| import OAI.Analysis.CircleDomains.Reflection.ReflectedCircleImages | |
| import OAI.Analysis.CircleDomains.Topology.HomeomorphRoundDiskImage | |
| namespace OAI | |
| /-! The actual invariant-coefficient straightening carries every closed | |
| analytic Jordan cap onto a positive-radius closed round disk. -/ | |
| open Set Function Metric | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| open FiniteTransfer | |
| namespace AnalyticReflectionData | |
| variable {I : Type*} [Finite I] {C : FiniteJordanConfiguration I} | |
| lemma exists_round_cap_images (D : AnalyticReflectionData C) : | |
| ∃s : I → ℝ,(∀i,0<s i) ∧ ∀i, | |
| (D.comparison.trans D.principal) '' closure (jordanInterior (C.hole i))= | |
| closedBall (D.principal (D.center i)) (s i) := by | |
| choose s hs he using D.exists_round_boundary_image | |
| refine ⟨s,hs,?_⟩ | |
| intro i | |
| change (D.principal ∘ D.comparison) '' closure (jordanInterior (C.hole i))=_ | |
| rw [image_comp,D.cap_image] | |
| exact homeomorph_image_closedBall_of_sphere D.principal (D.radius_pos i) (hs i) (he i) | |
| end AnalyticReflectionData | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |