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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 33 | import OAI.Analysis.CircleDomains.Reflection.MovingCircleReflection
namespace OAI
/-! Exact interior modulus control for the moving-circle reflection. -/
open Set Function Filter Metric Complex
open scoped Topology ComplexConjugate
noncomputable section
namespace CircleDomainRigidity
lemma circleReflectionExtension_inside_norm_le {U:Set ℂ} {f:ℂ→ℂ} {c:ℂ} {r:ℝ}
(hr:0<r) (hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U)
(hf:∀z∈U,1≤‖z‖ → r≤‖f z-c‖) {z:ℂ} (hzU:z∈U) (hzn:‖z‖<1) :
‖circleReflectionExtension c r f z-c‖≤r := by
let g := fun w => (f w-c)/(r:ℂ)
have hge : 1≤‖g (unitCircleReflection z)‖ := by
change 1≤‖(f (unitCircleReflection z)-c)/(r:ℂ)‖
rw [normalized_circle_norm c hr]
exact (one_le_div hr).mpr
(hf _ (hs z hzU) (unitCircleReflection_outer (hz z hzU) hzn.le))
have hsmall : ‖annularCircleReflect g z‖≤1 := by
rw [annularCircleReflect,ite_eq_right (not_le_of_gt hzn),unitCircleReflection_norm]
exact (inv_le_one₀ (zero_lt_one.trans_le hge)).mpr hge
change ‖c+(r:ℂ)*annularCircleReflect g z-c‖≤r
rw [add_sub_cancel_left,norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hr]
nlinarith
end CircleDomainRigidity
end
end OAI
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