| import Mathlib |
| import OAI.Geometry.Cannon.Boundary.BoundaryLocallyConnected |
| import OAI.Geometry.Cannon.PlaneTopology.PullChartUnionCompl |
| import OAI.Geometry.Cannon.Topology.ComponentRegionConnected |
| import OAI.Geometry.Cannon.Topology.Plane |
|
|
| namespace OAI |
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| section |
| section |
| section |
| noncomputable section |
| open Set unitInterval Topology |
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| noncomputable section |
| noncomputable section |
| noncomputable section |
| open Set Topology unitInterval |
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| noncomputable section |
| open Set Metric Topology unitInterval |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| noncomputable section |
| open Set unitInterval |
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| noncomputable section |
| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Metric Topology |
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology |
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| noncomputable section |
| open Set Topology |
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| namespace CannonIncidence |
| open Set Metric |
|
|
| variable {X ι : Type*} [MetricSpace X] [Fintype ι] |
|
|
| noncomputable def traceCard (E : ι → Set X) (S : Set X) : ℕ := by |
| classical |
| exact (Finset.univ.filter (fun i => (E i∩S).Nonempty)).card |
|
|
| omit [MetricSpace X] in |
| lemma traceCard_mono (E : ι → Set X) {S T : Set X} (h : S⊆T) : |
| traceCard E S≤traceCard E T := by |
| classical |
| apply Finset.card_le_card |
| intro i hi |
| obtain ⟨_,x,hxE,hxS⟩ := Finset.mem_filter.mp hi |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _,x,hxE,h hxS⟩ |
|
|
| omit [MetricSpace X] in |
| lemma traceCard_finite_cover (E : ι → Set X) (T : Finset X) (U : X → Set X) |
| (S : Set X) (hcover : S⊆⋃ x∈T,U x) (K : ℕ) |
| (hK : ∀ x∈T, traceCard E (U x)≤K) : traceCard E S≤T.card*K := by |
| classical |
| let IX (S : Set X) := Finset.univ.filter (fun i => (E i∩S).Nonempty) |
| have hsub : IX S⊆T.biUnion (fun x => IX (U x)) := by |
| intro i hi |
| obtain ⟨y,hyE,hyS⟩ := (Finset.mem_filter.mp hi).2 |
| obtain ⟨x,hx,hyU⟩ := mem_iUnion₂.mp (hcover hyS) |
| exact Finset.mem_biUnion.mpr ⟨x,hx,Finset.mem_filter.mpr ⟨Finset.mem_univ _,y,hyE,hyU⟩⟩ |
| calc |
| traceCard E S=(IX S).card := rfl |
| _≤(T.biUnion (fun x => IX (U x))).card := Finset.card_le_card hsub |
| _≤∑ x∈T,(IX (U x)).card := Finset.card_biUnion_le |
| _≤∑ _x∈T,K := Finset.sum_le_sum (fun x hx => hK x hx) |
| _=T.card*K := by simp |
|
|
| theorem uniform_larger_trace (Q : ℕ) (C R : ℝ) |
| (hpack : CannonPacking.PolynomialPacking (X:=X) Q C) (hC : 0≤C) (hR : 0<R) : |
| ∃ B : ℕ, ∀ (r A : ℝ), 0<r → 1≤A → (R+1)*r≤1 → |
| ∀ T : Finset X, (∀ x∈T, ∀ y∈T, x≠y → r≤dist x y) → |
| (∀ x : X, ∃ t∈T, dist x t≤r) → |
| ∀ {κ : Type*} [Fintype κ], ∀ E : κ → Set X, ∀ K : ℕ, (∀ x, traceCard E (ball x (2*A*r))≤K) → |
| ∀ x, traceCard E (ball x (R*r))≤B*K := by |
| classical |
| obtain ⟨B,hB⟩ := CannonPacking.exists_overlap_bound Q C (R+1) 1 hpack hC (by linarith) zero_lt_one |
| refine ⟨B,?_⟩ |
| intro r A hr hA hscale T hsep hcover κ _ E K hK x |
| let T' := T.filter (fun t => dist x t≤(R+1)*r) |
| have hTc : T'.card≤B := by |
| apply hB id T x r hr |
| · simpa [max_eq_left (by linarith : (1:ℝ)≤R+1)] using hscale |
| · simpa using hsep |
| have hcov : ball x (R*r)⊆⋃ t∈T',ball t (2*A*r) := by |
| intro y hy |
| obtain ⟨t,ht,hyT⟩ := hcover y |
| refine mem_iUnion₂.mpr ⟨t,Finset.mem_filter.mpr ⟨ht,?_⟩,?_⟩ |
| · have hh := dist_triangle x y t |
| have hyx : dist x y<R*r := by simpa only [mem_ball,dist_comm] using hy |
| linarith |
| · change dist y t<2*A*r |
| have hh : 2*r≤2*A*r := by nlinarith |
| linarith |
| exact (traceCard_finite_cover E T' (fun t => ball t (2*A*r)) _ hcov K |
| (fun t _ => hK t)).trans (Nat.mul_le_mul_right K hTc) |
| end CannonIncidence |
|
|
| namespace CannonRelease |
| open Set Metric Topology unitInterval CannonGraph CannonRegularDisk CannonEmbeddedArc CannonEmbeddedArc.Arc |
|
|
| CannonFaceIncidence CannonFaceLabels CannonPlaneLinear CannonIncidence |
| variable {G : Type*} [Group G] [TopologicalSpace G] [DiscreteTopology G] |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) |
|
|
| theorem boundary_uniform_controlled_faces (φ : Boundary D ≃ₜ SphereTwo) : |
| let _ := visualMetricSpace D hδ |
| ∃ A L r₀ : ℝ, 1≤A ∧ 1≤L ∧ 0<r₀ ∧ ∃ K J : ℕ, |
| ∀ r : ℝ, 0<r → r≤r₀ → ∀ T : Finset (Boundary D), |
| (∀ x∈T, ∀ y∈T, x≠y → r≤dist x y) → |
| (∀ x : Boundary D, ∃ y∈T, dist x y≤r) → |
| ∀ p : Boundary D, p∉T → |
| ∃ e : ℂ → Boundary D, IsOpenEmbedding e ∧ range e={p}ᶜ ∧ |
| ∃ Q : Finset (ℂ × ℂ), ∃ n : ℕ, 0<n ∧ ∃ F : Fin n → Arc (Boundary D), |
| Pairwise (fun i j => Compatible (F i) (F j)) ∧ |
| (∀ j, diam (range (F j))≤A*r) ∧ |
| (∀ x, intersectionCount F (ball x (2*A*r))≤K) ∧ |
| (∀ x, intersectionCount F (ball x ((L+1)*r))≤J) ∧ |
| IsPathConnected (⋃ j,range (F j)) ∧ (T:Set (Boundary D))⊆(⋃ j,range (F j)) ∧ |
| (⋃ j,range (F j))⊆e '' (⋃ q∈Q,segment ℝ q.1 q.2) ∧ |
| Finite (ConnectedComponents ((⋃ j,range (F j))ᶜ : Set (Boundary D))) ∧ |
| (∀ R : Set (Boundary D), IsPreconnected R → R⊆(⋃ j,range (F j))ᶜ → |
| ∀ x∈R, ∀ y∈R, dist x y<L*r) ∧ |
| (let Faces := ConnectedComponents ((⋃ j,range (F j))ᶜ : Set (Boundary D)) |
| let π : ((⋃ j,range (F j))ᶜ : Set (Boundary D)) → Faces := ConnectedComponents.mk |
| ∃ (S : Faces → Set (Fin n)) (labels : Fin n → Bool → Faces), |
| (∀ d, closure (region _ π d)=region _ π d∪⋃ i∈S d,range (F i)) ∧ |
| (∀ d i, i∈S d ↔ d∈range (labels i)) ∧ |
| (∀ d, Nat.card (S d)≤J)) := by |
| classical |
| let _ := visualMetricSpace D hδ |
| let _ : LocallyPathConnectedSpace SphereTwo := CannonPaths.sphere_locallyPathConnected |
| let _ : LocallyConnectedSpace (Boundary D) := φ.locallyConnectedSpace |
| let _ : PathConnectedSpace SphereTwo := CannonPaths.sphere_pathConnected |
| let _ : PathConnectedSpace (Boundary D) := φ.symm.surjective.pathConnectedSpace φ.symm.continuous |
| let s : SphereTwo := Classical.choice inferInstance |
| let _ : Nontrivial (Boundary D) := ⟨⟨φ.symm s,φ.symm (-s), |
| fun h => (ne_neg_of_mem_unit_sphere ℝ s) (φ.symm.injective h)⟩⟩ |
| obtain ⟨a,b,hab⟩ := exists_pair_ne (Boundary D) |
| have hdist : 0<dist a b := dist_pos.mpr hab |
| dsimp only |
| obtain ⟨A,L,rG,hA,hL,hrG,K,hgraph⟩ := boundary_uniform_finite_face_graph D hδ φ |
| obtain ⟨Q₀,C,hQ₀,hC,hpoly⟩ := relative_polynomial_packing D hδ |
| obtain ⟨B,hB⟩ := uniform_larger_trace Q₀ C (L+1) hpoly hC.le (by linarith) |
| refine ⟨A,L,min rG (min (dist a b/4) (1/(L+2))),hA,hL, |
| lt_min hrG (lt_min (by positivity) (by positivity)),K,B*K,?_⟩ |
| intro r hr hrs T hsep hcover p hpT |
| have hrG' : r≤rG := hrs.trans (min_le_left ..) |
| have hrD : r≤dist a b/4 := hrs.trans ((min_le_right ..).trans (min_le_left ..)) |
| have hrL : r≤1/(L+2) := hrs.trans ((min_le_right ..).trans (min_le_right ..)) |
| have hscale : (L+2)*r≤1 := by have hh := (le_div_iff₀ (by linarith : 0<L+2)).mp hrL; nlinarith |
| have hTnt : (T:Set (Boundary D)).Nontrivial := by |
| obtain ⟨u,hu,hau⟩ := hcover a |
| obtain ⟨v,hv,hbv⟩ := hcover b |
| refine ⟨u,hu,v,hv,?_⟩ |
| intro heq |
| subst v |
| have hh := dist_triangle a u b |
| rw [dist_comm u b] at hh |
| linarith |
| obtain ⟨e₀,he₀,he₀R,P,n,F,hFc,hFd,hFt,hconn,hFS,hfinite,hsmall⟩ := |
| hgraph r hr hrG' T hsep hcover p hpT |
| let V : Set (Boundary D) := ⋃ j,range (F j) |
| have hV : IsClosed V := isClosed_iUnion_of_finite (fun j => (F j).compact_range.isClosed) |
| have hTV : (T:Set (Boundary D))⊆V := CannonConnectedSupport.finite_extras_absorbed |
| hV T.finite_toSet hconn.isConnected.isPreconnected (hTnt.mono subset_union_right) |
| have heq : V∪(T:Set (Boundary D))=V := union_eq_left.mpr hTV |
| change IsPathConnected (V∪(T:Set (Boundary D))) at hconn |
| change V∪(T:Set (Boundary D))⊆e₀ '' _ at hFS |
| change Finite (ConnectedComponents ((V∪(T:Set (Boundary D)))ᶜ : Set (Boundary D))) at hfinite |
| change ∀ R : Set (Boundary D), IsPreconnected R → R⊆(V∪(T:Set (Boundary D)))ᶜ → |
| ∀ x∈R, ∀ y∈R, dist x y<L*r at hsmall |
| rw [heq] at hconn hFS hfinite hsmall |
| have hn : 0<n := by |
| by_contra hn |
| have hn0 : n=0 := Nat.eq_zero_of_not_pos hn |
| obtain ⟨x,hx⟩ := hTnt.nonempty |
| obtain ⟨i,hi⟩ := mem_iUnion.mp (hTV hx) |
| exact (Nat.not_lt_zero i.val) (hn0 ▸ i.isLt) |
| let e : ℂ → Boundary D := e₀ ∘ complexPlaneL |
| have he : IsOpenEmbedding e := he₀.comp complexPlaneL.toHomeomorph.isOpenEmbedding |
| have heR : range e={p}ᶜ := by rw [show range e=range e₀ from (complexPlaneL.surjective.range_comp e₀),he₀R] |
| obtain ⟨Q,hQ⟩ := finite_segments_pull P |
| have hVS : V⊆e '' (⋃ q∈Q,segment ℝ q.1 q.2) := by |
| intro x hx |
| obtain ⟨y,hy,rfl⟩ := hFS hx |
| refine ⟨complexPlaneL.symm y,?_,by simp [e,Function.comp_def]⟩ |
| rw [←hQ] |
| simpa using hy |
| have hrange (i : Fin n) : range (F i)⊆range e := |
| (subset_iUnion (fun j => range (F j)) i).trans (hVS.trans (image_subset_range ..)) |
| have hQs (i : Fin n) : e ⁻¹' range (F i)⊆⋃ q∈Q,segment ℝ q.1 q.2 := by |
| intro z hz |
| obtain ⟨w,hw,heq⟩ := hVS (mem_iUnion.mpr ⟨i,hz⟩) |
| exact he.injective heq ▸ hw |
| have hwide (x : Boundary D) : intersectionCount F (ball x ((L+1)*r))≤B*K := by |
| have hh := hB r A hr hA (by convert hscale using 1; ring) T hsep hcover |
| (fun i => range (F i)) K (fun x => by simpa only [traceCard,intersectionCount_eq_card] using hFt x) x |
| simpa only [traceCard,intersectionCount_eq_card] using hh |
| let Faces := ConnectedComponents (Vᶜ : Set (Boundary D)) |
| let _ : DiscreteTopology Faces := complement_components_discrete hV |
| let π : (Vᶜ : Set (Boundary D)) → Faces := ConnectedComponents.mk |
| have hπ : Continuous π := ConnectedComponents.continuous_coe |
| obtain ⟨S,labels,hcl,hlabels⟩ := chart_tame_face_structure e he F hrange hFc Q hQs hπ |
| refine ⟨e,he,heR,Q,n,hn,F,hFc,hFd,hFt,hwide,hconn,hTV,hVS,hfinite,hsmall, |
| S,labels,hcl,hlabels,?_⟩ |
| intro d |
| have hR := component_region_connected Vᶜ d |
| obtain ⟨x,hx⟩ := hR.nonempty |
| have hRd : region Vᶜ π d⊆closedBall x (L*r) := by |
| intro y hy |
| simpa only [mem_closedBall,dist_comm] using (hsmall _ hR.isPreconnected (region_subset Vᶜ π d) x hx y hy).le |
| have hclose : closure (region Vᶜ π d)⊆closedBall x (L*r) := |
| closure_minimal hRd isClosed_closedBall |
| have hsub : S d⊆{i | (range (F i)∩ball x ((L+1)*r)).Nonempty} := by |
| intro i hi |
| refine ⟨F i 0,mem_range_self 0,?_⟩ |
| have hh : F i 0∈closure (region Vᶜ π d) := by |
| rw [hcl d] |
| exact Or.inr (mem_iUnion₂.mpr ⟨i,hi,mem_range_self 0⟩) |
| have hd := hclose hh |
| change dist (F i 0) x≤L*r at hd |
| change dist (F i 0) x<(L+1)*r |
| nlinarith |
| calc |
| Nat.card (S d)≤Nat.card {i | (range (F i)∩ball x ((L+1)*r)).Nonempty} := |
| Nat.card_le_card_of_injective (Set.inclusion hsub) (Set.inclusion_injective hsub) |
| _=intersectionCount F (ball x ((L+1)*r)) := by |
| simp [intersectionCount_eq_card,Nat.card_eq_fintype_card,Fintype.card_subtype] |
| _≤B*K := hwide x |
| end CannonRelease |
| |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| end |
| |
| end OAI |
| |
