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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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | 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
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