openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /AnnularReflectionInjection.lean
| import OAI.Analysis.CircleDomains.Reflection.AnnularCircleReflection | |
| namespace OAI | |
| /-! The explicit annular reflection preserves injectivity and keeps | |
| uniform bounds from the exterior side. -/ | |
| open Set Function Filter Metric Complex | |
| open scoped Topology ComplexConjugate | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma annularCircleReflect_outer_iff {U:Set ℂ} {f:ℂ→ℂ} | |
| (hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U) | |
| (hp:∀z∈U,1<‖z‖→1<‖f z‖) | |
| (hb:∀z∈U,‖z‖=1→‖f z‖=1) {z:ℂ} (hzU:z∈U) : | |
| 1≤‖annularCircleReflect f z‖ ↔ 1≤‖z‖ := by | |
| by_cases hzo : 1≤‖z‖ | |
| · refine iff_of_true ?_ hzo | |
| rw [annularCircleReflect,ite_eq_left hzo] | |
| rcases eq_or_lt_of_le hzo with he|he | |
| · exact (hb z hzU he.symm).ge | |
| · exact (hp z hzU he).le | |
| · have hsmall : ‖z‖<1 := lt_of_not_ge hzo | |
| have href := unitCircleReflection_strict_outer (hz z hzU) hsmall | |
| have hf := hp _ (hs z hzU) href | |
| have hlt : ‖annularCircleReflect f z‖<1 := by | |
| rw [annularCircleReflect,ite_eq_right hzo,unitCircleReflection_norm] | |
| exact (inv_lt_one₀ (zero_lt_one.trans hf)).mpr hf | |
| exact iff_of_false (not_le_of_gt hlt) hzo | |
| lemma injOn_annularCircleReflect {U:Set ℂ} {f:ℂ→ℂ} | |
| (hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U) | |
| (hi:InjOn f (U∩{z | 1≤‖z‖})) | |
| (hp:∀z∈U,1<‖z‖→1<‖f z‖) | |
| (hb:∀z∈U,‖z‖=1→‖f z‖=1) : InjOn (annularCircleReflect f) U := by | |
| intro z hzU w hwU he | |
| have hside : (1≤‖z‖) ↔ 1≤‖w‖ := by | |
| rw [←annularCircleReflect_outer_iff hz hs hp hb hzU, | |
| ←annularCircleReflect_outer_iff hz hs hp hb hwU,he] | |
| by_cases hzo : 1≤‖z‖ | |
| · have hwo := hside.mp hzo | |
| rw [annularCircleReflect,ite_eq_left hzo,annularCircleReflect,ite_eq_left hwo] at he | |
| exact hi ⟨hzU,hzo⟩ ⟨hwU,hwo⟩ he | |
| · have hwo : ¬1≤‖w‖ := fun h => hzo (hside.mpr h) | |
| rw [annularCircleReflect,ite_eq_right hzo,annularCircleReflect,ite_eq_right hwo] at he | |
| have he' := unitCircleReflection_involutive.injective he | |
| have he'' := hi ⟨hs z hzU,unitCircleReflection_outer (hz z hzU) (lt_of_not_ge hzo).le⟩ | |
| ⟨hs w hwU,unitCircleReflection_outer (hz w hwU) (lt_of_not_ge hwo).le⟩ he' | |
| exact unitCircleReflection_involutive.injective he'' | |
| lemma annularCircleReflect_norm_le {U:Set ℂ} {f:ℂ→ℂ} {M:ℝ} | |
| (hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U) (hM:1≤M) | |
| (hf:∀z∈U,1≤‖z‖→1≤‖f z‖ ∧ ‖f z‖≤M) {z:ℂ} (hzU:z∈U) : | |
| ‖annularCircleReflect f z‖≤M := by | |
| by_cases hzo : 1≤‖z‖ | |
| · rw [annularCircleReflect,ite_eq_left hzo] | |
| exact (hf z hzU hzo).2 | |
| · rw [annularCircleReflect,ite_eq_right hzo,unitCircleReflection_norm] | |
| have hh := (hf _ (hs z hzU) | |
| (unitCircleReflection_outer (hz z hzU) (lt_of_not_ge hzo).le)).1 | |
| exact ((inv_le_one₀ (zero_lt_one.trans_le hh)).mpr hh).trans hM | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |