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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 34 35 36 37 38 39 40 41 42 43 44 45 | 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
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