openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /CompactReflectionCoefficient.lean
| import OAI.Analysis.CircleDomains.Reflection.RoundReflectionCoefficient | |
| namespace OAI | |
| /-! The reflection-invariant extension remains compactly supported: every | |
| non-base tile lies inside one of the original finitely many disks. -/ | |
| open Set Function MeasureTheory Metric Bornology | |
| open scoped Topology OnePoint | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| attribute [local instance] sphereMetric sphereBorel | |
| local instance : BorelSpace Sphere := ⟨rfl⟩ | |
| lemma exists_compact_reflection_coefficient {I : Type*} [Finite I] | |
| (c : I → ℂ) (r : I → ℝ) (hr : ∀i,0<r i) | |
| (hd : Pairwise (fun i j => Disjoint | |
| (((↑) : ℂ → Sphere) '' closedBall (c i) (r i)) | |
| (((↑) : ℂ → Sphere) '' closedBall (c j) (r j)))) | |
| (μ : ℂ → ℂ) (hm : Measurable μ) {k R : ℝ} | |
| (hk : 0≤k) (hb : ∀z,‖μ z‖≤k) (hz : ∀z,R≤‖z‖ → μ z=0) : | |
| ∃ (E : Sphere → ℂ) (S : ℝ),Measurable E ∧ 0<S ∧ (∀p,‖E p‖≤k) ∧ | |
| (∀z : ℂ,S≤‖z‖ → E (z : Sphere)=0) ∧ | |
| EqOn E (sphereCoefficient 0 μ) | |
| (⋃i,((↑) : ℂ → Sphere) '' closedBall (c i) (r i))ᶜ ∧ | |
| ∀i p,E p=roundReflectionCoefficient (c i) p | |
| (E (roundDiskReflection (c i) (r i) (hr i) p)) := by | |
| have hμb (p : Sphere) : ‖sphereCoefficient 0 μ p‖≤k := by | |
| induction p using OnePoint.rec with | |
| | infty => simpa only [sphereCoefficient_infty,norm_zero] using hk | |
| | coe z => exact hb z | |
| obtain ⟨E,hEm,hEb,hbase,hEinv⟩ := exists_round_reflection_coefficient c r hr hd | |
| (sphereCoefficient 0 μ) (sphereCoefficient_measurable 0 hm) hk hμb | |
| obtain ⟨B,hB,hbound⟩ := (isCompact_iUnion (fun i => isCompact_closedBall (c i) (r i))).isBounded.exists_pos_norm_lt | |
| let S := max B R | |
| have hS : 0<S := hB.trans_le (le_max_left B R) | |
| refine ⟨E,S,hEm,hS,hEb,?_,hbase,hEinv⟩ | |
| intro z hsz | |
| have hout : (z : Sphere)∉⋃i,((↑) : ℂ → Sphere) '' closedBall (c i) (r i) := by | |
| intro hz' | |
| obtain ⟨i,w,hw,he⟩ := mem_iUnion.mp hz' | |
| have hwz : w=z := OnePoint.coe_injective he | |
| subst w | |
| have hlt := hbound z (mem_iUnion.mpr ⟨i,hw⟩) | |
| exact (not_lt_of_ge ((le_max_left B R).trans hsz)) hlt | |
| rw [hbase hout,sphereCoefficient_coe,hz z ((le_max_right B R).trans hsz)] | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |