openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Uniformization /FiniteModelTerminalEstimates.lean
| import OAI.Analysis.CircleDomains.Transfer.TransportedTerminalBranches | |
| namespace OAI | |
| /-! | |
| # Terminal estimates on an actual finite circle model | |
| The terminal assignment is constructed from the smooth model. The same | |
| branches produced on physical terminal collars have the required summed | |
| cluster-oscillation bound at their assigned target components. | |
| -/ | |
| open Set Filter Function | |
| open scoped Topology ContDiff | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| open FiniteTransfer | |
| theorem exists_finite_model_terminal_estimates | |
| (hsard : SmoothPlanarSard) (hsch : PlanarSchoenflies) | |
| {I J : Type*} [Fintype I] [Fintype J] {n : ℕ} | |
| (T : I → ℂ ≃ₜ ℂ) (H : J → ℂ ≃ₜ ℂ) | |
| (hTd : Pairwise (Disjoint on fun i => closure (jordanInterior (T i)))) | |
| (hTH : ∀ i j, Disjoint (closure (jordanInterior (T i))) (closure (jordanInterior (H j)))) | |
| {φ φi : Sphere → Sphere} {V : Set Sphere} | |
| (heq : IsSmoothSphereEquivalence φ φi (openCompletedRegion T) V) | |
| (hV : IsOpen V) (hfix : φ OnePoint.infty = OnePoint.infty) | |
| {Ω : Set (ℝ × ℝ)} (O : I → Set (ℝ × ℝ)) (hO : ∀ i, IsOpen (O i)) | |
| (hTO : ∀ i, frontier (realJordanInterior (T i)) ⊆ O i) | |
| (M : Fin n → Set (ℝ × ℝ)) (f F : Fin n → (ℝ × ℝ) → UnitAddCircle) | |
| (χ : (ℝ × ℝ) → ℝ) (u : I → Fin n → (ℝ × ℝ) → ℝ) | |
| (S : Fin n → Finset I) (c : I → Fin n → ℝ) | |
| (hu : ∀ l i, i ∈ S l → ContDiffOn ℝ ∞ (u i l) (O i)) | |
| (hproj : ∀ l i, i ∈ S l → ∀ x ∈ O i, (u i l x : UnitAddCircle) = f l x) | |
| (hχone : ∀ i x, x ∈ O i → χ x = 1) | |
| (N : ℕ) {δ η : ℝ} (hδ : 0 < δ) (hδh : δ < reciprocalGridStep N / 4) | |
| (θ : Fin n → ℝ) | |
| (hF : IsMaskedFilledCircleFamily (fun j => realJordanInterior (H j)) Ω M | |
| (circleCutoffFamily f χ N δ θ) F) | |
| (hc : ∀ l i, i ∈ S l → 0 ≤ c i l) | |
| (hbound : ∀ l i, i ∈ S l → ∀ x ∈ frontier (jordanInterior (T i)), | |
| ∀ y ∈ frontier (jordanInterior (T i)), | |
| |shiftedStaircase (reciprocalGridStep N) δ (θ l) (u i l (Complex.equivRealProdCLM x)) - | |
| shiftedStaircase (reciprocalGridStep N) δ (θ l) (u i l (Complex.equivRealProdCLM y))| ≤ | |
| c i l + 4 * reciprocalGridStep N) | |
| (hsmall : reciprocalGridStep N * Real.sqrt (Fintype.card I : ℝ) < η) : | |
| ∃ (a : TerminalComponentAssignment T φ V) (w : I → Fin n → (ℝ × ℝ) → ℝ), | |
| (∀ l i, i ∈ S l → ContDiffOn ℝ ∞ (w i l) (O i) ∧ | |
| ∀ᶠ x in 𝓝ˢ (frontier (realJordanInterior (T i))), (w i l x : UnitAddCircle) = F l x) ∧ | |
| (∀ l, Real.sqrt (∑ i ∈ S l, | |
| max (Metric.diam (w i l '' frontier (realJordanInterior (T i))) - c i l) 0 ^ 2) < 4 * η) ∧ | |
| (∀ l, Real.sqrt (∑ i ∈ S l, max (Metric.diam | |
| (componentClusterValues V (a.point i) (sphereRealFunction (w i l) ∘ φi)) - c i l) 0 ^ 2) | |
| < 4 * η) := by | |
| obtain ⟨a⟩ := exists_terminal_component_assignment hsard hsch T hTd heq hV hfix | |
| obtain ⟨w, hw, hexcess⟩ := physical_masked_terminal_branches T H hTH O hO hTO | |
| M f F χ u S c hu hproj hχone N hδ hδh θ hF hc hbound hsmall | |
| refine ⟨a, w, hw, hexcess, ?_⟩ | |
| intro l | |
| exact (a.real_cluster_excess_le heq (S l) (fun i => w i l) (fun i => c i l) | |
| O hO hTO (fun i hi => (hw l i hi).1.continuousOn)).trans_lt (hexcess l) | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |