| import OAI.Combinatorics.CompleteCrossing.Ports |
| import OAI.Combinatorics.Crossing.RoundCircleChords |
| import OAI.Combinatorics.Crossing.SegmentIntersectionUnique |
| import OAI.Combinatorics.Crossing.VertexDiskSpokes |
|
|
| namespace OAI |
|
|
| /-! Interior disk passages, vertex ports, filling containment, and crossing uniqueness. -/ |
|
|
| section |
|
|
| /-! The actual internal marked passages and their two boundary endpoints. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.DrawingPassages |
| variable {D : FiniteIntersectionDrawing graph} |
|
|
| abbrev InnerIndex (C : DrawingPassages D) := |
| (e : Edge) × {i : Fin (C.count e+1) // 0 < i.val ∧ i.val < C.count e} |
|
|
| def innerMark (C : DrawingPassages D) (p : C.InnerIndex) : DrawingMark D := |
| ⟨D.edge p.1 (C.node p.1 p.2.val),C.node_mark p.1 p.2.val⟩ |
|
|
| def incomingArc (C : DrawingPassages D) (p : C.InnerIndex) : C.ArcIndex := |
| ⟨p.1,⟨p.2.val.val-1,by have h := p.2.property; omega⟩⟩ |
| |
| def outgoingArc (C : DrawingPassages D) (p : C.InnerIndex) : C.ArcIndex := |
| ⟨p.1,⟨p.2.val.val,p.2.property.2⟩⟩ |
|
|
| def incomingPort (C : DrawingPassages D) (p : C.InnerIndex) : C.Port := (C.incomingArc p,true) |
| def outgoingPort (C : DrawingPassages D) (p : C.InnerIndex) : C.Port := (C.outgoingArc p,false) |
|
|
| theorem incoming_node (C : DrawingPassages D) (p : C.InnerIndex) : |
| (C.incomingArc p).2.succ=p.2.val := by |
| apply Fin.ext |
| change p.2.val.val-1+1=p.2.val.val |
| have h := p.2.property.1 |
| omega |
|
|
| theorem outgoing_node (C : DrawingPassages D) (p : C.InnerIndex) : |
| (C.outgoingArc p).2.castSucc=p.2.val := rfl |
|
|
| theorem incomingPort_mark (C : DrawingPassages D) (p : C.InnerIndex) : |
| C.portMark (C.incomingPort p)=C.innerMark p := by |
| apply Subtype.ext |
| change D.edge p.1 (C.node p.1 (C.incomingArc p).2.succ)=D.edge p.1 (C.node p.1 p.2.val) |
| rw [C.incoming_node] |
|
|
| theorem outgoingPort_mark (C : DrawingPassages D) (p : C.InnerIndex) : |
| C.portMark (C.outgoingPort p)=C.innerMark p := rfl |
|
|
| theorem incomingPort_point (C : DrawingPassages D) (p : C.InnerIndex) : |
| C.portPoint (C.incomingPort p)=D.edge p.1 (C.passage p.1 p.2.val).enter := by |
| change D.edge p.1 (C.passage p.1 (C.incomingArc p).2.succ).enter=_ |
| rw [C.incoming_node] |
|
|
| theorem outgoingPort_point (C : DrawingPassages D) (p : C.InnerIndex) : |
| C.portPoint (C.outgoingPort p)=D.edge p.1 (C.passage p.1 p.2.val).leave := rfl |
|
|
| theorem inner_parameter_interior (C : DrawingPassages D) (p : C.InnerIndex) : |
| C.node p.1 p.2.val ∈ Set.Ioo (0 : unitInterval) 1 := by |
| constructor |
| · rw [← C.first p.1] |
| exact C.increasing p.1 (show (0 : Fin (C.count p.1+1)) < p.2.val from p.2.property.1) |
| · rw [← C.last p.1] |
| exact C.increasing p.1 (show p.2.val < Fin.last (C.count p.1) from p.2.property.2) |
|
|
| theorem innerMark_is_crossing (C : DrawingPassages D) (p : C.InnerIndex) : |
| (C.innerMark p).val ∈ Paper170.crossingPoints D.toContinuousDrawing := by |
| have hm := (C.innerMark p).property |
| rcases hm with ⟨v,hv⟩ | hcross |
| · exact False.elim (D.interior_avoids_vertices p.1 (C.node p.1 p.2.val) |
| (C.inner_parameter_interior p) v hv.symm) |
| · exact hcross |
|
|
| def crossingChord (C : DrawingPassages D) (p : C.InnerIndex) : |
| Path (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) := |
| Path.segment _ _ |
|
|
| theorem crossingChord_injective (C : DrawingPassages D) (p : C.InnerIndex) : |
| Function.Injective (C.crossingChord p) := by |
| apply Path.segment_injective_of_ne |
| intro he |
| rw [C.incomingPort_point,C.outgoingPort_point] at he |
| exact (C.passage p.1 p.2.val).enter_lt_leave.ne (D.edge_injective p.1 he) |
|
|
| end Paper170Geometry.DrawingPassages |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Actual replacement vertices and all incident straight-spoke directions. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.DrawingPassages |
| variable {D : FiniteIntersectionDrawing graph} |
|
|
| def vertexMark (_C : DrawingPassages D) (v : Vertex) : DrawingMark D := |
| ⟨D.vertex v,Or.inl ⟨v,rfl⟩⟩ |
|
|
| theorem vertexMark_injective (C : DrawingPassages D) : Function.Injective C.vertexMark := by |
| intro v w he |
| exact D.vertex_injective (congrArg Subtype.val he) |
|
|
| theorem exists_drawing_vertex_placement [Fintype Edge] (C : DrawingPassages D) : |
| ∃ v : Vertex → Plane, Function.Injective v ∧ |
| (∀ i, squaredLength (v i-D.vertex i) < C.radius^2) ∧ |
| (∀ i (p : C.PortsAt (C.vertexMark i)), v i ≠ C.portPoint p.val) ∧ |
| (∀ i (p q : C.PortsAt (C.vertexMark i)), p ≠ q → |
| det2 (C.portPoint p.val-v i) (C.portPoint q.val-v i) ≠ 0) := by |
| classical |
| have hex (i : Vertex) : ∃ v : Plane, |
| squaredLength (v-D.vertex i) < C.radius^2 ∧ |
| (∀ p : C.PortsAt (C.vertexMark i), v ≠ C.portPoint p.val) ∧ |
| (∀ p q : C.PortsAt (C.vertexMark i), p ≠ q → det2 (C.portPoint p.val-v) (C.portPoint q.val-v) ≠ 0) := |
| exists_vertex_disk_point (D.vertex i) C.radius C.radius_pos |
| (fun p : C.PortsAt (C.vertexMark i) => C.portPoint p.val) |
| (C.portPoint_injective.comp Subtype.val_injective) (C.portsAt_on_circle (C.vertexMark i)) |
| choose v hv hne hdet using hex |
| refine ⟨v,?_,hv,hne,hdet⟩ |
| intro i j he |
| by_contra hij |
| have hd := C.disks_disjoint (C.vertexMark i) (C.vertexMark j) (C.vertexMark_injective.ne hij) |
| have hi : v i ∈ closedRoundDisk (C.vertexMark i).val C.radius := (hv i).le |
| have hj : v j ∈ closedRoundDisk (C.vertexMark j).val C.radius := (hv j).le |
| exact Set.disjoint_left.mp hd hi (he.symm ▸ hj) |
|
|
| theorem vertex_spoke_injective (C : DrawingPassages D) (v : Vertex → Plane) |
| (hv : ∀ i, squaredLength (v i-D.vertex i) < C.radius^2) |
| (i : Vertex) (p : C.PortsAt (C.vertexMark i)) : |
| Function.Injective (Path.segment (v i) (C.portPoint p.val)) := by |
| apply Path.segment_injective_of_ne |
| intro he |
| have h := hv i |
| have hp : squaredLength (C.portPoint p.val-D.vertex i)=C.radius^2 := C.portsAt_on_circle (C.vertexMark i) p |
| rw [he,hp] at h |
| exact lt_irrefl _ h |
|
|
| theorem vertex_spoke_interior (C : DrawingPassages D) (v : Vertex → Plane) |
| (hv : ∀ i, squaredLength (v i-D.vertex i) < C.radius^2) |
| (i : Vertex) (p : C.PortsAt (C.vertexMark i)) (t : unitInterval) |
| (ht : t < 1) : squaredLength ((Path.segment (v i) (C.portPoint p.val)) t-D.vertex i) < C.radius^2 := by |
| simp only [Path.segment_apply,AffineMap.lineMap_apply_module] |
| exact spoke_in_open_disk (D.vertex i) (v i) (C.portPoint p.val) C.radius (hv i) |
| (C.portsAt_on_circle (C.vertexMark i) p) t.val ⟨t.property.1,ht⟩ |
|
|
| end Paper170Geometry.DrawingPassages |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Actual crossing-disk chord incidence, with no triples inherited from the original drawing. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.DrawingPassages |
| variable {D : FiniteIntersectionDrawing graph} |
|
|
| theorem innerIndex_eq_of_edge_mark (C : DrawingPassages D) (p q : C.InnerIndex) |
| (he : p.1=q.1) (hm : C.innerMark p=C.innerMark q) : p=q := by |
| rcases p with ⟨e,i⟩ |
| rcases q with ⟨f,j⟩ |
| change e=f at he |
| subst f |
| congr 1 |
| apply Subtype.ext |
| exact (C.increasing e).injective (D.edge_injective e (congrArg Subtype.val hm)) |
|
|
| theorem incomingPort_injective (C : DrawingPassages D) : Function.Injective C.incomingPort := by |
| intro p q he |
| apply C.innerIndex_eq_of_edge_mark p q (congrArg (fun x : C.Port => x.1.1) he) |
| have hh := congrArg C.portMark he |
| simpa only [C.incomingPort_mark] using hh |
|
|
| theorem outgoingPort_injective (C : DrawingPassages D) : Function.Injective C.outgoingPort := by |
| intro p q he |
| apply C.innerIndex_eq_of_edge_mark p q (congrArg (fun x : C.Port => x.1.1) he) |
| have hh := congrArg C.portMark he |
| simpa only [C.outgoingPort_mark] using hh |
|
|
| theorem incomingPort_ne_outgoingPort (C : DrawingPassages D) (p q : C.InnerIndex) : |
| C.incomingPort p ≠ C.outgoingPort q := by |
| intro he |
| have hh : true=false := congrArg Prod.snd he |
| cases hh |
|
|
| theorem incomingPoint_on_circle (C : DrawingPassages D) (p : C.InnerIndex) : |
| squaredLength (C.portPoint (C.incomingPort p)-(C.innerMark p).val)=C.radius^2 := by |
| simpa only [C.incomingPort_mark] using C.portPoint_on_circle (C.incomingPort p) |
|
|
| theorem outgoingPoint_on_circle (C : DrawingPassages D) (p : C.InnerIndex) : |
| squaredLength (C.portPoint (C.outgoingPort p)-(C.innerMark p).val)=C.radius^2 := by |
| simpa only [C.outgoingPort_mark] using C.portPoint_on_circle (C.outgoingPort p) |
|
|
| theorem crossing_chords_transverse_at_mark (C : DrawingPassages D) (p q : C.InnerIndex) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) : |
| SegmentPairTransverse (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) |
| (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q)) := by |
| apply round_circle_chords_transverse (C.innerMark p).val _ _ _ _ C.radius |
| · exact C.incomingPoint_on_circle p |
| · exact C.outgoingPoint_on_circle p |
| · simpa only [hm] using C.incomingPoint_on_circle q |
| · simpa only [hm] using C.outgoingPoint_on_circle q |
| · exact C.portPoint_injective.ne (C.incomingPort_ne_outgoingPort p p) |
| · exact C.portPoint_injective.ne (C.incomingPort_ne_outgoingPort q q) |
| · exact C.portPoint_injective.ne (C.incomingPort_injective.ne hpq) |
| · exact C.portPoint_injective.ne (C.incomingPort_ne_outgoingPort p q) |
| · exact C.portPoint_injective.ne (Ne.symm (C.incomingPort_ne_outgoingPort q p)) |
| · exact C.portPoint_injective.ne (C.outgoingPort_injective.ne hpq) |
|
|
| theorem innerMark_no_triples (C : DrawingPassages D) (p q r : C.InnerIndex) |
| (hpq : p ≠ q) (hpr : p ≠ r) (hqr : q ≠ r) |
| (hpqM : C.innerMark p=C.innerMark q) (hprM : C.innerMark p=C.innerMark r) : False := by |
| have heq : p.1 ≠ q.1 := fun he => hpq (C.innerIndex_eq_of_edge_mark p q he hpqM) |
| have her : p.1 ≠ r.1 := fun he => hpr (C.innerIndex_eq_of_edge_mark p r he hprM) |
| have hqrM : C.innerMark q=C.innerMark r := hpqM.symm.trans hprM |
| have hfr : q.1 ≠ r.1 := fun he => hqr (C.innerIndex_eq_of_edge_mark q r he hqrM) |
| have hp : (C.innerMark p).val ∈ Paper170.edgeInterior D.toContinuousDrawing p.1 := |
| ⟨C.node p.1 p.2.val,C.inner_parameter_interior p,rfl⟩ |
| have hq : (C.innerMark p).val ∈ Paper170.edgeInterior D.toContinuousDrawing q.1 := by |
| rw [hpqM] |
| exact ⟨C.node q.1 q.2.val,C.inner_parameter_interior q,rfl⟩ |
| have hr : (C.innerMark p).val ∈ Paper170.edgeInterior D.toContinuousDrawing r.1 := by |
| rw [hprM] |
| exact ⟨C.node r.1 r.2.val,C.inner_parameter_interior r,rfl⟩ |
| have hmem : (C.innerMark p).val ∈ Paper170.edgeInterior D.toContinuousDrawing p.1 ∩ |
| Paper170.edgeInterior D.toContinuousDrawing q.1 ∩ Paper170.edgeInterior D.toContinuousDrawing r.1 := ⟨⟨hp,hq⟩,hr⟩ |
| rw [D.no_triples p.1 q.1 r.1 heq her hfr] at hmem |
| exact hmem |
|
|
| end Paper170Geometry.DrawingPassages |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The actual first and last retained ports attach to the prescribed graph vertices. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.DrawingPassages |
| variable {D : FiniteIntersectionDrawing graph} |
|
|
| def firstArc (C : DrawingPassages D) (e : Edge) : C.ArcIndex := |
| ⟨e,⟨0,C.count_pos e⟩⟩ |
|
|
| def lastArc (C : DrawingPassages D) (e : Edge) : C.ArcIndex := |
| ⟨e,⟨C.count e-1,by have h := C.count_pos e; omega⟩⟩ |
| |
| theorem lastArc_node (C : DrawingPassages D) (e : Edge) : |
| (C.lastArc e).2.succ=Fin.last (C.count e) := by |
| apply Fin.ext |
| have h := C.count_pos e |
| change C.count e-1+1=C.count e |
| omega |
|
|
| theorem firstArc_mark (C : DrawingPassages D) (e : Edge) : |
| C.arcStartMark (C.firstArc e)=C.vertexMark (graph.src e) := by |
| apply Subtype.ext |
| change D.edge e (C.node e 0)=D.vertex (graph.src e) |
| rw [C.first,Path.source] |
|
|
| theorem lastArc_mark (C : DrawingPassages D) (e : Edge) : |
| C.arcEndMark (C.lastArc e)=C.vertexMark (graph.dst e) := by |
| apply Subtype.ext |
| change D.edge e (C.node e (C.lastArc e).2.succ)=D.vertex (graph.dst e) |
| rw [C.lastArc_node,C.last,Path.target] |
|
|
| def startPortAt (C : DrawingPassages D) (e : Edge) : C.PortsAt (C.vertexMark (graph.src e)) := |
| ⟨(C.firstArc e,false),C.firstArc_mark e⟩ |
|
|
| def endPortAt (C : DrawingPassages D) (e : Edge) : C.PortsAt (C.vertexMark (graph.dst e)) := |
| ⟨(C.lastArc e,true),C.lastArc_mark e⟩ |
|
|
| end Paper170Geometry.DrawingPassages |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Actual closed-disk containment and strict interior containment of disk fillings. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
|
|
| theorem round_chord_in_disk (z a b : Plane) (r : ℝ) |
| (ha : squaredLength (a-z)=r^2) (hb : squaredLength (b-z)=r^2) |
| (t : ℝ) (ht : t ∈ Set.Icc (0 : ℝ) 1) : |
| squaredLength (segmentPoint a b t-z) ≤ r^2 := by |
| have he : squaredLength (segmentPoint a b t-z)= |
| (1-t)*squaredLength (a-z)+t*squaredLength (b-z)-t*(1-t)*squaredLength (b-a) := by |
| dsimp [squaredLength,segmentPoint] |
| ring |
| have hnon : 0 ≤ t*(1-t)*squaredLength (b-a) := by |
| apply mul_nonneg (mul_nonneg ht.1 (sub_nonneg.mpr ht.2)) |
| dsimp [squaredLength] |
| positivity |
| rw [he,ha,hb] |
| linarith |
|
|
| namespace DrawingPassages |
| variable {D : FiniteIntersectionDrawing graph} |
|
|
| theorem crossingChord_range_subset_disk (C : DrawingPassages D) (p : C.InnerIndex) : |
| Set.range (C.crossingChord p) ⊆ closedRoundDisk (C.innerMark p).val C.radius := by |
| rintro x ⟨t,rfl⟩ |
| change squaredLength ((Path.segment _ _) t-(C.innerMark p).val) ≤ C.radius^2 |
| simp only [Path.segment_apply,AffineMap.lineMap_apply_module] |
| exact round_chord_in_disk _ _ _ _ (C.incomingPoint_on_circle p) (C.outgoingPoint_on_circle p) t.val t.property |
|
|
| theorem crossingChord_interior_in_disk (C : DrawingPassages D) (p : C.InnerIndex) |
| (t : unitInterval) (ht : t ∈ Set.Ioo (0 : unitInterval) 1) : |
| squaredLength (C.crossingChord p t-(C.innerMark p).val) < C.radius^2 := by |
| change squaredLength ((Path.segment _ _) t-(C.innerMark p).val) < C.radius^2 |
| simp only [Path.segment_apply,AffineMap.lineMap_apply_module] |
| exact round_chord_in_open_disk _ _ _ _ (C.incomingPoint_on_circle p) (C.outgoingPoint_on_circle p) |
| (C.portPoint_injective.ne (C.incomingPort_ne_outgoingPort p p)) t.val ht |
|
|
| theorem crossingChords_disjoint_of_marks_ne (C : DrawingPassages D) (p q : C.InnerIndex) |
| (hm : C.innerMark p ≠ C.innerMark q) : |
| Disjoint (Set.range (C.crossingChord p)) (Set.range (C.crossingChord q)) := |
| (C.disks_disjoint _ _ hm).mono (C.crossingChord_range_subset_disk p) (C.crossingChord_range_subset_disk q) |
|
|
| theorem portPoint_outside_other_disk (C : DrawingPassages D) (p : C.Port) (x : DrawingMark D) |
| (hx : C.portMark p ≠ x) : C.radius^2 < squaredLength (C.portPoint p-x.val) := by |
| apply lt_of_not_ge |
| intro hh |
| have hp : C.portPoint p ∈ closedRoundDisk (C.portMark p).val C.radius := (C.portPoint_on_circle p).le |
| exact Set.disjoint_left.mp (C.disks_disjoint _ _ hx) hp hh |
|
|
| theorem portPoint_outside_open_disks (C : DrawingPassages D) (p : C.Port) (x : DrawingMark D) : |
| C.radius^2 ≤ squaredLength (C.portPoint p-x.val) := by |
| by_cases he : C.portMark p=x |
| · have hp := C.portPoint_on_circle p |
| rw [he] at hp |
| exact hp.ge |
| · exact (C.portPoint_outside_other_disk p x he).le |
|
|
| theorem crossingChord_avoids_vertex_disks (C : DrawingPassages D) (p : C.InnerIndex) |
| (v : Vertex) : |
| Disjoint (Set.range (C.crossingChord p)) (closedRoundDisk (D.vertex v) C.radius) := by |
| have hn : C.innerMark p ≠ C.vertexMark v := by |
| intro he |
| exact crossing_not_vertex D (C.innerMark p).val (C.innerMark_is_crossing p) v (congrArg Subtype.val he) |
| exact (C.disks_disjoint _ _ hn).mono (C.crossingChord_range_subset_disk p) Set.Subset.rfl |
|
|
| end DrawingPassages |
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! At most one actual chord crossing point in each original crossing disk. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.DrawingPassages |
| variable {D : FiniteIntersectionDrawing graph} |
|
|
| theorem innerIndex_at_mark_pair (C : DrawingPassages D) (p q r : C.InnerIndex) |
| (hpq : p ≠ q) (hpqM : C.innerMark p=C.innerMark q) (hprM : C.innerMark p=C.innerMark r) : |
| r=p ∨ r=q := by |
| by_cases hrp : r=p |
| · exact Or.inl hrp |
| by_cases hrq : r=q |
| · exact Or.inr hrq |
| exact False.elim (C.innerMark_no_triples p q r hpq (Ne.symm hrp) (Ne.symm hrq) hpqM hprM) |
|
|
| def chordCrossingsAt (C : DrawingPassages D) (w : DrawingMark D) : Set Plane := |
| {x | ∃ p q : C.InnerIndex, p ≠ q ∧ C.innerMark p=w ∧ C.innerMark q=w ∧ |
| x ∈ Set.range (C.crossingChord p) ∧ x ∈ Set.range (C.crossingChord q)} |
|
|
| theorem chordCrossingsAt_subsingleton (C : DrawingPassages D) (w : DrawingMark D) : |
| (C.chordCrossingsAt w).Subsingleton := by |
| rintro x ⟨p,q,hpq,hp,hq,hxp,hxq⟩ y ⟨r,s,hrs,hr,hs,hyr,hys⟩ |
| have hpm : C.innerMark p=C.innerMark q := hp.trans hq.symm |
| have hrpair := C.innerIndex_at_mark_pair p q r hpq hpm (hp.trans hr.symm) |
| have hspair := C.innerIndex_at_mark_pair p q s hpq hpm (hp.trans hs.symm) |
| have hunique := transverse_segment_intersection_subsingleton _ _ _ _ |
| (C.crossing_chords_transverse_at_mark p q hpq hpm) |
| rcases hrpair with rfl | rfl <;> rcases hspair with rfl | rfl |
| · exact False.elim (hrs rfl) |
| · exact hunique ⟨hxp,hxq⟩ ⟨hyr,hys⟩ |
| · exact hunique ⟨hxp,hxq⟩ ⟨hys,hyr⟩ |
| · exact False.elim (hrs rfl) |
| |
| theorem chord_meeting_marks_eq (C : DrawingPassages D) (p q : C.InnerIndex) {x : Plane} |
| (hp : x ∈ Set.range (C.crossingChord p)) (hq : x ∈ Set.range (C.crossingChord q)) : |
| C.innerMark p=C.innerMark q := by |
| by_contra h |
| exact Set.disjoint_left.mp (C.crossingChords_disjoint_of_marks_ne p q h) hp hq |
|
|
| def chordCrossings (C : DrawingPassages D) : Set Plane := |
| {x | ∃ p q : C.InnerIndex, p ≠ q ∧ x ∈ Set.range (C.crossingChord p) ∧ x ∈ Set.range (C.crossingChord q)} |
|
|
| theorem chord_crossing_original_mark (C : DrawingPassages D) {x : Plane} (hx : x ∈ C.chordCrossings) : |
| ∃ w : DrawingMark D, w.val ∈ Paper170.crossingPoints D.toContinuousDrawing ∧ x ∈ C.chordCrossingsAt w := by |
| obtain ⟨p,q,hpq,hp,hq⟩ := hx |
| refine ⟨C.innerMark p,C.innerMark_is_crossing p,p,q,hpq,rfl,?_,hp,hq⟩ |
| exact (C.chord_meeting_marks_eq p q hp hq).symm |
|
|
| end Paper170Geometry.DrawingPassages |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Count actual spatial chord crossings by their original crossing disks. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.DrawingPassages |
| open scoped Classical |
| variable {D : FiniteIntersectionDrawing graph} |
|
|
| def chordCrossingOrigin (C : DrawingPassages D) (x : Plane) : Plane := |
| if hx : x ∈ C.chordCrossings then (C.chord_crossing_original_mark hx).choose.val else 0 |
|
|
| theorem chordCrossingOrigin_mapsTo (C : DrawingPassages D) : |
| Set.MapsTo C.chordCrossingOrigin C.chordCrossings (Paper170.crossingPoints D.toContinuousDrawing) := by |
| intro x hx |
| have hh := (C.chord_crossing_original_mark hx).choose_spec.1 |
| simpa only [chordCrossingOrigin,dite_eq_left hx] using hh |
|
|
| theorem chordCrossingOrigin_injOn (C : DrawingPassages D) : |
| Set.InjOn C.chordCrossingOrigin C.chordCrossings := by |
| intro x hx y hy he |
| let w : DrawingMark D := (C.chord_crossing_original_mark hx).choose |
| let v : DrawingMark D := (C.chord_crossing_original_mark hy).choose |
| have hwx : x ∈ C.chordCrossingsAt w := (C.chord_crossing_original_mark hx).choose_spec.2 |
| have hvy : y ∈ C.chordCrossingsAt v := (C.chord_crossing_original_mark hy).choose_spec.2 |
| have hvw : w=v := by |
| apply Subtype.ext |
| simpa only [chordCrossingOrigin,dite_eq_left hx,dite_eq_left hy] using he |
| rw [← hvw] at hvy |
| exact (C.chordCrossingsAt_subsingleton w) hwx hvy |
|
|
| theorem chordCrossings_finite (C : DrawingPassages D) : C.chordCrossings.Finite := |
| Set.Finite.of_injOn C.chordCrossingOrigin_mapsTo C.chordCrossingOrigin_injOn D.crossings_finite |
|
|
| /-- Different original crossing disks remain separately counted, even when |
| their two chords come from the same pair of graph edges. -/ |
| theorem chordCrossings_ncard_le (C : DrawingPassages D) : |
| C.chordCrossings.ncard ≤ finiteCrossingCount D := by |
| calc |
| C.chordCrossings.ncard ≤ (Paper170.crossingPoints D.toContinuousDrawing).ncard := |
| Set.ncard_le_ncard_of_injOn C.chordCrossingOrigin C.chordCrossingOrigin_mapsTo |
| C.chordCrossingOrigin_injOn D.crossings_finite |
| _ = finiteCrossingCount D := Set.ncard_eq_toFinset_card _ D.crossings_finite |
|
|
| end Paper170Geometry.DrawingPassages |
|
|
| end |
| end |
|
|
| end OAI |
|
|
