| import OAI.Combinatorics.CompleteCrossing.DartPaths |
| import OAI.Combinatorics.CompleteCrossing.VertexChords |
| import OAI.Combinatorics.Crossing.CyclicSumArithmetic |
| import OAI.Combinatorics.Crossing.StraightChainIntersections |
| import OAI.Combinatorics.Crossing.StraightChainTransversality |
|
|
| namespace OAI |
|
|
| /-! Signed dart and cycle intersections, passage pairing, and endpoint averaging. -/ |
|
|
| section |
|
|
| /-! Complete shifted edge paths have exactly the original signed edge intersections; the two actual connectors contribute zero. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| variable {D : Paper170.AdmissibleDrawing graph} {V : VertexTruncation D.toContinuousDrawing} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem exists_complete_edge_intersection_radius [Fintype Vertex] [Fintype Edge] (hnorm : NoAdjacentCrossings D) |
| (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D) : |
| ∃ δ : ℝ, 0<δ ∧ ∀ ε : ℝ, ε ≠ 0 → |ε|<δ → ∀ e f : Edge, |
| (V.trackChain e).Transverse (C.shiftedEdgeChain f ε) ∧ |
| polygonIntersection (V.trackChain e).vertices (C.shiftedEdgeChain f ε).vertices |
| (V.trackChain e).pieces (C.shiftedEdgeChain f ε).pieces=V.signedIntersection e f := by |
| obtain ⟨δ₀,hδ₀,hT⟩ := V.exists_all_edge_shift_radius hnorm hstraight htrans |
| obtain ⟨δ₁,hδ₁,hI⟩ := V.exists_uniform_track_intersections hnorm hstraight htrans |
| obtain ⟨δ₂,hδ₂,hC⟩ := C.exists_all_connector_track_radius hnorm |
| let δ := min δ₀ (min δ₁ δ₂) |
| have hδ : 0<δ := lt_min hδ₀ (lt_min hδ₁ hδ₂) |
| have hb₀ : δ≤δ₀ := min_le_left _ _ |
| have hb₁ : δ≤δ₁ := (min_le_right _ _).trans (min_le_left _ _) |
| have hb₂ : δ≤δ₂ := (min_le_right _ _).trans (min_le_right _ _) |
| refine ⟨δ,hδ,?_⟩ |
| intro ε hε hsmall e f |
| have hconn (j : Vertex) (g : IncidentEdges graph j) : |
| (V.trackChain e).Transverse (C.connectorChain j g ε) := by |
| apply PolygonalChain.transverse_single_right |
| intro i |
| exact segmentPairTransverse_of_not_meet _ _ _ _ (hC ε hε (hsmall.trans_le hb₂) j g e i) |
| have hmid : (V.trackChain e).Transverse (C.shiftedTrackChain f ε) := by |
| intro i j |
| exact (hT ε hε (hsmall.trans_le hb₀) e f i j).1 |
| have hzero (j : Vertex) (g : IncidentEdges graph j) : |
| polygonIntersection (V.trackChain e).vertices (C.connectorChain j g ε).vertices (V.trackChain e).pieces 1=0 := by |
| apply PolygonalChain.intersection_single_right_zero |
| intro i |
| exact hC ε hε (hsmall.trans_le hb₂) j g e i |
| constructor |
| · change (V.trackChain e).Transverse |
| (((C.connectorChain ((graph.src f)) (V.leftIncident f) ε).join (C.shiftedTrackChain f ε)).join |
| (C.connectorChain ((graph.dst f)) (V.rightIncident f) ε).reverse) |
| apply PolygonalChain.transverse_join_right |
| · exact PolygonalChain.transverse_join_right _ _ _ (hconn _ _) hmid |
| · exact (V.trackChain e).transverse_reverse_right _ (hconn _ _) |
| · rw [C.shiftedEdgeChain_intersection_right,hzero,hzero,zero_add,sub_zero] |
| exact hI ε hε (hsmall.trans_le hb₁) e f |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Complete edge paths and vertex chords have zero cross-type intersection sums, with all seams verified. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| variable {D : Paper170.AdmissibleDrawing graph} {V : VertexTruncation D.toContinuousDrawing} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem exists_chord_edge_radius [Fintype Vertex] [Fintype Edge] (hnorm : NoAdjacentCrossings D) : |
| ∃ δ : ℝ, 0<δ ∧ ∀ ε : ℝ, ε ≠ 0 → |ε|<δ → |
| (∀ j (a b : IncidentEdges graph j) e, |
| (V.chordChain j a b).Transverse (C.shiftedEdgeChain e ε) ∧ |
| polygonIntersection (V.chordChain j a b).vertices (C.shiftedEdgeChain e ε).vertices |
| (V.chordChain j a b).pieces (C.shiftedEdgeChain e ε).pieces=0) ∧ |
| (∀ j (a b : IncidentEdges graph j) e, |
| (V.trackChain e).Transverse (C.shiftedChordChain j a b ε) ∧ |
| polygonIntersection (V.trackChain e).vertices (C.shiftedChordChain j a b ε).vertices |
| (V.trackChain e).pieces (C.shiftedChordChain j a b ε).pieces=0) := by |
| obtain ⟨δ,hδ,hh⟩ := C.exists_cross_type_radius hnorm |
| refine ⟨δ,hδ,?_⟩ |
| intro ε hε hsmall |
| obtain ⟨hTC,hTChord,hChordT,hChordC,hChordChord,hOther⟩ := hh ε hε hsmall |
| constructor |
| · intro j a b e |
| have hconn (k : Vertex) (f : IncidentEdges graph k) : |
| (V.chordChain j a b).Transverse (C.connectorChain k f ε) := by |
| apply PolygonalChain.transverse_single_single |
| exact segmentPairTransverse_of_not_meet _ _ _ _ (hChordC j k a b f) |
| have hmid : (V.chordChain j a b).Transverse (C.shiftedTrackChain e ε) := by |
| apply PolygonalChain.transverse_single_left |
| intro i |
| exact segmentPairTransverse_of_not_meet _ _ _ _ (hChordT j a b e i) |
| have hzconn (k : Vertex) (f : IncidentEdges graph k) : |
| polygonIntersection (V.chordChain j a b).vertices (C.connectorChain k f ε).vertices 1 1=0 := by |
| rw [VertexTruncation.chordChain,connectorChain,PolygonalChain.intersection_single_single] |
| exact segmentCrossingSign_eq_zero_of_not_meet _ _ _ _ (hChordC j k a b f) |
| have hzmid : polygonIntersection (V.chordChain j a b).vertices |
| (shiftedTrack (V.vertices e) (V.pieces e) ε) 1 (V.pieces e)=0 := by |
| apply (C.shiftedTrackChain e ε).intersection_single_left_zero |
| intro i |
| exact hChordT j a b e i |
| constructor |
| · change (V.chordChain j a b).Transverse |
| (((C.connectorChain ((graph.src e)) (V.leftIncident e) ε).join (C.shiftedTrackChain e ε)).join |
| (C.connectorChain ((graph.dst e)) (V.rightIncident e) ε).reverse) |
| apply PolygonalChain.transverse_join_right |
| · exact PolygonalChain.transverse_join_right _ _ _ (hconn _ _) hmid |
| · exact (V.chordChain j a b).transverse_reverse_right _ (hconn _ _) |
| · change polygonIntersection (V.chordChain j a b).vertices (C.shiftedEdgeChain e ε).vertices 1 _=0 |
| rw [C.shiftedEdgeChain_intersection_right,hzconn,hzconn,hzmid] |
| norm_num |
| · intro j a b e |
| constructor |
| · apply PolygonalChain.transverse_single_right |
| intro i |
| exact segmentPairTransverse_of_not_meet _ _ _ _ (hTChord j a b e i) |
| · apply PolygonalChain.intersection_single_right_zero |
| intro i |
| exact hTChord j a b e i |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Actual vertex-chord chains have precisely the local two-sided averaged endpoint pairing. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| variable {D : Paper170.AdmissibleDrawing graph} {V : VertexTruncation D.toContinuousDrawing} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem chordChain_intersection (j k : Vertex) (a b : IncidentEdges graph j) |
| (c d : IncidentEdges graph k) (ε : ℝ) : |
| polygonIntersection (V.chordChain j a b).vertices (C.shiftedChordChain k c d ε).vertices 1 1= |
| segmentCrossingSign (V.spokePort j a) (V.spokePort j b) (C.shiftedPort k c ε) (C.shiftedPort k d ε) := by |
| rw [VertexTruncation.chordChain,shiftedChordChain,PolygonalChain.intersection_single_single] |
|
|
| theorem exists_vertex_chord_chain_radius [Fintype Vertex] [Fintype Edge] (hnorm : NoAdjacentCrossings D) : |
| ∃ δ : ℝ, 0<δ ∧ |
| (∀ ε : ℝ, ε ≠ 0 → |ε|<δ → ∀ j k (a b : IncidentEdges graph j) (c d : IncidentEdges graph k), |
| a ≠ b → c ≠ d → (V.chordChain j a b).Transverse (C.shiftedChordChain k c d ε)) ∧ |
| (∀ ε : ℝ, ε ≠ 0 → |ε|<δ → ∀ j k, j ≠ k → ∀ (a b : IncidentEdges graph j) (c d : IncidentEdges graph k), |
| polygonIntersection (V.chordChain j a b).vertices (C.shiftedChordChain k c d ε).vertices 1 1=0) ∧ |
