| import OAI.Combinatorics.CompleteCrossing.CrossingCuts |
| import OAI.Combinatorics.Crossing.PolygonalParametrizationCover |
|
|
| namespace OAI |
|
|
| /-! Reconnected paths and point-removing surgery on adjacent edges. -/ |
|
|
| section |
|
|
| /-! The two actual switched walks, before loop deletion. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| variable {D D' : Paper170.AdmissibleDrawing graph} |
| {C : DrawingPassages D.toFiniteIntersectionDrawing} {P : ∀ e, PolygonalParametrization (D'.edge e)} |
| {p q : C.InnerIndex} |
|
|
| theorem surgery_forward_adj : (crossingSurgeryGraph C P p q).Adj |
| (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort q)) := |
| crossingSurgeryGraph_reconnection_adj C P p q true |
|
|
| theorem surgery_backward_adj : (crossingSurgeryGraph C P p q).Adj |
| (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort p)) := |
| crossingSurgeryGraph_reconnection_adj C P p q false |
|
|
| def reconnectedForwardWalk (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) : |
| (crossingSurgeryGraph C P p q).Walk (D'.vertex ((graph.src p.1))) (D'.vertex ((graph.dst q.1))) := |
| walkThroughChord (Kp.prefixInSurgery p q rfl) |
| surgery_forward_adj (Kq.suffixInSurgery p q hm.symm) |
|
|
| def reconnectedBackwardWalk (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) : |
| (crossingSurgeryGraph C P p q).Walk (D'.vertex ((graph.src q.1))) (D'.vertex ((graph.dst p.1))) := |
| walkThroughChord (Kq.prefixInSurgery p q hm.symm) |
| surgery_backward_adj (Kp.suffixInSurgery p q rfl) |
|
|
| theorem reconnectedForwardWalk_trace (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) : |
| Set.range (graphWalkPath id (reconnectedForwardWalk Kp Kq hm))= |
| (Set.range (graphWalkPath id ((drawingEdgeWalk P p.1).take Kp.first.val)) ∪ |
| segment ℝ (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort q))) ∪ |
| Set.range (graphWalkPath id ((drawingEdgeWalk P q.1).drop Kq.last.val)) := by |
| rw [reconnectedForwardWalk,walkThroughChord_range, |
| ExteriorCrossingCut.prefixInSurgery_trace,ExteriorCrossingCut.suffixInSurgery_trace] |
|
|
| theorem reconnectedBackwardWalk_trace (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) : |
| Set.range (graphWalkPath id (reconnectedBackwardWalk Kp Kq hm))= |
| (Set.range (graphWalkPath id ((drawingEdgeWalk P q.1).take Kq.first.val)) ∪ |
| segment ℝ (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort p))) ∪ |
| Set.range (graphWalkPath id ((drawingEdgeWalk P p.1).drop Kp.last.val)) := by |
| rw [reconnectedBackwardWalk,walkThroughChord_range, |
| ExteriorCrossingCut.prefixInSurgery_trace,ExteriorCrossingCut.suffixInSurgery_trace] |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Actual simple polygonal paths extracted from the two switched walks. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| variable {D D' : Paper170.AdmissibleDrawing graph} |
| {C : DrawingPassages D.toFiniteIntersectionDrawing} {P : ∀ e, PolygonalParametrization (D'.edge e)} |
| {p q : C.InnerIndex} |
|
|
| def reconnectedForwardPath (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) : Path (D'.vertex ((graph.src p.1))) (D'.vertex ((graph.dst q.1))) := by |
| classical |
| exact graphWalkPath id (reconnectedForwardWalk Kp Kq hm).bypass |
|
|
| def reconnectedBackwardPath (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) : Path (D'.vertex ((graph.src q.1))) (D'.vertex ((graph.dst p.1))) := by |
| classical |
| exact graphWalkPath id (reconnectedBackwardWalk Kp Kq hm).bypass |
|
|
