openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /AnalyticConfigurationCollar.lean
| import OAI.Analysis.CircleDomains.Reflection.UnitAnalyticJordanParametrization | |
| import OAI.Analysis.CircleDomains.Transfer.GlobalCoreExtension | |
| namespace OAI | |
| /-! A fixed analytic source collar for a chosen actual hole. Its outer | |
| side lies in the multiply connected domain and avoids every other cap. -/ | |
| open Set Function Filter Metric Complex | |
| open scoped Topology OnePoint | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| open FiniteTransfer | |
| namespace FiniteJordanConfiguration | |
| variable {I:Type*} [Finite I] | |
| lemma exists_analytic_boundary_collar (C:FiniteJordanConfiguration I) (i:I) | |
| (ha:IsAnalyticJordanCap (C.hole i)) : | |
| ∃R:ℝ, ∃β:ℂ→ℂ, 1<R ∧ | |
| DifferentiableOn ℂ β (symmetricCircleAnnulus R) ∧ | |
| InjOn β (symmetricCircleAnnulus R) ∧ | |
| β '' sphere 0 1=frontier (jordanInterior (C.hole i)) ∧ | |
| (∀z∈symmetricCircleAnnulus R,1<‖z‖ → (β z:Sphere)∈C.region) ∧ | |
| (∀z∈symmetricCircleAnnulus R,1≤‖z‖ → (β z:Sphere)∈closure C.region) := by | |
| obtain ⟨S,β,hS,hDS,hβ,hi,hclosed,hboundary⟩ := | |
| exists_unit_analytic_jordan_parametrization ha | |
| let B : Set ℂ := ⋃j:{j:I // j≠i},closure (jordanInterior (C.hole j)) | |
| have hB : IsClosed B := isClosed_iUnion_of_finite fun _ => isClosed_closure | |
| let T := S∩β ⁻¹' Bᶜ | |
| have hT : IsOpen T := hβ.continuousOn.isOpen_inter_preimage hS hB.isOpen_compl | |
| have hcircle : sphere (0:ℂ) 1⊆T := by | |
| intro z hz | |
| refine ⟨hDS (sphere_subset_closedBall hz),?_⟩ | |
| intro hzB | |
| obtain ⟨j,hj⟩ := mem_iUnion.mp hzB | |
| have hzi : β z∈closure (jordanInterior (C.hole i)) := | |
| frontier_subset_closure (hboundary ▸ mem_image_of_mem β hz) | |
| exact Set.disjoint_left.mp (C.disjoint j.property.symm) hzi hj | |
| obtain ⟨R,hR,hRT⟩ := exists_symmetricCircleAnnulus_subset hT hcircle | |
| have hout : ∀z∈symmetricCircleAnnulus R,1<‖z‖ → (β z:Sphere)∈C.region := by | |
| intro z hz hzn hp | |
| obtain ⟨j,w,hw,he⟩ := mem_iUnion.mp hp | |
| have hwb : w=β z := OnePoint.coe_injective he | |
| subst w | |
| by_cases hji : j=i | |
| · subst j | |
| rw [←hclosed] at hw | |
| obtain ⟨w,hw,he⟩ := hw | |
| have hwz : w=z := hi (hDS hw) (hRT hz).1 he | |
| rw [hwz] at hw | |
| exact (not_le_of_gt hzn) (mem_closedBall_zero_iff.mp hw) | |
| · exact (hRT hz).2 (mem_iUnion.mpr ⟨⟨j,hji⟩,hw⟩) | |
| refine ⟨R,β,hR,hβ.mono (hRT.trans inter_subset_left), | |
| hi.mono (hRT.trans inter_subset_left),hboundary,hout,?_⟩ | |
| intro z hz hzn | |
| rcases eq_or_lt_of_le hzn with he|he | |
| · apply C.boundary_subset_closure_region i | |
| rw [frontier_closedTerminalCap] | |
| exact ⟨β z,hboundary ▸ mem_image_of_mem β (mem_sphere_zero_iff_norm.mpr he.symm),rfl⟩ | |
| · exact subset_closure (hout z hz he) | |
| end FiniteJordanConfiguration | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |