| import OAI.Combinatorics.CompleteCrossing.DrawingMarks | |
| import OAI.Combinatorics.Crossing.CurveDiskIsolation | |
| import OAI.Combinatorics.Crossing.FiniteDiskCollar | |
| import OAI.Combinatorics.Crossing.IntervalSegmentStrict | |
| import OAI.Combinatorics.Crossing.OrderedPassages | |
| namespace OAI | |
| /-! Isolated drawing disks, retained arcs, and boundary ports. -/ | |
| section | |
| /-! The actual finite vertex/crossing marks supply the normalization disk family. -/ | |
| noncomputable section | |
| open Zarankiewicz | |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} | |
| namespace Paper170Geometry | |
| abbrev DrawingMark (D : FiniteIntersectionDrawing graph) := ↥(drawingMarks D) | |
| theorem drawing_disk_isolation [Fintype Vertex] [Fintype Edge] (D : FiniteIntersectionDrawing graph) : | |
| ∃ δ : Edge → ℝ, ∃ r : ℝ, (∀ e, 0 < δ e) ∧ 0 < r ∧ | |
| (∀ x y : DrawingMark D, x ≠ y → | |
| Disjoint (closedRoundDisk x.val r) (closedRoundDisk y.val r)) ∧ | |
| (∀ e (x : DrawingMark D) t, D.edge e t ∈ closedRoundDisk x.val r → | |
| ∃ u : unitInterval, D.edge e u=x.val ∧ t ∈ parameterNeighborhood u (δ e)) ∧ | |
| (∀ e u v, D.edge e u ∈ drawingMarks D → D.edge e v ∈ drawingMarks D → u ≠ v → | |
| Disjoint (closure (parameterNeighborhood u (δ e))) | |
| (closure (parameterNeighborhood v (δ e)))) := by | |
| let : Fintype (DrawingMark D) := (drawingMarks_finite D).fintype | |
| let : DecidableEq (DrawingMark D) := Classical.decEq _ | |
| obtain ⟨δ,r,hδ,hr,hd,hv,hs⟩ := finite_injective_curve_disk_isolation | |
| (fun e => D.edge e) (fun e => (D.edge e).continuous) D.edge_injective | |
| (fun x : DrawingMark D => x.val) Subtype.val_injective | |
| refine ⟨δ,r,hδ,hr,hd,hv,?_⟩ | |
| intro e u v hu hv huv | |
| exact hs e u v ⟨⟨D.edge e u,hu⟩,rfl⟩ ⟨⟨D.edge e v,hv⟩,rfl⟩ huv | |
| end Paper170Geometry | |
| end | |
| end | |
| section | |
| /-! Constructed finite marked-passage decomposition of the given actual drawing. -/ | |
| noncomputable section | |
| open Zarankiewicz | |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} | |
| namespace Paper170Geometry | |
| /-- Geometric cut data for the original drawing. Existence is proved below; | |
| this record asserts no crossing bound or drawing-normalization conclusion. -/ | |
| structure DrawingPassages (D : FiniteIntersectionDrawing graph) where | |
| radius : ℝ | |
| radius_pos : 0 < radius | |
| disks_disjoint : ∀ x y : DrawingMark D, x ≠ y → | |
| Disjoint (closedRoundDisk x.val radius) (closedRoundDisk y.val radius) | |
| count : Edge → ℕ | |
| count_pos : ∀ e, 0 < count e | |
| node : ∀ e, Fin (count e+1) → unitInterval | |
| increasing : ∀ e, StrictMono (node e) | |
| first : ∀ e, node e 0=0 | |
| last : ∀ e, node e (Fin.last (count e))=1 | |
| node_range : ∀ e, Set.range (node e)=(D.edge e) ⁻¹' drawingMarks D | |
| passage : ∀ e i, DiskPassage (D.edge e) (D.edge e (node e i)) radius (node e i) | |
| separated : ∀ e i j, i < j → (passage e i).leave < (passage e j).enter | |
| visited_mark_on_edge : ∀ e (x : DrawingMark D) t, | |
| D.edge e t ∈ closedRoundDisk x.val radius → ∃ u, D.edge e u=x.val | |
| theorem nonempty_drawingPassages [Fintype Vertex] [Fintype Edge] (D : FiniteIntersectionDrawing graph) : Nonempty (DrawingPassages D) := by | |
| classical | |
| obtain ⟨δ,r,hδ,hr,hdisks,hvisits,hparam⟩ := drawing_disk_isolation D | |
| choose k hk τ hτ h0 h1 hR using (fun e => exists_sorted_drawing_parameters D e) | |
| have hnode (e : Edge) (i : Fin (k e+1)) : D.edge e (τ e i) ∈ drawingMarks D := by | |
| change τ e i ∈ (D.edge e) ⁻¹' drawingMarks D | |
| rw [← hR e] | |
| exact ⟨i,rfl⟩ | |
| have hex (e : Edge) (i : Fin (k e+1)) : | |
| Nonempty (DiskPassage (D.edge e) (D.edge e (τ e i)) r (τ e i)) := | |
| nonempty_diskPassage (D.edge e) (D.edge e (τ e i)) r (τ e i) | |
| (by simpa [squaredLength] using sq_pos_of_pos hr) | |
| let P (e : Edge) (i : Fin (k e+1)) := Classical.choice (hex e i) | |
| have hlocal (e : Edge) (i : Fin (k e+1)) (t : unitInterval) | |
| (ht : D.edge e t ∈ closedRoundDisk (D.edge e (τ e i)) r) : | |
| t ∈ parameterNeighborhood (τ e i) (δ e) := by | |
| obtain ⟨u,hu,hh⟩ := hvisits e ⟨D.edge e (τ e i),hnode e i⟩ t ht | |
| have he : u=τ e i := D.edge_injective e hu | |
| simpa [he] using hh | |
| have hsep (e : Edge) (i j : Fin (k e+1)) (hij : i < j) : | |
| (P e i).leave < (P e j).enter := by | |
| apply diskPassage_separated (P e i) (P e j) (hτ e hij) (hδ e) | |
| · exact hparam e (τ e i) (τ e j) (hnode e i) (hnode e j) ((hτ e).injective.ne (ne_of_lt hij)) | |
| · exact hlocal e i _ (P e i).leave_mem | |
| · exact hlocal e j _ (P e j).enter_mem | |
| exact ⟨{ | |
| radius := r | |
| radius_pos := hr | |
| disks_disjoint := hdisks | |
| count := k | |
| count_pos := hk | |
| node := τ | |
| increasing := hτ | |
| first := h0 | |
| last := h1 | |
| node_range := hR | |
| passage := P | |
| separated := hsep | |
| visited_mark_on_edge := fun e x t ht => by | |
| obtain ⟨u,hu,_⟩ := hvisits e x t ht | |
| exact ⟨u,hu⟩ }⟩ | |
| end Paper170Geometry | |
| end | |
| end | |
| section | |
| /-! Actual retained middle arcs: endpoint circles and avoidance of every disk and mark. -/ | |
| noncomputable section | |
| open Zarankiewicz | |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} | |
| namespace Paper170Geometry.DrawingPassages | |
| variable {D : FiniteIntersectionDrawing graph} | |
| abbrev ArcIndex (C : DrawingPassages D) := (e : Edge) × Fin (C.count e) | |
| theorem node_mark (C : DrawingPassages D) (e : Edge) (i : Fin (C.count e+1)) : | |
| D.edge e (C.node e i) ∈ drawingMarks D := by | |
| change C.node e i ∈ (D.edge e) ⁻¹' drawingMarks D | |
| rw [← C.node_range e] | |
| exact ⟨i,rfl⟩ | |
| def arcStartMark (C : DrawingPassages D) (i : C.ArcIndex) : DrawingMark D := | |
| ⟨D.edge i.1 (C.node i.1 i.2.castSucc),C.node_mark i.1 i.2.castSucc⟩ | |
| def arcEndMark (C : DrawingPassages D) (i : C.ArcIndex) : DrawingMark D := | |
| ⟨D.edge i.1 (C.node i.1 i.2.succ),C.node_mark i.1 i.2.succ⟩ | |
| def arcStart (C : DrawingPassages D) (i : C.ArcIndex) : Plane := | |
| D.edge i.1 (C.passage i.1 i.2.castSucc).leave | |
| def arcEnd (C : DrawingPassages D) (i : C.ArcIndex) : Plane := | |
| D.edge i.1 (C.passage i.1 i.2.succ).enter | |
| def arc (C : DrawingPassages D) (i : C.ArcIndex) : Path (C.arcStart i) (C.arcEnd i) := | |
| retainedSubarc (D.edge i.1) (C.node i.1) C.radius (C.passage i.1) i.2 | |
| theorem arc_gap (C : DrawingPassages D) (i : C.ArcIndex) : | |
| (C.passage i.1 i.2.castSucc).leave < (C.passage i.1 i.2.succ).enter := | |
| C.separated i.1 i.2.castSucc i.2.succ (by change i.2.val < i.2.val+1; omega) | |
| theorem arc_injective (C : DrawingPassages D) (i : C.ArcIndex) : Function.Injective (C.arc i) := | |
| pathSubarc_injective (D.edge i.1) (D.edge_injective i.1) _ _ (C.arc_gap i) | |
| theorem arc_endpoints_ne (C : DrawingPassages D) (i : C.ArcIndex) : C.arcStart i ≠ C.arcEnd i := by | |
| intro he | |
| exact (C.arc_gap i).ne (D.edge_injective i.1 he) | |
| theorem arc_range (C : DrawingPassages D) (i : C.ArcIndex) : | |
| Set.range (C.arc i)=D.edge i.1 '' Set.Icc (C.passage i.1 i.2.castSucc).leave (C.passage i.1 i.2.succ).enter := | |
| pathSubarc_range (D.edge i.1) _ _ (C.arc_gap i) | |
| theorem arc_avoids_marks (C : DrawingPassages D) (i : C.ArcIndex) (x : Plane) | |
| (hx : x ∈ Set.range (C.arc i)) : x ∉ drawingMarks D := by | |
| rw [C.arc_range] at hx | |
| obtain ⟨t,ht,rfl⟩ := hx | |
| intro hh | |
| have hp : t ∈ Set.range (C.node i.1) := by rw [C.node_range]; exact hh | |
| exact retained_interval_no_mark (D.edge i.1) (C.node i.1) (C.increasing i.1) | |
| (C.first i.1) (C.last i.1) C.radius (C.passage i.1) i.2 t ht hp | |
| theorem retained_parameter_outside (C : DrawingPassages D) (i : C.ArcIndex) (t : unitInterval) | |
| (ht : t ∈ Set.Ioo (C.passage i.1 i.2.castSucc).leave (C.passage i.1 i.2.succ).enter) : | |
| D.edge i.1 t ∈ outsideDisks (fun x : DrawingMark D => x.val) C.radius := by | |
| intro x | |
| apply lt_of_not_ge | |
| intro hh | |
| obtain ⟨u,hu⟩ := C.visited_mark_on_edge i.1 x t hh | |
| have hmu : u ∈ Set.range (C.node i.1) := by | |
| rw [C.node_range] | |
| change D.edge i.1 u ∈ drawingMarks D | |
| rw [hu] | |
| exact x.property | |
| obtain ⟨j,hj⟩ := hmu | |
| have hc : D.edge i.1 (C.node i.1 j)=x.val := (congrArg (D.edge i.1) hj).trans hu | |
| have hv := (C.passage i.1 j).all_visits t (by rw [hc]; exact hh) | |
| by_cases hji : j ≤ i.2.castSucc | |
| · have hm := (ordered_passage_leave_strictMono (D.edge i.1) (C.node i.1) C.radius | |
| (C.passage i.1) (C.separated i.1)).monotone hji | |
| exact ht.1.not_ge (hv.2.trans hm) | |
| · have hij : i.2.succ ≤ j := by | |
| change i.2.val+1 ≤ j.val | |
| have hn : ¬ j.val ≤ i.2.val := hji | |
| omega | |
| have hm := (ordered_passage_enter_strictMono (D.edge i.1) (C.node i.1) C.radius | |
| (C.passage i.1) (C.separated i.1)).monotone hij | |
| exact ht.2.not_ge (hm.trans hv.1) | |
| theorem arc_interior_outside (C : DrawingPassages D) (i : C.ArcIndex) (t : unitInterval) | |
| (ht : t ∈ Set.Ioo (0 : unitInterval) 1) : | |
| C.arc i t ∈ outsideDisks (fun x : DrawingMark D => x.val) C.radius := by | |
| exact C.retained_parameter_outside i _ (intervalSegment_strict_between _ _ t (C.arc_gap i) ht) | |
| theorem arc_start_on_circle (C : DrawingPassages D) (i : C.ArcIndex) : | |
| squaredLength (C.arcStart i-(C.arcStartMark i).val)=C.radius^2 := by | |
| exact (C.passage i.1 i.2.castSucc).leave_boundary | |
| ((C.arc_gap i).trans_le (show (C.passage i.1 i.2.succ).enter ≤ 1 from (C.passage i.1 i.2.succ).enter.property.2)) | |
| theorem arc_end_on_circle (C : DrawingPassages D) (i : C.ArcIndex) : | |
| squaredLength (C.arcEnd i-(C.arcEndMark i).val)=C.radius^2 := by | |
| exact (C.passage i.1 i.2.succ).enter_boundary | |
| ((show 0 ≤ (C.passage i.1 i.2.castSucc).leave from (C.passage i.1 i.2.castSucc).leave.property.1).trans_lt (C.arc_gap i)) | |
| end Paper170Geometry.DrawingPassages | |
| end | |
| end | |
| section | |
| /-! The retained actual compact arcs are pairwise disjoint, including on the same edge. -/ | |
