| import OAI.Combinatorics.CompleteCrossing.AdjacentSurgery |
|
|
| namespace OAI |
|
|
| /-! Surgery intersection containment, strict crossing reduction, and admissibility. -/ |
|
|
| section |
|
|
| /-! Actual new chords cannot meet any third normalized edge, including at ports. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
|
|
| theorem segment_disjoint_of_open_containment (a b : Plane) (U S : Set Plane) |
| (hin : openSegmentTrace a b ⊆ U) (hout : Disjoint S U) |
| (ha : a ∉ S) (hb : b ∉ S) : Disjoint (segment ℝ a b) S := by |
| apply Set.disjoint_left.mpr |
| intro x hx hs |
| rcases segment_endpoint_or_open a b x hx with rfl | rfl | hx |
| · exact ha hs |
| · exact hb hs |
| · exact Set.disjoint_left.mp hout hs (hin hx) |
|
|
| namespace PolygonalExterior |
| variable {D : Paper170.AdmissibleDrawing graph} {C : DrawingPassages D.toFiniteIntersectionDrawing} |
|
|
| theorem assembled_port_not_mem_other_edge (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (e : Edge) (r : C.Port) (her : e ≠ r.1.1) : |
| C.portPoint r ∉ Set.range (E.assembledEdge v e) := |
| fun hx => her (E.assembled_port_unique_edge v hv e r hx) |
|
|
| theorem reconnection_other_edge_disjoint (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (p q : C.InnerIndex) (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) |
| (i : Bool) (e : Edge) (hep : e ≠ p.1) (heq : e ≠ q.1) : |
| Disjoint (segment ℝ (C.reconnectionStart p q i) (C.reconnectionEnd p q i)) |
| (Set.range (E.assembledEdge v e)) := by |
| apply segment_disjoint_of_open_containment _ _ (C.crossingDisk p) _ |
| (C.reconnection_open_in_disk p q hm i) |
| (E.assembled_other_edge_avoids_crossingDisk v hv p q hpq hm e hep heq) |
| · cases i |
| · exact E.assembled_port_not_mem_other_edge v hv e (C.incomingPort q) heq |
| · exact E.assembled_port_not_mem_other_edge v hv e (C.incomingPort p) hep |
| · cases i |
| · exact E.assembled_port_not_mem_other_edge v hv e (C.outgoingPort p) hep |
| · exact E.assembled_port_not_mem_other_edge v hv e (C.outgoingPort q) heq |
|
|
| end PolygonalExterior |
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Each new chord avoids the retained cuts assigned to the other switched walk. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : Paper170.AdmissibleDrawing graph} {C : DrawingPassages D.toFiniteIntersectionDrawing} |
| (E : PolygonalExterior C) (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (P : ∀ e, PolygonalParametrization ((E.assembledAdmissible v hvinj hv hdet).edge e)) |
| {p q : C.InnerIndex} |
|
|
| theorem forward_chord_disjoint_opposite_prefix |
| (Kq : ExteriorCrossingCut C P q) (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) : |
| Disjoint (segment ℝ (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort q))) |
| (Set.range (graphWalkPath id ((drawingEdgeWalk P q.1).take Kq.first.val))) := by |
| have hqp : q.1 ≠ p.1 := fun he => hpq (C.innerIndex_eq_of_edge_mark p q he.symm hm) |
| apply segment_disjoint_of_open_containment _ _ (C.crossingDisk p) _ (C.reconnection_open_in_disk p q hm true) |
| · simpa only [DrawingPassages.crossingDisk,hm] using Kq.prefix_outside |
| · intro hx |
| exact E.assembled_port_not_mem_other_edge v hv q.1 (C.incomingPort p) hqp (Kq.prefix_subset_edge hx) |
| · exact Kq.outgoing_not_prefix |
|
|
| theorem forward_chord_disjoint_opposite_suffix |
| (Kp : ExteriorCrossingCut C P p) (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) : |
| Disjoint (segment ℝ (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort q))) |
| (Set.range (graphWalkPath id ((drawingEdgeWalk P p.1).drop Kp.last.val))) := by |
| have hpqe : p.1 ≠ q.1 := fun he => hpq (C.innerIndex_eq_of_edge_mark p q he hm) |
| apply segment_disjoint_of_open_containment _ _ (C.crossingDisk p) _ |
| (C.reconnection_open_in_disk p q hm true) Kp.suffix_outside |
| · exact Kp.incoming_not_suffix |
| · intro hx |
| exact E.assembled_port_not_mem_other_edge v hv p.1 (C.outgoingPort q) hpqe (Kp.suffix_subset_edge hx) |
|
|
| theorem backward_chord_disjoint_opposite_prefix |
| (Kp : ExteriorCrossingCut C P p) (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) : |
| Disjoint (segment ℝ (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort p))) |
| (Set.range (graphWalkPath id ((drawingEdgeWalk P p.1).take Kp.first.val))) := by |
| have hpqe : p.1 ≠ q.1 := fun he => hpq (C.innerIndex_eq_of_edge_mark p q he hm) |
| apply segment_disjoint_of_open_containment _ _ (C.crossingDisk p) _ |
| (C.reconnection_open_in_disk p q hm false) Kp.prefix_outside |
| · intro hx |
| exact E.assembled_port_not_mem_other_edge v hv p.1 (C.incomingPort q) hpqe (Kp.prefix_subset_edge hx) |
| · exact Kp.outgoing_not_prefix |
|
|
| theorem backward_chord_disjoint_opposite_suffix |
| (Kq : ExteriorCrossingCut C P q) (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) : |
| Disjoint (segment ℝ (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort p))) |
| (Set.range (graphWalkPath id ((drawingEdgeWalk P q.1).drop Kq.last.val))) := by |
| have hqp : q.1 ≠ p.1 := fun he => hpq (C.innerIndex_eq_of_edge_mark p q he.symm hm) |
| apply segment_disjoint_of_open_containment _ _ (C.crossingDisk p) _ (C.reconnection_open_in_disk p q hm false) |
| · simpa only [DrawingPassages.crossingDisk,hm] using Kq.suffix_outside |
| · exact Kq.incoming_not_suffix |
| · intro hx |
| exact E.assembled_port_not_mem_other_edge v hv q.1 (C.outgoingPort p) hqp (Kq.suffix_subset_edge hx) |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Exact retained-track cases for intersections of the two actual rerouted paths. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : Paper170.AdmissibleDrawing graph} {C : DrawingPassages D.toFiniteIntersectionDrawing} |
| (E : PolygonalExterior C) (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (P : ∀ e, PolygonalParametrization ((E.assembledAdmissible v hvinj hv hdet).edge e)) |
| {p q : C.InnerIndex} |
|
|
| theorem reconnected_paths_meeting_cases |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) |
| (hcross : SegmentsCrossProperly (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) |
| (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q))) |
| (x : Plane) (hx : x ∈ Set.range (reconnectedForwardPath Kp Kq hm)) |
| (hy : x ∈ Set.range (reconnectedBackwardPath Kp Kq hm)) : |
| (x ∈ Set.range (graphWalkPath id ((drawingEdgeWalk P p.1).take Kp.first.val)) ∧ |
| x ∈ Set.range (graphWalkPath id ((drawingEdgeWalk P q.1).take Kq.first.val))) ∨ |
| (x ∈ Set.range (graphWalkPath id ((drawingEdgeWalk P p.1).drop Kp.last.val)) ∧ |
| x ∈ Set.range (graphWalkPath id ((drawingEdgeWalk P q.1).drop Kq.last.val))) := by |
| have hFP := E.forward_chord_disjoint_opposite_prefix v hvinj hv hdet P Kq hpq hm |
| have hFS := E.forward_chord_disjoint_opposite_suffix v hvinj hv hdet P Kp hpq hm |
| have hBP := E.backward_chord_disjoint_opposite_prefix v hvinj hv hdet P Kp hpq hm |