| (∀ t : ℝ, 0<t → t<δ → ∀ j (a b c d : IncidentEdges graph j), a ≠ b → c ≠ d → |
| (polygonIntersection (V.chordChain j a b).vertices (C.shiftedChordChain j c d t).vertices 1 1+ |
| polygonIntersection (V.chordChain j a b).vertices (C.shiftedChordChain j c d (-t)).vertices 1 1)/2= |
| chordEndpointPairing (C.node j a) (C.node j b) (C.node j c) (C.node j d)) := by |
| obtain ⟨δ₀,hδ₀,hgeo⟩ := C.exists_cross_type_radius hnorm |
| obtain ⟨δ₁,hδ₁,havg⟩ := C.exists_uniform_average_radius |
| refine ⟨min δ₀ δ₁,lt_min hδ₀ hδ₁,?_,?_,?_⟩ |
| · intro ε hε hsmall j k a b c d hab hcd |
| have hh := (hgeo ε hε (hsmall.trans_le (min_le_left _ _))).2.2.2.2.1 j k a b c d hab hcd |
| exact PolygonalChain.transverse_single_single _ _ _ _ hh |
| · intro ε hε hsmall j k hjk a b c d |
| rw [C.chordChain_intersection] |
| exact segmentCrossingSign_eq_zero_of_not_meet _ _ _ _ |
| ((hgeo ε hε (hsmall.trans_le (min_le_left _ _))).2.2.2.2.2 j k hjk a b c d) |
| · intro t ht hsmall j a b c d hab hcd |
| rw [C.chordChain_intersection,C.chordChain_intersection] |
| exact havg t ht (hsmall.trans_le (min_le_right _ _)) j a b c d hab hcd |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Actual oriented dart intersections, including opposite traversals of shared edges. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| variable {D : Paper170.AdmissibleDrawing graph} {V : VertexTruncation D.toContinuousDrawing} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem dart_pair_intersection {ε : ℝ} (a b : OrientedDart graph) |
| (h : ∀ e f, (V.trackChain e).Transverse (C.shiftedEdgeChain f ε) ∧ |
| polygonIntersection (V.trackChain e).vertices (C.shiftedEdgeChain f ε).vertices |
| (V.trackChain e).pieces (C.shiftedEdgeChain f ε).pieces=V.signedIntersection e f) : |
| (V.dartTrackChain a).Transverse (C.shiftedDartChain b ε) ∧ |
| polygonIntersection (V.dartTrackChain a).vertices (C.shiftedDartChain b ε).vertices |
| (V.dartTrackChain a).pieces (C.shiftedDartChain b ε).pieces= |
| dartSign a*dartSign b*V.signedIntersection (OrientedDart.edge a) (OrientedDart.edge b) := by |
| rcases orientedDart_cases a with ⟨e,rfl⟩ | ⟨e,rfl⟩ <;> |
| rcases orientedDart_cases b with ⟨f,rfl⟩ | ⟨f,rfl⟩ |
| · obtain ⟨hT,hI⟩ := h e f |
| refine ⟨hT,?_⟩ |
| change polygonIntersection (V.trackChain e).vertices (C.shiftedEdgeChain f ε).vertices |
| (V.trackChain e).pieces (C.shiftedEdgeChain f ε).pieces=1*1*V.signedIntersection e f |
| simpa only [one_mul] using hI |
| · obtain ⟨hT,hI⟩ := h e f |
| refine ⟨(V.trackChain e).transverse_reverse_right _ hT,?_⟩ |
| change polygonIntersection (V.trackChain e).vertices (C.shiftedEdgeChain f ε).reverse.vertices |
| (V.trackChain e).pieces (C.shiftedEdgeChain f ε).reverse.pieces=1*(-1)*V.signedIntersection e f |
| rw [PolygonalChain.intersection_reverse_right,hI] |
| ring |
| · obtain ⟨hT,hI⟩ := h e f |
| refine ⟨(V.trackChain e).transverse_reverse_left _ hT,?_⟩ |
| change polygonIntersection (V.trackChain e).reverse.vertices (C.shiftedEdgeChain f ε).vertices |
| (V.trackChain e).reverse.pieces (C.shiftedEdgeChain f ε).pieces=(-1)*1*V.signedIntersection e f |
| rw [PolygonalChain.intersection_reverse_left,hI] |
| ring |
| · obtain ⟨hT,hI⟩ := h e f |
| refine ⟨(V.trackChain e).reverse.transverse_reverse_right _ |
| ((V.trackChain e).transverse_reverse_left _ hT),?_⟩ |
| change polygonIntersection (V.trackChain e).reverse.vertices (C.shiftedEdgeChain f ε).reverse.vertices |
| (V.trackChain e).reverse.pieces (C.shiftedEdgeChain f ε).reverse.pieces=(-1)*(-1)*V.signedIntersection e f |
| rw [PolygonalChain.intersection_reverse_left,PolygonalChain.intersection_reverse_right,hI] |
| ring |
|
|
| theorem chord_dart_intersection {ε : ℝ} (j : Vertex) (a b : IncidentEdges graph j) |
| (d : OrientedDart graph) |
| (h : ∀ e, (V.chordChain j a b).Transverse (C.shiftedEdgeChain e ε) ∧ |
| polygonIntersection (V.chordChain j a b).vertices (C.shiftedEdgeChain e ε).vertices |
| (V.chordChain j a b).pieces (C.shiftedEdgeChain e ε).pieces=0) : |
| (V.chordChain j a b).Transverse (C.shiftedDartChain d ε) ∧ |
| polygonIntersection (V.chordChain j a b).vertices (C.shiftedDartChain d ε).vertices |
| (V.chordChain j a b).pieces (C.shiftedDartChain d ε).pieces=0 := by |
| rcases orientedDart_cases d with ⟨e,rfl⟩ | ⟨e,rfl⟩ |
| · exact h e |
| · obtain ⟨hT,hI⟩ := h e |
| refine ⟨(V.chordChain j a b).transverse_reverse_right _ hT,?_⟩ |
| change polygonIntersection (V.chordChain j a b).vertices (C.shiftedEdgeChain e ε).reverse.vertices |
| (V.chordChain j a b).pieces (C.shiftedEdgeChain e ε).reverse.pieces=0 |
| rw [PolygonalChain.intersection_reverse_right,hI,neg_zero] |
|
|
| theorem dart_chord_intersection {ε : ℝ} (j : Vertex) (a b : IncidentEdges graph j) |
| (d : OrientedDart graph) |
| (h : ∀ e, (V.trackChain e).Transverse (C.shiftedChordChain j a b ε) ∧ |
| polygonIntersection (V.trackChain e).vertices (C.shiftedChordChain j a b ε).vertices |
| (V.trackChain e).pieces (C.shiftedChordChain j a b ε).pieces=0) : |
| (V.dartTrackChain d).Transverse (C.shiftedChordChain j a b ε) ∧ |
| polygonIntersection (V.dartTrackChain d).vertices (C.shiftedChordChain j a b ε).vertices |
| (V.dartTrackChain d).pieces (C.shiftedChordChain j a b ε).pieces=0 := by |
| rcases orientedDart_cases d with ⟨e,rfl⟩ | ⟨e,rfl⟩ |
| · exact h e |
| · obtain ⟨hT,hI⟩ := h e |
| refine ⟨(V.trackChain e).transverse_reverse_left _ hT,?_⟩ |
| change polygonIntersection (V.trackChain e).reverse.vertices (C.shiftedChordChain j a b ε).vertices |
| (V.trackChain e).reverse.pieces (C.shiftedChordChain j a b ε).pieces=0 |
| rw [PolygonalChain.intersection_reverse_left,hI,neg_zero] |
|
|
| theorem exists_dart_intersection_radius [Fintype Vertex] [Fintype Edge] (hnorm : NoAdjacentCrossings D) |
| (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ ε : ℝ, ε ≠ 0 → |ε| < δ → |
| (∀ a b : OrientedDart graph, |
| (V.dartTrackChain a).Transverse (C.shiftedDartChain b ε) ∧ |
| polygonIntersection (V.dartTrackChain a).vertices (C.shiftedDartChain b ε).vertices |
| (V.dartTrackChain a).pieces (C.shiftedDartChain b ε).pieces= |
| dartSign a*dartSign b*V.signedIntersection (OrientedDart.edge a) (OrientedDart.edge b)) ∧ |
| (∀ j (a b : IncidentEdges graph j) d, |
| (V.chordChain j a b).Transverse (C.shiftedDartChain d ε) ∧ |
| polygonIntersection (V.chordChain j a b).vertices (C.shiftedDartChain d ε).vertices |
| (V.chordChain j a b).pieces (C.shiftedDartChain d ε).pieces=0) ∧ |
| (∀ j (a b : IncidentEdges graph j) d, |
| (V.dartTrackChain d).Transverse (C.shiftedChordChain j a b ε) ∧ |
| polygonIntersection (V.dartTrackChain d).vertices (C.shiftedChordChain j a b ε).vertices |
| (V.dartTrackChain d).pieces (C.shiftedChordChain j a b ε).pieces=0) := by |
| obtain ⟨δ₀,hδ₀,h₀⟩ := C.exists_complete_edge_intersection_radius hnorm hstraight htrans |
| obtain ⟨δ₁,hδ₁,h₁⟩ := C.exists_chord_edge_radius hnorm |
| refine ⟨min δ₀ δ₁,lt_min hδ₀ hδ₁,?_⟩ |
| intro ε hε hs |
| have he := h₀ ε hε (hs.trans_le (min_le_left _ _)) |
| obtain ⟨hcd,hdc⟩ := h₁ ε hε (hs.trans_le (min_le_right _ _)) |
| exact ⟨fun a b => C.dart_pair_intersection a b he, |
| fun j a b d => C.chord_dart_intersection j a b d (hcd j a b), |
| fun j a b d => C.dart_chord_intersection j a b d (hdc j a b)⟩ |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Every actual cycle-block intersection splits into its edge term and its vertex-chord term. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry |
|
|
| namespace PolygonalChain |
| variable {O T : Set Plane} {a b c d e f : Plane} |
|
|
| theorem intersection_join_join (P : PolygonalChain O a b) (Q : PolygonalChain O b c) |
| (R : PolygonalChain T d e) (S : PolygonalChain T e f) : |
| polygonIntersection (P.join Q).vertices (R.join S).vertices (P.join Q).pieces (R.join S).pieces= |
| polygonIntersection P.vertices R.vertices P.pieces R.pieces+ |
| polygonIntersection P.vertices S.vertices P.pieces S.pieces+ |
| polygonIntersection Q.vertices R.vertices Q.pieces R.pieces+ |
| polygonIntersection Q.vertices S.vertices Q.pieces S.pieces := by |
| rw [Zarankiewicz.PolygonalChain.intersection_join_left, |
| Zarankiewicz.PolygonalChain.intersection_join_right, |
| Zarankiewicz.PolygonalChain.intersection_join_right] |
| ring |
|
|
| theorem transverse_join_join (P : PolygonalChain O a b) (Q : PolygonalChain O b c) |
| (R : PolygonalChain T d e) (S : PolygonalChain T e f) |
| (hPR : P.Transverse R) (hPS : P.Transverse S) (hQR : Q.Transverse R) (hQS : Q.Transverse S) : |
| (P.join Q).Transverse (R.join S) := |
| P.transverse_join_left Q (R.join S) |
| (R.transverse_join_right S P hPR hPS) (R.transverse_join_right S Q hQR hQS) |
|
|
| end PolygonalChain |
| namespace TruncatedSpokeCoordinates |
| variable {D : Paper170.AdmissibleDrawing graph} {V : VertexTruncation D.toContinuousDrawing} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| def passageIntersection (d f : ℕ → OrientedDart graph) |
| (hd : ∀ i, (d i).snd=(d (i+1)).fst) (hf : ∀ j, (f j).snd=(f (j+1)).fst) |
| (i j : ℕ) (ε : ℝ) : ℝ := |
| polygonIntersection (V.chordChain (d i).snd (dartTargetIncident (d i)) (dartPassageNext d hd i)).vertices |
| (C.shiftedChordChain (f j).snd (dartTargetIncident (f j)) (dartPassageNext f hf j) ε).vertices 1 1 |
|
|
| theorem exists_cycle_block_radius [Fintype Vertex] [Fintype Edge] (hnorm : NoAdjacentCrossings D) |
| (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ ε : ℝ, ε ≠ 0 → |ε| < δ → |
| ∀ (d f : ℕ → OrientedDart graph) |
| (hd : ∀ i, (d i).snd=(d (i+1)).fst) (hf : ∀ j, (f j).snd=(f (j+1)).fst) (i j : ℕ), |
| OrientedDart.edge (d (i+1)) ≠ OrientedDart.edge (d i) → |
| OrientedDart.edge (f (j+1)) ≠ OrientedDart.edge (f j) → |
| (V.dartCycleBlock d hd i).Transverse (C.shiftedCycleBlock f hf ε j) ∧ |
| polygonIntersection (V.dartCycleBlock d hd i).vertices (C.shiftedCycleBlock f hf ε j).vertices |
| (V.dartCycleBlock d hd i).pieces (C.shiftedCycleBlock f hf ε j).pieces= |
| dartSign (d i)*dartSign (f j)* |
| V.signedIntersection (OrientedDart.edge (d i)) (OrientedDart.edge (f j))+ |
| C.passageIntersection d f hd hf i j ε := by |
| obtain ⟨δ₀,hδ₀,h₀⟩ := C.exists_dart_intersection_radius hnorm hstraight htrans |
| obtain ⟨δ₁,hδ₁,h₁,-,-⟩ := C.exists_vertex_chord_chain_radius hnorm |
| refine ⟨min δ₀ δ₁,lt_min hδ₀ hδ₁,?_⟩ |
| intro ε hε hs d f hd hf i j hdi hfj |
| obtain ⟨hdd,hcd,hdc⟩ := h₀ ε hε (hs.trans_le (min_le_left _ _)) |
| have hcc := h₁ ε hε (hs.trans_le (min_le_right _ _)) |
| (d i).snd (f j).snd (dartTargetIncident (d i)) (dartPassageNext d hd i) |
| (dartTargetIncident (f j)) (dartPassageNext f hf j) |
| (dartPassage_ne d hd i hdi) (dartPassage_ne f hf j hfj) |
| obtain ⟨hDD,hDI⟩ := hdd (d i) (f j) |
| obtain ⟨hCD,hCI⟩ := hcd (d i).snd (dartTargetIncident (d i)) (dartPassageNext d hd i) (f j) |
| obtain ⟨hDC,hIC⟩ := hdc (f j).snd (dartTargetIncident (f j)) (dartPassageNext f hf j) (d i) |
| constructor |
| · exact PolygonalChain.transverse_join_join _ _ _ _ hDD hDC hCD hcc |
| · change polygonIntersection |
| ((V.dartTrackChain (d i)).join (V.chordChain (d i).snd (dartTargetIncident (d i)) (dartPassageNext d hd i))).vertices |