| theorem reconnectedForwardPath_injective (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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) : |
| Function.Injective (reconnectedForwardPath Kp Kq hm) := by |
| classical |
| exact graphWalkPath_injective id hemb (surgery_mixed_endpoints_ne C p q hadj) |
| _ (reconnectedForwardWalk Kp Kq hm).bypass_isPath |
|
|
| theorem reconnectedBackwardPath_injective (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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) : |
| Function.Injective (reconnectedBackwardPath Kp Kq hm) := by |
| classical |
| exact graphWalkPath_injective id hemb (surgery_mixed_endpoints_ne C q p (hadj.imp Eq.symm Eq.symm)) |
| _ (reconnectedBackwardWalk Kp Kq hm).bypass_isPath |
|
|
| theorem reconnectedForwardPath_hasFiniteAffineCover (Kp : ExteriorCrossingCut C P p) |
| (Kq : ExteriorCrossingCut C P q) (hm : C.innerMark p=C.innerMark q) : |
| HasFiniteAffineCover (reconnectedForwardPath Kp Kq hm) := by |
| classical |
| exact graphWalkPath_hasFiniteAffineCover id _ |
|
|
| theorem reconnectedBackwardPath_hasFiniteAffineCover (Kp : ExteriorCrossingCut C P p) |
| (Kq : ExteriorCrossingCut C P q) (hm : C.innerMark p=C.innerMark q) : |
| HasFiniteAffineCover (reconnectedBackwardPath Kp Kq hm) := by |
| classical |
| exact graphWalkPath_hasFiniteAffineCover id _ |
|
|
| theorem reconnectedForwardPath_trace (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) : |
| Set.range (reconnectedForwardPath Kp Kq hm) ⊆ |
| (Set.range (graphWalkPath id ((drawingEdgeWalk P p.1).take Kp.first.val)) ∪ |
| segment ℝ (C.portPoint (C.incomingPort p)) (C.portPoint (C.outgoingPort q))) ∪ |
| Set.range (graphWalkPath id ((drawingEdgeWalk P q.1).drop Kq.last.val)) := by |
| classical |
| exact (graphWalkPath_bypass_range_subset id (reconnectedForwardWalk Kp Kq hm)).trans |
| (by rw [reconnectedForwardWalk_trace]) |
|
|
| theorem reconnectedBackwardPath_trace (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) : |
| Set.range (reconnectedBackwardPath Kp Kq hm) ⊆ |
| (Set.range (graphWalkPath id ((drawingEdgeWalk P q.1).take Kq.first.val)) ∪ |
| segment ℝ (C.portPoint (C.incomingPort q)) (C.portPoint (C.outgoingPort p))) ∪ |
| Set.range (graphWalkPath id ((drawingEdgeWalk P p.1).drop Kp.last.val)) := by |
| classical |
| exact (graphWalkPath_bypass_range_subset id (reconnectedBackwardWalk Kp Kq hm)).trans |
| (by rw [reconnectedBackwardWalk_trace]) |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The selected spatial crossing point is absent from both actual bypassed paths. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| variable {D D' : Paper170.AdmissibleDrawing graph} |
| {C : DrawingPassages D.toFiniteIntersectionDrawing} {P : ∀ e, PolygonalParametrization (D'.edge e)} |
| {p q : C.InnerIndex} |
|
|
| theorem reconnected_paths_avoid_crossing (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) (hpq : p ≠ q) (x : Plane) |
| (hxp : x ∈ Set.range (C.crossingChord p)) (hxq : x ∈ Set.range (C.crossingChord q)) : |
| x ∉ Set.range (reconnectedForwardPath Kp Kq hm) ∧ |
| x ∉ Set.range (reconnectedBackwardPath Kp Kq hm) := by |
| obtain ⟨hxin,havoid⟩ := C.reconnection_avoids_chord_meeting p q hpq x hxp hxq |
| have hxinq : x ∈ C.crossingDisk q := by |
| simpa only [DrawingPassages.crossingDisk,hm] using hxin |
| constructor |
| · intro hx |
| rcases reconnectedForwardPath_trace Kp Kq hm hx with (hx | hx) | hx |
| · exact Set.disjoint_left.mp Kp.prefix_outside hx hxin |
| · exact havoid true hx |
| · exact Set.disjoint_left.mp Kq.suffix_outside hx hxinq |
| · intro hx |
