openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /MovingCircleReflectionSides.lean
| 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 | |