| ((C.shiftedDartChain (f j) ε).join (C.shiftedChordChain (f j).snd (dartTargetIncident (f j)) (dartPassageNext f hf j) ε)).vertices |
| ((V.dartTrackChain (d i)).join (V.chordChain (d i).snd (dartTargetIncident (d i)) (dartPassageNext d hd i))).pieces |
| ((C.shiftedDartChain (f j) ε).join (C.shiftedChordChain (f j).snd (dartTargetIncident (f j)) (dartPassageNext f hf j) ε)).pieces=_ |
| rw [PolygonalChain.intersection_join_join,hDI,hIC,hCI,add_zero,add_zero] |
| rfl |
|
|
| end TruncatedSpokeCoordinates |
| end Paper170Geometry |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The global sum is derived from the actual closed polygonal curves and every one of their traversals. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| open scoped BigOperators |
| variable {D : Paper170.AdmissibleDrawing graph} {V : VertexTruncation D.toContinuousDrawing} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem exists_closed_cycle_intersection_radius [Fintype Vertex] [Fintype Edge] (hnorm : NoAdjacentCrossings D) |
| (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ ε : ℝ, ε ≠ 0 → |ε| < δ → |
| ∀ (d f : ℕ → OrientedDart graph) |
| (hd : ∀ i, (d i).snd=(d (i+1)).fst) (hf : ∀ j, (f j).snd=(f (j+1)).fst) |
| (N K : ℕ), d N=d 0 → f K=f 0 → |
| (∀ i, i < N → OrientedDart.edge (d (i+1)) ≠ OrientedDart.edge (d i)) → |
| (∀ j, j < K → OrientedDart.edge (f (j+1)) ≠ OrientedDart.edge (f j)) → |
| ∑ i ∈ Finset.range N, ∑ j ∈ Finset.range K, |
| (dartSign (d i)*dartSign (f j)* |
| V.signedIntersection (OrientedDart.edge (d i)) (OrientedDart.edge (f j))+ |
| C.passageIntersection d f hd hf i j ε)=0 := by |
| obtain ⟨δ,hδ,hh⟩ := C.exists_cycle_block_radius hnorm hstraight htrans |
| refine ⟨δ,hδ,?_⟩ |
| intro ε hε hs d f hd hf N K hN hK hdi hfj |
| let P := V.dartCycleChain d hd N hN |
| let Q := C.shiftedCycleChain f hf K hK ε |
| have hblocks (i j : ℕ) (hi : i < N) (hj : j < K) := |
| hh ε hε hs d f hd hf i j (hdi i hi) (hfj j hj) |
| have hT : P.Transverse Q := by |
| change (finitePolygonalChain Set.univ (V.dartCycleVertex d) (Set.mem_univ _) (V.dartCycleBlock d hd) N).Transverse |
| (finitePolygonalChain Set.univ (C.shiftedCycleVertex f ε) (Set.mem_univ _) (C.shiftedCycleBlock f hf ε) K) |
| apply finitePolygonalChain_transverse_left |
| intro i hi |
| apply finitePolygonalChain_transverse_right |
| intro j hj |
| exact (hblocks i j hi hj).1 |
| have hzero := P.closed_intersection_zero Q hT |
| have hsplit : polygonIntersection P.vertices Q.vertices P.pieces Q.pieces= |
| ∑ i ∈ Finset.range N, ∑ j ∈ Finset.range K, |
| polygonIntersection (V.dartCycleBlock d hd i).vertices (C.shiftedCycleBlock f hf ε j).vertices |
| (V.dartCycleBlock d hd i).pieces (C.shiftedCycleBlock f hf ε j).pieces := |
| finitePolygonalChain_intersection_double _ _ _ _ _ _ _ _ |
| rw [hsplit] at hzero |
| calc |
| _ = ∑ i ∈ Finset.range N, ∑ j ∈ Finset.range K, |
| polygonIntersection (V.dartCycleBlock d hd i).vertices (C.shiftedCycleBlock f hf ε j).vertices |
| (V.dartCycleBlock d hd i).pieces (C.shiftedCycleBlock f hf ε j).pieces := by |
| apply Finset.sum_congr rfl |
| intro i hi |
| apply Finset.sum_congr rfl |
| intro j hj |
| exact (hblocks i j (Finset.mem_range.mp hi) (Finset.mem_range.mp hj)).2.symm |
| _ = 0 := hzero |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! The local pairing uses one fixed coordinate chart per vertex, including shared chord endpoints. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} [Fintype Vertex] [Fintype Edge] {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| variable {D : Paper170.AdmissibleDrawing graph} {V : VertexTruncation D.toContinuousDrawing} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem exists_vertex_pairing_radius (hnorm : NoAdjacentCrossings D) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ t : ℝ, 0 < t → t < δ → |