| rcases reconnectedBackwardPath_trace Kp Kq hm hx with (hx | hx) | hx |
| · exact Set.disjoint_left.mp Kq.prefix_outside hx hxinq |
| · exact havoid false hx |
| · exact Set.disjoint_left.mp Kp.suffix_outside hx hxin |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The actual rerouted simple paths have vertex-avoiding interiors. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| variable {D D' : Paper170.AdmissibleDrawing graph} |
| {C : DrawingPassages D.toFiniteIntersectionDrawing} {P : ∀ e, PolygonalParametrization (D'.edge e)} |
| {p q : C.InnerIndex} |
|
|
| theorem path_interior_avoids_vertices_of_range {V : Type*} (vertex : V → Plane) (a b : V) |
| (γ : Path (vertex a) (vertex b)) (hinj : Function.Injective γ) |
| (hrange : ∀ v, vertex v ∈ Set.range γ → v=a ∨ v=b) |
| (t : unitInterval) (ht : t ∈ Set.Ioo (0 : unitInterval) 1) (v : V) : γ t ≠ vertex v := by |
| intro he |
| rcases hrange v ⟨t,he⟩ with rfl | rfl |
| · exact ht.1.ne (hinj (he.trans γ.source.symm)).symm |
| · exact ht.2.ne (hinj (he.trans γ.target.symm)) |
|
|
| theorem reconnectedForwardPath_vertex_eq (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (j : Vertex) (hj : D'.vertex j ∈ Set.range (reconnectedForwardPath Kp Kq hm)) : |
| j=(graph.src p.1) ∨ j=(graph.dst q.1) := by |
| rcases reconnectedForwardPath_trace Kp Kq hm hj with (hj | hj) | hj |
| · exact Or.inl (Kp.prefix_vertex_eq j hj) |
| · exact False.elim (C.reconnection_segment_avoids_vertices p q hm D'.vertex hv true j hj) |
| · exact Or.inr (Kq.suffix_vertex_eq j hj) |
|
|
| theorem reconnectedBackwardPath_vertex_eq (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (j : Vertex) (hj : D'.vertex j ∈ Set.range (reconnectedBackwardPath Kp Kq hm)) : |
| j=(graph.src q.1) ∨ j=(graph.dst p.1) := by |
| rcases reconnectedBackwardPath_trace Kp Kq hm hj with (hj | hj) | hj |
| · exact Or.inl (Kq.prefix_vertex_eq j hj) |
| · exact False.elim (C.reconnection_segment_avoids_vertices p q hm D'.vertex hv false j hj) |
| · exact Or.inr (Kp.suffix_vertex_eq j hj) |
|
|
| theorem reconnectedForwardPath_interior_avoids_vertices |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (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) |
| (t : unitInterval) (ht : t ∈ Set.Ioo (0 : unitInterval) 1) (j : Vertex) : |
| reconnectedForwardPath Kp Kq hm t ≠ D'.vertex j := |
| path_interior_avoids_vertices_of_range D'.vertex _ _ _ |
| (reconnectedForwardPath_injective Kp Kq hm hadj hemb) |
| (reconnectedForwardPath_vertex_eq Kp Kq hm hv) t ht j |
|
|
| theorem reconnectedBackwardPath_interior_avoids_vertices |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (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) |
| (t : unitInterval) (ht : t ∈ Set.Ioo (0 : unitInterval) 1) (j : Vertex) : |
| reconnectedBackwardPath Kp Kq hm t ≠ D'.vertex j := |
| path_interior_avoids_vertices_of_range D'.vertex _ _ _ |
| (reconnectedBackwardPath_injective Kp Kq hm hadj hemb) |
| (reconnectedBackwardPath_vertex_eq Kp Kq hm hv) t ht j |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Replacement of two actual drawing routes with proved simple, vertex-avoiding paths. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170.ContinuousDrawing |
| def replaceTwoRoutes (D : Paper170.ContinuousDrawing graph) (e f : Edge) |
| (γ : Path (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (δ : Path (D.vertex ((graph.src f))) (D.vertex ((graph.dst f)))) |
| (g : Edge) : Path (D.vertex ((graph.src g))) (D.vertex ((graph.dst g))) := by |
| classical |
| exact if h : g=e then γ.cast (congrArg (fun k : Edge => D.vertex ((graph.src k))) h) |
