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8e4d533 | 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 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import Mathlib
import OAI.Geometry.Cannon.Conformal.SphereTile
namespace OAI
section
section
section
noncomputable section
namespace CannonMobius
open Set Metric Topology
open scoped OnePoint
theorem normalize_packing_centers {ν : Type*}
(z : ν → ℂ) (r : ν → ℝ) (hr : ∀ v,0<r v)
(hsep : ∀ u v,u≠v → r u+r v≤dist (z u) (z v))
(b : Fin 3 → ν) (hb : Function.Injective b) :
∃ N : Sphere ≃ₜ Sphere,N∈biholomorphic ∧
(∀ i,N ((z (b i):ℂ):Sphere)=std i) ∧
(∀ v,IsOpen (N '' ((↑) '' ball (z v) (r v) : Set Sphere))) ∧
Pairwise (fun u v=>Disjoint (N '' ((↑) '' ball (z u) (r u) : Set Sphere))
(N '' ((↑) '' ball (z v) (r v) : Set Sphere))) ∧
(∀ v,∃ i j : Fin 3,i≠j ∧
std i∉N '' ((↑) '' ball (z v) (r v) : Set Sphere) ∧
std j∉N '' ((↑) '' ball (z v) (r v) : Set Sphere)) := by
have hz : Function.Injective z := by
intro u v huv
by_contra hne
have hh:=hsep u v hne
rw [huv,dist_self] at hh
linarith [hr u,hr v]
let t : Triple:=⟨fun i=>(z (b i):Sphere),OnePoint.coe_injective.comp (hz.comp hb)⟩
obtain ⟨M,hM,hMt⟩:=exists_triple_normalizer t
let N : Sphere ≃ₜ Sphere:=M.symm
have hN (i : Fin 3) : N (z (b i):Sphere)=std i := by
have H : M (std i)=(z (b i):Sphere):=congrArg (fun t : Triple=>t.val i) hMt
dsimp only [N]
rw [←H,M.symm_apply_apply]
have hout {v : ν} {i : Fin 3} (hi : b i≠v) :
std i∉N '' ((↑) '' ball (z v) (r v) : Set Sphere) := by
rintro ⟨_,⟨w,hw,rfl⟩,hew⟩
have hcoe : (w:Sphere)=(z (b i):Sphere):=N.injective (hew.trans (hN i).symm)
have hew : w=z (b i):=OnePoint.coe_injective hcoe
rw [hew] at hw
have hh:=hsep (b i) v hi
have hh':=mem_ball.mp hw
linarith [hr (b i)]
refine ⟨N,biholomorphic.inv_mem hM,hN,?_,?_,?_⟩
· intro v
exact N.isOpenMap _ (OnePoint.isOpenMap_coe _ isOpen_ball)
· intro u v huv
apply disjoint_left.mpr
rintro q ⟨_,⟨w,hwu,rfl⟩,hwq⟩ ⟨_,⟨w',hwv,rfl⟩,hwq'⟩
have heq : w=w' := OnePoint.coe_injective (N.injective (hwq.trans hwq'.symm))
subst w'
have h1:=mem_ball.mp hwu
have h2:=mem_ball.mp hwv
have ht:=dist_triangle (z u) w (z v)
rw [dist_comm (z u) w] at ht
have hh:=hsep u v huv
linarith
· intro v
have hx : ∃ i j : Fin 3,i≠j ∧ b i≠v ∧ b j≠v := by
by_cases h0 : b 0=v
· refine ⟨1,2,by decide,?_,?_⟩ <;> intro h
· have hh:=hb (h.trans h0.symm)
exact (by decide : (1:Fin 3)≠0) hh
· have hh:=hb (h.trans h0.symm)
exact (by decide : (2:Fin 3)≠0) hh
· by_cases h1 : b 1=v
· refine ⟨0,2,by decide,h0,?_⟩
intro h
have hh:=hb (h.trans h1.symm)
exact (by decide : (2:Fin 3)≠1) hh
· exact ⟨0,1,by decide,h0,h1⟩
obtain ⟨i,j,hij,hi,hj⟩:=hx
exact ⟨i,j,hij,hout hi,hout hj⟩
end CannonMobius
end
end
end
end
end OAI
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