| noncomputable section | |
| open Zarankiewicz | |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} | |
| namespace Paper170Geometry.DrawingPassages | |
| variable {D : FiniteIntersectionDrawing graph} | |
| theorem arcs_pairwise_disjoint (C : DrawingPassages D) (p q : C.ArcIndex) (hpq : p ≠ q) : | |
| Disjoint (Set.range (C.arc p)) (Set.range (C.arc q)) := by | |
| apply Set.disjoint_left.mpr | |
| intro x hx hy | |
| have hxm := C.arc_avoids_marks p x hx | |
| rcases p with ⟨e,i⟩ | |
| rcases q with ⟨f,j⟩ | |
| rw [C.arc_range] at hx hy | |
| obtain ⟨s,hs,hsx⟩ := hx | |
| obtain ⟨t,ht,htx⟩ := hy | |
| have hef : e=f := unmarked_intersection_same_edge D e f s t (hsx.trans htx.symm) | |
| (hsx.symm ▸ hxm) | |
| subst f | |
| have hst : s=t := D.edge_injective e (hsx.trans htx.symm) | |
| rw [← hst] at ht | |
| have hij : i ≠ j := by | |
| intro he | |
| subst j | |
| exact hpq rfl | |
| rcases lt_or_gt_of_ne hij with hij | hji | |
| · exact Set.disjoint_left.mp | |
| (retained_interval_pair_disjoint (D.edge e) (C.node e) C.radius (C.passage e) (C.separated e) i j hij) hs ht | |
| · exact Set.disjoint_left.mp | |
| (retained_interval_pair_disjoint (D.edge e) (C.node e) C.radius (C.passage e) (C.separated e) j i hji) ht hs | |
| end Paper170Geometry.DrawingPassages | |
| end | |
| end | |
| section | |
| /-! The actual distinct disk-boundary endpoints of all retained arcs. -/ | |
| noncomputable section | |
| open Zarankiewicz | |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} | |
| namespace Paper170Geometry.DrawingPassages | |
| variable {D : FiniteIntersectionDrawing graph} | |
| abbrev Port (C : DrawingPassages D) := C.ArcIndex × Bool | |
| def portPoint (C : DrawingPassages D) (p : C.Port) : Plane := | |
| if p.2 then C.arcEnd p.1 else C.arcStart p.1 | |
| def portMark (C : DrawingPassages D) (p : C.Port) : DrawingMark D := | |
| if p.2 then C.arcEndMark p.1 else C.arcStartMark p.1 | |
| theorem portPoint_mem_arc (C : DrawingPassages D) (p : C.Port) : | |
| C.portPoint p ∈ Set.range (C.arc p.1) := by | |
| rcases p with ⟨i,flag⟩ | |
| cases flag | |
| · exact ⟨0,(C.arc i).source⟩ | |
| · exact ⟨1,(C.arc i).target⟩ | |
| theorem portPoint_injective (C : DrawingPassages D) : Function.Injective C.portPoint := by | |
| rintro ⟨i,s⟩ ⟨j,t⟩ he | |
| have hij : i=j := by | |
| by_contra h | |
| exact Set.disjoint_left.mp (C.arcs_pairwise_disjoint i j h) | |
| (C.portPoint_mem_arc (i,s)) (he.symm ▸ C.portPoint_mem_arc (j,t)) | |
| subst j | |
| have hst : s=t := by | |
| cases s <;> cases t | |
| · rfl | |
| · exact False.elim (C.arc_endpoints_ne i he) | |
| · exact False.elim (C.arc_endpoints_ne i he.symm) | |
| · rfl | |
| subst t | |
| rfl | |
| theorem portPoint_on_circle (C : DrawingPassages D) (p : C.Port) : | |
| squaredLength (C.portPoint p-(C.portMark p).val)=C.radius^2 := by | |
| rcases p with ⟨i,flag⟩ | |
| cases flag | |
| · exact C.arc_start_on_circle i | |
| · exact C.arc_end_on_circle i | |
| abbrev PortsAt (C : DrawingPassages D) (x : DrawingMark D) := {p : C.Port // C.portMark p=x} | |
| theorem portsAt_on_circle (C : DrawingPassages D) (x : DrawingMark D) (p : C.PortsAt x) : | |
| squaredLength (C.portPoint p.val-x.val)=C.radius^2 := by | |
| simpa only [p.property] using C.portPoint_on_circle p.val | |
| end Paper170Geometry.DrawingPassages | |
| end | |
| end | |
| end OAI | |