| (congrArg (fun k : Edge => D.vertex ((graph.dst k))) h) |
| else if h : g=f then δ.cast (congrArg (fun k : Edge => D.vertex ((graph.src k))) h) |
| (congrArg (fun k : Edge => D.vertex ((graph.dst k))) h) |
| else D.edge g |
|
|
| theorem replaceTwoRoutes_first (D : Paper170.ContinuousDrawing graph) (e f : Edge) |
| (γ : Path (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (δ : Path (D.vertex ((graph.src f))) (D.vertex ((graph.dst f)))) : |
| D.replaceTwoRoutes e f γ δ e=γ := by |
| simp only [replaceTwoRoutes,dite_eq_left rfl,Path.cast_rfl_rfl] |
|
|
| theorem replaceTwoRoutes_second (D : Paper170.ContinuousDrawing graph) (e f : Edge) (hef : e ≠ f) |
| (γ : Path (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (δ : Path (D.vertex ((graph.src f))) (D.vertex ((graph.dst f)))) : |
| D.replaceTwoRoutes e f γ δ f=δ := by |
| simp only [replaceTwoRoutes,dite_eq_right (Ne.symm hef),dite_eq_left rfl,Path.cast_rfl_rfl] |
|
|
| theorem replaceTwoRoutes_other (D : Paper170.ContinuousDrawing graph) (e f : Edge) |
| (γ : Path (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (δ : Path (D.vertex ((graph.src f))) (D.vertex ((graph.dst f)))) |
| (g : Edge) (hge : g ≠ e) (hgf : g ≠ f) : |
| D.replaceTwoRoutes e f γ δ g=D.edge g := by |
| simp only [replaceTwoRoutes,dite_eq_right hge,dite_eq_right hgf] |
|
|
| def replaceTwoEdges (D : Paper170.ContinuousDrawing graph) (e f : Edge) |
| (γ : Path (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (δ : Path (D.vertex ((graph.src f))) (D.vertex ((graph.dst f)))) |
| (hγ : Function.Injective γ) (hδ : Function.Injective δ) |
| (hvγ : ∀ t, t ∈ Set.Ioo (0 : unitInterval) 1 → ∀ j, γ t ≠ D.vertex j) |
| (hvδ : ∀ t, t ∈ Set.Ioo (0 : unitInterval) 1 → ∀ j, δ t ≠ D.vertex j) : |
| Paper170.ContinuousDrawing graph where |
| vertex := D.vertex |
| vertex_injective := D.vertex_injective |
| edge := D.replaceTwoRoutes e f γ δ |
| edge_injective := by |
| intro g |
| classical |
| by_cases hge : g=e |
| · subst g |
| rw [replaceTwoRoutes_first] |
| exact hγ |
| by_cases hgf : g=f |
| · subst g |
| rw [replaceTwoRoutes_second _ _ _ (Ne.symm hge)] |
| exact hδ |
| · rw [replaceTwoRoutes_other _ _ _ _ _ _ hge hgf] |
| exact D.edge_injective g |
| interior_avoids_vertices := by |
| intro g t ht j |
| classical |
| by_cases hge : g=e |
| · subst g |
| rw [replaceTwoRoutes_first] |
| exact hvγ t ht j |
| by_cases hgf : g=f |
| · subst g |
| rw [replaceTwoRoutes_second _ _ _ (Ne.symm hge)] |
| exact hvδ t ht j |
| · rw [replaceTwoRoutes_other _ _ _ _ _ _ hge hgf] |
| exact D.interior_avoids_vertices g t ht j |
|
|
| end Paper170.ContinuousDrawing |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Trace case analysis for the actual two-route replacement. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170.ContinuousDrawing |
| theorem replaceTwoRoutes_range_cases (D : Paper170.ContinuousDrawing graph) (e f : Edge) |
| (γ : Path (D.vertex ((graph.src e))) (D.vertex ((graph.dst e)))) |
| (δ : Path (D.vertex ((graph.src f))) (D.vertex ((graph.dst f)))) |
| (g : Edge) (x : Plane) (hx : x ∈ Set.range (D.replaceTwoRoutes e f γ δ g)) : |
| (g=e ∧ x ∈ Set.range γ) ∨ (g=f ∧ x ∈ Set.range δ) ∨ |
| (g ≠ e ∧ g ≠ f ∧ x ∈ Set.range (D.edge g)) := by |
| classical |
| by_cases hge : g=e |
| · subst g |
| exact Or.inl ⟨rfl,by simpa only [replaceTwoRoutes_first] using hx⟩ |
| by_cases hgf : g=f |
| · subst g |
| exact Or.inr (Or.inl ⟨rfl,by simpa only [replaceTwoRoutes_second D e f (Ne.symm hge)] using hx⟩) |
| · exact Or.inr (Or.inr ⟨hge,hgf,by simpa only [replaceTwoRoutes_other D e f γ δ g hge hgf] using hx⟩) |
