openAI-math-dataset / lean /OAI /Analysis /Conductivity /Geometry /LocalFamilyRegionCoordinates.lean
| import Mathlib | |
| import OAI.Analysis.Conductivity.Geometry.ParametricCorrectionRegionPiola | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace ScalarConductivity | |
| open Set Filter Topology | |
| open scoped Matrix.Norms.Elementwise | |
| variable {P : Type} [NormedAddCommGroup P] [NormedSpace ℝ P] [FiniteDimensional ℝ P] | |
| theorem exists_transported_vanishing_region | |
| {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) | |
| {U : Set Coord3} (hYU : Y.source⊆univ×ˢU) | |
| {x : Coord3} (hx : (p,x)∈Y.source) | |
| (hpot : ∀ z∈Y.source,v (z.1,(Y z).2)=u z) | |
| (hlocal : ∀ A : Set Coord3,IsOpen A → (Y (p,x)).2∈A → | |
| ∃ B : VanishingCorrectionRegion v p A,(Y (p,x)).2∈B.region) : | |
| ∃ B : VanishingCorrectionRegion u p U,x∈B.region := by | |
| let z := (Y (p,x)).2 | |
| have hpz : (p,z)∈Y.target := by | |
| have he : (p,z)=Y (p,x) := Prod.ext (hY p x).symm rfl | |
| rw [he] | |
| exact Y.map_source hx | |
| obtain ⟨V₀,A₀,hV₀,hA₀,hp₀,hz₀,hprod₀⟩ := exists_open_product_inside Y.open_target hpz | |
| obtain ⟨δ,hδ,hδball⟩ := Metric.isOpen_iff.mp hV₀ p hp₀ | |
| obtain ⟨η,hη,hηball⟩ := Metric.isOpen_iff.mp hA₀ z hz₀ | |
| let Kp := Metric.closedBall p (δ/2) | |
| let Ka := Metric.closedBall z (η/2) | |
| let A := Metric.ball z (η/2) | |
| have hpK : Kp⊆V₀ := fun q hq => hδball | |
| (lt_of_le_of_lt hq (by linarith : δ/2<δ)) | |
| have haK : Ka⊆A₀ := fun y hy => hηball | |
| (lt_of_le_of_lt hy (by linarith : η/2<η)) | |
| have hjoint : Kp×ˢKa⊆Y.target := fun q hq => hprod₀ ⟨hpK hq.1,haK hq.2⟩ | |
| obtain ⟨C,hC,hbound⟩ := parametric_chart_compact_bounds Y hY hsm hinv | |
| ((isCompact_closedBall _ _).prod (isCompact_closedBall _ _)) hjoint | |
| obtain ⟨B,hzB⟩ := hlocal A Metric.isOpen_ball (Metric.mem_ball_self (by positivity)) | |
| obtain ⟨τ,hτ,hτball⟩ := Metric.isOpen_iff.mp B.region_open z hzB | |
| let K := Metric.closedBall z (τ/2) | |
| have hKB : K⊆B.region := fun y hy => hτball | |
| (lt_of_le_of_lt hy (by linarith : τ/2<τ)) | |
| let O : Set (P×Coord3) := Y.source∩{q | (Y q).2∈Metric.ball z (τ/2)} | |
| have hO : IsOpen O := Y.open_source.inter | |
| (Metric.isOpen_ball.preimage (continuous_snd.comp hsm.continuous)) | |
| have hxO : (p,x)∈O := ⟨hx,Metric.mem_ball_self (by positivity)⟩ | |
| obtain ⟨V₁,W₁,hV₁,hW₁,hp₁,hx₁,hprod₁⟩ := exists_open_product_inside hO hxO | |
| let V := Metric.ball p (δ/2)∩V₁ | |
| let W := W₁∩Metric.ball x 1 | |
| have hVp : p∈V := ⟨Metric.mem_ball_self (by positivity),hp₁⟩ | |
| have hWx : x∈W := ⟨hx₁,Metric.mem_ball_self zero_lt_one⟩ | |
| have hWU : W⊆U := fun y hy => (hYU (hprod₁ (show (p,y)∈V₁×ˢW₁ from ⟨hp₁,hy.1⟩)).1).2 | |
| have hVA : V×ˢA⊆Y.target := fun q hq => hjoint | |
| ⟨Metric.ball_subset_closedBall hq.1.1,Metric.ball_subset_closedBall hq.2⟩ | |
| have hW : ∀ q∈V,W⊆(fiberSlice Y hY q).symm '' K := by | |
| intro q hq y hy | |
| have hh := hprod₁ (show (q,y)∈V₁×ˢW₁ from ⟨hq.2,hy.1⟩) | |
| exact ⟨(Y (q,y)).2,Metric.ball_subset_closedBall hh.2,(fiberSlice Y hY q).left_inv hh.1⟩ | |
| have himage : ∀ q∈V,(fiberSlice Y hY q).symm '' A⊆U := by | |
| rintro q hq _ ⟨y,hy,rfl⟩ | |
| exact (hYU ((fiberSlice Y hY q).map_target (hVA ⟨hq,hy⟩))).2 | |
| have hpot' : ∀ q∈V,∀ y∈A,u (q,(fiberSlice Y hY q).symm y)=v (q,y) := by | |
| intro q hq y hy | |
| have hyt := hVA (show (q,y)∈V×ˢA from ⟨hq,hy⟩) | |
| have hys := (fiberSlice Y hY q).map_target hyt | |
| have hh := hpot (q,(fiberSlice Y hY q).symm y) hys | |
| rw [show (Y (q,(fiberSlice Y hY q).symm y)).2=y from | |
| (fiberSlice Y hY q).right_inv hyt] at hh | |
| exact hh.symm | |
| let D := VanishingCorrectionRegion.piola hu hv p Y hY hsm hinv | |
| (Metric.isOpen_ball.inter hV₁) hVp (Metric.isBounded_ball.subset inter_subset_left) | |
| Metric.isOpen_ball B (isCompact_closedBall _ _) hKB hVA hW | |
| (hW₁.inter Metric.isOpen_ball) ⟨x,hWx⟩ | |
| (Metric.isBounded_ball.subset inter_subset_right) hWU himage hpot' hC | |
| (fun q hq y hy => (hbound (q,y) | |
| ⟨Metric.ball_subset_closedBall hq.1,Metric.ball_subset_closedBall hy⟩).2.2.2) | |
| exact ⟨D,hWx⟩ | |
| end ScalarConductivity | |
| end | |
| end | |
| end OAI | |