| import OAI.Analysis.CircleDomains.Topology.GraphFillingDisks |
| import OAI.Analysis.CircleDomains.Transfer.GeometricCompletion |
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|
| namespace OAI |
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|
| /-! Cover the remaining quotient dust using the actual closed level |
| sets of mixed real and period tests, after the physical collision |
| terminals have been selected. -/ |
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|
| open Set Function |
| open scoped Topology ContDiff Manifold |
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| noncomputable section |
| namespace CircleDomainRigidity.FiniteTransfer |
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|
| theorem exists_geometric_completion_for_graphs |
| (hsard : SmoothPlanarSard) (hschoenflies : PlanarSchoenflies) |
| (G : Set Sphere) (π : Sphere → Sphere) (hπ : Continuous π) |
| (hclosed : IsClosedMap π) (hsurj : Surjective π) |
| (hr : ∀ x y, π x = π y ↔ |
| x = y ∨ (x ∈ Gᶜ ∧ y ∈ connectedComponentIn Gᶜ x)) |
| {B C : Set Sphere} (hB : IsClosed B) (hBdef : B = π '' Gᶜ) |
| (hTD : IsTotallyDisconnected B) (hC : IsCompact C) (hCG : C ⊆ G) |
| (hinfty : OnePoint.infty ∈ C) |
| {n N : ℕ} {S : Type*} (R : Fin n → Set (Sphere × UnitAddCircle)) (hR : ∀ i, IsClosed (R i)) |
| (σ : S → Sphere) (hσB : ∀ s, σ s ∈ B) |
| (a b : Fin n → S → ℝ) (θ : Fin n → ℝ) (F : Set S) |
| (hband : ∀ i s, {t : UnitAddCircle | (σ s, t) ∈ R i} ⊆ |
| ((↑) : ℝ → UnitAddCircle) '' Icc (a i s) (b i s)) |
| (hfin : (gridTerminalIndices F (reciprocalGridStep N) upperPair a b θ).Finite) |
| (O : (gridTerminalIndices F (reciprocalGridStep N) upperPair a b θ) → Set Sphere) |
| (D : PhysicalTerminalDisks G π B (fun s : gridTerminalIndices F (reciprocalGridStep N) |
| upperPair a b θ => σ s) O) |
| (hDcore : ∀ i, Disjoint (((↑) : ℂ → Sphere) '' closure (jordanInterior (D.H i))) C) |
| (hfree : ∀ x ∈ terminalRemainder π B D.H, x ∉ range σ → ∀ i, |
| {t : UnitAddCircle | (x, t) ∈ R i}.Subsingleton) |
| (hcompat : OrderedGraphGridCompatibility R θ N (terminalRemainder π B D.H) σ) : |
| ∃ (m : ℕ) (H : Fin m → ℂ ≃ₜ ℂ) (γ : Fin m → Circle → ℂ), |
| Pairwise (Disjoint on fun j => closure (jordanInterior (H j))) ∧ |
| (∀ i j, Disjoint (closure (jordanInterior (D.H i))) (closure (jordanInterior (H j)))) ∧ |
| (∀ j, Disjoint (((↑) : ℂ → Sphere) '' closure (jordanInterior (H j))) C) ∧ |
| π ⁻¹' B ⊆ sphericalDiskUnion D.H ∪ sphericalDiskUnion H ∧ |
| C ⊆ completionCore D.H H ∧ completionCore D.H H ⊆ G ∧ |
| (∀ j, ContMDiff (𝓡 1) (𝓘(ℝ, ℂ)) ∞ (γ j) ∧ Topology.IsEmbedding (γ j) ∧ |
| frontier (jordanInterior (H j)) = range (γ j) ∧ |
| (∀ z ∈ frontier (jordanInterior (H j)), (z : Sphere) ∈ G ∧ π (z : Sphere) ∉ B) ∧ |
| ∃ k : ℕ, ∀ i : Fin n, i.val ≠ k → |
| Disjoint (frontier (jordanInterior (H j))) |
| ((fun z : ℂ => π (z : Sphere)) ⁻¹' (⋃ l, reciprocalGraphLevel (R i) (θ i) N l))) := by |
| classical |
| let : Fintype (gridTerminalIndices F (reciprocalGridStep N) upperPair a b θ) := hfin.fintype |
| have hBC : B ⊆ (π '' C)ᶜ := by |
| rw [hBdef] |
| exact (quotient_core_safe_open π hπ hC hCG hr).2.1 |
| obtain ⟨hA, hAB, hBA, hW, hAW, hAinf⟩ := |
| terminal_remainder_safe_region hπ hclosed hsurj hr hB hBC hC hinfty D |
| have hpinf (s : gridTerminalIndices F (reciprocalGridStep N) upperPair a b θ) : |
| σ s ≠ π OnePoint.infty := by |
| intro heq |
| exact hBC (hσB s) (heq ▸ (show π OnePoint.infty ∈ π '' C from |
| ⟨OnePoint.infty, hinfty, rfl⟩)) |
| have hnoterm : Disjoint (terminalRemainder π B D.H) |
| (σ '' gridTerminalIndices F (reciprocalGridStep N) upperPair a b θ) := by |
| apply Set.disjoint_left.mpr |
| rintro x hx ⟨s, hs, hσ⟩ |
| exact Set.disjoint_left.mp (D.remainder_avoids_indices hsurj hpinf) hx ⟨⟨s, hs⟩, hσ⟩ |
| obtain ⟨m, H, γ, hcover, hdisj, hcurves⟩ := exists_filling_disks_of_compatible_graphs |
| hsard hschoenflies G π hπ hclosed hsurj hr R hR σ a b θ |
| upperPair upperPair_covers F hband hB hTD hA hAB hBA hW hAW hAinf |
| (fun x hx => hBdef.symm ▸ (show π x ∈ π '' Gᶜ from ⟨x, hx, rfl⟩)) |
| hnoterm hfree hcompat |
| have hsafe := filling_closures_avoid_core_and_terminals D.H H |
| (fun j => (hcurves j).1) |
| have hfill := spherical_remainder_cover_of_physical_cover D.H H B π hAinf hcover |
| have htotal := dust_preimage_covered_of_remainder_cover D.H H G B π hsurj hr |
| (fun i z hz => (D.frontier_good i z hz).1) hfill |
| refine ⟨m, H, γ, hdisj, hsafe.2, hsafe.1, htotal, ?_, |
| completionCore_subset_domain D.H H G B π |
| (fun x hx => hBdef.symm ▸ (show π x ∈ π '' Gᶜ from ⟨x, hx, rfl⟩)) htotal, ?_⟩ |
| · intro x hx hbad |
| rcases hbad with ht | hf |
| · obtain ⟨i, z, hz, rfl⟩ := mem_iUnion.mp ht |
| exact Set.disjoint_left.mp (hDcore i) ⟨z, subset_closure hz, rfl⟩ hx |
| · obtain ⟨j, z, hz, rfl⟩ := mem_iUnion.mp hf |
| exact Set.disjoint_left.mp (hsafe.1 j) ⟨z, subset_closure hz, rfl⟩ hx |
| · intro j |
| exact (hcurves j).2 |
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| end CircleDomainRigidity.FiniteTransfer |
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| end |
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| end OAI |
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