|
|
| end Paper170.ContinuousDrawing |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Assigning the two switched paths to the original edges for either shared endpoint. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| variable {D D' : Paper170.AdmissibleDrawing graph} |
| {C : DrawingPassages D.toFiniteIntersectionDrawing} {P : ∀ e, PolygonalParametrization (D'.edge e)} |
| {p q : C.InnerIndex} |
|
|
| def adjacentSurgeryRouteP (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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)) : |
| Path (D'.vertex ((graph.src p.1))) (D'.vertex ((graph.dst p.1))) := by |
| classical |
| exact if h : (graph.src p.1)=(graph.src q.1) then (reconnectedBackwardPath Kp Kq hm).cast |
| (congrArg (fun i => D'.vertex (i)) h) rfl |
| else (reconnectedForwardPath Kp Kq hm).cast rfl |
| (congrArg (fun j => D'.vertex (j)) (hadj.resolve_left h)) |
|
|
| def adjacentSurgeryRouteQ (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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)) : |
| Path (D'.vertex ((graph.src q.1))) (D'.vertex ((graph.dst q.1))) := by |
| classical |
| exact if h : (graph.src p.1)=(graph.src q.1) then (reconnectedForwardPath Kp Kq hm).cast |
| (congrArg (fun i => D'.vertex (i)) h.symm) rfl |
| else (reconnectedBackwardPath Kp Kq hm).cast rfl |
| (congrArg (fun j => D'.vertex (j)) (hadj.resolve_left h).symm) |
|
|
| theorem adjacentSurgeryRouteP_injective (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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) : |
| Function.Injective (adjacentSurgeryRouteP Kp Kq hm hadj) := by |
| unfold adjacentSurgeryRouteP |
| split |
| · exact reconnectedBackwardPath_injective Kp Kq hm hadj hemb |
| · exact reconnectedForwardPath_injective Kp Kq hm hadj hemb |
|
|
| theorem adjacentSurgeryRouteQ_injective (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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) : |
| Function.Injective (adjacentSurgeryRouteQ Kp Kq hm hadj) := by |
| unfold adjacentSurgeryRouteQ |
| split |
| · exact reconnectedForwardPath_injective Kp Kq hm hadj hemb |
| · exact reconnectedBackwardPath_injective Kp Kq hm hadj hemb |
|
|
| theorem adjacentSurgeryRouteP_avoids_vertices |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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)) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) |
| (t : unitInterval) (ht : t ∈ Set.Ioo (0 : unitInterval) 1) (j : Vertex) : |
| adjacentSurgeryRouteP Kp Kq hm hadj t ≠ D'.vertex j := by |
| unfold adjacentSurgeryRouteP |
| split |
| · exact reconnectedBackwardPath_interior_avoids_vertices Kp Kq hm hv hadj hemb t ht j |
| · exact reconnectedForwardPath_interior_avoids_vertices Kp Kq hm hv hadj hemb t ht j |
|
|
| theorem adjacentSurgeryRouteQ_avoids_vertices |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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)) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) |
| (t : unitInterval) (ht : t ∈ Set.Ioo (0 : unitInterval) 1) (j : Vertex) : |
| adjacentSurgeryRouteQ Kp Kq hm hadj t ≠ D'.vertex j := by |
| unfold adjacentSurgeryRouteQ |
| split |
| · exact reconnectedForwardPath_interior_avoids_vertices Kp Kq hm hv hadj hemb t ht j |
| · exact reconnectedBackwardPath_interior_avoids_vertices Kp Kq hm hv hadj hemb t ht j |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! An actual continuous drawing with the two adjacent routes replaced. Admissibility is a later theorem. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| variable {D D' : Paper170.AdmissibleDrawing graph} |
