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9655094 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | 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
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