| ∀ j k (a b : IncidentEdges graph j) (c d : IncidentEdges graph k), a ≠ b → c ≠ d → |
| (polygonIntersection (V.chordChain j a b).vertices (C.shiftedChordChain k c d t).vertices 1 1+ |
| polygonIntersection (V.chordChain j a b).vertices (C.shiftedChordChain k c d (-t)).vertices 1 1)/2= |
| C.vertexChordPairing j k a b c d := by |
| obtain ⟨δ,hδ,-,hzero,havg⟩ := C.exists_vertex_chord_chain_radius hnorm |
| refine ⟨δ,hδ,?_⟩ |
| intro t ht hs j k a b c d hab hcd |
| by_cases h : k=j |
| · subst k |
| have hc : incidentCast (rfl : j=j) c=c := Subtype.ext rfl |
| have hd : incidentCast (rfl : j=j) d=d := Subtype.ext rfl |
| rw [vertexChordPairing,dite_eq_left rfl,hc,hd] |
| exact havg t ht hs j a b c d hab hcd |
| · have hzplus := hzero t (ne_of_gt ht) (by simpa only [abs_of_pos ht] using hs) j k (Ne.symm h) a b c d |
| have hzminus := hzero (-t) (neg_ne_zero.mpr (ne_of_gt ht)) (by simpa only [abs_neg,abs_of_pos ht] using hs) |
| j k (Ne.symm h) a b c d |
| rw [hzplus,hzminus,zero_add,zero_div] |
| simp only [vertexChordPairing,dite_eq_right h] |
|
|
| theorem exists_passage_pairing_radius (hnorm : NoAdjacentCrossings D) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ t : ℝ, 0 < t → t < δ → |
| ∀ (d f : ℕ → OrientedDart graph) |
| (hd : ∀ i, (d i).snd=(d (i+1)).fst) (hf : ∀ j, (f j).snd=(f (j+1)).fst) (i j : ℕ), |
| OrientedDart.edge (d (i+1)) ≠ OrientedDart.edge (d i) → |
| OrientedDart.edge (f (j+1)) ≠ OrientedDart.edge (f j) → |
| (C.passageIntersection d f hd hf i j t+C.passageIntersection d f hd hf i j (-t))/2= |
| C.passagePairing d f hd hf i j := by |
| obtain ⟨δ,hδ,hh⟩ := C.exists_vertex_pairing_radius hnorm |
| refine ⟨δ,hδ,?_⟩ |
| intro t ht hs d f hd hf i j hi hj |
| exact hh t ht hs _ _ _ _ _ _ (dartPassage_ne d hd i hi) (dartPassage_ne f hf j hj) |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| section |
|
|
| /-! Two actual physical shifts give the global averaged identity, with no restriction on shared edges between cycles. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} [Fintype Vertex] [Fintype Edge] {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| open scoped BigOperators |
| variable {D : Paper170.AdmissibleDrawing graph} {V : VertexTruncation D.toContinuousDrawing} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem closed_cycle_averaged_sum (hnorm : NoAdjacentCrossings D) |
| (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D) |
| (d f : ℕ → OrientedDart graph) |
| (hd : ∀ i, (d i).snd=(d (i+1)).fst) (hf : ∀ j, (f j).snd=(f (j+1)).fst) |
| (N K : ℕ) (hN : d N=d 0) (hK : f K=f 0) |
| (hdi : ∀ i, i < N → OrientedDart.edge (d (i+1)) ≠ OrientedDart.edge (d i)) |
| (hfj : ∀ j, j < K → OrientedDart.edge (f (j+1)) ≠ OrientedDart.edge (f j)) : |
| ∑ i ∈ Finset.range N, ∑ j ∈ Finset.range K, |
| (dartSign (d i)*dartSign (f j)* |
| V.signedIntersection (OrientedDart.edge (d i)) (OrientedDart.edge (f j))+ |
| C.passagePairing d f hd hf i j)=0 := by |
| obtain ⟨δ₀,hδ₀,hglobal⟩ := C.exists_closed_cycle_intersection_radius hnorm hstraight htrans |
| obtain ⟨δ₁,hδ₁,hlocal⟩ := C.exists_passage_pairing_radius hnorm |
| let t := min δ₀ δ₁/2 |
| have ht : 0 < t := half_pos (lt_min hδ₀ hδ₁) |
| have htmin : t < min δ₀ δ₁ := half_lt_self (lt_min hδ₀ hδ₁) |
| have ht₀ : t < δ₀ := htmin.trans_le (min_le_left _ _) |
| have ht₁ : t < δ₁ := htmin.trans_le (min_le_right _ _) |
| have hzplus := hglobal t (ne_of_gt ht) (by simpa only [abs_of_pos ht] using ht₀) d f hd hf N K hN hK hdi hfj |
| have hzminus := hglobal (-t) (neg_ne_zero.mpr (ne_of_gt ht)) |
| (by simpa only [abs_neg,abs_of_pos ht] using ht₀) d f hd hf N K hN hK hdi hfj |