| have hBS := E.backward_chord_disjoint_opposite_suffix v hvinj hv hdet P Kq hpq hm |
| have hchord := C.reconnection_chords_disjoint p q hcross true false (by decide) |
| rcases reconnectedForwardPath_trace Kp Kq hm hx with (hx | hx) | hx |
| · rcases reconnectedBackwardPath_trace Kp Kq hm hy with (hy | hy) | hy |
| · exact Or.inl ⟨hx,hy⟩ |
| · exact False.elim (Set.disjoint_left.mp hBP hy hx) |
| · exact False.elim (Set.disjoint_left.mp Kp.prefix_suffix_disjoint hx hy) |
| · rcases reconnectedBackwardPath_trace Kp Kq hm hy with (hy | hy) | hy |
| · exact False.elim (Set.disjoint_left.mp hFP hx hy) |
| · exact False.elim (Set.disjoint_left.mp hchord hx hy) |
| · exact False.elim (Set.disjoint_left.mp hFS hx hy) |
| · rcases reconnectedBackwardPath_trace Kp Kq hm hy with (hy | hy) | hy |
| · exact False.elim (Set.disjoint_left.mp Kq.prefix_suffix_disjoint hy hx) |
| · exact False.elim (Set.disjoint_left.mp hBS hy hx) |
| · exact Or.inr ⟨hy,hx⟩ |
|
|
| theorem reconnected_paths_meeting_old_edges |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) |
| (hcross : SegmentsCrossProperly (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) |
| (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q))) |
| (x : Plane) (hx : x ∈ Set.range (reconnectedForwardPath Kp Kq hm)) |
| (hy : x ∈ Set.range (reconnectedBackwardPath Kp Kq hm)) : |
| x ∈ Set.range (E.assembledEdge v p.1) ∧ x ∈ Set.range (E.assembledEdge v q.1) := by |
| rcases E.reconnected_paths_meeting_cases v hvinj hv hdet P Kp Kq hpq hm hcross x hx hy with h | h |
| · exact ⟨Kp.prefix_subset_edge h.1,Kq.prefix_subset_edge h.2⟩ |
| · exact ⟨Kp.suffix_subset_edge h.1,Kq.suffix_subset_edge h.2⟩ |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Every contact of rerouted edges is carried by the original drawing tracks. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : Paper170.AdmissibleDrawing graph} {C : DrawingPassages D.toFiniteIntersectionDrawing} |
| (E : PolygonalExterior C) (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (P : ∀ e, PolygonalParametrization ((E.assembledAdmissible v hvinj hv hdet).edge e)) |
| {p q : C.InnerIndex} |
|
|
| theorem reconnectedForwardPath_meeting_other |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) |
| (e : Edge) (hep : e ≠ p.1) (heq : e ≠ q.1) |
| (x : Plane) (hx : x ∈ Set.range (reconnectedForwardPath Kp Kq hm)) |
| (hy : x ∈ Set.range (E.assembledEdge v e)) : |
| x ∈ Set.range (E.assembledEdge v p.1) ∨ x ∈ Set.range (E.assembledEdge v q.1) := by |
| rcases reconnectedForwardPath_trace Kp Kq hm hx with (hx | hx) | hx |
| · exact Or.inl (Kp.prefix_subset_edge hx) |
| · exact False.elim (Set.disjoint_left.mp (E.reconnection_other_edge_disjoint v hv p q hpq hm true e hep heq) hx hy) |
| · exact Or.inr (Kq.suffix_subset_edge hx) |
|
|
| theorem reconnectedBackwardPath_meeting_other |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) |
| (e : Edge) (hep : e ≠ p.1) (heq : e ≠ q.1) |
| (x : Plane) (hx : x ∈ Set.range (reconnectedBackwardPath Kp Kq hm)) |
| (hy : x ∈ Set.range (E.assembledEdge v e)) : |
| x ∈ Set.range (E.assembledEdge v p.1) ∨ x ∈ Set.range (E.assembledEdge v q.1) := by |
| rcases reconnectedBackwardPath_trace Kp Kq hm hx with (hx | hx) | hx |
| · exact Or.inr (Kq.prefix_subset_edge hx) |
| · exact False.elim (Set.disjoint_left.mp (E.reconnection_other_edge_disjoint v hv p q hpq hm false e hep heq) hx hy) |
