| import OAI.Analysis.CircleDomains.Topology.FiniteJordanModelInput |
| import OAI.Analysis.CircleDomains.Transfer.RealTransferStage |
| import OAI.Analysis.CircleDomains.Selection.CountableMarks |
|
|
| namespace OAI |
|
|
| /-! |
| # Countably many real tests and mandatory marked ends |
|
|
| Every finite prefix is transferred by the actual assembled construction. |
| An increasing sequence of finite mandatory sets puts each fixed mark in |
| all sufficiently late models. Prescribed neighborhoods and source cores |
| are retained at each stage. |
| -/ |
|
|
| open Set Function Filter MeasureTheory |
| open scoped Topology ContDiff |
|
|
| noncomputable section |
|
|
| namespace CircleDomainRigidity.FiniteTransfer |
|
|
| structure CountableRealCoordinates {S : Type*} |
| (G : Set Sphere) (π : Sphere → Sphere) (σ : S → Sphere) where |
| source : Set Sphere |
| source_open : IsOpen source |
| coordinate : ℕ → Sphere → ℝ |
| coordinate_continuous : ∀ l, ContinuousOn (coordinate l) source |
| bound : ℕ → ℝ |
| coordinate_bounded : ∀ l x, x ∈ source → |coordinate l x| ≤ bound l |
| physical : ℕ → (ℝ × ℝ) → ℝ |
| physical_measurable : ∀ l, Measurable (physical l) |
| physical_smooth : ∀ l, ContDiffOn ℝ ∞ (physical l) (realChartDomain G) |
| physical_energy : ∀ l, MemLp (planarGradient (physical l)) 2 (volume.restrict (realChartDomain G)) |
| source_map : ∀ z : ℂ, (z : Sphere) ∈ G → π z ∈ source |
| pullback : ∀ l (z : ℂ), (z : Sphere) ∈ G → physical l (Complex.equivRealProdCLM z) = coordinate l (π z) |
| labels_injective : Injective σ |
| labels_dust : ∀ s, σ s ∈ π '' Gᶜ |
| labels_closure : ∀ s, σ s ∈ closure source |
| lower : ℕ → S → ℝ |
| upper : ℕ → S → ℝ |
| width_nonneg : ∀ l s, 0 ≤ upper l s - lower l s |
| width_summable : ∀ l, Summable (fun s => (upper l s - lower l s) ^ 2) |
| band : ∀ l s, {t : ℝ | (σ s, t) ∈ intervalFiberHull (clusterGraph source (coordinate l))} = |
| Icc (lower l s) (upper l s) |
| free_trace : ∀ x ∈ π '' Gᶜ, x ∉ range σ → ∀ l, |
| ∃ v : ℝ, {t : ℝ | (x, t) ∈ intervalFiberHull (clusterGraph source (coordinate l))} = {v} |
|
|
| def CountableRealCoordinates.NaturalCompatibility {S : Type*} |
| {G : Set Sphere} {π : Sphere → Sphere} {σ : S → Sphere} |
| (d : CountableRealCoordinates G π σ) : Prop := |
| ∀ (j n : ℕ) (labels : Fin n → ℕ), (∀ i, labels i < j) → |
| (volume.prod (Measure.pi (fun _ : Fin n => (volume : Measure UnitAddCircle)))) |
| (freeComponentFailure ((π '' Gᶜ) \ range σ) (circleBandGraph d.source (d.coordinate j)) |
| (fun k => circleBandGraph d.source (d.coordinate k)) labels) = 0 |
|
|
| abbrev CountableRealCoordinates.Stage {S : Type*} {G : Set Sphere} {π : Sphere → Sphere} |
| {σ : S → Sphere} (d : CountableRealCoordinates G π σ) (n : ℕ) |
| (Flag : Set S) (O : S → Set Sphere) (K₀ : Set (ℝ × ℝ)) (ε : ℝ) := |
| RealTransferStage G π σ (fun l : Fin (n + 1) => d.physical l) |
| (fun l => d.lower l) (fun l => d.upper l) Flag O K₀ ε |
|
|
| theorem CountableRealCoordinates.nonempty_stage |