| {C : DrawingPassages D.toFiniteIntersectionDrawing} {P : ∀ e, PolygonalParametrization (D'.edge e)} |
| {p q : C.InnerIndex} |
|
|
| def adjacentSurgeryDrawing (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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)) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) : Paper170.ContinuousDrawing graph := |
| D'.toContinuousDrawing.replaceTwoEdges p.1 q.1 |
| (adjacentSurgeryRouteP Kp Kq hm hadj) (adjacentSurgeryRouteQ Kp Kq hm hadj) |
| (adjacentSurgeryRouteP_injective Kp Kq hm hadj hemb) |
| (adjacentSurgeryRouteQ_injective Kp Kq hm hadj hemb) |
| (adjacentSurgeryRouteP_avoids_vertices Kp Kq hm hadj hv hemb) |
| (adjacentSurgeryRouteQ_avoids_vertices Kp Kq hm hadj hv hemb) |
|
|
| theorem adjacentSurgeryDrawing_vertex (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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)) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) : |
| (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).vertex=D'.vertex := rfl |
|
|
| theorem adjacentSurgeryDrawing_first (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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)) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) : |
| (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).edge p.1=adjacentSurgeryRouteP Kp Kq hm hadj := |
| Paper170.ContinuousDrawing.replaceTwoRoutes_first _ _ _ _ _ |
|
|
| theorem adjacentSurgeryDrawing_second (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) (hpq : p ≠ q) (hadj : (graph.src p.1)=(graph.src q.1) ∨ (graph.dst p.1)=(graph.dst q.1)) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) : |
| (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).edge q.1=adjacentSurgeryRouteQ Kp Kq hm hadj := |
| Paper170.ContinuousDrawing.replaceTwoRoutes_second _ _ _ (fun he => hpq (C.innerIndex_eq_of_edge_mark p q he hm)) _ _ |
|
|
| theorem adjacentSurgeryDrawing_other (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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)) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) |
| (e : Edge) (hep : e ≠ p.1) (heq : e ≠ q.1) : |
| (adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).edge e=D'.edge e := |
| Paper170.ContinuousDrawing.replaceTwoRoutes_other _ _ _ _ _ _ hep heq |
|
|
| theorem adjacentSurgeryRouteP_hasFiniteAffineCover |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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)) : |
| HasFiniteAffineCover (adjacentSurgeryRouteP Kp Kq hm hadj) := by |
| unfold adjacentSurgeryRouteP |
| split |
| · exact (reconnectedBackwardPath_hasFiniteAffineCover Kp Kq hm).cast _ _ |
| · exact (reconnectedForwardPath_hasFiniteAffineCover Kp Kq hm).cast _ _ |
|
|
| theorem adjacentSurgeryRouteQ_hasFiniteAffineCover |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C 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)) : |
| HasFiniteAffineCover (adjacentSurgeryRouteQ Kp Kq hm hadj) := by |
| unfold adjacentSurgeryRouteQ |
| split |
| · exact (reconnectedForwardPath_hasFiniteAffineCover Kp Kq hm).cast _ _ |
| · exact (reconnectedBackwardPath_hasFiniteAffineCover Kp Kq hm).cast _ _ |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Polygonality and actual removal of the selected point from every new drawing route. -/ |
| noncomputable section |
| open Zarankiewicz |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
|
|
| namespace Paper170Geometry |