| · exact Or.inl (Kp.suffix_subset_edge hx) |
|
|
| theorem adjacentSurgeryRoutes_meeting_other |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) (hadj : (graph.src p.1)=(graph.src q.1) ∨ (graph.dst p.1)=(graph.dst q.1)) |
| (e : Edge) (hep : e ≠ p.1) (heq : e ≠ q.1) |
| (x : Plane) (hx : x ∈ Set.range (adjacentSurgeryRouteP Kp Kq hm hadj) ∨ |
| x ∈ Set.range (adjacentSurgeryRouteQ Kp Kq hm hadj)) |
| (hy : x ∈ Set.range (E.assembledEdge v e)) : |
| x ∈ Set.range (E.assembledEdge v p.1) ∨ x ∈ Set.range (E.assembledEdge v q.1) := by |
| have hf := E.reconnectedForwardPath_meeting_other v hvinj hv hdet P Kp Kq hpq hm e hep heq x |
| have hb := E.reconnectedBackwardPath_meeting_other v hvinj hv hdet P Kp Kq hpq hm e hep heq x |
| rcases hx with hx | hx |
| · unfold adjacentSurgeryRouteP at hx |
| split at hx |
| · exact hb hx hy |
| · exact hf hx hy |
| · unfold adjacentSurgeryRouteQ at hx |
| split at hx |
| · exact hf hx hy |
| · exact hb hx hy |
|
|
| theorem adjacentSurgeryRoutes_meeting_each_other |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) (hadj : (graph.src p.1)=(graph.src q.1) ∨ (graph.dst p.1)=(graph.dst q.1)) |
| (hcross : SegmentsCrossProperly (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) |
| (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q))) |
| (x : Plane) (hx : x ∈ Set.range (adjacentSurgeryRouteP Kp Kq hm hadj)) |
| (hy : x ∈ Set.range (adjacentSurgeryRouteQ Kp Kq hm hadj)) : |
| x ∈ Set.range (E.assembledEdge v p.1) ∧ x ∈ Set.range (E.assembledEdge v q.1) := by |
| have hh := E.reconnected_paths_meeting_old_edges v hvinj hv hdet P Kp Kq hpq hm hcross x |
| by_cases h : (graph.src p.1)=(graph.src q.1) |
| · simp only [adjacentSurgeryRouteP,adjacentSurgeryRouteQ,dite_eq_left h] at hx hy |
| exact hh hy hx |
| · simp only [adjacentSurgeryRouteP,adjacentSurgeryRouteQ,dite_eq_right h] at hx hy |
| exact hh hx hy |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Every actual spatial crossing of the surgery drawing was already a crossing before surgery. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : Paper170.AdmissibleDrawing graph} {C : DrawingPassages D.toFiniteIntersectionDrawing} |
| (E : PolygonalExterior C) (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (P : ∀ e, PolygonalParametrization ((E.assembledAdmissible v hvinj hv hdet).edge e)) |
| {p q : C.InnerIndex} |
|
|
| theorem adjacentSurgeryDrawing_crossings_subset |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) (hadj : (graph.src p.1)=(graph.src q.1) ∨ (graph.dst p.1)=(graph.dst q.1)) |
| (hcross : SegmentsCrossProperly (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) |
| (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q))) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) : |
| Paper170.crossingPoints (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb) ⊆ |
| Paper170.crossingPoints (E.assembledAdmissible v hvinj hv hdet).toContinuousDrawing := by |
| let B := (E.assembledAdmissible v hvinj hv hdet).toContinuousDrawing |
| let A := adjacentSurgeryDrawing Kp Kq hm hadj hv hemb |
| intro x hx |
| obtain ⟨e,f,hef,hxe,hxf⟩ := hx |
| have havoid : ∀ j, x ≠ B.vertex j := edgeInterior_not_vertex A e x hxe |
| have hpair (hx : x ∈ Set.range (adjacentSurgeryRouteP Kp Kq hm hadj)) |
| (hy : x ∈ Set.range (adjacentSurgeryRouteQ Kp Kq hm hadj)) : x ∈ Paper170.crossingPoints B := by |
| have hh := E.adjacentSurgeryRoutes_meeting_each_other v hvinj hv hdet P Kp Kq hpq hm hadj hcross x hx hy |
| exact crossing_of_ranges_not_vertices B p.1 q.1 |
| (fun he => hpq (C.innerIndex_eq_of_edge_mark p q he hm)) x hh.1 hh.2 havoid |
| have hmixed (g : Edge) (hgp : g ≠ p.1) (hgq : g ≠ q.1) |
| (hx : x ∈ Set.range (adjacentSurgeryRouteP Kp Kq hm hadj) ∨ |
| x ∈ Set.range (adjacentSurgeryRouteQ Kp Kq hm hadj)) |
| (hy : x ∈ Set.range (B.edge g)) : x ∈ Paper170.crossingPoints B := by |
| rcases E.adjacentSurgeryRoutes_meeting_other v hvinj hv hdet P Kp Kq hpq hm hadj g hgp hgq x hx hy with h | h |
| · exact crossing_of_ranges_not_vertices B p.1 g (Ne.symm hgp) x h hy havoid |
| · exact crossing_of_ranges_not_vertices B q.1 g (Ne.symm hgq) x h hy havoid |
| have hcases (g : Edge) (hg : x ∈ Set.range (A.edge g)) := |
| B.replaceTwoRoutes_range_cases p.1 q.1 (adjacentSurgeryRouteP Kp Kq hm hadj) |
| (adjacentSurgeryRouteQ Kp Kq hm hadj) g x hg |
| rcases hcases e (edgeInterior_subset_range A e hxe) with ⟨hep,hx⟩ | ⟨heq,hx⟩ | ⟨hep,heq,hx⟩ |
| · rcases hcases f (edgeInterior_subset_range A f hxf) with ⟨hfp,hy⟩ | ⟨hfq,hy⟩ | ⟨hfp,hfq,hy⟩ |
| · exact False.elim (hef (hep.trans hfp.symm)) |
| · exact hpair hx hy |
| · exact hmixed f hfp hfq (Or.inl hx) hy |
| · rcases hcases f (edgeInterior_subset_range A f hxf) with ⟨hfp,hy⟩ | ⟨hfq,hy⟩ | ⟨hfp,hfq,hy⟩ |
| · exact hpair hy hx |
| · exact False.elim (hef (heq.trans hfq.symm)) |
| · exact hmixed f hfp hfq (Or.inr hx) hy |
| · rcases hcases f (edgeInterior_subset_range A f hxf) with ⟨hfp,hy⟩ | ⟨hfq,hy⟩ | ⟨hfp,hfq,hy⟩ |
| · exact hmixed e hep heq (Or.inl hy) hx |
| · exact hmixed e hep heq (Or.inr hy) hx |
| · exact crossing_of_ranges_not_vertices B e f hef x hx hy havoid |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! A strict decrease of the actual spatial crossing set; properness of the new drawing is still separate. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : Paper170.AdmissibleDrawing graph} {C : DrawingPassages D.toFiniteIntersectionDrawing} |
| (E : PolygonalExterior C) (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| {p q : C.InnerIndex} |
|
|
| theorem crossingChord_meeting_is_assembled_crossing (hpq : p ≠ q) (x : Plane) |
| (hxp : x ∈ Set.range (C.crossingChord p)) (hxq : x ∈ Set.range (C.crossingChord q)) : |
| x ∈ Paper170.crossingPoints (E.assembledAdmissible v hvinj hv hdet).toContinuousDrawing := by |
| have hm := C.chord_meeting_marks_eq p q hxp hxq |
| have hxin := (C.reconnection_avoids_chord_meeting p q hpq x hxp hxq).1 |
| have hxinq : x ∈ C.crossingDisk q := by |
| simpa only [DrawingPassages.crossingDisk,hm] using hxin |
| apply crossing_of_ranges_not_vertices (E.assembledAdmissible v hvinj hv hdet).toContinuousDrawing |
| p.1 q.1 (fun he => hpq (C.innerIndex_eq_of_edge_mark p q he hm)) x |
| ((E.assembled_range_in_crossingDisk v hv p x hxin).mpr hxp) |
| ((E.assembled_range_in_crossingDisk v hv q x hxinq).mpr hxq) |
| intro j he |
| exact Set.disjoint_left.mp (C.crossingDisk_disjoint_vertex_disk p j) hxin (he.symm ▸ (hv j).le) |
|
|
| theorem adjacentSurgeryDrawing_strict_crossing_ncard |
| (P : ∀ e, PolygonalParametrization ((E.assembledAdmissible v hvinj hv hdet).edge e)) |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) (hadj : (graph.src p.1)=(graph.src q.1) ∨ (graph.dst p.1)=(graph.dst q.1)) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) |
| (x : Plane) (hxp : x ∈ Set.range (C.crossingChord p)) (hxq : x ∈ Set.range (C.crossingChord q)) : |
| (Paper170.crossingPoints (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb)).Finite ∧ |
| (Paper170.crossingPoints (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb)).ncard < |
| (Paper170.crossingPoints (E.assembledAdmissible v hvinj hv hdet).toContinuousDrawing).ncard := by |
| have hcross := C.crossingChord_meeting_segmentsCrossProperly p q hpq x hxp hxq |
| have hsub := E.adjacentSurgeryDrawing_crossings_subset v hvinj hv hdet P Kp Kq hpq hm hadj hcross hemb |
| have hfinite := (E.assembledAdmissible v hvinj hv hdet).crossings_finite |
| have hremoved := adjacentSurgeryDrawing_avoids_crossing Kp Kq hm hpq hadj hv hemb x hxp hxq (by |
| intro e hep heq hx |
| exact Set.disjoint_left.mp (E.assembled_other_edge_avoids_crossingDisk v hv p q hpq hm e hep heq) |
| hx (C.reconnection_avoids_chord_meeting p q hpq x hxp hxq).1) |
| have hnot : x ∉ Paper170.crossingPoints (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb) := by |
| rintro ⟨e,f,hef,hxe,hxf⟩ |
| exact hremoved e (edgeInterior_subset_range _ e hxe) |
| refine ⟨hfinite.subset hsub,Set.ncard_lt_ncard ⟨hsub,?_⟩ hfinite⟩ |
| intro hrev |
| exact hnot (hrev (E.crossingChord_meeting_is_assembled_crossing v hvinj hv hdet hpq x hxp hxq)) |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The actual surgery drawing has no triple interior contacts. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : Paper170.AdmissibleDrawing graph} {C : DrawingPassages D.toFiniteIntersectionDrawing} |
| (E : PolygonalExterior C) (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (P : ∀ e, PolygonalParametrization ((E.assembledAdmissible v hvinj hv hdet).edge e)) |
| {p q : C.InnerIndex} |
|
|
| theorem adjacentSurgeryDrawing_no_triples |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) (hadj : (graph.src p.1)=(graph.src q.1) ∨ (graph.dst p.1)=(graph.dst q.1)) |
| (hcross : SegmentsCrossProperly (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort p)) |
| (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort q))) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) |
| (e f g : Edge) (hef : e ≠ f) (heg : e ≠ g) (hfg : f ≠ g) : |
| Paper170.edgeInterior (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb) e ∩ |
| Paper170.edgeInterior (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb) f ∩ |
| Paper170.edgeInterior (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb) g = ∅ := by |
| let B := E.assembledAdmissible v hvinj hv hdet |
| let A := adjacentSurgeryDrawing Kp Kq hm hadj hv hemb |
| apply Set.eq_empty_iff_forall_notMem.mpr |
| intro x hx |
| obtain ⟨⟨hxe,hxf⟩,hxg⟩ := hx |
| have havoid : ∀ j, x ≠ B.vertex j := edgeInterior_not_vertex A e x hxe |
| have hthree (a b c : Edge) (hab : a ≠ b) (hac : a ≠ c) (hbc : b ≠ c) |
| (ha : x ∈ Set.range (B.edge a)) (hb : x ∈ Set.range (B.edge b)) |
| (hc : x ∈ Set.range (B.edge c)) : False := by |
| have hmem : x ∈ Paper170.edgeInterior B.toContinuousDrawing a ∩ Paper170.edgeInterior B.toContinuousDrawing b ∩ |
| Paper170.edgeInterior B.toContinuousDrawing c := |
| ⟨⟨edgeInterior_of_range_not_vertices _ a x ha havoid, |
| edgeInterior_of_range_not_vertices _ b x hb havoid⟩,edgeInterior_of_range_not_vertices _ c x hc havoid⟩ |
| rw [B.no_triples a b c hab hac hbc] at hmem |
| exact hmem |
| have htwo (a : Edge) (hap : a ≠ p.1) (haq : a ≠ q.1) |
| (hp : x ∈ Set.range (adjacentSurgeryRouteP Kp Kq hm hadj)) |
| (hq : x ∈ Set.range (adjacentSurgeryRouteQ Kp Kq hm hadj)) |
| (ha : x ∈ Set.range (B.edge a)) : False := by |
| have hh := E.adjacentSurgeryRoutes_meeting_each_other v hvinj hv hdet P Kp Kq hpq hm hadj hcross x hp hq |
| exact hthree p.1 q.1 a (fun he => hpq (C.innerIndex_eq_of_edge_mark p q he hm)) |
| (Ne.symm hap) (Ne.symm haq) hh.1 hh.2 ha |
| have hone (a b : Edge) (hab : a ≠ b) |
| (hap : a ≠ p.1) (haq : a ≠ q.1) (hbp : b ≠ p.1) (hbq : b ≠ q.1) |
| (hh : x ∈ Set.range (adjacentSurgeryRouteP Kp Kq hm hadj) ∨ |
| x ∈ Set.range (adjacentSurgeryRouteQ Kp Kq hm hadj)) |
| (ha : x ∈ Set.range (B.edge a)) (hb : x ∈ Set.range (B.edge b)) : False := by |
| rcases E.adjacentSurgeryRoutes_meeting_other v hvinj hv hdet P Kp Kq hpq hm hadj a hap haq x hh ha with hp | hq |
| · exact hthree p.1 a b (Ne.symm hap) (Ne.symm hbp) hab hp ha hb |
| · exact hthree q.1 a b (Ne.symm haq) (Ne.symm hbq) hab hq ha hb |
| have hcases (a : Edge) (ha : x ∈ Set.range (A.edge a)) := |
| B.toContinuousDrawing.replaceTwoRoutes_range_cases p.1 q.1 (adjacentSurgeryRouteP Kp Kq hm hadj) |
| (adjacentSurgeryRouteQ Kp Kq hm hadj) a x ha |
| have hec := hcases e (edgeInterior_subset_range A e hxe) |
| have hfc := hcases f (edgeInterior_subset_range A f hxf) |
| have hgc := hcases g (edgeInterior_subset_range A g hxg) |
| rcases hec with ⟨hep,hx⟩ | ⟨heq,hx⟩ | ⟨hep,heq,hx⟩ |
| · rcases hfc with ⟨hfp,hy⟩ | ⟨hfq,hy⟩ | ⟨hfp,hfq,hy⟩ |
| · exact hef (hep.trans hfp.symm) |
| · rcases hgc with ⟨hgp,hz⟩ | ⟨hgq,hz⟩ | ⟨hgp,hgq,hz⟩ |
| · exact heg (hep.trans hgp.symm) |
| · exact hfg (hfq.trans hgq.symm) |
| · exact htwo g hgp hgq hx hy hz |
| · rcases hgc with ⟨hgp,hz⟩ | ⟨hgq,hz⟩ | ⟨hgp,hgq,hz⟩ |
| · exact heg (hep.trans hgp.symm) |
| · exact htwo f hfp hfq hx hz hy |
| · exact hone f g hfg hfp hfq hgp hgq (Or.inl hx) hy hz |
| · rcases hfc with ⟨hfp,hy⟩ | ⟨hfq,hy⟩ | ⟨hfp,hfq,hy⟩ |
| · rcases hgc with ⟨hgp,hz⟩ | ⟨hgq,hz⟩ | ⟨hgp,hgq,hz⟩ |
| · exact hfg (hfp.trans hgp.symm) |
| · exact heg (heq.trans hgq.symm) |
| · exact htwo g hgp hgq hy hx hz |
| · exact hef (heq.trans hfq.symm) |
| · rcases hgc with ⟨hgp,hz⟩ | ⟨hgq,hz⟩ | ⟨hgp,hgq,hz⟩ |
| · exact htwo f hfp hfq hz hx hy |
| · exact heg (heq.trans hgq.symm) |
| · exact hone f g hfg hfp hfq hgp hgq (Or.inr hx) hy hz |
| · rcases hfc with ⟨hfp,hy⟩ | ⟨hfq,hy⟩ | ⟨hfp,hfq,hy⟩ |
| · rcases hgc with ⟨hgp,hz⟩ | ⟨hgq,hz⟩ | ⟨hgp,hgq,hz⟩ |
| · exact hfg (hfp.trans hgp.symm) |
| · exact htwo e hep heq hy hz hx |
| · exact hone e g heg hep heq hgp hgq (Or.inl hy) hx hz |
| · rcases hgc with ⟨hgp,hz⟩ | ⟨hgq,hz⟩ | ⟨hgp,hgq,hz⟩ |
| · exact htwo e hep heq hz hy hx |
| · exact hfg (hfq.trans hgq.symm) |
| · exact hone e g heg hep heq hgp hgq (Or.inr hy) hx hz |
| · rcases hgc with ⟨hgp,hz⟩ | ⟨hgq,hz⟩ | ⟨hgp,hgq,hz⟩ |
| · exact hone e f hef hep heq hfp hfq (Or.inl hz) hx hy |
| · exact hone e f hef hep heq hfp hfq (Or.inr hz) hx hy |
| · exact hthree e f g hef heg hfg hx hy hz |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The constructed surgery drawing satisfies exactly the proved weaker normalization input. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : Paper170.AdmissibleDrawing graph} {C : DrawingPassages D.toFiniteIntersectionDrawing} |
| (E : PolygonalExterior C) (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (P : ∀ e, PolygonalParametrization ((E.assembledAdmissible v hvinj hv hdet).edge e)) |
| {p q : C.InnerIndex} |
|
|
