| import OAI.Analysis.CircleDomains.Selection.AnchoredDustBlock |
|
|
| namespace OAI |
|
|
| /-! A finite block of possibly repeated limiting dust junctions can be |
| replaced by one allowed true path. The original true-piece limits and one |
| paid toll per junction cover the replacement, up to any positive error. -/ |
|
|
| open Set Function Filter MeasureTheory |
| open scoped ENNReal Topology |
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|
| noncomputable section |
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|
| namespace CircleDomainRigidity |
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|
| attribute [local instance] sphereMetric sphereBorel |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
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|
| variable [MeasurableSpace C(PathTime, Sphere)] |
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|
| theorem exists_path_replacing_finite_limit_block |
| (hcoarea : WeightedPlanarScalarCoarea) (hsard : SmoothPlanarSard) |
| (hsch : PlanarSchoenflies) |
| (d : PathCodeData Sphere Sphere) (hU : IsOpen d.quotientDomain) |
| (hG : IsOpen d.physicalDomain) (hdisj : Disjoint d.quotientDomain d.dust) |
| (π : Sphere → Sphere) (hπ : Continuous π) (hclosed : IsClosedMap π) (hsurj : Surjective π) |
| (hr : ∀x y, π x = π y ↔ x = y ∨ |
| (x ∈ d.physicalDomainᶜ ∧ y ∈ connectedComponentIn d.physicalDomainᶜ x)) |
| (he : ∀x : d.physicalDomain, (d.domainEquiv x).val = π x.val) |
| {B : Set Sphere} (hB : IsClosed B) (hTD : IsTotallyDisconnected B) |
| (hbad : ∀x ∈ d.physicalDomainᶜ, π x ∈ B) |
| (hUB : Disjoint d.quotientDomain B) (hdense : Dense d.quotientDomain) |
| (ν : Measure Sphere) [NullSingletonClass ν] |
| (N : ∀n : ℕ, (Fin (n+1) → ℕ) → Set (Fin (n+1) → UnitAddCircle)) |
| (F H : Set Sphere) (hF : frontier F ⊆ d.quotientDomain) |
| (hH : frontier H ⊆ d.quotientDomain) |
| (n : ℕ) (p : Fin n → Sphere) |
| (hpB : ∀i, p i ∈ B) (hp : ∀i, p i ≠ π OnePoint.infty) |
| (Γ : Fin (n+1) → ℕ → TaggedPhysicalPath d) |
| (hΓ : ∀i k, AllowedTaggedPath d ν N ((Γ i k).1.val, (Γ i k).2)) |
| (hΓF : ∀k, (Γ 0 k).1.val ⊥ ∈ F ∩ d.quotientDomain) |
| (hΓH : ∀k, (Γ (Fin.last n) k).1.val ⊤ ∈ H ∩ d.quotientDomain) |
| (hstart : ∀i : Fin n, Tendsto (fun k => (Γ i.succ k).1.val ⊥) atTop (𝓝 (p i))) |
| (hend : ∀i : Fin n, Tendsto (fun k => (Γ i.castSucc k).1.val ⊤) atTop (𝓝 (p i))) |
| (g : Sphere → ℝ≥0∞) (hg : Measurable g) |
| (henergy : (∫⁻x, g x ^ 2 ∂sphericalArea) < ∞) |
| (m : Sphere → ℝ≥0∞) (hm : ∀x ∈ d.quotientDomain, m x = 0) |
| (A : Fin (n+1) → ℝ≥0∞) (hA : ∀i, A i < ∞) |
| (hcost : ∀i, Tendsto (fun k => physicalTollCost d g m (Γ i k).1) atTop (𝓝 (A i))) |
| (J : Fin n → ℝ≥0∞) (hpaid : ∀i, m (p i) ≤ J i) |
| (w : ℝ≥0∞) (hw : w ≠ 0) (hpos : ∀z ∈ d.physicalDomain, w ≤ g z) |
| (hpoint : ∀i z, volume {t | (t, z) ∈ d.levels i} ≤ d.levelWeights i z) |
| {η : ℝ} (hη : 0 < η) : |
| ∃ (ρ : FinitePhysicalPath d.domainEquiv d.basePoint) (θ : ℕ → TagSlot), |
| AllowedTaggedPath d ν N (ρ.val, θ) ∧ ρ.val ⊥ ∈ F ∩ d.quotientDomain ∧ ρ.val ⊤ ∈ H ∩ d.quotientDomain ∧ |
| physicalTollCost d g m ρ ≤ (∑i, A i) + (∑i, J i) + ENNReal.ofReal η := by |
| cases n with |
| | zero => |
| have hlt : A 0 < A 0 + ENNReal.ofReal η := |
| ENNReal.lt_add_right (hA 0).ne (ENNReal.ofReal_pos.mpr hη).ne' |
| obtain ⟨k, hk⟩ := ((hcost 0).eventually (gt_mem_nhds hlt)).exists |
| exact ⟨(Γ 0 k).1, (Γ 0 k).2, hΓ 0 k, hΓF k, hΓH k, by simpa using hk.le⟩ |
| | succ n => |
| exact exists_path_replacing_anchored_dust_block |
| hcoarea hsard hsch d hU hG hdisj π hπ hclosed hsurj hr he hB hTD hbad hUB hdense |
| ν N F H hF hH n p hpB hp Γ hΓ hΓF hΓH hstart hend |
| g hg henergy m hm A hA hcost J hpaid w hw hpos hpoint hη |
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| end CircleDomainRigidity |
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| end |
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| end OAI |
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|