| variable {D D' : Paper170.AdmissibleDrawing graph} |
| {C : DrawingPassages D.toFiniteIntersectionDrawing} {P : ∀ e, PolygonalParametrization (D'.edge e)} |
| {p q : C.InnerIndex} |
|
|
| theorem adjacentSurgeryDrawing_hasFiniteAffineCover |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) (hpq : p ≠ q) (hadj : (graph.src p.1)=(graph.src q.1) ∨ (graph.dst p.1)=(graph.dst q.1)) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) |
| (e : Edge) : HasFiniteAffineCover ((adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).edge e) := by |
| classical |
| by_cases hep : e=p.1 |
| · subst e |
| rw [adjacentSurgeryDrawing_first] |
| exact adjacentSurgeryRouteP_hasFiniteAffineCover Kp Kq hm hadj |
| by_cases heq : e=q.1 |
| · subst e |
| rw [adjacentSurgeryDrawing_second Kp Kq hm hpq] |
| exact adjacentSurgeryRouteQ_hasFiniteAffineCover Kp Kq hm hadj |
| · rw [adjacentSurgeryDrawing_other Kp Kq hm hadj hv hemb e hep heq] |
| exact (P e).hasFiniteAffineCover |
|
|
| theorem adjacentSurgeryRoutes_avoid_crossing |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) (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)) : |
| x ∉ Set.range (adjacentSurgeryRouteP Kp Kq hm hadj) ∧ |
| x ∉ Set.range (adjacentSurgeryRouteQ Kp Kq hm hadj) := by |
| have ha := reconnected_paths_avoid_crossing Kp Kq hm hpq x hxp hxq |
| constructor |
| · unfold adjacentSurgeryRouteP |
| split |
| · exact ha.2 |
| · exact ha.1 |
| · unfold adjacentSurgeryRouteQ |
| split |
| · exact ha.1 |
| · exact ha.2 |
|
|
| theorem adjacentSurgeryDrawing_avoids_crossing |
| (Kp : ExteriorCrossingCut C P p) (Kq : ExteriorCrossingCut C P q) |
| (hm : C.innerMark p=C.innerMark q) (hpq : p ≠ q) (hadj : (graph.src p.1)=(graph.src q.1) ∨ (graph.dst p.1)=(graph.dst q.1)) |
| (hv : ∀ j, squaredLength (D'.vertex j-D.vertex j) < C.radius^2) |
| (hemb : StraightEdgeEmbedding (crossingSurgeryGraph C P p q) id) |
| (x : Plane) (hxp : x ∈ Set.range (C.crossingChord p)) (hxq : x ∈ Set.range (C.crossingChord q)) |
| (hother : ∀ e, e ≠ p.1 → e ≠ q.1 → x ∉ Set.range (D'.edge e)) : |
| ∀ e, x ∉ Set.range ((adjacentSurgeryDrawing Kp Kq hm hadj hv hemb).edge e) := by |
| have ha := adjacentSurgeryRoutes_avoid_crossing Kp Kq hm hpq hadj x hxp hxq |
| intro e |
| classical |
| by_cases hep : e=p.1 |
| · subst e |
| rw [adjacentSurgeryDrawing_first] |
| exact ha.1 |
| by_cases heq : e=q.1 |
| · subst e |
| rw [adjacentSurgeryDrawing_second Kp Kq hm hpq] |
| exact ha.2 |
| · rw [adjacentSurgeryDrawing_other Kp Kq hm hadj hv hemb e hep heq] |
| exact hother e hep heq |
|
|
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! All cut and embedding data are constructed for the actual normalized drawing. -/ |
| 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_adjacentSurgery_continuousDrawing [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.ContinuousDrawing graph, A.vertex=v ∧ |
| (∀ e, HasFiniteAffineCover (A.edge e)) ∧ |
| (∀ e, x ∉ Set.range (A.edge e)) ∧ |
| (∀ e, e ≠ p.1 → e ≠ q.1 → ∀ t, A.edge e t=E.assembledEdge v e t) := 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 A := adjacentSurgeryDrawing Kp Kq hm hadj hv hemb |
| refine ⟨A,rfl,adjacentSurgeryDrawing_hasFiniteAffineCover Kp Kq hm hpq hadj hv hemb,?_,?_⟩ |
| · apply adjacentSurgeryDrawing_avoids_crossing Kp Kq hm hpq hadj hv hemb x hxp hxq |
| 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 |
| · intro e hep heq t |
| exact congrArg (fun γ => γ t) (adjacentSurgeryDrawing_other Kp Kq hm hadj hv hemb e hep heq) |
|
|
| end Paper170Geometry.PolygonalExterior |
|
|
| end |
| end |
|
|
| end OAI |
|
|