| def adjacentSurgeryFiniteDrawing |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) (hadj : (graph.src p.1)=(graph.src q.1) ∨ (graph.dst p.1)=(graph.dst q.1)) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) |
| (x : Plane) (hxp : x ∈ Set.range (C.crossingChord p)) (hxq : x ∈ Set.range (C.crossingChord q)) : |
| FiniteIntersectionDrawing graph where |
| toContinuousDrawing := adjacentSurgeryDrawing Kp Kq hm hadj hv hemb |
| crossings_finite := (E.adjacentSurgeryDrawing_strict_crossing_ncard v hvinj hv hdet P Kp Kq hpq hm hadj hemb x hxp hxq).1 |
| no_triples := E.adjacentSurgeryDrawing_no_triples v hvinj hv hdet P Kp Kq hpq hm hadj |
| (C.crossingChord_meeting_segmentsCrossProperly p q hpq x hxp hxq) hemb |
|
|
| theorem adjacentSurgeryFiniteDrawing_count_lt |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hpq : p ≠ q) (hm : C.innerMark p=C.innerMark q) (hadj : (graph.src p.1)=(graph.src q.1) ∨ (graph.dst p.1)=(graph.dst q.1)) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) |
| (x : Plane) (hxp : x ∈ Set.range (C.crossingChord p)) (hxq : x ∈ Set.range (C.crossingChord q)) : |
| finiteCrossingCount (E.adjacentSurgeryFiniteDrawing v hvinj hv hdet P Kp Kq hpq hm hadj hemb x hxp hxq) < |
| Paper170.crossingCount (E.assembledAdmissible v hvinj hv hdet) := by |
| rw [finiteCrossingCount_eq_ncard] |
| have h := (E.adjacentSurgeryDrawing_strict_crossing_ncard v hvinj hv hdet P Kp Kq hpq hm hadj hemb x hxp hxq).2 |
| exact h.trans_eq (Set.ncard_eq_toFinset_card _ (E.assembledAdmissible v hvinj hv hdet).crossings_finite) |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Strict adjacent-crossing improvement with actual admissibility restored by proved normalization. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry.PolygonalExterior |
| variable {D : Paper170.AdmissibleDrawing graph} {C : DrawingPassages D.toFiniteIntersectionDrawing} |
|
|
| theorem exists_admissible_adjacent_surgery [Fintype Vertex] [Fintype Edge] (E : PolygonalExterior C) |
| (v : Vertex → Plane) (hvinj : Function.Injective v) |
| (hv : ∀ j, squaredLength (v j-D.vertex j) < C.radius^2) |
| (hdet : ∀ j (p q : C.PortsAt (C.vertexMark j)), p ≠ q → |
| det2 (C.portPoint p.val-v j) (C.portPoint q.val-v j) ≠ 0) |
| (p q : C.InnerIndex) (hpq : p ≠ q) (hadj : (graph.src p.1)=(graph.src q.1) ∨ (graph.dst p.1)=(graph.dst q.1)) |
| (x : Plane) (hxp : x ∈ Set.range (C.crossingChord p)) (hxq : x ∈ Set.range (C.crossingChord q)) : |
| ∃ A : Paper170.AdmissibleDrawing graph, (∀ e, HasFiniteAffineCover (A.edge e)) ∧ |
| Paper170.crossingCount A < Paper170.crossingCount (E.assembledAdmissible v hvinj hv hdet) := by |
| have hm := C.chord_meeting_marks_eq p q hxp hxq |
| have hcross := C.crossingChord_meeting_segmentsCrossProperly p q hpq x hxp hxq |
| obtain ⟨P,hmark,hport⟩ := E.exists_assembled_port_subdivision v hvinj hv hdet |
| obtain ⟨Kp⟩ := E.nonempty_exteriorCrossingCut v hvinj hv hdet P hport p |
| obtain ⟨Kq⟩ := E.nonempty_exteriorCrossingCut v hvinj hv hdet P hport q |
| have hemb := crossingSurgeryGraph_embedding C P hmark hport p q hm hcross |
| let B := E.adjacentSurgeryFiniteDrawing v hvinj hv hdet P Kp Kq hpq hm hadj hemb x hxp hxq |
| obtain ⟨A,hpoly,hcount⟩ := finite_intersection_normalization B |
| exact ⟨A,hpoly,hcount.trans_lt |
| (E.adjacentSurgeryFiniteDrawing_count_lt v hvinj hv hdet P Kp Kq hpq hm hadj hemb x hxp hxq)⟩ |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| end OAI |
|
|