| let E : ℕ → ℕ → ℝ := fun i j => dartSign (d i)*dartSign (f j)* |
| V.signedIntersection (OrientedDart.edge (d i)) (OrientedDart.edge (f j)) |
| calc |
| _ = ∑ i ∈ Finset.range N, ∑ j ∈ Finset.range K, |
| ((E i j+C.passageIntersection d f hd hf i j t)+ |
| (E i j+C.passageIntersection d f hd hf i j (-t)))/2 := by |
| apply Finset.sum_congr rfl |
| intro i hi |
| apply Finset.sum_congr rfl |
| intro j hj |
| rw [← hlocal t ht ht₁ d f hd hf i j (hdi i (Finset.mem_range.mp hi)) (hfj j (Finset.mem_range.mp hj))] |
| dsimp only [E] |
| ring |
| _ = ((∑ i ∈ Finset.range N, ∑ j ∈ Finset.range K, (E i j+C.passageIntersection d f hd hf i j t))+ |
| (∑ i ∈ Finset.range N, ∑ j ∈ Finset.range K, (E i j+C.passageIntersection d f hd hf i j (-t))))/2 := by |
| simp only [div_eq_mul_inv,Finset.sum_add_distrib,Finset.sum_mul,add_mul] |
| _ = 0 := by rw [hzplus,hzminus]; norm_num |
| |
| end Paper170Geometry.TruncatedSpokeCoordinates |
| |
| end |
| end |
| |
| section |
| |
| /-! Every local passage endpoint is counted once at the corresponding graph-edge endpoint. -/ |
| noncomputable section |
| open Zarankiewicz |
| open scoped Classical |
| variable {Vertex Edge : Type} {graph : Paper170.Graph Vertex Edge} |
| namespace Paper170Geometry.TruncatedSpokeCoordinates |
| open scoped BigOperators |
| variable {D : Paper170.AdmissibleDrawing graph} {V : VertexTruncation D.toContinuousDrawing} |
| (C : TruncatedSpokeCoordinates V) |
|
|
| theorem passage_sum_eq_endpoint_sum (d f : ℕ → OrientedDart graph) |
| (hd : ∀ i, (d i).snd=(d (i+1)).fst) (hf : ∀ j, (f j).snd=(f (j+1)).fst) |
| (N K : ℕ) (hN : d N=d 0) (hK : f K=f 0) : |
| (∑ i ∈ Finset.range N, ∑ j ∈ Finset.range K, C.passagePairing d f hd hf i j)= |
| ∑ i ∈ Finset.range N, ∑ j ∈ Finset.range K, |
| dartSign (d i)*dartSign (f j)* |
| C.edgeEndpointPairing (OrientedDart.edge (d i)) (OrientedDart.edge (f j)) := by |
| have hTS := double_sum_cycle_shift_right d f N K hK |
| (fun a b => C.flagPairing (dartTargetFlag a) (dartSourceFlag b)) |
| have hST := double_sum_cycle_shift_left d f N K hN |
| (fun a b => C.flagPairing (dartSourceFlag a) (dartTargetFlag b)) |
| have hSS₀ := double_sum_cycle_shift_right (fun i => d (i+1)) f N K hK |
| (fun a b => C.flagPairing (dartSourceFlag a) (dartSourceFlag b)) |
| have hSS₁ := double_sum_cycle_shift_left d f N K hN |
| (fun a b => C.flagPairing (dartSourceFlag a) (dartSourceFlag b)) |
| simp_rw [C.passagePairing_flags,← C.dart_endpoint_pairing] |
| simp only [Finset.sum_add_distrib,Finset.sum_sub_distrib] |
| rw [hTS,hST,hSS₀,hSS₁] |
|
|
| def cycleIntersectionMatrix : Edge → Edge → ℝ := |
| fun e f => V.signedIntersection e f+C.edgeEndpointPairing e f |
|
|
| theorem closed_cycle_matrix_sum [Fintype Vertex] [Fintype Edge] (hnorm : NoAdjacentCrossings D) |
| (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D) |
| (d f : ℕ → OrientedDart graph) |
| (hd : ∀ i, (d i).snd=(d (i+1)).fst) (hf : ∀ j, (f j).snd=(f (j+1)).fst) |
| (N K : ℕ) (hN : d N=d 0) (hK : f K=f 0) |
| (hdi : ∀ i, i < N → OrientedDart.edge (d (i+1)) ≠ OrientedDart.edge (d i)) |
| (hfj : ∀ j, j < K → OrientedDart.edge (f (j+1)) ≠ OrientedDart.edge (f j)) : |
| ∑ i ∈ Finset.range N, ∑ j ∈ Finset.range K, |
| dartSign (d i)*dartSign (f j)* |
| C.cycleIntersectionMatrix (OrientedDart.edge (d i)) (OrientedDart.edge (f j))=0 := by |
| have hh := C.closed_cycle_averaged_sum hnorm hstraight htrans d f hd hf N K hN hK hdi hfj |
| simp only [Finset.sum_add_distrib] at hh |
| rw [C.passage_sum_eq_endpoint_sum d f hd hf N K hN hK] at hh |
| simpa only [cycleIntersectionMatrix,mul_add,Finset.sum_add_distrib] using hh |
|
|
| end Paper170Geometry.TruncatedSpokeCoordinates |
|
|
| end |
| end |
|
|
| end OAI |
|
|
