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a97875e | 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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import OAI.Analysis.CircleDomains.Topology.GraphFillingDisks
import OAI.Analysis.CircleDomains.Transfer.GeometricCompletion
namespace OAI
/-! 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. -/
open Set Function
open scoped Topology ContDiff Manifold
noncomputable section
namespace CircleDomainRigidity.FiniteTransfer
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
end CircleDomainRigidity.FiniteTransfer
end
end OAI
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