openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /MovingCircleReflectionInjection.lean
| import OAI.Analysis.CircleDomains.Reflection.MovingCircleReflection | |
| namespace OAI | |
| /-! Injectivity of the actual unnormalized reflection in a moving circle. -/ | |
| open Set Function Filter Metric Complex | |
| open scoped Topology | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma injOn_circleReflectionExtension {U:Set ℂ} {f:ℂ→ℂ} {c:ℂ} {r:ℝ} | |
| (hr:0<r) (hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U) | |
| (hi:InjOn f (U∩{z | 1≤‖z‖})) | |
| (ho:∀z∈U,1<‖z‖ → r<‖f z-c‖) | |
| (hb:∀z∈U,‖z‖=1 → ‖f z-c‖=r) : | |
| InjOn (circleReflectionExtension c r f) U := by | |
| let g := fun z => (f z-c)/(r:ℂ) | |
| have hrc : (r:ℂ)≠0 := by exact_mod_cast hr.ne' | |
| have hgi : InjOn g (U∩{z | 1≤‖z‖}) := by | |
| intro z hzU w hwU he | |
| apply hi hzU hwU | |
| have he' := congrArg (fun v:ℂ => v*(r:ℂ)) he | |
| change (f z-c)/(r:ℂ)*(r:ℂ)=(f w-c)/(r:ℂ)*(r:ℂ) at he' | |
| rw [div_mul_cancel₀ _ hrc,div_mul_cancel₀ _ hrc] at he' | |
| exact sub_left_injective he' | |
| have hgo : ∀z∈U,1<‖z‖ → 1<‖g z‖ := by | |
| intro z hzU hzn | |
| change 1<‖(f z-c)/(r:ℂ)‖ | |
| rw [normalized_circle_norm c hr] | |
| exact (one_lt_div hr).mpr (ho z hzU hzn) | |
| have hgb : ∀z∈U,‖z‖=1 → ‖g z‖=1 := by | |
| intro z hzU hzn | |
| change ‖(f z-c)/(r:ℂ)‖=1 | |
| rw [normalized_circle_norm c hr,hb z hzU hzn,div_self hr.ne'] | |
| intro z hzU w hwU he | |
| apply injOn_annularCircleReflect hz hs hgi hgo hgb hzU hwU | |
| change c+(r:ℂ)*annularCircleReflect g z=c+(r:ℂ)*annularCircleReflect g w at he | |
| exact mul_left_cancel₀ hrc (add_left_cancel he) | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |