| import Mathlib | |
| import OAI.Geometry.Cannon.PlanarGraphs.TraceCard | |
| import OAI.Geometry.Cannon.PlaneTopology.FaceVerticesCard | |
| namespace OAI | |
| section | |
| section | |
| section | |
| noncomputable section | |
| open Set unitInterval Topology | |
| noncomputable section | |
| noncomputable section | |
| noncomputable section | |
| open Set Topology unitInterval | |
| noncomputable section | |
| open Set Metric Topology unitInterval | |
| noncomputable section | |
| open Set Metric Topology | |
| noncomputable section | |
| noncomputable section | |
| open Set unitInterval | |
| noncomputable section | |
| noncomputable section | |
| open Set Metric Topology | |
| noncomputable section | |
| open Set Metric Topology | |
| noncomputable section | |
| open Set Metric Topology | |
| noncomputable section | |
| open Set Topology | |
| noncomputable section | |
| open Set Topology | |
| noncomputable section | |
| open Set Topology | |
| noncomputable section | |
| open Set Topology unitInterval | |
| noncomputable section | |
| namespace CannonRelease | |
| open Set Metric Topology unitInterval CannonGraph CannonRegularDisk CannonEmbeddedArc CannonEmbeddedArc.Arc | |
| CannonFaceIncidence CannonFaceLabels CannonPlaneLinear CannonIncidence CannonPolygonal | |
| variable {G : Type*} [Group G] [TopologicalSpace G] [DiscreteTopology G] | |
| variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ) | |
| theorem boundary_uniform_triangular_graphs (φ : Boundary D ≃ₜ SphereTwo) : | |
| let _ := visualMetricSpace D hδ | |
| ∃ M L r₀ : ℝ, 1≤M ∧ 1≤L ∧ 0<r₀ ∧ ∃ N : 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 ∧ | |
| ∃ (κ : Type) (H : κ → Arc ℂ), Finite κ ∧ Nonempty κ ∧ | |
| Pairwise (fun i j => Compatible (H i) (H j)) ∧ | |
| (∃ Q : Finset (ℂ × ℂ),∀ i,range (H i)⊆⋃ q∈Q,segment ℝ q.1 q.2) ∧ | |
| (T:Set (Boundary D))⊆e '' (⋃ i,range (H i)) ∧ | |
| (∀ q : ℂ,IsPreconnected ((⋃ i,range (H i))\{q})) ∧ | |
| (∀ i,∀ t s : I,dist (e (H i t)) (e (H i s))≤M*r) ∧ | |
| (∀ d,∀ z∈closure (face H d),∀ w∈closure (face H d),dist (e z) (e w)≤L*r) ∧ | |
| (∀ x : Boundary D,Nat.card {i : κ | ∃ t:I,dist (e (H i t)) x<2*M*r}≤N) ∧ | |
| (∀ d : Faces H,∃ i j k : κ,i≠j ∧ j≠k ∧ k≠i ∧ | |
| frontier (face H d)=range (H i)∪range (H j)∪range (H k) ∧ | |
| closure (face H d)=face H d∪(range (H i)∪range (H j)∪range (H k))) := by | |
| classical | |
| let _ := visualMetricSpace D hδ | |
| dsimp only | |
| obtain ⟨A,L,rG,hA,hL,hrG,K,J,hgraph⟩ := boundary_uniform_controlled_faces D hδ φ | |
| let M := max A L | |
| have hM : 1≤M := hA.trans (le_max_left _ _) | |
| let R := 2*M+2*L | |
| have hR : 0<R := by dsimp only [R]; linarith | |
| obtain ⟨Q₀,C,hQ₀,hC,hpoly⟩ := relative_polynomial_packing D hδ | |
| obtain ⟨B,hB⟩ := uniform_larger_trace Q₀ C R hpoly hC.le hR | |
| refine ⟨M,L,min rG (1/(R+1)),hM,hL,lt_min hrG (by positivity), | |
| (1+12*J)*(1+4*J)*(B*K),?_⟩ | |
| intro r hr hrs T hsep hcover p hpT | |
| have hrG' : r≤rG := hrs.trans (min_le_left ..) | |
| have hrR : r≤1/(R+1) := hrs.trans (min_le_right ..) | |
| have hscale : (R+1)*r≤1 := by | |
| have hh := (le_div_iff₀ (by linarith : 0<R+1)).mp hrR | |
| nlinarith | |
| obtain ⟨e,he,heR,Q,n,hn,F,hF,hFd,hFt,hFJ,hconn,hTV,hFS,hfinite,hsmall,hstructure⟩ := | |
| hgraph r hr hrG' T hsep hcover p hpT | |
| let _ : Nonempty (Fin n) := ⟨⟨0,hn⟩⟩ | |
| let V : Set (Boundary D) := ⋃ i,range (F i) | |
| have hrange (i : Fin n) : range (F i)⊆range e := | |
| (subset_iUnion (fun j => range (F j)) i).trans (hFS.trans (image_subset_range ..)) | |
| let Fc : Fin n → Arc ℂ := fun i => pullChart e he.isEmbedding (F i) (hrange i) | |
| have hmap (i : Fin n) (t : I) : e (Fc i t)=F i t := map_pullChart e he.isEmbedding (F i) (hrange i) t | |
| have hFc : Pairwise (fun i j => Compatible (Fc i) (Fc j)) := | |
| fun _ _ hij => pullChart_compatible e he.isEmbedding _ _ _ _ (hF hij) | |
| have hFcQ (i : Fin n) : range (Fc i)⊆⋃ q∈Q,segment ℝ q.1 q.2 := by | |
| rintro z ⟨t,rfl⟩ | |
| obtain ⟨w,hw,hew⟩ := hFS (mem_iUnion.mpr ⟨i,mem_range_self t⟩) | |
| exact he.injective (hew.trans (hmap i t).symm) ▸ hw | |
| have himage : e '' (⋃ i,range (Fc i))=V := by | |
| ext z | |
| constructor | |
| · rintro ⟨w,hw,rfl⟩ | |
| obtain ⟨i,t,rfl⟩ := mem_iUnion.mp hw | |
| exact mem_iUnion.mpr ⟨i,t,(hmap i t).symm⟩ | |
| · intro hz | |
| obtain ⟨i,t,rfl⟩ := mem_iUnion.mp hz | |
| exact ⟨Fc i t,mem_iUnion.mpr ⟨i,mem_range_self t⟩,hmap i t⟩ | |
| have hconnc : IsPreconnected (⋃ i,range (Fc i)) := by | |
| apply he.isEmbedding.isInducing.isPreconnected_image.mp | |
| rw [himage] | |
| exact hconn.isConnected.isPreconnected | |
| have hfcSmall (d : Faces Fc) : | |
| ∀ z∈closure (face Fc d),∀ w∈closure (face Fc d),dist (e z) (e w)≤L*r := by | |
| apply continuous_closure_diam e he.continuous | |
| have hc : IsPreconnected (e '' face Fc d) := | |
| (CannonFaceIncidence.component_region_connected _ d).isPreconnected.image e he.continuous.continuousOn | |
| have hs : e '' face Fc d⊆Vᶜ := by | |
| rintro z ⟨w,hw,rfl⟩ hv | |
| rw [←himage] at hv | |
| obtain ⟨u,hu,heu⟩ := hv | |
| have heq := he.injective heu | |
| subst u | |
| exact (region_subset _ ConnectedComponents.mk d hw) hu | |
| intro z hz w hw | |
| exact (hsmall _ hc hs (e z) (mem_image_of_mem _ hz) (e w) (mem_image_of_mem _ hw)).le | |
| have hfcE (i : Fin n) (t s : I) : dist (e (Fc i t)) (e (Fc i s))≤A*r := by | |
| rw [hmap,hmap] | |
| exact (dist_le_diam_of_mem (F i).compact_range.isBounded (mem_range_self _) (mem_range_self _)).trans (hFd i) | |
| let _ : LocallyConnectedSpace ((⋃ i,range (Fc i))ᶜ : Set ℂ) := | |
| (isClosed_iUnion_of_finite fun i => (Fc i).compact_range.isClosed).isOpen_compl.locallyConnectedSpace | |
| let _ : DiscreteTopology (ConnectedComponents ((⋃ i,range (Fc i))ᶜ : Set ℂ)) := inferInstance | |
| have hfcB (d : Faces Fc) : Nat.card (regionEdges Fc ConnectedComponents.mk d)≤J := by | |
| obtain ⟨x,hx⟩ := face_nonempty Fc d | |
| have hs : regionEdges Fc ConnectedComponents.mk d⊆ | |
| {i | (range (F i)∩ball (e x) ((L+1)*r)).Nonempty} := by | |
| intro i hi | |
| refine ⟨F i 0,mem_range_self 0,?_⟩ | |
| have hcl := region_edge_in_closure Fc hFc Q hFcQ ConnectedComponents.continuous_coe hi (mem_range_self (0:I)) | |
| have hd := hfcSmall d (Fc i 0) hcl x (subset_closure hx) | |
| rw [hmap] at hd | |
| change dist (F i 0) (e x)<(L+1)*r | |
| nlinarith | |
| calc | |
| Nat.card (regionEdges Fc ConnectedComponents.mk d)≤ | |
| Nat.card {i | (range (F i)∩ball (e x) ((L+1)*r)).Nonempty} := | |
| Nat.card_le_card_of_injective (Set.inclusion hs) (Set.inclusion_injective hs) | |
| _=intersectionCount F (ball (e x) ((L+1)*r)) := by | |
| simp [intersectionCount_eq_card,Nat.card_eq_fintype_card,Fintype.card_subtype] | |
| _≤J := hFJ (e x) | |
| obtain ⟨κ,H,hκ,hκn,hH,hHQ,hFH,hHK,hHE,hHD,hHC,hTri⟩ := | |
| quantitative_triangular_refinement Fc hFc Q hFcQ hconnc e (L*r) (A*r) | |
| (mul_nonneg (by linarith) hr.le) hfcSmall hfcE J hfcB | |
| let _ : Finite κ := hκ | |
| let _ : Nonempty κ := hκn | |
| refine ⟨e,he,heR,κ,H,hκ,hκn,hH,hHQ,?_,hHK,?_,hHD,?_,hTri⟩ | |
| · apply hTV.trans | |
| change V⊆e '' (⋃ i,range (H i)) | |
| rw [←himage] | |
| exact image_mono hFH | |
| · intro i t s | |
| have hm : max (A*r) (L*r)=M*r := by | |
| exact (max_mul_of_nonneg A L hr.le).symm | |
| exact hm ▸ hHE i t s | |
| · intro x | |
| have hlarge := hB r A hr hA hscale T hsep hcover (fun i => range (F i)) K | |
| (fun x => by simpa only [traceCard,intersectionCount_eq_card] using hFt x) x | |
| have hcount : Nat.card {i : Fin n | ∃ t:I,dist (e (Fc i t)) x<2*M*r+2*(L*r)}≤B*K := by | |
| have heq : 2*M*r+2*(L*r)=R*r := by dsimp only [R]; ring | |
| simp_rw [heq,hmap] | |
| have hp : {i : Fin n | ∃ t:I,dist (F i t) x<R*r}= | |
| {i : Fin n | (range (F i)∩ball x (R*r)).Nonempty} := by | |
| ext i | |
| constructor | |
| · rintro ⟨t,ht⟩ | |
| exact ⟨F i t,mem_range_self t,ht⟩ | |
| · rintro ⟨y,⟨t,rfl⟩,hy⟩ | |
| exact ⟨t,hy⟩ | |
| rw [hp] | |
| simpa only [Nat.card_eq_fintype_card,Fintype.card_subtype,traceCard,Set.mem_ofPred_eq] using hlarge | |
| exact (hHC x (2*M*r)).trans (Nat.mul_le_mul_left _ hcount) | |
| end CannonRelease | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end | |
| end OAI | |