| (hsard : SmoothPlanarSard) (hsch : PlanarSchoenflies) |
| (hcompact : SmoothCompactMetricSphereUniformization) (hfinite : FiniteJordanModelUniformization) |
| {G : Set Sphere} (hG : IsOpen G) (hinf : OnePoint.infty ∈ G) |
| {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)) |
| (hTD : IsTotallyDisconnected (π '' Gᶜ)) |
| {S : Type*} [Countable S] {σ : S → Sphere} (d : CountableRealCoordinates G π σ) |
| (hcompat : d.NaturalCompatibility) |
| (n : ℕ) (Flag : Set S) (hFlag : Flag.Finite) |
| (O : S → Set Sphere) (hO : ∀ s, O s ∈ 𝓝 (σ s)) |
| {K₀ : Set (ℝ × ℝ)} (hK₀ : IsClosed K₀) (hK₀G : K₀ ⊆ realChartDomain G) |
| {ε : ℝ} (hε : 0 < ε) : Nonempty (d.Stage n Flag O K₀ ε) := by |
| apply exists_real_transfer_stage hsard hsch hcompact hfinite hG hinf |
| π hπ hclosed hsurj hr hTD hK₀ hK₀G (Nat.succ_pos n) |
| (fun l : Fin (n + 1) => d.physical l) (fun l => d.physical_measurable l) |
| (fun l => d.physical_smooth l) (fun l => d.physical_energy l) |
| σ d.labels_injective d.labels_dust d.source d.source_open |
| (fun l => d.coordinate l) (fun l => d.coordinate_continuous l) (fun l => d.bound l) |
| (fun l => d.coordinate_bounded l) d.source_map (fun l => d.pullback l) |
| d.labels_closure (fun l => d.lower l) (fun l => d.upper l) |
| (fun l => d.width_nonneg l) (fun l => d.width_summable l) (fun l => d.band l) |
| Flag hFlag (fun x hx hfree l => d.free_trace x hx hfree l) ?_ O hO hε zero_lt_one 0 |
| intro j m labels hlabels |
| exact hcompat j m (fun i => labels i) hlabels |
|
|
| theorem exists_countable_real_transfer |
| (hsard : SmoothPlanarSard) (hsch : PlanarSchoenflies) |
| (hcompact : SmoothCompactMetricSphereUniformization) (hfinite : FiniteJordanModelUniformization) |
| {G : Set Sphere} (hG : IsOpen G) (hinf : OnePoint.infty ∈ G) |
| {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)) |
| (hTD : IsTotallyDisconnected (π '' Gᶜ)) |
| {S : Type*} [Countable S] {σ : S → Sphere} (d : CountableRealCoordinates G π σ) |
| (hcompat : d.NaturalCompatibility) |
| (K₀ : ℕ → Set (ℝ × ℝ)) (hK₀ : ∀ n, IsClosed (K₀ n)) |
| (hK₀G : ∀ n, K₀ n ⊆ realChartDomain G) |
| (O : ℕ → S → Set Sphere) (hO : ∀ n s, O n s ∈ 𝓝 (σ s)) |
| (ε : ℕ → ℝ) (hε : ∀ n, 0 < ε n) : |
| ∃ Flag : ℕ → Set S, (∀ n, (Flag n).Finite) ∧ Monotone Flag ∧ |
| (∀ n, Nonempty S → (Flag n).Nonempty) ∧ |
| ∃ M : ∀ n, d.Stage n (Flag n) (O n) (K₀ n) (ε n), |
| ∀ s, ∀ᶠ n in atTop, s ∈ (M n).indices := by |
| classical |
| obtain ⟨Flag, hfin, hmono, hne, hcover⟩ := exists_increasing_finite_marks_inhabited S |
| let M (n : ℕ) : d.Stage n (Flag n) (O n) (K₀ n) (ε n) := |
| Classical.choice (d.nonempty_stage hsard hsch hcompact hfinite hG hinf hπ hclosed |
| hsurj hr hTD hcompat n (Flag n) (hfin n) (O n) (hO n) (hK₀ n) (hK₀G n) (hε n)) |
| refine ⟨Flag, hfin, hmono, hne, M, ?_⟩ |
| intro s |
| exact (hcover s).mono (fun n hn => (M n).flags hn) |
|
|
| end CircleDomainRigidity.FiniteTransfer |
|
|
| end |
|
|
| end OAI |
|
|
