dataset / lean /OAI /Analysis /Conductivity /Geometry /ParametricCorrectionRegionPiola.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Sources.ParametricPullback | |
| import OAI.Analysis.Conductivity.Variational.LocalBoundedPiolaCorrection | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set Filter Topology Matrix MeasureTheory | |
| open scoped Matrix.Norms.Elementwise | |
| variable {P : Type} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P] | |
| def VanishingCorrectionRegion.piola | |
| {u v : P×Coord3 → Fin 2 → ℝ} | |
| (hu : ContDiff ℝ (↑(⊤:ℕ∞)) u) (hv : ContDiff ℝ (↑(⊤:ℕ∞)) v) (p : P) | |
| (Y : OpenPartialHomeomorph (P×Coord3) (P×Coord3)) | |
| (hY : ∀ q x,(Y (q,x)).1=q) | |
| (hsm : ContDiff ℝ (↑(⊤:ℕ∞)) Y) | |
| (hinv : ContDiffOn ℝ (↑(⊤:ℕ∞)) Y.symm Y.target) | |
| {V : Set P} (hV : IsOpen V) (hp : p∈V) (hVb : Bornology.IsBounded V) | |
| {A U W K : Set Coord3} (hA : IsOpen A) (B : VanishingCorrectionRegion v p A) | |
| (hK : IsCompact K) (hKB : K⊆B.region) (hVA : V×ˢA⊆Y.target) | |
| (hW : ∀ q∈V,W⊆(fiberSlice Y hY q).symm '' K) | |
| (hWo : IsOpen W) (hWn : W.Nonempty) (hWb : Bornology.IsBounded W) (hWU : W⊆U) | |
| (himage : ∀ q∈V,(fiberSlice Y hY q).symm '' A⊆U) | |
| (hpot : ∀ q∈V,∀ y∈A,u (q,(fiberSlice Y hY q).symm y)=v (q,y)) | |
| {C : ℝ} (hC : 0<C) | |
| (hbound : ∀ q∈V,∀ y∈A, | |
| let X := (fiberSlice Y hY q).symm | |
| 9*|(fderiv ℝ X y).det|⁻¹*‖operatorMatrix (fderiv ℝ X y)‖* | |
| ‖operatorMatrix (fderiv ℝ X y)‖≤C) : | |
| VanishingCorrectionRegion u p U := by | |
| refine ⟨W,hWo,hWn,hWU,hWb,?_⟩ | |
| intro r hr ε hε | |
| obtain ⟨R,hR,hRe⟩ := exists_parametric_source_pullback Y hY hsm hinv p hV hp hVb | |
| B.region_open B.region_bounded hK hKB (fun z hz => hVA ⟨hz.1,B.region_subset hz.2⟩) hW hr | |
| have hsolve := B.solve R hR (ε/C) (div_pos hε hC) | |
| filter_upwards [hRe,hsolve,hV.mem_nhds hp] with q he solve hq hm | |
| let X := (fiberSlice Y hY q).symm | |
| have hX : ContDiffOn ℝ (↑(⊤:ℕ∞)) X X.source := (fiberSlice_smooth Y hY hsm hinv q).2 | |
| have hXi : ContDiffOn ℝ (↑(⊤:ℕ∞)) X.symm X.target := | |
| (fiberSlice_smooth Y hY hsm hinv q).1.contDiffOn | |
| have hAs : A⊆X.source := fun y hy => hVA ⟨hq,hy⟩ | |
| have huq : ContDiff ℝ (↑(⊤:ℕ∞)) (fun y => u (q,y)) := hu.comp (contDiff_const.prodMk contDiff_id) | |
| have hvq : ContDiff ℝ (↑(⊤:ℕ∞)) (fun y => v (q,y)) := hv.comp (contDiff_const.prodMk contDiff_id) | |
| have hps : EqOn (fun y => u (q,y)) ((fun y => v (q,y))∘X.symm) (X '' A) := by | |
| rintro _ ⟨y,hy,rfl⟩ | |
| change u (q,X y)=v (q,X.symm (X y)) | |
| rw [X.left_inv (hAs hy)] | |
| exact hpot q hq y hy | |
| have hrX (j) : tsupport (fun y => r j (q,y))⊆X.target := | |
| (hr.supported q j).trans ((hW q hq).trans (by | |
| rintro _ ⟨y,hy,rfl⟩ | |
| exact X.map_source (hAs (B.region_subset (hKB hy))))) | |
| have hre : (fun j => localPiolaSource X (fun y => R j (q,y)))=(fun j y => r j (q,y)) := by | |
| funext j | |
| rw [congrFun he j] | |
| exact localPiolaSource_inverse X (hX.differentiableOn (by simp)) | |
| (hXi.differentiableOn (by simp)) _ ((subset_tsupport _).trans (hrX j)) | |
| have hRs : PairSupported (fun j y => R j (q,y)) A := (hR.supported q).mono B.region_subset | |
| have hRm : physicalSourceMoment (fun y => v (q,y)) (fun j y => R j (q,y))=0 := by | |
| have hh := physicalSourceMoment_piola_local X hX hXi huq hvq hAs hps (hR.compact q) hRs | |
| rw [hre,hm] at hh | |
| exact hh.symm | |
| have hh := (solve hRm).piola_local X hX hXi huq hvq hA hAs hps hC.le | |
| (hbound q hq) (hR.compact q) hRs | |
| rw [hre,mul_div_cancel₀ ε hC.ne'] at hh | |
| exact hh.region_mono (himage q hq) | |
| end ScalarConductivity | |
| end | |
| end | |
| end OAI | |